design & analysis of a tensioner for a belt-driven integrated starter
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DESIGN amp ANALYSIS OF A TENSIONER FOR A
BELT-DRIVEN INTEGRATED STARTER-
GENERATOR SYSTEM
OF MICRO-HYBRID VEHICLES
by
Adebukola O Olatunde
A Thesis submitted in conformity with the requirements
for the degree of Master of Applied Science
Graduate Department of Mechanical and Industrial Engineering
University of Toronto
copy Copyright by Adebukola O Olatunde 2008
ii
ABSTRACT
DESIGN AND ANALYSIS OF A TENSIONER FOR A BELT-DRIVEN INTEGRATED
STARTER-GENERATOR SYSTEM OF MICRO-HYBRID VEHICLES
Adebukola O Olatunde
Master of Applied Science
Graduate Department of Mechanical and Industrial Engineering
University of Toronto 2008
The thesis presents the design and analysis of a Twin Tensioner for a Belt-driven Integrated
Starter-generator (B-ISG) system The B-ISG is an emerging hybrid transmission closely
resembling conventional serpentine belt drives Models of the B-ISG system‟s geometric
properties and dynamic and static states are derived and simulated The objective is to reduce
the magnitudes of static tension in the belt for the ISG-driving phase A literature review of
hybrid systems serpentine belt drive modeling and automotive tensioners is included A
parametric study evaluates tensioner parameters with respect to their impact on static tensions
Design variables are selected from these for an optimization study The optimization uses a
genetic algorithm (GA) and a hybrid GA Results of the optimization indicate the optimal
system contains spans with static tensions that are significantly lower in magnitude than that of
the original design Implications of the research on future work are discussed in closing
iii
A testament unto the LORD God lsquowho answered me in the day of my distress and was with me in
the way which I wentrsquo
To my parents Joseph and Beatrice for your strength and persistent prayers
To my siblings Shade Charlene and Kevin for being a listener an editor and a relief when I
needed it
To my friends Samantha Esther and Yasmin who kept me motivated
amp
With love to my sweetheart Nana whose patience support and companionship has made life
sweeter
iv
ACKNOLOWEDGEMENTS
I would like to express deep gratitude to Dr Jean Zu for her guidance throughout the duration of
my studies and for providing me with the opportunity to conduct this thesis
I wish to thank the individuals of Litens Automotive who have provided guidance and data for
the research work Special thanks to Mike Clark Seeva Karuendiran and Dr Qiu for their time
and help
I thank my committee members Dr Naguib and Dr Sun for contributing their time to my
research work
My sincerest thanks to my research colleague David for his knowledge and support Many
thanks to my lab mates Qiming Hansong Ali Ming Andrew and Peyman for their guidance
I want to especially thank Dr Cleghorn Leslie Sinclair and Dr Zu for the opportunities to
teach These experiences have served to enrich my graduate studies As well thank you to Dr
Cleghorn for guidance in my research work
I am also in debt to my classmates and teaching colleagues throughout my time at the University
of Toronto especially Aaron and Mohammed for their support in my development as a graduate
researcher and teacher
v
CONTENTS
ABSTRACT ii
DEDICATION iii
ACKNOWLEDGEMENTS iv
CONTENTS v
LIST OF TABLES ix
LIST OF FIGURES xi
LIST OF SYMBOLS xvi
Chapter 1 INTRODUCTION 1
11 Background 1
12 Motivation 3
13 Thesis Objectives and Scope of Research 4
14 Organization and Content of Thesis 5
Chapter 2 LITERATURE REVIEW 7
21 Introduction 7
22 B-ISG System 8
221 ISG in Hybrids 8
2211 Full Hybrids 9
2212 Power Hybrids 10
2213 Mild Hybrids 11
2214 Micro Hybrids 11
222 B-ISG Structure Location and Function 13
2221 Structure and Location 13
2222 Functionalities 14
23 Belt Drive Modeling 15
24 Tensioners for B-ISG System 18
241 Tensioners Structures Function and Location 18
242 Systematic Review of Tensioner Designs for a B-ISG System 20
25 Summary 24
vi
Chapter 3 MODELING OF B-ISG SYSTEM 25
31 Overview 25
32 B-ISG Tensioner Design 25
33 Geometric Model of a B-ISG System with a Twin Tensioner 27
34 Equations of Motion for a B-ISG System with a Twin Tensioner 32
341 Dynamic Model of the B-ISG System 32
3411 Derivation of Equations of Motion 32
3412 Modeling of Phase Change 41
3413 Natural Frequencies Mode Shapes and Dynamic Responses 42
3414 Crankshaft Pulley Driving Torque Acceleration and Displacement 44
3415 ISG Pulley Driving Torque Acceleration and Displacement 46
3416 Tensioner Arms Dynamic Torques 48
3417 Dynamic Belt Span Tensions 49
342 Static Model of the B-ISG System 49
35 Simulations 50
351 Geometric Analysis 51
352 Dynamic Analysis 52
3521 Natural Frequency and Mode Shape 54
3522 Dynamic Response 58
3523 ISG Pulley and Crankshaft Pulley Torque Requirement 61
3524 Tensioner Arm Torque Requirement 62
3525 Dynamic Belt Span Tension 63
353 Static Analysis 66
36 Summary 69
Chapter 4 PARAMETRIC ANALYSIS OF A B-ISG TWIN TENSIONER 71
41 Introduction 71
42 Methodology 71
43 Results and Discussion 74
431 Influence of Tensioner Arm Stiffness on Static Tension 74
432 Influence of Tensioner Pulley Diameter on Static Tension 78
433 Influence of Tensioner Pulley 1 Coordinates on Static Tension 80
434 Influence of Tensioner Pulley 2 Coordinates on Static Tension 86
vii
44 Conclusion 92
Chapter 5 OPTIMIZATION OF A B-ISG TWIN TENSIONER 95
51 Optimization Problem 95
511 Selection of Design Variables 95
512 Objective Function amp Constraints 97
52 Optimization Method 100
521 Genetic Algorithm 100
522 Hybrid Optimization Algorithm 101
53 Results and Discussion 101
531 Parameter Settings amp Stopping Criteria for Simulations 101
532 Optimization Simulations 102
533 Discussion 106
54 Conclusion 109
Chapter 6 CONCLUSION AND RECOMMENDATIONS111
61 Summary 111
62 Conclusion 112
63 Recommendations for Future Work 113
REFERENCES 116
APPENDICIES 123
A Passive Dual Tensioner Designs from Patent Literature 123
B B-ISG Serpentine Belt Drive with Single Tensioner Equation of Motion 138
C MathCAD Scripts 145
C1 Geometric Analysis 145
C2 Dynamic Analysis 152
C3 Static Analysis 161
D MATLAB Functions amp Scripts 162
D1 Parametric Analysis 162
D11 TwinMainm 162
D12 TwinTenStaticTensionm 168
D2 Optimization 168
D21 OptimizationTwinm - Optimization Function 168
viii
D22 confunTwinm 169
D23 objfunTwinm 170
VITA 171
ix
LIST OF TABLES
21 Passive Dual Tensioner Designs from Patent Literature
31 Selected Contact Point Types on the ith Pulley and Types for the ith Belt Span
32 Coordinate Points for Pulley Centres and Twin Tensioner Pivot
33 Geometric Results of B-ISG System with Twin Tensioner
34 Data for Input Parameters used in Dynamic and Static Computations
35 Static Solution for Belt Span Tensions in Crankshaft and ISG Driving Cases for a B-ISG
Serpentine Belt Drive with a Single Tensioner
36 Static Solution for Belt Span Tensions in Crankshaft and ISG Driving Cases for a B-ISG
Serpentine Belt Drive with a Twin Tensioner
41 Initial Values Increments and Ranges for Parameters of Twin Tensioner
51 Summary of Parametric Analysis Data for Twin Tensioner Properties
52a GA Optimization Results for Twin Tensioner Parameters and Objective Function
52b Computations for Tensions and Angles from GA Optimization Results
53a Hybrid Optimization Results for Twin Tensioner Parameters and Objective Function
53b Computations for Tensions and Angles from Hybrid Optimization Results
54a Non-Weighted Optimization Results for Twin Tensioner Parameters and Objective
Function
54b Computations for Tensions and Angles from Non-Weighted Optimizations
x
55 Weighted Optimization Results for Static Tensions for Optimal B-ISG System with a
Twin Tensioner
56 Non-Weighted Optimization Results for Static Tensions for Optimal B-ISG System with a
Twin Tensioner
xi
LIST OF FIGURES
21 Hybrid Functions
31 Schematic of the Twin Tensioner
32 B-ISG Serpentine Belt Drive with Twin Tensioner
33 Angles Coordinates and Possible Belt Contact Points for the ith and i+1th Pulleys
34 Twin Tensioner Free Body Diagram of a Four-Degree of Freedom System
35 Free Body Diagram for Non-Tensioner Pulleys
36a ISG Driving Case Natural Frequency of System and Mode Shapes for Responsive Rigid
Bodies
36b ISG Driving Case First Mode Responses
36c ISG Driving Case Second Mode Responses
37a Crankshaft Driving Case Natural Frequency of System and Mode Shapes for Responsive
Rigid Bodies
37b Crankshaft Driving Case First Mode Responses
37c Crankshaft Driving Case Second Mode Responses
38 Crankshaft Pulley Dynamic Response (for crankshaft driven case)
39 ISG Pulley Dynamic Response (for ISG driven case)
310 Air Conditioner Pulley Dynamic Response
311 Tensioner Pulley 1 Dynamic Response
xii
312 Tensioner Pulley 2 Dynamic Response
313 Tensioner Arm 1 Dynamic Response
314 Tensioner Arm 2 Dynamic Response
315 Required Driving Torque for the ISG Pulley
316 Required Driving Torque for the Crankshaft Pulley
317 Dynamic Torque for Tensioner Arm 1
318 Dynamic Torque for Tensioner Arm 2
319 Span 1 (between crankshaft and air conditioner) Dynamic Belt Span Tension
320 Span 2 (between air conditioner and tensioner 1) Dynamic Belt Span Tension
321 Span 3 (between tensioner 1 and ISG) Dynamic Belt Span Tension
322 Span 4 (between ISG and tensioner 2) Dynamic Belt Span Tension
323 Span 5 (between tensioner 2 and crankshaft) Dynamic Belt Span Tension
324 B-ISG Serpentine Belt Drive with Single Tensioner
41a Regions 1 and 2 and their Associated Guidelines for Coordinates of Tensioner Pulleys 1
amp 2
41b Regions 1 and 2 in Cartesian Space
42 Parametric Analysis for Coupled Stiffness Arm Constant kt (N∙mrad)
43 Parametric Analysis for Stiffness of Arm 1 kt1 (N∙mrad)
44 Parametric Analysis for Stiffness of Arm 2 kt2 (N∙mrad)
xiii
45 Parametric Analysis for Pulley 1 Diameter D3 (m)
46 Parametric Analysis for Pulley 2 Diameter D5 (m)
47 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Tautest Span
Tension in Crankshaft Driving Case
48 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Slackest Span
Tension in Crankshaft Driving Case
49 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Tautest Span
Tension in ISG Driving Case
410 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Slackest Span
Tension in ISG Driving Case
411 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Tautest Span
Tension in Crankshaft Driving Case
412 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Slackest Span
Tension in Crankshaft Driving Case
413 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Tautest Span
Tension in ISG Driving Case
414 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Slackest Span
Tension in ISG Driving Case
51 Static Stability of the B-ISG Twin Tensioner Based on the Angular Displacement of
Tensioner Arms 1 and 2
A1 Proposed design by Bayerische Motoren Werke AG corresponding to patent nos
EP1420192-A2 and DE10253450-A1
A2a First of four proposed designs (and its various configurations) by Bosch GMBH
corresponding to patent no WO0026532-A1
A2b Second of four proposed designs by Bosch GMBH corresponding to patent no
WO0026532-A1
A2c Third of four proposed designs (and its various configurations) by Bosch GMBH
corresponding to patent no WO0026532-A1
A2d Fourth of four proposed designs (and its various configurations) by Bosch GMBH
corresponding to patent no WO0026532-A1
A3 Proposed design by Daimler Chrysler AG corresponding to patent no DE10324268-A1
A4 Proposed design by Dayco Products LLC corresponding to patent no US6942589-B2
xiv
A5 Proposed design by Gates Corp corresponding to patent no WO2003038309-A
A6 Proposed design by General Motors Corp corresponding to patent no US20060287146-
A1
A7 Proposed design by INA Schaeffler KG corresponding to patent no DE10044645-A1
A8a First of three proposed designs by INA Schaeffler KG corresponding to patent no
DE10159073-A1
A8b Second of three proposed designs by INA Schaeffler KG corresponding to patent no
DE10159073-A1
A8c Third of three proposed designs by INA Schaeffler KG corresponding to patent no
DE10159073-A1
A9 Proposed design by INA Schaeffler KG corresponding to patent no DE10359641-A1
A10 Proposed design by INA Schaeffler KG corresponding to patent no EP1723350-A1
A11 Proposed design by INA Schaeffler KG corresponding to patent no EP1738093-A1
A12 Proposed design by INA Schaeffler KG corresponding to patent no DE102004012395-
A1
A13 Proposed design by INA Schaeffler KG corresponding to patent nos DE102005017038-
A1and WO2006108461-A1
A14 Proposed design by Litens Automotive GMBH et al corresponding to patent no
US20010007839-A1
A15 Proposed design by Mitsubishi Jidosha Eng KK and Mitsubishi Motor Corp
corresponding to patent no JP2005083514-A
A16 Proposed design by Nissan Motor Co Ltd corresponding to patent no JP3565040-B2
A17 Proposed design by NTN Corp corresponding to patent no JP2006189073-A
A18 Proposed design by Valeo Equipment Electriques Moteur corresponding to patent nos
EP1658432 and WO2005015007
B1 Single Tensioner B-ISG System
B2 Free-body Diagram of ith Pulley
xv
B3 Free-body Diagram of Single Tensioner
C1 Schematic of B-ISG System with Twin Tensioner
C2 Possible Contact Points
xvi
LIST OF SYMBOLS
Latin Letters
A Belt cord cross-sectional area
C Damping matrix of the system
cb Belt damping
119888119894119887 Belt damping constant of the ith belt span
119914119946119946 Damping matrix element in the ith row and ith column
ct Damping acting between tensioner arms 1 and 2
cti Damping of the ith tensioner arm
DCS Diameter of crankshaft pulley
DISG Diameter of ISG pulley
ft Belt transition frequency
H(n) Phase change function
I Inertial matrix of the system
119920119938 Inertial matrix under ISG driving phase
119920119940 Inertial matrix under crankshaft driving phase
Ii Inertia of the ith pulley
Iti Inertia of the ith tensioner arm
119920120784120784 Submatrix of inertial matrix I
j Imaginary coordinate (ie (-1)12
)
K Stiffness matrix of the system
xvii
119896119887 Belt factor
119870119887 Belt cord stiffness
119896119894119887 Belt stiffness constant of the ith belt span
kt Spring stiffness acting between tensioner arms 1 and 2
kti Coil spring of the ith tensioner arm
119922120784120784 Submatrix of stiffness matrix K
Lfi Lbi Lengths of possible belt span connections from the ith pulley
Lti Length of the ith tensioner arm
Modeia Mode shape of the ith rigid body in the ISG driving phase
Modeic Mode shape of the ith rigid body in the crankshaft driving phase
n Engine speed
N Motor speed
nCS rpm of crankshaft pulley
NF Motor speed without load
nISG rpm of ISG pulley
Q Required torque matrix
qc Amplitude of the required crankshaft torque
QcsISG Required torque of the driving pulley (crankshaft or ISG)
Qm Required torque matrix of driven rigid bodies
Qti Dynamic torque of the ith tensioner arm
Ri Radius of the ith pulley
T Matrix of belt span static tensions
xviii
Trsquo Dynamic belt tension matrix
119931119940 Damping matrix due to the belt
119931119948 Stiffness matrix due to the belt
Ti Tension of the ith belt span
To Initial belt tension for the system
Ts Stall torque
Tti Tension for the neighbouring belt spans of the ith tensioner pulley
(XiYi) Coordinates of the ith pulley centre
XYfi XYbi XYfbi
XYbfi Possible connection points on the ith pulley leading to the ith belt span
XYf2i XYb2i
XYfb2i XYbf2i Possible connection points on the ith pulley leading to the (i-1)th belt span
Greek Letters
αi Angle between the datum and the line connecting the ith and (i+1)th pulley
centres
βji Angle of orientation for the ith belt span
120597θti(t) 120579 ti(t)
120579 ti(t)
Angular displacement velocity and acceleration (rotational coordinate) of the
ith tensioner arm
120637119938 General coordinate matrix under ISG driving phase
120637119940 General coordinate matrix under crankshaft driving phase
θfi θbi Angles between the datum and the belt connection spans with lengths Lfi and
Lbi respectively
Θi Amplitude of displacement of the ith pulley
xix
θi(t) 120579 i(t) 120579 i(t) Angular position velocity and acceleration (rotational coordinate) of the ith
pulley
θti Angle of the ith tensioner arm
θtoi Initial pivot angle of the ith tensioner arm
θm Angular displacement matrix of driven rigid bodies
Θm Amplitude of displacement of driven rigid bodies
ρ Belt cord density
120601119894 Belt wrap angle on the ith pulley
φmax Belt maximum phase angle
φ0deg Belt phase angle at zero frequency
ω Frequency of the system
ωcs Angular frequency of crankshaft pulley
ωISG Angular frequency of the ISG pulley
120654119951 Natural frequency of system
1
CHAPTER 1 INTRODUCTION
11 Background
Belt drive systems are the means of power transmission in conventional automobiles The
emergence of hybrid technologies specifically the Belt-driven Integrated Starter-generator (B-
ISG) has placed higher demands on belt drives than ever before The presence of an integrated
starter-generator (ISG) in a belt transmission places excessive strain on the belt leading to
premature belt failure This phenomenon has motivated automotive makers to design a tensioner
that is suitable for the B-ISG system
The belt drive is also known interchangeably as the front-end accessory drive-belt (FEAD) the
belt accessory-drive system (BAS) or the belt transmission system In a traditional setting the
role of this system is to transmit torque generated by an internal combustion engine (ICE) in
order to reliably drive multiple peripheral devices mounted on the engine block The high speed
torque is transmitted through a crankshaft pulley to a serpentine belt The serpentine belt is a
single continuous member that winds around the driving and driven accessory pulleys of the
drive system Serpentine belts used in automotive applications consist of several layers The
load-bearing layer is a flexible member consisting of high stiffness fibers [1] It is covered by a
protective layer to guard against mechanical damage and is bound below by a visco-elastic layer
that provides the required shock absorption and grip against the rigid pulleys [1] The accessory
devices may include an alternator power steering pump water pump and air conditioner
compressor among others
Introduction 2
The B-ISG system is a transmission system characteristic to micro-hybrid automobiles It is akin
to traditional belt drives differing in the fact that an electric motor called an integrated starter-
generator (ISG) replaces the original alternator re-starts the engine from idle speed and provides
braking regeneration [2] The re-start function of the micro-hybrid transmission is known as
stop-start In the B-ISG setting the ISG is mounted on the belt drive The ISG produces a speed
of approximately 2000 to 2500rpm in order to spin the engine at approximately 750rpm and
upwards to produce an instantaneous start in the start-stop process [3] The high rotations per
minute (rpm) produced by the ISG consistently places much higher tension requirements on the
belt than when the crankshaft is driving the belt It is preferable not to exceed a range of 600N to
800N of tension on the belt since this exceeds the safe operating conditions of belts used in most
traditional drive systems [4] The traditional belt drive system‟s tensioner a single-arm
tensioner does not suitably reduce the high belt tension nor provide enough tension in the slack
belts spans occurring in the ISG phase of operation for the B-ISG system
In order for the belt to transfer torque in a drive system its initial tension must be set to a value
that is sufficient to keep all spans rigid This value must not be too low as to allow any one span
to be slack during the drive‟s phases of operation Furthermore the belt must not be ldquoinstalled
with too high a tensionrdquo since this can lead to ldquopremature failure of the bearings supporting the
drive and driven pulleys and of the belt itselfrdquo [5] The presence of a tensioning mechanism in
an automotive belt drive allows for an enhanced belt life and performance since pre-tensioning
of the belt is normally not sufficient for all phases of belt drive operation A tensioner allows for
the system to cope with moderate to severe changes in belt span tensions
Introduction 3
Traditional automotive tensioners for belt drives of an ICE consist of a single spring-loaded
arm This type of tensioner is normally designed to provide a passive response to changes in belt
span tension The introduction of the ISG electric motor into the traditional belt drive with a
single-arm tensioner results in the presence of excessively slack spans and excessively tight
spans in the belt The tension requirements in the ISG-driving phase which differ from the
crankshaft-driving phase are poorly met by a traditional single-arm passive tensioner
Tensioners can be divided into two general classes passive and active In both classes the
single-arm tensioner design approach is the norm The passive class of tensioners employ purely
mechanical power to achieve tensioning of the belt while the active class also known as
automatic tensioners typically use some sort of electronic actuation Automatic tensioners have
been employed by various automotive manufacturers however ldquosuch devices add mass
complication and cost to each enginerdquo [5]
12 Motivation
The motivation for the research undertaken arises from the undesirable presence of high belt
tension in automotive belt drives Manufacturers of automotive belt drives have presented
numerous approaches for tension mechanism designs As mentioned in the preceding section
the automation of the traditional single-arm tensioner has disadvantages for manufacturers A
survey of the literature reveals that few quantitative investigations in comparison to the
qualitative investigations provided through patent literature have been conducted in the area of
passive and dual tensioner configurations As such the author of the research project has selected
to investigate the performance of a passive twin-arm tensioner design The theoretical tensioner
Introduction 4
configuration is motivated by research and developments of industry partner Litens
Automotivendash a manufacturer of automotive belt drive systems and components Litens‟
specialty in automotive tensioners has provided a basis for the research work conducted
13 Thesis Objectives and Scope of Research
The objective of this project is to model and investigate a system containing a passive twin-arm
tensioner in a B-ISG serpentine belt drive where the driving pulley alternates between a
crankshaft pulley and an ISG pulley The modeling of a serpentine belt drive system is in
continuation of the work done by post-doctoral fellow Zhen Mu in development of the priority
software known as FEAD at the University of Toronto Firstly for the B-ISG system with a
twin-arm tensioner the geometric state and its equations of motion (EOM) describing the
dynamic and static states are derived The modeling approach was verified by deriving the
geometric properties and the EOM of the system with a single tensioner arm and comparing its
crankshaft-phase‟s simulation results with FEAD software simulations This also provides
comparison of the new twin-arm tensioner belt drive model with the former single-arm tensioner
equipped belt drive model Secondly the model for the static system is investigated through
analysis of the tensioner parameters Thirdly the design variables selected from the parametric
analysis are used for optimization of the new system with respect to its criteria for desired
performance
Introduction 5
14 Organization and Content of the Thesis
This thesis presents the investigation of a passive twin-arm tensioner design in a B-ISG
serpentine belt drive system which is distinguished by having its driving pulley alternate
between a crankshaft pulley and an ISG pulley
Chapter 2 presents the literature reviewed relevant to the area of the thesis topic The context of
the research discusses the function and location of the ISG in hybrid technologies in order to
provide a background for the B-ISG system The attributes of the B-ISG are then discussed
Subsequently a description is given of the developments made in modeling belt drive systems
At the close of the chapter the prior art in tensioner designs and investigations are discussed
The third chapter describes the system models and theory for the B-ISG system with a twin-arm
tensioner Models for the geometric properties and the static and dynamic cases are derived The
simulation results of the system model are presented
Then the fourth chapter contains the parametric analysis The methodologies employed results
and a discussion are provided The design variables of the system to be considered in the
optimization are also discussed
The optimization of a B-ISG system with a passive twin-arm tensioner is presented in Chapter 5
The evaluation of optimization methods results of optimization and discussion of the results are
included Chapter 6 concludes the thesis work in summarizing the response to the thesis
Introduction 6
objectives and concluding the results of the investigation of the objectives Recommendations for
future work in the design and analysis of a B-ISG tensioner design are also described
7
CHAPTER 2 LITERATURE REVIEW
21 Introduction
This literature review justifies the study of the thesis research the significance of the topic and
provides the overall framework for the project The design of a tensioner for a Belt-driven
Integrated Starter-generator (B-ISG) system is a link in the chain of power transmission
developments in hybrid automobiles This chapter will begin with the context of the B-ISG
followed by a review of the hybrid classifications and the critical role of the ISG for each type
The function location and structure of the B-ISG system are then discussed Then a discussion
of the modeling of automotive belt transmissions is presented A systematic review of the prior
art and current state of tensioning mechanisms for B-ISG systems amalgamates the literature and
research evidence relevant to the thesis topic which is the design of a B-ISG tensioner
The Belt-driven Integrated Starter-generator (B-ISG) system is a part of a hybrid class that is
distinguished from other hybrid classes by the structure functions and location of its ISG The
B-ISG unit is a hybrid technology applied to traditional automotive belt drives The use of a B-
ISG system to achieve a start-stop function in the car engine is estimated to cut fuel consumption
in conventional automobiles by up to ten percent and thus reduce CO2 emissions [6]
Environmental and legislative standards for reducing CO2 emissions in vehicles have called for
carmakers to produce less polluting and more efficient vehicle powertrain systems [7] The
transition to bdquocleaner‟ cars makes room for the introduction of the ISG machine into conventional
automotive belt drives [8] The reduction of CO2 emissions and the similarity of the B-ISG
Literature Review 8
transmission to that of conventional cars provide the motivation for the thesis research
Consequently the micro-hybrid class of cars is especially discussed in the literature review since
it contains the B-ISG type of transmission system The micro-hybrid class is one of several
hybrid classes
A look at the performance of a belt-drive under the influence of an ISG is rooted in the
developments of hybrid technology The distinction of the ISG function and its location in each
hybrid class is discussed in the following section
22 B-ISG System
221 ISG in Hybrids
This section of the review discusses the standard classes of hybrid cars which are full power
mild and micro- hybrids Special attention is given to hybrid vehicle architectures involving
internal combustion engines (ICEs) as the main power source This is done for the sake of
comparison between hybrid classes since the ICE is the standard power source for B-ISG micro-
hybrids which is the focus of the research The term conventional car vehicle or automobile
henceforth refers to a vehicle powered solely by a gas or diesel ICE
A hybrid vehicle has a drive system that uses a combination of energy devices This may include
an ICE a battery and an electric motor typically an ISG Two systems exist in the classification
of hybrid vehicles The older system of classification separates hybrids into two classes series
hybrids and parallel hybrids In the older system many modern hybrid vehicles have modes of
operation matching both categories classifying them under either of the two classes [9] The
Literature Review 9
new system of classification has four classes full power mild and micro Under these classes
vehicles are more often under a sole category [9] In both systems an ICE may act as the primary
source of power otherwise it may be a fuel cell The fuel used by the ICE may be gas (petrol)
diesel or an alternative fuel such as ethanol bio-diesel or natural gas
2211 Full Hybrids
In a full hybrid car the ICE is used to power the integrated starter-generator (ISG) which stores
electrical energy in the batteries to be used to power an electric traction motor [8] The electric
traction motor is akin to a second ISG as it generates power and provides torque output It also
supplies an extra boost to the wheels during acceleration and drives up steep inclines A full
hybrid vehicle is able to move by electrical power only It can be driven by the ISG powering
the electric traction motor without the engine running This silent acceleration known as electric
launch is normally employed when accelerating from standstill [9] Full hybrids can generate
and consume energy at the same time Full hybrid vehicles also use regenerative braking [8]
The ISG allows this by converting from an electric traction motor to a generator when braking or
decelerating The kinetic energy from the car‟s motion is then turned into electricity and stored
in the batteries For full hybrids to achieve this they often use break-by-wire a form of
electronically controlled braking technology
A high-voltage (ie 36- or 42-volt) ISG is employed in full hybrids to start the ICE It spins the
engine more than 900 rpm whereas conventional 12-volt starter motors spin the engine at
approximately 250 rpm [9] Thus the full hybrid vehicle is able to have an instantaneous start In
full hybrids the ISG is placed in the position of the flywheel and can have its motion decoupled
Literature Review 10
from the engine [9] The ISG device also allows full hybrids to have engine start-stop also called
an idle-stop ability The idle-stop function refers to when the engine shuts down as soon as a
vehicle stops from its ICE driving mode which saves on the fuel it normally burns while idling
[8] The vehicle returns to the engine driving mode of operation by way of the ISG‟s start-up of
the crankshaft which restarts the engine in less than 300 milliseconds [9] In summary at
standstill the tachometer of the engine drops to 0 rpm since the engine has ceased the engine is
started only when needed which is often several seconds after acceleration has begun The
engine start-stop feature is achieved by way of an electronic control system that shuts off the ICE
when it is not needed to assist in driving the wheels or to produce electricity for recharging the
batteries The start-stop feature by itself is estimated to produce a ten percent fuel gain in hybrids
over conventional vehicles particularly in urban driving conditions [9] Since the ICE is
required to provide only the average horsepower used by the vehicle the engine is downsized in
comparison to a conventional automobile that obtains all its power from an ICE Frequently in
full hybrids the ICE uses an alternative operating strategy such as the Atkinson Cycle which has
a higher efficiency while having a lower power output Examples of full hybrids include the
Ford Escape and the Toyota Prius [9]
2212 Power Hybrids
Akin to the full hybrid the ISG of the power hybrid enables the same features electric launch
regenerative braking and engine idle-stop The distinguishing characteristic from full hybrids is
the ICE is not downsized to meet only the average power demand [9] Thus the engine of a
power hybrid is large and produces a high amount of horsepower compared to the former
Overall a power hybrid has the assist of a full size ICE and therefore has more torque and a
Literature Review 11
greater acceleration performance than a full hybrid or a conventional vehicle with the same size
ICE [9] The Lexus RX400h unit is an example of a power hybrid [9]
2213 Mild Hybrids
In the hybrid types discussed thus far the ISG is positioned between the engine and transmission
to provide traction for the wheels and for regenerative braking Often times the armature or rotor
of the electric motor-generator which is the ISG replaces the engine flywheel in full and power
hybrids [9] In the case of the mild hybrid the ISG is not decoupled from the ICE and hence it is
not able to drive the wheels apart from the engine It remains that the ISG shares the same shaft
with the ICE In this environment the electric launch feature does not exist since the ISG does
not turn the wheels independently of the engine and energy cannot be generated and consumed
at the same time However the ISG of the mild hybrid allows for the remaining features of the
full hybrid regenerative braking and engine idle-stop including the fact that the engine is
downsized to meet only the average demand for horsepower Mild hybrid vehicles include the
GMC Sierra pickup and 2003 to 2005 Honda Civic models [9]
2214 Micro Hybrids
Micro hybrid is the category of hybrids that can contain a B-ISG transmission and is also closest
to modern conventional vehicles This class normally features a gas or diesel ICE [9] The
conventional automobile is modified by installing an ISG unit on the mechanical drive in place
of or in addition to the starter motor The starter motor typically 12-volts is removed only in
the case that the ISG device passes cold start testing which is also dependent on the engine size
[10] Various mechanical drives that may be employed include chain gear or belt drives or a
Literature Review 12
clutchgear arrangement The majority of literature pertaining to mechanical driven ISG
applications does not pursue clutchgear arrangements since it is associated with greater costs
and increased speed issues Findings by Henry et al [11] show that the belt drive in
comparison to chain and gear drives has a decreased cost (especially if the ISG is mounted
directly to the accessory drive) has no need for lubrication has less restriction in the packaging
environment and produces very low noise Also mounting the ISG unit on a separate belt from
that linking the accessory pulleys is undesirable since applying the ISG directly to the accessory
belt drive requires less engine transmission or vehicle modifications
As with full power and mild hybrids the presence of the ISG allows for the start-stop feature
The automobile‟s electronic control unit (ECU) is calibrated or engine control circuitry (a
separate ECU) is added to the conventional car in order to shut down the engine when the
vehicle is stopped [12] The control system also controls the charge cycle of the ISG [9] This
entails that it dictates the field current by way of a microprocessor to allow the system to defer
battery charge cycles until the vehicle is decelerating [13] This produces electricity to recharge
the battery primarily during deceleration and braking The B-ISG transmission of a micro hybrid
and its various components are discussed in the subsequent section Examples of micro hybrid
vehicles are the PSA Group‟s Citroen C2 and C3 [14] Ford‟s Fiesta [14] and BMW‟s Mini
Cooper D and various others of BMW‟s European models [15]
Literature Review 13
Figure 21 Hybrid Functions
Source Dr Daniel Kok FFA July 2004 modified [16]
Figure 21 shows that the higher the voltage available to the ISG unit the more hybrid functions
it is capable of performing It is noted that B-ISG transmissions of the micro-hybrid class may
also exceed the typical functions of micro-hybrids For instance Ford‟s HyTrans van (developed
in partnership with Ricardo UK Ltd Valeo SA Gates Corporation and the UK Department for
Transport) uses a B-ISG system and a 42-volt battery The van is diesel-powered and has
characteristics of a mild hybrid such as cold cranks and engine assists [17]
222 B-ISG Structure Location and Function
2221 Structure and Location
The ISG is composed of an electrical machine normally of the inductive type which includes a
stator (stationary part of the ISG) and a rotor (non-stationary part of the ISG) and a converter
comprising of a regulator a modulator switches and filters There are various configurations to
integrate the ISG unit into an automobile power train One configuration situates the ISG
directly on the crankshaft in the place of the present flywheel [11] This set-up is more compact
however it results in a longer power train which becomes a potential concern for transverse-
Literature Review 14
mounted engines [18] An alternative set-up is to have a side-mounted ISG This term is used to
describe the configuration of mounting the electrical device on the side of the mechanical drive
[18] As mentioned in Section 2214 a belt drive is used as the mechanical drive for the thesis
research hence the ISG is belt-mounted and the transmission becomes a belt-driven ISG system
In this arrangement the ISG replaces the alternator [13] and in some cases the starter motor may
be removed This design allows for the functions of the ISG system mentioned in the description
of the ISG role in micro-hybrids [9] The side-mounted ISG specifically the belt-mounted ISG
is more evolutionary to the conventional car since it ldquoallows for a more traditional under-hood
layoutrdquo [11]
2222 Functionalities
The primary duty of the ISG in a micro hybrid specifically in a B-ISG setting is to bring the
engine from rest to normal operating speeds within a time span ranging from 250 to 400 ms [3]
and in some high voltage settings to provide cold starting
The cold starting operation of the ISG refers to starting the engine from its off mode rather than
idle mode andor when the engine is at a low temperature for example -29 to -50 degrees
Celsius [2] If the ISG is used for cold starting the peak torque is determined by the torque
requirement for the cold starting operation of the target vehicle since it is greater than the
nominal torque For this function the ldquomachine has to provide a breakaway torque about 15 [to]
18 times the nominal cranking torque to overcome static torque and rotate the engine from 0 to
[between] 10 [and] 20rpmrdquo [2] This remains to be a challenge for the ISG as the 12-volt
architecture most commonly found in vehicles does not supply sufficient voltage [2] The
introduction of the ISG machine and other electrical units in vehicles encourages a transition
Literature Review 15
from a 12-volt or 14-volt to a 42-volt electrical architecture [19] The transition to 42-volt
architecture brings ldquopotential higher-voltage functionalities that come with an ISG systemrdquo [20]
At present ldquowhen the [ISG] machine cannot provide enough torque for initial cold engine
cranking the conventional starter will [remain] in the system and perform only for the initial
cranking while the stop-start function is taken over by the [ISG] machinerdquo [2] The ISG‟s launch
assist torque the torque required to bring the engine from idle speed to the speed at which it can
develop a higher torque output is 2000 to 2500 rpm for most gas engines [3]
Delphi‟s Energen 5 High Output 12-volt Belt-alternator-starter (or B-ISG) was implemented by
researchers on a 53 L V-8 engine with an automatic transmission in a Chevrolet Silverado truck
[21] The ISG was applied in a belt-mounted configuration and was used only for warm engine
re-starts The results of Wezenbeek et al [21] showed that the starting torque for a re-start by the
12-Volt ISG was 42 Nm ISG‟s have also been used in 14V 36V and 42V architectures [13]
23 Belt Drive Modeling
The modeling of a serpentine belt drive and tensioning mechanism has typically involved the
application of Newtonian equilibrium equations to rigid bodies in order to derive the equations of
motion for the system There are two modes of motion in a serpentine belt drive transverse
motion and rotational motion The former can be viewed as the motion of the belt directed
normal to the direction of the beltpulley contact plane similar to the vibratory motion of a taut
string that is fixed at either end However the study of the rotational motion in a belt drive is the
focus of the thesis research
Literature Review 16
Much work on the mechanics of the belt drive was carried out by Firbank [22] Firbank‟s
models helped to understand belt performance and the influence of driving and driven pulleys on
the tension member The first description of a serpentine belt drive for automotive use was in
1979 by Cassidy et al [23] and since this time there has been an increasing body of knowledge
on the mathematical modeling of serpentine belt drives Ulsoy et al [24] presented a design
methodology to improve the dynamic performance of instability mechanisms for belt tensioner
systems The mathematical model developed by Ulsoy et al [24] coupled the equations of
motion that were obtained through a dynamic equilibrium of moments about a pivot point the
equations of motion for the transverse vibration of the belt and the equations of motion for the
belt tension variations appearing in the transverse vibrations This along with the boundary and
initial conditions were used to describe the vibration and stability of the coupled belt-tensioner
system Their system also considered the geometry of the belt drive and tensioner motion
Hereafter Beikmann et al [25] predicted the belt drive vibration for a system composed of a
driving pulley driven pulley and a dynamic tensioner The authors coupled the linear equations
of transverse motion for the respective belt spans with the equations of motion for pulleys and a
tensioner This was used to form the free response of the system and evaluate its response
through a closed-form solution of the system‟s natural frequencies and mode shapes
A complex modal analysis of a serpentine belt drive system was carried out by Kraver et al [26]
to determine the effect of damping on rotational vibration mode solutions The equations of
motion developed for a multi-pulley flat belt system with viscous damping and elastic
Literature Review 17
properties including the presence of a rotary tensioner were manipulated to carry out the modal
analysis
Beikmann et al [27] also derived a nonlinear model to predict the operating state of a belt-
tensioner system by way of nonlinear numerical methods and an approximated linear closed-
form method The authors used this strategy to develop a single design parameter referred to as
a tensioner constant to measure the effectiveness of the tensioning mechanism in relation to its
operating state from a reference state The authors considered the steady state tensions in belt
spans as a result of accessory loads belt drive geometry and tensioner properties
Zhang and Zu [28] conducted a modal analysis for the response of a linear serpentine belt drive
system A non-iterative approach was used to explicitly form the equations for the system‟s
natural frequencies An exact closed-form expression for the dynamic response of the system
using eigenfunction expansion was derived with the system under steady-state conditions and
subject to harmonic excitation
The work conducted by Balaji and Mockensturm [29] considered a front-end accessory drive
(FEAD) with a decoupler or isolator attached to a pulley The rotational response for the FEAD
was found analytically by considering the system to be piecewise linear about the equilibrium
angular deflections The effect of their nonlinear terms was considered through numerical
integration of the derived equations of motion by way of the iterative methodndash fourth order
Runge-Kutta The authors in this case considered the longitudinal (ie rotational) vibration of
the belt spans only
Literature Review 18
The first to carry out the analysis of a serpentine belt drive system containing a two-pulley
tensioner was Nouri in 2005 [30] Nouri found the closed-form analytical solution of a
serpentine belt drive with a two-pulley tensioner for the case of sinusoidal excitation He
employed Runge Kutta method as well to solve the equations of motion to find the response of
the system under a general input from the crankshaft The author‟s work also included the
optimization of the tensioner design in order to minimize belt span vibrations due to crankshaft
excitation Furthermore the author applied active control techniques to the tensioner in a belt
drive system
The works discussed have made significant contributions to the research and development into
tensioner systems for serpentine belt drives These lead into the requirements for the structure
function and location of tensioner systems particularly for B-ISG transmissions
24 Tensioners for B-ISG System
241 Tensioners Structure Function and Location
Literature shows that the improvement of a serpentine belt life in a B-ISG system centers on the
tensioning mechanism redesign This mechanism as shown by researchers including
Wezenbeek et al [21] and Henry et al [11] is crucial in establishing the least tension in the belt
(above a zero value) in order to guard against failure by way of slip due to slack spans in the belt
and oscillations during engine re-start It is noted by Firbank [22] that the mechanics of a belt-
drive ldquois based on the idea that belt behaviour is governed by the elastic extension or contraction
of the belt arising from tension variationsrdquo [22] these variations may be compensated for by an
adjustable tensioner
Literature Review 19
The two types of tensioners are passive and active tensioners The former permits an applied
initial tension and then acts as an idler and normally employs mechanical power and can include
passive hydraulic actuation This type is cheaper than the latter and easier to package The latter
type is capable of continually adjusting the belt tension since it permits a lower static tension
Active tensioners typically employ electric or magnetic-electric actuation andor a combination
of active and passive actuators such as electrical actuation of a hydraulic force
Conventional belt tensioners comprise of a single tensioner arm that is fitted with a sole idler
pulley to engage a serpentine belt [31] A radial bearing is used to rotatably connect the idler
pulley to the tensioner arm [31] The tensioner arm is mounted on a pivot pin that is wrapped by
a bushing and is free to rotate [31] The pin covered by the bushing is fixed to the engine
housing [31] A rotary spring is wrapped about the bearing pin and bushing to provide a pre-
tension force to the belt via the tensioner arm and idler pulley thus taking up the slack due to the
changes in belt length [31] When the belt undergoes stretch under a load the spring drives the
tensioner arm and idler pulley further into the belt [31] Belt tension changes under the modes of
operation which can include when the crankshaft (or driving pulley) abruptly decelerates from a
steady-state condition and auxiliary components continue to rotate still in their own inherent
inertia and thus become the primary drivers [31] These fluctuations in belt tension lead to belt
flutter or skip and slip that may damage other components present in the belt drive [31]
Locating the tensioner on the slack side of the belt is intended to lower the initial static tension
[11] In conventional vehicles the engine always drives the alternator so the tensioner is located
in the belt span that links the crankshaft and alternator pulleys In a B-ISG setting the slack span
Literature Review 20
of the belt alternates between the driving mode of the ISG and the driving mode of the crankshaft
[32] Research by Henry et al [11] and also the summary of prior art for tensioners in Table
21 show that placing the idlertensioner pulley in the slack span in the case that the ISG is
driving instead of in the slack span when the crankshaft is driving allows for easier packaging
and for the least static tension Designs shown in Table 21 place the tensioneridler pulley in the
same span as Henry et al [11] or in both the slack and taut spans if using a double
tensioneridler configuration
242 Systematic Review of Tensioner Designs for a B-ISG System
The proposals for belt tensioner devices to manage the issue of high peaks in belt tension for B-
ISG settings are largely in patent records as the re-design of a tensioner has been primarily a
concern of automotive makers thus far A systematic review of the patent literature has been
conducted in order to identify evaluate and collate relevant tensioning mechanism designs
applicable to a B-ISG setting Its research objective is to influence the selection of a tensioner
configuration for the thesis study
The predefined search strategy used by the researcher has been to consider patents dating only
post-2000 as many patents dating earlier are referred to in later patents as they are developed on
in most cases by the original inventor (eg an INA Schaeffler KG patent published in 2000 may
refer to its own earlier patent presented in 1999) Patents dating pre-2000 that do not have any
successor were also considered The inclusion and exclusion criteria and rationales that were
used to assess potential patents are as follows
Inclusion of
Literature Review 21
tensioner designs with two arms andor two pivots andor two pulleys
mechanical tensioners (ie exclusion of magnetic or electrical actuators or any
combination of active actuators) in order to minimize cost
tension devices that are an independent structure apart from the ISG structure in order to
reduce the required modification to the accessory belt drive of a conventional automobile
and
advanced designs that have not been further developed upon in a subsequent patent by the
inventor or an outside party
Table 21 provides a collation of the results for the systematic review based on the selection
criteria Illustrations of the collated patent designs may be seen in Appendix A It is noted that
the patent literature pertaining to these designs in most cases provides minimal numerical data
for belt tensions achieved by the tensioning mechanism In most cases only claims concerning
the outcome in belt performance achievable by the said tension device is stated in the patent
Table 21 Passive Dual Tensioner Designs from Patent Literature
Bayerische
Motoren Werke
AG
Patents EP1420192-A2 DE10253450-A1 [33]
Design Approach
2 tensioner pulleys (idlers) and 2 tension arms are mounted outside the periphery of the belt drive these form tiltable clamping arms around a common axis of rotation
A torsion spring is used at bearing bushings to mount tension arms at ISG shaft
Each tension arm cooperates with torsion spring mechanism to rotate through a damping
device in order to apply appropriate pressure to taut and slack spans of the belt in
different modes of operation
Bosch GMBH Patent WO0026532 et al [34]
Design Approach
2 tension pulleys each one is mounted on the return and load spans of the driven and
driving pulley respectively
Idlers (tension pulleys) each connect to a spring which is attached on one end to a fixed point
Literature Review 22
Idlers‟ motions are independent of each other and correspond to the tautness or
slackness in their respective spans
Or alternatively a spring connects the idler pulleys and one of the two idlers is fixed at
its axis of rotation
Daimler Chrysler
AG
Patents DE10324268-A1 [35]
Design Approach
2 idlers are given a working force by a self-aligning bearing
Bearing supports auxiliary unit (ISG) and is arranged concentrically with the axle
auxiliary unit pulley
Dayco Products
LLC
Patents US6942589-B2 et al [36]
Design Approach
2 tension arms are each rotatably coupled to an idler pulley
One idler pulley is on the tight belt span while the other idler pulley is on the slack belt
span
Tension arms maintain constant angle between one another
One arm forms a positive differential angle with the belt and the remaining arm forms a negative differential angle with the belt
Idler pulleys are on opposite sides of the ISG pulley
Gates Corporation Patents US20060249118-A1 WO2003038309-A [37]
Design Approach
A tensioner pulley contacts the belt at the slack span during start-up (ISG-driving mode)
A tensioner is asymmetrically biased in direction tending to cause power transmission
belt to be under tension
McVicar et al
(Firm General
Motors Corp)
Patent US20060287146-A1 [38]
Design Approach
2 tension pulleys and carrier arms with a central pivot are mounted to the engine
One tension arm and pulley moderately biases one side of belt run to take up slack
during engine start-up while other tension arm and pulley holds appropriate bias against
taut span of belt
A hydraulic strut is connected to one arm to provide moderate bias to belt during normal
engine operation and velocity sensitive resistance to increasing belt forces during engine
start-up
INA Schaeffler
KG et al
Patents DE10044645-A1 [39] DE10159073-A1 [40] EP1723350-A1 et al [41]
DE10359641-A1 et al [42] EP1738093-A1 et al [43] DE102004012395-A1 [44]
WO2006108461-A1 et al [45]
Design Approach
2 tension arms and 2 pulleys approach ndash o Mutually independent tensioning arms are supported for rotation in the same
plane of the housing part
o Idler pulley corresponding to each tensioning arm engages with different
sections of belt
o When high tension span alternates with slack span of belt drive one tension
arm will increase pressure on current slack span of belt and the other will
decrease pressure accordingly on taut span
o Or when the span under highest tension changes one tensioner arm moves out
of the belt drive periphery to a dead center due to a resulting force from the taut
span of the ISG starting mode
o Deflection of the taut span acts on associated pulley to apply a counter-moment to the other idler pulley on the slack span
Literature Review 23
o The 2 lever arms are of different lengths and each have an idler pulley of
different diameters and different wrap angles of belt (see DE10045143-A1 et
al)
1 tensioner arm and 2 pulleys approach ndash
o 2 idler pulleys are pinned to a beam arranged on a clamping arm that is tiltably
linked to the beam o The ISG machine is supported by a shock absorber
o During ISG start-up one idler pulley is induced to a dead center position while
it pulls the remaining idler pulley into a clamping position until force
equilibrium takes place
o A shock absorber is laid out such that its supporting spring action provides
necessary preloading at the idler pulley in the direction of the taut span during
ISG start-up mode
Litens Automotive
Group Ltd
Patents US6506137-B2 et al [46]
Design Approach
2 tension pulleys on opposite sides of the ISG pulley engage the belt
They are positioned such that their applied forces result in opposing directed moments with respect to the tension device‟s axis of pivot
The pivot axis varies relative to the force applied to each tension pulley
Diameters of the tensioner pulleys are approximately equal and belt wrap angles of the
tensioner pulleys are approximately equal
A limited swivel angle for the tensioner arms work cycle is permitted
Mitsubishi Jidosha
Eng KK
Mitsubishi Motor
Corp
Patents JP2005083514-A [47]
Design Approach
2 tensioners are used
1 tensioner is held on the slack span of the driving pulley in a locked condition and a
second tensioner is held on the slack side of the starting (driven) pulley in a free condition
Nissan Patents JP3565040-B2 et al [48]
Design Approach
A single tensioner is on the slack span once ISG pulley is in start-up mode
The tension device is comprised of a oil pressure tensioner and a half ratchet mechanism
(a plunger which performs retreat actuation according to the energizing force of the oil
pressure spring and load received from the ISG)
The tensioner is equipped with a relief valve to keep a predetermined load lower than the
maximum load added by the ISG device
NTN Corp Patent JP2006189073-A [49]
Design Approach
An automatic tensioner is equipped with a hydraulic damper mechanism comprised of a
screw bolt using saw-screwed teeth and a cylinder nut a return spring and a spring seat
in a pressure chamber (within the screw bolt) a rod seat (that is fitted to the lower end of
the cylinder nut) a spring support (arranged on varying diameter stepped recessed
sections of the rod seat) and a check valve with an openingclosing passage
The cylinder and screw bolt act as the rigidity buffer under excessive loads during ISG
start-up mode of operation
Valeo Equipment
Electriques
Moteur
Patents EP1658432 WO2005015007 [50]
Design Approach
ldquoThe invention relates to a system or a starter (10) in which a pulley (80) is rotationally mounted on a section (22) of a shaft which axially extends inside a pulley (80) and
Literature Review 24
forwards at least partially outside a support element (200) and is characterized in that
the free front end (23) of said shaft section (22) is carried by an arm (206) connected to
the support element (200)rdquo
The author notes that published patents and patent applications may retain patent numbers for multiple patent
offices (ie European Patent Office German Patent Office etc) In such cases the published patent number or in
the absence of such a number the published patent application number has been specified However published
patent documents in the above cases also served as the document (ie identical) to the published patent if available
Quoted from patent abstract as machine translation is poor
25 Summary
The research on tensioner designs from the patent literature demonstrates a lack of quantifiable
data for the performance of a twin tensioner particularly suited to a B-ISG system The review of
the literature for the modeling theory of serpentine belt drives and design of tensioners shows
few belt drive models that are specific to a B-ISG setting Hence the literature review supports
the thesis objective of modeling a B-ISG tensioner specifically one that has a passive twin
tensioner configuration and as well measuring the tensioner‟s performance The survey of
hybrid classes reveals that the micro-hybrid class is the only class employing a closely
conventional belt transmission and hence its B-ISG transmission is applicable for tensioner
investigation The patent designs for tensioners contribute to the development of the tensioner
design to be studied in the following chapter
25
CHAPTER 3 MODELING OF B-ISG SYSTEM
31 Overview
The derivation of a theoretical model for a B-ISG system uses real life data to explore the
conceptual system under realistic conditions The literature and prior art of tensioner designs
leads the researcher to make the following modeling contributions a proposed design for a
passive two-pulley tensioner computation of geometric attributes for a B-ISG system with the
proposed tensioner and derivation of the system‟s equations of motion (EOM) under dynamic
and static states as well as deriving the EOM for the B-ISG system with only a passive single-
pulley tensioner for comparison The principles of dynamic equilibrium are applied to the
conceptual system to derive the EOM
32 B-ISG Tensioner Design
The proposed design for a passive two pulley tensioner configures two tensioners about a single
fixed pivot point in the interior space of a serpentine belt drive One end of each tensioner arm
coincides with the centre point of a tensioner pulley and this point marks the axis of rotation of
the pulley The other end of each arm is pivoted about a point so that the arms share the same
axis of rotation This conceptual design henceforth is called a Twin Tensioner Figure 31 shows
a schematic for the proposed design
Modeling of B-ISG 26
Figure 31 Schematic of the Twin Tensioner
The tensioner pulley coordinates are described by (XiYi) their radii by Ri their arm lengths Lti
and their angles θti The rotation of the arms is resisted by stiffness kt of a coil spring acting
between the two arms and spring stiffness kti acting between each arm and the pivot point The
motion of each arm is dampened by dampers and akin to the springs a damper acts between the
two arms ct and a damper cti acts between each arm and the pivot point The result is a
tensioning mechanism with four degrees of freedom (DOF) that includes independent rotations
of the two pulleys and two arms
The following section relates the geometry of the rigid bodies in a B-ISG system equipped with a
Twin Tensioner to their respective motions
Modeling of B-ISG 27
33 Geometric Model of a B-ISG System with a Twin Tensioner
The B-ISG system with the Twin Tensioner is shown in Figure 32 The geometry of the drive
provides the lengths of the belt spans and angles of wrap for the belt and pulley contact surfaces
These variables are crucial to resolve the components of forces and moment arms acting on each
rigid body in the system and are used in the derivation of the EOM in section 34 Zhen Mu‟s
geometric modeling approach [51] used in the development of the software FEAD was applied
to the Twin Tensioner system to compute the system‟s unique geometric attributes
Figure 32 B-ISG Serpentine Belt Drive with Twin Tensioner
It is noted that in Figure 31 and Figure 32 showing the schematic of the Twin Tensioner and
the overall system respectively that for the purpose of the geometric computations the forward
direction follows the convention of the numbering order counterclockwise The numbering
order is in reverse to the actual direction of the belt motion which is in the clockwise direction in
this study The fourth pulley is identified as an ISG unit pulley However the properties used
for the ISG pulley‟s geometry inertia stiffness and damping is modeled as a conventional
Modeling of B-ISG 28
alternator pulley This pulley is conceptualized as an ISG when it is modeled as the driving
pulley at which point the requirements of the ISG are solved for and its non-inertia attributes
are not needed to be ascribed
Figure 33 shows the geometric attributes needed to resolve the wrap angle of the belt on each
pulley Variables (XiYi) and XYfi XYbi XYfbi and XYbfi are the ith pulley centre coordinates and
its possible belt connection points respectively Length Lfi is the length of the span connecting
the points XYfi and XYf(i+1) or XYbi and XYb(i+1) on the ith and (i+1)th pulleys respectively
Similarly Lbi is the length of the span between XYfbi and XYfb(i+1) or XYbfi and XYbf (i+1) on the
ith and (i+1)th pulleys respectively Angles αi θfi and θbi represent the angle between a line
connecting the ith and (i+1)th pulley centres and the angles of the belt connection spans with
lengths Lfi and Lbi respectively Ri is the radius of the ith pulley
Figure 33 Angles Coordinates and Possible Belt Contact Points for the ith and i+1th Pulleys
[modified] [51]
Modeling of B-ISG 29
The angle between the horizontal and the line connecting the ith and (i+1)th pulley centres αi is
calculated using Zhen‟s method [51] This method uses the pulley‟s coordinates and a cosine
trigonometric relation
i acos
Xi 1
Xi
Xi 1
Xi
2
Yi 1
Yi
2
Yi 1
Yi
if
(31a)
i 2 acos
Xi 1
Xi
Xi 1
Xi
2
Yi 1
Yi
2
Yi 1
Yi
if
(31b)
The lengths for connecting the possible belt spans are described by the variables Lfi and Lbi
The centre point coordinates and the radii of the pulleys are related through the solution of
triangles which they form to define values of the possible belt span lengths
Lfi
Xi 1
Xi
2
Yi 1
Yi
2
Ri 1
Ri
2
(32a)
Lbi
Xi 1
Xi
2
Yi 1
Yi
2
Ri 1
Ri
2
(32b)
The set of possible belt span lengths leads to the calculation of θfi and θbi the angles between the
line connecting the ith and (i+1)th pulley centres and the possible contact point on the pulley
perimeter
Modeling of B-ISG 30
(33a)
(33b)
The array of possible belt connection points comes about from the use of the pulley centre
coordinates and their radii and the sine of the sum or differences of αi and θfi or θbi The angle
αi is calculated in equations (31a) and (31b) and angles θfi and θbi are calculated in equations
(33a) and (33b) The formula to compute the array of points is shown in equations (34) and
(35) for the ith and (i+1)th pulleys Equation (34) describes the forward belt connection point
on the ith pulley which is in the span leading forward to the next (i+1)th pulley
(34a)
(34b)
(34c)
(34d)
bi atan
Lbi
Ri
Ri 1
Modeling of B-ISG 31
Equation (35) describes the backward belt connection point on the ith pulley This point sits on
the ith pulley in the contacting belt span which leads backward to connect with the (i-1)th
pulley
(35a)
(35b)
(35c)
(35d)
The selection of the coordinates from the array of possible connection points requires a graphic
user interface allowing for the points to be chosen based on observation This was achieved
using the MathCAD software package as demonstrated in the MathCAD scripts found in
Appendix C The belt connection points can be chosen so as to have a pulley on the interior or
exterior space of the serpentine belt drive The method used in the thesis research was to plot the
array of points in the MathCAD environment with distinct symbols used for each pair of points
and to select the belt connection points accordingly By observation of the selected point types
the type of belt span connection is also chosen Selected point and belt span types are shown in
Table 31
Modeling of B-ISG 32
Table 31 Selected Contact Point Types on the ith Pulley and Types for the ith Belt Span
Pulley Forward Contact
Point
Backwards Contact
Point
Belt Span
Connection
1 Crankshaft XYf1 XYbf21 Lf1
2 Air Conditioning XYfb2 XYf22 Lb2
3 Tensioner 1 XYbf3 XYfb23 Lb3
4 AlternatorISG XYfb4 XYbf24 Lb4
5 Tensioner 2 XYbf5 XYfb25 Lb5
The inscribed angles βji between the datum and the forward connection point on the ith pulley
and βji between the datum and its backward connection point are found through solving the
angle of the arc along the pulley circumference between the datum and specified point The
wrap angle ϕi is found as the difference between the two inscribed angles for each connection
point on the pulley The angle between each belt span and the horizontal as well as the initial
angle of the tensioner arms are found using arctangent relations Furthermore the total length of
the belt is determined by the sum of the lengths of the belt spans
34 Equations of Motion for a B-ISG System with a Twin Tensioner
341 Dynamic Model of the B-ISG System
3411 Derivation of Equations of Motion
This section derives the inertia damping stiffness and torque matrices for the entire system
Moment equilibrium equations are applied to each rigid body in the system and net force
equations are applied to each belt span From these two sets of equations the inertia damping
Modeling of B-ISG 33
and stiffness terms are grouped as factors against acceleration velocity and displacement
coordinates respectively and the torque matrix is resolved concurrently
A system whose motion can be described by n independent coordinates is called an n-DOF
system Consider the free body diagram of the Twin Tensioner in Figure 34 in which each
pulley of inertia Ii is supported on an arm of inertia Iti It is assumed that the pulleys are
constrained to rotate about their respective central axes and the arms are free to rotate about their
respective pivot points then at any time the position of each pulley can be described by a
rotational coordinate θi(t) and a coordinate θti(t) can denote the rotation of each arm Thus the
tensioner system comprises of four rigid bodies where each is described by one coordinate and
hence is a four-DOF system It is important to note that each rigid body is treated as a point
mass In addition inertial rotation in the positive direction is consistent with the direction of belt
motion The belt span tensions Ti and coupled radii Ri apply moments to the pulleys
Figure 34 Twin Tensioner Free Body Diagram of a Four-Degree of Freedom System
Modeling of B-ISG 34
For the serpentine belt system considered in the thesis research there are seven rigid bodies each
having a one-DOF of motion The EOM for a seven-DOF system form second-order coupled
differential equations meaning that each equation includes all of the general coordinates and
includes up to the second-order time derivatives of these coordinates The EOM can be
obtained by applying D‟Alembert‟s principle that the sum of the moments taken about any point
including the couples equals to zero Therefore the inertial couple the product of the inertia and
acceleration is equated to the moment sum as shown in equation (35)
I ∙ θ = ΣM (35)
The moment equilibrium equations for the Twin Tensioner in Figure 34 where the positive
direction is in the clockwise direction are shown in equations (36) through to (310) The
numbering convention used for each rigid body corresponds to the labeled serpentine belt drive
system shown in Figure 32 Qi represents the required torque of the ith rigid body ci is the
damping constant of the ith rigid body βji is the angle of orientation for the ith belt span and
120597120579119905119894 120579 119905119894 and 120579 119905119894 are the angular displacement angular velocity and angular acceleration of the ith
tensioner arm The initial angle of the ith tensioner arm is described by θtoi
minusI3 ∙ θ 3 = T3 ∙ R3 minus T2 ∙ R3 minus Q3 + c3 ∙ θ 3 (36)
minusI5 ∙ θ 5 = minusT4 ∙ R5 + T5 ∙ R5 minus Q5 + c5 ∙ θ 5 (37)
Modeling of B-ISG 35
It1 ∙ θ t1 = minusTt1 ∙ Lt1 ∙ sin θto 1 minus βj4 + sin θto 1 minus βj5 minus kt ∙ partθt1 minus partθt2 minus kt1 ∙
partθt1 minus ct ∙ partθ t1 minus partθ t2 minus ct1 ∙ partθ t1 (38)
It2 ∙ θ t2 = minusTt2 ∙ Lt2 ∙ sin θto 2 minus βj2 + sin θto 1 minus βj3 minus kt ∙ partθt2 minus partθt1 minus kt2 ∙ partθt2 minus
ct ∙ partθ t2 minus partθ t1 minus ct2 ∙ partθ t2 (39)
partθt1 = θt1 minus θto 1 (310a)
partθt2 = θt2 minus θto 2 (310b)
The free body diagrams for the remaining rigid bodies crankshaft pulley air conditioner pulley
and ISG pulley are in the general form of Figure 35 The sum of the moments about the axes of
rotation are taken for these structures in equations (311) through to (313)
Figure 35 Free Body Diagram for Non-Tensioner Pulleys
Modeling of B-ISG 36
I1 ∙ θ 1 = T5 ∙ R1 minus T1 ∙ R1 + Q1 minus c1 ∙ θ 1 (311)
I2 ∙ θ 2 = T1 ∙ R2 minus T2 ∙ R2 + Q2 minus c2 ∙ θ 2 (312)
I4 ∙ θ 4 = T3 ∙ R4 minus T4 ∙ R4 + Q4 minus c4 ∙ θ 4 (313)
The relationship between belt tensions and rigid body displacements is in the general form of
equation (314) where 119827119836 and 119827119844 are damping and stiffness matrices due to the belt respectively
with each factorized by a radial arm length This relationship is described for each span in
equations (315) through to (320) The belt damping constant for the ith belt span is cib
119827prime = 119827119836 ∙ 120521 + 119827119844 ∙ 120521 (314)
T1 = To + k1b ∙ R1 ∙ θ1 minus R2 ∙ θ2 + c1
b ∙ [R1 ∙ θ 1 minus R2 ∙ θ 2] (315)
T2 = To + k2b ∙ R2 ∙ θ2 minus R3 ∙ θ3 + Lt1 ∙ [sin θto 1 minus βj2 ] ∙ (θt1 minus θto 1) + c2
b ∙ [R2 ∙ θ 2 minus R3 ∙
θ 3 + Lt1 ∙ [sin θto 1 minus βj2 ] ∙ (θ t1)] (316)
T3 = To + k3b ∙ R3 ∙ θ3 minus R4 ∙ θ4 + Lt1 ∙ [sin θto 1 minus βj3 ] ∙ (θt1 minus θto 1) + c3
b ∙ [R3 ∙ θ 3 minus R4 ∙
θ 4 + Lt1 ∙ [sin θto 1 minus βj3 ] ∙ (θ t2)] (317)
Modeling of B-ISG 37
T4 = To + k4b ∙ R4 ∙ θ4 minus R5 ∙ θ5 + Lt2 ∙ [sin θto 2 minus βj4 ] ∙ (θt2 minus θto 2) + c4
b ∙ [R4 ∙ θ 4 minus R5 ∙
θ 5 + Lt2 ∙ [sin θto 2 minus βj4 ] ∙ (θ t1)] (318)
T5 = To + k5b ∙ R5 ∙ θ5 minus R1 ∙ θ1 + Lt2 ∙ [sin θto 2 minus βj5 ] ∙ (θt2 minus θto 2) + c5
b ∙ [R5 ∙ θ 5 minus R1 ∙
θ 1 + Lt2 ∙ [sin θto 2 minus βj5 ] ∙ (θ t2)] (319)
Tprime = Ti minus To (320)
Since the applied torques on the tensioner pulleys Q3 and Q4 are zero the static equilibrium
equation of the pulleys show that the adjacent spans of each tensioner pulley are equal to each
other Hence equations (321) and (322) are denoted as follows
Tt1 = T2 = T3 (321)
Tt2 = T4 = T5 (322)
Equations (310a) (310b) and (314) through to (322) are substituted into the EOMs described
in equations (36) to (39) and (311) to (313) The newly formed equations can be arranged
and written in matrix form as shown in equations (323) through to (328) The general
coordinate matrix 120521 and its first and second derivatives are shown in the EOM below
119816 ∙ 120521 + 119810 ∙ 120521 + 119818 ∙ 120521 = 119824 (323)
Modeling of B-ISG 38
The inertia matrix I includes the inertia of each rigid body in its diagonal elements The
damping matrix C includes variables 119888119894119887 the damping of the ith belt span 119877119894 its radius 120573119895119894 its
angle 119871119905119894 the ith tensioner arm‟s length 120579119905119900119894 its initial pivot angle and 119888119905 and 119888119905119894 the ith
tensioner arm viscous damping constants Stiffness matrix K contains 119896119894119887 the ith belt span
stiffness and 119896119905 and 119896119905119894 the ith tensioner arm stiffness constants and akin to the damping
matrix the variables 119877119894 119871119905119894 120579119905119900119894 and 120573119895119894 The belt span stiffness is computed in equation
(326b) where 119870119887 represents the belt cord stiffness 119896119887 is the belt factor obtained from
experimental data 120573119895119894 is the angle of orientation for the span between the jth and ith pulleys and
ϕi is the belt wrap angle on the ith pulley
Modeling of B-ISG 39
119816 =
I1 0 0 0 0 0 00 I2 0 0 0 0 00 0 I3 0 0 0 00 0 0 I4 0 0 00 0 0 0 I5 0 00 0 0 0 0 It1 00 0 0 0 0 0 It2
(324)
119810 =
c1
b ∙ R12 + c5
b ∙ R12 + c1 minusc1
b ∙ R1 ∙ R2 0 0 minusc5b ∙ R1 ∙ R5 0 c5
b ∙ R1 ∙ Lt2 ∙ sin θto 2 minus βj5
minusc1b ∙ R1 ∙ R2 c2
b ∙ R22 + c1
b ∙ R22 + c2 minusc2
b ∙ R2 ∙ R3 0 0 c2b ∙ R2 ∙ Lt1 ∙ sin θto 1 minus βj2 0
0 minusc2b ∙ R2 ∙ R3 c3
b ∙ R32 + c2
b ∙ R32 + c3 minusc3
b ∙ R3 ∙ R4 0 C36 0
0 0 minusc3b ∙ R3 ∙ R4 c4
b ∙ R42 + c3
b ∙ R42 + c4 minusc4
b ∙ R4 ∙ R5 minusc3b ∙ R4 ∙ Lt1 ∙ sin θto 1 minus βj3 c4
b ∙ R4 ∙ Lt2 ∙ sin θto 2 minus βj4
minusc5b ∙ R1 ∙ R5 0 0 minusc4
b ∙ R4 ∙ R5 c5b ∙ R5
2 + c4b ∙ R5
2 + c5 0 C57
0 0 0 0 0 ct +ct1 minusct
0 0 0 0 0 minusct ct +ct1
(325a)
C36 = 1198773 ∙ 1198711199051 ∙ [1198883119887 ∙ 119904119894119899 1205791199051199001 minus 1205731198953 minus 1198882
119887 ∙ 119904119894119899 1205791199051199001 minus 1205731198952 ] (325b)
C57 = 1198775 ∙ 1198711199052 ∙ [1198885119887 ∙ 119904119894119899 1205791199051199002 minus 1205731198955 minus 1198884
119887 ∙ 119904119894119899 1205791199051199002 minus 1205731198954 ] (325c)
Modeling of B-ISG 40
119818 =
k1
b ∙ R12 + k5
b ∙ R12 minusk1
b ∙ R1 ∙ R2 0 0 minusk5b ∙ R1 ∙ R5 0 k5
b ∙ R1 ∙ Lt2 ∙ sin θto 2 minus βj5
minusk1b ∙ R1 ∙ R2 k2
b ∙ R22 + k1
b ∙ R22 minusk2
b ∙ R2 ∙ R3 0 0 k2b ∙ R2 ∙ Lt1 ∙ sin θto 2 minus βj2 0
0 minusk2b ∙ R2 ∙ R3 k3
b ∙ R32 + k2
b ∙ R32 minusk3
b ∙ R3 ∙ R4 0 R3 ∙ Lt1 ∙ [k3b ∙ sin θto 1 minus βj3 minus k2
b ∙ sin θto 1 minus βj2 ] 0
0 0 minusk3b ∙ R3 ∙ R4 k4
b ∙ R42 + k3
b ∙ R42 minusk4
b ∙ R4 ∙ R5 minusk3b ∙ R4 ∙ Lt1 ∙ sin θto 1 minus βj3 k4
b ∙ R4 ∙ Lt2 ∙ sin θto 2 minus βj4
minusk5b ∙ R1 ∙ R5 0 0 minusk4
b ∙ R4 ∙ R5 k5b ∙ R5
2 + k4b ∙ R5
2 0 R5 ∙ Lt2 ∙ [k5b ∙ sin θto 2 minus βj5 minus k4
b ∙ sin θto 2 minus βj4 ]
0 0 0 0 0 kt +kt1 minuskt
0 0 0 0 0 minuskt kt +kt1
(326a)
k119894b =
Kb
Li + kb ∙ Ri ∙ϕi+1
2 + Ri ∙ϕi
2
(326b)
120521 =
θ1
θ2
θ3
θ4
θ5
partθt1
partθt2
(327)
119824 =
Q1
Q2
Q3
Q4
Q5
Qt1
Qt2
(328)
Modeling of B-ISG 41
3412 Modeling of Phase Change
The phase change from the crankshaft pulley being the driving pulley to the ISG pulley being the
driving pulley is described through a conditional equality based on a set of Boolean conditions
When the crankshaft is driving the rows and the columns of the EOM are swapped such that the
new order for rows and columns is 1 (crankshaft pulley) 4 (ISG pulley) 2 (air conditioner
pulley) 3 (tensioner 1 pulley) 5 (tensioner 2 pulley) 6 (tensioner arm 1) and 7 (tensioner arm 2)
When the ISG is driving the order is the same except that the second row and second column
terms relating to the ISG pulley become the first row and first column while the crankshaft
pulley terms (previously in the first row and first column) become the second row and second
column Hence the order for all rows and columns of the matrices making up the EOM in
equation (322) switches between 1423567 (when the crankshaft pulley is driving) and
4123567 (when the ISG pulley is driving) For example in the crankshaft driving and ISG
driving phases the general coordinate matrix and the inertia matrix become the following
120521119940 =
1205791
1205794
1205792
1205793
1205795
1205971205791199051
1205971205791199052
and 120521119938 =
1205794
1205791
1205792
1205793
1205795
1205971205791199051
1205971205791199052
(329a amp b)
119816119940 =
I1 0 0 0 0 0 00 I4 0 0 0 0 00 0 I2 0 0 0 00 0 0 I3 0 0 00 0 0 0 I5 0 00 0 0 0 0 It1 00 0 0 0 0 0 It2
and 119816119938 =
I4 0 0 0 0 0 00 I1 0 0 0 0 00 0 I2 0 0 0 00 0 0 I3 0 0 00 0 0 0 I5 0 00 0 0 0 0 It1 00 0 0 0 0 0 It2
(329c amp d)
Modeling of B-ISG 42
where subscripts c and a denote the crankshaft pulley driving phase and the ISG pulley driving
phase respectively
The condition for phase change is based on the engine speed n in units of rpm Equation (330)
demonstrates the phase change
H(n) = 1 119899 ge 750 (Crankshaft driving phase)0 119899 lt 750 (ISG driving phase)
(330)
When the crankshaft pulley is the driving pulley the ISG pulley becomes the driven pulley and
following suit when the ISG pulley is the driving pulley the crankshaft pulley becomes the
driven pulley These modes of operation mean that the system will predict two different sets of
natural frequencies and mode shapes Using a Boolean condition to allow for a swap between
the first and second rows as well as between the first and second columns of the EOM matrices
I C and K allows for a continuous plot of the dynamic response to be plotted for the ISG pulley
throughout its driving and driven phases as well as for that of the crankshaft pulley
3413 Natural Frequencies Mode Shapes and Dynamic Responses
Assuming the system undergoes simple harmonic motion its matrix of natural frequencies 120596119899
and modeshapes are found by solving the eigenvalue problem shown in equation (331a)
ωn ∙ 119816120784120784 minus 11981822 ∙ 120495m = 120782 (331a)
The displacement amplitude Θm is denoted implicitly in equation (331d)
Modeling of B-ISG 43
120521119846 = θ2 θ3 θ5 θ6 partθt1 partθt2 T for H n = 1 (331b)
120521119846 = θ1 θ3 θ5 θ6 partθt1 partθt2 T for H n = 0 (331c)
θm = 120495119846 ∙ sin(ω ∙ t) (331d)
I2 and K22 are submatrices of I and K respectively meaning the first row and column of each of
the original matrices are removed The eigenvalue problem is reached by considering the
undamped and unforced motion of the system Furthermore the dynamic responses are found by
knowing that the torque requirements in the matrixndash Qm for the driven pulleys and the tensioner
arms are zero in the dynamic case which signifies a response of the system to an input solely
from the driving pulley
I1 120782120782 119816120784120784
θ 1120521 119846
+ C11 119810120783120784119810120784120783 119810120784120784
θ 1120521 119846
+ K11 119818120783120784
119818120784120783 119818120784120784 θ1
120521119846 =
QCS ISG
119824119846 (332)
1
In the case of equation (331) θm is the submatrix identified in equations (331b) through to
(331d) Therein θ1 denotes the general coordinate for the driving pulley so that in the case the
phase change function H(n) is equal to zero θ1 becomes θ4 and the order of the rows and
columns for the remaining matrices correspond to the value of H(n) as mentioned earlier in
section 3412 For simple harmonic motion the motion of the driven pulleys are described as
1 The driving torque 119876119862119878119868119878119866 denotes the crankshaft torque 119876119862119878 when the crankshaft pulley is driving or the ISG
torque 119876119868119878119866 when the ISG pulley is in its driving function
Modeling of B-ISG 44
θm = 120495119846 ∙ sin(ω ∙ t) (333)
The dynamic response of the system to an input from the driving pulley under the assumption of
sinusoidal motion is expressed in equation (334)
120495119846 = [(119818120784120784 minusω2 ∙ 119816120784120784) + 119895ω ∙ 119810120784120784]minus1 ∙ (119818120784120783 + 119895ω ∙ 119810120784120783) ∙ Θ1 (334)
3414 Crankshaft Pulley Driving Torque Acceleration and Displacement
Subsequently the crankshaft pulley driving torque acceleration and displacement are firstly
discussed It is assumed in the thesis research for the purpose of modeling that the engine
serving the crankshaft is of the four cylinder type The input torque provided by a four-cylinder
engine is assumed to be dominated by two torque pulses per revolution of the crankshaft which
is represented by the factor of 2 on the steady component of the angular velocity in equation
(335) The torque requirement of the crankshaft pulley when it is the driving pulley is
Qc = qc ∙ sin(2 ∙ ωcs ∙ t) (335)
The amplitude of the required crankshaft torque qc is expressed in equation (336) and is
derived from equation (332)
qc = K11 minus ω2 ∙ I1 + 119895 ∙ ω ∙ C11 ∙ Θ1 + (119818120783120784 + 119895 ∙ ω ∙ 119810120783120784) ∙ 120495119846 (336)
Modeling of B-ISG 45
The angular frequency for the system in radians per second (rads) ω when the crankshaft
pulley is driving can be found as a function of the engine speed in rotations per minute (rpm) n
and by taking into account the double pulse per crankshaft revolution
ω = 2 ∙ ωcs = 4 ∙ π ∙ n
60
(337)
The system is considered when the amplitude of the crankshaft‟s angular acceleration is assumed
to be constant and equal to 650 rads2 during the crankshaft pulley driving phase The amplitude
of the excitation angular input from the engine is shown in equation (339b) and is found as a
result of (338)
θ 1CS = 650 ∙ sin(ω ∙ t) (338)
θ1CS = minus650
ω2sin(ω ∙ t)
(339a) where
Θ1CS = minus650
ω2
(339b)
Modeling of B-ISG 46
3415 ISG Pulley Driving Torque Acceleration and Displacement
Secondly the torque acceleration and the displacement of the ISG pulley in its driving phase is
discussed The torque for the ISG when it is in its driving function is assumed constant Ratings
for the ISG are taken from experiments performed by researchers Wezenbeek et al [21] on an
Energen 5 High Output Belt-alternator-starter (BAS) unit from Delphi The 12-Volt BAS which
can also be called a B-ISG was reported to have a maximum allowable speed of 18000 rpm [21]
As well it was noted that the ISG pulley was sized appropriately and the engine speed was
limited to ensure that an over-speed condition of the ISG pulley would not occur [21] The stall
torque rating for the Energen ISG was reported to be 48 Nm at the electric machine shaft [21]
The formula for the torque of a permanent magnet DC motor for any given speed (equation
(340)) is used to approximate the torque of the ISG in its driving mode[52]
QISG = Ts minus (N ∙ Ts divide NF) (340)2
Knowing the stall torque (the torque at 0 rpm) Ts and the maximum rpm of the motor when it is
not under load NF allows for the torque produced 119876119868119878119866 to be found for a given motor speed N
Experimental data from Litens Automotive Group [53] shows that for engine fire-up upon ISG
re-start the crankshaft must go from 0 rpm to an idle speed of approximately 750 rpm The
pulley installed on the ISG shaft in the case of the thesis research has a diameter of 6820 mm
(DISG) while that of the crankshaft has a diameter of 20065 mm (DCS) which makes the
2 The equation for the required driving torque for the ISG pulley may also be computed from the formula shown in
(336) Figure 315 for the driving torque of the ISG pulley shows that (336) and (340) produce similar results for
the required driving torque See Figure 315 for comparison of these results
Modeling of B-ISG 47
crankshaft to ISG pulley ratio approximately 2941 This ratio is used to determine the ISG
speed in equation (341)
nISG = nCS ∙DCS
DISG
(341)
For a crankshaft speed of 750 rpm the required ISG speed nISG is found from equation (341) to
be approximately 220656 rpm Thus the ISG torque during start-up is found from equation
(340) where N is equated to the value of nISG NF is assumed to be 18000 rpm and the stall
torque is allotted the value of 48 Nm The result is a required torque of approximately 42 Nm
for the ISG The acceleration of the ISG pulley is found by taking into account the torque
developed by the rotor and the polar moment of inertia of the pulley [54]
A1ISG = θ 1ISG = QISG IISG (342)
In torsional motion the function for angular displacement of input excitation is sinusoidal since
the electric motor is assumed to be resonating As a result of constant angular acceleration the
angular displacement of the ISG pulley in its driving mode is found in equation 343
θ1ISG = Θ1ISG ∙ sin(ωISG ∙ t) (343)
Knowing that acceleration is the second derivative of the displacement the amplitude of
displacement is solved subsequently [55]
Modeling of B-ISG 48
θ 1ISG = minusωISG2 ∙ Θ
1ISG ∙ sin(ωISG ∙ t) (344)
θ 1ISG = minusωISG2 ∙ Θ
1ISG
(345a)
Θ1ISG =minusQISG IISG
ωISG2
(345b)
In this case the angular frequency for the system 120596 is equivalent to 120596119868119878119866 that is the angular
frequency of the ISG pulley which can be expressed as a function of its speed in rpm
ω = ωISG =2 ∙ π ∙ nISG
60
(346a)
or in terms of the crankshaft rpm by substituting equation (341) into (346a)
ω =2 ∙ π
60∙ nCS ∙
DCS
DISG
(346b)
3416 Tensioner Arms Dynamic Torques
The dynamic torque for the tensioner arms are shown in equations (347) and (348)
Qt1 = kt + kt1 + 119895 ∙ ω ∙ (ct + ct1) ∙ (Θt1 ∙ Θ1) (347)
Modeling of B-ISG 49
Qt2 = kt + kt2 + 119895 ∙ ω ∙ (ct + ct2) ∙ (Θt2 ∙ Θ1) (348)
3417 Dynamic Belt Span Tensions
Furthermore the dynamic belt span tensions are derived from equation (314) and described in
matrix form in equations (349) and (350)
119827prime = 119895 ∙ ω ∙ 119827119836 + 119827119844 ∙ 120495119847 (349)
where
120495119847 = Θ1
120495119846 (350)
342 Static Model of the B-ISG System
It is fitting to pursue the derivation of the static model from the system using the dynamic EOM
For the system under static conditions equations (314) and (323) simplify to equations (351)
and (352) respectively
119827prime = 119827119844 ∙ 120521 (351)
119824 = 119818 ∙ 120521 (352)
Modeling of B-ISG 50
As noted in other chapters the focus of the B-ISG tensioner investigation especially for the
parametric and optimization studies in the subsequent chapters is to determine its effect on the
static belt span tensions Therein equations (351) and (352) are used to derive the expressions
for static tension in each belt span 119931prime is the tension solely due to deflection of the belt span
Equation (320) demonstrates the relationship between the tension due to belt response and the
initial tension also known as pre-tension The static tension 119931 is found by summing the initial
tension 1198790 with the expression for the dynamic tension shown in equations (315) through to
(319) and by substituting the expressions for the rigid bodies‟ displacements from equation
(352) and the relationship shown in equation (320) into equation (351)
119827 = 119827119844 ∙ (119818minus120783 ∙ 119824) + T0 (353)3
35 Simulations
The methods used to develop the geometric dynamic and static models of the Twin Tensioner B-
ISG system in the previous sections of this chapter were verified using the software FEAD The
input data for a single tensioner B-ISG system was entered into FEAD [51] to simulate the
crankshaft driving phase alone since the ISG phase is inapplicable in the FEAD [51] software
FEAD‟s [51] results agreed with those found in the simulation of the single tensioner system‟s
geometric model and EOMs in MathCAD software Furthermore the geometric simulation
3 For the purposes of the static tension the original order for the rows and columns of the stiffness matrix K and the
torque matrix Q are maintained as depicted in (326) and (328) In performing the inverse of K and its
multiplication with Q the first row and first column (in the case of the K matrix) are removed in the crankshaft
driving case whereas the fourth row and fourth column are removed in the ISG driving case Then the product for
the displacement120637 resulting from (119922minus120783 ∙ 119928) has a zero added to serve as the first element of the column matrix in
the crankshaft driving case or as the fourth element in the ISG driving case This is shown in detail in Appendix
C3 of MathCAD scripts
Modeling of B-ISG 51
results for both of the twin and single tensioner B-ISG systems were found to be in agreement as
well
351 Geometric Analysis
The initial coordinate inputs for the centre points of the five pulleys and the Twin Tensioner
pivot point are described as Cartesian coordinates and shown in Table 32 which also includes
the diameters for the pulleys
Table 32 Coordinate Points for Pulley Centres and Twin Tensioner Pivot [56]
Rigid Body Diameter [mm] Cartesian Coordinate [Xi Yi] [mm]
1Crankshaft Pulley 20065 [00]
2 Air Conditioner Pulley 10349 [224 -6395]
3 Tensioner Pulley 1 7240 [292761 87]
4 ISG Pulley 6820 [24759 16664]
5 Tensioner Pulley 2 7240 [12057 9193]
6 Tensioner Arm Pivot --- [201384 62516]
The geometric results for the B-ISG system are shown in Table 33
Table 33 Geometric Results of B-ISG System with Twin Tensioner
Pulley Forward
Connection Point
Backward
Connection Point
Wrap
Angle
ϕi (deg)
Angle of
Belt Span
βji (deg)
Length of
Belt Span
Li (mm)
1 Crankshaft [-6818-100093] [453889475] 202996 356103 227828
2 Air
Conditioning [275299-5717] [220484 -115575] 101425 277528 14064
3 Tensioner 1 [25887599735] [256873 82257] 28126 69403 58658
4 ISG [218374184225] [27951154644] 169554 58956 129513
5 Tensioner 2 [10419659645] [15158673262] 8585 333107 65949
Total Length of Belt (mm) 1243
Modeling of B-ISG 52
352 Dynamic Analysis
The dynamic results for the system include the natural frequencies mode shapes driven pulley
and tensioner arm responses the required torque for each driving pulley the dynamic torque for
each tensioner arm and the dynamic tension for each belt span These results for the model were
computed in equations (331a) through to (331d) for natural frequencies and mode shapes in
equation (334) for the driven pulley and tensioner arm responses in equation (336) for the
crankshaft pulley driving torque in equation (340) for the ISG pulley driving torque in
equations (347) and (348) for the tensioner arm torques and lastly in equation (349) for the
dynamic tension of each belt span Figures 36 through to 323 respectively display these
results The EOM simulations can also be contrasted with those of a similar system being a B-
ISG serpentine belt drive that is equipped with a single tensioner arm and single tensioner pulley
which interacts only in the span bridging the ISG and crankshaft pulleys The EOM for a B-ISG
with a single tensioner is presented in Appendix B
It is assumed for the sake of the dynamic and static computations that the system
does not have an isolator present on any pulley
has negligible rotational damping of the pulley shafts
has negligible belt span damping and that this damping does not differ amongst
spans (ie c1b = ∙∙∙ = ci
b = 0)
has quasi-static belt stretch where its belt experiences purely elastic deformation
has fixed axes for the pulley centres and tensioner pivot
has only one accessory pulley being modeled as an air conditioner pulley and
Modeling of B-ISG 53
has a rotational belt response that is decoupled from the transverse response of the
belt
The input parameter values of the dynamic (and static) computations as influenced by the above
assumptions for the present system equipped with a Twin Tensioner are shown in Table 34
Table 34 Data for Input Parameters used in Dynamic and Static Computations [56]
Rigid Body Data
Pulley Inertia
[kg∙mm2]
Damping
[N∙m∙srad]
Stiffness
[N∙mrad]
Required
Torque
[Nm]
Crankshaft 10 000 0 0 4
Air Conditioner 2 230 0 0 2
Tensioner 1 300 1x10-4
0 0
ISG 3000 0 0 5
Tensioner 2 300 1x10-4
0 0
Tensioner Arm 1 1500 1000 10314 0
Tensioner Arm 2 1500 1000 16502 0
Tensioner Arm
couple 1000 20626
Belt Data
Initial belt tension [N] To 300
Belt cord stiffness [Nmmmm] Kb 120 00000
Belt phase angle at zero frequency [deg] φ0deg 000
Belt transition frequency [Hz] ft 000
Belt maximum phase angle [deg] φmax 000
Belt factor [magnitude] kb 0500
Belt cord density [kgm3] ρ 1000
Belt cord cross-sectional area [mm2] A 693
Modeling of B-ISG 54
These values are for the driven cases for the ISG and crankshaft pulleys respectively In the
driving case for either pulley the inertia of the rigid body is defined as 1 kg∙mm2 and the driving
torque is determined in equations (335) and (340) for the crankshaft and ISG pulleys
respectively
It is noted that because of the belt data for the phase angle at zero frequency the transition
frequency and the maximum phase angle are all zero and hence the belt damping is assumed to
be constant between frequencies These three values are typically used to generate a phase angle
versus frequency curve for the belt where the phase angle is dependent on the frequency The
curve defined by equation (354) is normally symmetric with the lowest phase angle achieved at
0 Hz and the highest phase angle achieved at the prescribed transition frequency f The belt
damping would then be found by solving for cb in the following equation
tanφ = cb ∙ 2 ∙ π ∙ f (354)
Nevertheless the assumption for constant damping between frequencies is also in harmony with
the remaining assumptions which assume damping of the belt spans to be negligible and
constant between belt spans
3521 Natural Frequency and Mode Shape
The set of natural frequencies and mode shapes for the system are shown in Figures 36 and 37
under the cases of the ISG pulley driving and the crankshaft pulley driving The forcing
frequency for the system differs for each case due to the change in driving pulley Modeic and
Modeia denote the ith rigid body according to the numbering convention used in Figure 32 in
the crankshaft and ISG driving cases respectively
Modeling of B-ISG 55
Natural Frequency ωn [Hz]
Crankshaft Pulley ΔΘ4
Air Conditioner Pulley ΔΘ
Tensioner Pulley 1 ΔΘ
Tensioner Pulley 2 ΔΘ
Tensioner Arm 1 ΔΘ
Tensioner Arm 2 ΔΘ
Figure 36a ISG Driving Case Natural Frequency of System and Mode Shapes for Responsive
Rigid Bodies
Figure 36b ISG Driving Case First Mode Responses
4 ΔΘ signifies the term amplitude of angular displacement for the indicated rigid body
Modeling of B-ISG 56
Figure 36c ISG Driving Case Second Mode Responses
Natural Frequency ωn [Hz]
ISG Pulley ΔΘ5
Air Conditioner Pulley ΔΘ
Tensioner Pulley 1 ΔΘ
Tensioner Pulley 2 ΔΘ
Tensioner Arm 1 ΔΘ
Tensioner Arm 2 ΔΘ
Figure 37a Crankshaft Driving Case Natural Frequency of System and Mode Shapes for
Responsive Rigid Bodies
5 ΔΘ signifies the term amplitude of angular displacement for the indicated rigid body
Modeling of B-ISG 57
Figure 37b Crankshaft Driving Case First Mode Responses
Figure 37c Crankshaft Driving Case Second Mode Responses
Modeling of B-ISG 58
3522 Dynamic Response
The dynamic response specifically the magnitude of angular displacement for each rigid body is
plotted in Figures 38 through to 314 as a function of the crankshaft pulley speed n This is
fitting to the analysis since the crankshaft pulley‟s rpm decides the mode of operation for the
system in particular it determines whether the crankshaft pulley or ISG pulley is the driving
pulley
Figure 38 Crankshaft Pulley Dynamic Response (for crankshaft driven case)
Figure 39 ISG Pulley Dynamic Response (for ISG driven case)
Modeling of B-ISG 59
Figure 310 Air Conditioner Pulley Dynamic Response
Figure 311 Tensioner Pulley 1 Dynamic Response
Crankshaft Driving Phase ISG
Driving
Phase
Crankshaft Driving Phase ISG
Driving
Phase
Modeling of B-ISG 60
Figure 312 Tensioner Pulley 2 Dynamic Response
Figure 313 Tensioner Arm 1 Dynamic Response
Crankshaft Driving Phase ISG
Driving
Phase
Crankshaft Driving Phase ISG
Driving Phase
Modeling of B-ISG 61
Figure 314 Tensioner Arm 2 Dynamic Response
3523 ISG Pulley and Crankshaft Pulley Torque Requirement
Figures 315 and 316 respectively showcase the required torques for the ISG pulley in its driving
mode and the crankshaft pulley in its driving mode
Figure 315 Required Driving Torque for the ISG Pulley
Crankshaft Driving Phase ISG
Driving Phase
Modeling of B-ISG 62
Figure 315 shows two plots for the required driving torque of the ISG pulley The dashed line
labeled as Q(n) simulates the application of equation (340) which models the ISG torque as a
permanent magnet DC motor The additional solid line labeled as qamod uses the formula in
equation (336) which determines the load torque of the driving pulley based on the pulley
responses Figure 315 provides a comparison of the results
Figure 316 Required Driving Torque for the Crankshaft Pulley
3524 Tensioner Arms Torque Requirements
The torque for the tensioner arms are shown in Figures 317 and 318
Modeling of B-ISG 63
Figure 317 Dynamic Torque for Tensioner Arm 1
Figure 318 Dynamic Torque for Tensioner Arm 2
3525 Dynamic Belt Span Tension
The dynamic tensions for the belt spans are shown in Figures 319 through to 323 The values
plotted represent the magnitude of the dynamic tension
Crankshaft Driving Phase ISG
Driving Phase
Crankshaft Driving Phase ISG
Driving
Phase
Modeling of B-ISG 64
Figure 319 Span 1 (between crankshaft and air conditioner) Dynamic Belt Span Tension
Figure 320 Span 2 (between air conditioner and tensioner 1) Dynamic Belt Span Tension
Crankshaft Driving Phase ISG
Driving Phase
Crankshaft Driving Phase ISG
Driving
Phase
Modeling of B-ISG 65
Figure 321 Span 3 (between tensioner 1 and ISG) Dynamic Belt Span Tension
Figure 322 Span 4 (between ISG and tensioner 2) Dynamic Belt Span Tension
Crankshaft Driving Phase ISG
Driving
Phase
Crankshaft Driving Phase ISG
Driving Phase
Modeling of B-ISG 66
Figure 323 Span 5 (between tensioner 2 and crankshaft) Dynamic Belt Span Tension
The dynamic results for the system serve to show the conditions of the system for a set of input
parameters The following chapter targets the focus of the thesis research by analyzing the affect
of changing the input parameters on the static conditions of the system It is the static results that
are the focus of the thesis and is thus analyzed in Chapters 4 and 5 in the parametric and
optimization studies respectively The dynamic analysis has been used to complete the picture of
the system‟s state under set values for input parameters
353 Static Analysis
Before looking at the static results for the system under study in brevity the static results for a
B-ISG serpentine belt drive with a single tensioner are presented In this theoretical system the
tensioner arm and tensioner pulley that interacts with the span between the air conditioner and
ISG pulleys of the original system are removed as shown in Figure 324
Crankshaft Driving Phase ISG
Driving
Phase
Modeling of B-ISG 67
Figure 324 B-ISG Serpentine Belt Drive with Single Tensioner
The complete static model as well as the dynamic model for the system in Figure 324 is found
in Appendix B The results of the static tension for each belt span of the single tensioner system
when the crankshaft is driving and the ISG is driving are shown in Table 35
Table 35 Static Solution for Belt Span Tensions in Crankshaft and ISG Driving Cases for a B-
ISG Serpentine Belt Drive with a Single Tensioner
Belt Span between Pulleys Crankshaft Driving Case [N] ISG Driving Case [N]
Crankshaft ndash Air Conditioner 481239 -361076
Air Conditioner ndash ISG 442588 -399727
ISG ndash Tensioner 29596 316721
Tensioner ndash Crankshaft 29596 316721
The tensions in Table 35 are computed with an initial tension of 300N This value for pre-
tension allows the spans in the case that the crankshaft pulley is driving to be suitably tensioned
Modeling of B-ISG 68
Whereas in the case of the ISG pulley driving the first and second spans are excessively slack
Therein an additional pretension of approximately 400N would be required which would raise
the highest tension span to over 700N This leads to the motivation of the thesis researchndash to
reduce the static belt tensions when the ISG is driving As mentioned in Chapter 1 these
tensions should be minimized to prolong belt life preferably within the range of 600 to 800N
As well it is desirable to minimize the amount of pretension exerted on the belt The current
design uses a pre-tension of 300N The above results would lead to a required pre-tension of
more than 700N to keep all spans of the belt suitably in tension (well above 0N) in order to allow
the belt to exhibit high performance in power transmission and come near to the safe threshold
This is the rationale for investigating a Twin Tensioner configuration shown in Figure 32 for
the B-ISG serpentine belt drive under study For the theoretical system with a Twin Tensioner
the following static results in Table 36 are achieved
Table 36 Static Solution for Belt Span Tensions in Crankshaft and ISG Driving Cases for a B-
ISG Serpentine Belt Drive with a Twin Tensioner
Belt Span between Pulleys Crankshaft Driving Case [N] ISG Driving Case [N]
Crankshaft ndash Air Conditioner 465848 -284152
Air Conditioner ndash Tensioner 1 427197 -322803
Tensioner 1 ndash ISG 427197 -322803
ISG ndash Tensioner 2 28057 393645
Tensioner 2 ndash Crankshaft 28057 393645
The results in Table 36 show that the span following the ISG in the case between the Tensioner
1 and ISG pulleys is less slack than in the former single tensioner set-up However there
remains an excessive amount of pre-tension required to keep all spans suitably tensioned
Modeling of B-ISG 69
36 Summary
The simulation of the model for the B-ISG system with the Twin Tensioner shows that the mode
shapes of the rigid bodies within the system (Figures 36a to 37c) are greater in magnitude when
the ISG pulley is driving than when the crankshaft pulley is driving The dynamic responses of
the system as shown in Figures 38 and 310 to 314 is small for the crankshaft pulley and are
negligible for the remaining driven bodies when the ISG is driving For the crankshaft driving
phase there is greater dynamic response for the driven rigid bodies of the system including for
that of the ISG pulley
As the engine speed increases the torque requirement for the ISG was found to vary between
approximately 41Nm and 54Nm (before dropping steeply to approximately 3Nm at an engine
speed of about 720rpm) when modeled after equation (336) or between approximately 48Nm
and 34Nm when modeled after equation (340) In contrast the torque for the crankshaft peaks
at approximately 92Nm and 52Nm at an approximate engine speed of 1450rpm and 5000rpm
respectively The dynamic torque of the first tensioner arm was shown to peak at approximately
15Nm at the transition engine speed 750rpm and again at approximately 15Nm at an
approximate engine speed of about 1450rpm A small peak of about 3Nm was also seen at an
engine speed of 5000rpm Similarly for the second tensioner arm a torque peak of
approximately 20Nm was seen at 750rpm and 1450rpm and a smaller peak of about 8Nm was
seen at an engine speed of 5000rpm
The trend for the dynamic tensions is that the peaks are highest in the ISG driving portion of the
B-ISG operation in most cases and in a few cases they are seen to be close in magnitude to that
Modeling of B-ISG 70
of the highest peaks in the crankshaft driving portion The dynamic tension for the first belt span
peaked at approximately 780Nm 830Nm and 500Nm at engine speeds of 750rpm 1450rpm
5000rpm respectively For the dynamic tension of the second belt span peaks of approximately
1250Nm 675Nm and 760Nm were seen at the same respective engine speeds for the 3 peaks of
the former span At these same engine speeds the third belt span exhibited tension peaks at
approximately 1400Nm 650Nm and 890Nm The tension peaks of the fourth span were
approximately 165Nm 150Nm and 100Nm at engine speeds 750rpm 1450rpm and 5000rpm
The fifth span experienced peaks of approximately 165Nm 170Nm and 120Nm at the same
respective engine speeds of the fourth span
The simulation results for the static tension of the B-ISG system with the Twin Tensioner reveal
that taut spans of the crankshaft driving case are lower in the ISG driving case The largest
change is an approximate decrease of 750N in spans 1 through 3 while spans 4 and 5 increase
by approximately 113N It can be seen that the spans in highest tension (1 2 and 3) in the
crankshaft driving phase become excessively slack in the ISG driving phase There is a smaller
change between the tension values for the spans in the least tension in the crankshaft driving
phase and their corresponding span in the ISG driving phase
The summary of the simulation results are used as a benchmark for the optimized system shown
in Chapter 5 The static tension simulation results are investigated through a parametric study of
the Twin Tensioner system in Chapter 4 The optimization of the system is then based on the
selected design variables from the outcome of Chapter 4
71
CHAPTER 4 PARAMETRIC ANALYSIS OF A B-ISG
TWIN TENSIONER
41 Introduction
The parameters for the proposed Twin Tensioner for a Belt-driven Integrated Starter-generator
(B-ISG) system are investigated through a parametric analysis This analysis seeks to understand
how changing one parameter influences the static belt span tensions for the system Since the
thesis research focuses on the design of a tensioning mechanism to support static tension only
the parameters specific to the actual Twin Tensioner and applicable to the static case were
considered The parameters pertaining to accessory pulley properties such as radii or various
belt properties such as belt span stiffness are not considered In the analyses a single parameter
is varied over a prescribed range while all other parameters are held constant The pivot point
described by Cartesian Coordinates [X6Y6] for the tensioner arms is held constant in all cases
42 Methodology
The parametric study method applies to the general case of a function evaluated over changes in
one of its dependent variables The methodology is illustrated for the B-ISG system‟s function
for static tension which is evaluated for each change in one of its Twin Tensioner‟s parameters
The original data used for the system is based on sample vehicle data provided by Litens [56]
Table 41 provides the initial data for the parameters as well as the incremental change and
maxima and minima limits The increment Δi for the ith parameter is chosen arbitrarily Limits
for each parameter have been chosen to be plus or minus sixty percent of its initial value
Parametric Analysis 72
Table 41 Initial Values Increments and Ranges for Parameters of Twin Tensioner
Parameter Name Initial Value Increment (+- Δi) Minimum
value Maximum value
Coupled Spring
Stiffness kt
20626
N∙mrad 1238 N∙mrad 8250 N∙mrad 33002 N∙mrad
Tensioner Arm 1
Stiffness kt1
10314
N∙mrad 0619 N∙mrad 4126 N∙mrad 16502 N∙mrad
Tensioner Arm 2
Stiffness kt2
16502
N∙mrad 0990 N∙mrad 6601 N∙mrad 26403 N∙mrad
Tensioner Pulley 1
Diameter D3 007240 m 4344 ∙ 10
-3 m 00290 m 0116 m
Tensioner Pulley 2
Diameter D5 007240 m 4344 ∙ 10
-3 m 00290 m 0116 m
Tensioner Pulley 1
Initial Coordinates
[0292761
0087] m See Figure 41 for region of possible tensioner pulley
coordinates Tensioner Pulley 2
Initial Coordinates
[012057
009193] m
The mesh of possible points for the centre coordinates of tensioner pulley 1 and tensioner pulley
2 are designated as Region 1 and Region 2 respectively in Figures 41a and 41b
Figure 41a Regions 1 and 2 and their Associated Guidelines for Coordinates of Tensioner
Pulleys 1 amp 2
CS
AC
ISG
Ten 1
Ten 11
Region II
Region I
Parametric Analysis 73
Figure 41b Regions 1 and 2 in Cartesian Space
The selection for the minimum and maximum tensioner pulley centre coordinates and their
increments are not selected arbitrarily or without derivation as the other tensioner parameters
The coordinates for the pulley centres are identified using Intergraph‟s SmartSketch software a
graphing suite in MathCAD to model the regions of space The following descriptions are used
to describe the possible positions for the tensioner pulleys
Tensioner pulleys are situated such that they are exterior to the interior space created by
the serpentine belt thus they sit bdquooutside‟ the belt loop
The highest point on the tensioner pulley does not exceed the tangent line connecting the
upper hemispheres of the pulleys on either side of it
The tensioner pulleys may not overlap any other pulley
Parametric Analysis 74
Boundaries for regions described as Region 1 in span 2 and 3 and Region 2 in span 4
and 5 is selected based on the above criteria and their lower boundaries are selected
arbitrarily
These criteria were used to define the equation for each boundary line and leads to a set of
Boolean conditions that relate the x-coordinate and y-coordinate for each Cartesian pair The
density for the mesh of points in each region is arbitrarily selected as 101 x-points and 101 y-
points in each space for the purposes of the parametric analysis The outline of this method is
described in the MATLAB scripts contained in Appendix D
The results of the parametric analysis are shown for the slackest and tautest spans in each driving
case As was demonstrated in the literature review the tautest span immediately precedes the
driving pulley and the slackest span immediately follows the driving pulley in the direction of
the belt motion Thus in the case for the crankshaft driving the tautest span is in the first span
and the slackest span is in the fifth span Whereas in the ISG driving case the tautest span is in
the fourth span and the slackest span is in the third span Hence the parametric figures in this
chapter display only the tautest and slackest span values for both driving cases so as to describe
the maximum and minimum values for tension present in the given belt
43 Results amp Discussion
431 Influence of Tensioner Arm Stiffness on Static Tension
The parametric analysis begins with changing the stiffness value for the coil spring coupled
between tensioner arms 1 and 2 This stiffness value kt is changed over a range from sixty
percent less than its initial value kt0 to sixty percent more than its original value as shown in
Parametric Analysis 75
Table 41 The results of the static tension are shown in Figure 42 for the tautest and slackest
spans for both the crankshaft and ISG driving cases
Figure 42 Parametric Analysis for Coupled Stiffness Arm Constant kt (N∙mrad)
As kt increases in the crankshaft driving phase for the B-ISG system the highest tension
decreases from 4691N to 4646N while the lowest tension decreases from 2838N to 2793N
In the ISG driving phase the highest tension increases from 378N to 3998N and the lowest
tension increases from -3384N to -3167N Thus a change of approximately -45N is found in
the crankshaft driving case and approximately +22N is found in the ISG driving case for both the
tautest and slackest spans
Parametric Analysis 76
The second parameter analyzed is the stiffness value for tensioner arm 1 The results of this are
shown in Figure 43
Figure 43 Parametric Analysis for Stiffness of Arm 1 kt1 (N∙mrad)
In Figure 43 as kt1 increases an increase from 4628N to 4681N is observed for the tension of
the tautest span when the crankshaft is driving which is a change of +53N The same value for
net change is found in the slackest span for the same driving condition whose tension increases
from 2775N to 2828N For the case when the B-ISG system is in the ISG driving phase the
change is larger a value of -261N for the tautest span that changes from 4088N to 3827N and
for the slackest span that changes from -3077N to -3338N
Parametric Analysis 77
The change in static tension for the spans as the stiffness of arm 2 varies is demonstrated in
Figure 44
Figure 44 Parametric Analysis for Stiffness of Arm 2 kt2 (N∙mrad)
In this case it is observed that as kt2 increases the tautest span for the B-ISG system in the
crankshaft driving case decreases from 4675N to 4643N as well as the slackest span which
decreases from 2822N to 279N which is an overall change of -32N for both spans Whereas in
the ISG driving case a more noticeable change is once again found a difference of +144N
This is a result of the tautest span increasing from 3863N to 4007N and the slackest span
increasing from -3301N to -3157N
Parametric Analysis 78
432 Influence of Tensioner Pulley Diameter on Static Tension
The change in the diameter of tensioner pulley 1 D3 and its effect on static tension is shown in
Figure 45
Figure 45 Parametric Analysis for Pulley 1 Diameter D3 (m)
The change in the tautest and slackest spans for the B-ISG system‟s crankshaft driving case is
from 3248N to 425N and from 1395N to 240N respectively Peaks are seen at 4799N and
2946N for the respective spans This is a change of approximately +100N and a maximum
change of 1551N for both spans For the ISG driving case the tautest and slackest spans
decrease from 1083N to 6158N and 367N to -1006N Global minimums of 3246N and -391N
for the respective spans are seen This nets a change of approximately -467N and a maximum
change of approximately -759N
Parametric Analysis 79
The effect of changing the diameter of tensioner pulley 2 on the static tension is examined in
Figure 46
Figure 46 Parametric Analysis for Pulley 2 Diameter D5 (m)
The tautest and slackest spans in the crankshaft driving mode of the belt undergo a change from
4583N to 4721N and from 273N to 2869N respectively Therein as D5 increases the trend is
that for both spans there is an increase in tension of approximately 14N Contrastingly the spans
experience a decrease in the ISG driving case as D5 increases The tension of the tautest span
goes from 4296N to 3635N and that of the slackest span goes from -2866N to -3529N This
equals a decrease of approximately 66N for both spans
Parametric Analysis 80
433 Influence of Tensioner Pulley 1 Coordinates on Static Tension
The influence of the coordinates of tensioner pulley 1 on the value of tension in the tautest span
for the B-ISG system‟s crankshaft driving case is demonstrated in Figure 47
Figure 47 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Tautest Span
Tension in Crankshaft Driving Case
The region shown in Figure 47 corresponds to region 1 which is the realm of the positions for
tensioner pulley 1 The possible pulley coordinates in this case are represented by the non-blue
area reaching to the perimeter of the plot It is evident in the darkest red region of the plot
where the y-coordinate is between approximately 0m and 0075m and the x-coordinate is
(N)
Parametric Analysis 81
between approximately 026m and 031m that the highest value of tension is experienced in the
tautest span for the crankshaft driving case The range of tension for Region 1 in the tautest span
when the crankshaft is driving is between a maximum of approximately 500N and a minimum of
approximately 300N This equals an overall difference of 200N in tension for the tautest span by
moving the position of pulley 1 The lowest values for tension are obtained when the pulley
coordinates are approximately -0025m to 015m for the y-coordinate and approximately 031m
to 032m for the x-coordinate which corresponds to the yellow region An area of low tension is
also seen in the area where the y-coordinate is approximately 0m and the x-coordinate is
approximately between 026m and 027m
The changes in tension for the slackest span under the condition of the crankshaft pulley being
the driving pulley are shown in Figure 48
Parametric Analysis 82
Figure 48 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Slackest Span
Tension in Crankshaft Driving Case
Once again the possible coordinate points for tensioner pulley 1 in the B-ISG system are
represented by the non-blue region For the slackest span in the crankshaft driving case it is seen
that the lowest tension is approximately 125N while the highest tension is approximately 325N
This is an overall change of 200N that is achieved in the region The highest values are achieved
in the space where the y-coordinates are approximately 0m to 0075m and the x-coordinate
ranges from 026m to 031m which corresponds to the deep red region The lowest tension
values are achieved in the space where the y-coordinate ranges from approximately -0025m to
015m and the x-coordinate ranges from 031m to 032m which corresponds to the light blue-
green region of the plot The area containing a y-coordinate of approximately 0m and x-
(N)
Parametric Analysis 83
coordinates that are approximately between 026m and 027m also show minimum tension for
the slack span The regions of the x-y coordinates for the maximum and minimum tensions are
alike to the tautest span in Region 1 for the crankshaft driving case as well as was seen in Figure
47
The tension for the tautest span in the case that the ISG is driving in the B-ISG system is found
in Figure 49
Figure 49 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Tautest Span
Tension in ISG Driving Case
(N)
Parametric Analysis 84
Region 1 is represented by the coordinate values shown in the non-dark blue space of the plot in
Figure 49 The tautest span in the case of the ISG driving experiences a range of tension values
in Region 1 from 200N up to 1100N equaling a difference of 900N The minimum tension
values are achieved in the medium to light blue region This includes y-coordinates of
approximately 0m to 0075m and x-coordinates of approximately 026m to 03m The
maximum tension values are in the darkest red area inclusive of y-coordinates -0025m to 015m
and x-coordinates 031m to 032m in addition to y-coordinate of approximately 0m and x-
coordinates of approximately 026m to 027m It can be observed that aforementioned regions
for minimum and maximum tensions in Figure 49 are reverse to those seen in Figures 47 and
48 for the crankshaft driving case
The change in tension for the slackest span of the B-ISG system when it is driven by the ISG is
shown in Figure 410
Parametric Analysis 85
Figure 410 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Slackest Span
Tension in ISG Driving Case
Figure 410 exhibits the realm of possible points for tensioner pulley 1 for the case of the ISG
driving in the non-yellow-green area The minimum tension values are achieved in the darkest
blue area where the minimum tension is approximately -500N This area corresponds to y-
coordinates from approximately 0m to 005m and x-coordinates from approximately 026m to
03m The area of a maximum tension is approximately 400N and corresponds to the darkest red
area inclusive of y-coordinates -0025m to 015m and x-coordinates 031m to 032m as well as
the coordinates for y equaling approximately 0m and for x equaling approximately 026m to
027m The difference between maximum and minimum tensions in this case is approximately
900N It is noticed once again that the space of x- and y-coordinates containing the maximum
(N)
Parametric Analysis 86
tension is in the similar location to that of the described space for minimum tension in the
crankshaft driving case in Figure 47 and 48
434 Influence of Tensioner Pulley 2 Coordinates on Static Tension
The influence of pulley 2 coordinates on the tension value for the tautest span when the
crankshaft is driving the B-ISG system is shown in Figure 411 and is represented by the values
corresponding to the non-blue area
Figure 411 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Tautest Span
Tension in Crankshaft Driving Case
In Figure 411 the possible coordinates are contained within Region 2 The maximum tension
value is approximately 500N and is found in the darkest red space including approximately y-
(N)
Parametric Analysis 87
coordinates 004m to 014m and x-coordinates 0025m to 0175m and also y-coordinates 013m
to 02m corresponding to the x-coordinate at 0175m A minimum tension value of
approximately 350N is found in the yellow space and includes approximately y-coordinates
008m to 018m and x-coordinates 016m to 02m The difference in tension values is 150N
The analysis of the change in coordinates for tension pulley 2 on the value for tension in the
slackest span is shown in Figure 412 in the non-blue region
Figure 412 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Slackest Span
Tension in Crankshaft Driving Case
The value of 325N is the highest tension for the slack span in the crankshaft driving case of the
B-ISG system and is found in the deep-red region where the y-coordinates are between
(N)
Parametric Analysis 88
approximately 004m and 013m and the x-coordinates are approximately between 0025m and
016m as well as where y is between 013m and 02m and x is approximately 0175m The
lowest tension value for the slack span is approximately 150N and is found in the green-blue
space where y-coordinates are between approximately 01m and 022m and the x-coordinates
are between approximately 016m and 021m The overall difference in minimum and maximum
tension values is 175N The spaces for the maximum and minimum tension values are similar in
location to that found in Figure 411 for the tautest span in the crankshaft driving case
Figure 413 provides the theoretical data for the tension values of the tautest span as the position
of the B-ISG system‟s tensioner pulley 2 changes in the ISG driving case Possible points are in
the space of values which correspond to the non-dark-blue region in Figure 413
Parametric Analysis 89
Figure 413 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Tautest Span
Tension in ISG Driving Case
In Figure 413 the region for high tension reaches a value of approximately 950N and the region
for low tension reaches approximately 250N This equals a difference of 700N between
maximum and minimum tension values for the tautest span in the B-ISG system‟s ISG driving
case The coordinate points within the space that maximum tension is reached is in the dark red
region and includes y-coordinates from approximately 008m to 022m and x-coordinates from
approximately 016m to 021m The coordinate points within the space that minimum tension is
reached is in the blue-green region and includes y-coordinates from approximately 004m to
013m and the corresponding x-coordinates from approximately 0025m to 015m An additional
small region of minimum tension is seen in the area where the x-coordinate is approximately
(N)
Parametric Analysis 90
0175m and the y-coordinates are approximately between 013m and 02m The location for the
area of pulley centre points that achieve maximum and minimum tension values is approximately
located in the reverse positions on the plot when compared to that of the case for the crankshaft
driving in Figures 411 and 412 Therein the trend seen for pulley coordinates for the second
tensioner pulley follows suit with that of the first tensioner pulley which is that the area of points
for maximum tension in the crankshaft driving case becomes the approximate area of points for
minimum tension in the ISG driving case and vice versa
In Figure 414 the results of the parametric analysis on the coordinates of the second tensioner
pulley and its effect on the slackest span‟s tension in the ISG driving case is shown Similar to
earlier figures the non-dark yellow region represents Region 2 that contains the possible points
for the pulley‟s Cartesian coordinates
Parametric Analysis 91
Figure 414 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Slackest
Span Tension in ISG Driving Case
Figure 414 demonstrates a difference of approximately 725N between the highest and lowest
tension values for the slackest span of the B-ISG system in the ISG driving case The highest
tension values are approximately 225N The area of points that allow the second tension pulley
to achieve maximum tension in the belt span includes y-coordinates from approximately 01m to
022m and the corresponding x-coordinates from approximately 016m to 021m This
corresponds to the darkest red region in Figure 414 The coordinate values where the lowest
tension being approximately -500N is achieved include y-coordinate values from
approximately 004m to 013m and the corresponding x-coordinates from approximately 0025m
to 015m corresponding to the darkest blue region A dark blue region of lowest tension is also
(N)
Parametric Analysis 92
seen in the area where y is approximately between 013m and 02m and the x-coordinate is
approximately 0175m The regions for maximum and minimum tension values are observed to
be similar to those found in Figure 413 and alike to Figure 413 to be in reverse to those found
in Figure 411 and 412 for the tautest and slackest spans in the crankshaft driving case So as for
the changes in tensioner pulley 2 coordinates the areas for minimum tension in Region 2 of the
ISG driving case are similar to the areas for maximum tension in Region 2 of the crankshaft
driving case and vice versa for the maximum tension of the ISG driving case and the minimum
tension for the crankshaft driving case in Region 2
44 Conclusion
Overall the trend in the plots of Figures 47 48 411 and 412 indicate in the crankshaft driving
portion that the B-ISG system‟s belt span tensions experience the following effect
Minimum tension for the tautest span is achieved when tensioner pulley 1 centre
coordinates are located closer to the right side boundary and bottom left boundary of
Region 1 or when tensioner pulley 2 centre coordinates are within the upper right space
(near to the ISG pulley) and the space closer to the top boundary of Region 2
Maximum tension for the slackest span is achieved when the first tensioner pulley‟s
coordinates are located in the mid space and near to the bottom boundary of Region 1
and when the second tensioner pulley‟s coordinates are located near to the bottom left
boundary of Region 2 which is the boundary nearest to the crankshaft pulley
Parametric Analysis 93
The trend for minimizing the tautest span signifies that the tension for the slackest span is also
minimized at the same time As well maximizing the slackest span signifies that the tension for
the tautest span is also maximized at the same time too
The trend for the B-ISG system‟s ISG driving case as can be seen in Figures 49 410 413 and
414 is approximately in reverse to that of the crankshaft driving case for the system Wherein
points corresponding to minimum tension in Regions 1 and 2 in the ISG case are approximately
the same as points corresponding to maximum tension in the Regions for the crankshaft case and
vice versa for the ISG cases‟ areas of maximum tension
Minimum tension for the tautest span is present when the first tensioner pulley‟s
coordinates are near to mid to lower boundary of Region 1 and when the second
tensioner pulley‟s coordinates are close to the bottom left boundary of Region 2 which
is the furthest boundary from the ISG pulley and closest to the crankshaft pulley
Maximum tension for the slackest span is achieved when the first tensioner pulley is
located close to the right boundary of Region 1 and when the second tensioner pulley is
located near the right boundary and towards the top right boundary of Region 2
It is observed in Figures 47 to 414 and alike to Figures 42 to 46 the tautest and slackest
spans decrease or increase together Thus it can be assumed that the tension values in these
spans and likely the remaining spans outside of the tautest and slackest spans follow suit
Therein when parameters are changed to minimize one belt span‟s tension the remaining spans
will also have their tension values reduced Figures 42 through to 413 showed this clearly
where the overall change in the tension of the tautest and slackest spans changed by
Parametric Analysis 94
approximately the same values for each separate case of the crankshaft driving and the ISG
driving in the B-ISG system
Design variables are selected in the following chapter from the parameters that have been
analyzed in the present chapter The influence of changing parameters on the static tension
values for the various spans is further explored through an optimization study of the static belt
tension for the B-ISG system equipped with a Twin Tensioner in the following chapter Chapter
5
95
CHAPTER 5 OPTIMIZATION OF A B-ISG TENSIONER
The objective of the optimization analysis is to minimize the absolute magnitude of the static
tension in the ISG-operating mode of the serpentine belt drive The optimization seeks to
optimize the performance of the proposed Twin Tensioner design by using its properties as the
design variables for the objective function The optimization task begins with the selection of
these design variables for the objective function and then the selection of an optimization
method The results of the optimization will be compared with the results of the analytical
model for the static system and with the parametric analysis‟ data
51 Optimization Problem
511 Selection of Design Variables
The optimal system corresponds to the properties of the Twin Tensioner that result in minimized
magnitudes of static tension for the various belt spans Therein the design variables for the
optimization procedure are selected from amongst the Twin Tensioner‟s properties In the
parametric analysis of Chapter 4 the tensioner properties presented included
coupled stiffness kt
tensioner arm 1 stiffness kt1
tensioner arm 2 stiffness kt2
tensioner pulley 1 diameter D3
tensioner pulley 2 diameter D5
tensioner pulley 1 initial coordinates [X3Y3] and
Optimization 96
tensioner pulley 2 initial coordinates [X5Y5]
It was observed in the former chapter that perturbations of the stiffness and geometric parameters
caused a change between the lowest and highest values for the static tension especially in the
case of perturbations in the geometric parameters diameter and coordinates Table 51
summarizes the observed changes in the belt span tensions corresponding to the Twin Tensioner
parameters‟ maximum and minimum values
Table 51 Summary of Parametric Analysis Data for Twin Tensioner Properties
Parameter Symbol
Original Tensions in TautSlack Span (Crankshaft
Mode) [N]
Tension at
Min | Max Parameter6 for
Crankshaft Mode [N]
Percent Change from Original for
Min | Max Tensions []
Original Tension in TautSlack Span (ISG Mode)
[N]
Tension at
Min | Max Parameter Value in ISG Mode [N]
Percent Change from Original Tension for
Min | Max Tensions []
kt
465848 (taut) 4691 4646 07 -03 393645 (taut) 378 3998 -40 16
28057 (slack) 2838 2793 12 -05 -322803 (slack) -3384 -3167 -48 19
kt1
465848 (taut) 4628 4681 -07 05 393645 (taut) 4088 3827 38 -28
28057 (slack) 2775 2828 -11 08 -322803 (slack) -3077 -3338 47 -34
kt2
465848 (taut) 4675 4643 04 -03 393645 (taut) 3863 4007 -19 18
28057 (slack) 2822 279 06 -06 -322803 (slack) -3301 -3157 -23 22
D3 465848 (taut) 3248 425 -303 -88 393645 (taut) 1083 6158 1751 564
28057 (slack) 1395 240 -503 -145 -322803 (slack) 367 -1006 2137 688
D5 465848 (taut) 4583 4721 -16 13 393645 (taut) 4296 3635 91 -77
28057 (slack) 273 2869 -27 23 -322803 (slack) -2866 -3529 112 -93
[X3Y3] 465848 (taut) 300 500 -356 73 393645 (taut) 200 1100 -492 1794
28057 (slack) 125 325 -554 158 -322803 (slack) -500 400 -549 2239
6 The values for the tension for each of the taut and slack spans provided correspond to the minimum and maximum
values of the parameter listed in each case such that the columns of identical colour correspond to each other For
the coordinate parameters the minimum and maximum parameter value is inadmissible The tension values in these
cases are simply the minimum and maximum tension values achieved by the coordinate parameter listed
Optimization 97
[X5Y5] 465848 (taut) 350 500 -249 73 393645 (taut) 250 950 -365 1413
28057 (slack) 150 325 -465 158 -322803 (slack) -500 225 -549 1697
The results of the parametric analyses for the Twin Tensioner parameters show that there is a
noticeable percent change between the initial tensions and the tensions corresponding to each of
the minima and maxima parameter values or in the case of the coordinates between the
minimum and maximum tensions for the spans Thus the parametric data does not encourage
exclusion of any of the tensioner parameters from being selected as a design variable As a
theoretical experiment the optimization procedure seeks to find feasible physical solutions
Hence economic criteria are considered in the selection of the design variables from among the
Twin Tensioner‟s parameters Of the tensioner properties it is found that the diameter of the
tensioner pulleys has the largest impact on cost Adding mass to a tensioner pulley as a result of
increasing the diameter and consequently its inertia increases the cost of material Material cost
is most significant in the manufacture process of pulleys as their manufacturing is largely
automated [4] Furthermore varying the structure of a pulley requires retooling which also
increases the cost to manufacture As such the tensioner pulley diameters D3 and D5 are
excluded from being selected as design variables The remaining tensioner properties the
stiffness parameters and the initial coordinates of the pulley centres are selected as the design
variables for the objective function of the optimization process
512 Objective Function amp Constraints
In order to deal with two objective functions for a taut span and a slack span a weighted
approach was employed This emerges from the results of Chapter 3 for the static model and
Chapter 4 for the parametric study for the static system which show that a high tension span and
Optimization 98
a highly slack span exist in the ISG-driving phase of the B-ISG system Therein the first
objective function of equation (51a) is described as equaling fifty percent of the absolute tension
value of the tautest span and fifty percent of the absolute tension value of the slackest span for
the case of the ISG driving only The second objective function uses a non-weighted approach
and is described as the absolute tension of the slackest span when the ISG is driving A non-
weighted approach is motivated by the phenomenon of a fixed difference that is seen between
the slackest and tautest spans of the optimal designs found in the weighted optimization
simulations Equations (51a) through to (51c) display the objective functions
The limits for the design variables are expanded from those used in the parametric analysis for
the non-coordinate parameters kt kt1 and kt2 so that they are permitted to vary from
approximately 0 to approximately 200 of the initial value for each parameter kt0 kt10 and kt20
respectively In the case of the coordinate parameters [X3Y3] and [X5Y5] the x- and y-
coordinates are permitted to vary within the spaces Region 1 and Region 2 respectively which
were prescribed in Chapter 4 Figure 41a and 41b
Aside from the design variables design constraints on the system include the requirement for
static stability of the Twin Tensioner An optimal solution for the B-ISG system must achieve
the goal of the objective function which is to minimize the absolute tensions in the system
However for an optimal solution to be feasible the movement of the tensioner arm must remain
within an appropriate threshold In practice an automotive tensioner arm for the belt
transmission may be considered stable if its movement remains within a 10 degree range of
Optimization 99
motion [4] As such the angle of displacement for tensioner arms 1 and 2 are designated by θ t1
and θt2 respectively in the listed constraints
The optimization task is described in equations 51a to 52 Variables a through to g represent
scalar limits for the x-coordinate for corresponding ranges of the y-coordinate
Minimize 119879119908119890119894119892 119893119905119890119889 = 05 ∙ 119879119905119886119906119905 + 05 ∙ 119879119904119897119886119888119896
or119879119899119900119899 minus119908119890119894119892 119893119905119890119889 = 119879119904119897119886119888119896
(51a)
where
119879119905119886119906119905 = 119891119905119886119906119905 119896119905 1198961199051 1198961199052 1198833 1198843 1198835 1198845 (51b)
119879119904119897119886119888119896 = 119891119904119897119886119888119896 (119896119905 1198961199051 1198961199052 1198833 119884311988351198845) (51c)
Subject to
(1198961199050 minus 1 ∙ 1198961199050) le 119896119905 le (1198961199050 + 11198961199050)(11989611990510 minus 1 ∙ 11989611990510) le 1198961199051 le (11989611990510 + 111989611990510)(11989611990520 minus 1 ∙ 11989611990520) le 1198961199052 le (11989611990520 + 111989611990520)
119886 le 1198833 le 119888
1198931 1198833 le 1198843 le 1198933 1198833 119891119900119903 119886 le 1198833 lt 119887
1198932 1198833 le 1198843 le 1198933 1198833 119891119900119903 119887 le 1198833 le 119888119889 le 1198835 le 119892
1198934 1198835 le 1198845 le 1198937 1198835 for 119889 le 1198835 lt 1198901198935(1198835) le 1198845 le 1198937(1198835) for 119890 le 1198835 lt 119891
1198936 1198835 le 1198845 le 1198937 1198833 for 119891 le 1198833 le 119892 1205791199051 le 10deg 1205791199052 le 10deg
(52)
The functions for the taut and slack spans represent the fourth and third span respectively in the
case of the ISG driving The equations for the tensions of the aforementioned spans are shown
in equation 51a to 51c and are derived from equation 353 The constraints for the
optimization are described in equation 52
Optimization 100
52 Optimization Method
A twofold approach was used in the optimization method A global search alone and then a
hybrid search comprising of a global search and a local search The Genetic Algorithm is used
as the global search method and a Quadratic Sequential Programming algorithm is used for the
local search method
521 Genetic Algorithm
Genetic Algorithm (GA) is a part of the growing genre of evolutionary algorithms [57] The
optimization approach differs from classical search approaches by its ease of use and global
perspective [57] GA mimics biological evolution theory by using a ldquocross-over of heritable
information random mutation and selection on the basis of fitness between generationsrdquo [58] to
form a robust search algorithm that requires minimal problem information [57] The parameter
sets are represented as sample points modeled as bdquochromosomes‟ or data strings that are
measured against how well they allow the model to achieve the optimization task [58] GA is
stochastic which means that its algorithm uses random choices to generate subsequent sampling
points rather than using a set rule to generate the following sample This avoids the pitfall of
gradient-based techniques that may focus on local maxima or minima and end-up neglecting
regions containing higher peaks or lower valleys [57] Furthermore due to the randomness of
the GA‟s search strategy it is able to search a population (a region of possible parameter sets)
faster than other optimization techniques The GA approach is viewed as a universal
optimization approach while many classical methods viewed to be efficient for one optimization
problem may be seen as inefficient for others However because GA is a probabilistic algorithm
its solution for the objective function may only be near to a global optimum As such the current
Optimization 101
state of stochastic or global optimization methods has been to refine results of the GA with a
local search and optimization procedure
522 Hybrid Optimization Algorithm
In order to enhance the result of the objective function found by the GA a Hybrid optimization
function is implemented in MATLAB software The Hybrid optimization function combines a
global search GA with a local search Sequential Quadratic Programming (SQP) The hybrid
process refines the value of the objective function found through GA by using the final set of
points found by the algorithm as the initial point of the SQP algorithm The GA function
determines the region containing a global optimum and then the SQP algorithm uses a gradient
based technique to find a solution closer to the global optimum The MATLAB algorithm a
constrained minimization function known as fmincon uses an SQP method that approximates the
Hessian for the Lagrangian function (ie the second derivatives of the Lagrangian) by way of a
quasi-Newton approach to generate a quadratic program (QP) sub-problem [59] The solution
for the QP provides the search direction of the line search procedure used when each iteration is
performed [59]
53 Results and Discussion
531 Parameter Settings amp Stopping Criteria for Simulations
The parameter settings for the optimization procedure included setting the stall time limit to
200s This is the interval of time the GA is given to find an improvement in the value of the
objective function This is an increase from MATLAB‟s default of 20s Increasing the stall time
limit allows for the optimization search to consistently converge without being limited by time
Optimization 102
The population size used in finding the optimal solution is set at 100 This value was chosen
after varying the population size between 50 and 2000 showed no change in the value of the
objective function The max number of generations is set at 100 The time limit for the
algorithm is set at infinite The limiting factor serving as the stopping condition for the
optimization search was the function tolerance which is set at 1x10-6
This allows the program
to run until the ratio of the change in the objective function over the stall generations is less than
the value for function tolerance The stall generation setting is defined as the number of
generations since the last improvement of the objective function value and is limited to 50
532 Optimization Simulations
The results of the genetic algorithm optimization simulations performed in MATLAB are shown
in the following tables Table 52a and Table 52b
Table 52a GA Optimization Results for Twin Tensioner Parameters and Objective Function
Trial
No
Genetic Algorithm Optimization Method
Objective
Function
Value [N]
kt [N∙mrad]
kt1 [N∙mrad]
kt2 [N∙mrad]
[X3Y3]
[m] [X5Y5]
[m]
1 3582241 314069 204844 165020 [02928 00703] [01618 01036]
2 3582241 103646 205284 198901 [03009 00607] [01283 00809]
3 3582241 126431 204740 43549 [03010 00631] [01311 01147]
4 3582241 180285 206230 254870 [03095 00865] [01080 01675]
5 3582241 74757 204559 189077 [03084 00617] [01265 00718]
Original Design 20626 10314 16502 [02928 0087] [01206 00919]
Optimization 103
Table 52b Computations for Tensions and Angles from GA Optimization Results
Trial No
Genetic Algorithm Optimization Method
Slackest Tension [N] Tautest Tension [N] Tensioner Arm 1
Displacement [deg]
Tensioner Arm 2
Displacement [deg]
T3 T4 Θt1 Θt2
1 -1572307 5592176 -00025 -49748
2 -4054309 3110174 -00002 -20213
3 -3930858 3233624 -00004 -38370
4 -1309751 5854731 -00010 -49525
5 -4092446 3072036 -00000 -17703
Original Design -322803 393645 16410 -4571
For each trial above the GA function required 4 generations each consisting of 20 900 function
evaluations before finding no change in the optimal objective function value according to
stopping conditions
The results of the Hybrid function optimization are provided in Tables 53a and 53b below
Table 53a Hybrid Optimization Results for Twin Tensioner Parameters and Objective Function
Trial
No
Hybrid Optimization Method
Objective
Function
Value [N]
of
Function
Evals ( of
Iterations)
kt [N∙mrad]
kt1 [N∙mrad]
kt2 [N∙mrad]
[X3Y3]
[m] [X5Y5]
[m]
1 3582241 16 (1) 16065 205846 229494 [02780 00581] [01679 01288]
2 3582241 20 (1) 249227 205635 25218 [02901 00634] [01559 00870]
3 3582241 16 (1) 297295 204878 320479 [02962 00702] [01336 01447]
4 3582241 53 (1) 241433 204262 229683 [02912 00647] [00047 01465]
Optimization 104
5 3582241 21 (1) 379096 205548 188888 [02973 00703] [01206 01376]
Original Design 20626 10314 16502 [02928 0087] [01206 00919]
Table 53b Computations for Tensions and Angles from Hybrid Optimization Results
Trial No
Hybrid Algorithm Optimization Method
Slackest Tension [N] Tautest Tension [N] Tensioner Arm 1
Displacement [deg]
Tensioner Arm 2
Displacement [deg]
T3 T4 Θt1 Θt2
1 -2584641 4579841 -02430 67549
2 -3708747 3455736 -00023 -41068
3 -1707181 5457302 -00099 -43944
4 -269178 6895304 00006 -25366
5 -2982335 4182148 -00003 -41134
Original Design -322803 393645 16410 -4571
In Table 53a it can be seen that iterations of 16 20 21 or 53 were required for the local search
algorithm following the GA to find an optimal solution Once again the GA function
computed 4 generations which consisted of approximately 20 900 function evaluations before
securing an optimum solution
The simulation results of the non-weighted hybrid optimization approach are shown in tables
54a and 54b below
Optimization 105
Table 54a Non-Weighted Optimization Results for Twin Tensioner Parameters and Objective
Function
Trial
No
Objective
Function
Value [N]
of
Function
Evals ( of
Iterations)
kt [N∙mrad]
kt1 [N∙mrad]
kt2 [N∙mrad]
[X3Y3]
[m] [X5Y5]
[m]
Genetic Algorithm Optimization Method
1 33509e
-004 20900 (4) 321799 75530 212653 [02860 00602] [01082 01858]
Hybrid Optimization Method
1 73214e
-011 381 (13) 234881 14730 323358 [02952 00688] [00048 01466]
Original Design 20626 10314 16502 [02928 0087] [01206 00919]
Table 54b Computations for Tensions and Angles from Non-Weighted Optimizations
Trial No Slackest Tension [N] Tautest Tension [N]
Tensioner Arm 1
Displacement [deg]
Tensioner Arm 2
Displacement [deg]
T3 T4 Θt1 Θt2
Genetic Algorithm Optimization Method
1 -00003 7164479 -00588 -06213
Hybrid Optimization Method
1 -00000 7164482 15543 -16254
Original Design -322803 393645 16410 -4571
The weighted optimization data of Table 54a shows that the GA simulation again used 4
generations containing 20 900 function evaluations to conduct a global search for the optimal
system While the weighted Hybrid optimization used 13 iterations (consisting of 381 function
evaluations) after its GA run which used the same number of generations and function
evaluations as the GA run in the non-weighted simulations Tables 54a and 54b show the data
Optimization 106
for only one trial for each of the non-weighted GA and hybrid methods since only a single
optimal point exists in this case
533 Discussion
The optimal design from each search method can be selected based on the least amount of
additional pre-tension (corresponding to the largest magnitude of negative tension) that would
need to be added to the system This is in harmony with the goal of the optimization of the B-
ISG system as stated earlier to minimize the static tension for the tautest span and at the same
time minimize the absolute static tension of the slackest span for the ISG driving case As well
the angular displacements corresponding to each trial‟s results show that the Twin Tensioner is
under static stability Therein the optimal solution may be selected as the design parameters
corresponding to Trial 4 of the GA simulations to Trial 4 of the Hybrid simulations or to either
of the trials for the non-weighted optimization simulations
Given the ability of the Hybrid optimization to refine the results obtained in the GA
optimization the results of Trial 4 of the Hybrid simulations are selected as the most optimal
design from the weighted objective function approaches It is interesting to note that the Hybrid
case for the least slackness in belt span tension corresponds to a significantly larger number of
function evaluations than that of the remaining Hybrid cases This anomaly however does not
invalidate the other Hybrid cases since each still satisfy the design constraints Using the data
for the optimized system in Trial 4 (of the Hybrid optimization) the static tensions for the belt
spans in both of the B-ISG‟s phases of operation are as follows
Optimization 107
Table 55 Weighted Optimization Results for Static Tensions for Optimal B-ISG System with a
Twin Tensioner
Belt Span between Pulleys Crankshaft Driving Case [N] ISG Driving Case [N]
Optimized Original Optimized Original
Crankshaft ndash Air Conditioner 3926599 465848 117333 -284152
Air Conditioner ndash Tensioner 1 3540088 427197 -269178 -322803
Tensioner 1 ndash ISG 3540088 427197 -269178 -322803
ISG ndash Tensioner 2 2073813 28057 6895304 393645
Tensioner 2 ndash Crankshaft 2073813 28057 6895304 393645
Additional Pretension
Required (approximate) + 27000 +322803 + 27000 +322803
In Table 54b it is evident that the non-weighted class of optimization simulations achieves the
least amount of required pre-tension to be added to the system The computed tension results
corresponding to both of the non-weighted GA and Hybrid approaches are approximately
equivalent Therein either of their solution parameters may also be called the most optimal
design The Hybrid solution parameters are selected as the optimal design once again due to the
refinement of the GA output contained in the Hybrid approach and its corresponding belt
tensions are listed in Table 56 below
Optimization 108
Table 56 Non-Weighted Optimization Results for Static Tensions for Optimal B-ISG System
with a Twin Tensioner
Belt Span between Pulleys Crankshaft Driving Case [N] ISG Driving Case [N]
Optimized Original Optimized Original
Crankshaft ndash Air Conditioner 3891862 465848 386511 -284152
Air Conditioner ndash Tensioner 1 3505351 427197 -00000 -322803
Tensioner 1 ndash ISG 3505351 427197 -00000 -322803
ISG ndash Tensioner 2 2039076 28057 7164482 393645
Tensioner 2 ndash Crankshaft 2039076 28057 7164482 393645
Additional Pretension
Required (approximate) + 0000 +322803 + 00000 +322803
The results of the simulation experiments are limited by the following considerations
System equations are coupled so that a fixed difference remains between tautest and
slackest spans
A limited number of simulation trials have been performed
There are multiple optimal design points for the weighted optimization search methods
Remaining tensioner parameters tensioner pulley diameters and their stiffness have not
been included in the design variables for the experiments
The belt factor kb used in the modeling of the system‟s belt has been obtained
experimentally and may be open to further sources of error
Therein the conclusions obtained and interpretations of the simulation data can be limited by the
above noted comments on the optimization experiments
Optimization 109
54 Conclusion
The outcomes the trends in the experimental data and the optimal designs can be concluded
from the optimization simulations The simulation outcomes demonstrate that in all cases the
weighted optimization functions reached an identical value for the objective function whereas
the values reached for the parameters varied widely
The lowest tension values for the tautest and slackest span were achieved in Trial 5 of the GA
optimization approach In reiteration in the presence of slack spans the tension value of the
slackest span must be added to the initial static tension for the belt Therein for the former case
an amount of at least 409N would need to be added to the 300N of pre-tension already applied to
the system (see Table 34) The highest tension values for the spans were achieved in Trial 4 of
the weighted Hybrid optimization approach and in both trials of the non-weighted optimization
approaches In the former the weighted Hybrid trial the tension value achieved in the slackest
span was approximately -27N signifying that only at least 27N would need to be added to the
present pre-tension value for the system The tension of the slackest span in the non-weighted
approach was approximately 0N signifying that the minimum required additional tension is 0N
for the system
The optimized solutions for the tension values in each span show that there is consistently a fixed
difference of 716448N between the tautest and slackest span tension values as seen in Tables
52b 53b and 54b This difference is identical to the difference between the tautest and slackest
spans of the B-ISG system for the original values of the design parameters while in its ISG
mode As well the optimal stiffness parameters for the weighted Hybrid optimization case are
Optimization 110
greater than their original values except for that of the stiffness factor of tensioner arm 1
Likewise for the non-weighted Hybrid optimization case the stiffness parameters are above their
original values without exceptions The coordinates of the optimal solutions are within close
approximation to each other and also both match the regions for moderately low tension in
Regions 1 and 2 of the ISG driving case as is shown in Figures 49 410 413 and 414
The results of the non-weighted Hybrid optimization trial and Trial 4 of the weighted Hybrid
optimization simulations are selected as the most optimal designs for the B-ISG Twin Tensioner
In these designs the Twin Tensioner is shown in Table 53b and 54b to have static stability and
to maintain suitable tensions in the ISG driving phase The tensioner parameters for the optimal
designs allow for one of the lowest amounts of additional pre-tension to be added to the system
out of all the findings from the simulations which were conducted
111
CHAPTER 6 CONCLUSION
61 Summary
The primary aim of the thesis is to reduce the magnitude of static tension in the belt spans of a
Belt-driven Integrated Starter-generator (B-ISG) system by the design and investigation of a
Twin Tensioner It is established that the operating phases of the B-ISG system produced two
cases for static tension outcomes an ISG driving case and a crankshaft driving case The
approach taken in this thesis study includes the derivation of a system model for the geometric
properties as well as for the dynamic and static states of the B-ISG system The static state of a
B-ISG system with a single tensioner mechanism is highlighted for comparison with the static
state of the Twin Tensioner-equipped B-ISG system
It is observed that there is an overall reduction in the magnitudes of the static belt tensions with
the presence of a Twin Tensioner over that of a single tensioner The influences of the geometric
and stiffness properties of the Twin Tensioner affecting the static tensions in the system are
analyzed in a parametric study It is found that there is a notable change in the static tensions
produced as result of perturbations in each respective tensioner property This demonstrates
there are no reasons to not further consider a tensioner property based solely on its influence on
the B-ISG system‟s static tensions The phenomenon of higher magnitudes for static tensions in
the ISG mode of operation over that of the crankshaft mode of operation particularly in
excessively slack spans provides the motivation for optimizing the ISG case alone for static
tension The optimization method uses weighted and non-weighted approaches with genetic
algorithm (GA) and hybrid GA searches The most optimal design has been found to be one in
Conclusion 112
which the magnitude of tension in the excessively slack spans in the ISG driving case are
significantly lower than in that of the original B-ISG Twin Tensioner design
62 Conclusion
The conclusions that can be drawn from the study of a Twin Tensioner for a B-ISG system
include the following
1 The simulations of the dynamic model demonstrate that the mode shapes for the system
are greater in the ISG-phase of operation
2 It was observed in the output of the dynamic responses that the system‟s rigid bodies
experienced larger displacements when the crankshaft was driving over that of the ISG-
driving phase It was also noted that the transition speed marking the phase change from
the ISG driving to the crankshaft driving occurred before the system reached either of its
first natural frequencies
3 The magnitudes for static belt tensions as well as dynamic tensions for a B-ISG system
are consistently greater in its ISG operating phase than in its crankshaft operating phase
4 A Twin Tensioner is able to reduce the magnitudes of the static tension for the belt spans
of a B-ISG system in comparison to when only a single tensioner mechanism is present
5 The parametric study of the B-ISG system demonstrates that the slackest and tautest belt
spans decrease or increase together for either phase of operation
6 Perturbations in the Twin Tensioner‟s geometric and stiffness properties have a
significant influence on the magnitudes of the static tension of the slackest and tautest
belt spans The coordinate position of each pulley in the Twin Tensioner configuration
Conclusion 113
has the greatest influence on the belt span static tensions out of all the tensioner
properties considered
7 Optimization of the B-ISG system shows a fixed difference trend between the static
tension of the slackest and tautest belt spans for the B-ISG system
8 The values of the design variables for the most optimal system are found using a hybrid
algorithm approach The slackest span for the optimal system is significantly less slack
than that of the original design Therein less additional pretension is required to be added
to the system to compensate for slack spans in the ISG-driving phase of operation
63 Recommendation for Future Work
The investigation of the B-ISG Twin Tensioner encourages the following future work
1 The optimization of the B-ISG system with the inclusion of diametric Twin Tensioner
properties would provide a complete picture as to the highest possible performance
outcome that the Twin Tensioner is able to have with respect to the static tensions
achieved in the belt spans
2 A larger number of optimization trials using the genetic algorithm (GA) and hybrid GA
under weighted and other approaches would investigate the scope of optimal designs
available in the Twin Tensioner for the B-ISG system
3 A model of the system without the simplification of constant damping may produce
results that are more representative of realistic operating conditions of the serpentine belt
drive A finite element analysis of the Twin Tensioner B-ISG system may provide more
applicable findings
Conclusion 114
4 Investigation of the transverse motion coupled with the rotational belt motion in an
optimized B-ISG system equipped with a Twin Tensioner may also provide a closer look
at the system under realistic conditions In addition the affect of the Twin Tensioner on
transverse motion can determine whether significant improvements in the magnitudes of
static belt span tensions are still being achieved
5 The recommendation to conduct modal decoupling of the B-ISG system‟s static model is
motivated by the fixed difference trend between the slackest and tautest belt span
tensions shown in Chapter 5 The modal decoupling of the system would allow for its
matrices comprising the equations of motion to be diagonalized and therein to decouple
the system equations Modal analysis would transform the system from physical
coordinates into natural coordinates or modal coordinates which would lead to the
decoupling of system responses
6 An investigation and optimization of the dynamic belt span tensions for a B-ISG system
with a Twin Tensioner would increase understanding of the full impact of a Twin
Tensioner mechanism on the state of the B-ISG system It would be informative to
analyze the mode shapes of the first and second modes as well as the required torques of
the driving pulleys and the resulting torque of each of the tensioner arms The
observation of the dynamic belt span tensions would also give direction as to how
damping of the system may or may not be changed
7 Further comparison with the Twin Tensioner B-ISG system‟s dynamic and static states
including the Twin Tensioner‟s stability in each versus a B-ISG system with a single
tensioner would further demonstrate the improvements or dis-improvements in the Twin
Tensioner‟s performance on a B-ISG system
Conclusion 115
8 The influence of the belt properties on the dynamic and static tensions for a B-ISG
system with a Twin Tensioner can also be investigated This again will show the
evidence of improvements or dis-improvement in the Twin Tensioner‟s performance
within a B-ISG setting
9 Lastly an experimental apparatus of the B-ISG system with a Twin Tensioner can be
designed and constructed Suitable instrumentation can be set-up to measure belt span
tensions (both static and dynamic) belt motion and numerous other system qualities
This would provide extensive guidance as to finding the most appropriate theoretical
model for the system Experimental data would provide a bench mark for evaluating the
theoretical simulation results of the Twin Tensioner-equipped B-ISG system
116
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[14] National Alternative Fuels Training Consortium (NAFTC) (2005 Oct 3) Tech stuff
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[15] Green Car Congress BMW to Apply Start-Stop and Brake Regen to MINIs Up to 60
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[16] H Jeneacute E Scheid and H Kemper Hybrid electric vehicle (HEV) concepts - fuel savings
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[21] PJ Wezenbeek (Zytec Systems Ltd) D G Evans (General Motors Powertrain) D P
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(Delphi Corp) Combustion Assisted Belt-Cranking of a V-8 Engine at 12-Volts SAE
Technical Papers vol 113 pp 396-407 2004 Document no 2004-01-0569
[22] T C Firbank Mechanics of the Belt Drive International Journal of Mechanical
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[23] R L Cassidy S K Fan R S MacDonald and W F Samson Serpentine Extended Life
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[24] A G Ulsoy J E Whitesell and M D Hooven Design of Belt-Tensioner Systems for
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Transactions of the ASME vol 107 pp 282-290 July 1985
[25] R S Beikmann N C Perkins and A G Ulsoy Free Vibration of Serpentine Belt Drive
Systems Journal of Vibrations and Acoustics Transactions of the ASME vol 118 pp
406-413 1996
[26] T C Kraver G W Fan and J J Shah Complex Modal Analysis of a Flat Belt Pulley
System with Belt Damping and Coulomb-Damped Tensioner Journal of Mechanical
Design Transactions of the ASME vol 118 pp 306-311 Jun 1996
[27] R S Beikmann N C Perkins and A G Ulsoy Design and Analysis of Automotive
Serpentine Belt Drive Systems for Steady State Performance Journal of Mechanical
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[28] L Zhang and J W Zu Modal Analysis of Serpentine Belt Drive Systems Journal of
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[29] R Balaji and E M Mockensturm Dynamic analysis of a front-end accessory drive with a
decouplerisolator International Journal of Vehicle Design vol 39 pp 208-231 2005
[30] M Nouri Design Optimization and Active Control of Serpentine Belt Drive Systems with
Two-pulley Tensioners University of Toronto 2005
[31] G J Spicer (Litens Automotive Inc) Tensioner for use in eg belt drive system has
electronic actuator associated with clutch spring for engaging International
WO2005119089-A1 Jun 6 2005 2005
[32] Bando Chemical Industries Ltd and Litens Automotive GmbH About belt-type starter
system Feb 27 2002
[33] H Lemberger and R Jungjohann (Bayerische Motoren Werke AG) Tension device for an
envelope drive of a device especially a belt drive of a starter generator of an internal
combustion engine comprises a support part Europe EP1420192-A2 May 19 2004 2003
[34] P Ahner and M Ackermann (Bosch GMBH) Belt drive especially for internal
combustion engines to drive accessories in an automobile Germany DE19849886-A1
May 11 2000 1998
[35] N Freisinger K Hagemann J Sievert P Struebel and M Treusch (Daimler Chrysler AG)
Belt tensioning device for belt drive between engine and starter generator of motor
vehicle has self-aligning bearing that supports auxiliary unit and provides working force to
tensioners for tensioning belt Germany DE10324268 Dec 16 2004 2003
[36] C R Rogers (Dayco Products LLC) Offset starter generator drive system for a vehicle
engine has a dual arm pivoted tensioner United States US6942589-B2 Feb 8 2005 2002
[37] A Serkh and I Ali (Gates Corp) Internal combustion engine has belt drive system with
tensioner asymmetrically biased in direction tending to cause power transmission belt to be
under tension International WO2003038309-A1 May 8 2003 2002
References 120
[38] P J Mcvicar and C A Thurston (General Motors Corp) Belt alternator starter accessory
drive with dual tensioner United States US20060287146-A1 Dec 21 2006 2005
[39] W Petri and M Bogner (INA Schaeffler KG) Traction drive especially for driving
internal combustion engine units has arrangement for demand regulated setting of tension
consisting of unit with housing with limited rotation and pulley German DE10044645-
A1 Mar 21 2002 2000
[40] M Bogner (INA Schaeffler KG) Belt drive tensioner for a starter-generator of an IC
engine has locking system for locking tensioning element in an engine operating mode
locking system is directly connected to pivot arm follows arm control movements
German DE10159073-A1 Jun 12 2003 2001
[41] R Painta M Bogner and H Graf (INA Schaeffler KG) Traction mechanism drive esp
belt drive has belt tensioning pulley mounted on generator shaft and uncoupled from it via
freewheel to dampen load peaks Europe EP1723350-A1 Nov 22 2006 2005
[42] W Petri (INA Schaeffler KG) Drive unit for a combustion engine having a starter
generator and a belt drive has tensioner with spring and counter hydraulic force Germany
DE10359641-A1 Jul 28 2005 2003
[43] H Stief M Bogner B Hartmann T Kraft and M Schmid (INA Schaeffler KG) Traction
drive especially belt drive for short-duration driving of starter generator has tensioning
device with lever arm deflectable against restoring force and with end stop limiting
deflection travel Europe EP1738093-A1 Jan 3 2007 2005
[44] M Ulm (INA Schaeffler KG DE) Tension unit eg for drive in machine such as
combustion engine has belt or chain drive with wheels turning and connected with starter
generator and unit has two idlers arranged at clamping arm with machine stored by shock
absorber Germany DE102004012395-A1 Sep 29 2005 2004
[45] M Bogner (INA Schaeffler KG) Belt drive for starter motor-generator auxiliary assembly
has limited movement at the starter belt section tensioner roller bringing it into a dead point
position on starting the motor International WO2006108461-A1 Oct 19 2006 2006
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[46] W Guhr (Litens Automotive GMBH) Automotive motor and drive assembly includes
tension device positioned within belt drive system having combination starter United
States US2001007839-A1 Jul 12 2001 2001
[47] K Kuniaki K Masahiko H Kazuyuki I Shuichi and T Masaki (Mitsubishi Jidosha Eng
KK and Mitsubishi Motor Corp) Tension adjustment method of belt for starter generator
in vehicle involves shifting auto-tensioners between lock state and free state to adjust
tension of belt during driving of crank pulley Japan JP2005083514-A Mar 31 2005
2003
[48] Nissan Motor Co Ltd Winding gear for starting engine of hybrid motor vehicle has
tensioner tightening chain while cranking engine and slackens chain after start of engine
provided to span side of chain Japan JP3565040-B2 Sep 15 2004 1998
[49] S Sato and H Hayakawa (NTN Corp) Auto tensioner for ancillary drive belts has
cylinder nut and screw bolt in hydraulic damper mechanism provided in middle of cylinder
acting as start-up rigidity buffer component Japan JP2006189073-A Jul 20 2006 2005
[50] G Vadin-Michaud (Valeo Equip Electrique Moteur) Pulley and belt starting system for a
thermal engine for a motor vehicle Europe EP1658432 May 24 2006 2005
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2005
[52] W E Johns Notes on Motors [Electronic] 2003 [2008 June] Available at
httpwwwgizmologynetmotorshtm
[53] Litens Automotive Group Ltd DC BAS System - Conventional Start Input Profile Nov
23 2007
[54] Arnold Magnetic Technologies Corp General Motor Terminology [Electronic] pp 7
[2008 June] Available at httpwwwgrouparnoldcommtcpdfweb_motor_glossarypdf
[55] Douglas W Jones Stepping Motors University of Iowa - Department of Computer
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httpwwwcsuiowaedu~jonesstepphysicshtml
References 122
[56] Litens Automotive Group Ltd (2004 Jan 31) FEAD software input data for test project
[57] K Deb Multi-Objective Optimization using Evolutionary Algorithms Toronto John Wiley
amp Sons Ltd 2001 pp 81-85
[58] P E McSharry (2004 May 11) Department of Engineering Science University of Oxford
[httpwwwengoxacuksamppubsgawbreppdf]
[59] The MathWorks Inc MATLAB vol 750342 (R2007b) Aug 15 2007
123
APPENDIX A
Passive Dual Tensioner Designs from Patent Literature
Figure A1 Proposed design by Bayerische Motoren Werke AG corresponding to patent nos EP1420192-A2 and DE10253450-A1
Source European Patent Office espcenet (publication nos EP1420192-A2 and DE10253450-A1 accessed May 2007) epespacenetcom [33]
Figure A1 label identification 1 ndash tightner 2 ndash belt drive
3 ndash starter generator
4 ndash internal-combustion engine
4‟ ndash crankshaft-lateral drive disk
5 ndash generator housing
6 ndash common axis of rotation
7 ndash featherspring of tiltable clamping arms
8 ndash clamping arm
9 ndash clamping arm
10 11 ndash idlers
12 12‟ ndash Zugtrum 13 13‟ ndash Leertrum
14 ndash carry-hurries 15 ndash generator wave
16 ndash bush
17 ndash absorption mechanism
18 18‟ ndash support arms
19 19‟ ndash auxiliary straight lines
20 ndash pipe
21 ndash torsion bar
22 ndash breaking through
23 ndash featherspring
24 ndash friction disk
25 ndash screw connection 26 ndash Wellscheibe
(European Patent Office May 2007) [33]
Appendix A 124
Figure A2a First of four proposed designs (and its various configurations) by Bosch GMBH
corresponding to patent no WO0026532-A1
Source European Patent Office espcenet (publication no WO0026532-A1 accessed Jun 2007)
epespacenetcom [34]
Figure A2b Second of four proposed designs by Bosch GMBH corresponding to patent no
WO0026532-A1
Source European Patent Office espcenet (publication no WO0026532-A1 accessed Jun 2007) epespacenetcom [34]
Figure A2c Third of four proposed designs (and its various configurations) by Bosch GMBH
corresponding to patent no WO0026532-A1
Source European Patent Office espcenet (publication no WO0026532-A1 accessed Jun 2007)
epespacenetcom [34]
Appendix A 125
Figure A2d Fourth of four proposed designs (and its various configurations) by Bosch GMBH
corresponding to patent no WO0026532-A1
Source European Patent Office espcenet (publication no WO0026532-A1 accessed Jun 2007)
epespacenetcom [34]
Figure A2a through to A2d label identification 10 ndash engine wheel
11 ndash [generator] 13 ndash spring
14 ndash belt
16 17 ndash tensioning pulleys
18 19 ndash springs
20 21 ndash fixed points
25ab ndash carriers of idlers
25c ndash gang bolt
(European Patent Office June 2007) [34]
Figure A3 Proposed design by Daimler Chrysler AG corresponding to patent no DE10324268-A1
Source European Patent Office espcenet (publication no DE10324268-A1 accessed May 2007)
epespacenetcom [35]
Figure A3 label identification
Appendix A 126
10 12 ndash belt pulleys
14 ndash auxiliary unit
16 ndash belt
22-1 22-2 ndash belt tensioners
(European Patent Office May 2007) [35]
Figure A4 Proposed design by Dayco Products LLC corresponding to patent no US6942589-B2
Source European Patent Office espcenet (publication no US6942589-B2 accessed Jun 2007)
epespacenetcom [36]
Figure A4 label identification 12 ndash belt
14 ndash tensioner
16 ndash generator pulley
18 ndash crankshaft pulley
22 ndash slack span 24 ndash tight span
32 34 ndash arms
33 35 ndash pulley
(European Patent Office June 2007) [36]
Appendix A 127
Figure A5 Proposed design by Gates Corp corresponding to patent no WO2003038309-A
Source European Patent Office espcenet (publication no WO2003038309-A accessed Jun 2007)
epespacenetcom [37]
Figure A5 label identification 12 ndash motorgenerator
14 ndash motorgenerator pulley 26 ndash belt tensioner
28 ndash belt tensioner pulley
30 ndash transmission belt
(European Patent Office June 2007) [37]
Figure A6 Proposed design by General Motors Corp corresponding to patent no US20060287146-A1
Source European Patent Office espcenet (publication no US20060287146-A1 accessed Jun 2007)
epespacenetcom [38]
Appendix A 128
Figure A6 label identification 28 ndash tensioner
32 ndash carrier arm
34 ndash secondary carrier arm
46 ndash tensioner pulley
58 ndash secondary tensioner pulley
(European Patent Office June 2007) [38]
Figure A7 Proposed design by INA Schaeffler KG corresponding to patent no DE10044645-A1
Source European Patent Office espcenet (publication no DE10044645-A1 accessed Jun 2007)
epespacenetcom [39]
Figure A7 label identification 2 ndash internal combustion engine
3 ndash traction element
11 ndash housing with limited rotation 12 13 ndash direction changing pulleys
(European Patent Office June 2007) [39]
Appendix A 129
Figure A8a First of three proposed designs by INA Schaeffler KG corresponding to patent no
DE10159073-A1
Source European Patent Office espcenet (publication no DE10159073-A1 accessed May 2007)
epespacenetcom [40]
Figure A8b Second of three proposed designs by INA Schaeffler KG corresponding to patent no
DE10159073-A1
Source European Patent Office espcenet (publication no DE10159073-A1 accessed May 2007)
epespacenetcom [40]
Appendix A 130
Figure A8c Third of three proposed designs by INA Schaeffler KG corresponding to patent no
DE10159073-A1
Source European Patent Office espcenet (publication no DE10159073-A1 accessed May 2007)
epespacenetcom [40]
Figure A8a A8b and A8c label identification 1 ndash tightener [tensioner]
2 ndash idler
3 ndash drawing means
4 ndash swivel arm
5 ndash axis of rotation
6 ndash drawing means impulse [belt]
7 ndash crankshaft
8 ndash starter generator
9 ndash bolting volume 10a ndash bolting device system
10b ndash bolting device system
10c ndash bolting device system
11 ndash friction body
12 ndash lateral surface
13 ndash bolting tape end
14 ndash bolting tape end
15 ndash control member
16 ndash torsion bar
17 ndash base
18 ndash pylon
19 ndash hub
20 ndash annular gap
21 ndash Gleitlagerbuchse
23 ndash [nil]
23 ndash friction disk
24 ndash turning camps 25 ndash teeth
26 ndash elbow levers
27 ndash clamping wedge
28 ndash internal contour
29 ndash longitudinal guidance
30 ndash system
31 ndash sensor
32 ndash clamping gap
(European Patent Office May 2007) [40]
Appendix A 131
Figure A9 Proposed design by INA Schaeffler KG corresponding to patent no DE10359641-A1
Source European Patent Office espcenet (publication no DE10359641-A1 accessed Jun 2007)
epespacenetcom [42]
Figure A9 label identification 8 ndash cylinder
10 ndash rod
12 ndash spring plate
13 ndash spring
14 ndash pressure lead
(European Patent Office June 2007) [42]
Appendix A 132
Figure A10 Proposed design by INA Schaeffler KG corresponding to patent no EP1723350-A1
Source European Patent Office espcenet (publication no EP1723350-A1 accessed Jun 2007) epespacenetcom [41]
Figure A10 label identification 4 ndash pulley
5 ndash hydraulic element 11 ndash freewheel
12 ndash shaft
(European Patent Office June 2007) [41]
Figure A11 Proposed design by INA Schaeffler KG corresponding to patent no EP1738093-A1
Source European Patent Office espcenet (publication no EP1738093-A1 accessed Jun 2007)
epespacenetcom [43]
Figure A11 label identification 1 ndash traction drive
2 ndash traction belt
3 ndash starter generator
Appendix A 133
7 ndash tension device
9 ndash lever arm
10 ndash guide roller
16 ndash end stop
(European Patent Office June 2007) [43]
Figure A12 Proposed design by INA Schaeffler KG corresponding to patent no DE102004012395-A1
Source European Patent Office espcenet (publication no DE102004012395-A1 accessed May 2007) epespacenetcom [44]
Figure A12 label identification 1 ndash belt drive
2 ndash belts
3 ndash wheel of the internal-combustion engine
4 ndash wheel of a Nebenaggregats
5 ndash wheel of the starter generator
6 ndash clamping unit
7 ndash idler
8 ndash idler
9 ndash scale beams
10 ndash drive place
11 ndash clamping arm
12 ndash camps
13 ndash coupling point
14 ndash shock absorber element
15 ndash arrow
(European Patent Office May 2007) [44]
Appendix A 134
Figure A13 Proposed design by INA Schaeffler KG corresponding to patent nos DE102005017038-A1and WO2006108461-A1
Source European Patent Office espcenet (publication nos DE102005017038-A1and WO2006108461-A1 accessed May 2007) epespacenetcom [45]
Figure A13 label identification 1 ndash belt
2 ndash wheel of the crankshaft KW
3 ndash wheel of a climatic compressor AC
4 ndash wheel of a starter generator SG
5 ndash wheel of a water pump WP
6 ndash first clamping system
7 ndash first tension adjuster lever arm
8 ndash first tension adjuster role
9 ndash second clamping system
10 ndash second tension adjuster lever arm
11 ndash second tension adjuster role 12 ndash guide roller
13 ndash drive-conditioned Zugtrum
(generatorischer enterprise (GE))
13 ndash starter-conditioned Leertrum
(starter enterprise (SE))
14 ndash drive-conditioned Leertrum (GE)
14 ndash starter-conditioned Zugtrum (SE)
14a ndash drive-conditioned Leertrumast (GE)
14a ndash starter-conditioned Zugtrumast (SE)
14b ndash drive-conditioned Leertrumast (GE)
14b ndash starter-conditioned Zugtrumast (SE)
(European Patent Office May 2007) [45]
Figure A14 Proposed design by Litens Automotive GMBH et al corresponding to patent no
US20010007839-A1
Appendix A 135
Source European Patent Office espcenet (publication no US20010007839-A1 accessed Jun 2007)
epespacenetcom [46]
Figure A14 label identification E - belt
K - crankshaft
R1 ndash first tension pulley
R2 ndash second tension pulley
S ndash tension device
T ndash drive system
1 ndash belt pulley
4 ndash belt pulley
(European Patent Office June 2007) [46]
Figure A15 Proposed design by Mitsubishi Jidosha Eng KK and Mitsubishi Motor Corp corresponding
to patent no JP2005083514-A
Source Industrial Property Digital Library and Japanese Patent Office Patent amp Utility Model Gazette DB (document no A 2005-083514 accessed May 2007) wwwipdlinpitgojp [47]
Figure A15 label identification 1 ndash Pulley for Starting
2 ndash Crank Pulley
3 ndash AC Pulley
4a ndash 1st roller
4b ndash 2nd roller
5 ndash Idler Pulley
6 ndash Belt
7c ndash Starter generator control section
7d ndash Idle stop control means
8 ndash WP Pulley
9 ndash IG Switch
10 ndash Engine
11 ndash Starter Generator
12 ndash Driving Shaft
Appendix A 136
7 ndash ECU
7a ndash 1st auto tensioner control section (the 1st auto
tensioner control means)
7b ndash 2nd auto tensioner control section (the 2nd auto
tensioner control means)
13 ndash Air-conditioner Compressor
14a ndash 1st auto tensioner
14b ndash 2nd auto tensioner
18 ndash Water Pump
(Industrial Property Digital Library May 2007) [47]
Figure A16 Proposed design by Nissan Motor Co Ltd corresponding to patent no JP3565040-B2
Source European Patent Office espcenet (publication no JP3565040-B2 accessed Jun 2007) epespacenetcom [48]
Figure A16 label identification 3 ndash chain [or belt]
5 ndash tensioner
4 ndash belt pulley
(European Patent Office June 2007) [48]
Figure A17 Proposed design by NTN Corp corresponding to patent no JP2006189073-A
Appendix A 137
Source European Patent Office espcenet (publication no JP2006189073-A accessed Jun 2007)
epespacenetcom [49]
Figure A17 label identification 5d - flange
6 ndash tensile strength spring
10 ndash actuator
10c ndash cylinder
12 ndash rod
20 ndash hydraulic damper mechanism
21 ndash cylinder nut
22 ndash screw bolt
(European Patent Office June 2007) [49]
Figure A18 Proposed design by Valeo Equipment Electriques Moteur corresponding to patent nos
EP1658432 and WO2005015007
Source European Patent Office espcenet (publication nos EP1658432 and WO2005015007
accessed May 2007) epespacenetcom [50]
Figure A18 abbreviated list of label identifications
10 ndash starter
22 ndash shaft section
23 ndash free front end
80 ndash pulley
200 ndash support element
206 - arm
(European Patent Office May 2007) [50]
The author notes that the list of labels corresponding to Figures A1a through to A7 are generated
from machine translations translated from the documentrsquos original language (ie German)
Consequently words may be translated inaccurately or not at all
138
APPENDIX B
B-ISG Serpentine Belt Drive with Single Tensioner
Equation of Motion
The equations of motion (EOM) for a B-ISG serpentine belt drive with a single tensioner are
shown The EOM has been derived similarly to that of the same system with a twin tensioner
that was provided in Chapter 3 The assumptions for the twin tensioner B-ISG system are
applicable for the single tensioner B-ISG system as well
Figure B1 shows the B-ISG system with a single tensioner pulley and arm The pulleys are
numbered 1 through 4 and their associated belt spans are numbered accordingly
Figure B1 Single Tensioner B-ISG System
Appendix B 139
The free-body diagram for the ith non-tensioner pulley in the system shown above is found in
Figure B2 The moment of inertia for the ith pulley is designated as Ii while the angular
displacement velocity and acceleration is designated as 120579119905119894 120579 119905119894 and 120579 119905119894 respectively The
required torque is Qi the angular damping is Ci and the tension of the ith span is Ti
Figure B2 Free-body Diagram of ith Pulley
The positive motion designated is assumed to be in the clockwise direction The radius for the
ith pulley is represented by Ri The equilibrium equations for the ith pulley are as follows
I1 ∙ θ 1 = T4 ∙ R1 minus T1 ∙ R1 + Q1 minus c1 ∙ θ 1 (B1)
I2 ∙ θ 2 = T1 ∙ R2 minus T2 ∙ R2 + Q2 minus c2 ∙ θ 2 (B2)
I3 ∙ θ 3 = T2 ∙ R3 minus T3 ∙ R3 + Q3 minus c3 ∙ θ 3 (B3)
Appendix B 140
A free-body diagram for the single tensioner pulley is shown in Figure B3 The rotational
stiffness and damping for the tensioner arm is designated as kt and ct respectively The angular
rotation and velocity for the arm is 120579119905119894 and 120579 119905119894 respectively
Figure B3 Free-body Diagram of Single Tensioner
From figure B2 the equations of equilibrium are resolved for the tensioner pulley The angle of
orientation for the ith belt span is designated by 120573119895119894
minusI4 ∙ θ 4 = minusT3 ∙ R4 + T4 ∙ R4 + Q4 + c4 ∙ θ 4 (B4)
It ∙ θ t = minusTt ∙ Lt ∙ sin θto minus βj4 + sin θto 1 minus βj5 minus kt ∙ partθt minus ct ∙ partθ t
(B5)
Appendix B 141
partθt = θt minus θto (B6)
The dynamic tension matrix Trsquo is proportional to the damping (Tc) and stiffness (Tk) matrices
that are due to belt damping (119888119894119887 ) and belt stiffness (119896119894
119887 ) respectively
119827prime = 119827119836 ∙ 120521 + 119827119844 ∙ 120521 (B7)
The initial tension is represented by To and the initial angle of the tensioner arm is represented
by 120579119905119900 The equation for the tension of the ith span is shown in the following equations
T1 = To + k1b ∙ R1 ∙ θ1 minus R2 ∙ θ2 + c1
b ∙ [R1 ∙ θ 1 minus R2 ∙ θ 2] (B8)
T2 = To + k2b ∙ R2 ∙ θ2 minus R3 ∙ θ3 + c2
b ∙ [R2 ∙ θ 2 minus R3 ∙ θ 3)] (B9)
T3 = To + k3b ∙ R3 ∙ θ3 minus R4 ∙ θ4 + Lt ∙ [sin θto minus βj3 ] ∙ (θt minus θto ) + c3
b ∙ [R3 ∙ θ 3 minus R4 ∙
θ 4 + Lt ∙ [sin θto minus βj3 ] ∙ (θ t)] (B10)
T4 = To + k4b ∙ R4 ∙ θ4 minus R1 ∙ θ1 + Lt ∙ [sin θto minus βj4 ] ∙ (θt minus θto ) + c4
b ∙ [R4 ∙ θ 4 minus R1 ∙
θ 1 + Lt ∙ [sin θto minus βj4 ] ∙ (θ t)] (B11)
Tprime = Ti minus To (B12)
Tt = T3 = T4 (B13)
Appendix B 142
The EOM for the single tensioner B-ISG system is found by substitution of equations B8 to
B13 into B1 to B5 The matrices in the EOM include the inertial matrix I damping matrix C
stiffness matrix K and the required torque matrix Q as well as the angular displacement
velocity and acceleration matrices 120521 120521 and 120521 respectively
119816 ∙ 120521 + 119810 ∙ 120521 + 119818 ∙ 120521 = 119824 (B14)
119816 =
I1 0 0 0 00 I2 0 0 00 0 I3 0 00 0 0 I4 00 0 0 0 It1
(B15)
The stiffness matrix includes kb the belt factor Kb the belt cord stiffness 120601119894 the wrap angle of
the belt on the ith pulley and Kbi the stiffness factor of the ith belt span Cb represents the belt
damping for each span and βji is the angle of orientation for the span between the jth and ith
pulleys It is noted in the terms of the stiffness and damping matrices below that the numerical
subscripts refer to the (i+1)th pulley The term Qt may be found within the required torque
matrix and represents the required torque for the tensioner arm As well the term It1 represents
the moment of inertia for the tensioner arm
Appendix B 143
K =
(B16)
Kbi =Kb
Li + kb ∙ Ri ∙ϕi+1
2 + Ri ∙ϕi
2
(B17)
C =
(B18)
Appendix B 144
Appendix B 144
120521 =
θ1
θ2
θ3
θ4
partθt
(B19)
119824 =
Q1
Q2
Q3
Q4
Qt
(B20)
Simulations of the EOM for the B-ISG system with a single tensioner were performed in FEAD
[51] software for dynamic and static cases This allowed for the methodology for deriving the
EOM to be verified and then applied to the B-ISG system with a twin tensioner The natural
frequencies modes shapes dynamic responses tensioner arm torques as well as the crankshaft
required torque only and the dynamic tensions were solved from the EOM as described in (331)
to (339) of Chapter 3 and as well as for the static tension from (351) to (353) of Chapter 3
This permitted verification of the complete derivation methodology and allowed for comparison
of the static tension of the B-ISG system belt spans in the case that a single tensioner is present
and in the case that a Twin Tensioner is present [51]
145
APPENDIX C
MathCAD Scripts
C1 Geometric Analysis
1 - CS
2 - AC
4 - Alt
3 - Ten1
5 - Ten 2
6 - Ten Pivot
1
2
3
4
5
Figure C1 Schematic of B-ISG
System with Twin Tensioner
Coordinate Input DataXY1 0 0( ) XY4 24759 16664( )
XY2 224 6395( ) XY5 12057 9193( )
XY3 292761 87( ) XY6 201384 62516( )
Computations
Lt1 XY30 0
XY60 0
2
XY30 1
XY60 1
2
Lt2 XY50 0
XY60 0
2
XY50 1
XY60 1
2
t1 atan2 XY30 0
XY60 0
XY30 1
XY60 1
t2 atan2 XY50 0
XY60 0
XY50 1
XY60 1
XY
XY10 0
XY20 0
XY30 0
XY40 0
XY50 0
XY60 0
XY10 1
XY20 1
XY30 1
XY40 1
XY50 1
XY60 1
x XY
0 y XY
1
Appendix C 146
i - angle bw horizontal and l ine from ith pulley center to (i+1)th pulley center
Adjust last number in range variable p to correspond to number of pulleys
p 0 1 4
k p( ) p 1( ) p 4if
0 otherwise
condition1 p( ) acos
XYk p( ) 0
XYp 0
XYk p( ) 0
XYp 0
2
XYk p( ) 1
XYp 1
2
condition2 p( ) 2 acos
XYk p( ) 0
XYp 0
XYk p( ) 0
XYp 0
2
XYk p( ) 1
XYp 1
2
p( ) if XYk p( ) 1
XYp 1
condition1 p( ) condition2 p( )
Lfi Lbi - connection belt span lengths
D1 20065mm D2 10349mm D3 7240mm D4 6820mm D5 7240mm
D
D1
D2
D3
D4
D5
Lf p( ) XYk p( ) 0
XYp 0
2
XYk p( ) 1
XYp 1
2
1
mm
Dk p( )
2
Dp
2
2
Lb p( ) XYk p( ) 0
XYp 0
2
XYk p( ) 1
XYp 1
2
1
mm
Dk p( )
2
Dp
2
2
fi bi - angle bw ith pulley center connection l ine and contact points Pbfi (or Pfbi) and Pbi
(or Pfi) respecti vely l
f p( ) atanLf p( ) mm
Dp
2
Dk p( )
2
Dp
Dk p( )
if
atanLf p( ) mm
Dk p( )
2
Dp
2
Dp
Dk p( )
if
2D
pD
k p( )if
b p( ) atan
mmLb p( )
Dp
2
Dk p( )
2
Appendix C 147
XYfi XYbi XYfbi XYbfi - 4 possible contact points for i th pulley
XYf p( ) XYp 0
Dp
2 mmcos p( ) f p( )
XYp 1
Dp
2 mmsin p( ) f p( )
XYb p( ) XYp 0
Dp
2 mmcos p( ) f p( )
XYp 1
Dp
2 mmsin p( ) f p( )
XYfb p( ) XYp 0
Dp
2 mmcos p( ) b p( )
XYp 1
Dp
2 mmsin p( ) b p( )
XYbf p( ) XYp 0
Dp
2 mmcos p( ) b p( )
XYp 1
Dp
2 mmsin p( ) b p( )
XYfi+1 XYbi+1 XYfbi+1 XYbfi+1 - 4 possible contact points for i+1th pulley
XYf2 p( ) XYk p( ) 0
Dk p( )
2 mmcos p( ) f p( )
XYk p( ) 1
Dk p( )
2 mmsin p( ) f p( )
XYb2 p( ) XYk p( ) 0
Dk p( )
2 mmcos p( ) f p( )
XYk p( ) 1
Dk p( )
2 mmsin p( ) f p( )
XYfb2 p( ) XYk p( ) 0
Dk p( )
2 mmcos p( ) b p( ) XY
k p( ) 1
Dk p( )
2 mmsin p( ) b p( )
XYbf2 p( ) XYk p( ) 0
Dk p( )
2 mmcos p( ) b p( ) XY
k p( ) 1
Dk p( )
2 mmsin p( ) b p( )
Row 1 --gt Pulley 1 Row i --gt Pulley i
XYfi
XYf 0( )0 0
XYf 1( )0 0
XYf 2( )0 0
XYf 3( )0 0
XYf 4( )0 0
XYf 0( )0 1
XYf 1( )0 1
XYf 2( )0 1
XYf 3( )0 1
XYf 4( )0 1
XYfi
6818
269222
335325
251552
108978
100093
89099
60875
200509
207158
x1 XYfi0
y1 XYfi1
Appendix C 148
XYbi
XYb 0( )0 0
XYb 1( )0 0
XYb 2( )0 0
XYb 3( )0 0
XYb 4( )0 0
XYb 0( )0 1
XYb 1( )0 1
XYb 2( )0 1
XYb 3( )0 1
XYb 4( )0 1
XYbi
47054
18575
269403
244841
164847
88606
291
30965
132651
166182
x2 XYbi0
y2 XYbi1
XYfbi
XYfb 0( )0 0
XYfb 1( )0 0
XYfb 2( )0 0
XYfb 3( )0 0
XYfb 4( )0 0
XYfb 0( )0 1
XYfb 1( )0 1
XYfb 2( )0 1
XYfb 3( )0 1
XYfb 4( )0 1
XYfbi
42113
275543
322697
229969
9452
91058
59383
75509
195834
177002
x3 XYfbi0
y3 XYfbi1
XYbfi
XYbf 0( )0 0
XYbf 1( )0 0
XYbf 2( )0 0
XYbf 3( )0 0
XYbf 4( )0 0
XYbf 0( )0 1
XYbf 1( )0 1
XYbf 2( )0 1
XYbf 3( )0 1
XYbf 4( )0 1
XYbfi
8384
211903
266707
224592
140427
551
13639
50105
141463
143331
x4 XYbfi0
y4 XYbfi1
Row 1 --gt Pulley 2 Row i --gt Pulley i+1
XYf2i
XYf2 0( )0 0
XYf2 1( )0 0
XYf2 2( )0 0
XYf2 3( )0 0
XYf2 4( )0 0
XYf2 0( )0 1
XYf2 1( )0 1
XYf2 2( )0 1
XYf2 3( )0 1
XYf2 4( )0 1
XYf2x XYf2i0
XYf2y XYf2i1
XYb2i
XYb2 0( )0 0
XYb2 1( )0 0
XYb2 2( )0 0
XYb2 3( )0 0
XYb2 4( )0 0
XYb2 0( )0 1
XYb2 1( )0 1
XYb2 2( )0 1
XYb2 3( )0 1
XYb2 4( )0 1
XYb2x XYb2i0
XYb2y XYb2i1
Appendix C 149
XYfb2i
XYfb2 0( )0 0
XYfb2 1( )0 0
XYfb2 2( )0 0
XYfb2 3( )0 0
XYfb2 4( )0 0
XYfb2 0( )0 1
XYfb2 1( )0 1
XYfb2 2( )0 1
XYfb2 3( )0 1
XYfb2 4( )0 1
XYfb2x XYfb2i
0
XYfb2y XYfb2i1
XYbf2i
XYbf2 0( )0 0
XYbf2 1( )0 0
XYbf2 2( )0 0
XYbf2 3( )0 0
XYbf2 4( )0 0
XYbf2 0( )0 1
XYbf2 1( )0 1
XYbf2 2( )0 1
XYbf2 3( )0 1
XYbf2 4( )0 1
XYbf2x XYbf2i0
XYbf2y XYbf2i1
100 40 20 80 140 200 260 320 380 440 500150
110
70
30
10
50
90
130
170
210
250Figure C2 Possible Contact Points
250
150
y1
y2
y3
y4
y
XYf2y
XYb2y
XYfb2y
XYbf2y
500100 x1 x2 x3 x4 x XYf2x XYb2x XYfb2x XYbf2x
Appendix C 150
Xij Yij - selected contact point on ith pulley for span from ith pulley to jth pulley
XY15 XYbf2iT 4
XY12 XYfiT 0
Pulley 1 contact pts
XY21 XYf2iT 0
XY23 XYfbiT 1
Pulley 2 contact pts
XY32 XYfb2iT 1
XY34 XYbfiT 2
Pulley 3 contact pts
XY43 XYbf2iT 2
XY45 XYfbiT 3
Pulley 4 contact pts
XY54 XYfb2iT 3
XY51 XYbfiT 4
Pulley 5 contact pts
By observation the lengths of span i is the following
L1 Lf 0( ) L2 Lb 1( ) L3 Lb 2( ) L4 Lb 3( ) L5 Lb 4( ) Li
L1
L2
L3
L4
L5
mm
i Angle between horizontal and span of ith pulley
i
atan
XY121
XY211
XY12
0XY21
0
atan
XY231
XY321
XY23
0XY32
0
atan
XY341
XY431
XY34
0XY43
0
atan
XY451
XY541
XY45
0XY54
0
atan
XY511
XY151
XY51
0XY15
0
Appendix C 151
Pulley 1 Pulley 2 Pulley 3 Pulley 4 Pulley 5
12 i0 2 21 i0 32 i1 2 43 i2 54 i3
15 i4 2 23 i1 34 i2 45 i3 51 i4
15
21
32
43
54
12
23
34
45
51
Wrap angle i for the ith pulley
1 2 atan2 XY150
XY151
atan2 XY120
XY121
2 atan2 XY210
XY1 0
XY211
XY1 1
atan2 XY230
XY1 0
XY231
XY1 1
3 2 atan2 XY320
XY2 0
XY321
XY2 1
atan2 XY340
XY2 0
XY341
XY2 1
4 atan2 XY430
XY3 0
XY431
XY3 1
atan2 XY450
XY3 0
XY451
XY3 1
5 atan2 XY540
XY4 0
XY541
XY4 1
atan2 XY510
XY4 0
XY511
XY4 1
1
2
3
4
5
Lb length of belt
Lbelt Li1
2
0
4
p
Dpp
Input Data for B-ISG System
Kt 20626Nm
rad (spring stiffness between tensioner arms 1
and 2)
Kt1 10314Nm
rad (stiffness for spring attached at arm 1 only)
Kt2 16502Nm
rad (stiffness for spring attached at arm 2 only)
Appendix C 152
C2 Dynamic Analysis
I C K moment of inertia damping and stiffness matrices respectively
u 0 1 4 v 0 1 4 (new counter variables where final value = no of pulleys + no of ten arms)
RaD
2
Appendix C 153
RaD
2
Ii =gt moment of inertia for ith pulley where i-1 and i represent ten arms
Ii0
0
1
2
3
4
5
6
10000
2230
300
3000
300
1500
1500
I diag Ii( ) kg mm2
Ci =gt Rotational damping and belt damping for the ith pulley where i-1 and i represent tensioner arms
1000kg
m3
CrossArea 693mm2
0 M CrossArea Lbelt M 0086kg
cb 2 KbM
Lbelt
Cb
cb
cb
cb
cb
cb
Cri
0
0
010
0
010
N mmsec
rad
Ct 1000N mmsec
rad Ct1 1000 N mm
sec
rad Ct2 1000N mm
sec
rad
Cr
Cri0
0
0
0
0
0
0
0
Cri1
0
0
0
0
0
0
0
Cri2
0
0
0
0
0
0
0
Cri3
0
0
0
0
0
0
0
Cri4
0
0
0
0
0
0
0
Ct Ct1
Ct
0
0
0
0
0
Ct
Ct Ct2
Rt
Ra0
Ra1
0
0
0
0
0
0
Ra1
Ra2
0
0
Lt1 mm sin t1 32
0
0
0
Ra2
Ra3
0
Lt1 mm sin t1 34
0
0
0
0
Ra3
Ra4
0
Lt2 mm sin t2 54
Ra0
0
0
0
Ra4
0
Lt2 mm sin t2 51
Appendix C 154
Kr
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Kt Kt1
Kt
0
0
0
0
0
Kt
Kt Kt2
Tk
Kbi 0( ) Ra0
0
0
0
Kbi 4( ) Ra0
Kbi 0( ) Ra1
Kbi 1( ) Ra1
0
0
0
0
Kbi 1( ) Ra2
Kbi 2( ) Ra2
0
0
0
0
Kbi 2( ) Ra3
Kbi 3( ) Ra3
0
0
0
0
Kbi 3( ) Ra4
Kbi 4( ) Ra4
0
Kbi 1( ) Lt1 mm sin t1 32
Kbi 2( ) Lt1 mm sin t1 34
0
0
0
0
0
Kbi 3( ) Lt2 mm sin t2 54
Kbi 4( ) Lt2 mm sin t2 51
Tc
Cb0
Ra0
0
0
0
Cb4
Ra0
Cb0
Ra1
Cb1
Ra1
0
0
0
0
Cb1
Ra2
Cb2
Ra2
0
0
0
0
Cb2
Ra3
Cb3
Ra3
0
0
0
0
Cb3
Ra4
Cb4
Ra4
0
Cb1
Lt1 mm sin t1 32
Cb2
Lt1 mm sin t1 34
0
0
0
0
0
Cb3
Lt2 mm sin t2 54
Cb4
Lt2 mm sin t2 51
C matrix
C Cr Rt Tc
K matrix
K Kr Rt Tk
New Equations of Motion for Dual Drive System
I K amp C matricies rearranged to place driving pulley in 1st row + 1st column and driven in 2nd row + 2nd column
IA augment I3
I0
I1
I2
I4
I5
I6
IC augment I0
I3
I1
I2
I4
I5
I6
I1kgmm2 1 106
kg m2
0 0 0 0 0 0
Ia stack I1kgmm2 IAT 0
T
IAT 1
T
IAT 2
T
IAT 4
T
IAT 5
T
IAT 6
T
Ic stack I1kgmm2 ICT 3
T
ICT 1
T
ICT 2
T
ICT 4
T
ICT 5
T
ICT 6
T
Appendix C 155
RtA augment Rt3
Rt0
Rt1
Rt2
Rt4
RtC augment Rt0
Rt3
Rt1
Rt2
Rt4
Rta stack RtAT 3
T
RtAT 0
T
RtAT 1
T
RtAT 2
T
RtAT 4
T
RtAT 5
T
RtAT 6
T
Rtc stack RtCT 0
T
RtCT 3
T
RtCT 1
T
RtCT 2
T
RtCT 4
T
RtCT 5
T
RtCT 6
T
TkA augment Tk3
Tk0
Tk1
Tk2
Tk4
Tk5
Tk6
Tka stack TkAT 3
T
TkAT 0
T
TkAT 1
T
TkAT 2
T
TkAT 4
T
TkC augment Tk0
Tk3
Tk1
Tk2
Tk4
Tk5
Tk6
Tkc stack TkCT 0
T
TkCT 3
T
TkCT 1
T
TkCT 2
T
TkCT 4
T
TcA augment Tc3
Tc0
Tc1
Tc2
Tc4
Tc5
Tc6
Tca stack TcAT 3
T
TcAT 0
T
TcAT 1
T
TcAT 2
T
TcAT 4
T
TcC augment Tc0
Tc3
Tc1
Tc2
Tc4
Tc5
Tc6
Tcc stack TcAT 0
T
TcAT 3
T
TcAT 1
T
TcAT 2
T
TcAT 4
T
Ka Kr Rta Tka Kc Kr Rtc Tkc Ca Cr Rta Tca Cc Cr Rtc Tcc
CHECK
KA augment K3
K0
K1
K2
K4
K5
K6
KC augment K0
K3
K1
K2
K4
K5
K6
CA augment C3
C0
C1
C2
C4
C5
C6
CC augment C0
C3
C1
C2
C4
C5
C6
Appendix C 156
Kacheck stack KAT 3
T
KAT 0
T
KAT 1
T
KAT 2
T
KAT 4
T
KAT 5
T
KAT 6
T
Kccheck stack KCT 0
T
KCT 3
T
KCT 1
T
KCT 2
T
KCT 4
T
KCT 5
T
KCT 6
T
Cacheck stack CAT 3
T
CAT 0
T
CAT 1
T
CAT 2
T
CAT 4
T
CAT 5
T
CAT 6
T
Cccheck stack CCT 0
T
CCT 3
T
CCT 1
T
CCT 2
T
CCT 4
T
CCT 5
T
CCT 6
T
Results for System switching from ISG as DRIVING pulley to Crankshaft as Drivi ng Pulley
Modified Submatricies for ISG Driving Phase --gt CS Driving Phase
Unit step function to provide shift from crankshaft DRIVING case (ie ISG driven case) to crankshaft DRIVEN
case (ie ISG driving case)
H n( ) 1 n 750if
0 n 750if
lt-- crankshaft DRIVING case (Phase change bw 2 cases occurs when n
reaches start speed)
I11mod n( ) Ic0 0
H n( ) 1if
Ia0 0
H n( ) 0if
I22mod n( )submatrix Ic 1 6 1 6( )
UnitsOf I( )H n( ) 1if
submatrix Ia 1 6 1 6( )
UnitsOf I( )H n( ) 0if
K11mod n( )
Kc0 0
UnitsOf K( )H n( ) 1if
Ka0 0
UnitsOf K( )H n( ) 0if
C11modn( )
Cc0 0
UnitsOf C( )H n( ) 1if
Ca0 0
UnitsOf C( )H n( ) 0if
K22mod n( )submatrix Kc 1 6 1 6( )
UnitsOf K( )H n( ) 1if
submatrix Ka 1 6 1 6( )
UnitsOf K( )H n( ) 0if
C22modn( )submatrix Cc 1 6 1 6( )
UnitsOf C( )H n( ) 1if
submatrix Ca 1 6 1 6( )
UnitsOf C( )H n( ) 0if
K21mod n( )submatrix Kc 1 6 0 0( )
UnitsOf K( )H n( ) 1if
submatrix Ka 1 6 0 0( )
UnitsOf K( )H n( ) 0if
C21modn( )submatrix Cc 1 6 0 0( )
UnitsOf C( )H n( ) 1if
submatrix Ca 1 6 0 0( )
UnitsOf C( )H n( ) 0if
K12mod n( )submatrix Kc 0 0 1 6( )
UnitsOf K( )H n( ) 1if
submatrix Ka 0 0 1 6( )
UnitsOf K( )H n( ) 0if
C12modn( )submatrix Cc 0 0 1 6( )
UnitsOf C( )H n( ) 1if
submatrix Ca 0 0 1 6( )
UnitsOf C( )H n( ) 0if
Appendix C 157
2mod n( ) I22mod n( )1
K22mod n( ) mod n( ) sort eigenvals 2mod n( ) nmod n( )mod n( )
2
EVmodn( ) augmenteigenvec 2mod n( ) mod n( )0
max eigenvec 2mod n( ) mod n( )0
eigenvec 2mod n( ) mod n( )1
max eigenvec 2mod n( ) mod n( )1
eigenvec 2mod n( ) mod n( )2
max eigenvec 2mod n( ) mod n( )2
eigenvec 2mod n( ) mod n( )3
max eigenvec 2mod n( ) mod n( )3
eigenvec 2mod n( ) mod n( )4
max eigenvec 2mod n( ) mod n( )4
eigenvec 2mod n( ) mod n( )5
max eigenvec 2mod n( ) mod n( )5
modeshapesmod n( ) stack nmod n( )T
EVmodn( )
t 0 0001 1
mode1a t( ) EVmod100( )0
sin nmod 100( )0 t mode2a t( ) EVmod100( )1
sin nmod 100( )1 t
mode1c t( ) EVmod800( )0
sin nmod 800( )0 t mode2c t( ) EVmod800( )1
sin nmod 800( )1 t
Pulley responses amp torque requirement for crankshaft amp alternator pulleys pulley1 and 4 respectively
The system equation becomes
I14q14 -double-dot + C1144 q14 -dot + K1144 q14 + C12qm-dot + K12qm = Qc
I2qm-double-dot + C22qm-dot + K22qm + C21q1-dot + K21q1 = 0
Pulley responses
Qm = - [(K22 - 2I2) + jC22 ]-1(K21 + jC21 )Q1
Torque requirement for crank shaft Pulley 1
qc = [(K11 -2I1) + jC11 ]Q1 + (K12 + jC12 )Qm
Torque requirement for alternator shaft Pulley 4
qa = [(K44 -2I4) + jC44 ]Q4 + (K12 + jC12 )Qm
Appendix C 158
Let DRIVING pulley have a unit amplitude 1 = 1 and let the system frequency be calculated based on
engine speed n
n 60 90 6000 n( )4n
60 a n( )
2n Ra0
60 Ra3
mod n( ) n( ) H n( ) 1if
a n( ) H n( ) 0if
Ymod n( ) K22mod n( ) mod n( ) 2 I22mod n( )
j mod n( ) C22modn( )
mmod n( ) Ymod n( )( )1
K21mod n( ) j mod n( ) C21modn( )
Crankshaft amp ISG required torques
Let input from DRIVING pulley be an angular displacement with constant amplitude of angular acceleration
Ac n( ) 650 1 n( )Ac n( )
n( ) 2
Let Qm = QmQ1(n) for n lt 750
and Qm = QmQ4(n) for n gt 750
Aa n( )42
I3 3
1a n( )Aa n( )
a n( ) 2
Qc0 4
qcmod n( ) K11mod n( ) mod n( ) 2
I11mod n( )
j mod n( ) C11modn( )
1 n( ) K12mod n( ) j mod n( ) C12modn( ) mmod n( ) 1 n( )
H n( ) 1if
Qc0 H n( ) 0if
qamod n( ) K11mod n( ) mod n( ) 2
I11mod n( )
j mod n( ) C11modn( )
1a n( ) K12mod n( ) jmod n( ) C12modn( ) mmod n( ) 1a n( ) Qc0
H n( ) 0if
0 H n( ) 1if
Q n( ) 48 n
Ra0
Ra3
48
18000
(ISG torque requirement alternate equation)
Appendix C 159
Dynamic tensioner arm torques
Qtt1mod n( )Kt Kt1
UnitsOf Kt( )j mod n( )
Ct Ct1
UnitsOf Cr( )
mmod n( )4 1 n( )
H n( ) 1if
Kt Kt1
UnitsOf Kt( )j mod n( )
Ct Ct1
UnitsOf Cr( )
mmod n( )4 1a n( )
H n( ) 0if
Qtt2mod n( )Kt Kt2
UnitsOf Kt( )j mod n( )
Ct Ct2
UnitsOf Cr( )
mmod n( )5 1 n( )
H n( ) 1if
Kt Kt2
UnitsOf Kt( )j mod n( )
Ct Ct2
UnitsOf Cr( )
mmod n( )5 1a n( )
H n( ) 0if
Appendix C 160
Dynamic belt span tensions
d n( ) 1 n( ) H n( ) 1if
1a n( ) H n( ) 0if
mod n( )
d n( )
mmod n( ) d n( ) 0 0
mmod n( ) d n( ) 1 0
mmod n( ) d n( ) 2 0
mmod n( ) d n( ) 3 0
mmod n( ) d n( ) 4 0
mmod n( ) d n( ) 5 0
Tm n( ) j n( )Tcc
UnitsOf Tcc( )
Tkc
UnitsOf Tkc( )
mod n( )
H n( ) 1if
j n( )Tca
UnitsOf Tca( )
Tka
UnitsOf Tka( )
mod n( )
H n( ) 0if
Tm n( ) j n( )Tcc
UnitsOf Tcc( )
Tkc
UnitsOf Tkc( )
mod n( )
H n( ) 1if
j n( )Tca
UnitsOf Tca( )
Tka
UnitsOf Tka( )
mod n( )
H n( ) 0if
(tensions for driving pulley belt spans)
Appendix C 161
C3 Static Analysis
Static Analysis using K Tk amp Q matricies amp Ts
For static case K = Q
Tension T = T0 + Tks
Thus T = K-1QTks + T0
Q1 68N m Qt1 0N m Qt2 0N m Ts 300N
Qc
Q4
Q2
Q3
Q5
Qt1
Qt2
Qc
5
2
0
0
0
0
J Qa
Q1
Q2
Q3
Q5
Qt1
Qt2
Qa
68
2
0
0
0
0
N m
cK22mod 900( )( )
1
N mQc A
K22mod 600( )1
N mQa
a
A0
A1
A2
0
A3
A4
A5
0
c1
c2
c0
c3
c4
c5
Tc Tk Ts Ta Tk a Ts
162
APPENDIX D
MATLAB Functions amp Scripts
D1 Parametric Analysis
D11 TwinMainm
The following function script performs the parametric analysis for the B-ISG system with a Twin
Tensioner It calls the function TwinTenStaticTensionm The parametric analysis perturbs a
single input parameter for the called function TwinTenStaticTensionm The main function takes
an initial input value for the Twin Tensioner‟s stiffness parameters Kto Kt1o Kt2o and
geometric parameters D3o D5o X3o Y3o X5o and Y5o An input parameter is allowed to
increment by six percent over a range from sixty percent below its initial value to sixty percent
above its initial value The coordinate parameters are incremented through a mesh of Cartesian
points with prescribed boundaries The TwinMainm function plots the parametric results
______________________________________________________________________________
clc
clear all
Static tension for single tensioner system for CS and Alt driving
Initial Conditions
Kto = 20626
Kt1o = 10314
Kt2o = 16502
D3o = 007240
D5o = 007240
X3o =0292761
Y3o =087
X5o =12057
Y5o =09193
Pertubations of initial parameters
Kt = (Kto-060Kto)006Kto(Kto+060Kto)
Kt1 = (Kt1o-060Kt1o)006Kt1o(Kt1o+060Kt1o)
Kt2 = (Kt2o-060Kt2o)006Kt2o(Kt2o+060Kt2o)
D3 = (D3o-060D3o)006D3o(D3o+060D3o)
D5 = (D5o-060D5o)006D5o(D5o+060D5o)
No of data points
s = 21
T = zeros(5s)
Ta = zeros(5s)
Parametric Plots
for i = 1s
Appendix D 163
[T(i)Ta(i)] = TwinTenStaticTension(Kt(i)Kt1oKt2oD3oD5oX3oY3oX5oY5o)
end
figure
[AXH1H2] = plotyy(Kt()T(1)Kt()Ta(4)plot) hold on
H3 = line(Kt()T(5)ParentAX(1)) hold on
H4 = line(Kt()Ta(3)Parent AX(2)) hold off
set(AXYLimModeautoYTickModeauto)
title(Static Tension vs Coupled Stiffness bw Arms 1 amp 2)
xlabel(Kt (Nmrad))
set(get(AX(1)Ylabel)StringCS Span Tension (N))
set(get(AX(2)Ylabel)StringISG Span Tension (N))
set(H1LineStyle-Colorb)
set(H2LineStyle-Colorg)
set(H3LineStyleColorget(H1Color))
set(H4LineStyleColorget(H2Color))
legend([H1H2H3H4]CS Tautest SpanISG Tautest SpanCS Slackest SpanISG
Slackest SpanLocationBest)
for i = 1s
[T(i)Ta(i)] = TwinTenStaticTension(KtoKt1(i)Kt2oD3oD5oX3oY3oX5oY5o)
end
figure
[AXH1H2] = plotyy(Kt1()T(1)Kt1()Ta(4)plot) hold on
H3 = line(Kt1()T(5)ParentAX(1)) hold on
H4 = line(Kt1()Ta(3)Parent AX(2)) hold off
set(AXYLimModeautoYTickModeauto)
title(Static Tension vs Tensioner Arm 1 Stiffness)
xlabel(Kt1 (Nmrad))
set(get(AX(1)Ylabel)StringCS Span Tension (N))
set(get(AX(2)Ylabel)StringISG Span Tension (N))
set(H1LineStyle-Colorb)
set(H2LineStyle-Colorg)
set(H3LineStyleColorget(H1Color))
set(H4LineStyleColorget(H2Color))
legend([H1H2H3H4]CS Tautest SpanISG Tautest SpanCS Slackest SpanISG
Slackest SpanLocationBest)
for i = 1s
[T(i)Ta(i)] = TwinTenStaticTension(KtoKt1oKt2(i)D3oD5oX3oY3oX5oY5o)
end
figure
[AXH1H2] = plotyy(Kt2()T(1)Kt2()Ta(4)plot) hold on
H3 = line(Kt2()T(5)ParentAX(1)) hold on
H4 = line(Kt2()Ta(3)Parent AX(2)) hold off
set(AXYLimModeautoYTickModeauto)
title(Static Tension vs Tensioner Arm 2 Stiffness)
xlabel(Kt2 (Nmrad))
set(get(AX(1)Ylabel)StringCS Span Tension (N))
Appendix D 164
set(get(AX(2)Ylabel)StringISG Span Tension (N))
set(H1LineStyle-Colorb)
set(H2LineStyle-Colorg)
set(H3LineStyleColorget(H1Color))
set(H4LineStyleColorget(H2Color))
legend([H1H2H3H4]CS Tautest SpanISG Tautest SpanCS Slackest SpanISG
Slackest SpanLocationBest)
for i = 1s
[T(i)Ta(i)] = TwinTenStaticTension(KtoKt1oKt2oD3(i)D5oX3oY3oX5oY5o)
end
figure
[AXH1H2] = plotyy(D3()T(1)D3()Ta(4)plot) hold on
H3 = line(D3()T(5)ParentAX(1)) hold on
H4 = line(D3()Ta(3)Parent AX(2)) hold off
set(AXYLimModeautoYTickModeauto)
title(Static Tension vs Tensioner Pulley 1 Diameter)
xlabel(D3 (m))
set(get(AX(1)Ylabel)StringCS Span Tension (N))
set(get(AX(2)Ylabel)StringISG Span Tension (N))
set(H1LineStyle-Colorb)
set(H2LineStyle-Colorg)
set(H3LineStyleColorget(H1Color))
set(H4LineStyleColorget(H2Color))
legend([H1H2H3H4]CS Tautest SpanISG Tautest SpanCS Slackest SpanISG
Slackest SpanLocationBest)
for i = 1s
[T(i)Ta(i)] = TwinTenStaticTension(KtoKt1oKt2oD3oD5(i)X3oY3oX5oY5o)
end
figure
[AXH1H2] = plotyy(D5()T(1)D5()Ta(4)plot) hold on
H3 = line(D5()T(5)ParentAX(1)) hold on
H4 = line(D5()Ta(3)Parent AX(2)) hold off
set(AXYLimModeautoYTickModeauto)
title(Static Tension vs Tensioner Pulley 2 Diameter)
xlabel(D5 (m))
set(get(AX(1)Ylabel)StringCS Span Tension (N))
set(get(AX(2)Ylabel)StringISG Span Tension (N))
set(H1LineStyle-Colorb)
set(H2LineStyle-Colorg)
set(H3LineStyleColorget(H1Color))
set(H4LineStyleColorget(H2Color))
legend([H1H2H3H4]CS Tautest SpanISG Tautest SpanCS Slackest SpanISG
Slackest SpanLocationBest)
Mesh points
m = 101
n = 101
Appendix D 165
T = zeros(5nm)
Ta = zeros(5nm)
[ixxiyy] = meshgrid(1m1n)
minX3 = 0260200
maxX3 = 0317677
minY3 = -0056640
maxY3 = 0228456
midX3 = 0311641
X3 = minX3 + (ixx-1)(maxX3-minX3)(m-1)
Y3 = minY3 + (iyy-1)(maxY3-minY3)(n-1)
for i = 1n
for j = 1m
if ((X3(ij)lt midX3)ampamp(Y3(ij)gt=(sqrt((0087945^2)-((X3(ij)-0224)^2)-
006395)))ampamp(Y3(ij)lt=(-1sqrt(((00703^2)-((X3(ij)-
024759)^2)))+016664)))||((X3(ij)gt=midX3)ampamp(Y3(ij)gt=(35548X3(ij)-
11134868))ampamp(Y3(ij)lt=(-1(sqrt(((00703^2)-((X3(ij)-024759)^2))))+016664))) mx+b
lt= y lt= circle4
[T(ij)Ta(ij)] =
TwinTenStaticTension(KtoKt1oKt2oD3oD5oX3(ij)Y3(ij)X5oY5o)
else
T(ij) = zeros(511)
Ta(ij) = zeros(511)
end
end
end
figure
Z1 = squeeze(T(1))
surf(X3Y3real(Z1))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(CS DRIVING Tautest Span Tension vs Tensioner Pulley 1 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(CS Span Tension (N))
figure
Z5 = squeeze(T(5))
surf(X3Y3real(Z5))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(CS DRIVING Slackest Span Tension vs Tensioner Pulley 1 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
Appendix D 166
zlabel(CS Span Tension (N))
figure
Za4 = squeeze(Ta(4))
surf(X3Y3real(Za4))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(ISG DRIVING Tautest Span Tension vs Tensioner Pulley 1 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(ISG Span Tension (N))
figure
Za3 = squeeze(Ta(3))
surf(X3Y3real(Za3))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(ISG DRIVING Slackest Span Tension vs Tensioner Pulley 1 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(ISG Span Tension (N))
minX5 = -0037093
maxX5 = 0212509
minY5 = 00362
maxY5 = 0228456
midX5a = 0131965
midX5b = 017729
X5 = minX5 + (ixx-1)(maxX5-minX5)(m-1)
Y5 = minY5 + (iyy-1)(maxY5-minY5)(n-1)
for i = 1n
for j = 1m
if
(X5(ij)ltmidX5a)ampamp(Y5(ij)lt=(0386X5(ij)+0146468))ampamp(Y5(ij)gt=(sqrt((0136525^2)-
(X5(ij)^2))))
[T(ij)Ta(ij)] =
TwinTenStaticTension(KtoKt1oKt2oD3oD5oX3oY3oX5(ij)Y5(ij))
elseif
((X5(ij)gt=midX5a)ampamp(X5(ij)ltmidX5b))ampamp(Y5(ij)gt=00362)ampamp(Y5(ij)lt=(0386X5(ij)+0
146468))
[T(ij)Ta(ij)] =
TwinTenStaticTension(KtoKt1oKt2oD3oD5oX3oY3oX5(ij)Y5(ij))
elseif (X5(ij)gt=midX5b)ampamp(Y5(ij)gt=(sqrt((00703^2)-(((X5(ij)-
024759)^2)))+016664))ampamp(Y5(ij)lt=(0386X5(ij)+0146468))
Appendix D 167
[T(ij)Ta(ij)] =
TwinTenStaticTension(KtoKt1oKt2oD3oD5oX3oY3oX5(ij)Y5(ij))
else
T(ij) = zeros(511)
Ta(ij) = zeros(511)
end
end
end
figure
Z1 = squeeze(T(1))
surf(X5Y5real(Z1))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(CS DRIVING Tautest Span Tension vs Tensioner Pulley 2 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(CS Span Tension (N))
figure
Z5 = squeeze(T(5))
surf(X5Y5real(Z5))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(CS DRIVING Slackest Span Tension vs Tensioner Pulley 2 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(CS Span Tension (N))
figure
Za4 = squeeze(Ta(4))
surf(X5Y5real(Za4))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(ISG DRIVING Tautest Span Tension vs Tensioner Pulley 2 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(ISG Span Tension (N))
figure
Za3 = squeeze(Ta(3))
surf(X5Y5real(Za3))
ZLim([50 500])
axis tight
Appendix D 168
colormap jet
colorbar
title(ISG DRIVING Slackest Span Tension vs Tensioner Pulley 2 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(ISG Span Tension (N))
D12 TwinTenStaticTensionm
The function TwinTenStaticTensionm simulates the static model of the B-ISG system with a
Twin Tensioner This function returns 3 vectors the static tension of each belt span in the
crankshaft- and ISG-driving phases of operation and the angle of displacement of each rigid
body in the ISG- driving phase It takes the input parameters kt kt1 kt2 for the tensioner arm
stiffness D3 and D5 for the tensioner pulley diameters and X3Y3 X5 and Y5 for the tensioner
arm pulley coordinates The function is called in the parametric analysis solution script
TwinMainm and in the optimization solution script OptimizationTwinm
D2 Optimization
D21 OptimizationTwinm
The following script is for the main function OptimizationTwinm It performs an optimization
search for the B-ISG system with a Twin Tensioner It takes an input for a parameter vector
containing values for the design variables The program calls the objective function
objfunTwinm and the constraint function confunTwinm The program can perform a genetic
algorithm (GA) optimization search or a hybrid GA optimization that includes a localized search
The optimal solution vector corresponding to the design variables and the optimal objective
function value is returned The program inputs the optimized values for the design variables into
TwinTenStaticTensionm This called function returns the optimized static state of tensions for
the crankshaft- and ISG- driving phases and for the angle of displacement of the rigid bodies in
the ISG driving phase
______________________________________________________________________________
clc
clear all
Initial values for variables
Kto = 20626
Kt1o = 10314
Kt2o = 16502
X3o = 0292761
Y3o = 0087
X5o = 012057
Appendix D 169
Y5o = 009193
w0 =[Kto Kt1o Kt2o X3o Y3o X5o Y5o] Start Point (row vector)
Variable ranges
minKt = Kto - 1Kto
maxKt = Kto + 1Kto
minKt1 = Kt1o - 1Kt1o
maxKt1 = Kt1o + 1Kt1o
minKt2 = Kt2o - 1Kt2o
maxKt2 = Kt2o + 1Kt2o
minX3 = 0260200
maxX3 = 0317677
minY3 = -0056640
maxY3 = 0228456
minX5 = -0037093
maxX5 = 0212509
minY5 = 00362
maxY5 = 0228456
ObjectiveFunction = objfunTwin
nvars = 7 Number of variables
ConstraintFunction = confunTwin
Uncomment next two lines (and comment the two functions after them) to use GA algorithm
for optimization
options = gaoptimset(InitialPopulationw0PopInitRange[minKt minKt1 minKt2 minX3
minY3 minX5 minY5 maxKt maxKt1 maxKt2 maxX3 maxY3 maxX5
maxY5]PopulationSize100Displayfinal)
[wfvalexitflagoutput] =
ga(ObjectiveFunctionnvars[][][][][][]ConstraintFunctionoptions)
fminconOptions = optimset(DisplayiterLargeScaleoff) Largescale off since gradient not
provided
options = gaoptimset(InitialPopulationw0PopInitRange[minKt minKt1 minKt2 minX3
minY3 minX5 minY5 maxKt maxKt1 maxKt2 maxX3 maxY3 maxX5
maxY5]PopulationSize100HybridFcnfmincon fminconOptions)
[wfvalexitflagoutput] = ga(ObjectiveFunctionnvars[][][][][][]ConstraintFunctionoptions)
[TTaThetaDegA] = TwinTenStaticTension(w(1)w(2)w(3)w(4)w(5)w(6)w(7))
D22 confunTwinm
The constraint function confunTwinm is used by the main optimization program to ensure
input values are constrained within the prescribed regions The function makes use of inequality
constraints for seven constrained variables corresponding to the design variables It takes an
input vector corresponding to the design variables and returns a data set of the vector values that
satisfy the prescribed constraints
Appendix D 170
D23 objfunTwinm
This function is the objective function for the main optimization program It outputs a value for
a weighted objective function or a non-weighted objective function relating the optimization of
the static tension The program takes an input vector containing a set of values for the design
variables that are within prescribed constraints The description of the function is similar to
TwinTenStaticTensionm but differs in the fact that it only returns a scalar value which is the
value of the objective function
171
VITA
ADEBUKOLA OLATUNDE
Email adebukolaolatundegmailcom
Adebukola Olatunde is a graduate research student at the University of Toronto in Toronto
Ontario Canada She obtained a Bachelor‟s Degree in Mechanical Engineering from McMaster
University in Hamilton Ontario Canada in 2002 Upon graduation she pursued a graduate
degree in mechanical engineering at the University of Toronto with a specialization in
mechanical systems dynamics and vibrations and environmental engineering In September
2008 she completed the requirements for the Master of Applied Science degree in Mechanical
Engineering She has held the position of teaching assistant for undergraduate courses in
dynamics and vibrations Adebukola has completed course work in professional education She
is a registered member of professional engineering organizations including the Professional
Engineer‟s of Ontario Engineer-in-Training program the Canadian Society of Mechanical
Engineers and the National Society of Black Engineers She intends to practice as a professional
engineering consultant in mechanical design
ii
ABSTRACT
DESIGN AND ANALYSIS OF A TENSIONER FOR A BELT-DRIVEN INTEGRATED
STARTER-GENERATOR SYSTEM OF MICRO-HYBRID VEHICLES
Adebukola O Olatunde
Master of Applied Science
Graduate Department of Mechanical and Industrial Engineering
University of Toronto 2008
The thesis presents the design and analysis of a Twin Tensioner for a Belt-driven Integrated
Starter-generator (B-ISG) system The B-ISG is an emerging hybrid transmission closely
resembling conventional serpentine belt drives Models of the B-ISG system‟s geometric
properties and dynamic and static states are derived and simulated The objective is to reduce
the magnitudes of static tension in the belt for the ISG-driving phase A literature review of
hybrid systems serpentine belt drive modeling and automotive tensioners is included A
parametric study evaluates tensioner parameters with respect to their impact on static tensions
Design variables are selected from these for an optimization study The optimization uses a
genetic algorithm (GA) and a hybrid GA Results of the optimization indicate the optimal
system contains spans with static tensions that are significantly lower in magnitude than that of
the original design Implications of the research on future work are discussed in closing
iii
A testament unto the LORD God lsquowho answered me in the day of my distress and was with me in
the way which I wentrsquo
To my parents Joseph and Beatrice for your strength and persistent prayers
To my siblings Shade Charlene and Kevin for being a listener an editor and a relief when I
needed it
To my friends Samantha Esther and Yasmin who kept me motivated
amp
With love to my sweetheart Nana whose patience support and companionship has made life
sweeter
iv
ACKNOLOWEDGEMENTS
I would like to express deep gratitude to Dr Jean Zu for her guidance throughout the duration of
my studies and for providing me with the opportunity to conduct this thesis
I wish to thank the individuals of Litens Automotive who have provided guidance and data for
the research work Special thanks to Mike Clark Seeva Karuendiran and Dr Qiu for their time
and help
I thank my committee members Dr Naguib and Dr Sun for contributing their time to my
research work
My sincerest thanks to my research colleague David for his knowledge and support Many
thanks to my lab mates Qiming Hansong Ali Ming Andrew and Peyman for their guidance
I want to especially thank Dr Cleghorn Leslie Sinclair and Dr Zu for the opportunities to
teach These experiences have served to enrich my graduate studies As well thank you to Dr
Cleghorn for guidance in my research work
I am also in debt to my classmates and teaching colleagues throughout my time at the University
of Toronto especially Aaron and Mohammed for their support in my development as a graduate
researcher and teacher
v
CONTENTS
ABSTRACT ii
DEDICATION iii
ACKNOWLEDGEMENTS iv
CONTENTS v
LIST OF TABLES ix
LIST OF FIGURES xi
LIST OF SYMBOLS xvi
Chapter 1 INTRODUCTION 1
11 Background 1
12 Motivation 3
13 Thesis Objectives and Scope of Research 4
14 Organization and Content of Thesis 5
Chapter 2 LITERATURE REVIEW 7
21 Introduction 7
22 B-ISG System 8
221 ISG in Hybrids 8
2211 Full Hybrids 9
2212 Power Hybrids 10
2213 Mild Hybrids 11
2214 Micro Hybrids 11
222 B-ISG Structure Location and Function 13
2221 Structure and Location 13
2222 Functionalities 14
23 Belt Drive Modeling 15
24 Tensioners for B-ISG System 18
241 Tensioners Structures Function and Location 18
242 Systematic Review of Tensioner Designs for a B-ISG System 20
25 Summary 24
vi
Chapter 3 MODELING OF B-ISG SYSTEM 25
31 Overview 25
32 B-ISG Tensioner Design 25
33 Geometric Model of a B-ISG System with a Twin Tensioner 27
34 Equations of Motion for a B-ISG System with a Twin Tensioner 32
341 Dynamic Model of the B-ISG System 32
3411 Derivation of Equations of Motion 32
3412 Modeling of Phase Change 41
3413 Natural Frequencies Mode Shapes and Dynamic Responses 42
3414 Crankshaft Pulley Driving Torque Acceleration and Displacement 44
3415 ISG Pulley Driving Torque Acceleration and Displacement 46
3416 Tensioner Arms Dynamic Torques 48
3417 Dynamic Belt Span Tensions 49
342 Static Model of the B-ISG System 49
35 Simulations 50
351 Geometric Analysis 51
352 Dynamic Analysis 52
3521 Natural Frequency and Mode Shape 54
3522 Dynamic Response 58
3523 ISG Pulley and Crankshaft Pulley Torque Requirement 61
3524 Tensioner Arm Torque Requirement 62
3525 Dynamic Belt Span Tension 63
353 Static Analysis 66
36 Summary 69
Chapter 4 PARAMETRIC ANALYSIS OF A B-ISG TWIN TENSIONER 71
41 Introduction 71
42 Methodology 71
43 Results and Discussion 74
431 Influence of Tensioner Arm Stiffness on Static Tension 74
432 Influence of Tensioner Pulley Diameter on Static Tension 78
433 Influence of Tensioner Pulley 1 Coordinates on Static Tension 80
434 Influence of Tensioner Pulley 2 Coordinates on Static Tension 86
vii
44 Conclusion 92
Chapter 5 OPTIMIZATION OF A B-ISG TWIN TENSIONER 95
51 Optimization Problem 95
511 Selection of Design Variables 95
512 Objective Function amp Constraints 97
52 Optimization Method 100
521 Genetic Algorithm 100
522 Hybrid Optimization Algorithm 101
53 Results and Discussion 101
531 Parameter Settings amp Stopping Criteria for Simulations 101
532 Optimization Simulations 102
533 Discussion 106
54 Conclusion 109
Chapter 6 CONCLUSION AND RECOMMENDATIONS111
61 Summary 111
62 Conclusion 112
63 Recommendations for Future Work 113
REFERENCES 116
APPENDICIES 123
A Passive Dual Tensioner Designs from Patent Literature 123
B B-ISG Serpentine Belt Drive with Single Tensioner Equation of Motion 138
C MathCAD Scripts 145
C1 Geometric Analysis 145
C2 Dynamic Analysis 152
C3 Static Analysis 161
D MATLAB Functions amp Scripts 162
D1 Parametric Analysis 162
D11 TwinMainm 162
D12 TwinTenStaticTensionm 168
D2 Optimization 168
D21 OptimizationTwinm - Optimization Function 168
viii
D22 confunTwinm 169
D23 objfunTwinm 170
VITA 171
ix
LIST OF TABLES
21 Passive Dual Tensioner Designs from Patent Literature
31 Selected Contact Point Types on the ith Pulley and Types for the ith Belt Span
32 Coordinate Points for Pulley Centres and Twin Tensioner Pivot
33 Geometric Results of B-ISG System with Twin Tensioner
34 Data for Input Parameters used in Dynamic and Static Computations
35 Static Solution for Belt Span Tensions in Crankshaft and ISG Driving Cases for a B-ISG
Serpentine Belt Drive with a Single Tensioner
36 Static Solution for Belt Span Tensions in Crankshaft and ISG Driving Cases for a B-ISG
Serpentine Belt Drive with a Twin Tensioner
41 Initial Values Increments and Ranges for Parameters of Twin Tensioner
51 Summary of Parametric Analysis Data for Twin Tensioner Properties
52a GA Optimization Results for Twin Tensioner Parameters and Objective Function
52b Computations for Tensions and Angles from GA Optimization Results
53a Hybrid Optimization Results for Twin Tensioner Parameters and Objective Function
53b Computations for Tensions and Angles from Hybrid Optimization Results
54a Non-Weighted Optimization Results for Twin Tensioner Parameters and Objective
Function
54b Computations for Tensions and Angles from Non-Weighted Optimizations
x
55 Weighted Optimization Results for Static Tensions for Optimal B-ISG System with a
Twin Tensioner
56 Non-Weighted Optimization Results for Static Tensions for Optimal B-ISG System with a
Twin Tensioner
xi
LIST OF FIGURES
21 Hybrid Functions
31 Schematic of the Twin Tensioner
32 B-ISG Serpentine Belt Drive with Twin Tensioner
33 Angles Coordinates and Possible Belt Contact Points for the ith and i+1th Pulleys
34 Twin Tensioner Free Body Diagram of a Four-Degree of Freedom System
35 Free Body Diagram for Non-Tensioner Pulleys
36a ISG Driving Case Natural Frequency of System and Mode Shapes for Responsive Rigid
Bodies
36b ISG Driving Case First Mode Responses
36c ISG Driving Case Second Mode Responses
37a Crankshaft Driving Case Natural Frequency of System and Mode Shapes for Responsive
Rigid Bodies
37b Crankshaft Driving Case First Mode Responses
37c Crankshaft Driving Case Second Mode Responses
38 Crankshaft Pulley Dynamic Response (for crankshaft driven case)
39 ISG Pulley Dynamic Response (for ISG driven case)
310 Air Conditioner Pulley Dynamic Response
311 Tensioner Pulley 1 Dynamic Response
xii
312 Tensioner Pulley 2 Dynamic Response
313 Tensioner Arm 1 Dynamic Response
314 Tensioner Arm 2 Dynamic Response
315 Required Driving Torque for the ISG Pulley
316 Required Driving Torque for the Crankshaft Pulley
317 Dynamic Torque for Tensioner Arm 1
318 Dynamic Torque for Tensioner Arm 2
319 Span 1 (between crankshaft and air conditioner) Dynamic Belt Span Tension
320 Span 2 (between air conditioner and tensioner 1) Dynamic Belt Span Tension
321 Span 3 (between tensioner 1 and ISG) Dynamic Belt Span Tension
322 Span 4 (between ISG and tensioner 2) Dynamic Belt Span Tension
323 Span 5 (between tensioner 2 and crankshaft) Dynamic Belt Span Tension
324 B-ISG Serpentine Belt Drive with Single Tensioner
41a Regions 1 and 2 and their Associated Guidelines for Coordinates of Tensioner Pulleys 1
amp 2
41b Regions 1 and 2 in Cartesian Space
42 Parametric Analysis for Coupled Stiffness Arm Constant kt (N∙mrad)
43 Parametric Analysis for Stiffness of Arm 1 kt1 (N∙mrad)
44 Parametric Analysis for Stiffness of Arm 2 kt2 (N∙mrad)
xiii
45 Parametric Analysis for Pulley 1 Diameter D3 (m)
46 Parametric Analysis for Pulley 2 Diameter D5 (m)
47 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Tautest Span
Tension in Crankshaft Driving Case
48 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Slackest Span
Tension in Crankshaft Driving Case
49 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Tautest Span
Tension in ISG Driving Case
410 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Slackest Span
Tension in ISG Driving Case
411 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Tautest Span
Tension in Crankshaft Driving Case
412 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Slackest Span
Tension in Crankshaft Driving Case
413 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Tautest Span
Tension in ISG Driving Case
414 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Slackest Span
Tension in ISG Driving Case
51 Static Stability of the B-ISG Twin Tensioner Based on the Angular Displacement of
Tensioner Arms 1 and 2
A1 Proposed design by Bayerische Motoren Werke AG corresponding to patent nos
EP1420192-A2 and DE10253450-A1
A2a First of four proposed designs (and its various configurations) by Bosch GMBH
corresponding to patent no WO0026532-A1
A2b Second of four proposed designs by Bosch GMBH corresponding to patent no
WO0026532-A1
A2c Third of four proposed designs (and its various configurations) by Bosch GMBH
corresponding to patent no WO0026532-A1
A2d Fourth of four proposed designs (and its various configurations) by Bosch GMBH
corresponding to patent no WO0026532-A1
A3 Proposed design by Daimler Chrysler AG corresponding to patent no DE10324268-A1
A4 Proposed design by Dayco Products LLC corresponding to patent no US6942589-B2
xiv
A5 Proposed design by Gates Corp corresponding to patent no WO2003038309-A
A6 Proposed design by General Motors Corp corresponding to patent no US20060287146-
A1
A7 Proposed design by INA Schaeffler KG corresponding to patent no DE10044645-A1
A8a First of three proposed designs by INA Schaeffler KG corresponding to patent no
DE10159073-A1
A8b Second of three proposed designs by INA Schaeffler KG corresponding to patent no
DE10159073-A1
A8c Third of three proposed designs by INA Schaeffler KG corresponding to patent no
DE10159073-A1
A9 Proposed design by INA Schaeffler KG corresponding to patent no DE10359641-A1
A10 Proposed design by INA Schaeffler KG corresponding to patent no EP1723350-A1
A11 Proposed design by INA Schaeffler KG corresponding to patent no EP1738093-A1
A12 Proposed design by INA Schaeffler KG corresponding to patent no DE102004012395-
A1
A13 Proposed design by INA Schaeffler KG corresponding to patent nos DE102005017038-
A1and WO2006108461-A1
A14 Proposed design by Litens Automotive GMBH et al corresponding to patent no
US20010007839-A1
A15 Proposed design by Mitsubishi Jidosha Eng KK and Mitsubishi Motor Corp
corresponding to patent no JP2005083514-A
A16 Proposed design by Nissan Motor Co Ltd corresponding to patent no JP3565040-B2
A17 Proposed design by NTN Corp corresponding to patent no JP2006189073-A
A18 Proposed design by Valeo Equipment Electriques Moteur corresponding to patent nos
EP1658432 and WO2005015007
B1 Single Tensioner B-ISG System
B2 Free-body Diagram of ith Pulley
xv
B3 Free-body Diagram of Single Tensioner
C1 Schematic of B-ISG System with Twin Tensioner
C2 Possible Contact Points
xvi
LIST OF SYMBOLS
Latin Letters
A Belt cord cross-sectional area
C Damping matrix of the system
cb Belt damping
119888119894119887 Belt damping constant of the ith belt span
119914119946119946 Damping matrix element in the ith row and ith column
ct Damping acting between tensioner arms 1 and 2
cti Damping of the ith tensioner arm
DCS Diameter of crankshaft pulley
DISG Diameter of ISG pulley
ft Belt transition frequency
H(n) Phase change function
I Inertial matrix of the system
119920119938 Inertial matrix under ISG driving phase
119920119940 Inertial matrix under crankshaft driving phase
Ii Inertia of the ith pulley
Iti Inertia of the ith tensioner arm
119920120784120784 Submatrix of inertial matrix I
j Imaginary coordinate (ie (-1)12
)
K Stiffness matrix of the system
xvii
119896119887 Belt factor
119870119887 Belt cord stiffness
119896119894119887 Belt stiffness constant of the ith belt span
kt Spring stiffness acting between tensioner arms 1 and 2
kti Coil spring of the ith tensioner arm
119922120784120784 Submatrix of stiffness matrix K
Lfi Lbi Lengths of possible belt span connections from the ith pulley
Lti Length of the ith tensioner arm
Modeia Mode shape of the ith rigid body in the ISG driving phase
Modeic Mode shape of the ith rigid body in the crankshaft driving phase
n Engine speed
N Motor speed
nCS rpm of crankshaft pulley
NF Motor speed without load
nISG rpm of ISG pulley
Q Required torque matrix
qc Amplitude of the required crankshaft torque
QcsISG Required torque of the driving pulley (crankshaft or ISG)
Qm Required torque matrix of driven rigid bodies
Qti Dynamic torque of the ith tensioner arm
Ri Radius of the ith pulley
T Matrix of belt span static tensions
xviii
Trsquo Dynamic belt tension matrix
119931119940 Damping matrix due to the belt
119931119948 Stiffness matrix due to the belt
Ti Tension of the ith belt span
To Initial belt tension for the system
Ts Stall torque
Tti Tension for the neighbouring belt spans of the ith tensioner pulley
(XiYi) Coordinates of the ith pulley centre
XYfi XYbi XYfbi
XYbfi Possible connection points on the ith pulley leading to the ith belt span
XYf2i XYb2i
XYfb2i XYbf2i Possible connection points on the ith pulley leading to the (i-1)th belt span
Greek Letters
αi Angle between the datum and the line connecting the ith and (i+1)th pulley
centres
βji Angle of orientation for the ith belt span
120597θti(t) 120579 ti(t)
120579 ti(t)
Angular displacement velocity and acceleration (rotational coordinate) of the
ith tensioner arm
120637119938 General coordinate matrix under ISG driving phase
120637119940 General coordinate matrix under crankshaft driving phase
θfi θbi Angles between the datum and the belt connection spans with lengths Lfi and
Lbi respectively
Θi Amplitude of displacement of the ith pulley
xix
θi(t) 120579 i(t) 120579 i(t) Angular position velocity and acceleration (rotational coordinate) of the ith
pulley
θti Angle of the ith tensioner arm
θtoi Initial pivot angle of the ith tensioner arm
θm Angular displacement matrix of driven rigid bodies
Θm Amplitude of displacement of driven rigid bodies
ρ Belt cord density
120601119894 Belt wrap angle on the ith pulley
φmax Belt maximum phase angle
φ0deg Belt phase angle at zero frequency
ω Frequency of the system
ωcs Angular frequency of crankshaft pulley
ωISG Angular frequency of the ISG pulley
120654119951 Natural frequency of system
1
CHAPTER 1 INTRODUCTION
11 Background
Belt drive systems are the means of power transmission in conventional automobiles The
emergence of hybrid technologies specifically the Belt-driven Integrated Starter-generator (B-
ISG) has placed higher demands on belt drives than ever before The presence of an integrated
starter-generator (ISG) in a belt transmission places excessive strain on the belt leading to
premature belt failure This phenomenon has motivated automotive makers to design a tensioner
that is suitable for the B-ISG system
The belt drive is also known interchangeably as the front-end accessory drive-belt (FEAD) the
belt accessory-drive system (BAS) or the belt transmission system In a traditional setting the
role of this system is to transmit torque generated by an internal combustion engine (ICE) in
order to reliably drive multiple peripheral devices mounted on the engine block The high speed
torque is transmitted through a crankshaft pulley to a serpentine belt The serpentine belt is a
single continuous member that winds around the driving and driven accessory pulleys of the
drive system Serpentine belts used in automotive applications consist of several layers The
load-bearing layer is a flexible member consisting of high stiffness fibers [1] It is covered by a
protective layer to guard against mechanical damage and is bound below by a visco-elastic layer
that provides the required shock absorption and grip against the rigid pulleys [1] The accessory
devices may include an alternator power steering pump water pump and air conditioner
compressor among others
Introduction 2
The B-ISG system is a transmission system characteristic to micro-hybrid automobiles It is akin
to traditional belt drives differing in the fact that an electric motor called an integrated starter-
generator (ISG) replaces the original alternator re-starts the engine from idle speed and provides
braking regeneration [2] The re-start function of the micro-hybrid transmission is known as
stop-start In the B-ISG setting the ISG is mounted on the belt drive The ISG produces a speed
of approximately 2000 to 2500rpm in order to spin the engine at approximately 750rpm and
upwards to produce an instantaneous start in the start-stop process [3] The high rotations per
minute (rpm) produced by the ISG consistently places much higher tension requirements on the
belt than when the crankshaft is driving the belt It is preferable not to exceed a range of 600N to
800N of tension on the belt since this exceeds the safe operating conditions of belts used in most
traditional drive systems [4] The traditional belt drive system‟s tensioner a single-arm
tensioner does not suitably reduce the high belt tension nor provide enough tension in the slack
belts spans occurring in the ISG phase of operation for the B-ISG system
In order for the belt to transfer torque in a drive system its initial tension must be set to a value
that is sufficient to keep all spans rigid This value must not be too low as to allow any one span
to be slack during the drive‟s phases of operation Furthermore the belt must not be ldquoinstalled
with too high a tensionrdquo since this can lead to ldquopremature failure of the bearings supporting the
drive and driven pulleys and of the belt itselfrdquo [5] The presence of a tensioning mechanism in
an automotive belt drive allows for an enhanced belt life and performance since pre-tensioning
of the belt is normally not sufficient for all phases of belt drive operation A tensioner allows for
the system to cope with moderate to severe changes in belt span tensions
Introduction 3
Traditional automotive tensioners for belt drives of an ICE consist of a single spring-loaded
arm This type of tensioner is normally designed to provide a passive response to changes in belt
span tension The introduction of the ISG electric motor into the traditional belt drive with a
single-arm tensioner results in the presence of excessively slack spans and excessively tight
spans in the belt The tension requirements in the ISG-driving phase which differ from the
crankshaft-driving phase are poorly met by a traditional single-arm passive tensioner
Tensioners can be divided into two general classes passive and active In both classes the
single-arm tensioner design approach is the norm The passive class of tensioners employ purely
mechanical power to achieve tensioning of the belt while the active class also known as
automatic tensioners typically use some sort of electronic actuation Automatic tensioners have
been employed by various automotive manufacturers however ldquosuch devices add mass
complication and cost to each enginerdquo [5]
12 Motivation
The motivation for the research undertaken arises from the undesirable presence of high belt
tension in automotive belt drives Manufacturers of automotive belt drives have presented
numerous approaches for tension mechanism designs As mentioned in the preceding section
the automation of the traditional single-arm tensioner has disadvantages for manufacturers A
survey of the literature reveals that few quantitative investigations in comparison to the
qualitative investigations provided through patent literature have been conducted in the area of
passive and dual tensioner configurations As such the author of the research project has selected
to investigate the performance of a passive twin-arm tensioner design The theoretical tensioner
Introduction 4
configuration is motivated by research and developments of industry partner Litens
Automotivendash a manufacturer of automotive belt drive systems and components Litens‟
specialty in automotive tensioners has provided a basis for the research work conducted
13 Thesis Objectives and Scope of Research
The objective of this project is to model and investigate a system containing a passive twin-arm
tensioner in a B-ISG serpentine belt drive where the driving pulley alternates between a
crankshaft pulley and an ISG pulley The modeling of a serpentine belt drive system is in
continuation of the work done by post-doctoral fellow Zhen Mu in development of the priority
software known as FEAD at the University of Toronto Firstly for the B-ISG system with a
twin-arm tensioner the geometric state and its equations of motion (EOM) describing the
dynamic and static states are derived The modeling approach was verified by deriving the
geometric properties and the EOM of the system with a single tensioner arm and comparing its
crankshaft-phase‟s simulation results with FEAD software simulations This also provides
comparison of the new twin-arm tensioner belt drive model with the former single-arm tensioner
equipped belt drive model Secondly the model for the static system is investigated through
analysis of the tensioner parameters Thirdly the design variables selected from the parametric
analysis are used for optimization of the new system with respect to its criteria for desired
performance
Introduction 5
14 Organization and Content of the Thesis
This thesis presents the investigation of a passive twin-arm tensioner design in a B-ISG
serpentine belt drive system which is distinguished by having its driving pulley alternate
between a crankshaft pulley and an ISG pulley
Chapter 2 presents the literature reviewed relevant to the area of the thesis topic The context of
the research discusses the function and location of the ISG in hybrid technologies in order to
provide a background for the B-ISG system The attributes of the B-ISG are then discussed
Subsequently a description is given of the developments made in modeling belt drive systems
At the close of the chapter the prior art in tensioner designs and investigations are discussed
The third chapter describes the system models and theory for the B-ISG system with a twin-arm
tensioner Models for the geometric properties and the static and dynamic cases are derived The
simulation results of the system model are presented
Then the fourth chapter contains the parametric analysis The methodologies employed results
and a discussion are provided The design variables of the system to be considered in the
optimization are also discussed
The optimization of a B-ISG system with a passive twin-arm tensioner is presented in Chapter 5
The evaluation of optimization methods results of optimization and discussion of the results are
included Chapter 6 concludes the thesis work in summarizing the response to the thesis
Introduction 6
objectives and concluding the results of the investigation of the objectives Recommendations for
future work in the design and analysis of a B-ISG tensioner design are also described
7
CHAPTER 2 LITERATURE REVIEW
21 Introduction
This literature review justifies the study of the thesis research the significance of the topic and
provides the overall framework for the project The design of a tensioner for a Belt-driven
Integrated Starter-generator (B-ISG) system is a link in the chain of power transmission
developments in hybrid automobiles This chapter will begin with the context of the B-ISG
followed by a review of the hybrid classifications and the critical role of the ISG for each type
The function location and structure of the B-ISG system are then discussed Then a discussion
of the modeling of automotive belt transmissions is presented A systematic review of the prior
art and current state of tensioning mechanisms for B-ISG systems amalgamates the literature and
research evidence relevant to the thesis topic which is the design of a B-ISG tensioner
The Belt-driven Integrated Starter-generator (B-ISG) system is a part of a hybrid class that is
distinguished from other hybrid classes by the structure functions and location of its ISG The
B-ISG unit is a hybrid technology applied to traditional automotive belt drives The use of a B-
ISG system to achieve a start-stop function in the car engine is estimated to cut fuel consumption
in conventional automobiles by up to ten percent and thus reduce CO2 emissions [6]
Environmental and legislative standards for reducing CO2 emissions in vehicles have called for
carmakers to produce less polluting and more efficient vehicle powertrain systems [7] The
transition to bdquocleaner‟ cars makes room for the introduction of the ISG machine into conventional
automotive belt drives [8] The reduction of CO2 emissions and the similarity of the B-ISG
Literature Review 8
transmission to that of conventional cars provide the motivation for the thesis research
Consequently the micro-hybrid class of cars is especially discussed in the literature review since
it contains the B-ISG type of transmission system The micro-hybrid class is one of several
hybrid classes
A look at the performance of a belt-drive under the influence of an ISG is rooted in the
developments of hybrid technology The distinction of the ISG function and its location in each
hybrid class is discussed in the following section
22 B-ISG System
221 ISG in Hybrids
This section of the review discusses the standard classes of hybrid cars which are full power
mild and micro- hybrids Special attention is given to hybrid vehicle architectures involving
internal combustion engines (ICEs) as the main power source This is done for the sake of
comparison between hybrid classes since the ICE is the standard power source for B-ISG micro-
hybrids which is the focus of the research The term conventional car vehicle or automobile
henceforth refers to a vehicle powered solely by a gas or diesel ICE
A hybrid vehicle has a drive system that uses a combination of energy devices This may include
an ICE a battery and an electric motor typically an ISG Two systems exist in the classification
of hybrid vehicles The older system of classification separates hybrids into two classes series
hybrids and parallel hybrids In the older system many modern hybrid vehicles have modes of
operation matching both categories classifying them under either of the two classes [9] The
Literature Review 9
new system of classification has four classes full power mild and micro Under these classes
vehicles are more often under a sole category [9] In both systems an ICE may act as the primary
source of power otherwise it may be a fuel cell The fuel used by the ICE may be gas (petrol)
diesel or an alternative fuel such as ethanol bio-diesel or natural gas
2211 Full Hybrids
In a full hybrid car the ICE is used to power the integrated starter-generator (ISG) which stores
electrical energy in the batteries to be used to power an electric traction motor [8] The electric
traction motor is akin to a second ISG as it generates power and provides torque output It also
supplies an extra boost to the wheels during acceleration and drives up steep inclines A full
hybrid vehicle is able to move by electrical power only It can be driven by the ISG powering
the electric traction motor without the engine running This silent acceleration known as electric
launch is normally employed when accelerating from standstill [9] Full hybrids can generate
and consume energy at the same time Full hybrid vehicles also use regenerative braking [8]
The ISG allows this by converting from an electric traction motor to a generator when braking or
decelerating The kinetic energy from the car‟s motion is then turned into electricity and stored
in the batteries For full hybrids to achieve this they often use break-by-wire a form of
electronically controlled braking technology
A high-voltage (ie 36- or 42-volt) ISG is employed in full hybrids to start the ICE It spins the
engine more than 900 rpm whereas conventional 12-volt starter motors spin the engine at
approximately 250 rpm [9] Thus the full hybrid vehicle is able to have an instantaneous start In
full hybrids the ISG is placed in the position of the flywheel and can have its motion decoupled
Literature Review 10
from the engine [9] The ISG device also allows full hybrids to have engine start-stop also called
an idle-stop ability The idle-stop function refers to when the engine shuts down as soon as a
vehicle stops from its ICE driving mode which saves on the fuel it normally burns while idling
[8] The vehicle returns to the engine driving mode of operation by way of the ISG‟s start-up of
the crankshaft which restarts the engine in less than 300 milliseconds [9] In summary at
standstill the tachometer of the engine drops to 0 rpm since the engine has ceased the engine is
started only when needed which is often several seconds after acceleration has begun The
engine start-stop feature is achieved by way of an electronic control system that shuts off the ICE
when it is not needed to assist in driving the wheels or to produce electricity for recharging the
batteries The start-stop feature by itself is estimated to produce a ten percent fuel gain in hybrids
over conventional vehicles particularly in urban driving conditions [9] Since the ICE is
required to provide only the average horsepower used by the vehicle the engine is downsized in
comparison to a conventional automobile that obtains all its power from an ICE Frequently in
full hybrids the ICE uses an alternative operating strategy such as the Atkinson Cycle which has
a higher efficiency while having a lower power output Examples of full hybrids include the
Ford Escape and the Toyota Prius [9]
2212 Power Hybrids
Akin to the full hybrid the ISG of the power hybrid enables the same features electric launch
regenerative braking and engine idle-stop The distinguishing characteristic from full hybrids is
the ICE is not downsized to meet only the average power demand [9] Thus the engine of a
power hybrid is large and produces a high amount of horsepower compared to the former
Overall a power hybrid has the assist of a full size ICE and therefore has more torque and a
Literature Review 11
greater acceleration performance than a full hybrid or a conventional vehicle with the same size
ICE [9] The Lexus RX400h unit is an example of a power hybrid [9]
2213 Mild Hybrids
In the hybrid types discussed thus far the ISG is positioned between the engine and transmission
to provide traction for the wheels and for regenerative braking Often times the armature or rotor
of the electric motor-generator which is the ISG replaces the engine flywheel in full and power
hybrids [9] In the case of the mild hybrid the ISG is not decoupled from the ICE and hence it is
not able to drive the wheels apart from the engine It remains that the ISG shares the same shaft
with the ICE In this environment the electric launch feature does not exist since the ISG does
not turn the wheels independently of the engine and energy cannot be generated and consumed
at the same time However the ISG of the mild hybrid allows for the remaining features of the
full hybrid regenerative braking and engine idle-stop including the fact that the engine is
downsized to meet only the average demand for horsepower Mild hybrid vehicles include the
GMC Sierra pickup and 2003 to 2005 Honda Civic models [9]
2214 Micro Hybrids
Micro hybrid is the category of hybrids that can contain a B-ISG transmission and is also closest
to modern conventional vehicles This class normally features a gas or diesel ICE [9] The
conventional automobile is modified by installing an ISG unit on the mechanical drive in place
of or in addition to the starter motor The starter motor typically 12-volts is removed only in
the case that the ISG device passes cold start testing which is also dependent on the engine size
[10] Various mechanical drives that may be employed include chain gear or belt drives or a
Literature Review 12
clutchgear arrangement The majority of literature pertaining to mechanical driven ISG
applications does not pursue clutchgear arrangements since it is associated with greater costs
and increased speed issues Findings by Henry et al [11] show that the belt drive in
comparison to chain and gear drives has a decreased cost (especially if the ISG is mounted
directly to the accessory drive) has no need for lubrication has less restriction in the packaging
environment and produces very low noise Also mounting the ISG unit on a separate belt from
that linking the accessory pulleys is undesirable since applying the ISG directly to the accessory
belt drive requires less engine transmission or vehicle modifications
As with full power and mild hybrids the presence of the ISG allows for the start-stop feature
The automobile‟s electronic control unit (ECU) is calibrated or engine control circuitry (a
separate ECU) is added to the conventional car in order to shut down the engine when the
vehicle is stopped [12] The control system also controls the charge cycle of the ISG [9] This
entails that it dictates the field current by way of a microprocessor to allow the system to defer
battery charge cycles until the vehicle is decelerating [13] This produces electricity to recharge
the battery primarily during deceleration and braking The B-ISG transmission of a micro hybrid
and its various components are discussed in the subsequent section Examples of micro hybrid
vehicles are the PSA Group‟s Citroen C2 and C3 [14] Ford‟s Fiesta [14] and BMW‟s Mini
Cooper D and various others of BMW‟s European models [15]
Literature Review 13
Figure 21 Hybrid Functions
Source Dr Daniel Kok FFA July 2004 modified [16]
Figure 21 shows that the higher the voltage available to the ISG unit the more hybrid functions
it is capable of performing It is noted that B-ISG transmissions of the micro-hybrid class may
also exceed the typical functions of micro-hybrids For instance Ford‟s HyTrans van (developed
in partnership with Ricardo UK Ltd Valeo SA Gates Corporation and the UK Department for
Transport) uses a B-ISG system and a 42-volt battery The van is diesel-powered and has
characteristics of a mild hybrid such as cold cranks and engine assists [17]
222 B-ISG Structure Location and Function
2221 Structure and Location
The ISG is composed of an electrical machine normally of the inductive type which includes a
stator (stationary part of the ISG) and a rotor (non-stationary part of the ISG) and a converter
comprising of a regulator a modulator switches and filters There are various configurations to
integrate the ISG unit into an automobile power train One configuration situates the ISG
directly on the crankshaft in the place of the present flywheel [11] This set-up is more compact
however it results in a longer power train which becomes a potential concern for transverse-
Literature Review 14
mounted engines [18] An alternative set-up is to have a side-mounted ISG This term is used to
describe the configuration of mounting the electrical device on the side of the mechanical drive
[18] As mentioned in Section 2214 a belt drive is used as the mechanical drive for the thesis
research hence the ISG is belt-mounted and the transmission becomes a belt-driven ISG system
In this arrangement the ISG replaces the alternator [13] and in some cases the starter motor may
be removed This design allows for the functions of the ISG system mentioned in the description
of the ISG role in micro-hybrids [9] The side-mounted ISG specifically the belt-mounted ISG
is more evolutionary to the conventional car since it ldquoallows for a more traditional under-hood
layoutrdquo [11]
2222 Functionalities
The primary duty of the ISG in a micro hybrid specifically in a B-ISG setting is to bring the
engine from rest to normal operating speeds within a time span ranging from 250 to 400 ms [3]
and in some high voltage settings to provide cold starting
The cold starting operation of the ISG refers to starting the engine from its off mode rather than
idle mode andor when the engine is at a low temperature for example -29 to -50 degrees
Celsius [2] If the ISG is used for cold starting the peak torque is determined by the torque
requirement for the cold starting operation of the target vehicle since it is greater than the
nominal torque For this function the ldquomachine has to provide a breakaway torque about 15 [to]
18 times the nominal cranking torque to overcome static torque and rotate the engine from 0 to
[between] 10 [and] 20rpmrdquo [2] This remains to be a challenge for the ISG as the 12-volt
architecture most commonly found in vehicles does not supply sufficient voltage [2] The
introduction of the ISG machine and other electrical units in vehicles encourages a transition
Literature Review 15
from a 12-volt or 14-volt to a 42-volt electrical architecture [19] The transition to 42-volt
architecture brings ldquopotential higher-voltage functionalities that come with an ISG systemrdquo [20]
At present ldquowhen the [ISG] machine cannot provide enough torque for initial cold engine
cranking the conventional starter will [remain] in the system and perform only for the initial
cranking while the stop-start function is taken over by the [ISG] machinerdquo [2] The ISG‟s launch
assist torque the torque required to bring the engine from idle speed to the speed at which it can
develop a higher torque output is 2000 to 2500 rpm for most gas engines [3]
Delphi‟s Energen 5 High Output 12-volt Belt-alternator-starter (or B-ISG) was implemented by
researchers on a 53 L V-8 engine with an automatic transmission in a Chevrolet Silverado truck
[21] The ISG was applied in a belt-mounted configuration and was used only for warm engine
re-starts The results of Wezenbeek et al [21] showed that the starting torque for a re-start by the
12-Volt ISG was 42 Nm ISG‟s have also been used in 14V 36V and 42V architectures [13]
23 Belt Drive Modeling
The modeling of a serpentine belt drive and tensioning mechanism has typically involved the
application of Newtonian equilibrium equations to rigid bodies in order to derive the equations of
motion for the system There are two modes of motion in a serpentine belt drive transverse
motion and rotational motion The former can be viewed as the motion of the belt directed
normal to the direction of the beltpulley contact plane similar to the vibratory motion of a taut
string that is fixed at either end However the study of the rotational motion in a belt drive is the
focus of the thesis research
Literature Review 16
Much work on the mechanics of the belt drive was carried out by Firbank [22] Firbank‟s
models helped to understand belt performance and the influence of driving and driven pulleys on
the tension member The first description of a serpentine belt drive for automotive use was in
1979 by Cassidy et al [23] and since this time there has been an increasing body of knowledge
on the mathematical modeling of serpentine belt drives Ulsoy et al [24] presented a design
methodology to improve the dynamic performance of instability mechanisms for belt tensioner
systems The mathematical model developed by Ulsoy et al [24] coupled the equations of
motion that were obtained through a dynamic equilibrium of moments about a pivot point the
equations of motion for the transverse vibration of the belt and the equations of motion for the
belt tension variations appearing in the transverse vibrations This along with the boundary and
initial conditions were used to describe the vibration and stability of the coupled belt-tensioner
system Their system also considered the geometry of the belt drive and tensioner motion
Hereafter Beikmann et al [25] predicted the belt drive vibration for a system composed of a
driving pulley driven pulley and a dynamic tensioner The authors coupled the linear equations
of transverse motion for the respective belt spans with the equations of motion for pulleys and a
tensioner This was used to form the free response of the system and evaluate its response
through a closed-form solution of the system‟s natural frequencies and mode shapes
A complex modal analysis of a serpentine belt drive system was carried out by Kraver et al [26]
to determine the effect of damping on rotational vibration mode solutions The equations of
motion developed for a multi-pulley flat belt system with viscous damping and elastic
Literature Review 17
properties including the presence of a rotary tensioner were manipulated to carry out the modal
analysis
Beikmann et al [27] also derived a nonlinear model to predict the operating state of a belt-
tensioner system by way of nonlinear numerical methods and an approximated linear closed-
form method The authors used this strategy to develop a single design parameter referred to as
a tensioner constant to measure the effectiveness of the tensioning mechanism in relation to its
operating state from a reference state The authors considered the steady state tensions in belt
spans as a result of accessory loads belt drive geometry and tensioner properties
Zhang and Zu [28] conducted a modal analysis for the response of a linear serpentine belt drive
system A non-iterative approach was used to explicitly form the equations for the system‟s
natural frequencies An exact closed-form expression for the dynamic response of the system
using eigenfunction expansion was derived with the system under steady-state conditions and
subject to harmonic excitation
The work conducted by Balaji and Mockensturm [29] considered a front-end accessory drive
(FEAD) with a decoupler or isolator attached to a pulley The rotational response for the FEAD
was found analytically by considering the system to be piecewise linear about the equilibrium
angular deflections The effect of their nonlinear terms was considered through numerical
integration of the derived equations of motion by way of the iterative methodndash fourth order
Runge-Kutta The authors in this case considered the longitudinal (ie rotational) vibration of
the belt spans only
Literature Review 18
The first to carry out the analysis of a serpentine belt drive system containing a two-pulley
tensioner was Nouri in 2005 [30] Nouri found the closed-form analytical solution of a
serpentine belt drive with a two-pulley tensioner for the case of sinusoidal excitation He
employed Runge Kutta method as well to solve the equations of motion to find the response of
the system under a general input from the crankshaft The author‟s work also included the
optimization of the tensioner design in order to minimize belt span vibrations due to crankshaft
excitation Furthermore the author applied active control techniques to the tensioner in a belt
drive system
The works discussed have made significant contributions to the research and development into
tensioner systems for serpentine belt drives These lead into the requirements for the structure
function and location of tensioner systems particularly for B-ISG transmissions
24 Tensioners for B-ISG System
241 Tensioners Structure Function and Location
Literature shows that the improvement of a serpentine belt life in a B-ISG system centers on the
tensioning mechanism redesign This mechanism as shown by researchers including
Wezenbeek et al [21] and Henry et al [11] is crucial in establishing the least tension in the belt
(above a zero value) in order to guard against failure by way of slip due to slack spans in the belt
and oscillations during engine re-start It is noted by Firbank [22] that the mechanics of a belt-
drive ldquois based on the idea that belt behaviour is governed by the elastic extension or contraction
of the belt arising from tension variationsrdquo [22] these variations may be compensated for by an
adjustable tensioner
Literature Review 19
The two types of tensioners are passive and active tensioners The former permits an applied
initial tension and then acts as an idler and normally employs mechanical power and can include
passive hydraulic actuation This type is cheaper than the latter and easier to package The latter
type is capable of continually adjusting the belt tension since it permits a lower static tension
Active tensioners typically employ electric or magnetic-electric actuation andor a combination
of active and passive actuators such as electrical actuation of a hydraulic force
Conventional belt tensioners comprise of a single tensioner arm that is fitted with a sole idler
pulley to engage a serpentine belt [31] A radial bearing is used to rotatably connect the idler
pulley to the tensioner arm [31] The tensioner arm is mounted on a pivot pin that is wrapped by
a bushing and is free to rotate [31] The pin covered by the bushing is fixed to the engine
housing [31] A rotary spring is wrapped about the bearing pin and bushing to provide a pre-
tension force to the belt via the tensioner arm and idler pulley thus taking up the slack due to the
changes in belt length [31] When the belt undergoes stretch under a load the spring drives the
tensioner arm and idler pulley further into the belt [31] Belt tension changes under the modes of
operation which can include when the crankshaft (or driving pulley) abruptly decelerates from a
steady-state condition and auxiliary components continue to rotate still in their own inherent
inertia and thus become the primary drivers [31] These fluctuations in belt tension lead to belt
flutter or skip and slip that may damage other components present in the belt drive [31]
Locating the tensioner on the slack side of the belt is intended to lower the initial static tension
[11] In conventional vehicles the engine always drives the alternator so the tensioner is located
in the belt span that links the crankshaft and alternator pulleys In a B-ISG setting the slack span
Literature Review 20
of the belt alternates between the driving mode of the ISG and the driving mode of the crankshaft
[32] Research by Henry et al [11] and also the summary of prior art for tensioners in Table
21 show that placing the idlertensioner pulley in the slack span in the case that the ISG is
driving instead of in the slack span when the crankshaft is driving allows for easier packaging
and for the least static tension Designs shown in Table 21 place the tensioneridler pulley in the
same span as Henry et al [11] or in both the slack and taut spans if using a double
tensioneridler configuration
242 Systematic Review of Tensioner Designs for a B-ISG System
The proposals for belt tensioner devices to manage the issue of high peaks in belt tension for B-
ISG settings are largely in patent records as the re-design of a tensioner has been primarily a
concern of automotive makers thus far A systematic review of the patent literature has been
conducted in order to identify evaluate and collate relevant tensioning mechanism designs
applicable to a B-ISG setting Its research objective is to influence the selection of a tensioner
configuration for the thesis study
The predefined search strategy used by the researcher has been to consider patents dating only
post-2000 as many patents dating earlier are referred to in later patents as they are developed on
in most cases by the original inventor (eg an INA Schaeffler KG patent published in 2000 may
refer to its own earlier patent presented in 1999) Patents dating pre-2000 that do not have any
successor were also considered The inclusion and exclusion criteria and rationales that were
used to assess potential patents are as follows
Inclusion of
Literature Review 21
tensioner designs with two arms andor two pivots andor two pulleys
mechanical tensioners (ie exclusion of magnetic or electrical actuators or any
combination of active actuators) in order to minimize cost
tension devices that are an independent structure apart from the ISG structure in order to
reduce the required modification to the accessory belt drive of a conventional automobile
and
advanced designs that have not been further developed upon in a subsequent patent by the
inventor or an outside party
Table 21 provides a collation of the results for the systematic review based on the selection
criteria Illustrations of the collated patent designs may be seen in Appendix A It is noted that
the patent literature pertaining to these designs in most cases provides minimal numerical data
for belt tensions achieved by the tensioning mechanism In most cases only claims concerning
the outcome in belt performance achievable by the said tension device is stated in the patent
Table 21 Passive Dual Tensioner Designs from Patent Literature
Bayerische
Motoren Werke
AG
Patents EP1420192-A2 DE10253450-A1 [33]
Design Approach
2 tensioner pulleys (idlers) and 2 tension arms are mounted outside the periphery of the belt drive these form tiltable clamping arms around a common axis of rotation
A torsion spring is used at bearing bushings to mount tension arms at ISG shaft
Each tension arm cooperates with torsion spring mechanism to rotate through a damping
device in order to apply appropriate pressure to taut and slack spans of the belt in
different modes of operation
Bosch GMBH Patent WO0026532 et al [34]
Design Approach
2 tension pulleys each one is mounted on the return and load spans of the driven and
driving pulley respectively
Idlers (tension pulleys) each connect to a spring which is attached on one end to a fixed point
Literature Review 22
Idlers‟ motions are independent of each other and correspond to the tautness or
slackness in their respective spans
Or alternatively a spring connects the idler pulleys and one of the two idlers is fixed at
its axis of rotation
Daimler Chrysler
AG
Patents DE10324268-A1 [35]
Design Approach
2 idlers are given a working force by a self-aligning bearing
Bearing supports auxiliary unit (ISG) and is arranged concentrically with the axle
auxiliary unit pulley
Dayco Products
LLC
Patents US6942589-B2 et al [36]
Design Approach
2 tension arms are each rotatably coupled to an idler pulley
One idler pulley is on the tight belt span while the other idler pulley is on the slack belt
span
Tension arms maintain constant angle between one another
One arm forms a positive differential angle with the belt and the remaining arm forms a negative differential angle with the belt
Idler pulleys are on opposite sides of the ISG pulley
Gates Corporation Patents US20060249118-A1 WO2003038309-A [37]
Design Approach
A tensioner pulley contacts the belt at the slack span during start-up (ISG-driving mode)
A tensioner is asymmetrically biased in direction tending to cause power transmission
belt to be under tension
McVicar et al
(Firm General
Motors Corp)
Patent US20060287146-A1 [38]
Design Approach
2 tension pulleys and carrier arms with a central pivot are mounted to the engine
One tension arm and pulley moderately biases one side of belt run to take up slack
during engine start-up while other tension arm and pulley holds appropriate bias against
taut span of belt
A hydraulic strut is connected to one arm to provide moderate bias to belt during normal
engine operation and velocity sensitive resistance to increasing belt forces during engine
start-up
INA Schaeffler
KG et al
Patents DE10044645-A1 [39] DE10159073-A1 [40] EP1723350-A1 et al [41]
DE10359641-A1 et al [42] EP1738093-A1 et al [43] DE102004012395-A1 [44]
WO2006108461-A1 et al [45]
Design Approach
2 tension arms and 2 pulleys approach ndash o Mutually independent tensioning arms are supported for rotation in the same
plane of the housing part
o Idler pulley corresponding to each tensioning arm engages with different
sections of belt
o When high tension span alternates with slack span of belt drive one tension
arm will increase pressure on current slack span of belt and the other will
decrease pressure accordingly on taut span
o Or when the span under highest tension changes one tensioner arm moves out
of the belt drive periphery to a dead center due to a resulting force from the taut
span of the ISG starting mode
o Deflection of the taut span acts on associated pulley to apply a counter-moment to the other idler pulley on the slack span
Literature Review 23
o The 2 lever arms are of different lengths and each have an idler pulley of
different diameters and different wrap angles of belt (see DE10045143-A1 et
al)
1 tensioner arm and 2 pulleys approach ndash
o 2 idler pulleys are pinned to a beam arranged on a clamping arm that is tiltably
linked to the beam o The ISG machine is supported by a shock absorber
o During ISG start-up one idler pulley is induced to a dead center position while
it pulls the remaining idler pulley into a clamping position until force
equilibrium takes place
o A shock absorber is laid out such that its supporting spring action provides
necessary preloading at the idler pulley in the direction of the taut span during
ISG start-up mode
Litens Automotive
Group Ltd
Patents US6506137-B2 et al [46]
Design Approach
2 tension pulleys on opposite sides of the ISG pulley engage the belt
They are positioned such that their applied forces result in opposing directed moments with respect to the tension device‟s axis of pivot
The pivot axis varies relative to the force applied to each tension pulley
Diameters of the tensioner pulleys are approximately equal and belt wrap angles of the
tensioner pulleys are approximately equal
A limited swivel angle for the tensioner arms work cycle is permitted
Mitsubishi Jidosha
Eng KK
Mitsubishi Motor
Corp
Patents JP2005083514-A [47]
Design Approach
2 tensioners are used
1 tensioner is held on the slack span of the driving pulley in a locked condition and a
second tensioner is held on the slack side of the starting (driven) pulley in a free condition
Nissan Patents JP3565040-B2 et al [48]
Design Approach
A single tensioner is on the slack span once ISG pulley is in start-up mode
The tension device is comprised of a oil pressure tensioner and a half ratchet mechanism
(a plunger which performs retreat actuation according to the energizing force of the oil
pressure spring and load received from the ISG)
The tensioner is equipped with a relief valve to keep a predetermined load lower than the
maximum load added by the ISG device
NTN Corp Patent JP2006189073-A [49]
Design Approach
An automatic tensioner is equipped with a hydraulic damper mechanism comprised of a
screw bolt using saw-screwed teeth and a cylinder nut a return spring and a spring seat
in a pressure chamber (within the screw bolt) a rod seat (that is fitted to the lower end of
the cylinder nut) a spring support (arranged on varying diameter stepped recessed
sections of the rod seat) and a check valve with an openingclosing passage
The cylinder and screw bolt act as the rigidity buffer under excessive loads during ISG
start-up mode of operation
Valeo Equipment
Electriques
Moteur
Patents EP1658432 WO2005015007 [50]
Design Approach
ldquoThe invention relates to a system or a starter (10) in which a pulley (80) is rotationally mounted on a section (22) of a shaft which axially extends inside a pulley (80) and
Literature Review 24
forwards at least partially outside a support element (200) and is characterized in that
the free front end (23) of said shaft section (22) is carried by an arm (206) connected to
the support element (200)rdquo
The author notes that published patents and patent applications may retain patent numbers for multiple patent
offices (ie European Patent Office German Patent Office etc) In such cases the published patent number or in
the absence of such a number the published patent application number has been specified However published
patent documents in the above cases also served as the document (ie identical) to the published patent if available
Quoted from patent abstract as machine translation is poor
25 Summary
The research on tensioner designs from the patent literature demonstrates a lack of quantifiable
data for the performance of a twin tensioner particularly suited to a B-ISG system The review of
the literature for the modeling theory of serpentine belt drives and design of tensioners shows
few belt drive models that are specific to a B-ISG setting Hence the literature review supports
the thesis objective of modeling a B-ISG tensioner specifically one that has a passive twin
tensioner configuration and as well measuring the tensioner‟s performance The survey of
hybrid classes reveals that the micro-hybrid class is the only class employing a closely
conventional belt transmission and hence its B-ISG transmission is applicable for tensioner
investigation The patent designs for tensioners contribute to the development of the tensioner
design to be studied in the following chapter
25
CHAPTER 3 MODELING OF B-ISG SYSTEM
31 Overview
The derivation of a theoretical model for a B-ISG system uses real life data to explore the
conceptual system under realistic conditions The literature and prior art of tensioner designs
leads the researcher to make the following modeling contributions a proposed design for a
passive two-pulley tensioner computation of geometric attributes for a B-ISG system with the
proposed tensioner and derivation of the system‟s equations of motion (EOM) under dynamic
and static states as well as deriving the EOM for the B-ISG system with only a passive single-
pulley tensioner for comparison The principles of dynamic equilibrium are applied to the
conceptual system to derive the EOM
32 B-ISG Tensioner Design
The proposed design for a passive two pulley tensioner configures two tensioners about a single
fixed pivot point in the interior space of a serpentine belt drive One end of each tensioner arm
coincides with the centre point of a tensioner pulley and this point marks the axis of rotation of
the pulley The other end of each arm is pivoted about a point so that the arms share the same
axis of rotation This conceptual design henceforth is called a Twin Tensioner Figure 31 shows
a schematic for the proposed design
Modeling of B-ISG 26
Figure 31 Schematic of the Twin Tensioner
The tensioner pulley coordinates are described by (XiYi) their radii by Ri their arm lengths Lti
and their angles θti The rotation of the arms is resisted by stiffness kt of a coil spring acting
between the two arms and spring stiffness kti acting between each arm and the pivot point The
motion of each arm is dampened by dampers and akin to the springs a damper acts between the
two arms ct and a damper cti acts between each arm and the pivot point The result is a
tensioning mechanism with four degrees of freedom (DOF) that includes independent rotations
of the two pulleys and two arms
The following section relates the geometry of the rigid bodies in a B-ISG system equipped with a
Twin Tensioner to their respective motions
Modeling of B-ISG 27
33 Geometric Model of a B-ISG System with a Twin Tensioner
The B-ISG system with the Twin Tensioner is shown in Figure 32 The geometry of the drive
provides the lengths of the belt spans and angles of wrap for the belt and pulley contact surfaces
These variables are crucial to resolve the components of forces and moment arms acting on each
rigid body in the system and are used in the derivation of the EOM in section 34 Zhen Mu‟s
geometric modeling approach [51] used in the development of the software FEAD was applied
to the Twin Tensioner system to compute the system‟s unique geometric attributes
Figure 32 B-ISG Serpentine Belt Drive with Twin Tensioner
It is noted that in Figure 31 and Figure 32 showing the schematic of the Twin Tensioner and
the overall system respectively that for the purpose of the geometric computations the forward
direction follows the convention of the numbering order counterclockwise The numbering
order is in reverse to the actual direction of the belt motion which is in the clockwise direction in
this study The fourth pulley is identified as an ISG unit pulley However the properties used
for the ISG pulley‟s geometry inertia stiffness and damping is modeled as a conventional
Modeling of B-ISG 28
alternator pulley This pulley is conceptualized as an ISG when it is modeled as the driving
pulley at which point the requirements of the ISG are solved for and its non-inertia attributes
are not needed to be ascribed
Figure 33 shows the geometric attributes needed to resolve the wrap angle of the belt on each
pulley Variables (XiYi) and XYfi XYbi XYfbi and XYbfi are the ith pulley centre coordinates and
its possible belt connection points respectively Length Lfi is the length of the span connecting
the points XYfi and XYf(i+1) or XYbi and XYb(i+1) on the ith and (i+1)th pulleys respectively
Similarly Lbi is the length of the span between XYfbi and XYfb(i+1) or XYbfi and XYbf (i+1) on the
ith and (i+1)th pulleys respectively Angles αi θfi and θbi represent the angle between a line
connecting the ith and (i+1)th pulley centres and the angles of the belt connection spans with
lengths Lfi and Lbi respectively Ri is the radius of the ith pulley
Figure 33 Angles Coordinates and Possible Belt Contact Points for the ith and i+1th Pulleys
[modified] [51]
Modeling of B-ISG 29
The angle between the horizontal and the line connecting the ith and (i+1)th pulley centres αi is
calculated using Zhen‟s method [51] This method uses the pulley‟s coordinates and a cosine
trigonometric relation
i acos
Xi 1
Xi
Xi 1
Xi
2
Yi 1
Yi
2
Yi 1
Yi
if
(31a)
i 2 acos
Xi 1
Xi
Xi 1
Xi
2
Yi 1
Yi
2
Yi 1
Yi
if
(31b)
The lengths for connecting the possible belt spans are described by the variables Lfi and Lbi
The centre point coordinates and the radii of the pulleys are related through the solution of
triangles which they form to define values of the possible belt span lengths
Lfi
Xi 1
Xi
2
Yi 1
Yi
2
Ri 1
Ri
2
(32a)
Lbi
Xi 1
Xi
2
Yi 1
Yi
2
Ri 1
Ri
2
(32b)
The set of possible belt span lengths leads to the calculation of θfi and θbi the angles between the
line connecting the ith and (i+1)th pulley centres and the possible contact point on the pulley
perimeter
Modeling of B-ISG 30
(33a)
(33b)
The array of possible belt connection points comes about from the use of the pulley centre
coordinates and their radii and the sine of the sum or differences of αi and θfi or θbi The angle
αi is calculated in equations (31a) and (31b) and angles θfi and θbi are calculated in equations
(33a) and (33b) The formula to compute the array of points is shown in equations (34) and
(35) for the ith and (i+1)th pulleys Equation (34) describes the forward belt connection point
on the ith pulley which is in the span leading forward to the next (i+1)th pulley
(34a)
(34b)
(34c)
(34d)
bi atan
Lbi
Ri
Ri 1
Modeling of B-ISG 31
Equation (35) describes the backward belt connection point on the ith pulley This point sits on
the ith pulley in the contacting belt span which leads backward to connect with the (i-1)th
pulley
(35a)
(35b)
(35c)
(35d)
The selection of the coordinates from the array of possible connection points requires a graphic
user interface allowing for the points to be chosen based on observation This was achieved
using the MathCAD software package as demonstrated in the MathCAD scripts found in
Appendix C The belt connection points can be chosen so as to have a pulley on the interior or
exterior space of the serpentine belt drive The method used in the thesis research was to plot the
array of points in the MathCAD environment with distinct symbols used for each pair of points
and to select the belt connection points accordingly By observation of the selected point types
the type of belt span connection is also chosen Selected point and belt span types are shown in
Table 31
Modeling of B-ISG 32
Table 31 Selected Contact Point Types on the ith Pulley and Types for the ith Belt Span
Pulley Forward Contact
Point
Backwards Contact
Point
Belt Span
Connection
1 Crankshaft XYf1 XYbf21 Lf1
2 Air Conditioning XYfb2 XYf22 Lb2
3 Tensioner 1 XYbf3 XYfb23 Lb3
4 AlternatorISG XYfb4 XYbf24 Lb4
5 Tensioner 2 XYbf5 XYfb25 Lb5
The inscribed angles βji between the datum and the forward connection point on the ith pulley
and βji between the datum and its backward connection point are found through solving the
angle of the arc along the pulley circumference between the datum and specified point The
wrap angle ϕi is found as the difference between the two inscribed angles for each connection
point on the pulley The angle between each belt span and the horizontal as well as the initial
angle of the tensioner arms are found using arctangent relations Furthermore the total length of
the belt is determined by the sum of the lengths of the belt spans
34 Equations of Motion for a B-ISG System with a Twin Tensioner
341 Dynamic Model of the B-ISG System
3411 Derivation of Equations of Motion
This section derives the inertia damping stiffness and torque matrices for the entire system
Moment equilibrium equations are applied to each rigid body in the system and net force
equations are applied to each belt span From these two sets of equations the inertia damping
Modeling of B-ISG 33
and stiffness terms are grouped as factors against acceleration velocity and displacement
coordinates respectively and the torque matrix is resolved concurrently
A system whose motion can be described by n independent coordinates is called an n-DOF
system Consider the free body diagram of the Twin Tensioner in Figure 34 in which each
pulley of inertia Ii is supported on an arm of inertia Iti It is assumed that the pulleys are
constrained to rotate about their respective central axes and the arms are free to rotate about their
respective pivot points then at any time the position of each pulley can be described by a
rotational coordinate θi(t) and a coordinate θti(t) can denote the rotation of each arm Thus the
tensioner system comprises of four rigid bodies where each is described by one coordinate and
hence is a four-DOF system It is important to note that each rigid body is treated as a point
mass In addition inertial rotation in the positive direction is consistent with the direction of belt
motion The belt span tensions Ti and coupled radii Ri apply moments to the pulleys
Figure 34 Twin Tensioner Free Body Diagram of a Four-Degree of Freedom System
Modeling of B-ISG 34
For the serpentine belt system considered in the thesis research there are seven rigid bodies each
having a one-DOF of motion The EOM for a seven-DOF system form second-order coupled
differential equations meaning that each equation includes all of the general coordinates and
includes up to the second-order time derivatives of these coordinates The EOM can be
obtained by applying D‟Alembert‟s principle that the sum of the moments taken about any point
including the couples equals to zero Therefore the inertial couple the product of the inertia and
acceleration is equated to the moment sum as shown in equation (35)
I ∙ θ = ΣM (35)
The moment equilibrium equations for the Twin Tensioner in Figure 34 where the positive
direction is in the clockwise direction are shown in equations (36) through to (310) The
numbering convention used for each rigid body corresponds to the labeled serpentine belt drive
system shown in Figure 32 Qi represents the required torque of the ith rigid body ci is the
damping constant of the ith rigid body βji is the angle of orientation for the ith belt span and
120597120579119905119894 120579 119905119894 and 120579 119905119894 are the angular displacement angular velocity and angular acceleration of the ith
tensioner arm The initial angle of the ith tensioner arm is described by θtoi
minusI3 ∙ θ 3 = T3 ∙ R3 minus T2 ∙ R3 minus Q3 + c3 ∙ θ 3 (36)
minusI5 ∙ θ 5 = minusT4 ∙ R5 + T5 ∙ R5 minus Q5 + c5 ∙ θ 5 (37)
Modeling of B-ISG 35
It1 ∙ θ t1 = minusTt1 ∙ Lt1 ∙ sin θto 1 minus βj4 + sin θto 1 minus βj5 minus kt ∙ partθt1 minus partθt2 minus kt1 ∙
partθt1 minus ct ∙ partθ t1 minus partθ t2 minus ct1 ∙ partθ t1 (38)
It2 ∙ θ t2 = minusTt2 ∙ Lt2 ∙ sin θto 2 minus βj2 + sin θto 1 minus βj3 minus kt ∙ partθt2 minus partθt1 minus kt2 ∙ partθt2 minus
ct ∙ partθ t2 minus partθ t1 minus ct2 ∙ partθ t2 (39)
partθt1 = θt1 minus θto 1 (310a)
partθt2 = θt2 minus θto 2 (310b)
The free body diagrams for the remaining rigid bodies crankshaft pulley air conditioner pulley
and ISG pulley are in the general form of Figure 35 The sum of the moments about the axes of
rotation are taken for these structures in equations (311) through to (313)
Figure 35 Free Body Diagram for Non-Tensioner Pulleys
Modeling of B-ISG 36
I1 ∙ θ 1 = T5 ∙ R1 minus T1 ∙ R1 + Q1 minus c1 ∙ θ 1 (311)
I2 ∙ θ 2 = T1 ∙ R2 minus T2 ∙ R2 + Q2 minus c2 ∙ θ 2 (312)
I4 ∙ θ 4 = T3 ∙ R4 minus T4 ∙ R4 + Q4 minus c4 ∙ θ 4 (313)
The relationship between belt tensions and rigid body displacements is in the general form of
equation (314) where 119827119836 and 119827119844 are damping and stiffness matrices due to the belt respectively
with each factorized by a radial arm length This relationship is described for each span in
equations (315) through to (320) The belt damping constant for the ith belt span is cib
119827prime = 119827119836 ∙ 120521 + 119827119844 ∙ 120521 (314)
T1 = To + k1b ∙ R1 ∙ θ1 minus R2 ∙ θ2 + c1
b ∙ [R1 ∙ θ 1 minus R2 ∙ θ 2] (315)
T2 = To + k2b ∙ R2 ∙ θ2 minus R3 ∙ θ3 + Lt1 ∙ [sin θto 1 minus βj2 ] ∙ (θt1 minus θto 1) + c2
b ∙ [R2 ∙ θ 2 minus R3 ∙
θ 3 + Lt1 ∙ [sin θto 1 minus βj2 ] ∙ (θ t1)] (316)
T3 = To + k3b ∙ R3 ∙ θ3 minus R4 ∙ θ4 + Lt1 ∙ [sin θto 1 minus βj3 ] ∙ (θt1 minus θto 1) + c3
b ∙ [R3 ∙ θ 3 minus R4 ∙
θ 4 + Lt1 ∙ [sin θto 1 minus βj3 ] ∙ (θ t2)] (317)
Modeling of B-ISG 37
T4 = To + k4b ∙ R4 ∙ θ4 minus R5 ∙ θ5 + Lt2 ∙ [sin θto 2 minus βj4 ] ∙ (θt2 minus θto 2) + c4
b ∙ [R4 ∙ θ 4 minus R5 ∙
θ 5 + Lt2 ∙ [sin θto 2 minus βj4 ] ∙ (θ t1)] (318)
T5 = To + k5b ∙ R5 ∙ θ5 minus R1 ∙ θ1 + Lt2 ∙ [sin θto 2 minus βj5 ] ∙ (θt2 minus θto 2) + c5
b ∙ [R5 ∙ θ 5 minus R1 ∙
θ 1 + Lt2 ∙ [sin θto 2 minus βj5 ] ∙ (θ t2)] (319)
Tprime = Ti minus To (320)
Since the applied torques on the tensioner pulleys Q3 and Q4 are zero the static equilibrium
equation of the pulleys show that the adjacent spans of each tensioner pulley are equal to each
other Hence equations (321) and (322) are denoted as follows
Tt1 = T2 = T3 (321)
Tt2 = T4 = T5 (322)
Equations (310a) (310b) and (314) through to (322) are substituted into the EOMs described
in equations (36) to (39) and (311) to (313) The newly formed equations can be arranged
and written in matrix form as shown in equations (323) through to (328) The general
coordinate matrix 120521 and its first and second derivatives are shown in the EOM below
119816 ∙ 120521 + 119810 ∙ 120521 + 119818 ∙ 120521 = 119824 (323)
Modeling of B-ISG 38
The inertia matrix I includes the inertia of each rigid body in its diagonal elements The
damping matrix C includes variables 119888119894119887 the damping of the ith belt span 119877119894 its radius 120573119895119894 its
angle 119871119905119894 the ith tensioner arm‟s length 120579119905119900119894 its initial pivot angle and 119888119905 and 119888119905119894 the ith
tensioner arm viscous damping constants Stiffness matrix K contains 119896119894119887 the ith belt span
stiffness and 119896119905 and 119896119905119894 the ith tensioner arm stiffness constants and akin to the damping
matrix the variables 119877119894 119871119905119894 120579119905119900119894 and 120573119895119894 The belt span stiffness is computed in equation
(326b) where 119870119887 represents the belt cord stiffness 119896119887 is the belt factor obtained from
experimental data 120573119895119894 is the angle of orientation for the span between the jth and ith pulleys and
ϕi is the belt wrap angle on the ith pulley
Modeling of B-ISG 39
119816 =
I1 0 0 0 0 0 00 I2 0 0 0 0 00 0 I3 0 0 0 00 0 0 I4 0 0 00 0 0 0 I5 0 00 0 0 0 0 It1 00 0 0 0 0 0 It2
(324)
119810 =
c1
b ∙ R12 + c5
b ∙ R12 + c1 minusc1
b ∙ R1 ∙ R2 0 0 minusc5b ∙ R1 ∙ R5 0 c5
b ∙ R1 ∙ Lt2 ∙ sin θto 2 minus βj5
minusc1b ∙ R1 ∙ R2 c2
b ∙ R22 + c1
b ∙ R22 + c2 minusc2
b ∙ R2 ∙ R3 0 0 c2b ∙ R2 ∙ Lt1 ∙ sin θto 1 minus βj2 0
0 minusc2b ∙ R2 ∙ R3 c3
b ∙ R32 + c2
b ∙ R32 + c3 minusc3
b ∙ R3 ∙ R4 0 C36 0
0 0 minusc3b ∙ R3 ∙ R4 c4
b ∙ R42 + c3
b ∙ R42 + c4 minusc4
b ∙ R4 ∙ R5 minusc3b ∙ R4 ∙ Lt1 ∙ sin θto 1 minus βj3 c4
b ∙ R4 ∙ Lt2 ∙ sin θto 2 minus βj4
minusc5b ∙ R1 ∙ R5 0 0 minusc4
b ∙ R4 ∙ R5 c5b ∙ R5
2 + c4b ∙ R5
2 + c5 0 C57
0 0 0 0 0 ct +ct1 minusct
0 0 0 0 0 minusct ct +ct1
(325a)
C36 = 1198773 ∙ 1198711199051 ∙ [1198883119887 ∙ 119904119894119899 1205791199051199001 minus 1205731198953 minus 1198882
119887 ∙ 119904119894119899 1205791199051199001 minus 1205731198952 ] (325b)
C57 = 1198775 ∙ 1198711199052 ∙ [1198885119887 ∙ 119904119894119899 1205791199051199002 minus 1205731198955 minus 1198884
119887 ∙ 119904119894119899 1205791199051199002 minus 1205731198954 ] (325c)
Modeling of B-ISG 40
119818 =
k1
b ∙ R12 + k5
b ∙ R12 minusk1
b ∙ R1 ∙ R2 0 0 minusk5b ∙ R1 ∙ R5 0 k5
b ∙ R1 ∙ Lt2 ∙ sin θto 2 minus βj5
minusk1b ∙ R1 ∙ R2 k2
b ∙ R22 + k1
b ∙ R22 minusk2
b ∙ R2 ∙ R3 0 0 k2b ∙ R2 ∙ Lt1 ∙ sin θto 2 minus βj2 0
0 minusk2b ∙ R2 ∙ R3 k3
b ∙ R32 + k2
b ∙ R32 minusk3
b ∙ R3 ∙ R4 0 R3 ∙ Lt1 ∙ [k3b ∙ sin θto 1 minus βj3 minus k2
b ∙ sin θto 1 minus βj2 ] 0
0 0 minusk3b ∙ R3 ∙ R4 k4
b ∙ R42 + k3
b ∙ R42 minusk4
b ∙ R4 ∙ R5 minusk3b ∙ R4 ∙ Lt1 ∙ sin θto 1 minus βj3 k4
b ∙ R4 ∙ Lt2 ∙ sin θto 2 minus βj4
minusk5b ∙ R1 ∙ R5 0 0 minusk4
b ∙ R4 ∙ R5 k5b ∙ R5
2 + k4b ∙ R5
2 0 R5 ∙ Lt2 ∙ [k5b ∙ sin θto 2 minus βj5 minus k4
b ∙ sin θto 2 minus βj4 ]
0 0 0 0 0 kt +kt1 minuskt
0 0 0 0 0 minuskt kt +kt1
(326a)
k119894b =
Kb
Li + kb ∙ Ri ∙ϕi+1
2 + Ri ∙ϕi
2
(326b)
120521 =
θ1
θ2
θ3
θ4
θ5
partθt1
partθt2
(327)
119824 =
Q1
Q2
Q3
Q4
Q5
Qt1
Qt2
(328)
Modeling of B-ISG 41
3412 Modeling of Phase Change
The phase change from the crankshaft pulley being the driving pulley to the ISG pulley being the
driving pulley is described through a conditional equality based on a set of Boolean conditions
When the crankshaft is driving the rows and the columns of the EOM are swapped such that the
new order for rows and columns is 1 (crankshaft pulley) 4 (ISG pulley) 2 (air conditioner
pulley) 3 (tensioner 1 pulley) 5 (tensioner 2 pulley) 6 (tensioner arm 1) and 7 (tensioner arm 2)
When the ISG is driving the order is the same except that the second row and second column
terms relating to the ISG pulley become the first row and first column while the crankshaft
pulley terms (previously in the first row and first column) become the second row and second
column Hence the order for all rows and columns of the matrices making up the EOM in
equation (322) switches between 1423567 (when the crankshaft pulley is driving) and
4123567 (when the ISG pulley is driving) For example in the crankshaft driving and ISG
driving phases the general coordinate matrix and the inertia matrix become the following
120521119940 =
1205791
1205794
1205792
1205793
1205795
1205971205791199051
1205971205791199052
and 120521119938 =
1205794
1205791
1205792
1205793
1205795
1205971205791199051
1205971205791199052
(329a amp b)
119816119940 =
I1 0 0 0 0 0 00 I4 0 0 0 0 00 0 I2 0 0 0 00 0 0 I3 0 0 00 0 0 0 I5 0 00 0 0 0 0 It1 00 0 0 0 0 0 It2
and 119816119938 =
I4 0 0 0 0 0 00 I1 0 0 0 0 00 0 I2 0 0 0 00 0 0 I3 0 0 00 0 0 0 I5 0 00 0 0 0 0 It1 00 0 0 0 0 0 It2
(329c amp d)
Modeling of B-ISG 42
where subscripts c and a denote the crankshaft pulley driving phase and the ISG pulley driving
phase respectively
The condition for phase change is based on the engine speed n in units of rpm Equation (330)
demonstrates the phase change
H(n) = 1 119899 ge 750 (Crankshaft driving phase)0 119899 lt 750 (ISG driving phase)
(330)
When the crankshaft pulley is the driving pulley the ISG pulley becomes the driven pulley and
following suit when the ISG pulley is the driving pulley the crankshaft pulley becomes the
driven pulley These modes of operation mean that the system will predict two different sets of
natural frequencies and mode shapes Using a Boolean condition to allow for a swap between
the first and second rows as well as between the first and second columns of the EOM matrices
I C and K allows for a continuous plot of the dynamic response to be plotted for the ISG pulley
throughout its driving and driven phases as well as for that of the crankshaft pulley
3413 Natural Frequencies Mode Shapes and Dynamic Responses
Assuming the system undergoes simple harmonic motion its matrix of natural frequencies 120596119899
and modeshapes are found by solving the eigenvalue problem shown in equation (331a)
ωn ∙ 119816120784120784 minus 11981822 ∙ 120495m = 120782 (331a)
The displacement amplitude Θm is denoted implicitly in equation (331d)
Modeling of B-ISG 43
120521119846 = θ2 θ3 θ5 θ6 partθt1 partθt2 T for H n = 1 (331b)
120521119846 = θ1 θ3 θ5 θ6 partθt1 partθt2 T for H n = 0 (331c)
θm = 120495119846 ∙ sin(ω ∙ t) (331d)
I2 and K22 are submatrices of I and K respectively meaning the first row and column of each of
the original matrices are removed The eigenvalue problem is reached by considering the
undamped and unforced motion of the system Furthermore the dynamic responses are found by
knowing that the torque requirements in the matrixndash Qm for the driven pulleys and the tensioner
arms are zero in the dynamic case which signifies a response of the system to an input solely
from the driving pulley
I1 120782120782 119816120784120784
θ 1120521 119846
+ C11 119810120783120784119810120784120783 119810120784120784
θ 1120521 119846
+ K11 119818120783120784
119818120784120783 119818120784120784 θ1
120521119846 =
QCS ISG
119824119846 (332)
1
In the case of equation (331) θm is the submatrix identified in equations (331b) through to
(331d) Therein θ1 denotes the general coordinate for the driving pulley so that in the case the
phase change function H(n) is equal to zero θ1 becomes θ4 and the order of the rows and
columns for the remaining matrices correspond to the value of H(n) as mentioned earlier in
section 3412 For simple harmonic motion the motion of the driven pulleys are described as
1 The driving torque 119876119862119878119868119878119866 denotes the crankshaft torque 119876119862119878 when the crankshaft pulley is driving or the ISG
torque 119876119868119878119866 when the ISG pulley is in its driving function
Modeling of B-ISG 44
θm = 120495119846 ∙ sin(ω ∙ t) (333)
The dynamic response of the system to an input from the driving pulley under the assumption of
sinusoidal motion is expressed in equation (334)
120495119846 = [(119818120784120784 minusω2 ∙ 119816120784120784) + 119895ω ∙ 119810120784120784]minus1 ∙ (119818120784120783 + 119895ω ∙ 119810120784120783) ∙ Θ1 (334)
3414 Crankshaft Pulley Driving Torque Acceleration and Displacement
Subsequently the crankshaft pulley driving torque acceleration and displacement are firstly
discussed It is assumed in the thesis research for the purpose of modeling that the engine
serving the crankshaft is of the four cylinder type The input torque provided by a four-cylinder
engine is assumed to be dominated by two torque pulses per revolution of the crankshaft which
is represented by the factor of 2 on the steady component of the angular velocity in equation
(335) The torque requirement of the crankshaft pulley when it is the driving pulley is
Qc = qc ∙ sin(2 ∙ ωcs ∙ t) (335)
The amplitude of the required crankshaft torque qc is expressed in equation (336) and is
derived from equation (332)
qc = K11 minus ω2 ∙ I1 + 119895 ∙ ω ∙ C11 ∙ Θ1 + (119818120783120784 + 119895 ∙ ω ∙ 119810120783120784) ∙ 120495119846 (336)
Modeling of B-ISG 45
The angular frequency for the system in radians per second (rads) ω when the crankshaft
pulley is driving can be found as a function of the engine speed in rotations per minute (rpm) n
and by taking into account the double pulse per crankshaft revolution
ω = 2 ∙ ωcs = 4 ∙ π ∙ n
60
(337)
The system is considered when the amplitude of the crankshaft‟s angular acceleration is assumed
to be constant and equal to 650 rads2 during the crankshaft pulley driving phase The amplitude
of the excitation angular input from the engine is shown in equation (339b) and is found as a
result of (338)
θ 1CS = 650 ∙ sin(ω ∙ t) (338)
θ1CS = minus650
ω2sin(ω ∙ t)
(339a) where
Θ1CS = minus650
ω2
(339b)
Modeling of B-ISG 46
3415 ISG Pulley Driving Torque Acceleration and Displacement
Secondly the torque acceleration and the displacement of the ISG pulley in its driving phase is
discussed The torque for the ISG when it is in its driving function is assumed constant Ratings
for the ISG are taken from experiments performed by researchers Wezenbeek et al [21] on an
Energen 5 High Output Belt-alternator-starter (BAS) unit from Delphi The 12-Volt BAS which
can also be called a B-ISG was reported to have a maximum allowable speed of 18000 rpm [21]
As well it was noted that the ISG pulley was sized appropriately and the engine speed was
limited to ensure that an over-speed condition of the ISG pulley would not occur [21] The stall
torque rating for the Energen ISG was reported to be 48 Nm at the electric machine shaft [21]
The formula for the torque of a permanent magnet DC motor for any given speed (equation
(340)) is used to approximate the torque of the ISG in its driving mode[52]
QISG = Ts minus (N ∙ Ts divide NF) (340)2
Knowing the stall torque (the torque at 0 rpm) Ts and the maximum rpm of the motor when it is
not under load NF allows for the torque produced 119876119868119878119866 to be found for a given motor speed N
Experimental data from Litens Automotive Group [53] shows that for engine fire-up upon ISG
re-start the crankshaft must go from 0 rpm to an idle speed of approximately 750 rpm The
pulley installed on the ISG shaft in the case of the thesis research has a diameter of 6820 mm
(DISG) while that of the crankshaft has a diameter of 20065 mm (DCS) which makes the
2 The equation for the required driving torque for the ISG pulley may also be computed from the formula shown in
(336) Figure 315 for the driving torque of the ISG pulley shows that (336) and (340) produce similar results for
the required driving torque See Figure 315 for comparison of these results
Modeling of B-ISG 47
crankshaft to ISG pulley ratio approximately 2941 This ratio is used to determine the ISG
speed in equation (341)
nISG = nCS ∙DCS
DISG
(341)
For a crankshaft speed of 750 rpm the required ISG speed nISG is found from equation (341) to
be approximately 220656 rpm Thus the ISG torque during start-up is found from equation
(340) where N is equated to the value of nISG NF is assumed to be 18000 rpm and the stall
torque is allotted the value of 48 Nm The result is a required torque of approximately 42 Nm
for the ISG The acceleration of the ISG pulley is found by taking into account the torque
developed by the rotor and the polar moment of inertia of the pulley [54]
A1ISG = θ 1ISG = QISG IISG (342)
In torsional motion the function for angular displacement of input excitation is sinusoidal since
the electric motor is assumed to be resonating As a result of constant angular acceleration the
angular displacement of the ISG pulley in its driving mode is found in equation 343
θ1ISG = Θ1ISG ∙ sin(ωISG ∙ t) (343)
Knowing that acceleration is the second derivative of the displacement the amplitude of
displacement is solved subsequently [55]
Modeling of B-ISG 48
θ 1ISG = minusωISG2 ∙ Θ
1ISG ∙ sin(ωISG ∙ t) (344)
θ 1ISG = minusωISG2 ∙ Θ
1ISG
(345a)
Θ1ISG =minusQISG IISG
ωISG2
(345b)
In this case the angular frequency for the system 120596 is equivalent to 120596119868119878119866 that is the angular
frequency of the ISG pulley which can be expressed as a function of its speed in rpm
ω = ωISG =2 ∙ π ∙ nISG
60
(346a)
or in terms of the crankshaft rpm by substituting equation (341) into (346a)
ω =2 ∙ π
60∙ nCS ∙
DCS
DISG
(346b)
3416 Tensioner Arms Dynamic Torques
The dynamic torque for the tensioner arms are shown in equations (347) and (348)
Qt1 = kt + kt1 + 119895 ∙ ω ∙ (ct + ct1) ∙ (Θt1 ∙ Θ1) (347)
Modeling of B-ISG 49
Qt2 = kt + kt2 + 119895 ∙ ω ∙ (ct + ct2) ∙ (Θt2 ∙ Θ1) (348)
3417 Dynamic Belt Span Tensions
Furthermore the dynamic belt span tensions are derived from equation (314) and described in
matrix form in equations (349) and (350)
119827prime = 119895 ∙ ω ∙ 119827119836 + 119827119844 ∙ 120495119847 (349)
where
120495119847 = Θ1
120495119846 (350)
342 Static Model of the B-ISG System
It is fitting to pursue the derivation of the static model from the system using the dynamic EOM
For the system under static conditions equations (314) and (323) simplify to equations (351)
and (352) respectively
119827prime = 119827119844 ∙ 120521 (351)
119824 = 119818 ∙ 120521 (352)
Modeling of B-ISG 50
As noted in other chapters the focus of the B-ISG tensioner investigation especially for the
parametric and optimization studies in the subsequent chapters is to determine its effect on the
static belt span tensions Therein equations (351) and (352) are used to derive the expressions
for static tension in each belt span 119931prime is the tension solely due to deflection of the belt span
Equation (320) demonstrates the relationship between the tension due to belt response and the
initial tension also known as pre-tension The static tension 119931 is found by summing the initial
tension 1198790 with the expression for the dynamic tension shown in equations (315) through to
(319) and by substituting the expressions for the rigid bodies‟ displacements from equation
(352) and the relationship shown in equation (320) into equation (351)
119827 = 119827119844 ∙ (119818minus120783 ∙ 119824) + T0 (353)3
35 Simulations
The methods used to develop the geometric dynamic and static models of the Twin Tensioner B-
ISG system in the previous sections of this chapter were verified using the software FEAD The
input data for a single tensioner B-ISG system was entered into FEAD [51] to simulate the
crankshaft driving phase alone since the ISG phase is inapplicable in the FEAD [51] software
FEAD‟s [51] results agreed with those found in the simulation of the single tensioner system‟s
geometric model and EOMs in MathCAD software Furthermore the geometric simulation
3 For the purposes of the static tension the original order for the rows and columns of the stiffness matrix K and the
torque matrix Q are maintained as depicted in (326) and (328) In performing the inverse of K and its
multiplication with Q the first row and first column (in the case of the K matrix) are removed in the crankshaft
driving case whereas the fourth row and fourth column are removed in the ISG driving case Then the product for
the displacement120637 resulting from (119922minus120783 ∙ 119928) has a zero added to serve as the first element of the column matrix in
the crankshaft driving case or as the fourth element in the ISG driving case This is shown in detail in Appendix
C3 of MathCAD scripts
Modeling of B-ISG 51
results for both of the twin and single tensioner B-ISG systems were found to be in agreement as
well
351 Geometric Analysis
The initial coordinate inputs for the centre points of the five pulleys and the Twin Tensioner
pivot point are described as Cartesian coordinates and shown in Table 32 which also includes
the diameters for the pulleys
Table 32 Coordinate Points for Pulley Centres and Twin Tensioner Pivot [56]
Rigid Body Diameter [mm] Cartesian Coordinate [Xi Yi] [mm]
1Crankshaft Pulley 20065 [00]
2 Air Conditioner Pulley 10349 [224 -6395]
3 Tensioner Pulley 1 7240 [292761 87]
4 ISG Pulley 6820 [24759 16664]
5 Tensioner Pulley 2 7240 [12057 9193]
6 Tensioner Arm Pivot --- [201384 62516]
The geometric results for the B-ISG system are shown in Table 33
Table 33 Geometric Results of B-ISG System with Twin Tensioner
Pulley Forward
Connection Point
Backward
Connection Point
Wrap
Angle
ϕi (deg)
Angle of
Belt Span
βji (deg)
Length of
Belt Span
Li (mm)
1 Crankshaft [-6818-100093] [453889475] 202996 356103 227828
2 Air
Conditioning [275299-5717] [220484 -115575] 101425 277528 14064
3 Tensioner 1 [25887599735] [256873 82257] 28126 69403 58658
4 ISG [218374184225] [27951154644] 169554 58956 129513
5 Tensioner 2 [10419659645] [15158673262] 8585 333107 65949
Total Length of Belt (mm) 1243
Modeling of B-ISG 52
352 Dynamic Analysis
The dynamic results for the system include the natural frequencies mode shapes driven pulley
and tensioner arm responses the required torque for each driving pulley the dynamic torque for
each tensioner arm and the dynamic tension for each belt span These results for the model were
computed in equations (331a) through to (331d) for natural frequencies and mode shapes in
equation (334) for the driven pulley and tensioner arm responses in equation (336) for the
crankshaft pulley driving torque in equation (340) for the ISG pulley driving torque in
equations (347) and (348) for the tensioner arm torques and lastly in equation (349) for the
dynamic tension of each belt span Figures 36 through to 323 respectively display these
results The EOM simulations can also be contrasted with those of a similar system being a B-
ISG serpentine belt drive that is equipped with a single tensioner arm and single tensioner pulley
which interacts only in the span bridging the ISG and crankshaft pulleys The EOM for a B-ISG
with a single tensioner is presented in Appendix B
It is assumed for the sake of the dynamic and static computations that the system
does not have an isolator present on any pulley
has negligible rotational damping of the pulley shafts
has negligible belt span damping and that this damping does not differ amongst
spans (ie c1b = ∙∙∙ = ci
b = 0)
has quasi-static belt stretch where its belt experiences purely elastic deformation
has fixed axes for the pulley centres and tensioner pivot
has only one accessory pulley being modeled as an air conditioner pulley and
Modeling of B-ISG 53
has a rotational belt response that is decoupled from the transverse response of the
belt
The input parameter values of the dynamic (and static) computations as influenced by the above
assumptions for the present system equipped with a Twin Tensioner are shown in Table 34
Table 34 Data for Input Parameters used in Dynamic and Static Computations [56]
Rigid Body Data
Pulley Inertia
[kg∙mm2]
Damping
[N∙m∙srad]
Stiffness
[N∙mrad]
Required
Torque
[Nm]
Crankshaft 10 000 0 0 4
Air Conditioner 2 230 0 0 2
Tensioner 1 300 1x10-4
0 0
ISG 3000 0 0 5
Tensioner 2 300 1x10-4
0 0
Tensioner Arm 1 1500 1000 10314 0
Tensioner Arm 2 1500 1000 16502 0
Tensioner Arm
couple 1000 20626
Belt Data
Initial belt tension [N] To 300
Belt cord stiffness [Nmmmm] Kb 120 00000
Belt phase angle at zero frequency [deg] φ0deg 000
Belt transition frequency [Hz] ft 000
Belt maximum phase angle [deg] φmax 000
Belt factor [magnitude] kb 0500
Belt cord density [kgm3] ρ 1000
Belt cord cross-sectional area [mm2] A 693
Modeling of B-ISG 54
These values are for the driven cases for the ISG and crankshaft pulleys respectively In the
driving case for either pulley the inertia of the rigid body is defined as 1 kg∙mm2 and the driving
torque is determined in equations (335) and (340) for the crankshaft and ISG pulleys
respectively
It is noted that because of the belt data for the phase angle at zero frequency the transition
frequency and the maximum phase angle are all zero and hence the belt damping is assumed to
be constant between frequencies These three values are typically used to generate a phase angle
versus frequency curve for the belt where the phase angle is dependent on the frequency The
curve defined by equation (354) is normally symmetric with the lowest phase angle achieved at
0 Hz and the highest phase angle achieved at the prescribed transition frequency f The belt
damping would then be found by solving for cb in the following equation
tanφ = cb ∙ 2 ∙ π ∙ f (354)
Nevertheless the assumption for constant damping between frequencies is also in harmony with
the remaining assumptions which assume damping of the belt spans to be negligible and
constant between belt spans
3521 Natural Frequency and Mode Shape
The set of natural frequencies and mode shapes for the system are shown in Figures 36 and 37
under the cases of the ISG pulley driving and the crankshaft pulley driving The forcing
frequency for the system differs for each case due to the change in driving pulley Modeic and
Modeia denote the ith rigid body according to the numbering convention used in Figure 32 in
the crankshaft and ISG driving cases respectively
Modeling of B-ISG 55
Natural Frequency ωn [Hz]
Crankshaft Pulley ΔΘ4
Air Conditioner Pulley ΔΘ
Tensioner Pulley 1 ΔΘ
Tensioner Pulley 2 ΔΘ
Tensioner Arm 1 ΔΘ
Tensioner Arm 2 ΔΘ
Figure 36a ISG Driving Case Natural Frequency of System and Mode Shapes for Responsive
Rigid Bodies
Figure 36b ISG Driving Case First Mode Responses
4 ΔΘ signifies the term amplitude of angular displacement for the indicated rigid body
Modeling of B-ISG 56
Figure 36c ISG Driving Case Second Mode Responses
Natural Frequency ωn [Hz]
ISG Pulley ΔΘ5
Air Conditioner Pulley ΔΘ
Tensioner Pulley 1 ΔΘ
Tensioner Pulley 2 ΔΘ
Tensioner Arm 1 ΔΘ
Tensioner Arm 2 ΔΘ
Figure 37a Crankshaft Driving Case Natural Frequency of System and Mode Shapes for
Responsive Rigid Bodies
5 ΔΘ signifies the term amplitude of angular displacement for the indicated rigid body
Modeling of B-ISG 57
Figure 37b Crankshaft Driving Case First Mode Responses
Figure 37c Crankshaft Driving Case Second Mode Responses
Modeling of B-ISG 58
3522 Dynamic Response
The dynamic response specifically the magnitude of angular displacement for each rigid body is
plotted in Figures 38 through to 314 as a function of the crankshaft pulley speed n This is
fitting to the analysis since the crankshaft pulley‟s rpm decides the mode of operation for the
system in particular it determines whether the crankshaft pulley or ISG pulley is the driving
pulley
Figure 38 Crankshaft Pulley Dynamic Response (for crankshaft driven case)
Figure 39 ISG Pulley Dynamic Response (for ISG driven case)
Modeling of B-ISG 59
Figure 310 Air Conditioner Pulley Dynamic Response
Figure 311 Tensioner Pulley 1 Dynamic Response
Crankshaft Driving Phase ISG
Driving
Phase
Crankshaft Driving Phase ISG
Driving
Phase
Modeling of B-ISG 60
Figure 312 Tensioner Pulley 2 Dynamic Response
Figure 313 Tensioner Arm 1 Dynamic Response
Crankshaft Driving Phase ISG
Driving
Phase
Crankshaft Driving Phase ISG
Driving Phase
Modeling of B-ISG 61
Figure 314 Tensioner Arm 2 Dynamic Response
3523 ISG Pulley and Crankshaft Pulley Torque Requirement
Figures 315 and 316 respectively showcase the required torques for the ISG pulley in its driving
mode and the crankshaft pulley in its driving mode
Figure 315 Required Driving Torque for the ISG Pulley
Crankshaft Driving Phase ISG
Driving Phase
Modeling of B-ISG 62
Figure 315 shows two plots for the required driving torque of the ISG pulley The dashed line
labeled as Q(n) simulates the application of equation (340) which models the ISG torque as a
permanent magnet DC motor The additional solid line labeled as qamod uses the formula in
equation (336) which determines the load torque of the driving pulley based on the pulley
responses Figure 315 provides a comparison of the results
Figure 316 Required Driving Torque for the Crankshaft Pulley
3524 Tensioner Arms Torque Requirements
The torque for the tensioner arms are shown in Figures 317 and 318
Modeling of B-ISG 63
Figure 317 Dynamic Torque for Tensioner Arm 1
Figure 318 Dynamic Torque for Tensioner Arm 2
3525 Dynamic Belt Span Tension
The dynamic tensions for the belt spans are shown in Figures 319 through to 323 The values
plotted represent the magnitude of the dynamic tension
Crankshaft Driving Phase ISG
Driving Phase
Crankshaft Driving Phase ISG
Driving
Phase
Modeling of B-ISG 64
Figure 319 Span 1 (between crankshaft and air conditioner) Dynamic Belt Span Tension
Figure 320 Span 2 (between air conditioner and tensioner 1) Dynamic Belt Span Tension
Crankshaft Driving Phase ISG
Driving Phase
Crankshaft Driving Phase ISG
Driving
Phase
Modeling of B-ISG 65
Figure 321 Span 3 (between tensioner 1 and ISG) Dynamic Belt Span Tension
Figure 322 Span 4 (between ISG and tensioner 2) Dynamic Belt Span Tension
Crankshaft Driving Phase ISG
Driving
Phase
Crankshaft Driving Phase ISG
Driving Phase
Modeling of B-ISG 66
Figure 323 Span 5 (between tensioner 2 and crankshaft) Dynamic Belt Span Tension
The dynamic results for the system serve to show the conditions of the system for a set of input
parameters The following chapter targets the focus of the thesis research by analyzing the affect
of changing the input parameters on the static conditions of the system It is the static results that
are the focus of the thesis and is thus analyzed in Chapters 4 and 5 in the parametric and
optimization studies respectively The dynamic analysis has been used to complete the picture of
the system‟s state under set values for input parameters
353 Static Analysis
Before looking at the static results for the system under study in brevity the static results for a
B-ISG serpentine belt drive with a single tensioner are presented In this theoretical system the
tensioner arm and tensioner pulley that interacts with the span between the air conditioner and
ISG pulleys of the original system are removed as shown in Figure 324
Crankshaft Driving Phase ISG
Driving
Phase
Modeling of B-ISG 67
Figure 324 B-ISG Serpentine Belt Drive with Single Tensioner
The complete static model as well as the dynamic model for the system in Figure 324 is found
in Appendix B The results of the static tension for each belt span of the single tensioner system
when the crankshaft is driving and the ISG is driving are shown in Table 35
Table 35 Static Solution for Belt Span Tensions in Crankshaft and ISG Driving Cases for a B-
ISG Serpentine Belt Drive with a Single Tensioner
Belt Span between Pulleys Crankshaft Driving Case [N] ISG Driving Case [N]
Crankshaft ndash Air Conditioner 481239 -361076
Air Conditioner ndash ISG 442588 -399727
ISG ndash Tensioner 29596 316721
Tensioner ndash Crankshaft 29596 316721
The tensions in Table 35 are computed with an initial tension of 300N This value for pre-
tension allows the spans in the case that the crankshaft pulley is driving to be suitably tensioned
Modeling of B-ISG 68
Whereas in the case of the ISG pulley driving the first and second spans are excessively slack
Therein an additional pretension of approximately 400N would be required which would raise
the highest tension span to over 700N This leads to the motivation of the thesis researchndash to
reduce the static belt tensions when the ISG is driving As mentioned in Chapter 1 these
tensions should be minimized to prolong belt life preferably within the range of 600 to 800N
As well it is desirable to minimize the amount of pretension exerted on the belt The current
design uses a pre-tension of 300N The above results would lead to a required pre-tension of
more than 700N to keep all spans of the belt suitably in tension (well above 0N) in order to allow
the belt to exhibit high performance in power transmission and come near to the safe threshold
This is the rationale for investigating a Twin Tensioner configuration shown in Figure 32 for
the B-ISG serpentine belt drive under study For the theoretical system with a Twin Tensioner
the following static results in Table 36 are achieved
Table 36 Static Solution for Belt Span Tensions in Crankshaft and ISG Driving Cases for a B-
ISG Serpentine Belt Drive with a Twin Tensioner
Belt Span between Pulleys Crankshaft Driving Case [N] ISG Driving Case [N]
Crankshaft ndash Air Conditioner 465848 -284152
Air Conditioner ndash Tensioner 1 427197 -322803
Tensioner 1 ndash ISG 427197 -322803
ISG ndash Tensioner 2 28057 393645
Tensioner 2 ndash Crankshaft 28057 393645
The results in Table 36 show that the span following the ISG in the case between the Tensioner
1 and ISG pulleys is less slack than in the former single tensioner set-up However there
remains an excessive amount of pre-tension required to keep all spans suitably tensioned
Modeling of B-ISG 69
36 Summary
The simulation of the model for the B-ISG system with the Twin Tensioner shows that the mode
shapes of the rigid bodies within the system (Figures 36a to 37c) are greater in magnitude when
the ISG pulley is driving than when the crankshaft pulley is driving The dynamic responses of
the system as shown in Figures 38 and 310 to 314 is small for the crankshaft pulley and are
negligible for the remaining driven bodies when the ISG is driving For the crankshaft driving
phase there is greater dynamic response for the driven rigid bodies of the system including for
that of the ISG pulley
As the engine speed increases the torque requirement for the ISG was found to vary between
approximately 41Nm and 54Nm (before dropping steeply to approximately 3Nm at an engine
speed of about 720rpm) when modeled after equation (336) or between approximately 48Nm
and 34Nm when modeled after equation (340) In contrast the torque for the crankshaft peaks
at approximately 92Nm and 52Nm at an approximate engine speed of 1450rpm and 5000rpm
respectively The dynamic torque of the first tensioner arm was shown to peak at approximately
15Nm at the transition engine speed 750rpm and again at approximately 15Nm at an
approximate engine speed of about 1450rpm A small peak of about 3Nm was also seen at an
engine speed of 5000rpm Similarly for the second tensioner arm a torque peak of
approximately 20Nm was seen at 750rpm and 1450rpm and a smaller peak of about 8Nm was
seen at an engine speed of 5000rpm
The trend for the dynamic tensions is that the peaks are highest in the ISG driving portion of the
B-ISG operation in most cases and in a few cases they are seen to be close in magnitude to that
Modeling of B-ISG 70
of the highest peaks in the crankshaft driving portion The dynamic tension for the first belt span
peaked at approximately 780Nm 830Nm and 500Nm at engine speeds of 750rpm 1450rpm
5000rpm respectively For the dynamic tension of the second belt span peaks of approximately
1250Nm 675Nm and 760Nm were seen at the same respective engine speeds for the 3 peaks of
the former span At these same engine speeds the third belt span exhibited tension peaks at
approximately 1400Nm 650Nm and 890Nm The tension peaks of the fourth span were
approximately 165Nm 150Nm and 100Nm at engine speeds 750rpm 1450rpm and 5000rpm
The fifth span experienced peaks of approximately 165Nm 170Nm and 120Nm at the same
respective engine speeds of the fourth span
The simulation results for the static tension of the B-ISG system with the Twin Tensioner reveal
that taut spans of the crankshaft driving case are lower in the ISG driving case The largest
change is an approximate decrease of 750N in spans 1 through 3 while spans 4 and 5 increase
by approximately 113N It can be seen that the spans in highest tension (1 2 and 3) in the
crankshaft driving phase become excessively slack in the ISG driving phase There is a smaller
change between the tension values for the spans in the least tension in the crankshaft driving
phase and their corresponding span in the ISG driving phase
The summary of the simulation results are used as a benchmark for the optimized system shown
in Chapter 5 The static tension simulation results are investigated through a parametric study of
the Twin Tensioner system in Chapter 4 The optimization of the system is then based on the
selected design variables from the outcome of Chapter 4
71
CHAPTER 4 PARAMETRIC ANALYSIS OF A B-ISG
TWIN TENSIONER
41 Introduction
The parameters for the proposed Twin Tensioner for a Belt-driven Integrated Starter-generator
(B-ISG) system are investigated through a parametric analysis This analysis seeks to understand
how changing one parameter influences the static belt span tensions for the system Since the
thesis research focuses on the design of a tensioning mechanism to support static tension only
the parameters specific to the actual Twin Tensioner and applicable to the static case were
considered The parameters pertaining to accessory pulley properties such as radii or various
belt properties such as belt span stiffness are not considered In the analyses a single parameter
is varied over a prescribed range while all other parameters are held constant The pivot point
described by Cartesian Coordinates [X6Y6] for the tensioner arms is held constant in all cases
42 Methodology
The parametric study method applies to the general case of a function evaluated over changes in
one of its dependent variables The methodology is illustrated for the B-ISG system‟s function
for static tension which is evaluated for each change in one of its Twin Tensioner‟s parameters
The original data used for the system is based on sample vehicle data provided by Litens [56]
Table 41 provides the initial data for the parameters as well as the incremental change and
maxima and minima limits The increment Δi for the ith parameter is chosen arbitrarily Limits
for each parameter have been chosen to be plus or minus sixty percent of its initial value
Parametric Analysis 72
Table 41 Initial Values Increments and Ranges for Parameters of Twin Tensioner
Parameter Name Initial Value Increment (+- Δi) Minimum
value Maximum value
Coupled Spring
Stiffness kt
20626
N∙mrad 1238 N∙mrad 8250 N∙mrad 33002 N∙mrad
Tensioner Arm 1
Stiffness kt1
10314
N∙mrad 0619 N∙mrad 4126 N∙mrad 16502 N∙mrad
Tensioner Arm 2
Stiffness kt2
16502
N∙mrad 0990 N∙mrad 6601 N∙mrad 26403 N∙mrad
Tensioner Pulley 1
Diameter D3 007240 m 4344 ∙ 10
-3 m 00290 m 0116 m
Tensioner Pulley 2
Diameter D5 007240 m 4344 ∙ 10
-3 m 00290 m 0116 m
Tensioner Pulley 1
Initial Coordinates
[0292761
0087] m See Figure 41 for region of possible tensioner pulley
coordinates Tensioner Pulley 2
Initial Coordinates
[012057
009193] m
The mesh of possible points for the centre coordinates of tensioner pulley 1 and tensioner pulley
2 are designated as Region 1 and Region 2 respectively in Figures 41a and 41b
Figure 41a Regions 1 and 2 and their Associated Guidelines for Coordinates of Tensioner
Pulleys 1 amp 2
CS
AC
ISG
Ten 1
Ten 11
Region II
Region I
Parametric Analysis 73
Figure 41b Regions 1 and 2 in Cartesian Space
The selection for the minimum and maximum tensioner pulley centre coordinates and their
increments are not selected arbitrarily or without derivation as the other tensioner parameters
The coordinates for the pulley centres are identified using Intergraph‟s SmartSketch software a
graphing suite in MathCAD to model the regions of space The following descriptions are used
to describe the possible positions for the tensioner pulleys
Tensioner pulleys are situated such that they are exterior to the interior space created by
the serpentine belt thus they sit bdquooutside‟ the belt loop
The highest point on the tensioner pulley does not exceed the tangent line connecting the
upper hemispheres of the pulleys on either side of it
The tensioner pulleys may not overlap any other pulley
Parametric Analysis 74
Boundaries for regions described as Region 1 in span 2 and 3 and Region 2 in span 4
and 5 is selected based on the above criteria and their lower boundaries are selected
arbitrarily
These criteria were used to define the equation for each boundary line and leads to a set of
Boolean conditions that relate the x-coordinate and y-coordinate for each Cartesian pair The
density for the mesh of points in each region is arbitrarily selected as 101 x-points and 101 y-
points in each space for the purposes of the parametric analysis The outline of this method is
described in the MATLAB scripts contained in Appendix D
The results of the parametric analysis are shown for the slackest and tautest spans in each driving
case As was demonstrated in the literature review the tautest span immediately precedes the
driving pulley and the slackest span immediately follows the driving pulley in the direction of
the belt motion Thus in the case for the crankshaft driving the tautest span is in the first span
and the slackest span is in the fifth span Whereas in the ISG driving case the tautest span is in
the fourth span and the slackest span is in the third span Hence the parametric figures in this
chapter display only the tautest and slackest span values for both driving cases so as to describe
the maximum and minimum values for tension present in the given belt
43 Results amp Discussion
431 Influence of Tensioner Arm Stiffness on Static Tension
The parametric analysis begins with changing the stiffness value for the coil spring coupled
between tensioner arms 1 and 2 This stiffness value kt is changed over a range from sixty
percent less than its initial value kt0 to sixty percent more than its original value as shown in
Parametric Analysis 75
Table 41 The results of the static tension are shown in Figure 42 for the tautest and slackest
spans for both the crankshaft and ISG driving cases
Figure 42 Parametric Analysis for Coupled Stiffness Arm Constant kt (N∙mrad)
As kt increases in the crankshaft driving phase for the B-ISG system the highest tension
decreases from 4691N to 4646N while the lowest tension decreases from 2838N to 2793N
In the ISG driving phase the highest tension increases from 378N to 3998N and the lowest
tension increases from -3384N to -3167N Thus a change of approximately -45N is found in
the crankshaft driving case and approximately +22N is found in the ISG driving case for both the
tautest and slackest spans
Parametric Analysis 76
The second parameter analyzed is the stiffness value for tensioner arm 1 The results of this are
shown in Figure 43
Figure 43 Parametric Analysis for Stiffness of Arm 1 kt1 (N∙mrad)
In Figure 43 as kt1 increases an increase from 4628N to 4681N is observed for the tension of
the tautest span when the crankshaft is driving which is a change of +53N The same value for
net change is found in the slackest span for the same driving condition whose tension increases
from 2775N to 2828N For the case when the B-ISG system is in the ISG driving phase the
change is larger a value of -261N for the tautest span that changes from 4088N to 3827N and
for the slackest span that changes from -3077N to -3338N
Parametric Analysis 77
The change in static tension for the spans as the stiffness of arm 2 varies is demonstrated in
Figure 44
Figure 44 Parametric Analysis for Stiffness of Arm 2 kt2 (N∙mrad)
In this case it is observed that as kt2 increases the tautest span for the B-ISG system in the
crankshaft driving case decreases from 4675N to 4643N as well as the slackest span which
decreases from 2822N to 279N which is an overall change of -32N for both spans Whereas in
the ISG driving case a more noticeable change is once again found a difference of +144N
This is a result of the tautest span increasing from 3863N to 4007N and the slackest span
increasing from -3301N to -3157N
Parametric Analysis 78
432 Influence of Tensioner Pulley Diameter on Static Tension
The change in the diameter of tensioner pulley 1 D3 and its effect on static tension is shown in
Figure 45
Figure 45 Parametric Analysis for Pulley 1 Diameter D3 (m)
The change in the tautest and slackest spans for the B-ISG system‟s crankshaft driving case is
from 3248N to 425N and from 1395N to 240N respectively Peaks are seen at 4799N and
2946N for the respective spans This is a change of approximately +100N and a maximum
change of 1551N for both spans For the ISG driving case the tautest and slackest spans
decrease from 1083N to 6158N and 367N to -1006N Global minimums of 3246N and -391N
for the respective spans are seen This nets a change of approximately -467N and a maximum
change of approximately -759N
Parametric Analysis 79
The effect of changing the diameter of tensioner pulley 2 on the static tension is examined in
Figure 46
Figure 46 Parametric Analysis for Pulley 2 Diameter D5 (m)
The tautest and slackest spans in the crankshaft driving mode of the belt undergo a change from
4583N to 4721N and from 273N to 2869N respectively Therein as D5 increases the trend is
that for both spans there is an increase in tension of approximately 14N Contrastingly the spans
experience a decrease in the ISG driving case as D5 increases The tension of the tautest span
goes from 4296N to 3635N and that of the slackest span goes from -2866N to -3529N This
equals a decrease of approximately 66N for both spans
Parametric Analysis 80
433 Influence of Tensioner Pulley 1 Coordinates on Static Tension
The influence of the coordinates of tensioner pulley 1 on the value of tension in the tautest span
for the B-ISG system‟s crankshaft driving case is demonstrated in Figure 47
Figure 47 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Tautest Span
Tension in Crankshaft Driving Case
The region shown in Figure 47 corresponds to region 1 which is the realm of the positions for
tensioner pulley 1 The possible pulley coordinates in this case are represented by the non-blue
area reaching to the perimeter of the plot It is evident in the darkest red region of the plot
where the y-coordinate is between approximately 0m and 0075m and the x-coordinate is
(N)
Parametric Analysis 81
between approximately 026m and 031m that the highest value of tension is experienced in the
tautest span for the crankshaft driving case The range of tension for Region 1 in the tautest span
when the crankshaft is driving is between a maximum of approximately 500N and a minimum of
approximately 300N This equals an overall difference of 200N in tension for the tautest span by
moving the position of pulley 1 The lowest values for tension are obtained when the pulley
coordinates are approximately -0025m to 015m for the y-coordinate and approximately 031m
to 032m for the x-coordinate which corresponds to the yellow region An area of low tension is
also seen in the area where the y-coordinate is approximately 0m and the x-coordinate is
approximately between 026m and 027m
The changes in tension for the slackest span under the condition of the crankshaft pulley being
the driving pulley are shown in Figure 48
Parametric Analysis 82
Figure 48 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Slackest Span
Tension in Crankshaft Driving Case
Once again the possible coordinate points for tensioner pulley 1 in the B-ISG system are
represented by the non-blue region For the slackest span in the crankshaft driving case it is seen
that the lowest tension is approximately 125N while the highest tension is approximately 325N
This is an overall change of 200N that is achieved in the region The highest values are achieved
in the space where the y-coordinates are approximately 0m to 0075m and the x-coordinate
ranges from 026m to 031m which corresponds to the deep red region The lowest tension
values are achieved in the space where the y-coordinate ranges from approximately -0025m to
015m and the x-coordinate ranges from 031m to 032m which corresponds to the light blue-
green region of the plot The area containing a y-coordinate of approximately 0m and x-
(N)
Parametric Analysis 83
coordinates that are approximately between 026m and 027m also show minimum tension for
the slack span The regions of the x-y coordinates for the maximum and minimum tensions are
alike to the tautest span in Region 1 for the crankshaft driving case as well as was seen in Figure
47
The tension for the tautest span in the case that the ISG is driving in the B-ISG system is found
in Figure 49
Figure 49 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Tautest Span
Tension in ISG Driving Case
(N)
Parametric Analysis 84
Region 1 is represented by the coordinate values shown in the non-dark blue space of the plot in
Figure 49 The tautest span in the case of the ISG driving experiences a range of tension values
in Region 1 from 200N up to 1100N equaling a difference of 900N The minimum tension
values are achieved in the medium to light blue region This includes y-coordinates of
approximately 0m to 0075m and x-coordinates of approximately 026m to 03m The
maximum tension values are in the darkest red area inclusive of y-coordinates -0025m to 015m
and x-coordinates 031m to 032m in addition to y-coordinate of approximately 0m and x-
coordinates of approximately 026m to 027m It can be observed that aforementioned regions
for minimum and maximum tensions in Figure 49 are reverse to those seen in Figures 47 and
48 for the crankshaft driving case
The change in tension for the slackest span of the B-ISG system when it is driven by the ISG is
shown in Figure 410
Parametric Analysis 85
Figure 410 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Slackest Span
Tension in ISG Driving Case
Figure 410 exhibits the realm of possible points for tensioner pulley 1 for the case of the ISG
driving in the non-yellow-green area The minimum tension values are achieved in the darkest
blue area where the minimum tension is approximately -500N This area corresponds to y-
coordinates from approximately 0m to 005m and x-coordinates from approximately 026m to
03m The area of a maximum tension is approximately 400N and corresponds to the darkest red
area inclusive of y-coordinates -0025m to 015m and x-coordinates 031m to 032m as well as
the coordinates for y equaling approximately 0m and for x equaling approximately 026m to
027m The difference between maximum and minimum tensions in this case is approximately
900N It is noticed once again that the space of x- and y-coordinates containing the maximum
(N)
Parametric Analysis 86
tension is in the similar location to that of the described space for minimum tension in the
crankshaft driving case in Figure 47 and 48
434 Influence of Tensioner Pulley 2 Coordinates on Static Tension
The influence of pulley 2 coordinates on the tension value for the tautest span when the
crankshaft is driving the B-ISG system is shown in Figure 411 and is represented by the values
corresponding to the non-blue area
Figure 411 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Tautest Span
Tension in Crankshaft Driving Case
In Figure 411 the possible coordinates are contained within Region 2 The maximum tension
value is approximately 500N and is found in the darkest red space including approximately y-
(N)
Parametric Analysis 87
coordinates 004m to 014m and x-coordinates 0025m to 0175m and also y-coordinates 013m
to 02m corresponding to the x-coordinate at 0175m A minimum tension value of
approximately 350N is found in the yellow space and includes approximately y-coordinates
008m to 018m and x-coordinates 016m to 02m The difference in tension values is 150N
The analysis of the change in coordinates for tension pulley 2 on the value for tension in the
slackest span is shown in Figure 412 in the non-blue region
Figure 412 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Slackest Span
Tension in Crankshaft Driving Case
The value of 325N is the highest tension for the slack span in the crankshaft driving case of the
B-ISG system and is found in the deep-red region where the y-coordinates are between
(N)
Parametric Analysis 88
approximately 004m and 013m and the x-coordinates are approximately between 0025m and
016m as well as where y is between 013m and 02m and x is approximately 0175m The
lowest tension value for the slack span is approximately 150N and is found in the green-blue
space where y-coordinates are between approximately 01m and 022m and the x-coordinates
are between approximately 016m and 021m The overall difference in minimum and maximum
tension values is 175N The spaces for the maximum and minimum tension values are similar in
location to that found in Figure 411 for the tautest span in the crankshaft driving case
Figure 413 provides the theoretical data for the tension values of the tautest span as the position
of the B-ISG system‟s tensioner pulley 2 changes in the ISG driving case Possible points are in
the space of values which correspond to the non-dark-blue region in Figure 413
Parametric Analysis 89
Figure 413 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Tautest Span
Tension in ISG Driving Case
In Figure 413 the region for high tension reaches a value of approximately 950N and the region
for low tension reaches approximately 250N This equals a difference of 700N between
maximum and minimum tension values for the tautest span in the B-ISG system‟s ISG driving
case The coordinate points within the space that maximum tension is reached is in the dark red
region and includes y-coordinates from approximately 008m to 022m and x-coordinates from
approximately 016m to 021m The coordinate points within the space that minimum tension is
reached is in the blue-green region and includes y-coordinates from approximately 004m to
013m and the corresponding x-coordinates from approximately 0025m to 015m An additional
small region of minimum tension is seen in the area where the x-coordinate is approximately
(N)
Parametric Analysis 90
0175m and the y-coordinates are approximately between 013m and 02m The location for the
area of pulley centre points that achieve maximum and minimum tension values is approximately
located in the reverse positions on the plot when compared to that of the case for the crankshaft
driving in Figures 411 and 412 Therein the trend seen for pulley coordinates for the second
tensioner pulley follows suit with that of the first tensioner pulley which is that the area of points
for maximum tension in the crankshaft driving case becomes the approximate area of points for
minimum tension in the ISG driving case and vice versa
In Figure 414 the results of the parametric analysis on the coordinates of the second tensioner
pulley and its effect on the slackest span‟s tension in the ISG driving case is shown Similar to
earlier figures the non-dark yellow region represents Region 2 that contains the possible points
for the pulley‟s Cartesian coordinates
Parametric Analysis 91
Figure 414 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Slackest
Span Tension in ISG Driving Case
Figure 414 demonstrates a difference of approximately 725N between the highest and lowest
tension values for the slackest span of the B-ISG system in the ISG driving case The highest
tension values are approximately 225N The area of points that allow the second tension pulley
to achieve maximum tension in the belt span includes y-coordinates from approximately 01m to
022m and the corresponding x-coordinates from approximately 016m to 021m This
corresponds to the darkest red region in Figure 414 The coordinate values where the lowest
tension being approximately -500N is achieved include y-coordinate values from
approximately 004m to 013m and the corresponding x-coordinates from approximately 0025m
to 015m corresponding to the darkest blue region A dark blue region of lowest tension is also
(N)
Parametric Analysis 92
seen in the area where y is approximately between 013m and 02m and the x-coordinate is
approximately 0175m The regions for maximum and minimum tension values are observed to
be similar to those found in Figure 413 and alike to Figure 413 to be in reverse to those found
in Figure 411 and 412 for the tautest and slackest spans in the crankshaft driving case So as for
the changes in tensioner pulley 2 coordinates the areas for minimum tension in Region 2 of the
ISG driving case are similar to the areas for maximum tension in Region 2 of the crankshaft
driving case and vice versa for the maximum tension of the ISG driving case and the minimum
tension for the crankshaft driving case in Region 2
44 Conclusion
Overall the trend in the plots of Figures 47 48 411 and 412 indicate in the crankshaft driving
portion that the B-ISG system‟s belt span tensions experience the following effect
Minimum tension for the tautest span is achieved when tensioner pulley 1 centre
coordinates are located closer to the right side boundary and bottom left boundary of
Region 1 or when tensioner pulley 2 centre coordinates are within the upper right space
(near to the ISG pulley) and the space closer to the top boundary of Region 2
Maximum tension for the slackest span is achieved when the first tensioner pulley‟s
coordinates are located in the mid space and near to the bottom boundary of Region 1
and when the second tensioner pulley‟s coordinates are located near to the bottom left
boundary of Region 2 which is the boundary nearest to the crankshaft pulley
Parametric Analysis 93
The trend for minimizing the tautest span signifies that the tension for the slackest span is also
minimized at the same time As well maximizing the slackest span signifies that the tension for
the tautest span is also maximized at the same time too
The trend for the B-ISG system‟s ISG driving case as can be seen in Figures 49 410 413 and
414 is approximately in reverse to that of the crankshaft driving case for the system Wherein
points corresponding to minimum tension in Regions 1 and 2 in the ISG case are approximately
the same as points corresponding to maximum tension in the Regions for the crankshaft case and
vice versa for the ISG cases‟ areas of maximum tension
Minimum tension for the tautest span is present when the first tensioner pulley‟s
coordinates are near to mid to lower boundary of Region 1 and when the second
tensioner pulley‟s coordinates are close to the bottom left boundary of Region 2 which
is the furthest boundary from the ISG pulley and closest to the crankshaft pulley
Maximum tension for the slackest span is achieved when the first tensioner pulley is
located close to the right boundary of Region 1 and when the second tensioner pulley is
located near the right boundary and towards the top right boundary of Region 2
It is observed in Figures 47 to 414 and alike to Figures 42 to 46 the tautest and slackest
spans decrease or increase together Thus it can be assumed that the tension values in these
spans and likely the remaining spans outside of the tautest and slackest spans follow suit
Therein when parameters are changed to minimize one belt span‟s tension the remaining spans
will also have their tension values reduced Figures 42 through to 413 showed this clearly
where the overall change in the tension of the tautest and slackest spans changed by
Parametric Analysis 94
approximately the same values for each separate case of the crankshaft driving and the ISG
driving in the B-ISG system
Design variables are selected in the following chapter from the parameters that have been
analyzed in the present chapter The influence of changing parameters on the static tension
values for the various spans is further explored through an optimization study of the static belt
tension for the B-ISG system equipped with a Twin Tensioner in the following chapter Chapter
5
95
CHAPTER 5 OPTIMIZATION OF A B-ISG TENSIONER
The objective of the optimization analysis is to minimize the absolute magnitude of the static
tension in the ISG-operating mode of the serpentine belt drive The optimization seeks to
optimize the performance of the proposed Twin Tensioner design by using its properties as the
design variables for the objective function The optimization task begins with the selection of
these design variables for the objective function and then the selection of an optimization
method The results of the optimization will be compared with the results of the analytical
model for the static system and with the parametric analysis‟ data
51 Optimization Problem
511 Selection of Design Variables
The optimal system corresponds to the properties of the Twin Tensioner that result in minimized
magnitudes of static tension for the various belt spans Therein the design variables for the
optimization procedure are selected from amongst the Twin Tensioner‟s properties In the
parametric analysis of Chapter 4 the tensioner properties presented included
coupled stiffness kt
tensioner arm 1 stiffness kt1
tensioner arm 2 stiffness kt2
tensioner pulley 1 diameter D3
tensioner pulley 2 diameter D5
tensioner pulley 1 initial coordinates [X3Y3] and
Optimization 96
tensioner pulley 2 initial coordinates [X5Y5]
It was observed in the former chapter that perturbations of the stiffness and geometric parameters
caused a change between the lowest and highest values for the static tension especially in the
case of perturbations in the geometric parameters diameter and coordinates Table 51
summarizes the observed changes in the belt span tensions corresponding to the Twin Tensioner
parameters‟ maximum and minimum values
Table 51 Summary of Parametric Analysis Data for Twin Tensioner Properties
Parameter Symbol
Original Tensions in TautSlack Span (Crankshaft
Mode) [N]
Tension at
Min | Max Parameter6 for
Crankshaft Mode [N]
Percent Change from Original for
Min | Max Tensions []
Original Tension in TautSlack Span (ISG Mode)
[N]
Tension at
Min | Max Parameter Value in ISG Mode [N]
Percent Change from Original Tension for
Min | Max Tensions []
kt
465848 (taut) 4691 4646 07 -03 393645 (taut) 378 3998 -40 16
28057 (slack) 2838 2793 12 -05 -322803 (slack) -3384 -3167 -48 19
kt1
465848 (taut) 4628 4681 -07 05 393645 (taut) 4088 3827 38 -28
28057 (slack) 2775 2828 -11 08 -322803 (slack) -3077 -3338 47 -34
kt2
465848 (taut) 4675 4643 04 -03 393645 (taut) 3863 4007 -19 18
28057 (slack) 2822 279 06 -06 -322803 (slack) -3301 -3157 -23 22
D3 465848 (taut) 3248 425 -303 -88 393645 (taut) 1083 6158 1751 564
28057 (slack) 1395 240 -503 -145 -322803 (slack) 367 -1006 2137 688
D5 465848 (taut) 4583 4721 -16 13 393645 (taut) 4296 3635 91 -77
28057 (slack) 273 2869 -27 23 -322803 (slack) -2866 -3529 112 -93
[X3Y3] 465848 (taut) 300 500 -356 73 393645 (taut) 200 1100 -492 1794
28057 (slack) 125 325 -554 158 -322803 (slack) -500 400 -549 2239
6 The values for the tension for each of the taut and slack spans provided correspond to the minimum and maximum
values of the parameter listed in each case such that the columns of identical colour correspond to each other For
the coordinate parameters the minimum and maximum parameter value is inadmissible The tension values in these
cases are simply the minimum and maximum tension values achieved by the coordinate parameter listed
Optimization 97
[X5Y5] 465848 (taut) 350 500 -249 73 393645 (taut) 250 950 -365 1413
28057 (slack) 150 325 -465 158 -322803 (slack) -500 225 -549 1697
The results of the parametric analyses for the Twin Tensioner parameters show that there is a
noticeable percent change between the initial tensions and the tensions corresponding to each of
the minima and maxima parameter values or in the case of the coordinates between the
minimum and maximum tensions for the spans Thus the parametric data does not encourage
exclusion of any of the tensioner parameters from being selected as a design variable As a
theoretical experiment the optimization procedure seeks to find feasible physical solutions
Hence economic criteria are considered in the selection of the design variables from among the
Twin Tensioner‟s parameters Of the tensioner properties it is found that the diameter of the
tensioner pulleys has the largest impact on cost Adding mass to a tensioner pulley as a result of
increasing the diameter and consequently its inertia increases the cost of material Material cost
is most significant in the manufacture process of pulleys as their manufacturing is largely
automated [4] Furthermore varying the structure of a pulley requires retooling which also
increases the cost to manufacture As such the tensioner pulley diameters D3 and D5 are
excluded from being selected as design variables The remaining tensioner properties the
stiffness parameters and the initial coordinates of the pulley centres are selected as the design
variables for the objective function of the optimization process
512 Objective Function amp Constraints
In order to deal with two objective functions for a taut span and a slack span a weighted
approach was employed This emerges from the results of Chapter 3 for the static model and
Chapter 4 for the parametric study for the static system which show that a high tension span and
Optimization 98
a highly slack span exist in the ISG-driving phase of the B-ISG system Therein the first
objective function of equation (51a) is described as equaling fifty percent of the absolute tension
value of the tautest span and fifty percent of the absolute tension value of the slackest span for
the case of the ISG driving only The second objective function uses a non-weighted approach
and is described as the absolute tension of the slackest span when the ISG is driving A non-
weighted approach is motivated by the phenomenon of a fixed difference that is seen between
the slackest and tautest spans of the optimal designs found in the weighted optimization
simulations Equations (51a) through to (51c) display the objective functions
The limits for the design variables are expanded from those used in the parametric analysis for
the non-coordinate parameters kt kt1 and kt2 so that they are permitted to vary from
approximately 0 to approximately 200 of the initial value for each parameter kt0 kt10 and kt20
respectively In the case of the coordinate parameters [X3Y3] and [X5Y5] the x- and y-
coordinates are permitted to vary within the spaces Region 1 and Region 2 respectively which
were prescribed in Chapter 4 Figure 41a and 41b
Aside from the design variables design constraints on the system include the requirement for
static stability of the Twin Tensioner An optimal solution for the B-ISG system must achieve
the goal of the objective function which is to minimize the absolute tensions in the system
However for an optimal solution to be feasible the movement of the tensioner arm must remain
within an appropriate threshold In practice an automotive tensioner arm for the belt
transmission may be considered stable if its movement remains within a 10 degree range of
Optimization 99
motion [4] As such the angle of displacement for tensioner arms 1 and 2 are designated by θ t1
and θt2 respectively in the listed constraints
The optimization task is described in equations 51a to 52 Variables a through to g represent
scalar limits for the x-coordinate for corresponding ranges of the y-coordinate
Minimize 119879119908119890119894119892 119893119905119890119889 = 05 ∙ 119879119905119886119906119905 + 05 ∙ 119879119904119897119886119888119896
or119879119899119900119899 minus119908119890119894119892 119893119905119890119889 = 119879119904119897119886119888119896
(51a)
where
119879119905119886119906119905 = 119891119905119886119906119905 119896119905 1198961199051 1198961199052 1198833 1198843 1198835 1198845 (51b)
119879119904119897119886119888119896 = 119891119904119897119886119888119896 (119896119905 1198961199051 1198961199052 1198833 119884311988351198845) (51c)
Subject to
(1198961199050 minus 1 ∙ 1198961199050) le 119896119905 le (1198961199050 + 11198961199050)(11989611990510 minus 1 ∙ 11989611990510) le 1198961199051 le (11989611990510 + 111989611990510)(11989611990520 minus 1 ∙ 11989611990520) le 1198961199052 le (11989611990520 + 111989611990520)
119886 le 1198833 le 119888
1198931 1198833 le 1198843 le 1198933 1198833 119891119900119903 119886 le 1198833 lt 119887
1198932 1198833 le 1198843 le 1198933 1198833 119891119900119903 119887 le 1198833 le 119888119889 le 1198835 le 119892
1198934 1198835 le 1198845 le 1198937 1198835 for 119889 le 1198835 lt 1198901198935(1198835) le 1198845 le 1198937(1198835) for 119890 le 1198835 lt 119891
1198936 1198835 le 1198845 le 1198937 1198833 for 119891 le 1198833 le 119892 1205791199051 le 10deg 1205791199052 le 10deg
(52)
The functions for the taut and slack spans represent the fourth and third span respectively in the
case of the ISG driving The equations for the tensions of the aforementioned spans are shown
in equation 51a to 51c and are derived from equation 353 The constraints for the
optimization are described in equation 52
Optimization 100
52 Optimization Method
A twofold approach was used in the optimization method A global search alone and then a
hybrid search comprising of a global search and a local search The Genetic Algorithm is used
as the global search method and a Quadratic Sequential Programming algorithm is used for the
local search method
521 Genetic Algorithm
Genetic Algorithm (GA) is a part of the growing genre of evolutionary algorithms [57] The
optimization approach differs from classical search approaches by its ease of use and global
perspective [57] GA mimics biological evolution theory by using a ldquocross-over of heritable
information random mutation and selection on the basis of fitness between generationsrdquo [58] to
form a robust search algorithm that requires minimal problem information [57] The parameter
sets are represented as sample points modeled as bdquochromosomes‟ or data strings that are
measured against how well they allow the model to achieve the optimization task [58] GA is
stochastic which means that its algorithm uses random choices to generate subsequent sampling
points rather than using a set rule to generate the following sample This avoids the pitfall of
gradient-based techniques that may focus on local maxima or minima and end-up neglecting
regions containing higher peaks or lower valleys [57] Furthermore due to the randomness of
the GA‟s search strategy it is able to search a population (a region of possible parameter sets)
faster than other optimization techniques The GA approach is viewed as a universal
optimization approach while many classical methods viewed to be efficient for one optimization
problem may be seen as inefficient for others However because GA is a probabilistic algorithm
its solution for the objective function may only be near to a global optimum As such the current
Optimization 101
state of stochastic or global optimization methods has been to refine results of the GA with a
local search and optimization procedure
522 Hybrid Optimization Algorithm
In order to enhance the result of the objective function found by the GA a Hybrid optimization
function is implemented in MATLAB software The Hybrid optimization function combines a
global search GA with a local search Sequential Quadratic Programming (SQP) The hybrid
process refines the value of the objective function found through GA by using the final set of
points found by the algorithm as the initial point of the SQP algorithm The GA function
determines the region containing a global optimum and then the SQP algorithm uses a gradient
based technique to find a solution closer to the global optimum The MATLAB algorithm a
constrained minimization function known as fmincon uses an SQP method that approximates the
Hessian for the Lagrangian function (ie the second derivatives of the Lagrangian) by way of a
quasi-Newton approach to generate a quadratic program (QP) sub-problem [59] The solution
for the QP provides the search direction of the line search procedure used when each iteration is
performed [59]
53 Results and Discussion
531 Parameter Settings amp Stopping Criteria for Simulations
The parameter settings for the optimization procedure included setting the stall time limit to
200s This is the interval of time the GA is given to find an improvement in the value of the
objective function This is an increase from MATLAB‟s default of 20s Increasing the stall time
limit allows for the optimization search to consistently converge without being limited by time
Optimization 102
The population size used in finding the optimal solution is set at 100 This value was chosen
after varying the population size between 50 and 2000 showed no change in the value of the
objective function The max number of generations is set at 100 The time limit for the
algorithm is set at infinite The limiting factor serving as the stopping condition for the
optimization search was the function tolerance which is set at 1x10-6
This allows the program
to run until the ratio of the change in the objective function over the stall generations is less than
the value for function tolerance The stall generation setting is defined as the number of
generations since the last improvement of the objective function value and is limited to 50
532 Optimization Simulations
The results of the genetic algorithm optimization simulations performed in MATLAB are shown
in the following tables Table 52a and Table 52b
Table 52a GA Optimization Results for Twin Tensioner Parameters and Objective Function
Trial
No
Genetic Algorithm Optimization Method
Objective
Function
Value [N]
kt [N∙mrad]
kt1 [N∙mrad]
kt2 [N∙mrad]
[X3Y3]
[m] [X5Y5]
[m]
1 3582241 314069 204844 165020 [02928 00703] [01618 01036]
2 3582241 103646 205284 198901 [03009 00607] [01283 00809]
3 3582241 126431 204740 43549 [03010 00631] [01311 01147]
4 3582241 180285 206230 254870 [03095 00865] [01080 01675]
5 3582241 74757 204559 189077 [03084 00617] [01265 00718]
Original Design 20626 10314 16502 [02928 0087] [01206 00919]
Optimization 103
Table 52b Computations for Tensions and Angles from GA Optimization Results
Trial No
Genetic Algorithm Optimization Method
Slackest Tension [N] Tautest Tension [N] Tensioner Arm 1
Displacement [deg]
Tensioner Arm 2
Displacement [deg]
T3 T4 Θt1 Θt2
1 -1572307 5592176 -00025 -49748
2 -4054309 3110174 -00002 -20213
3 -3930858 3233624 -00004 -38370
4 -1309751 5854731 -00010 -49525
5 -4092446 3072036 -00000 -17703
Original Design -322803 393645 16410 -4571
For each trial above the GA function required 4 generations each consisting of 20 900 function
evaluations before finding no change in the optimal objective function value according to
stopping conditions
The results of the Hybrid function optimization are provided in Tables 53a and 53b below
Table 53a Hybrid Optimization Results for Twin Tensioner Parameters and Objective Function
Trial
No
Hybrid Optimization Method
Objective
Function
Value [N]
of
Function
Evals ( of
Iterations)
kt [N∙mrad]
kt1 [N∙mrad]
kt2 [N∙mrad]
[X3Y3]
[m] [X5Y5]
[m]
1 3582241 16 (1) 16065 205846 229494 [02780 00581] [01679 01288]
2 3582241 20 (1) 249227 205635 25218 [02901 00634] [01559 00870]
3 3582241 16 (1) 297295 204878 320479 [02962 00702] [01336 01447]
4 3582241 53 (1) 241433 204262 229683 [02912 00647] [00047 01465]
Optimization 104
5 3582241 21 (1) 379096 205548 188888 [02973 00703] [01206 01376]
Original Design 20626 10314 16502 [02928 0087] [01206 00919]
Table 53b Computations for Tensions and Angles from Hybrid Optimization Results
Trial No
Hybrid Algorithm Optimization Method
Slackest Tension [N] Tautest Tension [N] Tensioner Arm 1
Displacement [deg]
Tensioner Arm 2
Displacement [deg]
T3 T4 Θt1 Θt2
1 -2584641 4579841 -02430 67549
2 -3708747 3455736 -00023 -41068
3 -1707181 5457302 -00099 -43944
4 -269178 6895304 00006 -25366
5 -2982335 4182148 -00003 -41134
Original Design -322803 393645 16410 -4571
In Table 53a it can be seen that iterations of 16 20 21 or 53 were required for the local search
algorithm following the GA to find an optimal solution Once again the GA function
computed 4 generations which consisted of approximately 20 900 function evaluations before
securing an optimum solution
The simulation results of the non-weighted hybrid optimization approach are shown in tables
54a and 54b below
Optimization 105
Table 54a Non-Weighted Optimization Results for Twin Tensioner Parameters and Objective
Function
Trial
No
Objective
Function
Value [N]
of
Function
Evals ( of
Iterations)
kt [N∙mrad]
kt1 [N∙mrad]
kt2 [N∙mrad]
[X3Y3]
[m] [X5Y5]
[m]
Genetic Algorithm Optimization Method
1 33509e
-004 20900 (4) 321799 75530 212653 [02860 00602] [01082 01858]
Hybrid Optimization Method
1 73214e
-011 381 (13) 234881 14730 323358 [02952 00688] [00048 01466]
Original Design 20626 10314 16502 [02928 0087] [01206 00919]
Table 54b Computations for Tensions and Angles from Non-Weighted Optimizations
Trial No Slackest Tension [N] Tautest Tension [N]
Tensioner Arm 1
Displacement [deg]
Tensioner Arm 2
Displacement [deg]
T3 T4 Θt1 Θt2
Genetic Algorithm Optimization Method
1 -00003 7164479 -00588 -06213
Hybrid Optimization Method
1 -00000 7164482 15543 -16254
Original Design -322803 393645 16410 -4571
The weighted optimization data of Table 54a shows that the GA simulation again used 4
generations containing 20 900 function evaluations to conduct a global search for the optimal
system While the weighted Hybrid optimization used 13 iterations (consisting of 381 function
evaluations) after its GA run which used the same number of generations and function
evaluations as the GA run in the non-weighted simulations Tables 54a and 54b show the data
Optimization 106
for only one trial for each of the non-weighted GA and hybrid methods since only a single
optimal point exists in this case
533 Discussion
The optimal design from each search method can be selected based on the least amount of
additional pre-tension (corresponding to the largest magnitude of negative tension) that would
need to be added to the system This is in harmony with the goal of the optimization of the B-
ISG system as stated earlier to minimize the static tension for the tautest span and at the same
time minimize the absolute static tension of the slackest span for the ISG driving case As well
the angular displacements corresponding to each trial‟s results show that the Twin Tensioner is
under static stability Therein the optimal solution may be selected as the design parameters
corresponding to Trial 4 of the GA simulations to Trial 4 of the Hybrid simulations or to either
of the trials for the non-weighted optimization simulations
Given the ability of the Hybrid optimization to refine the results obtained in the GA
optimization the results of Trial 4 of the Hybrid simulations are selected as the most optimal
design from the weighted objective function approaches It is interesting to note that the Hybrid
case for the least slackness in belt span tension corresponds to a significantly larger number of
function evaluations than that of the remaining Hybrid cases This anomaly however does not
invalidate the other Hybrid cases since each still satisfy the design constraints Using the data
for the optimized system in Trial 4 (of the Hybrid optimization) the static tensions for the belt
spans in both of the B-ISG‟s phases of operation are as follows
Optimization 107
Table 55 Weighted Optimization Results for Static Tensions for Optimal B-ISG System with a
Twin Tensioner
Belt Span between Pulleys Crankshaft Driving Case [N] ISG Driving Case [N]
Optimized Original Optimized Original
Crankshaft ndash Air Conditioner 3926599 465848 117333 -284152
Air Conditioner ndash Tensioner 1 3540088 427197 -269178 -322803
Tensioner 1 ndash ISG 3540088 427197 -269178 -322803
ISG ndash Tensioner 2 2073813 28057 6895304 393645
Tensioner 2 ndash Crankshaft 2073813 28057 6895304 393645
Additional Pretension
Required (approximate) + 27000 +322803 + 27000 +322803
In Table 54b it is evident that the non-weighted class of optimization simulations achieves the
least amount of required pre-tension to be added to the system The computed tension results
corresponding to both of the non-weighted GA and Hybrid approaches are approximately
equivalent Therein either of their solution parameters may also be called the most optimal
design The Hybrid solution parameters are selected as the optimal design once again due to the
refinement of the GA output contained in the Hybrid approach and its corresponding belt
tensions are listed in Table 56 below
Optimization 108
Table 56 Non-Weighted Optimization Results for Static Tensions for Optimal B-ISG System
with a Twin Tensioner
Belt Span between Pulleys Crankshaft Driving Case [N] ISG Driving Case [N]
Optimized Original Optimized Original
Crankshaft ndash Air Conditioner 3891862 465848 386511 -284152
Air Conditioner ndash Tensioner 1 3505351 427197 -00000 -322803
Tensioner 1 ndash ISG 3505351 427197 -00000 -322803
ISG ndash Tensioner 2 2039076 28057 7164482 393645
Tensioner 2 ndash Crankshaft 2039076 28057 7164482 393645
Additional Pretension
Required (approximate) + 0000 +322803 + 00000 +322803
The results of the simulation experiments are limited by the following considerations
System equations are coupled so that a fixed difference remains between tautest and
slackest spans
A limited number of simulation trials have been performed
There are multiple optimal design points for the weighted optimization search methods
Remaining tensioner parameters tensioner pulley diameters and their stiffness have not
been included in the design variables for the experiments
The belt factor kb used in the modeling of the system‟s belt has been obtained
experimentally and may be open to further sources of error
Therein the conclusions obtained and interpretations of the simulation data can be limited by the
above noted comments on the optimization experiments
Optimization 109
54 Conclusion
The outcomes the trends in the experimental data and the optimal designs can be concluded
from the optimization simulations The simulation outcomes demonstrate that in all cases the
weighted optimization functions reached an identical value for the objective function whereas
the values reached for the parameters varied widely
The lowest tension values for the tautest and slackest span were achieved in Trial 5 of the GA
optimization approach In reiteration in the presence of slack spans the tension value of the
slackest span must be added to the initial static tension for the belt Therein for the former case
an amount of at least 409N would need to be added to the 300N of pre-tension already applied to
the system (see Table 34) The highest tension values for the spans were achieved in Trial 4 of
the weighted Hybrid optimization approach and in both trials of the non-weighted optimization
approaches In the former the weighted Hybrid trial the tension value achieved in the slackest
span was approximately -27N signifying that only at least 27N would need to be added to the
present pre-tension value for the system The tension of the slackest span in the non-weighted
approach was approximately 0N signifying that the minimum required additional tension is 0N
for the system
The optimized solutions for the tension values in each span show that there is consistently a fixed
difference of 716448N between the tautest and slackest span tension values as seen in Tables
52b 53b and 54b This difference is identical to the difference between the tautest and slackest
spans of the B-ISG system for the original values of the design parameters while in its ISG
mode As well the optimal stiffness parameters for the weighted Hybrid optimization case are
Optimization 110
greater than their original values except for that of the stiffness factor of tensioner arm 1
Likewise for the non-weighted Hybrid optimization case the stiffness parameters are above their
original values without exceptions The coordinates of the optimal solutions are within close
approximation to each other and also both match the regions for moderately low tension in
Regions 1 and 2 of the ISG driving case as is shown in Figures 49 410 413 and 414
The results of the non-weighted Hybrid optimization trial and Trial 4 of the weighted Hybrid
optimization simulations are selected as the most optimal designs for the B-ISG Twin Tensioner
In these designs the Twin Tensioner is shown in Table 53b and 54b to have static stability and
to maintain suitable tensions in the ISG driving phase The tensioner parameters for the optimal
designs allow for one of the lowest amounts of additional pre-tension to be added to the system
out of all the findings from the simulations which were conducted
111
CHAPTER 6 CONCLUSION
61 Summary
The primary aim of the thesis is to reduce the magnitude of static tension in the belt spans of a
Belt-driven Integrated Starter-generator (B-ISG) system by the design and investigation of a
Twin Tensioner It is established that the operating phases of the B-ISG system produced two
cases for static tension outcomes an ISG driving case and a crankshaft driving case The
approach taken in this thesis study includes the derivation of a system model for the geometric
properties as well as for the dynamic and static states of the B-ISG system The static state of a
B-ISG system with a single tensioner mechanism is highlighted for comparison with the static
state of the Twin Tensioner-equipped B-ISG system
It is observed that there is an overall reduction in the magnitudes of the static belt tensions with
the presence of a Twin Tensioner over that of a single tensioner The influences of the geometric
and stiffness properties of the Twin Tensioner affecting the static tensions in the system are
analyzed in a parametric study It is found that there is a notable change in the static tensions
produced as result of perturbations in each respective tensioner property This demonstrates
there are no reasons to not further consider a tensioner property based solely on its influence on
the B-ISG system‟s static tensions The phenomenon of higher magnitudes for static tensions in
the ISG mode of operation over that of the crankshaft mode of operation particularly in
excessively slack spans provides the motivation for optimizing the ISG case alone for static
tension The optimization method uses weighted and non-weighted approaches with genetic
algorithm (GA) and hybrid GA searches The most optimal design has been found to be one in
Conclusion 112
which the magnitude of tension in the excessively slack spans in the ISG driving case are
significantly lower than in that of the original B-ISG Twin Tensioner design
62 Conclusion
The conclusions that can be drawn from the study of a Twin Tensioner for a B-ISG system
include the following
1 The simulations of the dynamic model demonstrate that the mode shapes for the system
are greater in the ISG-phase of operation
2 It was observed in the output of the dynamic responses that the system‟s rigid bodies
experienced larger displacements when the crankshaft was driving over that of the ISG-
driving phase It was also noted that the transition speed marking the phase change from
the ISG driving to the crankshaft driving occurred before the system reached either of its
first natural frequencies
3 The magnitudes for static belt tensions as well as dynamic tensions for a B-ISG system
are consistently greater in its ISG operating phase than in its crankshaft operating phase
4 A Twin Tensioner is able to reduce the magnitudes of the static tension for the belt spans
of a B-ISG system in comparison to when only a single tensioner mechanism is present
5 The parametric study of the B-ISG system demonstrates that the slackest and tautest belt
spans decrease or increase together for either phase of operation
6 Perturbations in the Twin Tensioner‟s geometric and stiffness properties have a
significant influence on the magnitudes of the static tension of the slackest and tautest
belt spans The coordinate position of each pulley in the Twin Tensioner configuration
Conclusion 113
has the greatest influence on the belt span static tensions out of all the tensioner
properties considered
7 Optimization of the B-ISG system shows a fixed difference trend between the static
tension of the slackest and tautest belt spans for the B-ISG system
8 The values of the design variables for the most optimal system are found using a hybrid
algorithm approach The slackest span for the optimal system is significantly less slack
than that of the original design Therein less additional pretension is required to be added
to the system to compensate for slack spans in the ISG-driving phase of operation
63 Recommendation for Future Work
The investigation of the B-ISG Twin Tensioner encourages the following future work
1 The optimization of the B-ISG system with the inclusion of diametric Twin Tensioner
properties would provide a complete picture as to the highest possible performance
outcome that the Twin Tensioner is able to have with respect to the static tensions
achieved in the belt spans
2 A larger number of optimization trials using the genetic algorithm (GA) and hybrid GA
under weighted and other approaches would investigate the scope of optimal designs
available in the Twin Tensioner for the B-ISG system
3 A model of the system without the simplification of constant damping may produce
results that are more representative of realistic operating conditions of the serpentine belt
drive A finite element analysis of the Twin Tensioner B-ISG system may provide more
applicable findings
Conclusion 114
4 Investigation of the transverse motion coupled with the rotational belt motion in an
optimized B-ISG system equipped with a Twin Tensioner may also provide a closer look
at the system under realistic conditions In addition the affect of the Twin Tensioner on
transverse motion can determine whether significant improvements in the magnitudes of
static belt span tensions are still being achieved
5 The recommendation to conduct modal decoupling of the B-ISG system‟s static model is
motivated by the fixed difference trend between the slackest and tautest belt span
tensions shown in Chapter 5 The modal decoupling of the system would allow for its
matrices comprising the equations of motion to be diagonalized and therein to decouple
the system equations Modal analysis would transform the system from physical
coordinates into natural coordinates or modal coordinates which would lead to the
decoupling of system responses
6 An investigation and optimization of the dynamic belt span tensions for a B-ISG system
with a Twin Tensioner would increase understanding of the full impact of a Twin
Tensioner mechanism on the state of the B-ISG system It would be informative to
analyze the mode shapes of the first and second modes as well as the required torques of
the driving pulleys and the resulting torque of each of the tensioner arms The
observation of the dynamic belt span tensions would also give direction as to how
damping of the system may or may not be changed
7 Further comparison with the Twin Tensioner B-ISG system‟s dynamic and static states
including the Twin Tensioner‟s stability in each versus a B-ISG system with a single
tensioner would further demonstrate the improvements or dis-improvements in the Twin
Tensioner‟s performance on a B-ISG system
Conclusion 115
8 The influence of the belt properties on the dynamic and static tensions for a B-ISG
system with a Twin Tensioner can also be investigated This again will show the
evidence of improvements or dis-improvement in the Twin Tensioner‟s performance
within a B-ISG setting
9 Lastly an experimental apparatus of the B-ISG system with a Twin Tensioner can be
designed and constructed Suitable instrumentation can be set-up to measure belt span
tensions (both static and dynamic) belt motion and numerous other system qualities
This would provide extensive guidance as to finding the most appropriate theoretical
model for the system Experimental data would provide a bench mark for evaluating the
theoretical simulation results of the Twin Tensioner-equipped B-ISG system
116
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[14] National Alternative Fuels Training Consortium (NAFTC) (2005 Oct 3) Tech stuff
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[15] Green Car Congress BMW to Apply Start-Stop and Brake Regen to MINIs Up to 60
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[21] PJ Wezenbeek (Zytec Systems Ltd) D G Evans (General Motors Powertrain) D P
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(Delphi Corp) Combustion Assisted Belt-Cranking of a V-8 Engine at 12-Volts SAE
Technical Papers vol 113 pp 396-407 2004 Document no 2004-01-0569
[22] T C Firbank Mechanics of the Belt Drive International Journal of Mechanical
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[23] R L Cassidy S K Fan R S MacDonald and W F Samson Serpentine Extended Life
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[24] A G Ulsoy J E Whitesell and M D Hooven Design of Belt-Tensioner Systems for
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Transactions of the ASME vol 107 pp 282-290 July 1985
[25] R S Beikmann N C Perkins and A G Ulsoy Free Vibration of Serpentine Belt Drive
Systems Journal of Vibrations and Acoustics Transactions of the ASME vol 118 pp
406-413 1996
[26] T C Kraver G W Fan and J J Shah Complex Modal Analysis of a Flat Belt Pulley
System with Belt Damping and Coulomb-Damped Tensioner Journal of Mechanical
Design Transactions of the ASME vol 118 pp 306-311 Jun 1996
[27] R S Beikmann N C Perkins and A G Ulsoy Design and Analysis of Automotive
Serpentine Belt Drive Systems for Steady State Performance Journal of Mechanical
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[28] L Zhang and J W Zu Modal Analysis of Serpentine Belt Drive Systems Journal of
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[29] R Balaji and E M Mockensturm Dynamic analysis of a front-end accessory drive with a
decouplerisolator International Journal of Vehicle Design vol 39 pp 208-231 2005
[30] M Nouri Design Optimization and Active Control of Serpentine Belt Drive Systems with
Two-pulley Tensioners University of Toronto 2005
[31] G J Spicer (Litens Automotive Inc) Tensioner for use in eg belt drive system has
electronic actuator associated with clutch spring for engaging International
WO2005119089-A1 Jun 6 2005 2005
[32] Bando Chemical Industries Ltd and Litens Automotive GmbH About belt-type starter
system Feb 27 2002
[33] H Lemberger and R Jungjohann (Bayerische Motoren Werke AG) Tension device for an
envelope drive of a device especially a belt drive of a starter generator of an internal
combustion engine comprises a support part Europe EP1420192-A2 May 19 2004 2003
[34] P Ahner and M Ackermann (Bosch GMBH) Belt drive especially for internal
combustion engines to drive accessories in an automobile Germany DE19849886-A1
May 11 2000 1998
[35] N Freisinger K Hagemann J Sievert P Struebel and M Treusch (Daimler Chrysler AG)
Belt tensioning device for belt drive between engine and starter generator of motor
vehicle has self-aligning bearing that supports auxiliary unit and provides working force to
tensioners for tensioning belt Germany DE10324268 Dec 16 2004 2003
[36] C R Rogers (Dayco Products LLC) Offset starter generator drive system for a vehicle
engine has a dual arm pivoted tensioner United States US6942589-B2 Feb 8 2005 2002
[37] A Serkh and I Ali (Gates Corp) Internal combustion engine has belt drive system with
tensioner asymmetrically biased in direction tending to cause power transmission belt to be
under tension International WO2003038309-A1 May 8 2003 2002
References 120
[38] P J Mcvicar and C A Thurston (General Motors Corp) Belt alternator starter accessory
drive with dual tensioner United States US20060287146-A1 Dec 21 2006 2005
[39] W Petri and M Bogner (INA Schaeffler KG) Traction drive especially for driving
internal combustion engine units has arrangement for demand regulated setting of tension
consisting of unit with housing with limited rotation and pulley German DE10044645-
A1 Mar 21 2002 2000
[40] M Bogner (INA Schaeffler KG) Belt drive tensioner for a starter-generator of an IC
engine has locking system for locking tensioning element in an engine operating mode
locking system is directly connected to pivot arm follows arm control movements
German DE10159073-A1 Jun 12 2003 2001
[41] R Painta M Bogner and H Graf (INA Schaeffler KG) Traction mechanism drive esp
belt drive has belt tensioning pulley mounted on generator shaft and uncoupled from it via
freewheel to dampen load peaks Europe EP1723350-A1 Nov 22 2006 2005
[42] W Petri (INA Schaeffler KG) Drive unit for a combustion engine having a starter
generator and a belt drive has tensioner with spring and counter hydraulic force Germany
DE10359641-A1 Jul 28 2005 2003
[43] H Stief M Bogner B Hartmann T Kraft and M Schmid (INA Schaeffler KG) Traction
drive especially belt drive for short-duration driving of starter generator has tensioning
device with lever arm deflectable against restoring force and with end stop limiting
deflection travel Europe EP1738093-A1 Jan 3 2007 2005
[44] M Ulm (INA Schaeffler KG DE) Tension unit eg for drive in machine such as
combustion engine has belt or chain drive with wheels turning and connected with starter
generator and unit has two idlers arranged at clamping arm with machine stored by shock
absorber Germany DE102004012395-A1 Sep 29 2005 2004
[45] M Bogner (INA Schaeffler KG) Belt drive for starter motor-generator auxiliary assembly
has limited movement at the starter belt section tensioner roller bringing it into a dead point
position on starting the motor International WO2006108461-A1 Oct 19 2006 2006
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[46] W Guhr (Litens Automotive GMBH) Automotive motor and drive assembly includes
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[47] K Kuniaki K Masahiko H Kazuyuki I Shuichi and T Masaki (Mitsubishi Jidosha Eng
KK and Mitsubishi Motor Corp) Tension adjustment method of belt for starter generator
in vehicle involves shifting auto-tensioners between lock state and free state to adjust
tension of belt during driving of crank pulley Japan JP2005083514-A Mar 31 2005
2003
[48] Nissan Motor Co Ltd Winding gear for starting engine of hybrid motor vehicle has
tensioner tightening chain while cranking engine and slackens chain after start of engine
provided to span side of chain Japan JP3565040-B2 Sep 15 2004 1998
[49] S Sato and H Hayakawa (NTN Corp) Auto tensioner for ancillary drive belts has
cylinder nut and screw bolt in hydraulic damper mechanism provided in middle of cylinder
acting as start-up rigidity buffer component Japan JP2006189073-A Jul 20 2006 2005
[50] G Vadin-Michaud (Valeo Equip Electrique Moteur) Pulley and belt starting system for a
thermal engine for a motor vehicle Europe EP1658432 May 24 2006 2005
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2005
[52] W E Johns Notes on Motors [Electronic] 2003 [2008 June] Available at
httpwwwgizmologynetmotorshtm
[53] Litens Automotive Group Ltd DC BAS System - Conventional Start Input Profile Nov
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[54] Arnold Magnetic Technologies Corp General Motor Terminology [Electronic] pp 7
[2008 June] Available at httpwwwgrouparnoldcommtcpdfweb_motor_glossarypdf
[55] Douglas W Jones Stepping Motors University of Iowa - Department of Computer
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httpwwwcsuiowaedu~jonesstepphysicshtml
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[56] Litens Automotive Group Ltd (2004 Jan 31) FEAD software input data for test project
[57] K Deb Multi-Objective Optimization using Evolutionary Algorithms Toronto John Wiley
amp Sons Ltd 2001 pp 81-85
[58] P E McSharry (2004 May 11) Department of Engineering Science University of Oxford
[httpwwwengoxacuksamppubsgawbreppdf]
[59] The MathWorks Inc MATLAB vol 750342 (R2007b) Aug 15 2007
123
APPENDIX A
Passive Dual Tensioner Designs from Patent Literature
Figure A1 Proposed design by Bayerische Motoren Werke AG corresponding to patent nos EP1420192-A2 and DE10253450-A1
Source European Patent Office espcenet (publication nos EP1420192-A2 and DE10253450-A1 accessed May 2007) epespacenetcom [33]
Figure A1 label identification 1 ndash tightner 2 ndash belt drive
3 ndash starter generator
4 ndash internal-combustion engine
4‟ ndash crankshaft-lateral drive disk
5 ndash generator housing
6 ndash common axis of rotation
7 ndash featherspring of tiltable clamping arms
8 ndash clamping arm
9 ndash clamping arm
10 11 ndash idlers
12 12‟ ndash Zugtrum 13 13‟ ndash Leertrum
14 ndash carry-hurries 15 ndash generator wave
16 ndash bush
17 ndash absorption mechanism
18 18‟ ndash support arms
19 19‟ ndash auxiliary straight lines
20 ndash pipe
21 ndash torsion bar
22 ndash breaking through
23 ndash featherspring
24 ndash friction disk
25 ndash screw connection 26 ndash Wellscheibe
(European Patent Office May 2007) [33]
Appendix A 124
Figure A2a First of four proposed designs (and its various configurations) by Bosch GMBH
corresponding to patent no WO0026532-A1
Source European Patent Office espcenet (publication no WO0026532-A1 accessed Jun 2007)
epespacenetcom [34]
Figure A2b Second of four proposed designs by Bosch GMBH corresponding to patent no
WO0026532-A1
Source European Patent Office espcenet (publication no WO0026532-A1 accessed Jun 2007) epespacenetcom [34]
Figure A2c Third of four proposed designs (and its various configurations) by Bosch GMBH
corresponding to patent no WO0026532-A1
Source European Patent Office espcenet (publication no WO0026532-A1 accessed Jun 2007)
epespacenetcom [34]
Appendix A 125
Figure A2d Fourth of four proposed designs (and its various configurations) by Bosch GMBH
corresponding to patent no WO0026532-A1
Source European Patent Office espcenet (publication no WO0026532-A1 accessed Jun 2007)
epespacenetcom [34]
Figure A2a through to A2d label identification 10 ndash engine wheel
11 ndash [generator] 13 ndash spring
14 ndash belt
16 17 ndash tensioning pulleys
18 19 ndash springs
20 21 ndash fixed points
25ab ndash carriers of idlers
25c ndash gang bolt
(European Patent Office June 2007) [34]
Figure A3 Proposed design by Daimler Chrysler AG corresponding to patent no DE10324268-A1
Source European Patent Office espcenet (publication no DE10324268-A1 accessed May 2007)
epespacenetcom [35]
Figure A3 label identification
Appendix A 126
10 12 ndash belt pulleys
14 ndash auxiliary unit
16 ndash belt
22-1 22-2 ndash belt tensioners
(European Patent Office May 2007) [35]
Figure A4 Proposed design by Dayco Products LLC corresponding to patent no US6942589-B2
Source European Patent Office espcenet (publication no US6942589-B2 accessed Jun 2007)
epespacenetcom [36]
Figure A4 label identification 12 ndash belt
14 ndash tensioner
16 ndash generator pulley
18 ndash crankshaft pulley
22 ndash slack span 24 ndash tight span
32 34 ndash arms
33 35 ndash pulley
(European Patent Office June 2007) [36]
Appendix A 127
Figure A5 Proposed design by Gates Corp corresponding to patent no WO2003038309-A
Source European Patent Office espcenet (publication no WO2003038309-A accessed Jun 2007)
epespacenetcom [37]
Figure A5 label identification 12 ndash motorgenerator
14 ndash motorgenerator pulley 26 ndash belt tensioner
28 ndash belt tensioner pulley
30 ndash transmission belt
(European Patent Office June 2007) [37]
Figure A6 Proposed design by General Motors Corp corresponding to patent no US20060287146-A1
Source European Patent Office espcenet (publication no US20060287146-A1 accessed Jun 2007)
epespacenetcom [38]
Appendix A 128
Figure A6 label identification 28 ndash tensioner
32 ndash carrier arm
34 ndash secondary carrier arm
46 ndash tensioner pulley
58 ndash secondary tensioner pulley
(European Patent Office June 2007) [38]
Figure A7 Proposed design by INA Schaeffler KG corresponding to patent no DE10044645-A1
Source European Patent Office espcenet (publication no DE10044645-A1 accessed Jun 2007)
epespacenetcom [39]
Figure A7 label identification 2 ndash internal combustion engine
3 ndash traction element
11 ndash housing with limited rotation 12 13 ndash direction changing pulleys
(European Patent Office June 2007) [39]
Appendix A 129
Figure A8a First of three proposed designs by INA Schaeffler KG corresponding to patent no
DE10159073-A1
Source European Patent Office espcenet (publication no DE10159073-A1 accessed May 2007)
epespacenetcom [40]
Figure A8b Second of three proposed designs by INA Schaeffler KG corresponding to patent no
DE10159073-A1
Source European Patent Office espcenet (publication no DE10159073-A1 accessed May 2007)
epespacenetcom [40]
Appendix A 130
Figure A8c Third of three proposed designs by INA Schaeffler KG corresponding to patent no
DE10159073-A1
Source European Patent Office espcenet (publication no DE10159073-A1 accessed May 2007)
epespacenetcom [40]
Figure A8a A8b and A8c label identification 1 ndash tightener [tensioner]
2 ndash idler
3 ndash drawing means
4 ndash swivel arm
5 ndash axis of rotation
6 ndash drawing means impulse [belt]
7 ndash crankshaft
8 ndash starter generator
9 ndash bolting volume 10a ndash bolting device system
10b ndash bolting device system
10c ndash bolting device system
11 ndash friction body
12 ndash lateral surface
13 ndash bolting tape end
14 ndash bolting tape end
15 ndash control member
16 ndash torsion bar
17 ndash base
18 ndash pylon
19 ndash hub
20 ndash annular gap
21 ndash Gleitlagerbuchse
23 ndash [nil]
23 ndash friction disk
24 ndash turning camps 25 ndash teeth
26 ndash elbow levers
27 ndash clamping wedge
28 ndash internal contour
29 ndash longitudinal guidance
30 ndash system
31 ndash sensor
32 ndash clamping gap
(European Patent Office May 2007) [40]
Appendix A 131
Figure A9 Proposed design by INA Schaeffler KG corresponding to patent no DE10359641-A1
Source European Patent Office espcenet (publication no DE10359641-A1 accessed Jun 2007)
epespacenetcom [42]
Figure A9 label identification 8 ndash cylinder
10 ndash rod
12 ndash spring plate
13 ndash spring
14 ndash pressure lead
(European Patent Office June 2007) [42]
Appendix A 132
Figure A10 Proposed design by INA Schaeffler KG corresponding to patent no EP1723350-A1
Source European Patent Office espcenet (publication no EP1723350-A1 accessed Jun 2007) epespacenetcom [41]
Figure A10 label identification 4 ndash pulley
5 ndash hydraulic element 11 ndash freewheel
12 ndash shaft
(European Patent Office June 2007) [41]
Figure A11 Proposed design by INA Schaeffler KG corresponding to patent no EP1738093-A1
Source European Patent Office espcenet (publication no EP1738093-A1 accessed Jun 2007)
epespacenetcom [43]
Figure A11 label identification 1 ndash traction drive
2 ndash traction belt
3 ndash starter generator
Appendix A 133
7 ndash tension device
9 ndash lever arm
10 ndash guide roller
16 ndash end stop
(European Patent Office June 2007) [43]
Figure A12 Proposed design by INA Schaeffler KG corresponding to patent no DE102004012395-A1
Source European Patent Office espcenet (publication no DE102004012395-A1 accessed May 2007) epespacenetcom [44]
Figure A12 label identification 1 ndash belt drive
2 ndash belts
3 ndash wheel of the internal-combustion engine
4 ndash wheel of a Nebenaggregats
5 ndash wheel of the starter generator
6 ndash clamping unit
7 ndash idler
8 ndash idler
9 ndash scale beams
10 ndash drive place
11 ndash clamping arm
12 ndash camps
13 ndash coupling point
14 ndash shock absorber element
15 ndash arrow
(European Patent Office May 2007) [44]
Appendix A 134
Figure A13 Proposed design by INA Schaeffler KG corresponding to patent nos DE102005017038-A1and WO2006108461-A1
Source European Patent Office espcenet (publication nos DE102005017038-A1and WO2006108461-A1 accessed May 2007) epespacenetcom [45]
Figure A13 label identification 1 ndash belt
2 ndash wheel of the crankshaft KW
3 ndash wheel of a climatic compressor AC
4 ndash wheel of a starter generator SG
5 ndash wheel of a water pump WP
6 ndash first clamping system
7 ndash first tension adjuster lever arm
8 ndash first tension adjuster role
9 ndash second clamping system
10 ndash second tension adjuster lever arm
11 ndash second tension adjuster role 12 ndash guide roller
13 ndash drive-conditioned Zugtrum
(generatorischer enterprise (GE))
13 ndash starter-conditioned Leertrum
(starter enterprise (SE))
14 ndash drive-conditioned Leertrum (GE)
14 ndash starter-conditioned Zugtrum (SE)
14a ndash drive-conditioned Leertrumast (GE)
14a ndash starter-conditioned Zugtrumast (SE)
14b ndash drive-conditioned Leertrumast (GE)
14b ndash starter-conditioned Zugtrumast (SE)
(European Patent Office May 2007) [45]
Figure A14 Proposed design by Litens Automotive GMBH et al corresponding to patent no
US20010007839-A1
Appendix A 135
Source European Patent Office espcenet (publication no US20010007839-A1 accessed Jun 2007)
epespacenetcom [46]
Figure A14 label identification E - belt
K - crankshaft
R1 ndash first tension pulley
R2 ndash second tension pulley
S ndash tension device
T ndash drive system
1 ndash belt pulley
4 ndash belt pulley
(European Patent Office June 2007) [46]
Figure A15 Proposed design by Mitsubishi Jidosha Eng KK and Mitsubishi Motor Corp corresponding
to patent no JP2005083514-A
Source Industrial Property Digital Library and Japanese Patent Office Patent amp Utility Model Gazette DB (document no A 2005-083514 accessed May 2007) wwwipdlinpitgojp [47]
Figure A15 label identification 1 ndash Pulley for Starting
2 ndash Crank Pulley
3 ndash AC Pulley
4a ndash 1st roller
4b ndash 2nd roller
5 ndash Idler Pulley
6 ndash Belt
7c ndash Starter generator control section
7d ndash Idle stop control means
8 ndash WP Pulley
9 ndash IG Switch
10 ndash Engine
11 ndash Starter Generator
12 ndash Driving Shaft
Appendix A 136
7 ndash ECU
7a ndash 1st auto tensioner control section (the 1st auto
tensioner control means)
7b ndash 2nd auto tensioner control section (the 2nd auto
tensioner control means)
13 ndash Air-conditioner Compressor
14a ndash 1st auto tensioner
14b ndash 2nd auto tensioner
18 ndash Water Pump
(Industrial Property Digital Library May 2007) [47]
Figure A16 Proposed design by Nissan Motor Co Ltd corresponding to patent no JP3565040-B2
Source European Patent Office espcenet (publication no JP3565040-B2 accessed Jun 2007) epespacenetcom [48]
Figure A16 label identification 3 ndash chain [or belt]
5 ndash tensioner
4 ndash belt pulley
(European Patent Office June 2007) [48]
Figure A17 Proposed design by NTN Corp corresponding to patent no JP2006189073-A
Appendix A 137
Source European Patent Office espcenet (publication no JP2006189073-A accessed Jun 2007)
epespacenetcom [49]
Figure A17 label identification 5d - flange
6 ndash tensile strength spring
10 ndash actuator
10c ndash cylinder
12 ndash rod
20 ndash hydraulic damper mechanism
21 ndash cylinder nut
22 ndash screw bolt
(European Patent Office June 2007) [49]
Figure A18 Proposed design by Valeo Equipment Electriques Moteur corresponding to patent nos
EP1658432 and WO2005015007
Source European Patent Office espcenet (publication nos EP1658432 and WO2005015007
accessed May 2007) epespacenetcom [50]
Figure A18 abbreviated list of label identifications
10 ndash starter
22 ndash shaft section
23 ndash free front end
80 ndash pulley
200 ndash support element
206 - arm
(European Patent Office May 2007) [50]
The author notes that the list of labels corresponding to Figures A1a through to A7 are generated
from machine translations translated from the documentrsquos original language (ie German)
Consequently words may be translated inaccurately or not at all
138
APPENDIX B
B-ISG Serpentine Belt Drive with Single Tensioner
Equation of Motion
The equations of motion (EOM) for a B-ISG serpentine belt drive with a single tensioner are
shown The EOM has been derived similarly to that of the same system with a twin tensioner
that was provided in Chapter 3 The assumptions for the twin tensioner B-ISG system are
applicable for the single tensioner B-ISG system as well
Figure B1 shows the B-ISG system with a single tensioner pulley and arm The pulleys are
numbered 1 through 4 and their associated belt spans are numbered accordingly
Figure B1 Single Tensioner B-ISG System
Appendix B 139
The free-body diagram for the ith non-tensioner pulley in the system shown above is found in
Figure B2 The moment of inertia for the ith pulley is designated as Ii while the angular
displacement velocity and acceleration is designated as 120579119905119894 120579 119905119894 and 120579 119905119894 respectively The
required torque is Qi the angular damping is Ci and the tension of the ith span is Ti
Figure B2 Free-body Diagram of ith Pulley
The positive motion designated is assumed to be in the clockwise direction The radius for the
ith pulley is represented by Ri The equilibrium equations for the ith pulley are as follows
I1 ∙ θ 1 = T4 ∙ R1 minus T1 ∙ R1 + Q1 minus c1 ∙ θ 1 (B1)
I2 ∙ θ 2 = T1 ∙ R2 minus T2 ∙ R2 + Q2 minus c2 ∙ θ 2 (B2)
I3 ∙ θ 3 = T2 ∙ R3 minus T3 ∙ R3 + Q3 minus c3 ∙ θ 3 (B3)
Appendix B 140
A free-body diagram for the single tensioner pulley is shown in Figure B3 The rotational
stiffness and damping for the tensioner arm is designated as kt and ct respectively The angular
rotation and velocity for the arm is 120579119905119894 and 120579 119905119894 respectively
Figure B3 Free-body Diagram of Single Tensioner
From figure B2 the equations of equilibrium are resolved for the tensioner pulley The angle of
orientation for the ith belt span is designated by 120573119895119894
minusI4 ∙ θ 4 = minusT3 ∙ R4 + T4 ∙ R4 + Q4 + c4 ∙ θ 4 (B4)
It ∙ θ t = minusTt ∙ Lt ∙ sin θto minus βj4 + sin θto 1 minus βj5 minus kt ∙ partθt minus ct ∙ partθ t
(B5)
Appendix B 141
partθt = θt minus θto (B6)
The dynamic tension matrix Trsquo is proportional to the damping (Tc) and stiffness (Tk) matrices
that are due to belt damping (119888119894119887 ) and belt stiffness (119896119894
119887 ) respectively
119827prime = 119827119836 ∙ 120521 + 119827119844 ∙ 120521 (B7)
The initial tension is represented by To and the initial angle of the tensioner arm is represented
by 120579119905119900 The equation for the tension of the ith span is shown in the following equations
T1 = To + k1b ∙ R1 ∙ θ1 minus R2 ∙ θ2 + c1
b ∙ [R1 ∙ θ 1 minus R2 ∙ θ 2] (B8)
T2 = To + k2b ∙ R2 ∙ θ2 minus R3 ∙ θ3 + c2
b ∙ [R2 ∙ θ 2 minus R3 ∙ θ 3)] (B9)
T3 = To + k3b ∙ R3 ∙ θ3 minus R4 ∙ θ4 + Lt ∙ [sin θto minus βj3 ] ∙ (θt minus θto ) + c3
b ∙ [R3 ∙ θ 3 minus R4 ∙
θ 4 + Lt ∙ [sin θto minus βj3 ] ∙ (θ t)] (B10)
T4 = To + k4b ∙ R4 ∙ θ4 minus R1 ∙ θ1 + Lt ∙ [sin θto minus βj4 ] ∙ (θt minus θto ) + c4
b ∙ [R4 ∙ θ 4 minus R1 ∙
θ 1 + Lt ∙ [sin θto minus βj4 ] ∙ (θ t)] (B11)
Tprime = Ti minus To (B12)
Tt = T3 = T4 (B13)
Appendix B 142
The EOM for the single tensioner B-ISG system is found by substitution of equations B8 to
B13 into B1 to B5 The matrices in the EOM include the inertial matrix I damping matrix C
stiffness matrix K and the required torque matrix Q as well as the angular displacement
velocity and acceleration matrices 120521 120521 and 120521 respectively
119816 ∙ 120521 + 119810 ∙ 120521 + 119818 ∙ 120521 = 119824 (B14)
119816 =
I1 0 0 0 00 I2 0 0 00 0 I3 0 00 0 0 I4 00 0 0 0 It1
(B15)
The stiffness matrix includes kb the belt factor Kb the belt cord stiffness 120601119894 the wrap angle of
the belt on the ith pulley and Kbi the stiffness factor of the ith belt span Cb represents the belt
damping for each span and βji is the angle of orientation for the span between the jth and ith
pulleys It is noted in the terms of the stiffness and damping matrices below that the numerical
subscripts refer to the (i+1)th pulley The term Qt may be found within the required torque
matrix and represents the required torque for the tensioner arm As well the term It1 represents
the moment of inertia for the tensioner arm
Appendix B 143
K =
(B16)
Kbi =Kb
Li + kb ∙ Ri ∙ϕi+1
2 + Ri ∙ϕi
2
(B17)
C =
(B18)
Appendix B 144
Appendix B 144
120521 =
θ1
θ2
θ3
θ4
partθt
(B19)
119824 =
Q1
Q2
Q3
Q4
Qt
(B20)
Simulations of the EOM for the B-ISG system with a single tensioner were performed in FEAD
[51] software for dynamic and static cases This allowed for the methodology for deriving the
EOM to be verified and then applied to the B-ISG system with a twin tensioner The natural
frequencies modes shapes dynamic responses tensioner arm torques as well as the crankshaft
required torque only and the dynamic tensions were solved from the EOM as described in (331)
to (339) of Chapter 3 and as well as for the static tension from (351) to (353) of Chapter 3
This permitted verification of the complete derivation methodology and allowed for comparison
of the static tension of the B-ISG system belt spans in the case that a single tensioner is present
and in the case that a Twin Tensioner is present [51]
145
APPENDIX C
MathCAD Scripts
C1 Geometric Analysis
1 - CS
2 - AC
4 - Alt
3 - Ten1
5 - Ten 2
6 - Ten Pivot
1
2
3
4
5
Figure C1 Schematic of B-ISG
System with Twin Tensioner
Coordinate Input DataXY1 0 0( ) XY4 24759 16664( )
XY2 224 6395( ) XY5 12057 9193( )
XY3 292761 87( ) XY6 201384 62516( )
Computations
Lt1 XY30 0
XY60 0
2
XY30 1
XY60 1
2
Lt2 XY50 0
XY60 0
2
XY50 1
XY60 1
2
t1 atan2 XY30 0
XY60 0
XY30 1
XY60 1
t2 atan2 XY50 0
XY60 0
XY50 1
XY60 1
XY
XY10 0
XY20 0
XY30 0
XY40 0
XY50 0
XY60 0
XY10 1
XY20 1
XY30 1
XY40 1
XY50 1
XY60 1
x XY
0 y XY
1
Appendix C 146
i - angle bw horizontal and l ine from ith pulley center to (i+1)th pulley center
Adjust last number in range variable p to correspond to number of pulleys
p 0 1 4
k p( ) p 1( ) p 4if
0 otherwise
condition1 p( ) acos
XYk p( ) 0
XYp 0
XYk p( ) 0
XYp 0
2
XYk p( ) 1
XYp 1
2
condition2 p( ) 2 acos
XYk p( ) 0
XYp 0
XYk p( ) 0
XYp 0
2
XYk p( ) 1
XYp 1
2
p( ) if XYk p( ) 1
XYp 1
condition1 p( ) condition2 p( )
Lfi Lbi - connection belt span lengths
D1 20065mm D2 10349mm D3 7240mm D4 6820mm D5 7240mm
D
D1
D2
D3
D4
D5
Lf p( ) XYk p( ) 0
XYp 0
2
XYk p( ) 1
XYp 1
2
1
mm
Dk p( )
2
Dp
2
2
Lb p( ) XYk p( ) 0
XYp 0
2
XYk p( ) 1
XYp 1
2
1
mm
Dk p( )
2
Dp
2
2
fi bi - angle bw ith pulley center connection l ine and contact points Pbfi (or Pfbi) and Pbi
(or Pfi) respecti vely l
f p( ) atanLf p( ) mm
Dp
2
Dk p( )
2
Dp
Dk p( )
if
atanLf p( ) mm
Dk p( )
2
Dp
2
Dp
Dk p( )
if
2D
pD
k p( )if
b p( ) atan
mmLb p( )
Dp
2
Dk p( )
2
Appendix C 147
XYfi XYbi XYfbi XYbfi - 4 possible contact points for i th pulley
XYf p( ) XYp 0
Dp
2 mmcos p( ) f p( )
XYp 1
Dp
2 mmsin p( ) f p( )
XYb p( ) XYp 0
Dp
2 mmcos p( ) f p( )
XYp 1
Dp
2 mmsin p( ) f p( )
XYfb p( ) XYp 0
Dp
2 mmcos p( ) b p( )
XYp 1
Dp
2 mmsin p( ) b p( )
XYbf p( ) XYp 0
Dp
2 mmcos p( ) b p( )
XYp 1
Dp
2 mmsin p( ) b p( )
XYfi+1 XYbi+1 XYfbi+1 XYbfi+1 - 4 possible contact points for i+1th pulley
XYf2 p( ) XYk p( ) 0
Dk p( )
2 mmcos p( ) f p( )
XYk p( ) 1
Dk p( )
2 mmsin p( ) f p( )
XYb2 p( ) XYk p( ) 0
Dk p( )
2 mmcos p( ) f p( )
XYk p( ) 1
Dk p( )
2 mmsin p( ) f p( )
XYfb2 p( ) XYk p( ) 0
Dk p( )
2 mmcos p( ) b p( ) XY
k p( ) 1
Dk p( )
2 mmsin p( ) b p( )
XYbf2 p( ) XYk p( ) 0
Dk p( )
2 mmcos p( ) b p( ) XY
k p( ) 1
Dk p( )
2 mmsin p( ) b p( )
Row 1 --gt Pulley 1 Row i --gt Pulley i
XYfi
XYf 0( )0 0
XYf 1( )0 0
XYf 2( )0 0
XYf 3( )0 0
XYf 4( )0 0
XYf 0( )0 1
XYf 1( )0 1
XYf 2( )0 1
XYf 3( )0 1
XYf 4( )0 1
XYfi
6818
269222
335325
251552
108978
100093
89099
60875
200509
207158
x1 XYfi0
y1 XYfi1
Appendix C 148
XYbi
XYb 0( )0 0
XYb 1( )0 0
XYb 2( )0 0
XYb 3( )0 0
XYb 4( )0 0
XYb 0( )0 1
XYb 1( )0 1
XYb 2( )0 1
XYb 3( )0 1
XYb 4( )0 1
XYbi
47054
18575
269403
244841
164847
88606
291
30965
132651
166182
x2 XYbi0
y2 XYbi1
XYfbi
XYfb 0( )0 0
XYfb 1( )0 0
XYfb 2( )0 0
XYfb 3( )0 0
XYfb 4( )0 0
XYfb 0( )0 1
XYfb 1( )0 1
XYfb 2( )0 1
XYfb 3( )0 1
XYfb 4( )0 1
XYfbi
42113
275543
322697
229969
9452
91058
59383
75509
195834
177002
x3 XYfbi0
y3 XYfbi1
XYbfi
XYbf 0( )0 0
XYbf 1( )0 0
XYbf 2( )0 0
XYbf 3( )0 0
XYbf 4( )0 0
XYbf 0( )0 1
XYbf 1( )0 1
XYbf 2( )0 1
XYbf 3( )0 1
XYbf 4( )0 1
XYbfi
8384
211903
266707
224592
140427
551
13639
50105
141463
143331
x4 XYbfi0
y4 XYbfi1
Row 1 --gt Pulley 2 Row i --gt Pulley i+1
XYf2i
XYf2 0( )0 0
XYf2 1( )0 0
XYf2 2( )0 0
XYf2 3( )0 0
XYf2 4( )0 0
XYf2 0( )0 1
XYf2 1( )0 1
XYf2 2( )0 1
XYf2 3( )0 1
XYf2 4( )0 1
XYf2x XYf2i0
XYf2y XYf2i1
XYb2i
XYb2 0( )0 0
XYb2 1( )0 0
XYb2 2( )0 0
XYb2 3( )0 0
XYb2 4( )0 0
XYb2 0( )0 1
XYb2 1( )0 1
XYb2 2( )0 1
XYb2 3( )0 1
XYb2 4( )0 1
XYb2x XYb2i0
XYb2y XYb2i1
Appendix C 149
XYfb2i
XYfb2 0( )0 0
XYfb2 1( )0 0
XYfb2 2( )0 0
XYfb2 3( )0 0
XYfb2 4( )0 0
XYfb2 0( )0 1
XYfb2 1( )0 1
XYfb2 2( )0 1
XYfb2 3( )0 1
XYfb2 4( )0 1
XYfb2x XYfb2i
0
XYfb2y XYfb2i1
XYbf2i
XYbf2 0( )0 0
XYbf2 1( )0 0
XYbf2 2( )0 0
XYbf2 3( )0 0
XYbf2 4( )0 0
XYbf2 0( )0 1
XYbf2 1( )0 1
XYbf2 2( )0 1
XYbf2 3( )0 1
XYbf2 4( )0 1
XYbf2x XYbf2i0
XYbf2y XYbf2i1
100 40 20 80 140 200 260 320 380 440 500150
110
70
30
10
50
90
130
170
210
250Figure C2 Possible Contact Points
250
150
y1
y2
y3
y4
y
XYf2y
XYb2y
XYfb2y
XYbf2y
500100 x1 x2 x3 x4 x XYf2x XYb2x XYfb2x XYbf2x
Appendix C 150
Xij Yij - selected contact point on ith pulley for span from ith pulley to jth pulley
XY15 XYbf2iT 4
XY12 XYfiT 0
Pulley 1 contact pts
XY21 XYf2iT 0
XY23 XYfbiT 1
Pulley 2 contact pts
XY32 XYfb2iT 1
XY34 XYbfiT 2
Pulley 3 contact pts
XY43 XYbf2iT 2
XY45 XYfbiT 3
Pulley 4 contact pts
XY54 XYfb2iT 3
XY51 XYbfiT 4
Pulley 5 contact pts
By observation the lengths of span i is the following
L1 Lf 0( ) L2 Lb 1( ) L3 Lb 2( ) L4 Lb 3( ) L5 Lb 4( ) Li
L1
L2
L3
L4
L5
mm
i Angle between horizontal and span of ith pulley
i
atan
XY121
XY211
XY12
0XY21
0
atan
XY231
XY321
XY23
0XY32
0
atan
XY341
XY431
XY34
0XY43
0
atan
XY451
XY541
XY45
0XY54
0
atan
XY511
XY151
XY51
0XY15
0
Appendix C 151
Pulley 1 Pulley 2 Pulley 3 Pulley 4 Pulley 5
12 i0 2 21 i0 32 i1 2 43 i2 54 i3
15 i4 2 23 i1 34 i2 45 i3 51 i4
15
21
32
43
54
12
23
34
45
51
Wrap angle i for the ith pulley
1 2 atan2 XY150
XY151
atan2 XY120
XY121
2 atan2 XY210
XY1 0
XY211
XY1 1
atan2 XY230
XY1 0
XY231
XY1 1
3 2 atan2 XY320
XY2 0
XY321
XY2 1
atan2 XY340
XY2 0
XY341
XY2 1
4 atan2 XY430
XY3 0
XY431
XY3 1
atan2 XY450
XY3 0
XY451
XY3 1
5 atan2 XY540
XY4 0
XY541
XY4 1
atan2 XY510
XY4 0
XY511
XY4 1
1
2
3
4
5
Lb length of belt
Lbelt Li1
2
0
4
p
Dpp
Input Data for B-ISG System
Kt 20626Nm
rad (spring stiffness between tensioner arms 1
and 2)
Kt1 10314Nm
rad (stiffness for spring attached at arm 1 only)
Kt2 16502Nm
rad (stiffness for spring attached at arm 2 only)
Appendix C 152
C2 Dynamic Analysis
I C K moment of inertia damping and stiffness matrices respectively
u 0 1 4 v 0 1 4 (new counter variables where final value = no of pulleys + no of ten arms)
RaD
2
Appendix C 153
RaD
2
Ii =gt moment of inertia for ith pulley where i-1 and i represent ten arms
Ii0
0
1
2
3
4
5
6
10000
2230
300
3000
300
1500
1500
I diag Ii( ) kg mm2
Ci =gt Rotational damping and belt damping for the ith pulley where i-1 and i represent tensioner arms
1000kg
m3
CrossArea 693mm2
0 M CrossArea Lbelt M 0086kg
cb 2 KbM
Lbelt
Cb
cb
cb
cb
cb
cb
Cri
0
0
010
0
010
N mmsec
rad
Ct 1000N mmsec
rad Ct1 1000 N mm
sec
rad Ct2 1000N mm
sec
rad
Cr
Cri0
0
0
0
0
0
0
0
Cri1
0
0
0
0
0
0
0
Cri2
0
0
0
0
0
0
0
Cri3
0
0
0
0
0
0
0
Cri4
0
0
0
0
0
0
0
Ct Ct1
Ct
0
0
0
0
0
Ct
Ct Ct2
Rt
Ra0
Ra1
0
0
0
0
0
0
Ra1
Ra2
0
0
Lt1 mm sin t1 32
0
0
0
Ra2
Ra3
0
Lt1 mm sin t1 34
0
0
0
0
Ra3
Ra4
0
Lt2 mm sin t2 54
Ra0
0
0
0
Ra4
0
Lt2 mm sin t2 51
Appendix C 154
Kr
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Kt Kt1
Kt
0
0
0
0
0
Kt
Kt Kt2
Tk
Kbi 0( ) Ra0
0
0
0
Kbi 4( ) Ra0
Kbi 0( ) Ra1
Kbi 1( ) Ra1
0
0
0
0
Kbi 1( ) Ra2
Kbi 2( ) Ra2
0
0
0
0
Kbi 2( ) Ra3
Kbi 3( ) Ra3
0
0
0
0
Kbi 3( ) Ra4
Kbi 4( ) Ra4
0
Kbi 1( ) Lt1 mm sin t1 32
Kbi 2( ) Lt1 mm sin t1 34
0
0
0
0
0
Kbi 3( ) Lt2 mm sin t2 54
Kbi 4( ) Lt2 mm sin t2 51
Tc
Cb0
Ra0
0
0
0
Cb4
Ra0
Cb0
Ra1
Cb1
Ra1
0
0
0
0
Cb1
Ra2
Cb2
Ra2
0
0
0
0
Cb2
Ra3
Cb3
Ra3
0
0
0
0
Cb3
Ra4
Cb4
Ra4
0
Cb1
Lt1 mm sin t1 32
Cb2
Lt1 mm sin t1 34
0
0
0
0
0
Cb3
Lt2 mm sin t2 54
Cb4
Lt2 mm sin t2 51
C matrix
C Cr Rt Tc
K matrix
K Kr Rt Tk
New Equations of Motion for Dual Drive System
I K amp C matricies rearranged to place driving pulley in 1st row + 1st column and driven in 2nd row + 2nd column
IA augment I3
I0
I1
I2
I4
I5
I6
IC augment I0
I3
I1
I2
I4
I5
I6
I1kgmm2 1 106
kg m2
0 0 0 0 0 0
Ia stack I1kgmm2 IAT 0
T
IAT 1
T
IAT 2
T
IAT 4
T
IAT 5
T
IAT 6
T
Ic stack I1kgmm2 ICT 3
T
ICT 1
T
ICT 2
T
ICT 4
T
ICT 5
T
ICT 6
T
Appendix C 155
RtA augment Rt3
Rt0
Rt1
Rt2
Rt4
RtC augment Rt0
Rt3
Rt1
Rt2
Rt4
Rta stack RtAT 3
T
RtAT 0
T
RtAT 1
T
RtAT 2
T
RtAT 4
T
RtAT 5
T
RtAT 6
T
Rtc stack RtCT 0
T
RtCT 3
T
RtCT 1
T
RtCT 2
T
RtCT 4
T
RtCT 5
T
RtCT 6
T
TkA augment Tk3
Tk0
Tk1
Tk2
Tk4
Tk5
Tk6
Tka stack TkAT 3
T
TkAT 0
T
TkAT 1
T
TkAT 2
T
TkAT 4
T
TkC augment Tk0
Tk3
Tk1
Tk2
Tk4
Tk5
Tk6
Tkc stack TkCT 0
T
TkCT 3
T
TkCT 1
T
TkCT 2
T
TkCT 4
T
TcA augment Tc3
Tc0
Tc1
Tc2
Tc4
Tc5
Tc6
Tca stack TcAT 3
T
TcAT 0
T
TcAT 1
T
TcAT 2
T
TcAT 4
T
TcC augment Tc0
Tc3
Tc1
Tc2
Tc4
Tc5
Tc6
Tcc stack TcAT 0
T
TcAT 3
T
TcAT 1
T
TcAT 2
T
TcAT 4
T
Ka Kr Rta Tka Kc Kr Rtc Tkc Ca Cr Rta Tca Cc Cr Rtc Tcc
CHECK
KA augment K3
K0
K1
K2
K4
K5
K6
KC augment K0
K3
K1
K2
K4
K5
K6
CA augment C3
C0
C1
C2
C4
C5
C6
CC augment C0
C3
C1
C2
C4
C5
C6
Appendix C 156
Kacheck stack KAT 3
T
KAT 0
T
KAT 1
T
KAT 2
T
KAT 4
T
KAT 5
T
KAT 6
T
Kccheck stack KCT 0
T
KCT 3
T
KCT 1
T
KCT 2
T
KCT 4
T
KCT 5
T
KCT 6
T
Cacheck stack CAT 3
T
CAT 0
T
CAT 1
T
CAT 2
T
CAT 4
T
CAT 5
T
CAT 6
T
Cccheck stack CCT 0
T
CCT 3
T
CCT 1
T
CCT 2
T
CCT 4
T
CCT 5
T
CCT 6
T
Results for System switching from ISG as DRIVING pulley to Crankshaft as Drivi ng Pulley
Modified Submatricies for ISG Driving Phase --gt CS Driving Phase
Unit step function to provide shift from crankshaft DRIVING case (ie ISG driven case) to crankshaft DRIVEN
case (ie ISG driving case)
H n( ) 1 n 750if
0 n 750if
lt-- crankshaft DRIVING case (Phase change bw 2 cases occurs when n
reaches start speed)
I11mod n( ) Ic0 0
H n( ) 1if
Ia0 0
H n( ) 0if
I22mod n( )submatrix Ic 1 6 1 6( )
UnitsOf I( )H n( ) 1if
submatrix Ia 1 6 1 6( )
UnitsOf I( )H n( ) 0if
K11mod n( )
Kc0 0
UnitsOf K( )H n( ) 1if
Ka0 0
UnitsOf K( )H n( ) 0if
C11modn( )
Cc0 0
UnitsOf C( )H n( ) 1if
Ca0 0
UnitsOf C( )H n( ) 0if
K22mod n( )submatrix Kc 1 6 1 6( )
UnitsOf K( )H n( ) 1if
submatrix Ka 1 6 1 6( )
UnitsOf K( )H n( ) 0if
C22modn( )submatrix Cc 1 6 1 6( )
UnitsOf C( )H n( ) 1if
submatrix Ca 1 6 1 6( )
UnitsOf C( )H n( ) 0if
K21mod n( )submatrix Kc 1 6 0 0( )
UnitsOf K( )H n( ) 1if
submatrix Ka 1 6 0 0( )
UnitsOf K( )H n( ) 0if
C21modn( )submatrix Cc 1 6 0 0( )
UnitsOf C( )H n( ) 1if
submatrix Ca 1 6 0 0( )
UnitsOf C( )H n( ) 0if
K12mod n( )submatrix Kc 0 0 1 6( )
UnitsOf K( )H n( ) 1if
submatrix Ka 0 0 1 6( )
UnitsOf K( )H n( ) 0if
C12modn( )submatrix Cc 0 0 1 6( )
UnitsOf C( )H n( ) 1if
submatrix Ca 0 0 1 6( )
UnitsOf C( )H n( ) 0if
Appendix C 157
2mod n( ) I22mod n( )1
K22mod n( ) mod n( ) sort eigenvals 2mod n( ) nmod n( )mod n( )
2
EVmodn( ) augmenteigenvec 2mod n( ) mod n( )0
max eigenvec 2mod n( ) mod n( )0
eigenvec 2mod n( ) mod n( )1
max eigenvec 2mod n( ) mod n( )1
eigenvec 2mod n( ) mod n( )2
max eigenvec 2mod n( ) mod n( )2
eigenvec 2mod n( ) mod n( )3
max eigenvec 2mod n( ) mod n( )3
eigenvec 2mod n( ) mod n( )4
max eigenvec 2mod n( ) mod n( )4
eigenvec 2mod n( ) mod n( )5
max eigenvec 2mod n( ) mod n( )5
modeshapesmod n( ) stack nmod n( )T
EVmodn( )
t 0 0001 1
mode1a t( ) EVmod100( )0
sin nmod 100( )0 t mode2a t( ) EVmod100( )1
sin nmod 100( )1 t
mode1c t( ) EVmod800( )0
sin nmod 800( )0 t mode2c t( ) EVmod800( )1
sin nmod 800( )1 t
Pulley responses amp torque requirement for crankshaft amp alternator pulleys pulley1 and 4 respectively
The system equation becomes
I14q14 -double-dot + C1144 q14 -dot + K1144 q14 + C12qm-dot + K12qm = Qc
I2qm-double-dot + C22qm-dot + K22qm + C21q1-dot + K21q1 = 0
Pulley responses
Qm = - [(K22 - 2I2) + jC22 ]-1(K21 + jC21 )Q1
Torque requirement for crank shaft Pulley 1
qc = [(K11 -2I1) + jC11 ]Q1 + (K12 + jC12 )Qm
Torque requirement for alternator shaft Pulley 4
qa = [(K44 -2I4) + jC44 ]Q4 + (K12 + jC12 )Qm
Appendix C 158
Let DRIVING pulley have a unit amplitude 1 = 1 and let the system frequency be calculated based on
engine speed n
n 60 90 6000 n( )4n
60 a n( )
2n Ra0
60 Ra3
mod n( ) n( ) H n( ) 1if
a n( ) H n( ) 0if
Ymod n( ) K22mod n( ) mod n( ) 2 I22mod n( )
j mod n( ) C22modn( )
mmod n( ) Ymod n( )( )1
K21mod n( ) j mod n( ) C21modn( )
Crankshaft amp ISG required torques
Let input from DRIVING pulley be an angular displacement with constant amplitude of angular acceleration
Ac n( ) 650 1 n( )Ac n( )
n( ) 2
Let Qm = QmQ1(n) for n lt 750
and Qm = QmQ4(n) for n gt 750
Aa n( )42
I3 3
1a n( )Aa n( )
a n( ) 2
Qc0 4
qcmod n( ) K11mod n( ) mod n( ) 2
I11mod n( )
j mod n( ) C11modn( )
1 n( ) K12mod n( ) j mod n( ) C12modn( ) mmod n( ) 1 n( )
H n( ) 1if
Qc0 H n( ) 0if
qamod n( ) K11mod n( ) mod n( ) 2
I11mod n( )
j mod n( ) C11modn( )
1a n( ) K12mod n( ) jmod n( ) C12modn( ) mmod n( ) 1a n( ) Qc0
H n( ) 0if
0 H n( ) 1if
Q n( ) 48 n
Ra0
Ra3
48
18000
(ISG torque requirement alternate equation)
Appendix C 159
Dynamic tensioner arm torques
Qtt1mod n( )Kt Kt1
UnitsOf Kt( )j mod n( )
Ct Ct1
UnitsOf Cr( )
mmod n( )4 1 n( )
H n( ) 1if
Kt Kt1
UnitsOf Kt( )j mod n( )
Ct Ct1
UnitsOf Cr( )
mmod n( )4 1a n( )
H n( ) 0if
Qtt2mod n( )Kt Kt2
UnitsOf Kt( )j mod n( )
Ct Ct2
UnitsOf Cr( )
mmod n( )5 1 n( )
H n( ) 1if
Kt Kt2
UnitsOf Kt( )j mod n( )
Ct Ct2
UnitsOf Cr( )
mmod n( )5 1a n( )
H n( ) 0if
Appendix C 160
Dynamic belt span tensions
d n( ) 1 n( ) H n( ) 1if
1a n( ) H n( ) 0if
mod n( )
d n( )
mmod n( ) d n( ) 0 0
mmod n( ) d n( ) 1 0
mmod n( ) d n( ) 2 0
mmod n( ) d n( ) 3 0
mmod n( ) d n( ) 4 0
mmod n( ) d n( ) 5 0
Tm n( ) j n( )Tcc
UnitsOf Tcc( )
Tkc
UnitsOf Tkc( )
mod n( )
H n( ) 1if
j n( )Tca
UnitsOf Tca( )
Tka
UnitsOf Tka( )
mod n( )
H n( ) 0if
Tm n( ) j n( )Tcc
UnitsOf Tcc( )
Tkc
UnitsOf Tkc( )
mod n( )
H n( ) 1if
j n( )Tca
UnitsOf Tca( )
Tka
UnitsOf Tka( )
mod n( )
H n( ) 0if
(tensions for driving pulley belt spans)
Appendix C 161
C3 Static Analysis
Static Analysis using K Tk amp Q matricies amp Ts
For static case K = Q
Tension T = T0 + Tks
Thus T = K-1QTks + T0
Q1 68N m Qt1 0N m Qt2 0N m Ts 300N
Qc
Q4
Q2
Q3
Q5
Qt1
Qt2
Qc
5
2
0
0
0
0
J Qa
Q1
Q2
Q3
Q5
Qt1
Qt2
Qa
68
2
0
0
0
0
N m
cK22mod 900( )( )
1
N mQc A
K22mod 600( )1
N mQa
a
A0
A1
A2
0
A3
A4
A5
0
c1
c2
c0
c3
c4
c5
Tc Tk Ts Ta Tk a Ts
162
APPENDIX D
MATLAB Functions amp Scripts
D1 Parametric Analysis
D11 TwinMainm
The following function script performs the parametric analysis for the B-ISG system with a Twin
Tensioner It calls the function TwinTenStaticTensionm The parametric analysis perturbs a
single input parameter for the called function TwinTenStaticTensionm The main function takes
an initial input value for the Twin Tensioner‟s stiffness parameters Kto Kt1o Kt2o and
geometric parameters D3o D5o X3o Y3o X5o and Y5o An input parameter is allowed to
increment by six percent over a range from sixty percent below its initial value to sixty percent
above its initial value The coordinate parameters are incremented through a mesh of Cartesian
points with prescribed boundaries The TwinMainm function plots the parametric results
______________________________________________________________________________
clc
clear all
Static tension for single tensioner system for CS and Alt driving
Initial Conditions
Kto = 20626
Kt1o = 10314
Kt2o = 16502
D3o = 007240
D5o = 007240
X3o =0292761
Y3o =087
X5o =12057
Y5o =09193
Pertubations of initial parameters
Kt = (Kto-060Kto)006Kto(Kto+060Kto)
Kt1 = (Kt1o-060Kt1o)006Kt1o(Kt1o+060Kt1o)
Kt2 = (Kt2o-060Kt2o)006Kt2o(Kt2o+060Kt2o)
D3 = (D3o-060D3o)006D3o(D3o+060D3o)
D5 = (D5o-060D5o)006D5o(D5o+060D5o)
No of data points
s = 21
T = zeros(5s)
Ta = zeros(5s)
Parametric Plots
for i = 1s
Appendix D 163
[T(i)Ta(i)] = TwinTenStaticTension(Kt(i)Kt1oKt2oD3oD5oX3oY3oX5oY5o)
end
figure
[AXH1H2] = plotyy(Kt()T(1)Kt()Ta(4)plot) hold on
H3 = line(Kt()T(5)ParentAX(1)) hold on
H4 = line(Kt()Ta(3)Parent AX(2)) hold off
set(AXYLimModeautoYTickModeauto)
title(Static Tension vs Coupled Stiffness bw Arms 1 amp 2)
xlabel(Kt (Nmrad))
set(get(AX(1)Ylabel)StringCS Span Tension (N))
set(get(AX(2)Ylabel)StringISG Span Tension (N))
set(H1LineStyle-Colorb)
set(H2LineStyle-Colorg)
set(H3LineStyleColorget(H1Color))
set(H4LineStyleColorget(H2Color))
legend([H1H2H3H4]CS Tautest SpanISG Tautest SpanCS Slackest SpanISG
Slackest SpanLocationBest)
for i = 1s
[T(i)Ta(i)] = TwinTenStaticTension(KtoKt1(i)Kt2oD3oD5oX3oY3oX5oY5o)
end
figure
[AXH1H2] = plotyy(Kt1()T(1)Kt1()Ta(4)plot) hold on
H3 = line(Kt1()T(5)ParentAX(1)) hold on
H4 = line(Kt1()Ta(3)Parent AX(2)) hold off
set(AXYLimModeautoYTickModeauto)
title(Static Tension vs Tensioner Arm 1 Stiffness)
xlabel(Kt1 (Nmrad))
set(get(AX(1)Ylabel)StringCS Span Tension (N))
set(get(AX(2)Ylabel)StringISG Span Tension (N))
set(H1LineStyle-Colorb)
set(H2LineStyle-Colorg)
set(H3LineStyleColorget(H1Color))
set(H4LineStyleColorget(H2Color))
legend([H1H2H3H4]CS Tautest SpanISG Tautest SpanCS Slackest SpanISG
Slackest SpanLocationBest)
for i = 1s
[T(i)Ta(i)] = TwinTenStaticTension(KtoKt1oKt2(i)D3oD5oX3oY3oX5oY5o)
end
figure
[AXH1H2] = plotyy(Kt2()T(1)Kt2()Ta(4)plot) hold on
H3 = line(Kt2()T(5)ParentAX(1)) hold on
H4 = line(Kt2()Ta(3)Parent AX(2)) hold off
set(AXYLimModeautoYTickModeauto)
title(Static Tension vs Tensioner Arm 2 Stiffness)
xlabel(Kt2 (Nmrad))
set(get(AX(1)Ylabel)StringCS Span Tension (N))
Appendix D 164
set(get(AX(2)Ylabel)StringISG Span Tension (N))
set(H1LineStyle-Colorb)
set(H2LineStyle-Colorg)
set(H3LineStyleColorget(H1Color))
set(H4LineStyleColorget(H2Color))
legend([H1H2H3H4]CS Tautest SpanISG Tautest SpanCS Slackest SpanISG
Slackest SpanLocationBest)
for i = 1s
[T(i)Ta(i)] = TwinTenStaticTension(KtoKt1oKt2oD3(i)D5oX3oY3oX5oY5o)
end
figure
[AXH1H2] = plotyy(D3()T(1)D3()Ta(4)plot) hold on
H3 = line(D3()T(5)ParentAX(1)) hold on
H4 = line(D3()Ta(3)Parent AX(2)) hold off
set(AXYLimModeautoYTickModeauto)
title(Static Tension vs Tensioner Pulley 1 Diameter)
xlabel(D3 (m))
set(get(AX(1)Ylabel)StringCS Span Tension (N))
set(get(AX(2)Ylabel)StringISG Span Tension (N))
set(H1LineStyle-Colorb)
set(H2LineStyle-Colorg)
set(H3LineStyleColorget(H1Color))
set(H4LineStyleColorget(H2Color))
legend([H1H2H3H4]CS Tautest SpanISG Tautest SpanCS Slackest SpanISG
Slackest SpanLocationBest)
for i = 1s
[T(i)Ta(i)] = TwinTenStaticTension(KtoKt1oKt2oD3oD5(i)X3oY3oX5oY5o)
end
figure
[AXH1H2] = plotyy(D5()T(1)D5()Ta(4)plot) hold on
H3 = line(D5()T(5)ParentAX(1)) hold on
H4 = line(D5()Ta(3)Parent AX(2)) hold off
set(AXYLimModeautoYTickModeauto)
title(Static Tension vs Tensioner Pulley 2 Diameter)
xlabel(D5 (m))
set(get(AX(1)Ylabel)StringCS Span Tension (N))
set(get(AX(2)Ylabel)StringISG Span Tension (N))
set(H1LineStyle-Colorb)
set(H2LineStyle-Colorg)
set(H3LineStyleColorget(H1Color))
set(H4LineStyleColorget(H2Color))
legend([H1H2H3H4]CS Tautest SpanISG Tautest SpanCS Slackest SpanISG
Slackest SpanLocationBest)
Mesh points
m = 101
n = 101
Appendix D 165
T = zeros(5nm)
Ta = zeros(5nm)
[ixxiyy] = meshgrid(1m1n)
minX3 = 0260200
maxX3 = 0317677
minY3 = -0056640
maxY3 = 0228456
midX3 = 0311641
X3 = minX3 + (ixx-1)(maxX3-minX3)(m-1)
Y3 = minY3 + (iyy-1)(maxY3-minY3)(n-1)
for i = 1n
for j = 1m
if ((X3(ij)lt midX3)ampamp(Y3(ij)gt=(sqrt((0087945^2)-((X3(ij)-0224)^2)-
006395)))ampamp(Y3(ij)lt=(-1sqrt(((00703^2)-((X3(ij)-
024759)^2)))+016664)))||((X3(ij)gt=midX3)ampamp(Y3(ij)gt=(35548X3(ij)-
11134868))ampamp(Y3(ij)lt=(-1(sqrt(((00703^2)-((X3(ij)-024759)^2))))+016664))) mx+b
lt= y lt= circle4
[T(ij)Ta(ij)] =
TwinTenStaticTension(KtoKt1oKt2oD3oD5oX3(ij)Y3(ij)X5oY5o)
else
T(ij) = zeros(511)
Ta(ij) = zeros(511)
end
end
end
figure
Z1 = squeeze(T(1))
surf(X3Y3real(Z1))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(CS DRIVING Tautest Span Tension vs Tensioner Pulley 1 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(CS Span Tension (N))
figure
Z5 = squeeze(T(5))
surf(X3Y3real(Z5))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(CS DRIVING Slackest Span Tension vs Tensioner Pulley 1 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
Appendix D 166
zlabel(CS Span Tension (N))
figure
Za4 = squeeze(Ta(4))
surf(X3Y3real(Za4))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(ISG DRIVING Tautest Span Tension vs Tensioner Pulley 1 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(ISG Span Tension (N))
figure
Za3 = squeeze(Ta(3))
surf(X3Y3real(Za3))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(ISG DRIVING Slackest Span Tension vs Tensioner Pulley 1 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(ISG Span Tension (N))
minX5 = -0037093
maxX5 = 0212509
minY5 = 00362
maxY5 = 0228456
midX5a = 0131965
midX5b = 017729
X5 = minX5 + (ixx-1)(maxX5-minX5)(m-1)
Y5 = minY5 + (iyy-1)(maxY5-minY5)(n-1)
for i = 1n
for j = 1m
if
(X5(ij)ltmidX5a)ampamp(Y5(ij)lt=(0386X5(ij)+0146468))ampamp(Y5(ij)gt=(sqrt((0136525^2)-
(X5(ij)^2))))
[T(ij)Ta(ij)] =
TwinTenStaticTension(KtoKt1oKt2oD3oD5oX3oY3oX5(ij)Y5(ij))
elseif
((X5(ij)gt=midX5a)ampamp(X5(ij)ltmidX5b))ampamp(Y5(ij)gt=00362)ampamp(Y5(ij)lt=(0386X5(ij)+0
146468))
[T(ij)Ta(ij)] =
TwinTenStaticTension(KtoKt1oKt2oD3oD5oX3oY3oX5(ij)Y5(ij))
elseif (X5(ij)gt=midX5b)ampamp(Y5(ij)gt=(sqrt((00703^2)-(((X5(ij)-
024759)^2)))+016664))ampamp(Y5(ij)lt=(0386X5(ij)+0146468))
Appendix D 167
[T(ij)Ta(ij)] =
TwinTenStaticTension(KtoKt1oKt2oD3oD5oX3oY3oX5(ij)Y5(ij))
else
T(ij) = zeros(511)
Ta(ij) = zeros(511)
end
end
end
figure
Z1 = squeeze(T(1))
surf(X5Y5real(Z1))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(CS DRIVING Tautest Span Tension vs Tensioner Pulley 2 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(CS Span Tension (N))
figure
Z5 = squeeze(T(5))
surf(X5Y5real(Z5))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(CS DRIVING Slackest Span Tension vs Tensioner Pulley 2 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(CS Span Tension (N))
figure
Za4 = squeeze(Ta(4))
surf(X5Y5real(Za4))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(ISG DRIVING Tautest Span Tension vs Tensioner Pulley 2 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(ISG Span Tension (N))
figure
Za3 = squeeze(Ta(3))
surf(X5Y5real(Za3))
ZLim([50 500])
axis tight
Appendix D 168
colormap jet
colorbar
title(ISG DRIVING Slackest Span Tension vs Tensioner Pulley 2 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(ISG Span Tension (N))
D12 TwinTenStaticTensionm
The function TwinTenStaticTensionm simulates the static model of the B-ISG system with a
Twin Tensioner This function returns 3 vectors the static tension of each belt span in the
crankshaft- and ISG-driving phases of operation and the angle of displacement of each rigid
body in the ISG- driving phase It takes the input parameters kt kt1 kt2 for the tensioner arm
stiffness D3 and D5 for the tensioner pulley diameters and X3Y3 X5 and Y5 for the tensioner
arm pulley coordinates The function is called in the parametric analysis solution script
TwinMainm and in the optimization solution script OptimizationTwinm
D2 Optimization
D21 OptimizationTwinm
The following script is for the main function OptimizationTwinm It performs an optimization
search for the B-ISG system with a Twin Tensioner It takes an input for a parameter vector
containing values for the design variables The program calls the objective function
objfunTwinm and the constraint function confunTwinm The program can perform a genetic
algorithm (GA) optimization search or a hybrid GA optimization that includes a localized search
The optimal solution vector corresponding to the design variables and the optimal objective
function value is returned The program inputs the optimized values for the design variables into
TwinTenStaticTensionm This called function returns the optimized static state of tensions for
the crankshaft- and ISG- driving phases and for the angle of displacement of the rigid bodies in
the ISG driving phase
______________________________________________________________________________
clc
clear all
Initial values for variables
Kto = 20626
Kt1o = 10314
Kt2o = 16502
X3o = 0292761
Y3o = 0087
X5o = 012057
Appendix D 169
Y5o = 009193
w0 =[Kto Kt1o Kt2o X3o Y3o X5o Y5o] Start Point (row vector)
Variable ranges
minKt = Kto - 1Kto
maxKt = Kto + 1Kto
minKt1 = Kt1o - 1Kt1o
maxKt1 = Kt1o + 1Kt1o
minKt2 = Kt2o - 1Kt2o
maxKt2 = Kt2o + 1Kt2o
minX3 = 0260200
maxX3 = 0317677
minY3 = -0056640
maxY3 = 0228456
minX5 = -0037093
maxX5 = 0212509
minY5 = 00362
maxY5 = 0228456
ObjectiveFunction = objfunTwin
nvars = 7 Number of variables
ConstraintFunction = confunTwin
Uncomment next two lines (and comment the two functions after them) to use GA algorithm
for optimization
options = gaoptimset(InitialPopulationw0PopInitRange[minKt minKt1 minKt2 minX3
minY3 minX5 minY5 maxKt maxKt1 maxKt2 maxX3 maxY3 maxX5
maxY5]PopulationSize100Displayfinal)
[wfvalexitflagoutput] =
ga(ObjectiveFunctionnvars[][][][][][]ConstraintFunctionoptions)
fminconOptions = optimset(DisplayiterLargeScaleoff) Largescale off since gradient not
provided
options = gaoptimset(InitialPopulationw0PopInitRange[minKt minKt1 minKt2 minX3
minY3 minX5 minY5 maxKt maxKt1 maxKt2 maxX3 maxY3 maxX5
maxY5]PopulationSize100HybridFcnfmincon fminconOptions)
[wfvalexitflagoutput] = ga(ObjectiveFunctionnvars[][][][][][]ConstraintFunctionoptions)
[TTaThetaDegA] = TwinTenStaticTension(w(1)w(2)w(3)w(4)w(5)w(6)w(7))
D22 confunTwinm
The constraint function confunTwinm is used by the main optimization program to ensure
input values are constrained within the prescribed regions The function makes use of inequality
constraints for seven constrained variables corresponding to the design variables It takes an
input vector corresponding to the design variables and returns a data set of the vector values that
satisfy the prescribed constraints
Appendix D 170
D23 objfunTwinm
This function is the objective function for the main optimization program It outputs a value for
a weighted objective function or a non-weighted objective function relating the optimization of
the static tension The program takes an input vector containing a set of values for the design
variables that are within prescribed constraints The description of the function is similar to
TwinTenStaticTensionm but differs in the fact that it only returns a scalar value which is the
value of the objective function
171
VITA
ADEBUKOLA OLATUNDE
Email adebukolaolatundegmailcom
Adebukola Olatunde is a graduate research student at the University of Toronto in Toronto
Ontario Canada She obtained a Bachelor‟s Degree in Mechanical Engineering from McMaster
University in Hamilton Ontario Canada in 2002 Upon graduation she pursued a graduate
degree in mechanical engineering at the University of Toronto with a specialization in
mechanical systems dynamics and vibrations and environmental engineering In September
2008 she completed the requirements for the Master of Applied Science degree in Mechanical
Engineering She has held the position of teaching assistant for undergraduate courses in
dynamics and vibrations Adebukola has completed course work in professional education She
is a registered member of professional engineering organizations including the Professional
Engineer‟s of Ontario Engineer-in-Training program the Canadian Society of Mechanical
Engineers and the National Society of Black Engineers She intends to practice as a professional
engineering consultant in mechanical design
iii
A testament unto the LORD God lsquowho answered me in the day of my distress and was with me in
the way which I wentrsquo
To my parents Joseph and Beatrice for your strength and persistent prayers
To my siblings Shade Charlene and Kevin for being a listener an editor and a relief when I
needed it
To my friends Samantha Esther and Yasmin who kept me motivated
amp
With love to my sweetheart Nana whose patience support and companionship has made life
sweeter
iv
ACKNOLOWEDGEMENTS
I would like to express deep gratitude to Dr Jean Zu for her guidance throughout the duration of
my studies and for providing me with the opportunity to conduct this thesis
I wish to thank the individuals of Litens Automotive who have provided guidance and data for
the research work Special thanks to Mike Clark Seeva Karuendiran and Dr Qiu for their time
and help
I thank my committee members Dr Naguib and Dr Sun for contributing their time to my
research work
My sincerest thanks to my research colleague David for his knowledge and support Many
thanks to my lab mates Qiming Hansong Ali Ming Andrew and Peyman for their guidance
I want to especially thank Dr Cleghorn Leslie Sinclair and Dr Zu for the opportunities to
teach These experiences have served to enrich my graduate studies As well thank you to Dr
Cleghorn for guidance in my research work
I am also in debt to my classmates and teaching colleagues throughout my time at the University
of Toronto especially Aaron and Mohammed for their support in my development as a graduate
researcher and teacher
v
CONTENTS
ABSTRACT ii
DEDICATION iii
ACKNOWLEDGEMENTS iv
CONTENTS v
LIST OF TABLES ix
LIST OF FIGURES xi
LIST OF SYMBOLS xvi
Chapter 1 INTRODUCTION 1
11 Background 1
12 Motivation 3
13 Thesis Objectives and Scope of Research 4
14 Organization and Content of Thesis 5
Chapter 2 LITERATURE REVIEW 7
21 Introduction 7
22 B-ISG System 8
221 ISG in Hybrids 8
2211 Full Hybrids 9
2212 Power Hybrids 10
2213 Mild Hybrids 11
2214 Micro Hybrids 11
222 B-ISG Structure Location and Function 13
2221 Structure and Location 13
2222 Functionalities 14
23 Belt Drive Modeling 15
24 Tensioners for B-ISG System 18
241 Tensioners Structures Function and Location 18
242 Systematic Review of Tensioner Designs for a B-ISG System 20
25 Summary 24
vi
Chapter 3 MODELING OF B-ISG SYSTEM 25
31 Overview 25
32 B-ISG Tensioner Design 25
33 Geometric Model of a B-ISG System with a Twin Tensioner 27
34 Equations of Motion for a B-ISG System with a Twin Tensioner 32
341 Dynamic Model of the B-ISG System 32
3411 Derivation of Equations of Motion 32
3412 Modeling of Phase Change 41
3413 Natural Frequencies Mode Shapes and Dynamic Responses 42
3414 Crankshaft Pulley Driving Torque Acceleration and Displacement 44
3415 ISG Pulley Driving Torque Acceleration and Displacement 46
3416 Tensioner Arms Dynamic Torques 48
3417 Dynamic Belt Span Tensions 49
342 Static Model of the B-ISG System 49
35 Simulations 50
351 Geometric Analysis 51
352 Dynamic Analysis 52
3521 Natural Frequency and Mode Shape 54
3522 Dynamic Response 58
3523 ISG Pulley and Crankshaft Pulley Torque Requirement 61
3524 Tensioner Arm Torque Requirement 62
3525 Dynamic Belt Span Tension 63
353 Static Analysis 66
36 Summary 69
Chapter 4 PARAMETRIC ANALYSIS OF A B-ISG TWIN TENSIONER 71
41 Introduction 71
42 Methodology 71
43 Results and Discussion 74
431 Influence of Tensioner Arm Stiffness on Static Tension 74
432 Influence of Tensioner Pulley Diameter on Static Tension 78
433 Influence of Tensioner Pulley 1 Coordinates on Static Tension 80
434 Influence of Tensioner Pulley 2 Coordinates on Static Tension 86
vii
44 Conclusion 92
Chapter 5 OPTIMIZATION OF A B-ISG TWIN TENSIONER 95
51 Optimization Problem 95
511 Selection of Design Variables 95
512 Objective Function amp Constraints 97
52 Optimization Method 100
521 Genetic Algorithm 100
522 Hybrid Optimization Algorithm 101
53 Results and Discussion 101
531 Parameter Settings amp Stopping Criteria for Simulations 101
532 Optimization Simulations 102
533 Discussion 106
54 Conclusion 109
Chapter 6 CONCLUSION AND RECOMMENDATIONS111
61 Summary 111
62 Conclusion 112
63 Recommendations for Future Work 113
REFERENCES 116
APPENDICIES 123
A Passive Dual Tensioner Designs from Patent Literature 123
B B-ISG Serpentine Belt Drive with Single Tensioner Equation of Motion 138
C MathCAD Scripts 145
C1 Geometric Analysis 145
C2 Dynamic Analysis 152
C3 Static Analysis 161
D MATLAB Functions amp Scripts 162
D1 Parametric Analysis 162
D11 TwinMainm 162
D12 TwinTenStaticTensionm 168
D2 Optimization 168
D21 OptimizationTwinm - Optimization Function 168
viii
D22 confunTwinm 169
D23 objfunTwinm 170
VITA 171
ix
LIST OF TABLES
21 Passive Dual Tensioner Designs from Patent Literature
31 Selected Contact Point Types on the ith Pulley and Types for the ith Belt Span
32 Coordinate Points for Pulley Centres and Twin Tensioner Pivot
33 Geometric Results of B-ISG System with Twin Tensioner
34 Data for Input Parameters used in Dynamic and Static Computations
35 Static Solution for Belt Span Tensions in Crankshaft and ISG Driving Cases for a B-ISG
Serpentine Belt Drive with a Single Tensioner
36 Static Solution for Belt Span Tensions in Crankshaft and ISG Driving Cases for a B-ISG
Serpentine Belt Drive with a Twin Tensioner
41 Initial Values Increments and Ranges for Parameters of Twin Tensioner
51 Summary of Parametric Analysis Data for Twin Tensioner Properties
52a GA Optimization Results for Twin Tensioner Parameters and Objective Function
52b Computations for Tensions and Angles from GA Optimization Results
53a Hybrid Optimization Results for Twin Tensioner Parameters and Objective Function
53b Computations for Tensions and Angles from Hybrid Optimization Results
54a Non-Weighted Optimization Results for Twin Tensioner Parameters and Objective
Function
54b Computations for Tensions and Angles from Non-Weighted Optimizations
x
55 Weighted Optimization Results for Static Tensions for Optimal B-ISG System with a
Twin Tensioner
56 Non-Weighted Optimization Results for Static Tensions for Optimal B-ISG System with a
Twin Tensioner
xi
LIST OF FIGURES
21 Hybrid Functions
31 Schematic of the Twin Tensioner
32 B-ISG Serpentine Belt Drive with Twin Tensioner
33 Angles Coordinates and Possible Belt Contact Points for the ith and i+1th Pulleys
34 Twin Tensioner Free Body Diagram of a Four-Degree of Freedom System
35 Free Body Diagram for Non-Tensioner Pulleys
36a ISG Driving Case Natural Frequency of System and Mode Shapes for Responsive Rigid
Bodies
36b ISG Driving Case First Mode Responses
36c ISG Driving Case Second Mode Responses
37a Crankshaft Driving Case Natural Frequency of System and Mode Shapes for Responsive
Rigid Bodies
37b Crankshaft Driving Case First Mode Responses
37c Crankshaft Driving Case Second Mode Responses
38 Crankshaft Pulley Dynamic Response (for crankshaft driven case)
39 ISG Pulley Dynamic Response (for ISG driven case)
310 Air Conditioner Pulley Dynamic Response
311 Tensioner Pulley 1 Dynamic Response
xii
312 Tensioner Pulley 2 Dynamic Response
313 Tensioner Arm 1 Dynamic Response
314 Tensioner Arm 2 Dynamic Response
315 Required Driving Torque for the ISG Pulley
316 Required Driving Torque for the Crankshaft Pulley
317 Dynamic Torque for Tensioner Arm 1
318 Dynamic Torque for Tensioner Arm 2
319 Span 1 (between crankshaft and air conditioner) Dynamic Belt Span Tension
320 Span 2 (between air conditioner and tensioner 1) Dynamic Belt Span Tension
321 Span 3 (between tensioner 1 and ISG) Dynamic Belt Span Tension
322 Span 4 (between ISG and tensioner 2) Dynamic Belt Span Tension
323 Span 5 (between tensioner 2 and crankshaft) Dynamic Belt Span Tension
324 B-ISG Serpentine Belt Drive with Single Tensioner
41a Regions 1 and 2 and their Associated Guidelines for Coordinates of Tensioner Pulleys 1
amp 2
41b Regions 1 and 2 in Cartesian Space
42 Parametric Analysis for Coupled Stiffness Arm Constant kt (N∙mrad)
43 Parametric Analysis for Stiffness of Arm 1 kt1 (N∙mrad)
44 Parametric Analysis for Stiffness of Arm 2 kt2 (N∙mrad)
xiii
45 Parametric Analysis for Pulley 1 Diameter D3 (m)
46 Parametric Analysis for Pulley 2 Diameter D5 (m)
47 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Tautest Span
Tension in Crankshaft Driving Case
48 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Slackest Span
Tension in Crankshaft Driving Case
49 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Tautest Span
Tension in ISG Driving Case
410 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Slackest Span
Tension in ISG Driving Case
411 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Tautest Span
Tension in Crankshaft Driving Case
412 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Slackest Span
Tension in Crankshaft Driving Case
413 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Tautest Span
Tension in ISG Driving Case
414 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Slackest Span
Tension in ISG Driving Case
51 Static Stability of the B-ISG Twin Tensioner Based on the Angular Displacement of
Tensioner Arms 1 and 2
A1 Proposed design by Bayerische Motoren Werke AG corresponding to patent nos
EP1420192-A2 and DE10253450-A1
A2a First of four proposed designs (and its various configurations) by Bosch GMBH
corresponding to patent no WO0026532-A1
A2b Second of four proposed designs by Bosch GMBH corresponding to patent no
WO0026532-A1
A2c Third of four proposed designs (and its various configurations) by Bosch GMBH
corresponding to patent no WO0026532-A1
A2d Fourth of four proposed designs (and its various configurations) by Bosch GMBH
corresponding to patent no WO0026532-A1
A3 Proposed design by Daimler Chrysler AG corresponding to patent no DE10324268-A1
A4 Proposed design by Dayco Products LLC corresponding to patent no US6942589-B2
xiv
A5 Proposed design by Gates Corp corresponding to patent no WO2003038309-A
A6 Proposed design by General Motors Corp corresponding to patent no US20060287146-
A1
A7 Proposed design by INA Schaeffler KG corresponding to patent no DE10044645-A1
A8a First of three proposed designs by INA Schaeffler KG corresponding to patent no
DE10159073-A1
A8b Second of three proposed designs by INA Schaeffler KG corresponding to patent no
DE10159073-A1
A8c Third of three proposed designs by INA Schaeffler KG corresponding to patent no
DE10159073-A1
A9 Proposed design by INA Schaeffler KG corresponding to patent no DE10359641-A1
A10 Proposed design by INA Schaeffler KG corresponding to patent no EP1723350-A1
A11 Proposed design by INA Schaeffler KG corresponding to patent no EP1738093-A1
A12 Proposed design by INA Schaeffler KG corresponding to patent no DE102004012395-
A1
A13 Proposed design by INA Schaeffler KG corresponding to patent nos DE102005017038-
A1and WO2006108461-A1
A14 Proposed design by Litens Automotive GMBH et al corresponding to patent no
US20010007839-A1
A15 Proposed design by Mitsubishi Jidosha Eng KK and Mitsubishi Motor Corp
corresponding to patent no JP2005083514-A
A16 Proposed design by Nissan Motor Co Ltd corresponding to patent no JP3565040-B2
A17 Proposed design by NTN Corp corresponding to patent no JP2006189073-A
A18 Proposed design by Valeo Equipment Electriques Moteur corresponding to patent nos
EP1658432 and WO2005015007
B1 Single Tensioner B-ISG System
B2 Free-body Diagram of ith Pulley
xv
B3 Free-body Diagram of Single Tensioner
C1 Schematic of B-ISG System with Twin Tensioner
C2 Possible Contact Points
xvi
LIST OF SYMBOLS
Latin Letters
A Belt cord cross-sectional area
C Damping matrix of the system
cb Belt damping
119888119894119887 Belt damping constant of the ith belt span
119914119946119946 Damping matrix element in the ith row and ith column
ct Damping acting between tensioner arms 1 and 2
cti Damping of the ith tensioner arm
DCS Diameter of crankshaft pulley
DISG Diameter of ISG pulley
ft Belt transition frequency
H(n) Phase change function
I Inertial matrix of the system
119920119938 Inertial matrix under ISG driving phase
119920119940 Inertial matrix under crankshaft driving phase
Ii Inertia of the ith pulley
Iti Inertia of the ith tensioner arm
119920120784120784 Submatrix of inertial matrix I
j Imaginary coordinate (ie (-1)12
)
K Stiffness matrix of the system
xvii
119896119887 Belt factor
119870119887 Belt cord stiffness
119896119894119887 Belt stiffness constant of the ith belt span
kt Spring stiffness acting between tensioner arms 1 and 2
kti Coil spring of the ith tensioner arm
119922120784120784 Submatrix of stiffness matrix K
Lfi Lbi Lengths of possible belt span connections from the ith pulley
Lti Length of the ith tensioner arm
Modeia Mode shape of the ith rigid body in the ISG driving phase
Modeic Mode shape of the ith rigid body in the crankshaft driving phase
n Engine speed
N Motor speed
nCS rpm of crankshaft pulley
NF Motor speed without load
nISG rpm of ISG pulley
Q Required torque matrix
qc Amplitude of the required crankshaft torque
QcsISG Required torque of the driving pulley (crankshaft or ISG)
Qm Required torque matrix of driven rigid bodies
Qti Dynamic torque of the ith tensioner arm
Ri Radius of the ith pulley
T Matrix of belt span static tensions
xviii
Trsquo Dynamic belt tension matrix
119931119940 Damping matrix due to the belt
119931119948 Stiffness matrix due to the belt
Ti Tension of the ith belt span
To Initial belt tension for the system
Ts Stall torque
Tti Tension for the neighbouring belt spans of the ith tensioner pulley
(XiYi) Coordinates of the ith pulley centre
XYfi XYbi XYfbi
XYbfi Possible connection points on the ith pulley leading to the ith belt span
XYf2i XYb2i
XYfb2i XYbf2i Possible connection points on the ith pulley leading to the (i-1)th belt span
Greek Letters
αi Angle between the datum and the line connecting the ith and (i+1)th pulley
centres
βji Angle of orientation for the ith belt span
120597θti(t) 120579 ti(t)
120579 ti(t)
Angular displacement velocity and acceleration (rotational coordinate) of the
ith tensioner arm
120637119938 General coordinate matrix under ISG driving phase
120637119940 General coordinate matrix under crankshaft driving phase
θfi θbi Angles between the datum and the belt connection spans with lengths Lfi and
Lbi respectively
Θi Amplitude of displacement of the ith pulley
xix
θi(t) 120579 i(t) 120579 i(t) Angular position velocity and acceleration (rotational coordinate) of the ith
pulley
θti Angle of the ith tensioner arm
θtoi Initial pivot angle of the ith tensioner arm
θm Angular displacement matrix of driven rigid bodies
Θm Amplitude of displacement of driven rigid bodies
ρ Belt cord density
120601119894 Belt wrap angle on the ith pulley
φmax Belt maximum phase angle
φ0deg Belt phase angle at zero frequency
ω Frequency of the system
ωcs Angular frequency of crankshaft pulley
ωISG Angular frequency of the ISG pulley
120654119951 Natural frequency of system
1
CHAPTER 1 INTRODUCTION
11 Background
Belt drive systems are the means of power transmission in conventional automobiles The
emergence of hybrid technologies specifically the Belt-driven Integrated Starter-generator (B-
ISG) has placed higher demands on belt drives than ever before The presence of an integrated
starter-generator (ISG) in a belt transmission places excessive strain on the belt leading to
premature belt failure This phenomenon has motivated automotive makers to design a tensioner
that is suitable for the B-ISG system
The belt drive is also known interchangeably as the front-end accessory drive-belt (FEAD) the
belt accessory-drive system (BAS) or the belt transmission system In a traditional setting the
role of this system is to transmit torque generated by an internal combustion engine (ICE) in
order to reliably drive multiple peripheral devices mounted on the engine block The high speed
torque is transmitted through a crankshaft pulley to a serpentine belt The serpentine belt is a
single continuous member that winds around the driving and driven accessory pulleys of the
drive system Serpentine belts used in automotive applications consist of several layers The
load-bearing layer is a flexible member consisting of high stiffness fibers [1] It is covered by a
protective layer to guard against mechanical damage and is bound below by a visco-elastic layer
that provides the required shock absorption and grip against the rigid pulleys [1] The accessory
devices may include an alternator power steering pump water pump and air conditioner
compressor among others
Introduction 2
The B-ISG system is a transmission system characteristic to micro-hybrid automobiles It is akin
to traditional belt drives differing in the fact that an electric motor called an integrated starter-
generator (ISG) replaces the original alternator re-starts the engine from idle speed and provides
braking regeneration [2] The re-start function of the micro-hybrid transmission is known as
stop-start In the B-ISG setting the ISG is mounted on the belt drive The ISG produces a speed
of approximately 2000 to 2500rpm in order to spin the engine at approximately 750rpm and
upwards to produce an instantaneous start in the start-stop process [3] The high rotations per
minute (rpm) produced by the ISG consistently places much higher tension requirements on the
belt than when the crankshaft is driving the belt It is preferable not to exceed a range of 600N to
800N of tension on the belt since this exceeds the safe operating conditions of belts used in most
traditional drive systems [4] The traditional belt drive system‟s tensioner a single-arm
tensioner does not suitably reduce the high belt tension nor provide enough tension in the slack
belts spans occurring in the ISG phase of operation for the B-ISG system
In order for the belt to transfer torque in a drive system its initial tension must be set to a value
that is sufficient to keep all spans rigid This value must not be too low as to allow any one span
to be slack during the drive‟s phases of operation Furthermore the belt must not be ldquoinstalled
with too high a tensionrdquo since this can lead to ldquopremature failure of the bearings supporting the
drive and driven pulleys and of the belt itselfrdquo [5] The presence of a tensioning mechanism in
an automotive belt drive allows for an enhanced belt life and performance since pre-tensioning
of the belt is normally not sufficient for all phases of belt drive operation A tensioner allows for
the system to cope with moderate to severe changes in belt span tensions
Introduction 3
Traditional automotive tensioners for belt drives of an ICE consist of a single spring-loaded
arm This type of tensioner is normally designed to provide a passive response to changes in belt
span tension The introduction of the ISG electric motor into the traditional belt drive with a
single-arm tensioner results in the presence of excessively slack spans and excessively tight
spans in the belt The tension requirements in the ISG-driving phase which differ from the
crankshaft-driving phase are poorly met by a traditional single-arm passive tensioner
Tensioners can be divided into two general classes passive and active In both classes the
single-arm tensioner design approach is the norm The passive class of tensioners employ purely
mechanical power to achieve tensioning of the belt while the active class also known as
automatic tensioners typically use some sort of electronic actuation Automatic tensioners have
been employed by various automotive manufacturers however ldquosuch devices add mass
complication and cost to each enginerdquo [5]
12 Motivation
The motivation for the research undertaken arises from the undesirable presence of high belt
tension in automotive belt drives Manufacturers of automotive belt drives have presented
numerous approaches for tension mechanism designs As mentioned in the preceding section
the automation of the traditional single-arm tensioner has disadvantages for manufacturers A
survey of the literature reveals that few quantitative investigations in comparison to the
qualitative investigations provided through patent literature have been conducted in the area of
passive and dual tensioner configurations As such the author of the research project has selected
to investigate the performance of a passive twin-arm tensioner design The theoretical tensioner
Introduction 4
configuration is motivated by research and developments of industry partner Litens
Automotivendash a manufacturer of automotive belt drive systems and components Litens‟
specialty in automotive tensioners has provided a basis for the research work conducted
13 Thesis Objectives and Scope of Research
The objective of this project is to model and investigate a system containing a passive twin-arm
tensioner in a B-ISG serpentine belt drive where the driving pulley alternates between a
crankshaft pulley and an ISG pulley The modeling of a serpentine belt drive system is in
continuation of the work done by post-doctoral fellow Zhen Mu in development of the priority
software known as FEAD at the University of Toronto Firstly for the B-ISG system with a
twin-arm tensioner the geometric state and its equations of motion (EOM) describing the
dynamic and static states are derived The modeling approach was verified by deriving the
geometric properties and the EOM of the system with a single tensioner arm and comparing its
crankshaft-phase‟s simulation results with FEAD software simulations This also provides
comparison of the new twin-arm tensioner belt drive model with the former single-arm tensioner
equipped belt drive model Secondly the model for the static system is investigated through
analysis of the tensioner parameters Thirdly the design variables selected from the parametric
analysis are used for optimization of the new system with respect to its criteria for desired
performance
Introduction 5
14 Organization and Content of the Thesis
This thesis presents the investigation of a passive twin-arm tensioner design in a B-ISG
serpentine belt drive system which is distinguished by having its driving pulley alternate
between a crankshaft pulley and an ISG pulley
Chapter 2 presents the literature reviewed relevant to the area of the thesis topic The context of
the research discusses the function and location of the ISG in hybrid technologies in order to
provide a background for the B-ISG system The attributes of the B-ISG are then discussed
Subsequently a description is given of the developments made in modeling belt drive systems
At the close of the chapter the prior art in tensioner designs and investigations are discussed
The third chapter describes the system models and theory for the B-ISG system with a twin-arm
tensioner Models for the geometric properties and the static and dynamic cases are derived The
simulation results of the system model are presented
Then the fourth chapter contains the parametric analysis The methodologies employed results
and a discussion are provided The design variables of the system to be considered in the
optimization are also discussed
The optimization of a B-ISG system with a passive twin-arm tensioner is presented in Chapter 5
The evaluation of optimization methods results of optimization and discussion of the results are
included Chapter 6 concludes the thesis work in summarizing the response to the thesis
Introduction 6
objectives and concluding the results of the investigation of the objectives Recommendations for
future work in the design and analysis of a B-ISG tensioner design are also described
7
CHAPTER 2 LITERATURE REVIEW
21 Introduction
This literature review justifies the study of the thesis research the significance of the topic and
provides the overall framework for the project The design of a tensioner for a Belt-driven
Integrated Starter-generator (B-ISG) system is a link in the chain of power transmission
developments in hybrid automobiles This chapter will begin with the context of the B-ISG
followed by a review of the hybrid classifications and the critical role of the ISG for each type
The function location and structure of the B-ISG system are then discussed Then a discussion
of the modeling of automotive belt transmissions is presented A systematic review of the prior
art and current state of tensioning mechanisms for B-ISG systems amalgamates the literature and
research evidence relevant to the thesis topic which is the design of a B-ISG tensioner
The Belt-driven Integrated Starter-generator (B-ISG) system is a part of a hybrid class that is
distinguished from other hybrid classes by the structure functions and location of its ISG The
B-ISG unit is a hybrid technology applied to traditional automotive belt drives The use of a B-
ISG system to achieve a start-stop function in the car engine is estimated to cut fuel consumption
in conventional automobiles by up to ten percent and thus reduce CO2 emissions [6]
Environmental and legislative standards for reducing CO2 emissions in vehicles have called for
carmakers to produce less polluting and more efficient vehicle powertrain systems [7] The
transition to bdquocleaner‟ cars makes room for the introduction of the ISG machine into conventional
automotive belt drives [8] The reduction of CO2 emissions and the similarity of the B-ISG
Literature Review 8
transmission to that of conventional cars provide the motivation for the thesis research
Consequently the micro-hybrid class of cars is especially discussed in the literature review since
it contains the B-ISG type of transmission system The micro-hybrid class is one of several
hybrid classes
A look at the performance of a belt-drive under the influence of an ISG is rooted in the
developments of hybrid technology The distinction of the ISG function and its location in each
hybrid class is discussed in the following section
22 B-ISG System
221 ISG in Hybrids
This section of the review discusses the standard classes of hybrid cars which are full power
mild and micro- hybrids Special attention is given to hybrid vehicle architectures involving
internal combustion engines (ICEs) as the main power source This is done for the sake of
comparison between hybrid classes since the ICE is the standard power source for B-ISG micro-
hybrids which is the focus of the research The term conventional car vehicle or automobile
henceforth refers to a vehicle powered solely by a gas or diesel ICE
A hybrid vehicle has a drive system that uses a combination of energy devices This may include
an ICE a battery and an electric motor typically an ISG Two systems exist in the classification
of hybrid vehicles The older system of classification separates hybrids into two classes series
hybrids and parallel hybrids In the older system many modern hybrid vehicles have modes of
operation matching both categories classifying them under either of the two classes [9] The
Literature Review 9
new system of classification has four classes full power mild and micro Under these classes
vehicles are more often under a sole category [9] In both systems an ICE may act as the primary
source of power otherwise it may be a fuel cell The fuel used by the ICE may be gas (petrol)
diesel or an alternative fuel such as ethanol bio-diesel or natural gas
2211 Full Hybrids
In a full hybrid car the ICE is used to power the integrated starter-generator (ISG) which stores
electrical energy in the batteries to be used to power an electric traction motor [8] The electric
traction motor is akin to a second ISG as it generates power and provides torque output It also
supplies an extra boost to the wheels during acceleration and drives up steep inclines A full
hybrid vehicle is able to move by electrical power only It can be driven by the ISG powering
the electric traction motor without the engine running This silent acceleration known as electric
launch is normally employed when accelerating from standstill [9] Full hybrids can generate
and consume energy at the same time Full hybrid vehicles also use regenerative braking [8]
The ISG allows this by converting from an electric traction motor to a generator when braking or
decelerating The kinetic energy from the car‟s motion is then turned into electricity and stored
in the batteries For full hybrids to achieve this they often use break-by-wire a form of
electronically controlled braking technology
A high-voltage (ie 36- or 42-volt) ISG is employed in full hybrids to start the ICE It spins the
engine more than 900 rpm whereas conventional 12-volt starter motors spin the engine at
approximately 250 rpm [9] Thus the full hybrid vehicle is able to have an instantaneous start In
full hybrids the ISG is placed in the position of the flywheel and can have its motion decoupled
Literature Review 10
from the engine [9] The ISG device also allows full hybrids to have engine start-stop also called
an idle-stop ability The idle-stop function refers to when the engine shuts down as soon as a
vehicle stops from its ICE driving mode which saves on the fuel it normally burns while idling
[8] The vehicle returns to the engine driving mode of operation by way of the ISG‟s start-up of
the crankshaft which restarts the engine in less than 300 milliseconds [9] In summary at
standstill the tachometer of the engine drops to 0 rpm since the engine has ceased the engine is
started only when needed which is often several seconds after acceleration has begun The
engine start-stop feature is achieved by way of an electronic control system that shuts off the ICE
when it is not needed to assist in driving the wheels or to produce electricity for recharging the
batteries The start-stop feature by itself is estimated to produce a ten percent fuel gain in hybrids
over conventional vehicles particularly in urban driving conditions [9] Since the ICE is
required to provide only the average horsepower used by the vehicle the engine is downsized in
comparison to a conventional automobile that obtains all its power from an ICE Frequently in
full hybrids the ICE uses an alternative operating strategy such as the Atkinson Cycle which has
a higher efficiency while having a lower power output Examples of full hybrids include the
Ford Escape and the Toyota Prius [9]
2212 Power Hybrids
Akin to the full hybrid the ISG of the power hybrid enables the same features electric launch
regenerative braking and engine idle-stop The distinguishing characteristic from full hybrids is
the ICE is not downsized to meet only the average power demand [9] Thus the engine of a
power hybrid is large and produces a high amount of horsepower compared to the former
Overall a power hybrid has the assist of a full size ICE and therefore has more torque and a
Literature Review 11
greater acceleration performance than a full hybrid or a conventional vehicle with the same size
ICE [9] The Lexus RX400h unit is an example of a power hybrid [9]
2213 Mild Hybrids
In the hybrid types discussed thus far the ISG is positioned between the engine and transmission
to provide traction for the wheels and for regenerative braking Often times the armature or rotor
of the electric motor-generator which is the ISG replaces the engine flywheel in full and power
hybrids [9] In the case of the mild hybrid the ISG is not decoupled from the ICE and hence it is
not able to drive the wheels apart from the engine It remains that the ISG shares the same shaft
with the ICE In this environment the electric launch feature does not exist since the ISG does
not turn the wheels independently of the engine and energy cannot be generated and consumed
at the same time However the ISG of the mild hybrid allows for the remaining features of the
full hybrid regenerative braking and engine idle-stop including the fact that the engine is
downsized to meet only the average demand for horsepower Mild hybrid vehicles include the
GMC Sierra pickup and 2003 to 2005 Honda Civic models [9]
2214 Micro Hybrids
Micro hybrid is the category of hybrids that can contain a B-ISG transmission and is also closest
to modern conventional vehicles This class normally features a gas or diesel ICE [9] The
conventional automobile is modified by installing an ISG unit on the mechanical drive in place
of or in addition to the starter motor The starter motor typically 12-volts is removed only in
the case that the ISG device passes cold start testing which is also dependent on the engine size
[10] Various mechanical drives that may be employed include chain gear or belt drives or a
Literature Review 12
clutchgear arrangement The majority of literature pertaining to mechanical driven ISG
applications does not pursue clutchgear arrangements since it is associated with greater costs
and increased speed issues Findings by Henry et al [11] show that the belt drive in
comparison to chain and gear drives has a decreased cost (especially if the ISG is mounted
directly to the accessory drive) has no need for lubrication has less restriction in the packaging
environment and produces very low noise Also mounting the ISG unit on a separate belt from
that linking the accessory pulleys is undesirable since applying the ISG directly to the accessory
belt drive requires less engine transmission or vehicle modifications
As with full power and mild hybrids the presence of the ISG allows for the start-stop feature
The automobile‟s electronic control unit (ECU) is calibrated or engine control circuitry (a
separate ECU) is added to the conventional car in order to shut down the engine when the
vehicle is stopped [12] The control system also controls the charge cycle of the ISG [9] This
entails that it dictates the field current by way of a microprocessor to allow the system to defer
battery charge cycles until the vehicle is decelerating [13] This produces electricity to recharge
the battery primarily during deceleration and braking The B-ISG transmission of a micro hybrid
and its various components are discussed in the subsequent section Examples of micro hybrid
vehicles are the PSA Group‟s Citroen C2 and C3 [14] Ford‟s Fiesta [14] and BMW‟s Mini
Cooper D and various others of BMW‟s European models [15]
Literature Review 13
Figure 21 Hybrid Functions
Source Dr Daniel Kok FFA July 2004 modified [16]
Figure 21 shows that the higher the voltage available to the ISG unit the more hybrid functions
it is capable of performing It is noted that B-ISG transmissions of the micro-hybrid class may
also exceed the typical functions of micro-hybrids For instance Ford‟s HyTrans van (developed
in partnership with Ricardo UK Ltd Valeo SA Gates Corporation and the UK Department for
Transport) uses a B-ISG system and a 42-volt battery The van is diesel-powered and has
characteristics of a mild hybrid such as cold cranks and engine assists [17]
222 B-ISG Structure Location and Function
2221 Structure and Location
The ISG is composed of an electrical machine normally of the inductive type which includes a
stator (stationary part of the ISG) and a rotor (non-stationary part of the ISG) and a converter
comprising of a regulator a modulator switches and filters There are various configurations to
integrate the ISG unit into an automobile power train One configuration situates the ISG
directly on the crankshaft in the place of the present flywheel [11] This set-up is more compact
however it results in a longer power train which becomes a potential concern for transverse-
Literature Review 14
mounted engines [18] An alternative set-up is to have a side-mounted ISG This term is used to
describe the configuration of mounting the electrical device on the side of the mechanical drive
[18] As mentioned in Section 2214 a belt drive is used as the mechanical drive for the thesis
research hence the ISG is belt-mounted and the transmission becomes a belt-driven ISG system
In this arrangement the ISG replaces the alternator [13] and in some cases the starter motor may
be removed This design allows for the functions of the ISG system mentioned in the description
of the ISG role in micro-hybrids [9] The side-mounted ISG specifically the belt-mounted ISG
is more evolutionary to the conventional car since it ldquoallows for a more traditional under-hood
layoutrdquo [11]
2222 Functionalities
The primary duty of the ISG in a micro hybrid specifically in a B-ISG setting is to bring the
engine from rest to normal operating speeds within a time span ranging from 250 to 400 ms [3]
and in some high voltage settings to provide cold starting
The cold starting operation of the ISG refers to starting the engine from its off mode rather than
idle mode andor when the engine is at a low temperature for example -29 to -50 degrees
Celsius [2] If the ISG is used for cold starting the peak torque is determined by the torque
requirement for the cold starting operation of the target vehicle since it is greater than the
nominal torque For this function the ldquomachine has to provide a breakaway torque about 15 [to]
18 times the nominal cranking torque to overcome static torque and rotate the engine from 0 to
[between] 10 [and] 20rpmrdquo [2] This remains to be a challenge for the ISG as the 12-volt
architecture most commonly found in vehicles does not supply sufficient voltage [2] The
introduction of the ISG machine and other electrical units in vehicles encourages a transition
Literature Review 15
from a 12-volt or 14-volt to a 42-volt electrical architecture [19] The transition to 42-volt
architecture brings ldquopotential higher-voltage functionalities that come with an ISG systemrdquo [20]
At present ldquowhen the [ISG] machine cannot provide enough torque for initial cold engine
cranking the conventional starter will [remain] in the system and perform only for the initial
cranking while the stop-start function is taken over by the [ISG] machinerdquo [2] The ISG‟s launch
assist torque the torque required to bring the engine from idle speed to the speed at which it can
develop a higher torque output is 2000 to 2500 rpm for most gas engines [3]
Delphi‟s Energen 5 High Output 12-volt Belt-alternator-starter (or B-ISG) was implemented by
researchers on a 53 L V-8 engine with an automatic transmission in a Chevrolet Silverado truck
[21] The ISG was applied in a belt-mounted configuration and was used only for warm engine
re-starts The results of Wezenbeek et al [21] showed that the starting torque for a re-start by the
12-Volt ISG was 42 Nm ISG‟s have also been used in 14V 36V and 42V architectures [13]
23 Belt Drive Modeling
The modeling of a serpentine belt drive and tensioning mechanism has typically involved the
application of Newtonian equilibrium equations to rigid bodies in order to derive the equations of
motion for the system There are two modes of motion in a serpentine belt drive transverse
motion and rotational motion The former can be viewed as the motion of the belt directed
normal to the direction of the beltpulley contact plane similar to the vibratory motion of a taut
string that is fixed at either end However the study of the rotational motion in a belt drive is the
focus of the thesis research
Literature Review 16
Much work on the mechanics of the belt drive was carried out by Firbank [22] Firbank‟s
models helped to understand belt performance and the influence of driving and driven pulleys on
the tension member The first description of a serpentine belt drive for automotive use was in
1979 by Cassidy et al [23] and since this time there has been an increasing body of knowledge
on the mathematical modeling of serpentine belt drives Ulsoy et al [24] presented a design
methodology to improve the dynamic performance of instability mechanisms for belt tensioner
systems The mathematical model developed by Ulsoy et al [24] coupled the equations of
motion that were obtained through a dynamic equilibrium of moments about a pivot point the
equations of motion for the transverse vibration of the belt and the equations of motion for the
belt tension variations appearing in the transverse vibrations This along with the boundary and
initial conditions were used to describe the vibration and stability of the coupled belt-tensioner
system Their system also considered the geometry of the belt drive and tensioner motion
Hereafter Beikmann et al [25] predicted the belt drive vibration for a system composed of a
driving pulley driven pulley and a dynamic tensioner The authors coupled the linear equations
of transverse motion for the respective belt spans with the equations of motion for pulleys and a
tensioner This was used to form the free response of the system and evaluate its response
through a closed-form solution of the system‟s natural frequencies and mode shapes
A complex modal analysis of a serpentine belt drive system was carried out by Kraver et al [26]
to determine the effect of damping on rotational vibration mode solutions The equations of
motion developed for a multi-pulley flat belt system with viscous damping and elastic
Literature Review 17
properties including the presence of a rotary tensioner were manipulated to carry out the modal
analysis
Beikmann et al [27] also derived a nonlinear model to predict the operating state of a belt-
tensioner system by way of nonlinear numerical methods and an approximated linear closed-
form method The authors used this strategy to develop a single design parameter referred to as
a tensioner constant to measure the effectiveness of the tensioning mechanism in relation to its
operating state from a reference state The authors considered the steady state tensions in belt
spans as a result of accessory loads belt drive geometry and tensioner properties
Zhang and Zu [28] conducted a modal analysis for the response of a linear serpentine belt drive
system A non-iterative approach was used to explicitly form the equations for the system‟s
natural frequencies An exact closed-form expression for the dynamic response of the system
using eigenfunction expansion was derived with the system under steady-state conditions and
subject to harmonic excitation
The work conducted by Balaji and Mockensturm [29] considered a front-end accessory drive
(FEAD) with a decoupler or isolator attached to a pulley The rotational response for the FEAD
was found analytically by considering the system to be piecewise linear about the equilibrium
angular deflections The effect of their nonlinear terms was considered through numerical
integration of the derived equations of motion by way of the iterative methodndash fourth order
Runge-Kutta The authors in this case considered the longitudinal (ie rotational) vibration of
the belt spans only
Literature Review 18
The first to carry out the analysis of a serpentine belt drive system containing a two-pulley
tensioner was Nouri in 2005 [30] Nouri found the closed-form analytical solution of a
serpentine belt drive with a two-pulley tensioner for the case of sinusoidal excitation He
employed Runge Kutta method as well to solve the equations of motion to find the response of
the system under a general input from the crankshaft The author‟s work also included the
optimization of the tensioner design in order to minimize belt span vibrations due to crankshaft
excitation Furthermore the author applied active control techniques to the tensioner in a belt
drive system
The works discussed have made significant contributions to the research and development into
tensioner systems for serpentine belt drives These lead into the requirements for the structure
function and location of tensioner systems particularly for B-ISG transmissions
24 Tensioners for B-ISG System
241 Tensioners Structure Function and Location
Literature shows that the improvement of a serpentine belt life in a B-ISG system centers on the
tensioning mechanism redesign This mechanism as shown by researchers including
Wezenbeek et al [21] and Henry et al [11] is crucial in establishing the least tension in the belt
(above a zero value) in order to guard against failure by way of slip due to slack spans in the belt
and oscillations during engine re-start It is noted by Firbank [22] that the mechanics of a belt-
drive ldquois based on the idea that belt behaviour is governed by the elastic extension or contraction
of the belt arising from tension variationsrdquo [22] these variations may be compensated for by an
adjustable tensioner
Literature Review 19
The two types of tensioners are passive and active tensioners The former permits an applied
initial tension and then acts as an idler and normally employs mechanical power and can include
passive hydraulic actuation This type is cheaper than the latter and easier to package The latter
type is capable of continually adjusting the belt tension since it permits a lower static tension
Active tensioners typically employ electric or magnetic-electric actuation andor a combination
of active and passive actuators such as electrical actuation of a hydraulic force
Conventional belt tensioners comprise of a single tensioner arm that is fitted with a sole idler
pulley to engage a serpentine belt [31] A radial bearing is used to rotatably connect the idler
pulley to the tensioner arm [31] The tensioner arm is mounted on a pivot pin that is wrapped by
a bushing and is free to rotate [31] The pin covered by the bushing is fixed to the engine
housing [31] A rotary spring is wrapped about the bearing pin and bushing to provide a pre-
tension force to the belt via the tensioner arm and idler pulley thus taking up the slack due to the
changes in belt length [31] When the belt undergoes stretch under a load the spring drives the
tensioner arm and idler pulley further into the belt [31] Belt tension changes under the modes of
operation which can include when the crankshaft (or driving pulley) abruptly decelerates from a
steady-state condition and auxiliary components continue to rotate still in their own inherent
inertia and thus become the primary drivers [31] These fluctuations in belt tension lead to belt
flutter or skip and slip that may damage other components present in the belt drive [31]
Locating the tensioner on the slack side of the belt is intended to lower the initial static tension
[11] In conventional vehicles the engine always drives the alternator so the tensioner is located
in the belt span that links the crankshaft and alternator pulleys In a B-ISG setting the slack span
Literature Review 20
of the belt alternates between the driving mode of the ISG and the driving mode of the crankshaft
[32] Research by Henry et al [11] and also the summary of prior art for tensioners in Table
21 show that placing the idlertensioner pulley in the slack span in the case that the ISG is
driving instead of in the slack span when the crankshaft is driving allows for easier packaging
and for the least static tension Designs shown in Table 21 place the tensioneridler pulley in the
same span as Henry et al [11] or in both the slack and taut spans if using a double
tensioneridler configuration
242 Systematic Review of Tensioner Designs for a B-ISG System
The proposals for belt tensioner devices to manage the issue of high peaks in belt tension for B-
ISG settings are largely in patent records as the re-design of a tensioner has been primarily a
concern of automotive makers thus far A systematic review of the patent literature has been
conducted in order to identify evaluate and collate relevant tensioning mechanism designs
applicable to a B-ISG setting Its research objective is to influence the selection of a tensioner
configuration for the thesis study
The predefined search strategy used by the researcher has been to consider patents dating only
post-2000 as many patents dating earlier are referred to in later patents as they are developed on
in most cases by the original inventor (eg an INA Schaeffler KG patent published in 2000 may
refer to its own earlier patent presented in 1999) Patents dating pre-2000 that do not have any
successor were also considered The inclusion and exclusion criteria and rationales that were
used to assess potential patents are as follows
Inclusion of
Literature Review 21
tensioner designs with two arms andor two pivots andor two pulleys
mechanical tensioners (ie exclusion of magnetic or electrical actuators or any
combination of active actuators) in order to minimize cost
tension devices that are an independent structure apart from the ISG structure in order to
reduce the required modification to the accessory belt drive of a conventional automobile
and
advanced designs that have not been further developed upon in a subsequent patent by the
inventor or an outside party
Table 21 provides a collation of the results for the systematic review based on the selection
criteria Illustrations of the collated patent designs may be seen in Appendix A It is noted that
the patent literature pertaining to these designs in most cases provides minimal numerical data
for belt tensions achieved by the tensioning mechanism In most cases only claims concerning
the outcome in belt performance achievable by the said tension device is stated in the patent
Table 21 Passive Dual Tensioner Designs from Patent Literature
Bayerische
Motoren Werke
AG
Patents EP1420192-A2 DE10253450-A1 [33]
Design Approach
2 tensioner pulleys (idlers) and 2 tension arms are mounted outside the periphery of the belt drive these form tiltable clamping arms around a common axis of rotation
A torsion spring is used at bearing bushings to mount tension arms at ISG shaft
Each tension arm cooperates with torsion spring mechanism to rotate through a damping
device in order to apply appropriate pressure to taut and slack spans of the belt in
different modes of operation
Bosch GMBH Patent WO0026532 et al [34]
Design Approach
2 tension pulleys each one is mounted on the return and load spans of the driven and
driving pulley respectively
Idlers (tension pulleys) each connect to a spring which is attached on one end to a fixed point
Literature Review 22
Idlers‟ motions are independent of each other and correspond to the tautness or
slackness in their respective spans
Or alternatively a spring connects the idler pulleys and one of the two idlers is fixed at
its axis of rotation
Daimler Chrysler
AG
Patents DE10324268-A1 [35]
Design Approach
2 idlers are given a working force by a self-aligning bearing
Bearing supports auxiliary unit (ISG) and is arranged concentrically with the axle
auxiliary unit pulley
Dayco Products
LLC
Patents US6942589-B2 et al [36]
Design Approach
2 tension arms are each rotatably coupled to an idler pulley
One idler pulley is on the tight belt span while the other idler pulley is on the slack belt
span
Tension arms maintain constant angle between one another
One arm forms a positive differential angle with the belt and the remaining arm forms a negative differential angle with the belt
Idler pulleys are on opposite sides of the ISG pulley
Gates Corporation Patents US20060249118-A1 WO2003038309-A [37]
Design Approach
A tensioner pulley contacts the belt at the slack span during start-up (ISG-driving mode)
A tensioner is asymmetrically biased in direction tending to cause power transmission
belt to be under tension
McVicar et al
(Firm General
Motors Corp)
Patent US20060287146-A1 [38]
Design Approach
2 tension pulleys and carrier arms with a central pivot are mounted to the engine
One tension arm and pulley moderately biases one side of belt run to take up slack
during engine start-up while other tension arm and pulley holds appropriate bias against
taut span of belt
A hydraulic strut is connected to one arm to provide moderate bias to belt during normal
engine operation and velocity sensitive resistance to increasing belt forces during engine
start-up
INA Schaeffler
KG et al
Patents DE10044645-A1 [39] DE10159073-A1 [40] EP1723350-A1 et al [41]
DE10359641-A1 et al [42] EP1738093-A1 et al [43] DE102004012395-A1 [44]
WO2006108461-A1 et al [45]
Design Approach
2 tension arms and 2 pulleys approach ndash o Mutually independent tensioning arms are supported for rotation in the same
plane of the housing part
o Idler pulley corresponding to each tensioning arm engages with different
sections of belt
o When high tension span alternates with slack span of belt drive one tension
arm will increase pressure on current slack span of belt and the other will
decrease pressure accordingly on taut span
o Or when the span under highest tension changes one tensioner arm moves out
of the belt drive periphery to a dead center due to a resulting force from the taut
span of the ISG starting mode
o Deflection of the taut span acts on associated pulley to apply a counter-moment to the other idler pulley on the slack span
Literature Review 23
o The 2 lever arms are of different lengths and each have an idler pulley of
different diameters and different wrap angles of belt (see DE10045143-A1 et
al)
1 tensioner arm and 2 pulleys approach ndash
o 2 idler pulleys are pinned to a beam arranged on a clamping arm that is tiltably
linked to the beam o The ISG machine is supported by a shock absorber
o During ISG start-up one idler pulley is induced to a dead center position while
it pulls the remaining idler pulley into a clamping position until force
equilibrium takes place
o A shock absorber is laid out such that its supporting spring action provides
necessary preloading at the idler pulley in the direction of the taut span during
ISG start-up mode
Litens Automotive
Group Ltd
Patents US6506137-B2 et al [46]
Design Approach
2 tension pulleys on opposite sides of the ISG pulley engage the belt
They are positioned such that their applied forces result in opposing directed moments with respect to the tension device‟s axis of pivot
The pivot axis varies relative to the force applied to each tension pulley
Diameters of the tensioner pulleys are approximately equal and belt wrap angles of the
tensioner pulleys are approximately equal
A limited swivel angle for the tensioner arms work cycle is permitted
Mitsubishi Jidosha
Eng KK
Mitsubishi Motor
Corp
Patents JP2005083514-A [47]
Design Approach
2 tensioners are used
1 tensioner is held on the slack span of the driving pulley in a locked condition and a
second tensioner is held on the slack side of the starting (driven) pulley in a free condition
Nissan Patents JP3565040-B2 et al [48]
Design Approach
A single tensioner is on the slack span once ISG pulley is in start-up mode
The tension device is comprised of a oil pressure tensioner and a half ratchet mechanism
(a plunger which performs retreat actuation according to the energizing force of the oil
pressure spring and load received from the ISG)
The tensioner is equipped with a relief valve to keep a predetermined load lower than the
maximum load added by the ISG device
NTN Corp Patent JP2006189073-A [49]
Design Approach
An automatic tensioner is equipped with a hydraulic damper mechanism comprised of a
screw bolt using saw-screwed teeth and a cylinder nut a return spring and a spring seat
in a pressure chamber (within the screw bolt) a rod seat (that is fitted to the lower end of
the cylinder nut) a spring support (arranged on varying diameter stepped recessed
sections of the rod seat) and a check valve with an openingclosing passage
The cylinder and screw bolt act as the rigidity buffer under excessive loads during ISG
start-up mode of operation
Valeo Equipment
Electriques
Moteur
Patents EP1658432 WO2005015007 [50]
Design Approach
ldquoThe invention relates to a system or a starter (10) in which a pulley (80) is rotationally mounted on a section (22) of a shaft which axially extends inside a pulley (80) and
Literature Review 24
forwards at least partially outside a support element (200) and is characterized in that
the free front end (23) of said shaft section (22) is carried by an arm (206) connected to
the support element (200)rdquo
The author notes that published patents and patent applications may retain patent numbers for multiple patent
offices (ie European Patent Office German Patent Office etc) In such cases the published patent number or in
the absence of such a number the published patent application number has been specified However published
patent documents in the above cases also served as the document (ie identical) to the published patent if available
Quoted from patent abstract as machine translation is poor
25 Summary
The research on tensioner designs from the patent literature demonstrates a lack of quantifiable
data for the performance of a twin tensioner particularly suited to a B-ISG system The review of
the literature for the modeling theory of serpentine belt drives and design of tensioners shows
few belt drive models that are specific to a B-ISG setting Hence the literature review supports
the thesis objective of modeling a B-ISG tensioner specifically one that has a passive twin
tensioner configuration and as well measuring the tensioner‟s performance The survey of
hybrid classes reveals that the micro-hybrid class is the only class employing a closely
conventional belt transmission and hence its B-ISG transmission is applicable for tensioner
investigation The patent designs for tensioners contribute to the development of the tensioner
design to be studied in the following chapter
25
CHAPTER 3 MODELING OF B-ISG SYSTEM
31 Overview
The derivation of a theoretical model for a B-ISG system uses real life data to explore the
conceptual system under realistic conditions The literature and prior art of tensioner designs
leads the researcher to make the following modeling contributions a proposed design for a
passive two-pulley tensioner computation of geometric attributes for a B-ISG system with the
proposed tensioner and derivation of the system‟s equations of motion (EOM) under dynamic
and static states as well as deriving the EOM for the B-ISG system with only a passive single-
pulley tensioner for comparison The principles of dynamic equilibrium are applied to the
conceptual system to derive the EOM
32 B-ISG Tensioner Design
The proposed design for a passive two pulley tensioner configures two tensioners about a single
fixed pivot point in the interior space of a serpentine belt drive One end of each tensioner arm
coincides with the centre point of a tensioner pulley and this point marks the axis of rotation of
the pulley The other end of each arm is pivoted about a point so that the arms share the same
axis of rotation This conceptual design henceforth is called a Twin Tensioner Figure 31 shows
a schematic for the proposed design
Modeling of B-ISG 26
Figure 31 Schematic of the Twin Tensioner
The tensioner pulley coordinates are described by (XiYi) their radii by Ri their arm lengths Lti
and their angles θti The rotation of the arms is resisted by stiffness kt of a coil spring acting
between the two arms and spring stiffness kti acting between each arm and the pivot point The
motion of each arm is dampened by dampers and akin to the springs a damper acts between the
two arms ct and a damper cti acts between each arm and the pivot point The result is a
tensioning mechanism with four degrees of freedom (DOF) that includes independent rotations
of the two pulleys and two arms
The following section relates the geometry of the rigid bodies in a B-ISG system equipped with a
Twin Tensioner to their respective motions
Modeling of B-ISG 27
33 Geometric Model of a B-ISG System with a Twin Tensioner
The B-ISG system with the Twin Tensioner is shown in Figure 32 The geometry of the drive
provides the lengths of the belt spans and angles of wrap for the belt and pulley contact surfaces
These variables are crucial to resolve the components of forces and moment arms acting on each
rigid body in the system and are used in the derivation of the EOM in section 34 Zhen Mu‟s
geometric modeling approach [51] used in the development of the software FEAD was applied
to the Twin Tensioner system to compute the system‟s unique geometric attributes
Figure 32 B-ISG Serpentine Belt Drive with Twin Tensioner
It is noted that in Figure 31 and Figure 32 showing the schematic of the Twin Tensioner and
the overall system respectively that for the purpose of the geometric computations the forward
direction follows the convention of the numbering order counterclockwise The numbering
order is in reverse to the actual direction of the belt motion which is in the clockwise direction in
this study The fourth pulley is identified as an ISG unit pulley However the properties used
for the ISG pulley‟s geometry inertia stiffness and damping is modeled as a conventional
Modeling of B-ISG 28
alternator pulley This pulley is conceptualized as an ISG when it is modeled as the driving
pulley at which point the requirements of the ISG are solved for and its non-inertia attributes
are not needed to be ascribed
Figure 33 shows the geometric attributes needed to resolve the wrap angle of the belt on each
pulley Variables (XiYi) and XYfi XYbi XYfbi and XYbfi are the ith pulley centre coordinates and
its possible belt connection points respectively Length Lfi is the length of the span connecting
the points XYfi and XYf(i+1) or XYbi and XYb(i+1) on the ith and (i+1)th pulleys respectively
Similarly Lbi is the length of the span between XYfbi and XYfb(i+1) or XYbfi and XYbf (i+1) on the
ith and (i+1)th pulleys respectively Angles αi θfi and θbi represent the angle between a line
connecting the ith and (i+1)th pulley centres and the angles of the belt connection spans with
lengths Lfi and Lbi respectively Ri is the radius of the ith pulley
Figure 33 Angles Coordinates and Possible Belt Contact Points for the ith and i+1th Pulleys
[modified] [51]
Modeling of B-ISG 29
The angle between the horizontal and the line connecting the ith and (i+1)th pulley centres αi is
calculated using Zhen‟s method [51] This method uses the pulley‟s coordinates and a cosine
trigonometric relation
i acos
Xi 1
Xi
Xi 1
Xi
2
Yi 1
Yi
2
Yi 1
Yi
if
(31a)
i 2 acos
Xi 1
Xi
Xi 1
Xi
2
Yi 1
Yi
2
Yi 1
Yi
if
(31b)
The lengths for connecting the possible belt spans are described by the variables Lfi and Lbi
The centre point coordinates and the radii of the pulleys are related through the solution of
triangles which they form to define values of the possible belt span lengths
Lfi
Xi 1
Xi
2
Yi 1
Yi
2
Ri 1
Ri
2
(32a)
Lbi
Xi 1
Xi
2
Yi 1
Yi
2
Ri 1
Ri
2
(32b)
The set of possible belt span lengths leads to the calculation of θfi and θbi the angles between the
line connecting the ith and (i+1)th pulley centres and the possible contact point on the pulley
perimeter
Modeling of B-ISG 30
(33a)
(33b)
The array of possible belt connection points comes about from the use of the pulley centre
coordinates and their radii and the sine of the sum or differences of αi and θfi or θbi The angle
αi is calculated in equations (31a) and (31b) and angles θfi and θbi are calculated in equations
(33a) and (33b) The formula to compute the array of points is shown in equations (34) and
(35) for the ith and (i+1)th pulleys Equation (34) describes the forward belt connection point
on the ith pulley which is in the span leading forward to the next (i+1)th pulley
(34a)
(34b)
(34c)
(34d)
bi atan
Lbi
Ri
Ri 1
Modeling of B-ISG 31
Equation (35) describes the backward belt connection point on the ith pulley This point sits on
the ith pulley in the contacting belt span which leads backward to connect with the (i-1)th
pulley
(35a)
(35b)
(35c)
(35d)
The selection of the coordinates from the array of possible connection points requires a graphic
user interface allowing for the points to be chosen based on observation This was achieved
using the MathCAD software package as demonstrated in the MathCAD scripts found in
Appendix C The belt connection points can be chosen so as to have a pulley on the interior or
exterior space of the serpentine belt drive The method used in the thesis research was to plot the
array of points in the MathCAD environment with distinct symbols used for each pair of points
and to select the belt connection points accordingly By observation of the selected point types
the type of belt span connection is also chosen Selected point and belt span types are shown in
Table 31
Modeling of B-ISG 32
Table 31 Selected Contact Point Types on the ith Pulley and Types for the ith Belt Span
Pulley Forward Contact
Point
Backwards Contact
Point
Belt Span
Connection
1 Crankshaft XYf1 XYbf21 Lf1
2 Air Conditioning XYfb2 XYf22 Lb2
3 Tensioner 1 XYbf3 XYfb23 Lb3
4 AlternatorISG XYfb4 XYbf24 Lb4
5 Tensioner 2 XYbf5 XYfb25 Lb5
The inscribed angles βji between the datum and the forward connection point on the ith pulley
and βji between the datum and its backward connection point are found through solving the
angle of the arc along the pulley circumference between the datum and specified point The
wrap angle ϕi is found as the difference between the two inscribed angles for each connection
point on the pulley The angle between each belt span and the horizontal as well as the initial
angle of the tensioner arms are found using arctangent relations Furthermore the total length of
the belt is determined by the sum of the lengths of the belt spans
34 Equations of Motion for a B-ISG System with a Twin Tensioner
341 Dynamic Model of the B-ISG System
3411 Derivation of Equations of Motion
This section derives the inertia damping stiffness and torque matrices for the entire system
Moment equilibrium equations are applied to each rigid body in the system and net force
equations are applied to each belt span From these two sets of equations the inertia damping
Modeling of B-ISG 33
and stiffness terms are grouped as factors against acceleration velocity and displacement
coordinates respectively and the torque matrix is resolved concurrently
A system whose motion can be described by n independent coordinates is called an n-DOF
system Consider the free body diagram of the Twin Tensioner in Figure 34 in which each
pulley of inertia Ii is supported on an arm of inertia Iti It is assumed that the pulleys are
constrained to rotate about their respective central axes and the arms are free to rotate about their
respective pivot points then at any time the position of each pulley can be described by a
rotational coordinate θi(t) and a coordinate θti(t) can denote the rotation of each arm Thus the
tensioner system comprises of four rigid bodies where each is described by one coordinate and
hence is a four-DOF system It is important to note that each rigid body is treated as a point
mass In addition inertial rotation in the positive direction is consistent with the direction of belt
motion The belt span tensions Ti and coupled radii Ri apply moments to the pulleys
Figure 34 Twin Tensioner Free Body Diagram of a Four-Degree of Freedom System
Modeling of B-ISG 34
For the serpentine belt system considered in the thesis research there are seven rigid bodies each
having a one-DOF of motion The EOM for a seven-DOF system form second-order coupled
differential equations meaning that each equation includes all of the general coordinates and
includes up to the second-order time derivatives of these coordinates The EOM can be
obtained by applying D‟Alembert‟s principle that the sum of the moments taken about any point
including the couples equals to zero Therefore the inertial couple the product of the inertia and
acceleration is equated to the moment sum as shown in equation (35)
I ∙ θ = ΣM (35)
The moment equilibrium equations for the Twin Tensioner in Figure 34 where the positive
direction is in the clockwise direction are shown in equations (36) through to (310) The
numbering convention used for each rigid body corresponds to the labeled serpentine belt drive
system shown in Figure 32 Qi represents the required torque of the ith rigid body ci is the
damping constant of the ith rigid body βji is the angle of orientation for the ith belt span and
120597120579119905119894 120579 119905119894 and 120579 119905119894 are the angular displacement angular velocity and angular acceleration of the ith
tensioner arm The initial angle of the ith tensioner arm is described by θtoi
minusI3 ∙ θ 3 = T3 ∙ R3 minus T2 ∙ R3 minus Q3 + c3 ∙ θ 3 (36)
minusI5 ∙ θ 5 = minusT4 ∙ R5 + T5 ∙ R5 minus Q5 + c5 ∙ θ 5 (37)
Modeling of B-ISG 35
It1 ∙ θ t1 = minusTt1 ∙ Lt1 ∙ sin θto 1 minus βj4 + sin θto 1 minus βj5 minus kt ∙ partθt1 minus partθt2 minus kt1 ∙
partθt1 minus ct ∙ partθ t1 minus partθ t2 minus ct1 ∙ partθ t1 (38)
It2 ∙ θ t2 = minusTt2 ∙ Lt2 ∙ sin θto 2 minus βj2 + sin θto 1 minus βj3 minus kt ∙ partθt2 minus partθt1 minus kt2 ∙ partθt2 minus
ct ∙ partθ t2 minus partθ t1 minus ct2 ∙ partθ t2 (39)
partθt1 = θt1 minus θto 1 (310a)
partθt2 = θt2 minus θto 2 (310b)
The free body diagrams for the remaining rigid bodies crankshaft pulley air conditioner pulley
and ISG pulley are in the general form of Figure 35 The sum of the moments about the axes of
rotation are taken for these structures in equations (311) through to (313)
Figure 35 Free Body Diagram for Non-Tensioner Pulleys
Modeling of B-ISG 36
I1 ∙ θ 1 = T5 ∙ R1 minus T1 ∙ R1 + Q1 minus c1 ∙ θ 1 (311)
I2 ∙ θ 2 = T1 ∙ R2 minus T2 ∙ R2 + Q2 minus c2 ∙ θ 2 (312)
I4 ∙ θ 4 = T3 ∙ R4 minus T4 ∙ R4 + Q4 minus c4 ∙ θ 4 (313)
The relationship between belt tensions and rigid body displacements is in the general form of
equation (314) where 119827119836 and 119827119844 are damping and stiffness matrices due to the belt respectively
with each factorized by a radial arm length This relationship is described for each span in
equations (315) through to (320) The belt damping constant for the ith belt span is cib
119827prime = 119827119836 ∙ 120521 + 119827119844 ∙ 120521 (314)
T1 = To + k1b ∙ R1 ∙ θ1 minus R2 ∙ θ2 + c1
b ∙ [R1 ∙ θ 1 minus R2 ∙ θ 2] (315)
T2 = To + k2b ∙ R2 ∙ θ2 minus R3 ∙ θ3 + Lt1 ∙ [sin θto 1 minus βj2 ] ∙ (θt1 minus θto 1) + c2
b ∙ [R2 ∙ θ 2 minus R3 ∙
θ 3 + Lt1 ∙ [sin θto 1 minus βj2 ] ∙ (θ t1)] (316)
T3 = To + k3b ∙ R3 ∙ θ3 minus R4 ∙ θ4 + Lt1 ∙ [sin θto 1 minus βj3 ] ∙ (θt1 minus θto 1) + c3
b ∙ [R3 ∙ θ 3 minus R4 ∙
θ 4 + Lt1 ∙ [sin θto 1 minus βj3 ] ∙ (θ t2)] (317)
Modeling of B-ISG 37
T4 = To + k4b ∙ R4 ∙ θ4 minus R5 ∙ θ5 + Lt2 ∙ [sin θto 2 minus βj4 ] ∙ (θt2 minus θto 2) + c4
b ∙ [R4 ∙ θ 4 minus R5 ∙
θ 5 + Lt2 ∙ [sin θto 2 minus βj4 ] ∙ (θ t1)] (318)
T5 = To + k5b ∙ R5 ∙ θ5 minus R1 ∙ θ1 + Lt2 ∙ [sin θto 2 minus βj5 ] ∙ (θt2 minus θto 2) + c5
b ∙ [R5 ∙ θ 5 minus R1 ∙
θ 1 + Lt2 ∙ [sin θto 2 minus βj5 ] ∙ (θ t2)] (319)
Tprime = Ti minus To (320)
Since the applied torques on the tensioner pulleys Q3 and Q4 are zero the static equilibrium
equation of the pulleys show that the adjacent spans of each tensioner pulley are equal to each
other Hence equations (321) and (322) are denoted as follows
Tt1 = T2 = T3 (321)
Tt2 = T4 = T5 (322)
Equations (310a) (310b) and (314) through to (322) are substituted into the EOMs described
in equations (36) to (39) and (311) to (313) The newly formed equations can be arranged
and written in matrix form as shown in equations (323) through to (328) The general
coordinate matrix 120521 and its first and second derivatives are shown in the EOM below
119816 ∙ 120521 + 119810 ∙ 120521 + 119818 ∙ 120521 = 119824 (323)
Modeling of B-ISG 38
The inertia matrix I includes the inertia of each rigid body in its diagonal elements The
damping matrix C includes variables 119888119894119887 the damping of the ith belt span 119877119894 its radius 120573119895119894 its
angle 119871119905119894 the ith tensioner arm‟s length 120579119905119900119894 its initial pivot angle and 119888119905 and 119888119905119894 the ith
tensioner arm viscous damping constants Stiffness matrix K contains 119896119894119887 the ith belt span
stiffness and 119896119905 and 119896119905119894 the ith tensioner arm stiffness constants and akin to the damping
matrix the variables 119877119894 119871119905119894 120579119905119900119894 and 120573119895119894 The belt span stiffness is computed in equation
(326b) where 119870119887 represents the belt cord stiffness 119896119887 is the belt factor obtained from
experimental data 120573119895119894 is the angle of orientation for the span between the jth and ith pulleys and
ϕi is the belt wrap angle on the ith pulley
Modeling of B-ISG 39
119816 =
I1 0 0 0 0 0 00 I2 0 0 0 0 00 0 I3 0 0 0 00 0 0 I4 0 0 00 0 0 0 I5 0 00 0 0 0 0 It1 00 0 0 0 0 0 It2
(324)
119810 =
c1
b ∙ R12 + c5
b ∙ R12 + c1 minusc1
b ∙ R1 ∙ R2 0 0 minusc5b ∙ R1 ∙ R5 0 c5
b ∙ R1 ∙ Lt2 ∙ sin θto 2 minus βj5
minusc1b ∙ R1 ∙ R2 c2
b ∙ R22 + c1
b ∙ R22 + c2 minusc2
b ∙ R2 ∙ R3 0 0 c2b ∙ R2 ∙ Lt1 ∙ sin θto 1 minus βj2 0
0 minusc2b ∙ R2 ∙ R3 c3
b ∙ R32 + c2
b ∙ R32 + c3 minusc3
b ∙ R3 ∙ R4 0 C36 0
0 0 minusc3b ∙ R3 ∙ R4 c4
b ∙ R42 + c3
b ∙ R42 + c4 minusc4
b ∙ R4 ∙ R5 minusc3b ∙ R4 ∙ Lt1 ∙ sin θto 1 minus βj3 c4
b ∙ R4 ∙ Lt2 ∙ sin θto 2 minus βj4
minusc5b ∙ R1 ∙ R5 0 0 minusc4
b ∙ R4 ∙ R5 c5b ∙ R5
2 + c4b ∙ R5
2 + c5 0 C57
0 0 0 0 0 ct +ct1 minusct
0 0 0 0 0 minusct ct +ct1
(325a)
C36 = 1198773 ∙ 1198711199051 ∙ [1198883119887 ∙ 119904119894119899 1205791199051199001 minus 1205731198953 minus 1198882
119887 ∙ 119904119894119899 1205791199051199001 minus 1205731198952 ] (325b)
C57 = 1198775 ∙ 1198711199052 ∙ [1198885119887 ∙ 119904119894119899 1205791199051199002 minus 1205731198955 minus 1198884
119887 ∙ 119904119894119899 1205791199051199002 minus 1205731198954 ] (325c)
Modeling of B-ISG 40
119818 =
k1
b ∙ R12 + k5
b ∙ R12 minusk1
b ∙ R1 ∙ R2 0 0 minusk5b ∙ R1 ∙ R5 0 k5
b ∙ R1 ∙ Lt2 ∙ sin θto 2 minus βj5
minusk1b ∙ R1 ∙ R2 k2
b ∙ R22 + k1
b ∙ R22 minusk2
b ∙ R2 ∙ R3 0 0 k2b ∙ R2 ∙ Lt1 ∙ sin θto 2 minus βj2 0
0 minusk2b ∙ R2 ∙ R3 k3
b ∙ R32 + k2
b ∙ R32 minusk3
b ∙ R3 ∙ R4 0 R3 ∙ Lt1 ∙ [k3b ∙ sin θto 1 minus βj3 minus k2
b ∙ sin θto 1 minus βj2 ] 0
0 0 minusk3b ∙ R3 ∙ R4 k4
b ∙ R42 + k3
b ∙ R42 minusk4
b ∙ R4 ∙ R5 minusk3b ∙ R4 ∙ Lt1 ∙ sin θto 1 minus βj3 k4
b ∙ R4 ∙ Lt2 ∙ sin θto 2 minus βj4
minusk5b ∙ R1 ∙ R5 0 0 minusk4
b ∙ R4 ∙ R5 k5b ∙ R5
2 + k4b ∙ R5
2 0 R5 ∙ Lt2 ∙ [k5b ∙ sin θto 2 minus βj5 minus k4
b ∙ sin θto 2 minus βj4 ]
0 0 0 0 0 kt +kt1 minuskt
0 0 0 0 0 minuskt kt +kt1
(326a)
k119894b =
Kb
Li + kb ∙ Ri ∙ϕi+1
2 + Ri ∙ϕi
2
(326b)
120521 =
θ1
θ2
θ3
θ4
θ5
partθt1
partθt2
(327)
119824 =
Q1
Q2
Q3
Q4
Q5
Qt1
Qt2
(328)
Modeling of B-ISG 41
3412 Modeling of Phase Change
The phase change from the crankshaft pulley being the driving pulley to the ISG pulley being the
driving pulley is described through a conditional equality based on a set of Boolean conditions
When the crankshaft is driving the rows and the columns of the EOM are swapped such that the
new order for rows and columns is 1 (crankshaft pulley) 4 (ISG pulley) 2 (air conditioner
pulley) 3 (tensioner 1 pulley) 5 (tensioner 2 pulley) 6 (tensioner arm 1) and 7 (tensioner arm 2)
When the ISG is driving the order is the same except that the second row and second column
terms relating to the ISG pulley become the first row and first column while the crankshaft
pulley terms (previously in the first row and first column) become the second row and second
column Hence the order for all rows and columns of the matrices making up the EOM in
equation (322) switches between 1423567 (when the crankshaft pulley is driving) and
4123567 (when the ISG pulley is driving) For example in the crankshaft driving and ISG
driving phases the general coordinate matrix and the inertia matrix become the following
120521119940 =
1205791
1205794
1205792
1205793
1205795
1205971205791199051
1205971205791199052
and 120521119938 =
1205794
1205791
1205792
1205793
1205795
1205971205791199051
1205971205791199052
(329a amp b)
119816119940 =
I1 0 0 0 0 0 00 I4 0 0 0 0 00 0 I2 0 0 0 00 0 0 I3 0 0 00 0 0 0 I5 0 00 0 0 0 0 It1 00 0 0 0 0 0 It2
and 119816119938 =
I4 0 0 0 0 0 00 I1 0 0 0 0 00 0 I2 0 0 0 00 0 0 I3 0 0 00 0 0 0 I5 0 00 0 0 0 0 It1 00 0 0 0 0 0 It2
(329c amp d)
Modeling of B-ISG 42
where subscripts c and a denote the crankshaft pulley driving phase and the ISG pulley driving
phase respectively
The condition for phase change is based on the engine speed n in units of rpm Equation (330)
demonstrates the phase change
H(n) = 1 119899 ge 750 (Crankshaft driving phase)0 119899 lt 750 (ISG driving phase)
(330)
When the crankshaft pulley is the driving pulley the ISG pulley becomes the driven pulley and
following suit when the ISG pulley is the driving pulley the crankshaft pulley becomes the
driven pulley These modes of operation mean that the system will predict two different sets of
natural frequencies and mode shapes Using a Boolean condition to allow for a swap between
the first and second rows as well as between the first and second columns of the EOM matrices
I C and K allows for a continuous plot of the dynamic response to be plotted for the ISG pulley
throughout its driving and driven phases as well as for that of the crankshaft pulley
3413 Natural Frequencies Mode Shapes and Dynamic Responses
Assuming the system undergoes simple harmonic motion its matrix of natural frequencies 120596119899
and modeshapes are found by solving the eigenvalue problem shown in equation (331a)
ωn ∙ 119816120784120784 minus 11981822 ∙ 120495m = 120782 (331a)
The displacement amplitude Θm is denoted implicitly in equation (331d)
Modeling of B-ISG 43
120521119846 = θ2 θ3 θ5 θ6 partθt1 partθt2 T for H n = 1 (331b)
120521119846 = θ1 θ3 θ5 θ6 partθt1 partθt2 T for H n = 0 (331c)
θm = 120495119846 ∙ sin(ω ∙ t) (331d)
I2 and K22 are submatrices of I and K respectively meaning the first row and column of each of
the original matrices are removed The eigenvalue problem is reached by considering the
undamped and unforced motion of the system Furthermore the dynamic responses are found by
knowing that the torque requirements in the matrixndash Qm for the driven pulleys and the tensioner
arms are zero in the dynamic case which signifies a response of the system to an input solely
from the driving pulley
I1 120782120782 119816120784120784
θ 1120521 119846
+ C11 119810120783120784119810120784120783 119810120784120784
θ 1120521 119846
+ K11 119818120783120784
119818120784120783 119818120784120784 θ1
120521119846 =
QCS ISG
119824119846 (332)
1
In the case of equation (331) θm is the submatrix identified in equations (331b) through to
(331d) Therein θ1 denotes the general coordinate for the driving pulley so that in the case the
phase change function H(n) is equal to zero θ1 becomes θ4 and the order of the rows and
columns for the remaining matrices correspond to the value of H(n) as mentioned earlier in
section 3412 For simple harmonic motion the motion of the driven pulleys are described as
1 The driving torque 119876119862119878119868119878119866 denotes the crankshaft torque 119876119862119878 when the crankshaft pulley is driving or the ISG
torque 119876119868119878119866 when the ISG pulley is in its driving function
Modeling of B-ISG 44
θm = 120495119846 ∙ sin(ω ∙ t) (333)
The dynamic response of the system to an input from the driving pulley under the assumption of
sinusoidal motion is expressed in equation (334)
120495119846 = [(119818120784120784 minusω2 ∙ 119816120784120784) + 119895ω ∙ 119810120784120784]minus1 ∙ (119818120784120783 + 119895ω ∙ 119810120784120783) ∙ Θ1 (334)
3414 Crankshaft Pulley Driving Torque Acceleration and Displacement
Subsequently the crankshaft pulley driving torque acceleration and displacement are firstly
discussed It is assumed in the thesis research for the purpose of modeling that the engine
serving the crankshaft is of the four cylinder type The input torque provided by a four-cylinder
engine is assumed to be dominated by two torque pulses per revolution of the crankshaft which
is represented by the factor of 2 on the steady component of the angular velocity in equation
(335) The torque requirement of the crankshaft pulley when it is the driving pulley is
Qc = qc ∙ sin(2 ∙ ωcs ∙ t) (335)
The amplitude of the required crankshaft torque qc is expressed in equation (336) and is
derived from equation (332)
qc = K11 minus ω2 ∙ I1 + 119895 ∙ ω ∙ C11 ∙ Θ1 + (119818120783120784 + 119895 ∙ ω ∙ 119810120783120784) ∙ 120495119846 (336)
Modeling of B-ISG 45
The angular frequency for the system in radians per second (rads) ω when the crankshaft
pulley is driving can be found as a function of the engine speed in rotations per minute (rpm) n
and by taking into account the double pulse per crankshaft revolution
ω = 2 ∙ ωcs = 4 ∙ π ∙ n
60
(337)
The system is considered when the amplitude of the crankshaft‟s angular acceleration is assumed
to be constant and equal to 650 rads2 during the crankshaft pulley driving phase The amplitude
of the excitation angular input from the engine is shown in equation (339b) and is found as a
result of (338)
θ 1CS = 650 ∙ sin(ω ∙ t) (338)
θ1CS = minus650
ω2sin(ω ∙ t)
(339a) where
Θ1CS = minus650
ω2
(339b)
Modeling of B-ISG 46
3415 ISG Pulley Driving Torque Acceleration and Displacement
Secondly the torque acceleration and the displacement of the ISG pulley in its driving phase is
discussed The torque for the ISG when it is in its driving function is assumed constant Ratings
for the ISG are taken from experiments performed by researchers Wezenbeek et al [21] on an
Energen 5 High Output Belt-alternator-starter (BAS) unit from Delphi The 12-Volt BAS which
can also be called a B-ISG was reported to have a maximum allowable speed of 18000 rpm [21]
As well it was noted that the ISG pulley was sized appropriately and the engine speed was
limited to ensure that an over-speed condition of the ISG pulley would not occur [21] The stall
torque rating for the Energen ISG was reported to be 48 Nm at the electric machine shaft [21]
The formula for the torque of a permanent magnet DC motor for any given speed (equation
(340)) is used to approximate the torque of the ISG in its driving mode[52]
QISG = Ts minus (N ∙ Ts divide NF) (340)2
Knowing the stall torque (the torque at 0 rpm) Ts and the maximum rpm of the motor when it is
not under load NF allows for the torque produced 119876119868119878119866 to be found for a given motor speed N
Experimental data from Litens Automotive Group [53] shows that for engine fire-up upon ISG
re-start the crankshaft must go from 0 rpm to an idle speed of approximately 750 rpm The
pulley installed on the ISG shaft in the case of the thesis research has a diameter of 6820 mm
(DISG) while that of the crankshaft has a diameter of 20065 mm (DCS) which makes the
2 The equation for the required driving torque for the ISG pulley may also be computed from the formula shown in
(336) Figure 315 for the driving torque of the ISG pulley shows that (336) and (340) produce similar results for
the required driving torque See Figure 315 for comparison of these results
Modeling of B-ISG 47
crankshaft to ISG pulley ratio approximately 2941 This ratio is used to determine the ISG
speed in equation (341)
nISG = nCS ∙DCS
DISG
(341)
For a crankshaft speed of 750 rpm the required ISG speed nISG is found from equation (341) to
be approximately 220656 rpm Thus the ISG torque during start-up is found from equation
(340) where N is equated to the value of nISG NF is assumed to be 18000 rpm and the stall
torque is allotted the value of 48 Nm The result is a required torque of approximately 42 Nm
for the ISG The acceleration of the ISG pulley is found by taking into account the torque
developed by the rotor and the polar moment of inertia of the pulley [54]
A1ISG = θ 1ISG = QISG IISG (342)
In torsional motion the function for angular displacement of input excitation is sinusoidal since
the electric motor is assumed to be resonating As a result of constant angular acceleration the
angular displacement of the ISG pulley in its driving mode is found in equation 343
θ1ISG = Θ1ISG ∙ sin(ωISG ∙ t) (343)
Knowing that acceleration is the second derivative of the displacement the amplitude of
displacement is solved subsequently [55]
Modeling of B-ISG 48
θ 1ISG = minusωISG2 ∙ Θ
1ISG ∙ sin(ωISG ∙ t) (344)
θ 1ISG = minusωISG2 ∙ Θ
1ISG
(345a)
Θ1ISG =minusQISG IISG
ωISG2
(345b)
In this case the angular frequency for the system 120596 is equivalent to 120596119868119878119866 that is the angular
frequency of the ISG pulley which can be expressed as a function of its speed in rpm
ω = ωISG =2 ∙ π ∙ nISG
60
(346a)
or in terms of the crankshaft rpm by substituting equation (341) into (346a)
ω =2 ∙ π
60∙ nCS ∙
DCS
DISG
(346b)
3416 Tensioner Arms Dynamic Torques
The dynamic torque for the tensioner arms are shown in equations (347) and (348)
Qt1 = kt + kt1 + 119895 ∙ ω ∙ (ct + ct1) ∙ (Θt1 ∙ Θ1) (347)
Modeling of B-ISG 49
Qt2 = kt + kt2 + 119895 ∙ ω ∙ (ct + ct2) ∙ (Θt2 ∙ Θ1) (348)
3417 Dynamic Belt Span Tensions
Furthermore the dynamic belt span tensions are derived from equation (314) and described in
matrix form in equations (349) and (350)
119827prime = 119895 ∙ ω ∙ 119827119836 + 119827119844 ∙ 120495119847 (349)
where
120495119847 = Θ1
120495119846 (350)
342 Static Model of the B-ISG System
It is fitting to pursue the derivation of the static model from the system using the dynamic EOM
For the system under static conditions equations (314) and (323) simplify to equations (351)
and (352) respectively
119827prime = 119827119844 ∙ 120521 (351)
119824 = 119818 ∙ 120521 (352)
Modeling of B-ISG 50
As noted in other chapters the focus of the B-ISG tensioner investigation especially for the
parametric and optimization studies in the subsequent chapters is to determine its effect on the
static belt span tensions Therein equations (351) and (352) are used to derive the expressions
for static tension in each belt span 119931prime is the tension solely due to deflection of the belt span
Equation (320) demonstrates the relationship between the tension due to belt response and the
initial tension also known as pre-tension The static tension 119931 is found by summing the initial
tension 1198790 with the expression for the dynamic tension shown in equations (315) through to
(319) and by substituting the expressions for the rigid bodies‟ displacements from equation
(352) and the relationship shown in equation (320) into equation (351)
119827 = 119827119844 ∙ (119818minus120783 ∙ 119824) + T0 (353)3
35 Simulations
The methods used to develop the geometric dynamic and static models of the Twin Tensioner B-
ISG system in the previous sections of this chapter were verified using the software FEAD The
input data for a single tensioner B-ISG system was entered into FEAD [51] to simulate the
crankshaft driving phase alone since the ISG phase is inapplicable in the FEAD [51] software
FEAD‟s [51] results agreed with those found in the simulation of the single tensioner system‟s
geometric model and EOMs in MathCAD software Furthermore the geometric simulation
3 For the purposes of the static tension the original order for the rows and columns of the stiffness matrix K and the
torque matrix Q are maintained as depicted in (326) and (328) In performing the inverse of K and its
multiplication with Q the first row and first column (in the case of the K matrix) are removed in the crankshaft
driving case whereas the fourth row and fourth column are removed in the ISG driving case Then the product for
the displacement120637 resulting from (119922minus120783 ∙ 119928) has a zero added to serve as the first element of the column matrix in
the crankshaft driving case or as the fourth element in the ISG driving case This is shown in detail in Appendix
C3 of MathCAD scripts
Modeling of B-ISG 51
results for both of the twin and single tensioner B-ISG systems were found to be in agreement as
well
351 Geometric Analysis
The initial coordinate inputs for the centre points of the five pulleys and the Twin Tensioner
pivot point are described as Cartesian coordinates and shown in Table 32 which also includes
the diameters for the pulleys
Table 32 Coordinate Points for Pulley Centres and Twin Tensioner Pivot [56]
Rigid Body Diameter [mm] Cartesian Coordinate [Xi Yi] [mm]
1Crankshaft Pulley 20065 [00]
2 Air Conditioner Pulley 10349 [224 -6395]
3 Tensioner Pulley 1 7240 [292761 87]
4 ISG Pulley 6820 [24759 16664]
5 Tensioner Pulley 2 7240 [12057 9193]
6 Tensioner Arm Pivot --- [201384 62516]
The geometric results for the B-ISG system are shown in Table 33
Table 33 Geometric Results of B-ISG System with Twin Tensioner
Pulley Forward
Connection Point
Backward
Connection Point
Wrap
Angle
ϕi (deg)
Angle of
Belt Span
βji (deg)
Length of
Belt Span
Li (mm)
1 Crankshaft [-6818-100093] [453889475] 202996 356103 227828
2 Air
Conditioning [275299-5717] [220484 -115575] 101425 277528 14064
3 Tensioner 1 [25887599735] [256873 82257] 28126 69403 58658
4 ISG [218374184225] [27951154644] 169554 58956 129513
5 Tensioner 2 [10419659645] [15158673262] 8585 333107 65949
Total Length of Belt (mm) 1243
Modeling of B-ISG 52
352 Dynamic Analysis
The dynamic results for the system include the natural frequencies mode shapes driven pulley
and tensioner arm responses the required torque for each driving pulley the dynamic torque for
each tensioner arm and the dynamic tension for each belt span These results for the model were
computed in equations (331a) through to (331d) for natural frequencies and mode shapes in
equation (334) for the driven pulley and tensioner arm responses in equation (336) for the
crankshaft pulley driving torque in equation (340) for the ISG pulley driving torque in
equations (347) and (348) for the tensioner arm torques and lastly in equation (349) for the
dynamic tension of each belt span Figures 36 through to 323 respectively display these
results The EOM simulations can also be contrasted with those of a similar system being a B-
ISG serpentine belt drive that is equipped with a single tensioner arm and single tensioner pulley
which interacts only in the span bridging the ISG and crankshaft pulleys The EOM for a B-ISG
with a single tensioner is presented in Appendix B
It is assumed for the sake of the dynamic and static computations that the system
does not have an isolator present on any pulley
has negligible rotational damping of the pulley shafts
has negligible belt span damping and that this damping does not differ amongst
spans (ie c1b = ∙∙∙ = ci
b = 0)
has quasi-static belt stretch where its belt experiences purely elastic deformation
has fixed axes for the pulley centres and tensioner pivot
has only one accessory pulley being modeled as an air conditioner pulley and
Modeling of B-ISG 53
has a rotational belt response that is decoupled from the transverse response of the
belt
The input parameter values of the dynamic (and static) computations as influenced by the above
assumptions for the present system equipped with a Twin Tensioner are shown in Table 34
Table 34 Data for Input Parameters used in Dynamic and Static Computations [56]
Rigid Body Data
Pulley Inertia
[kg∙mm2]
Damping
[N∙m∙srad]
Stiffness
[N∙mrad]
Required
Torque
[Nm]
Crankshaft 10 000 0 0 4
Air Conditioner 2 230 0 0 2
Tensioner 1 300 1x10-4
0 0
ISG 3000 0 0 5
Tensioner 2 300 1x10-4
0 0
Tensioner Arm 1 1500 1000 10314 0
Tensioner Arm 2 1500 1000 16502 0
Tensioner Arm
couple 1000 20626
Belt Data
Initial belt tension [N] To 300
Belt cord stiffness [Nmmmm] Kb 120 00000
Belt phase angle at zero frequency [deg] φ0deg 000
Belt transition frequency [Hz] ft 000
Belt maximum phase angle [deg] φmax 000
Belt factor [magnitude] kb 0500
Belt cord density [kgm3] ρ 1000
Belt cord cross-sectional area [mm2] A 693
Modeling of B-ISG 54
These values are for the driven cases for the ISG and crankshaft pulleys respectively In the
driving case for either pulley the inertia of the rigid body is defined as 1 kg∙mm2 and the driving
torque is determined in equations (335) and (340) for the crankshaft and ISG pulleys
respectively
It is noted that because of the belt data for the phase angle at zero frequency the transition
frequency and the maximum phase angle are all zero and hence the belt damping is assumed to
be constant between frequencies These three values are typically used to generate a phase angle
versus frequency curve for the belt where the phase angle is dependent on the frequency The
curve defined by equation (354) is normally symmetric with the lowest phase angle achieved at
0 Hz and the highest phase angle achieved at the prescribed transition frequency f The belt
damping would then be found by solving for cb in the following equation
tanφ = cb ∙ 2 ∙ π ∙ f (354)
Nevertheless the assumption for constant damping between frequencies is also in harmony with
the remaining assumptions which assume damping of the belt spans to be negligible and
constant between belt spans
3521 Natural Frequency and Mode Shape
The set of natural frequencies and mode shapes for the system are shown in Figures 36 and 37
under the cases of the ISG pulley driving and the crankshaft pulley driving The forcing
frequency for the system differs for each case due to the change in driving pulley Modeic and
Modeia denote the ith rigid body according to the numbering convention used in Figure 32 in
the crankshaft and ISG driving cases respectively
Modeling of B-ISG 55
Natural Frequency ωn [Hz]
Crankshaft Pulley ΔΘ4
Air Conditioner Pulley ΔΘ
Tensioner Pulley 1 ΔΘ
Tensioner Pulley 2 ΔΘ
Tensioner Arm 1 ΔΘ
Tensioner Arm 2 ΔΘ
Figure 36a ISG Driving Case Natural Frequency of System and Mode Shapes for Responsive
Rigid Bodies
Figure 36b ISG Driving Case First Mode Responses
4 ΔΘ signifies the term amplitude of angular displacement for the indicated rigid body
Modeling of B-ISG 56
Figure 36c ISG Driving Case Second Mode Responses
Natural Frequency ωn [Hz]
ISG Pulley ΔΘ5
Air Conditioner Pulley ΔΘ
Tensioner Pulley 1 ΔΘ
Tensioner Pulley 2 ΔΘ
Tensioner Arm 1 ΔΘ
Tensioner Arm 2 ΔΘ
Figure 37a Crankshaft Driving Case Natural Frequency of System and Mode Shapes for
Responsive Rigid Bodies
5 ΔΘ signifies the term amplitude of angular displacement for the indicated rigid body
Modeling of B-ISG 57
Figure 37b Crankshaft Driving Case First Mode Responses
Figure 37c Crankshaft Driving Case Second Mode Responses
Modeling of B-ISG 58
3522 Dynamic Response
The dynamic response specifically the magnitude of angular displacement for each rigid body is
plotted in Figures 38 through to 314 as a function of the crankshaft pulley speed n This is
fitting to the analysis since the crankshaft pulley‟s rpm decides the mode of operation for the
system in particular it determines whether the crankshaft pulley or ISG pulley is the driving
pulley
Figure 38 Crankshaft Pulley Dynamic Response (for crankshaft driven case)
Figure 39 ISG Pulley Dynamic Response (for ISG driven case)
Modeling of B-ISG 59
Figure 310 Air Conditioner Pulley Dynamic Response
Figure 311 Tensioner Pulley 1 Dynamic Response
Crankshaft Driving Phase ISG
Driving
Phase
Crankshaft Driving Phase ISG
Driving
Phase
Modeling of B-ISG 60
Figure 312 Tensioner Pulley 2 Dynamic Response
Figure 313 Tensioner Arm 1 Dynamic Response
Crankshaft Driving Phase ISG
Driving
Phase
Crankshaft Driving Phase ISG
Driving Phase
Modeling of B-ISG 61
Figure 314 Tensioner Arm 2 Dynamic Response
3523 ISG Pulley and Crankshaft Pulley Torque Requirement
Figures 315 and 316 respectively showcase the required torques for the ISG pulley in its driving
mode and the crankshaft pulley in its driving mode
Figure 315 Required Driving Torque for the ISG Pulley
Crankshaft Driving Phase ISG
Driving Phase
Modeling of B-ISG 62
Figure 315 shows two plots for the required driving torque of the ISG pulley The dashed line
labeled as Q(n) simulates the application of equation (340) which models the ISG torque as a
permanent magnet DC motor The additional solid line labeled as qamod uses the formula in
equation (336) which determines the load torque of the driving pulley based on the pulley
responses Figure 315 provides a comparison of the results
Figure 316 Required Driving Torque for the Crankshaft Pulley
3524 Tensioner Arms Torque Requirements
The torque for the tensioner arms are shown in Figures 317 and 318
Modeling of B-ISG 63
Figure 317 Dynamic Torque for Tensioner Arm 1
Figure 318 Dynamic Torque for Tensioner Arm 2
3525 Dynamic Belt Span Tension
The dynamic tensions for the belt spans are shown in Figures 319 through to 323 The values
plotted represent the magnitude of the dynamic tension
Crankshaft Driving Phase ISG
Driving Phase
Crankshaft Driving Phase ISG
Driving
Phase
Modeling of B-ISG 64
Figure 319 Span 1 (between crankshaft and air conditioner) Dynamic Belt Span Tension
Figure 320 Span 2 (between air conditioner and tensioner 1) Dynamic Belt Span Tension
Crankshaft Driving Phase ISG
Driving Phase
Crankshaft Driving Phase ISG
Driving
Phase
Modeling of B-ISG 65
Figure 321 Span 3 (between tensioner 1 and ISG) Dynamic Belt Span Tension
Figure 322 Span 4 (between ISG and tensioner 2) Dynamic Belt Span Tension
Crankshaft Driving Phase ISG
Driving
Phase
Crankshaft Driving Phase ISG
Driving Phase
Modeling of B-ISG 66
Figure 323 Span 5 (between tensioner 2 and crankshaft) Dynamic Belt Span Tension
The dynamic results for the system serve to show the conditions of the system for a set of input
parameters The following chapter targets the focus of the thesis research by analyzing the affect
of changing the input parameters on the static conditions of the system It is the static results that
are the focus of the thesis and is thus analyzed in Chapters 4 and 5 in the parametric and
optimization studies respectively The dynamic analysis has been used to complete the picture of
the system‟s state under set values for input parameters
353 Static Analysis
Before looking at the static results for the system under study in brevity the static results for a
B-ISG serpentine belt drive with a single tensioner are presented In this theoretical system the
tensioner arm and tensioner pulley that interacts with the span between the air conditioner and
ISG pulleys of the original system are removed as shown in Figure 324
Crankshaft Driving Phase ISG
Driving
Phase
Modeling of B-ISG 67
Figure 324 B-ISG Serpentine Belt Drive with Single Tensioner
The complete static model as well as the dynamic model for the system in Figure 324 is found
in Appendix B The results of the static tension for each belt span of the single tensioner system
when the crankshaft is driving and the ISG is driving are shown in Table 35
Table 35 Static Solution for Belt Span Tensions in Crankshaft and ISG Driving Cases for a B-
ISG Serpentine Belt Drive with a Single Tensioner
Belt Span between Pulleys Crankshaft Driving Case [N] ISG Driving Case [N]
Crankshaft ndash Air Conditioner 481239 -361076
Air Conditioner ndash ISG 442588 -399727
ISG ndash Tensioner 29596 316721
Tensioner ndash Crankshaft 29596 316721
The tensions in Table 35 are computed with an initial tension of 300N This value for pre-
tension allows the spans in the case that the crankshaft pulley is driving to be suitably tensioned
Modeling of B-ISG 68
Whereas in the case of the ISG pulley driving the first and second spans are excessively slack
Therein an additional pretension of approximately 400N would be required which would raise
the highest tension span to over 700N This leads to the motivation of the thesis researchndash to
reduce the static belt tensions when the ISG is driving As mentioned in Chapter 1 these
tensions should be minimized to prolong belt life preferably within the range of 600 to 800N
As well it is desirable to minimize the amount of pretension exerted on the belt The current
design uses a pre-tension of 300N The above results would lead to a required pre-tension of
more than 700N to keep all spans of the belt suitably in tension (well above 0N) in order to allow
the belt to exhibit high performance in power transmission and come near to the safe threshold
This is the rationale for investigating a Twin Tensioner configuration shown in Figure 32 for
the B-ISG serpentine belt drive under study For the theoretical system with a Twin Tensioner
the following static results in Table 36 are achieved
Table 36 Static Solution for Belt Span Tensions in Crankshaft and ISG Driving Cases for a B-
ISG Serpentine Belt Drive with a Twin Tensioner
Belt Span between Pulleys Crankshaft Driving Case [N] ISG Driving Case [N]
Crankshaft ndash Air Conditioner 465848 -284152
Air Conditioner ndash Tensioner 1 427197 -322803
Tensioner 1 ndash ISG 427197 -322803
ISG ndash Tensioner 2 28057 393645
Tensioner 2 ndash Crankshaft 28057 393645
The results in Table 36 show that the span following the ISG in the case between the Tensioner
1 and ISG pulleys is less slack than in the former single tensioner set-up However there
remains an excessive amount of pre-tension required to keep all spans suitably tensioned
Modeling of B-ISG 69
36 Summary
The simulation of the model for the B-ISG system with the Twin Tensioner shows that the mode
shapes of the rigid bodies within the system (Figures 36a to 37c) are greater in magnitude when
the ISG pulley is driving than when the crankshaft pulley is driving The dynamic responses of
the system as shown in Figures 38 and 310 to 314 is small for the crankshaft pulley and are
negligible for the remaining driven bodies when the ISG is driving For the crankshaft driving
phase there is greater dynamic response for the driven rigid bodies of the system including for
that of the ISG pulley
As the engine speed increases the torque requirement for the ISG was found to vary between
approximately 41Nm and 54Nm (before dropping steeply to approximately 3Nm at an engine
speed of about 720rpm) when modeled after equation (336) or between approximately 48Nm
and 34Nm when modeled after equation (340) In contrast the torque for the crankshaft peaks
at approximately 92Nm and 52Nm at an approximate engine speed of 1450rpm and 5000rpm
respectively The dynamic torque of the first tensioner arm was shown to peak at approximately
15Nm at the transition engine speed 750rpm and again at approximately 15Nm at an
approximate engine speed of about 1450rpm A small peak of about 3Nm was also seen at an
engine speed of 5000rpm Similarly for the second tensioner arm a torque peak of
approximately 20Nm was seen at 750rpm and 1450rpm and a smaller peak of about 8Nm was
seen at an engine speed of 5000rpm
The trend for the dynamic tensions is that the peaks are highest in the ISG driving portion of the
B-ISG operation in most cases and in a few cases they are seen to be close in magnitude to that
Modeling of B-ISG 70
of the highest peaks in the crankshaft driving portion The dynamic tension for the first belt span
peaked at approximately 780Nm 830Nm and 500Nm at engine speeds of 750rpm 1450rpm
5000rpm respectively For the dynamic tension of the second belt span peaks of approximately
1250Nm 675Nm and 760Nm were seen at the same respective engine speeds for the 3 peaks of
the former span At these same engine speeds the third belt span exhibited tension peaks at
approximately 1400Nm 650Nm and 890Nm The tension peaks of the fourth span were
approximately 165Nm 150Nm and 100Nm at engine speeds 750rpm 1450rpm and 5000rpm
The fifth span experienced peaks of approximately 165Nm 170Nm and 120Nm at the same
respective engine speeds of the fourth span
The simulation results for the static tension of the B-ISG system with the Twin Tensioner reveal
that taut spans of the crankshaft driving case are lower in the ISG driving case The largest
change is an approximate decrease of 750N in spans 1 through 3 while spans 4 and 5 increase
by approximately 113N It can be seen that the spans in highest tension (1 2 and 3) in the
crankshaft driving phase become excessively slack in the ISG driving phase There is a smaller
change between the tension values for the spans in the least tension in the crankshaft driving
phase and their corresponding span in the ISG driving phase
The summary of the simulation results are used as a benchmark for the optimized system shown
in Chapter 5 The static tension simulation results are investigated through a parametric study of
the Twin Tensioner system in Chapter 4 The optimization of the system is then based on the
selected design variables from the outcome of Chapter 4
71
CHAPTER 4 PARAMETRIC ANALYSIS OF A B-ISG
TWIN TENSIONER
41 Introduction
The parameters for the proposed Twin Tensioner for a Belt-driven Integrated Starter-generator
(B-ISG) system are investigated through a parametric analysis This analysis seeks to understand
how changing one parameter influences the static belt span tensions for the system Since the
thesis research focuses on the design of a tensioning mechanism to support static tension only
the parameters specific to the actual Twin Tensioner and applicable to the static case were
considered The parameters pertaining to accessory pulley properties such as radii or various
belt properties such as belt span stiffness are not considered In the analyses a single parameter
is varied over a prescribed range while all other parameters are held constant The pivot point
described by Cartesian Coordinates [X6Y6] for the tensioner arms is held constant in all cases
42 Methodology
The parametric study method applies to the general case of a function evaluated over changes in
one of its dependent variables The methodology is illustrated for the B-ISG system‟s function
for static tension which is evaluated for each change in one of its Twin Tensioner‟s parameters
The original data used for the system is based on sample vehicle data provided by Litens [56]
Table 41 provides the initial data for the parameters as well as the incremental change and
maxima and minima limits The increment Δi for the ith parameter is chosen arbitrarily Limits
for each parameter have been chosen to be plus or minus sixty percent of its initial value
Parametric Analysis 72
Table 41 Initial Values Increments and Ranges for Parameters of Twin Tensioner
Parameter Name Initial Value Increment (+- Δi) Minimum
value Maximum value
Coupled Spring
Stiffness kt
20626
N∙mrad 1238 N∙mrad 8250 N∙mrad 33002 N∙mrad
Tensioner Arm 1
Stiffness kt1
10314
N∙mrad 0619 N∙mrad 4126 N∙mrad 16502 N∙mrad
Tensioner Arm 2
Stiffness kt2
16502
N∙mrad 0990 N∙mrad 6601 N∙mrad 26403 N∙mrad
Tensioner Pulley 1
Diameter D3 007240 m 4344 ∙ 10
-3 m 00290 m 0116 m
Tensioner Pulley 2
Diameter D5 007240 m 4344 ∙ 10
-3 m 00290 m 0116 m
Tensioner Pulley 1
Initial Coordinates
[0292761
0087] m See Figure 41 for region of possible tensioner pulley
coordinates Tensioner Pulley 2
Initial Coordinates
[012057
009193] m
The mesh of possible points for the centre coordinates of tensioner pulley 1 and tensioner pulley
2 are designated as Region 1 and Region 2 respectively in Figures 41a and 41b
Figure 41a Regions 1 and 2 and their Associated Guidelines for Coordinates of Tensioner
Pulleys 1 amp 2
CS
AC
ISG
Ten 1
Ten 11
Region II
Region I
Parametric Analysis 73
Figure 41b Regions 1 and 2 in Cartesian Space
The selection for the minimum and maximum tensioner pulley centre coordinates and their
increments are not selected arbitrarily or without derivation as the other tensioner parameters
The coordinates for the pulley centres are identified using Intergraph‟s SmartSketch software a
graphing suite in MathCAD to model the regions of space The following descriptions are used
to describe the possible positions for the tensioner pulleys
Tensioner pulleys are situated such that they are exterior to the interior space created by
the serpentine belt thus they sit bdquooutside‟ the belt loop
The highest point on the tensioner pulley does not exceed the tangent line connecting the
upper hemispheres of the pulleys on either side of it
The tensioner pulleys may not overlap any other pulley
Parametric Analysis 74
Boundaries for regions described as Region 1 in span 2 and 3 and Region 2 in span 4
and 5 is selected based on the above criteria and their lower boundaries are selected
arbitrarily
These criteria were used to define the equation for each boundary line and leads to a set of
Boolean conditions that relate the x-coordinate and y-coordinate for each Cartesian pair The
density for the mesh of points in each region is arbitrarily selected as 101 x-points and 101 y-
points in each space for the purposes of the parametric analysis The outline of this method is
described in the MATLAB scripts contained in Appendix D
The results of the parametric analysis are shown for the slackest and tautest spans in each driving
case As was demonstrated in the literature review the tautest span immediately precedes the
driving pulley and the slackest span immediately follows the driving pulley in the direction of
the belt motion Thus in the case for the crankshaft driving the tautest span is in the first span
and the slackest span is in the fifth span Whereas in the ISG driving case the tautest span is in
the fourth span and the slackest span is in the third span Hence the parametric figures in this
chapter display only the tautest and slackest span values for both driving cases so as to describe
the maximum and minimum values for tension present in the given belt
43 Results amp Discussion
431 Influence of Tensioner Arm Stiffness on Static Tension
The parametric analysis begins with changing the stiffness value for the coil spring coupled
between tensioner arms 1 and 2 This stiffness value kt is changed over a range from sixty
percent less than its initial value kt0 to sixty percent more than its original value as shown in
Parametric Analysis 75
Table 41 The results of the static tension are shown in Figure 42 for the tautest and slackest
spans for both the crankshaft and ISG driving cases
Figure 42 Parametric Analysis for Coupled Stiffness Arm Constant kt (N∙mrad)
As kt increases in the crankshaft driving phase for the B-ISG system the highest tension
decreases from 4691N to 4646N while the lowest tension decreases from 2838N to 2793N
In the ISG driving phase the highest tension increases from 378N to 3998N and the lowest
tension increases from -3384N to -3167N Thus a change of approximately -45N is found in
the crankshaft driving case and approximately +22N is found in the ISG driving case for both the
tautest and slackest spans
Parametric Analysis 76
The second parameter analyzed is the stiffness value for tensioner arm 1 The results of this are
shown in Figure 43
Figure 43 Parametric Analysis for Stiffness of Arm 1 kt1 (N∙mrad)
In Figure 43 as kt1 increases an increase from 4628N to 4681N is observed for the tension of
the tautest span when the crankshaft is driving which is a change of +53N The same value for
net change is found in the slackest span for the same driving condition whose tension increases
from 2775N to 2828N For the case when the B-ISG system is in the ISG driving phase the
change is larger a value of -261N for the tautest span that changes from 4088N to 3827N and
for the slackest span that changes from -3077N to -3338N
Parametric Analysis 77
The change in static tension for the spans as the stiffness of arm 2 varies is demonstrated in
Figure 44
Figure 44 Parametric Analysis for Stiffness of Arm 2 kt2 (N∙mrad)
In this case it is observed that as kt2 increases the tautest span for the B-ISG system in the
crankshaft driving case decreases from 4675N to 4643N as well as the slackest span which
decreases from 2822N to 279N which is an overall change of -32N for both spans Whereas in
the ISG driving case a more noticeable change is once again found a difference of +144N
This is a result of the tautest span increasing from 3863N to 4007N and the slackest span
increasing from -3301N to -3157N
Parametric Analysis 78
432 Influence of Tensioner Pulley Diameter on Static Tension
The change in the diameter of tensioner pulley 1 D3 and its effect on static tension is shown in
Figure 45
Figure 45 Parametric Analysis for Pulley 1 Diameter D3 (m)
The change in the tautest and slackest spans for the B-ISG system‟s crankshaft driving case is
from 3248N to 425N and from 1395N to 240N respectively Peaks are seen at 4799N and
2946N for the respective spans This is a change of approximately +100N and a maximum
change of 1551N for both spans For the ISG driving case the tautest and slackest spans
decrease from 1083N to 6158N and 367N to -1006N Global minimums of 3246N and -391N
for the respective spans are seen This nets a change of approximately -467N and a maximum
change of approximately -759N
Parametric Analysis 79
The effect of changing the diameter of tensioner pulley 2 on the static tension is examined in
Figure 46
Figure 46 Parametric Analysis for Pulley 2 Diameter D5 (m)
The tautest and slackest spans in the crankshaft driving mode of the belt undergo a change from
4583N to 4721N and from 273N to 2869N respectively Therein as D5 increases the trend is
that for both spans there is an increase in tension of approximately 14N Contrastingly the spans
experience a decrease in the ISG driving case as D5 increases The tension of the tautest span
goes from 4296N to 3635N and that of the slackest span goes from -2866N to -3529N This
equals a decrease of approximately 66N for both spans
Parametric Analysis 80
433 Influence of Tensioner Pulley 1 Coordinates on Static Tension
The influence of the coordinates of tensioner pulley 1 on the value of tension in the tautest span
for the B-ISG system‟s crankshaft driving case is demonstrated in Figure 47
Figure 47 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Tautest Span
Tension in Crankshaft Driving Case
The region shown in Figure 47 corresponds to region 1 which is the realm of the positions for
tensioner pulley 1 The possible pulley coordinates in this case are represented by the non-blue
area reaching to the perimeter of the plot It is evident in the darkest red region of the plot
where the y-coordinate is between approximately 0m and 0075m and the x-coordinate is
(N)
Parametric Analysis 81
between approximately 026m and 031m that the highest value of tension is experienced in the
tautest span for the crankshaft driving case The range of tension for Region 1 in the tautest span
when the crankshaft is driving is between a maximum of approximately 500N and a minimum of
approximately 300N This equals an overall difference of 200N in tension for the tautest span by
moving the position of pulley 1 The lowest values for tension are obtained when the pulley
coordinates are approximately -0025m to 015m for the y-coordinate and approximately 031m
to 032m for the x-coordinate which corresponds to the yellow region An area of low tension is
also seen in the area where the y-coordinate is approximately 0m and the x-coordinate is
approximately between 026m and 027m
The changes in tension for the slackest span under the condition of the crankshaft pulley being
the driving pulley are shown in Figure 48
Parametric Analysis 82
Figure 48 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Slackest Span
Tension in Crankshaft Driving Case
Once again the possible coordinate points for tensioner pulley 1 in the B-ISG system are
represented by the non-blue region For the slackest span in the crankshaft driving case it is seen
that the lowest tension is approximately 125N while the highest tension is approximately 325N
This is an overall change of 200N that is achieved in the region The highest values are achieved
in the space where the y-coordinates are approximately 0m to 0075m and the x-coordinate
ranges from 026m to 031m which corresponds to the deep red region The lowest tension
values are achieved in the space where the y-coordinate ranges from approximately -0025m to
015m and the x-coordinate ranges from 031m to 032m which corresponds to the light blue-
green region of the plot The area containing a y-coordinate of approximately 0m and x-
(N)
Parametric Analysis 83
coordinates that are approximately between 026m and 027m also show minimum tension for
the slack span The regions of the x-y coordinates for the maximum and minimum tensions are
alike to the tautest span in Region 1 for the crankshaft driving case as well as was seen in Figure
47
The tension for the tautest span in the case that the ISG is driving in the B-ISG system is found
in Figure 49
Figure 49 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Tautest Span
Tension in ISG Driving Case
(N)
Parametric Analysis 84
Region 1 is represented by the coordinate values shown in the non-dark blue space of the plot in
Figure 49 The tautest span in the case of the ISG driving experiences a range of tension values
in Region 1 from 200N up to 1100N equaling a difference of 900N The minimum tension
values are achieved in the medium to light blue region This includes y-coordinates of
approximately 0m to 0075m and x-coordinates of approximately 026m to 03m The
maximum tension values are in the darkest red area inclusive of y-coordinates -0025m to 015m
and x-coordinates 031m to 032m in addition to y-coordinate of approximately 0m and x-
coordinates of approximately 026m to 027m It can be observed that aforementioned regions
for minimum and maximum tensions in Figure 49 are reverse to those seen in Figures 47 and
48 for the crankshaft driving case
The change in tension for the slackest span of the B-ISG system when it is driven by the ISG is
shown in Figure 410
Parametric Analysis 85
Figure 410 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Slackest Span
Tension in ISG Driving Case
Figure 410 exhibits the realm of possible points for tensioner pulley 1 for the case of the ISG
driving in the non-yellow-green area The minimum tension values are achieved in the darkest
blue area where the minimum tension is approximately -500N This area corresponds to y-
coordinates from approximately 0m to 005m and x-coordinates from approximately 026m to
03m The area of a maximum tension is approximately 400N and corresponds to the darkest red
area inclusive of y-coordinates -0025m to 015m and x-coordinates 031m to 032m as well as
the coordinates for y equaling approximately 0m and for x equaling approximately 026m to
027m The difference between maximum and minimum tensions in this case is approximately
900N It is noticed once again that the space of x- and y-coordinates containing the maximum
(N)
Parametric Analysis 86
tension is in the similar location to that of the described space for minimum tension in the
crankshaft driving case in Figure 47 and 48
434 Influence of Tensioner Pulley 2 Coordinates on Static Tension
The influence of pulley 2 coordinates on the tension value for the tautest span when the
crankshaft is driving the B-ISG system is shown in Figure 411 and is represented by the values
corresponding to the non-blue area
Figure 411 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Tautest Span
Tension in Crankshaft Driving Case
In Figure 411 the possible coordinates are contained within Region 2 The maximum tension
value is approximately 500N and is found in the darkest red space including approximately y-
(N)
Parametric Analysis 87
coordinates 004m to 014m and x-coordinates 0025m to 0175m and also y-coordinates 013m
to 02m corresponding to the x-coordinate at 0175m A minimum tension value of
approximately 350N is found in the yellow space and includes approximately y-coordinates
008m to 018m and x-coordinates 016m to 02m The difference in tension values is 150N
The analysis of the change in coordinates for tension pulley 2 on the value for tension in the
slackest span is shown in Figure 412 in the non-blue region
Figure 412 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Slackest Span
Tension in Crankshaft Driving Case
The value of 325N is the highest tension for the slack span in the crankshaft driving case of the
B-ISG system and is found in the deep-red region where the y-coordinates are between
(N)
Parametric Analysis 88
approximately 004m and 013m and the x-coordinates are approximately between 0025m and
016m as well as where y is between 013m and 02m and x is approximately 0175m The
lowest tension value for the slack span is approximately 150N and is found in the green-blue
space where y-coordinates are between approximately 01m and 022m and the x-coordinates
are between approximately 016m and 021m The overall difference in minimum and maximum
tension values is 175N The spaces for the maximum and minimum tension values are similar in
location to that found in Figure 411 for the tautest span in the crankshaft driving case
Figure 413 provides the theoretical data for the tension values of the tautest span as the position
of the B-ISG system‟s tensioner pulley 2 changes in the ISG driving case Possible points are in
the space of values which correspond to the non-dark-blue region in Figure 413
Parametric Analysis 89
Figure 413 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Tautest Span
Tension in ISG Driving Case
In Figure 413 the region for high tension reaches a value of approximately 950N and the region
for low tension reaches approximately 250N This equals a difference of 700N between
maximum and minimum tension values for the tautest span in the B-ISG system‟s ISG driving
case The coordinate points within the space that maximum tension is reached is in the dark red
region and includes y-coordinates from approximately 008m to 022m and x-coordinates from
approximately 016m to 021m The coordinate points within the space that minimum tension is
reached is in the blue-green region and includes y-coordinates from approximately 004m to
013m and the corresponding x-coordinates from approximately 0025m to 015m An additional
small region of minimum tension is seen in the area where the x-coordinate is approximately
(N)
Parametric Analysis 90
0175m and the y-coordinates are approximately between 013m and 02m The location for the
area of pulley centre points that achieve maximum and minimum tension values is approximately
located in the reverse positions on the plot when compared to that of the case for the crankshaft
driving in Figures 411 and 412 Therein the trend seen for pulley coordinates for the second
tensioner pulley follows suit with that of the first tensioner pulley which is that the area of points
for maximum tension in the crankshaft driving case becomes the approximate area of points for
minimum tension in the ISG driving case and vice versa
In Figure 414 the results of the parametric analysis on the coordinates of the second tensioner
pulley and its effect on the slackest span‟s tension in the ISG driving case is shown Similar to
earlier figures the non-dark yellow region represents Region 2 that contains the possible points
for the pulley‟s Cartesian coordinates
Parametric Analysis 91
Figure 414 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Slackest
Span Tension in ISG Driving Case
Figure 414 demonstrates a difference of approximately 725N between the highest and lowest
tension values for the slackest span of the B-ISG system in the ISG driving case The highest
tension values are approximately 225N The area of points that allow the second tension pulley
to achieve maximum tension in the belt span includes y-coordinates from approximately 01m to
022m and the corresponding x-coordinates from approximately 016m to 021m This
corresponds to the darkest red region in Figure 414 The coordinate values where the lowest
tension being approximately -500N is achieved include y-coordinate values from
approximately 004m to 013m and the corresponding x-coordinates from approximately 0025m
to 015m corresponding to the darkest blue region A dark blue region of lowest tension is also
(N)
Parametric Analysis 92
seen in the area where y is approximately between 013m and 02m and the x-coordinate is
approximately 0175m The regions for maximum and minimum tension values are observed to
be similar to those found in Figure 413 and alike to Figure 413 to be in reverse to those found
in Figure 411 and 412 for the tautest and slackest spans in the crankshaft driving case So as for
the changes in tensioner pulley 2 coordinates the areas for minimum tension in Region 2 of the
ISG driving case are similar to the areas for maximum tension in Region 2 of the crankshaft
driving case and vice versa for the maximum tension of the ISG driving case and the minimum
tension for the crankshaft driving case in Region 2
44 Conclusion
Overall the trend in the plots of Figures 47 48 411 and 412 indicate in the crankshaft driving
portion that the B-ISG system‟s belt span tensions experience the following effect
Minimum tension for the tautest span is achieved when tensioner pulley 1 centre
coordinates are located closer to the right side boundary and bottom left boundary of
Region 1 or when tensioner pulley 2 centre coordinates are within the upper right space
(near to the ISG pulley) and the space closer to the top boundary of Region 2
Maximum tension for the slackest span is achieved when the first tensioner pulley‟s
coordinates are located in the mid space and near to the bottom boundary of Region 1
and when the second tensioner pulley‟s coordinates are located near to the bottom left
boundary of Region 2 which is the boundary nearest to the crankshaft pulley
Parametric Analysis 93
The trend for minimizing the tautest span signifies that the tension for the slackest span is also
minimized at the same time As well maximizing the slackest span signifies that the tension for
the tautest span is also maximized at the same time too
The trend for the B-ISG system‟s ISG driving case as can be seen in Figures 49 410 413 and
414 is approximately in reverse to that of the crankshaft driving case for the system Wherein
points corresponding to minimum tension in Regions 1 and 2 in the ISG case are approximately
the same as points corresponding to maximum tension in the Regions for the crankshaft case and
vice versa for the ISG cases‟ areas of maximum tension
Minimum tension for the tautest span is present when the first tensioner pulley‟s
coordinates are near to mid to lower boundary of Region 1 and when the second
tensioner pulley‟s coordinates are close to the bottom left boundary of Region 2 which
is the furthest boundary from the ISG pulley and closest to the crankshaft pulley
Maximum tension for the slackest span is achieved when the first tensioner pulley is
located close to the right boundary of Region 1 and when the second tensioner pulley is
located near the right boundary and towards the top right boundary of Region 2
It is observed in Figures 47 to 414 and alike to Figures 42 to 46 the tautest and slackest
spans decrease or increase together Thus it can be assumed that the tension values in these
spans and likely the remaining spans outside of the tautest and slackest spans follow suit
Therein when parameters are changed to minimize one belt span‟s tension the remaining spans
will also have their tension values reduced Figures 42 through to 413 showed this clearly
where the overall change in the tension of the tautest and slackest spans changed by
Parametric Analysis 94
approximately the same values for each separate case of the crankshaft driving and the ISG
driving in the B-ISG system
Design variables are selected in the following chapter from the parameters that have been
analyzed in the present chapter The influence of changing parameters on the static tension
values for the various spans is further explored through an optimization study of the static belt
tension for the B-ISG system equipped with a Twin Tensioner in the following chapter Chapter
5
95
CHAPTER 5 OPTIMIZATION OF A B-ISG TENSIONER
The objective of the optimization analysis is to minimize the absolute magnitude of the static
tension in the ISG-operating mode of the serpentine belt drive The optimization seeks to
optimize the performance of the proposed Twin Tensioner design by using its properties as the
design variables for the objective function The optimization task begins with the selection of
these design variables for the objective function and then the selection of an optimization
method The results of the optimization will be compared with the results of the analytical
model for the static system and with the parametric analysis‟ data
51 Optimization Problem
511 Selection of Design Variables
The optimal system corresponds to the properties of the Twin Tensioner that result in minimized
magnitudes of static tension for the various belt spans Therein the design variables for the
optimization procedure are selected from amongst the Twin Tensioner‟s properties In the
parametric analysis of Chapter 4 the tensioner properties presented included
coupled stiffness kt
tensioner arm 1 stiffness kt1
tensioner arm 2 stiffness kt2
tensioner pulley 1 diameter D3
tensioner pulley 2 diameter D5
tensioner pulley 1 initial coordinates [X3Y3] and
Optimization 96
tensioner pulley 2 initial coordinates [X5Y5]
It was observed in the former chapter that perturbations of the stiffness and geometric parameters
caused a change between the lowest and highest values for the static tension especially in the
case of perturbations in the geometric parameters diameter and coordinates Table 51
summarizes the observed changes in the belt span tensions corresponding to the Twin Tensioner
parameters‟ maximum and minimum values
Table 51 Summary of Parametric Analysis Data for Twin Tensioner Properties
Parameter Symbol
Original Tensions in TautSlack Span (Crankshaft
Mode) [N]
Tension at
Min | Max Parameter6 for
Crankshaft Mode [N]
Percent Change from Original for
Min | Max Tensions []
Original Tension in TautSlack Span (ISG Mode)
[N]
Tension at
Min | Max Parameter Value in ISG Mode [N]
Percent Change from Original Tension for
Min | Max Tensions []
kt
465848 (taut) 4691 4646 07 -03 393645 (taut) 378 3998 -40 16
28057 (slack) 2838 2793 12 -05 -322803 (slack) -3384 -3167 -48 19
kt1
465848 (taut) 4628 4681 -07 05 393645 (taut) 4088 3827 38 -28
28057 (slack) 2775 2828 -11 08 -322803 (slack) -3077 -3338 47 -34
kt2
465848 (taut) 4675 4643 04 -03 393645 (taut) 3863 4007 -19 18
28057 (slack) 2822 279 06 -06 -322803 (slack) -3301 -3157 -23 22
D3 465848 (taut) 3248 425 -303 -88 393645 (taut) 1083 6158 1751 564
28057 (slack) 1395 240 -503 -145 -322803 (slack) 367 -1006 2137 688
D5 465848 (taut) 4583 4721 -16 13 393645 (taut) 4296 3635 91 -77
28057 (slack) 273 2869 -27 23 -322803 (slack) -2866 -3529 112 -93
[X3Y3] 465848 (taut) 300 500 -356 73 393645 (taut) 200 1100 -492 1794
28057 (slack) 125 325 -554 158 -322803 (slack) -500 400 -549 2239
6 The values for the tension for each of the taut and slack spans provided correspond to the minimum and maximum
values of the parameter listed in each case such that the columns of identical colour correspond to each other For
the coordinate parameters the minimum and maximum parameter value is inadmissible The tension values in these
cases are simply the minimum and maximum tension values achieved by the coordinate parameter listed
Optimization 97
[X5Y5] 465848 (taut) 350 500 -249 73 393645 (taut) 250 950 -365 1413
28057 (slack) 150 325 -465 158 -322803 (slack) -500 225 -549 1697
The results of the parametric analyses for the Twin Tensioner parameters show that there is a
noticeable percent change between the initial tensions and the tensions corresponding to each of
the minima and maxima parameter values or in the case of the coordinates between the
minimum and maximum tensions for the spans Thus the parametric data does not encourage
exclusion of any of the tensioner parameters from being selected as a design variable As a
theoretical experiment the optimization procedure seeks to find feasible physical solutions
Hence economic criteria are considered in the selection of the design variables from among the
Twin Tensioner‟s parameters Of the tensioner properties it is found that the diameter of the
tensioner pulleys has the largest impact on cost Adding mass to a tensioner pulley as a result of
increasing the diameter and consequently its inertia increases the cost of material Material cost
is most significant in the manufacture process of pulleys as their manufacturing is largely
automated [4] Furthermore varying the structure of a pulley requires retooling which also
increases the cost to manufacture As such the tensioner pulley diameters D3 and D5 are
excluded from being selected as design variables The remaining tensioner properties the
stiffness parameters and the initial coordinates of the pulley centres are selected as the design
variables for the objective function of the optimization process
512 Objective Function amp Constraints
In order to deal with two objective functions for a taut span and a slack span a weighted
approach was employed This emerges from the results of Chapter 3 for the static model and
Chapter 4 for the parametric study for the static system which show that a high tension span and
Optimization 98
a highly slack span exist in the ISG-driving phase of the B-ISG system Therein the first
objective function of equation (51a) is described as equaling fifty percent of the absolute tension
value of the tautest span and fifty percent of the absolute tension value of the slackest span for
the case of the ISG driving only The second objective function uses a non-weighted approach
and is described as the absolute tension of the slackest span when the ISG is driving A non-
weighted approach is motivated by the phenomenon of a fixed difference that is seen between
the slackest and tautest spans of the optimal designs found in the weighted optimization
simulations Equations (51a) through to (51c) display the objective functions
The limits for the design variables are expanded from those used in the parametric analysis for
the non-coordinate parameters kt kt1 and kt2 so that they are permitted to vary from
approximately 0 to approximately 200 of the initial value for each parameter kt0 kt10 and kt20
respectively In the case of the coordinate parameters [X3Y3] and [X5Y5] the x- and y-
coordinates are permitted to vary within the spaces Region 1 and Region 2 respectively which
were prescribed in Chapter 4 Figure 41a and 41b
Aside from the design variables design constraints on the system include the requirement for
static stability of the Twin Tensioner An optimal solution for the B-ISG system must achieve
the goal of the objective function which is to minimize the absolute tensions in the system
However for an optimal solution to be feasible the movement of the tensioner arm must remain
within an appropriate threshold In practice an automotive tensioner arm for the belt
transmission may be considered stable if its movement remains within a 10 degree range of
Optimization 99
motion [4] As such the angle of displacement for tensioner arms 1 and 2 are designated by θ t1
and θt2 respectively in the listed constraints
The optimization task is described in equations 51a to 52 Variables a through to g represent
scalar limits for the x-coordinate for corresponding ranges of the y-coordinate
Minimize 119879119908119890119894119892 119893119905119890119889 = 05 ∙ 119879119905119886119906119905 + 05 ∙ 119879119904119897119886119888119896
or119879119899119900119899 minus119908119890119894119892 119893119905119890119889 = 119879119904119897119886119888119896
(51a)
where
119879119905119886119906119905 = 119891119905119886119906119905 119896119905 1198961199051 1198961199052 1198833 1198843 1198835 1198845 (51b)
119879119904119897119886119888119896 = 119891119904119897119886119888119896 (119896119905 1198961199051 1198961199052 1198833 119884311988351198845) (51c)
Subject to
(1198961199050 minus 1 ∙ 1198961199050) le 119896119905 le (1198961199050 + 11198961199050)(11989611990510 minus 1 ∙ 11989611990510) le 1198961199051 le (11989611990510 + 111989611990510)(11989611990520 minus 1 ∙ 11989611990520) le 1198961199052 le (11989611990520 + 111989611990520)
119886 le 1198833 le 119888
1198931 1198833 le 1198843 le 1198933 1198833 119891119900119903 119886 le 1198833 lt 119887
1198932 1198833 le 1198843 le 1198933 1198833 119891119900119903 119887 le 1198833 le 119888119889 le 1198835 le 119892
1198934 1198835 le 1198845 le 1198937 1198835 for 119889 le 1198835 lt 1198901198935(1198835) le 1198845 le 1198937(1198835) for 119890 le 1198835 lt 119891
1198936 1198835 le 1198845 le 1198937 1198833 for 119891 le 1198833 le 119892 1205791199051 le 10deg 1205791199052 le 10deg
(52)
The functions for the taut and slack spans represent the fourth and third span respectively in the
case of the ISG driving The equations for the tensions of the aforementioned spans are shown
in equation 51a to 51c and are derived from equation 353 The constraints for the
optimization are described in equation 52
Optimization 100
52 Optimization Method
A twofold approach was used in the optimization method A global search alone and then a
hybrid search comprising of a global search and a local search The Genetic Algorithm is used
as the global search method and a Quadratic Sequential Programming algorithm is used for the
local search method
521 Genetic Algorithm
Genetic Algorithm (GA) is a part of the growing genre of evolutionary algorithms [57] The
optimization approach differs from classical search approaches by its ease of use and global
perspective [57] GA mimics biological evolution theory by using a ldquocross-over of heritable
information random mutation and selection on the basis of fitness between generationsrdquo [58] to
form a robust search algorithm that requires minimal problem information [57] The parameter
sets are represented as sample points modeled as bdquochromosomes‟ or data strings that are
measured against how well they allow the model to achieve the optimization task [58] GA is
stochastic which means that its algorithm uses random choices to generate subsequent sampling
points rather than using a set rule to generate the following sample This avoids the pitfall of
gradient-based techniques that may focus on local maxima or minima and end-up neglecting
regions containing higher peaks or lower valleys [57] Furthermore due to the randomness of
the GA‟s search strategy it is able to search a population (a region of possible parameter sets)
faster than other optimization techniques The GA approach is viewed as a universal
optimization approach while many classical methods viewed to be efficient for one optimization
problem may be seen as inefficient for others However because GA is a probabilistic algorithm
its solution for the objective function may only be near to a global optimum As such the current
Optimization 101
state of stochastic or global optimization methods has been to refine results of the GA with a
local search and optimization procedure
522 Hybrid Optimization Algorithm
In order to enhance the result of the objective function found by the GA a Hybrid optimization
function is implemented in MATLAB software The Hybrid optimization function combines a
global search GA with a local search Sequential Quadratic Programming (SQP) The hybrid
process refines the value of the objective function found through GA by using the final set of
points found by the algorithm as the initial point of the SQP algorithm The GA function
determines the region containing a global optimum and then the SQP algorithm uses a gradient
based technique to find a solution closer to the global optimum The MATLAB algorithm a
constrained minimization function known as fmincon uses an SQP method that approximates the
Hessian for the Lagrangian function (ie the second derivatives of the Lagrangian) by way of a
quasi-Newton approach to generate a quadratic program (QP) sub-problem [59] The solution
for the QP provides the search direction of the line search procedure used when each iteration is
performed [59]
53 Results and Discussion
531 Parameter Settings amp Stopping Criteria for Simulations
The parameter settings for the optimization procedure included setting the stall time limit to
200s This is the interval of time the GA is given to find an improvement in the value of the
objective function This is an increase from MATLAB‟s default of 20s Increasing the stall time
limit allows for the optimization search to consistently converge without being limited by time
Optimization 102
The population size used in finding the optimal solution is set at 100 This value was chosen
after varying the population size between 50 and 2000 showed no change in the value of the
objective function The max number of generations is set at 100 The time limit for the
algorithm is set at infinite The limiting factor serving as the stopping condition for the
optimization search was the function tolerance which is set at 1x10-6
This allows the program
to run until the ratio of the change in the objective function over the stall generations is less than
the value for function tolerance The stall generation setting is defined as the number of
generations since the last improvement of the objective function value and is limited to 50
532 Optimization Simulations
The results of the genetic algorithm optimization simulations performed in MATLAB are shown
in the following tables Table 52a and Table 52b
Table 52a GA Optimization Results for Twin Tensioner Parameters and Objective Function
Trial
No
Genetic Algorithm Optimization Method
Objective
Function
Value [N]
kt [N∙mrad]
kt1 [N∙mrad]
kt2 [N∙mrad]
[X3Y3]
[m] [X5Y5]
[m]
1 3582241 314069 204844 165020 [02928 00703] [01618 01036]
2 3582241 103646 205284 198901 [03009 00607] [01283 00809]
3 3582241 126431 204740 43549 [03010 00631] [01311 01147]
4 3582241 180285 206230 254870 [03095 00865] [01080 01675]
5 3582241 74757 204559 189077 [03084 00617] [01265 00718]
Original Design 20626 10314 16502 [02928 0087] [01206 00919]
Optimization 103
Table 52b Computations for Tensions and Angles from GA Optimization Results
Trial No
Genetic Algorithm Optimization Method
Slackest Tension [N] Tautest Tension [N] Tensioner Arm 1
Displacement [deg]
Tensioner Arm 2
Displacement [deg]
T3 T4 Θt1 Θt2
1 -1572307 5592176 -00025 -49748
2 -4054309 3110174 -00002 -20213
3 -3930858 3233624 -00004 -38370
4 -1309751 5854731 -00010 -49525
5 -4092446 3072036 -00000 -17703
Original Design -322803 393645 16410 -4571
For each trial above the GA function required 4 generations each consisting of 20 900 function
evaluations before finding no change in the optimal objective function value according to
stopping conditions
The results of the Hybrid function optimization are provided in Tables 53a and 53b below
Table 53a Hybrid Optimization Results for Twin Tensioner Parameters and Objective Function
Trial
No
Hybrid Optimization Method
Objective
Function
Value [N]
of
Function
Evals ( of
Iterations)
kt [N∙mrad]
kt1 [N∙mrad]
kt2 [N∙mrad]
[X3Y3]
[m] [X5Y5]
[m]
1 3582241 16 (1) 16065 205846 229494 [02780 00581] [01679 01288]
2 3582241 20 (1) 249227 205635 25218 [02901 00634] [01559 00870]
3 3582241 16 (1) 297295 204878 320479 [02962 00702] [01336 01447]
4 3582241 53 (1) 241433 204262 229683 [02912 00647] [00047 01465]
Optimization 104
5 3582241 21 (1) 379096 205548 188888 [02973 00703] [01206 01376]
Original Design 20626 10314 16502 [02928 0087] [01206 00919]
Table 53b Computations for Tensions and Angles from Hybrid Optimization Results
Trial No
Hybrid Algorithm Optimization Method
Slackest Tension [N] Tautest Tension [N] Tensioner Arm 1
Displacement [deg]
Tensioner Arm 2
Displacement [deg]
T3 T4 Θt1 Θt2
1 -2584641 4579841 -02430 67549
2 -3708747 3455736 -00023 -41068
3 -1707181 5457302 -00099 -43944
4 -269178 6895304 00006 -25366
5 -2982335 4182148 -00003 -41134
Original Design -322803 393645 16410 -4571
In Table 53a it can be seen that iterations of 16 20 21 or 53 were required for the local search
algorithm following the GA to find an optimal solution Once again the GA function
computed 4 generations which consisted of approximately 20 900 function evaluations before
securing an optimum solution
The simulation results of the non-weighted hybrid optimization approach are shown in tables
54a and 54b below
Optimization 105
Table 54a Non-Weighted Optimization Results for Twin Tensioner Parameters and Objective
Function
Trial
No
Objective
Function
Value [N]
of
Function
Evals ( of
Iterations)
kt [N∙mrad]
kt1 [N∙mrad]
kt2 [N∙mrad]
[X3Y3]
[m] [X5Y5]
[m]
Genetic Algorithm Optimization Method
1 33509e
-004 20900 (4) 321799 75530 212653 [02860 00602] [01082 01858]
Hybrid Optimization Method
1 73214e
-011 381 (13) 234881 14730 323358 [02952 00688] [00048 01466]
Original Design 20626 10314 16502 [02928 0087] [01206 00919]
Table 54b Computations for Tensions and Angles from Non-Weighted Optimizations
Trial No Slackest Tension [N] Tautest Tension [N]
Tensioner Arm 1
Displacement [deg]
Tensioner Arm 2
Displacement [deg]
T3 T4 Θt1 Θt2
Genetic Algorithm Optimization Method
1 -00003 7164479 -00588 -06213
Hybrid Optimization Method
1 -00000 7164482 15543 -16254
Original Design -322803 393645 16410 -4571
The weighted optimization data of Table 54a shows that the GA simulation again used 4
generations containing 20 900 function evaluations to conduct a global search for the optimal
system While the weighted Hybrid optimization used 13 iterations (consisting of 381 function
evaluations) after its GA run which used the same number of generations and function
evaluations as the GA run in the non-weighted simulations Tables 54a and 54b show the data
Optimization 106
for only one trial for each of the non-weighted GA and hybrid methods since only a single
optimal point exists in this case
533 Discussion
The optimal design from each search method can be selected based on the least amount of
additional pre-tension (corresponding to the largest magnitude of negative tension) that would
need to be added to the system This is in harmony with the goal of the optimization of the B-
ISG system as stated earlier to minimize the static tension for the tautest span and at the same
time minimize the absolute static tension of the slackest span for the ISG driving case As well
the angular displacements corresponding to each trial‟s results show that the Twin Tensioner is
under static stability Therein the optimal solution may be selected as the design parameters
corresponding to Trial 4 of the GA simulations to Trial 4 of the Hybrid simulations or to either
of the trials for the non-weighted optimization simulations
Given the ability of the Hybrid optimization to refine the results obtained in the GA
optimization the results of Trial 4 of the Hybrid simulations are selected as the most optimal
design from the weighted objective function approaches It is interesting to note that the Hybrid
case for the least slackness in belt span tension corresponds to a significantly larger number of
function evaluations than that of the remaining Hybrid cases This anomaly however does not
invalidate the other Hybrid cases since each still satisfy the design constraints Using the data
for the optimized system in Trial 4 (of the Hybrid optimization) the static tensions for the belt
spans in both of the B-ISG‟s phases of operation are as follows
Optimization 107
Table 55 Weighted Optimization Results for Static Tensions for Optimal B-ISG System with a
Twin Tensioner
Belt Span between Pulleys Crankshaft Driving Case [N] ISG Driving Case [N]
Optimized Original Optimized Original
Crankshaft ndash Air Conditioner 3926599 465848 117333 -284152
Air Conditioner ndash Tensioner 1 3540088 427197 -269178 -322803
Tensioner 1 ndash ISG 3540088 427197 -269178 -322803
ISG ndash Tensioner 2 2073813 28057 6895304 393645
Tensioner 2 ndash Crankshaft 2073813 28057 6895304 393645
Additional Pretension
Required (approximate) + 27000 +322803 + 27000 +322803
In Table 54b it is evident that the non-weighted class of optimization simulations achieves the
least amount of required pre-tension to be added to the system The computed tension results
corresponding to both of the non-weighted GA and Hybrid approaches are approximately
equivalent Therein either of their solution parameters may also be called the most optimal
design The Hybrid solution parameters are selected as the optimal design once again due to the
refinement of the GA output contained in the Hybrid approach and its corresponding belt
tensions are listed in Table 56 below
Optimization 108
Table 56 Non-Weighted Optimization Results for Static Tensions for Optimal B-ISG System
with a Twin Tensioner
Belt Span between Pulleys Crankshaft Driving Case [N] ISG Driving Case [N]
Optimized Original Optimized Original
Crankshaft ndash Air Conditioner 3891862 465848 386511 -284152
Air Conditioner ndash Tensioner 1 3505351 427197 -00000 -322803
Tensioner 1 ndash ISG 3505351 427197 -00000 -322803
ISG ndash Tensioner 2 2039076 28057 7164482 393645
Tensioner 2 ndash Crankshaft 2039076 28057 7164482 393645
Additional Pretension
Required (approximate) + 0000 +322803 + 00000 +322803
The results of the simulation experiments are limited by the following considerations
System equations are coupled so that a fixed difference remains between tautest and
slackest spans
A limited number of simulation trials have been performed
There are multiple optimal design points for the weighted optimization search methods
Remaining tensioner parameters tensioner pulley diameters and their stiffness have not
been included in the design variables for the experiments
The belt factor kb used in the modeling of the system‟s belt has been obtained
experimentally and may be open to further sources of error
Therein the conclusions obtained and interpretations of the simulation data can be limited by the
above noted comments on the optimization experiments
Optimization 109
54 Conclusion
The outcomes the trends in the experimental data and the optimal designs can be concluded
from the optimization simulations The simulation outcomes demonstrate that in all cases the
weighted optimization functions reached an identical value for the objective function whereas
the values reached for the parameters varied widely
The lowest tension values for the tautest and slackest span were achieved in Trial 5 of the GA
optimization approach In reiteration in the presence of slack spans the tension value of the
slackest span must be added to the initial static tension for the belt Therein for the former case
an amount of at least 409N would need to be added to the 300N of pre-tension already applied to
the system (see Table 34) The highest tension values for the spans were achieved in Trial 4 of
the weighted Hybrid optimization approach and in both trials of the non-weighted optimization
approaches In the former the weighted Hybrid trial the tension value achieved in the slackest
span was approximately -27N signifying that only at least 27N would need to be added to the
present pre-tension value for the system The tension of the slackest span in the non-weighted
approach was approximately 0N signifying that the minimum required additional tension is 0N
for the system
The optimized solutions for the tension values in each span show that there is consistently a fixed
difference of 716448N between the tautest and slackest span tension values as seen in Tables
52b 53b and 54b This difference is identical to the difference between the tautest and slackest
spans of the B-ISG system for the original values of the design parameters while in its ISG
mode As well the optimal stiffness parameters for the weighted Hybrid optimization case are
Optimization 110
greater than their original values except for that of the stiffness factor of tensioner arm 1
Likewise for the non-weighted Hybrid optimization case the stiffness parameters are above their
original values without exceptions The coordinates of the optimal solutions are within close
approximation to each other and also both match the regions for moderately low tension in
Regions 1 and 2 of the ISG driving case as is shown in Figures 49 410 413 and 414
The results of the non-weighted Hybrid optimization trial and Trial 4 of the weighted Hybrid
optimization simulations are selected as the most optimal designs for the B-ISG Twin Tensioner
In these designs the Twin Tensioner is shown in Table 53b and 54b to have static stability and
to maintain suitable tensions in the ISG driving phase The tensioner parameters for the optimal
designs allow for one of the lowest amounts of additional pre-tension to be added to the system
out of all the findings from the simulations which were conducted
111
CHAPTER 6 CONCLUSION
61 Summary
The primary aim of the thesis is to reduce the magnitude of static tension in the belt spans of a
Belt-driven Integrated Starter-generator (B-ISG) system by the design and investigation of a
Twin Tensioner It is established that the operating phases of the B-ISG system produced two
cases for static tension outcomes an ISG driving case and a crankshaft driving case The
approach taken in this thesis study includes the derivation of a system model for the geometric
properties as well as for the dynamic and static states of the B-ISG system The static state of a
B-ISG system with a single tensioner mechanism is highlighted for comparison with the static
state of the Twin Tensioner-equipped B-ISG system
It is observed that there is an overall reduction in the magnitudes of the static belt tensions with
the presence of a Twin Tensioner over that of a single tensioner The influences of the geometric
and stiffness properties of the Twin Tensioner affecting the static tensions in the system are
analyzed in a parametric study It is found that there is a notable change in the static tensions
produced as result of perturbations in each respective tensioner property This demonstrates
there are no reasons to not further consider a tensioner property based solely on its influence on
the B-ISG system‟s static tensions The phenomenon of higher magnitudes for static tensions in
the ISG mode of operation over that of the crankshaft mode of operation particularly in
excessively slack spans provides the motivation for optimizing the ISG case alone for static
tension The optimization method uses weighted and non-weighted approaches with genetic
algorithm (GA) and hybrid GA searches The most optimal design has been found to be one in
Conclusion 112
which the magnitude of tension in the excessively slack spans in the ISG driving case are
significantly lower than in that of the original B-ISG Twin Tensioner design
62 Conclusion
The conclusions that can be drawn from the study of a Twin Tensioner for a B-ISG system
include the following
1 The simulations of the dynamic model demonstrate that the mode shapes for the system
are greater in the ISG-phase of operation
2 It was observed in the output of the dynamic responses that the system‟s rigid bodies
experienced larger displacements when the crankshaft was driving over that of the ISG-
driving phase It was also noted that the transition speed marking the phase change from
the ISG driving to the crankshaft driving occurred before the system reached either of its
first natural frequencies
3 The magnitudes for static belt tensions as well as dynamic tensions for a B-ISG system
are consistently greater in its ISG operating phase than in its crankshaft operating phase
4 A Twin Tensioner is able to reduce the magnitudes of the static tension for the belt spans
of a B-ISG system in comparison to when only a single tensioner mechanism is present
5 The parametric study of the B-ISG system demonstrates that the slackest and tautest belt
spans decrease or increase together for either phase of operation
6 Perturbations in the Twin Tensioner‟s geometric and stiffness properties have a
significant influence on the magnitudes of the static tension of the slackest and tautest
belt spans The coordinate position of each pulley in the Twin Tensioner configuration
Conclusion 113
has the greatest influence on the belt span static tensions out of all the tensioner
properties considered
7 Optimization of the B-ISG system shows a fixed difference trend between the static
tension of the slackest and tautest belt spans for the B-ISG system
8 The values of the design variables for the most optimal system are found using a hybrid
algorithm approach The slackest span for the optimal system is significantly less slack
than that of the original design Therein less additional pretension is required to be added
to the system to compensate for slack spans in the ISG-driving phase of operation
63 Recommendation for Future Work
The investigation of the B-ISG Twin Tensioner encourages the following future work
1 The optimization of the B-ISG system with the inclusion of diametric Twin Tensioner
properties would provide a complete picture as to the highest possible performance
outcome that the Twin Tensioner is able to have with respect to the static tensions
achieved in the belt spans
2 A larger number of optimization trials using the genetic algorithm (GA) and hybrid GA
under weighted and other approaches would investigate the scope of optimal designs
available in the Twin Tensioner for the B-ISG system
3 A model of the system without the simplification of constant damping may produce
results that are more representative of realistic operating conditions of the serpentine belt
drive A finite element analysis of the Twin Tensioner B-ISG system may provide more
applicable findings
Conclusion 114
4 Investigation of the transverse motion coupled with the rotational belt motion in an
optimized B-ISG system equipped with a Twin Tensioner may also provide a closer look
at the system under realistic conditions In addition the affect of the Twin Tensioner on
transverse motion can determine whether significant improvements in the magnitudes of
static belt span tensions are still being achieved
5 The recommendation to conduct modal decoupling of the B-ISG system‟s static model is
motivated by the fixed difference trend between the slackest and tautest belt span
tensions shown in Chapter 5 The modal decoupling of the system would allow for its
matrices comprising the equations of motion to be diagonalized and therein to decouple
the system equations Modal analysis would transform the system from physical
coordinates into natural coordinates or modal coordinates which would lead to the
decoupling of system responses
6 An investigation and optimization of the dynamic belt span tensions for a B-ISG system
with a Twin Tensioner would increase understanding of the full impact of a Twin
Tensioner mechanism on the state of the B-ISG system It would be informative to
analyze the mode shapes of the first and second modes as well as the required torques of
the driving pulleys and the resulting torque of each of the tensioner arms The
observation of the dynamic belt span tensions would also give direction as to how
damping of the system may or may not be changed
7 Further comparison with the Twin Tensioner B-ISG system‟s dynamic and static states
including the Twin Tensioner‟s stability in each versus a B-ISG system with a single
tensioner would further demonstrate the improvements or dis-improvements in the Twin
Tensioner‟s performance on a B-ISG system
Conclusion 115
8 The influence of the belt properties on the dynamic and static tensions for a B-ISG
system with a Twin Tensioner can also be investigated This again will show the
evidence of improvements or dis-improvement in the Twin Tensioner‟s performance
within a B-ISG setting
9 Lastly an experimental apparatus of the B-ISG system with a Twin Tensioner can be
designed and constructed Suitable instrumentation can be set-up to measure belt span
tensions (both static and dynamic) belt motion and numerous other system qualities
This would provide extensive guidance as to finding the most appropriate theoretical
model for the system Experimental data would provide a bench mark for evaluating the
theoretical simulation results of the Twin Tensioner-equipped B-ISG system
116
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[14] National Alternative Fuels Training Consortium (NAFTC) (2005 Oct 3) Tech stuff
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[15] Green Car Congress BMW to Apply Start-Stop and Brake Regen to MINIs Up to 60
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[21] PJ Wezenbeek (Zytec Systems Ltd) D G Evans (General Motors Powertrain) D P
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(Delphi Corp) Combustion Assisted Belt-Cranking of a V-8 Engine at 12-Volts SAE
Technical Papers vol 113 pp 396-407 2004 Document no 2004-01-0569
[22] T C Firbank Mechanics of the Belt Drive International Journal of Mechanical
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[23] R L Cassidy S K Fan R S MacDonald and W F Samson Serpentine Extended Life
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[24] A G Ulsoy J E Whitesell and M D Hooven Design of Belt-Tensioner Systems for
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Transactions of the ASME vol 107 pp 282-290 July 1985
[25] R S Beikmann N C Perkins and A G Ulsoy Free Vibration of Serpentine Belt Drive
Systems Journal of Vibrations and Acoustics Transactions of the ASME vol 118 pp
406-413 1996
[26] T C Kraver G W Fan and J J Shah Complex Modal Analysis of a Flat Belt Pulley
System with Belt Damping and Coulomb-Damped Tensioner Journal of Mechanical
Design Transactions of the ASME vol 118 pp 306-311 Jun 1996
[27] R S Beikmann N C Perkins and A G Ulsoy Design and Analysis of Automotive
Serpentine Belt Drive Systems for Steady State Performance Journal of Mechanical
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[28] L Zhang and J W Zu Modal Analysis of Serpentine Belt Drive Systems Journal of
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[29] R Balaji and E M Mockensturm Dynamic analysis of a front-end accessory drive with a
decouplerisolator International Journal of Vehicle Design vol 39 pp 208-231 2005
[30] M Nouri Design Optimization and Active Control of Serpentine Belt Drive Systems with
Two-pulley Tensioners University of Toronto 2005
[31] G J Spicer (Litens Automotive Inc) Tensioner for use in eg belt drive system has
electronic actuator associated with clutch spring for engaging International
WO2005119089-A1 Jun 6 2005 2005
[32] Bando Chemical Industries Ltd and Litens Automotive GmbH About belt-type starter
system Feb 27 2002
[33] H Lemberger and R Jungjohann (Bayerische Motoren Werke AG) Tension device for an
envelope drive of a device especially a belt drive of a starter generator of an internal
combustion engine comprises a support part Europe EP1420192-A2 May 19 2004 2003
[34] P Ahner and M Ackermann (Bosch GMBH) Belt drive especially for internal
combustion engines to drive accessories in an automobile Germany DE19849886-A1
May 11 2000 1998
[35] N Freisinger K Hagemann J Sievert P Struebel and M Treusch (Daimler Chrysler AG)
Belt tensioning device for belt drive between engine and starter generator of motor
vehicle has self-aligning bearing that supports auxiliary unit and provides working force to
tensioners for tensioning belt Germany DE10324268 Dec 16 2004 2003
[36] C R Rogers (Dayco Products LLC) Offset starter generator drive system for a vehicle
engine has a dual arm pivoted tensioner United States US6942589-B2 Feb 8 2005 2002
[37] A Serkh and I Ali (Gates Corp) Internal combustion engine has belt drive system with
tensioner asymmetrically biased in direction tending to cause power transmission belt to be
under tension International WO2003038309-A1 May 8 2003 2002
References 120
[38] P J Mcvicar and C A Thurston (General Motors Corp) Belt alternator starter accessory
drive with dual tensioner United States US20060287146-A1 Dec 21 2006 2005
[39] W Petri and M Bogner (INA Schaeffler KG) Traction drive especially for driving
internal combustion engine units has arrangement for demand regulated setting of tension
consisting of unit with housing with limited rotation and pulley German DE10044645-
A1 Mar 21 2002 2000
[40] M Bogner (INA Schaeffler KG) Belt drive tensioner for a starter-generator of an IC
engine has locking system for locking tensioning element in an engine operating mode
locking system is directly connected to pivot arm follows arm control movements
German DE10159073-A1 Jun 12 2003 2001
[41] R Painta M Bogner and H Graf (INA Schaeffler KG) Traction mechanism drive esp
belt drive has belt tensioning pulley mounted on generator shaft and uncoupled from it via
freewheel to dampen load peaks Europe EP1723350-A1 Nov 22 2006 2005
[42] W Petri (INA Schaeffler KG) Drive unit for a combustion engine having a starter
generator and a belt drive has tensioner with spring and counter hydraulic force Germany
DE10359641-A1 Jul 28 2005 2003
[43] H Stief M Bogner B Hartmann T Kraft and M Schmid (INA Schaeffler KG) Traction
drive especially belt drive for short-duration driving of starter generator has tensioning
device with lever arm deflectable against restoring force and with end stop limiting
deflection travel Europe EP1738093-A1 Jan 3 2007 2005
[44] M Ulm (INA Schaeffler KG DE) Tension unit eg for drive in machine such as
combustion engine has belt or chain drive with wheels turning and connected with starter
generator and unit has two idlers arranged at clamping arm with machine stored by shock
absorber Germany DE102004012395-A1 Sep 29 2005 2004
[45] M Bogner (INA Schaeffler KG) Belt drive for starter motor-generator auxiliary assembly
has limited movement at the starter belt section tensioner roller bringing it into a dead point
position on starting the motor International WO2006108461-A1 Oct 19 2006 2006
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[46] W Guhr (Litens Automotive GMBH) Automotive motor and drive assembly includes
tension device positioned within belt drive system having combination starter United
States US2001007839-A1 Jul 12 2001 2001
[47] K Kuniaki K Masahiko H Kazuyuki I Shuichi and T Masaki (Mitsubishi Jidosha Eng
KK and Mitsubishi Motor Corp) Tension adjustment method of belt for starter generator
in vehicle involves shifting auto-tensioners between lock state and free state to adjust
tension of belt during driving of crank pulley Japan JP2005083514-A Mar 31 2005
2003
[48] Nissan Motor Co Ltd Winding gear for starting engine of hybrid motor vehicle has
tensioner tightening chain while cranking engine and slackens chain after start of engine
provided to span side of chain Japan JP3565040-B2 Sep 15 2004 1998
[49] S Sato and H Hayakawa (NTN Corp) Auto tensioner for ancillary drive belts has
cylinder nut and screw bolt in hydraulic damper mechanism provided in middle of cylinder
acting as start-up rigidity buffer component Japan JP2006189073-A Jul 20 2006 2005
[50] G Vadin-Michaud (Valeo Equip Electrique Moteur) Pulley and belt starting system for a
thermal engine for a motor vehicle Europe EP1658432 May 24 2006 2005
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2005
[52] W E Johns Notes on Motors [Electronic] 2003 [2008 June] Available at
httpwwwgizmologynetmotorshtm
[53] Litens Automotive Group Ltd DC BAS System - Conventional Start Input Profile Nov
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[54] Arnold Magnetic Technologies Corp General Motor Terminology [Electronic] pp 7
[2008 June] Available at httpwwwgrouparnoldcommtcpdfweb_motor_glossarypdf
[55] Douglas W Jones Stepping Motors University of Iowa - Department of Computer
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httpwwwcsuiowaedu~jonesstepphysicshtml
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[56] Litens Automotive Group Ltd (2004 Jan 31) FEAD software input data for test project
[57] K Deb Multi-Objective Optimization using Evolutionary Algorithms Toronto John Wiley
amp Sons Ltd 2001 pp 81-85
[58] P E McSharry (2004 May 11) Department of Engineering Science University of Oxford
[httpwwwengoxacuksamppubsgawbreppdf]
[59] The MathWorks Inc MATLAB vol 750342 (R2007b) Aug 15 2007
123
APPENDIX A
Passive Dual Tensioner Designs from Patent Literature
Figure A1 Proposed design by Bayerische Motoren Werke AG corresponding to patent nos EP1420192-A2 and DE10253450-A1
Source European Patent Office espcenet (publication nos EP1420192-A2 and DE10253450-A1 accessed May 2007) epespacenetcom [33]
Figure A1 label identification 1 ndash tightner 2 ndash belt drive
3 ndash starter generator
4 ndash internal-combustion engine
4‟ ndash crankshaft-lateral drive disk
5 ndash generator housing
6 ndash common axis of rotation
7 ndash featherspring of tiltable clamping arms
8 ndash clamping arm
9 ndash clamping arm
10 11 ndash idlers
12 12‟ ndash Zugtrum 13 13‟ ndash Leertrum
14 ndash carry-hurries 15 ndash generator wave
16 ndash bush
17 ndash absorption mechanism
18 18‟ ndash support arms
19 19‟ ndash auxiliary straight lines
20 ndash pipe
21 ndash torsion bar
22 ndash breaking through
23 ndash featherspring
24 ndash friction disk
25 ndash screw connection 26 ndash Wellscheibe
(European Patent Office May 2007) [33]
Appendix A 124
Figure A2a First of four proposed designs (and its various configurations) by Bosch GMBH
corresponding to patent no WO0026532-A1
Source European Patent Office espcenet (publication no WO0026532-A1 accessed Jun 2007)
epespacenetcom [34]
Figure A2b Second of four proposed designs by Bosch GMBH corresponding to patent no
WO0026532-A1
Source European Patent Office espcenet (publication no WO0026532-A1 accessed Jun 2007) epespacenetcom [34]
Figure A2c Third of four proposed designs (and its various configurations) by Bosch GMBH
corresponding to patent no WO0026532-A1
Source European Patent Office espcenet (publication no WO0026532-A1 accessed Jun 2007)
epespacenetcom [34]
Appendix A 125
Figure A2d Fourth of four proposed designs (and its various configurations) by Bosch GMBH
corresponding to patent no WO0026532-A1
Source European Patent Office espcenet (publication no WO0026532-A1 accessed Jun 2007)
epespacenetcom [34]
Figure A2a through to A2d label identification 10 ndash engine wheel
11 ndash [generator] 13 ndash spring
14 ndash belt
16 17 ndash tensioning pulleys
18 19 ndash springs
20 21 ndash fixed points
25ab ndash carriers of idlers
25c ndash gang bolt
(European Patent Office June 2007) [34]
Figure A3 Proposed design by Daimler Chrysler AG corresponding to patent no DE10324268-A1
Source European Patent Office espcenet (publication no DE10324268-A1 accessed May 2007)
epespacenetcom [35]
Figure A3 label identification
Appendix A 126
10 12 ndash belt pulleys
14 ndash auxiliary unit
16 ndash belt
22-1 22-2 ndash belt tensioners
(European Patent Office May 2007) [35]
Figure A4 Proposed design by Dayco Products LLC corresponding to patent no US6942589-B2
Source European Patent Office espcenet (publication no US6942589-B2 accessed Jun 2007)
epespacenetcom [36]
Figure A4 label identification 12 ndash belt
14 ndash tensioner
16 ndash generator pulley
18 ndash crankshaft pulley
22 ndash slack span 24 ndash tight span
32 34 ndash arms
33 35 ndash pulley
(European Patent Office June 2007) [36]
Appendix A 127
Figure A5 Proposed design by Gates Corp corresponding to patent no WO2003038309-A
Source European Patent Office espcenet (publication no WO2003038309-A accessed Jun 2007)
epespacenetcom [37]
Figure A5 label identification 12 ndash motorgenerator
14 ndash motorgenerator pulley 26 ndash belt tensioner
28 ndash belt tensioner pulley
30 ndash transmission belt
(European Patent Office June 2007) [37]
Figure A6 Proposed design by General Motors Corp corresponding to patent no US20060287146-A1
Source European Patent Office espcenet (publication no US20060287146-A1 accessed Jun 2007)
epespacenetcom [38]
Appendix A 128
Figure A6 label identification 28 ndash tensioner
32 ndash carrier arm
34 ndash secondary carrier arm
46 ndash tensioner pulley
58 ndash secondary tensioner pulley
(European Patent Office June 2007) [38]
Figure A7 Proposed design by INA Schaeffler KG corresponding to patent no DE10044645-A1
Source European Patent Office espcenet (publication no DE10044645-A1 accessed Jun 2007)
epespacenetcom [39]
Figure A7 label identification 2 ndash internal combustion engine
3 ndash traction element
11 ndash housing with limited rotation 12 13 ndash direction changing pulleys
(European Patent Office June 2007) [39]
Appendix A 129
Figure A8a First of three proposed designs by INA Schaeffler KG corresponding to patent no
DE10159073-A1
Source European Patent Office espcenet (publication no DE10159073-A1 accessed May 2007)
epespacenetcom [40]
Figure A8b Second of three proposed designs by INA Schaeffler KG corresponding to patent no
DE10159073-A1
Source European Patent Office espcenet (publication no DE10159073-A1 accessed May 2007)
epespacenetcom [40]
Appendix A 130
Figure A8c Third of three proposed designs by INA Schaeffler KG corresponding to patent no
DE10159073-A1
Source European Patent Office espcenet (publication no DE10159073-A1 accessed May 2007)
epespacenetcom [40]
Figure A8a A8b and A8c label identification 1 ndash tightener [tensioner]
2 ndash idler
3 ndash drawing means
4 ndash swivel arm
5 ndash axis of rotation
6 ndash drawing means impulse [belt]
7 ndash crankshaft
8 ndash starter generator
9 ndash bolting volume 10a ndash bolting device system
10b ndash bolting device system
10c ndash bolting device system
11 ndash friction body
12 ndash lateral surface
13 ndash bolting tape end
14 ndash bolting tape end
15 ndash control member
16 ndash torsion bar
17 ndash base
18 ndash pylon
19 ndash hub
20 ndash annular gap
21 ndash Gleitlagerbuchse
23 ndash [nil]
23 ndash friction disk
24 ndash turning camps 25 ndash teeth
26 ndash elbow levers
27 ndash clamping wedge
28 ndash internal contour
29 ndash longitudinal guidance
30 ndash system
31 ndash sensor
32 ndash clamping gap
(European Patent Office May 2007) [40]
Appendix A 131
Figure A9 Proposed design by INA Schaeffler KG corresponding to patent no DE10359641-A1
Source European Patent Office espcenet (publication no DE10359641-A1 accessed Jun 2007)
epespacenetcom [42]
Figure A9 label identification 8 ndash cylinder
10 ndash rod
12 ndash spring plate
13 ndash spring
14 ndash pressure lead
(European Patent Office June 2007) [42]
Appendix A 132
Figure A10 Proposed design by INA Schaeffler KG corresponding to patent no EP1723350-A1
Source European Patent Office espcenet (publication no EP1723350-A1 accessed Jun 2007) epespacenetcom [41]
Figure A10 label identification 4 ndash pulley
5 ndash hydraulic element 11 ndash freewheel
12 ndash shaft
(European Patent Office June 2007) [41]
Figure A11 Proposed design by INA Schaeffler KG corresponding to patent no EP1738093-A1
Source European Patent Office espcenet (publication no EP1738093-A1 accessed Jun 2007)
epespacenetcom [43]
Figure A11 label identification 1 ndash traction drive
2 ndash traction belt
3 ndash starter generator
Appendix A 133
7 ndash tension device
9 ndash lever arm
10 ndash guide roller
16 ndash end stop
(European Patent Office June 2007) [43]
Figure A12 Proposed design by INA Schaeffler KG corresponding to patent no DE102004012395-A1
Source European Patent Office espcenet (publication no DE102004012395-A1 accessed May 2007) epespacenetcom [44]
Figure A12 label identification 1 ndash belt drive
2 ndash belts
3 ndash wheel of the internal-combustion engine
4 ndash wheel of a Nebenaggregats
5 ndash wheel of the starter generator
6 ndash clamping unit
7 ndash idler
8 ndash idler
9 ndash scale beams
10 ndash drive place
11 ndash clamping arm
12 ndash camps
13 ndash coupling point
14 ndash shock absorber element
15 ndash arrow
(European Patent Office May 2007) [44]
Appendix A 134
Figure A13 Proposed design by INA Schaeffler KG corresponding to patent nos DE102005017038-A1and WO2006108461-A1
Source European Patent Office espcenet (publication nos DE102005017038-A1and WO2006108461-A1 accessed May 2007) epespacenetcom [45]
Figure A13 label identification 1 ndash belt
2 ndash wheel of the crankshaft KW
3 ndash wheel of a climatic compressor AC
4 ndash wheel of a starter generator SG
5 ndash wheel of a water pump WP
6 ndash first clamping system
7 ndash first tension adjuster lever arm
8 ndash first tension adjuster role
9 ndash second clamping system
10 ndash second tension adjuster lever arm
11 ndash second tension adjuster role 12 ndash guide roller
13 ndash drive-conditioned Zugtrum
(generatorischer enterprise (GE))
13 ndash starter-conditioned Leertrum
(starter enterprise (SE))
14 ndash drive-conditioned Leertrum (GE)
14 ndash starter-conditioned Zugtrum (SE)
14a ndash drive-conditioned Leertrumast (GE)
14a ndash starter-conditioned Zugtrumast (SE)
14b ndash drive-conditioned Leertrumast (GE)
14b ndash starter-conditioned Zugtrumast (SE)
(European Patent Office May 2007) [45]
Figure A14 Proposed design by Litens Automotive GMBH et al corresponding to patent no
US20010007839-A1
Appendix A 135
Source European Patent Office espcenet (publication no US20010007839-A1 accessed Jun 2007)
epespacenetcom [46]
Figure A14 label identification E - belt
K - crankshaft
R1 ndash first tension pulley
R2 ndash second tension pulley
S ndash tension device
T ndash drive system
1 ndash belt pulley
4 ndash belt pulley
(European Patent Office June 2007) [46]
Figure A15 Proposed design by Mitsubishi Jidosha Eng KK and Mitsubishi Motor Corp corresponding
to patent no JP2005083514-A
Source Industrial Property Digital Library and Japanese Patent Office Patent amp Utility Model Gazette DB (document no A 2005-083514 accessed May 2007) wwwipdlinpitgojp [47]
Figure A15 label identification 1 ndash Pulley for Starting
2 ndash Crank Pulley
3 ndash AC Pulley
4a ndash 1st roller
4b ndash 2nd roller
5 ndash Idler Pulley
6 ndash Belt
7c ndash Starter generator control section
7d ndash Idle stop control means
8 ndash WP Pulley
9 ndash IG Switch
10 ndash Engine
11 ndash Starter Generator
12 ndash Driving Shaft
Appendix A 136
7 ndash ECU
7a ndash 1st auto tensioner control section (the 1st auto
tensioner control means)
7b ndash 2nd auto tensioner control section (the 2nd auto
tensioner control means)
13 ndash Air-conditioner Compressor
14a ndash 1st auto tensioner
14b ndash 2nd auto tensioner
18 ndash Water Pump
(Industrial Property Digital Library May 2007) [47]
Figure A16 Proposed design by Nissan Motor Co Ltd corresponding to patent no JP3565040-B2
Source European Patent Office espcenet (publication no JP3565040-B2 accessed Jun 2007) epespacenetcom [48]
Figure A16 label identification 3 ndash chain [or belt]
5 ndash tensioner
4 ndash belt pulley
(European Patent Office June 2007) [48]
Figure A17 Proposed design by NTN Corp corresponding to patent no JP2006189073-A
Appendix A 137
Source European Patent Office espcenet (publication no JP2006189073-A accessed Jun 2007)
epespacenetcom [49]
Figure A17 label identification 5d - flange
6 ndash tensile strength spring
10 ndash actuator
10c ndash cylinder
12 ndash rod
20 ndash hydraulic damper mechanism
21 ndash cylinder nut
22 ndash screw bolt
(European Patent Office June 2007) [49]
Figure A18 Proposed design by Valeo Equipment Electriques Moteur corresponding to patent nos
EP1658432 and WO2005015007
Source European Patent Office espcenet (publication nos EP1658432 and WO2005015007
accessed May 2007) epespacenetcom [50]
Figure A18 abbreviated list of label identifications
10 ndash starter
22 ndash shaft section
23 ndash free front end
80 ndash pulley
200 ndash support element
206 - arm
(European Patent Office May 2007) [50]
The author notes that the list of labels corresponding to Figures A1a through to A7 are generated
from machine translations translated from the documentrsquos original language (ie German)
Consequently words may be translated inaccurately or not at all
138
APPENDIX B
B-ISG Serpentine Belt Drive with Single Tensioner
Equation of Motion
The equations of motion (EOM) for a B-ISG serpentine belt drive with a single tensioner are
shown The EOM has been derived similarly to that of the same system with a twin tensioner
that was provided in Chapter 3 The assumptions for the twin tensioner B-ISG system are
applicable for the single tensioner B-ISG system as well
Figure B1 shows the B-ISG system with a single tensioner pulley and arm The pulleys are
numbered 1 through 4 and their associated belt spans are numbered accordingly
Figure B1 Single Tensioner B-ISG System
Appendix B 139
The free-body diagram for the ith non-tensioner pulley in the system shown above is found in
Figure B2 The moment of inertia for the ith pulley is designated as Ii while the angular
displacement velocity and acceleration is designated as 120579119905119894 120579 119905119894 and 120579 119905119894 respectively The
required torque is Qi the angular damping is Ci and the tension of the ith span is Ti
Figure B2 Free-body Diagram of ith Pulley
The positive motion designated is assumed to be in the clockwise direction The radius for the
ith pulley is represented by Ri The equilibrium equations for the ith pulley are as follows
I1 ∙ θ 1 = T4 ∙ R1 minus T1 ∙ R1 + Q1 minus c1 ∙ θ 1 (B1)
I2 ∙ θ 2 = T1 ∙ R2 minus T2 ∙ R2 + Q2 minus c2 ∙ θ 2 (B2)
I3 ∙ θ 3 = T2 ∙ R3 minus T3 ∙ R3 + Q3 minus c3 ∙ θ 3 (B3)
Appendix B 140
A free-body diagram for the single tensioner pulley is shown in Figure B3 The rotational
stiffness and damping for the tensioner arm is designated as kt and ct respectively The angular
rotation and velocity for the arm is 120579119905119894 and 120579 119905119894 respectively
Figure B3 Free-body Diagram of Single Tensioner
From figure B2 the equations of equilibrium are resolved for the tensioner pulley The angle of
orientation for the ith belt span is designated by 120573119895119894
minusI4 ∙ θ 4 = minusT3 ∙ R4 + T4 ∙ R4 + Q4 + c4 ∙ θ 4 (B4)
It ∙ θ t = minusTt ∙ Lt ∙ sin θto minus βj4 + sin θto 1 minus βj5 minus kt ∙ partθt minus ct ∙ partθ t
(B5)
Appendix B 141
partθt = θt minus θto (B6)
The dynamic tension matrix Trsquo is proportional to the damping (Tc) and stiffness (Tk) matrices
that are due to belt damping (119888119894119887 ) and belt stiffness (119896119894
119887 ) respectively
119827prime = 119827119836 ∙ 120521 + 119827119844 ∙ 120521 (B7)
The initial tension is represented by To and the initial angle of the tensioner arm is represented
by 120579119905119900 The equation for the tension of the ith span is shown in the following equations
T1 = To + k1b ∙ R1 ∙ θ1 minus R2 ∙ θ2 + c1
b ∙ [R1 ∙ θ 1 minus R2 ∙ θ 2] (B8)
T2 = To + k2b ∙ R2 ∙ θ2 minus R3 ∙ θ3 + c2
b ∙ [R2 ∙ θ 2 minus R3 ∙ θ 3)] (B9)
T3 = To + k3b ∙ R3 ∙ θ3 minus R4 ∙ θ4 + Lt ∙ [sin θto minus βj3 ] ∙ (θt minus θto ) + c3
b ∙ [R3 ∙ θ 3 minus R4 ∙
θ 4 + Lt ∙ [sin θto minus βj3 ] ∙ (θ t)] (B10)
T4 = To + k4b ∙ R4 ∙ θ4 minus R1 ∙ θ1 + Lt ∙ [sin θto minus βj4 ] ∙ (θt minus θto ) + c4
b ∙ [R4 ∙ θ 4 minus R1 ∙
θ 1 + Lt ∙ [sin θto minus βj4 ] ∙ (θ t)] (B11)
Tprime = Ti minus To (B12)
Tt = T3 = T4 (B13)
Appendix B 142
The EOM for the single tensioner B-ISG system is found by substitution of equations B8 to
B13 into B1 to B5 The matrices in the EOM include the inertial matrix I damping matrix C
stiffness matrix K and the required torque matrix Q as well as the angular displacement
velocity and acceleration matrices 120521 120521 and 120521 respectively
119816 ∙ 120521 + 119810 ∙ 120521 + 119818 ∙ 120521 = 119824 (B14)
119816 =
I1 0 0 0 00 I2 0 0 00 0 I3 0 00 0 0 I4 00 0 0 0 It1
(B15)
The stiffness matrix includes kb the belt factor Kb the belt cord stiffness 120601119894 the wrap angle of
the belt on the ith pulley and Kbi the stiffness factor of the ith belt span Cb represents the belt
damping for each span and βji is the angle of orientation for the span between the jth and ith
pulleys It is noted in the terms of the stiffness and damping matrices below that the numerical
subscripts refer to the (i+1)th pulley The term Qt may be found within the required torque
matrix and represents the required torque for the tensioner arm As well the term It1 represents
the moment of inertia for the tensioner arm
Appendix B 143
K =
(B16)
Kbi =Kb
Li + kb ∙ Ri ∙ϕi+1
2 + Ri ∙ϕi
2
(B17)
C =
(B18)
Appendix B 144
Appendix B 144
120521 =
θ1
θ2
θ3
θ4
partθt
(B19)
119824 =
Q1
Q2
Q3
Q4
Qt
(B20)
Simulations of the EOM for the B-ISG system with a single tensioner were performed in FEAD
[51] software for dynamic and static cases This allowed for the methodology for deriving the
EOM to be verified and then applied to the B-ISG system with a twin tensioner The natural
frequencies modes shapes dynamic responses tensioner arm torques as well as the crankshaft
required torque only and the dynamic tensions were solved from the EOM as described in (331)
to (339) of Chapter 3 and as well as for the static tension from (351) to (353) of Chapter 3
This permitted verification of the complete derivation methodology and allowed for comparison
of the static tension of the B-ISG system belt spans in the case that a single tensioner is present
and in the case that a Twin Tensioner is present [51]
145
APPENDIX C
MathCAD Scripts
C1 Geometric Analysis
1 - CS
2 - AC
4 - Alt
3 - Ten1
5 - Ten 2
6 - Ten Pivot
1
2
3
4
5
Figure C1 Schematic of B-ISG
System with Twin Tensioner
Coordinate Input DataXY1 0 0( ) XY4 24759 16664( )
XY2 224 6395( ) XY5 12057 9193( )
XY3 292761 87( ) XY6 201384 62516( )
Computations
Lt1 XY30 0
XY60 0
2
XY30 1
XY60 1
2
Lt2 XY50 0
XY60 0
2
XY50 1
XY60 1
2
t1 atan2 XY30 0
XY60 0
XY30 1
XY60 1
t2 atan2 XY50 0
XY60 0
XY50 1
XY60 1
XY
XY10 0
XY20 0
XY30 0
XY40 0
XY50 0
XY60 0
XY10 1
XY20 1
XY30 1
XY40 1
XY50 1
XY60 1
x XY
0 y XY
1
Appendix C 146
i - angle bw horizontal and l ine from ith pulley center to (i+1)th pulley center
Adjust last number in range variable p to correspond to number of pulleys
p 0 1 4
k p( ) p 1( ) p 4if
0 otherwise
condition1 p( ) acos
XYk p( ) 0
XYp 0
XYk p( ) 0
XYp 0
2
XYk p( ) 1
XYp 1
2
condition2 p( ) 2 acos
XYk p( ) 0
XYp 0
XYk p( ) 0
XYp 0
2
XYk p( ) 1
XYp 1
2
p( ) if XYk p( ) 1
XYp 1
condition1 p( ) condition2 p( )
Lfi Lbi - connection belt span lengths
D1 20065mm D2 10349mm D3 7240mm D4 6820mm D5 7240mm
D
D1
D2
D3
D4
D5
Lf p( ) XYk p( ) 0
XYp 0
2
XYk p( ) 1
XYp 1
2
1
mm
Dk p( )
2
Dp
2
2
Lb p( ) XYk p( ) 0
XYp 0
2
XYk p( ) 1
XYp 1
2
1
mm
Dk p( )
2
Dp
2
2
fi bi - angle bw ith pulley center connection l ine and contact points Pbfi (or Pfbi) and Pbi
(or Pfi) respecti vely l
f p( ) atanLf p( ) mm
Dp
2
Dk p( )
2
Dp
Dk p( )
if
atanLf p( ) mm
Dk p( )
2
Dp
2
Dp
Dk p( )
if
2D
pD
k p( )if
b p( ) atan
mmLb p( )
Dp
2
Dk p( )
2
Appendix C 147
XYfi XYbi XYfbi XYbfi - 4 possible contact points for i th pulley
XYf p( ) XYp 0
Dp
2 mmcos p( ) f p( )
XYp 1
Dp
2 mmsin p( ) f p( )
XYb p( ) XYp 0
Dp
2 mmcos p( ) f p( )
XYp 1
Dp
2 mmsin p( ) f p( )
XYfb p( ) XYp 0
Dp
2 mmcos p( ) b p( )
XYp 1
Dp
2 mmsin p( ) b p( )
XYbf p( ) XYp 0
Dp
2 mmcos p( ) b p( )
XYp 1
Dp
2 mmsin p( ) b p( )
XYfi+1 XYbi+1 XYfbi+1 XYbfi+1 - 4 possible contact points for i+1th pulley
XYf2 p( ) XYk p( ) 0
Dk p( )
2 mmcos p( ) f p( )
XYk p( ) 1
Dk p( )
2 mmsin p( ) f p( )
XYb2 p( ) XYk p( ) 0
Dk p( )
2 mmcos p( ) f p( )
XYk p( ) 1
Dk p( )
2 mmsin p( ) f p( )
XYfb2 p( ) XYk p( ) 0
Dk p( )
2 mmcos p( ) b p( ) XY
k p( ) 1
Dk p( )
2 mmsin p( ) b p( )
XYbf2 p( ) XYk p( ) 0
Dk p( )
2 mmcos p( ) b p( ) XY
k p( ) 1
Dk p( )
2 mmsin p( ) b p( )
Row 1 --gt Pulley 1 Row i --gt Pulley i
XYfi
XYf 0( )0 0
XYf 1( )0 0
XYf 2( )0 0
XYf 3( )0 0
XYf 4( )0 0
XYf 0( )0 1
XYf 1( )0 1
XYf 2( )0 1
XYf 3( )0 1
XYf 4( )0 1
XYfi
6818
269222
335325
251552
108978
100093
89099
60875
200509
207158
x1 XYfi0
y1 XYfi1
Appendix C 148
XYbi
XYb 0( )0 0
XYb 1( )0 0
XYb 2( )0 0
XYb 3( )0 0
XYb 4( )0 0
XYb 0( )0 1
XYb 1( )0 1
XYb 2( )0 1
XYb 3( )0 1
XYb 4( )0 1
XYbi
47054
18575
269403
244841
164847
88606
291
30965
132651
166182
x2 XYbi0
y2 XYbi1
XYfbi
XYfb 0( )0 0
XYfb 1( )0 0
XYfb 2( )0 0
XYfb 3( )0 0
XYfb 4( )0 0
XYfb 0( )0 1
XYfb 1( )0 1
XYfb 2( )0 1
XYfb 3( )0 1
XYfb 4( )0 1
XYfbi
42113
275543
322697
229969
9452
91058
59383
75509
195834
177002
x3 XYfbi0
y3 XYfbi1
XYbfi
XYbf 0( )0 0
XYbf 1( )0 0
XYbf 2( )0 0
XYbf 3( )0 0
XYbf 4( )0 0
XYbf 0( )0 1
XYbf 1( )0 1
XYbf 2( )0 1
XYbf 3( )0 1
XYbf 4( )0 1
XYbfi
8384
211903
266707
224592
140427
551
13639
50105
141463
143331
x4 XYbfi0
y4 XYbfi1
Row 1 --gt Pulley 2 Row i --gt Pulley i+1
XYf2i
XYf2 0( )0 0
XYf2 1( )0 0
XYf2 2( )0 0
XYf2 3( )0 0
XYf2 4( )0 0
XYf2 0( )0 1
XYf2 1( )0 1
XYf2 2( )0 1
XYf2 3( )0 1
XYf2 4( )0 1
XYf2x XYf2i0
XYf2y XYf2i1
XYb2i
XYb2 0( )0 0
XYb2 1( )0 0
XYb2 2( )0 0
XYb2 3( )0 0
XYb2 4( )0 0
XYb2 0( )0 1
XYb2 1( )0 1
XYb2 2( )0 1
XYb2 3( )0 1
XYb2 4( )0 1
XYb2x XYb2i0
XYb2y XYb2i1
Appendix C 149
XYfb2i
XYfb2 0( )0 0
XYfb2 1( )0 0
XYfb2 2( )0 0
XYfb2 3( )0 0
XYfb2 4( )0 0
XYfb2 0( )0 1
XYfb2 1( )0 1
XYfb2 2( )0 1
XYfb2 3( )0 1
XYfb2 4( )0 1
XYfb2x XYfb2i
0
XYfb2y XYfb2i1
XYbf2i
XYbf2 0( )0 0
XYbf2 1( )0 0
XYbf2 2( )0 0
XYbf2 3( )0 0
XYbf2 4( )0 0
XYbf2 0( )0 1
XYbf2 1( )0 1
XYbf2 2( )0 1
XYbf2 3( )0 1
XYbf2 4( )0 1
XYbf2x XYbf2i0
XYbf2y XYbf2i1
100 40 20 80 140 200 260 320 380 440 500150
110
70
30
10
50
90
130
170
210
250Figure C2 Possible Contact Points
250
150
y1
y2
y3
y4
y
XYf2y
XYb2y
XYfb2y
XYbf2y
500100 x1 x2 x3 x4 x XYf2x XYb2x XYfb2x XYbf2x
Appendix C 150
Xij Yij - selected contact point on ith pulley for span from ith pulley to jth pulley
XY15 XYbf2iT 4
XY12 XYfiT 0
Pulley 1 contact pts
XY21 XYf2iT 0
XY23 XYfbiT 1
Pulley 2 contact pts
XY32 XYfb2iT 1
XY34 XYbfiT 2
Pulley 3 contact pts
XY43 XYbf2iT 2
XY45 XYfbiT 3
Pulley 4 contact pts
XY54 XYfb2iT 3
XY51 XYbfiT 4
Pulley 5 contact pts
By observation the lengths of span i is the following
L1 Lf 0( ) L2 Lb 1( ) L3 Lb 2( ) L4 Lb 3( ) L5 Lb 4( ) Li
L1
L2
L3
L4
L5
mm
i Angle between horizontal and span of ith pulley
i
atan
XY121
XY211
XY12
0XY21
0
atan
XY231
XY321
XY23
0XY32
0
atan
XY341
XY431
XY34
0XY43
0
atan
XY451
XY541
XY45
0XY54
0
atan
XY511
XY151
XY51
0XY15
0
Appendix C 151
Pulley 1 Pulley 2 Pulley 3 Pulley 4 Pulley 5
12 i0 2 21 i0 32 i1 2 43 i2 54 i3
15 i4 2 23 i1 34 i2 45 i3 51 i4
15
21
32
43
54
12
23
34
45
51
Wrap angle i for the ith pulley
1 2 atan2 XY150
XY151
atan2 XY120
XY121
2 atan2 XY210
XY1 0
XY211
XY1 1
atan2 XY230
XY1 0
XY231
XY1 1
3 2 atan2 XY320
XY2 0
XY321
XY2 1
atan2 XY340
XY2 0
XY341
XY2 1
4 atan2 XY430
XY3 0
XY431
XY3 1
atan2 XY450
XY3 0
XY451
XY3 1
5 atan2 XY540
XY4 0
XY541
XY4 1
atan2 XY510
XY4 0
XY511
XY4 1
1
2
3
4
5
Lb length of belt
Lbelt Li1
2
0
4
p
Dpp
Input Data for B-ISG System
Kt 20626Nm
rad (spring stiffness between tensioner arms 1
and 2)
Kt1 10314Nm
rad (stiffness for spring attached at arm 1 only)
Kt2 16502Nm
rad (stiffness for spring attached at arm 2 only)
Appendix C 152
C2 Dynamic Analysis
I C K moment of inertia damping and stiffness matrices respectively
u 0 1 4 v 0 1 4 (new counter variables where final value = no of pulleys + no of ten arms)
RaD
2
Appendix C 153
RaD
2
Ii =gt moment of inertia for ith pulley where i-1 and i represent ten arms
Ii0
0
1
2
3
4
5
6
10000
2230
300
3000
300
1500
1500
I diag Ii( ) kg mm2
Ci =gt Rotational damping and belt damping for the ith pulley where i-1 and i represent tensioner arms
1000kg
m3
CrossArea 693mm2
0 M CrossArea Lbelt M 0086kg
cb 2 KbM
Lbelt
Cb
cb
cb
cb
cb
cb
Cri
0
0
010
0
010
N mmsec
rad
Ct 1000N mmsec
rad Ct1 1000 N mm
sec
rad Ct2 1000N mm
sec
rad
Cr
Cri0
0
0
0
0
0
0
0
Cri1
0
0
0
0
0
0
0
Cri2
0
0
0
0
0
0
0
Cri3
0
0
0
0
0
0
0
Cri4
0
0
0
0
0
0
0
Ct Ct1
Ct
0
0
0
0
0
Ct
Ct Ct2
Rt
Ra0
Ra1
0
0
0
0
0
0
Ra1
Ra2
0
0
Lt1 mm sin t1 32
0
0
0
Ra2
Ra3
0
Lt1 mm sin t1 34
0
0
0
0
Ra3
Ra4
0
Lt2 mm sin t2 54
Ra0
0
0
0
Ra4
0
Lt2 mm sin t2 51
Appendix C 154
Kr
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Kt Kt1
Kt
0
0
0
0
0
Kt
Kt Kt2
Tk
Kbi 0( ) Ra0
0
0
0
Kbi 4( ) Ra0
Kbi 0( ) Ra1
Kbi 1( ) Ra1
0
0
0
0
Kbi 1( ) Ra2
Kbi 2( ) Ra2
0
0
0
0
Kbi 2( ) Ra3
Kbi 3( ) Ra3
0
0
0
0
Kbi 3( ) Ra4
Kbi 4( ) Ra4
0
Kbi 1( ) Lt1 mm sin t1 32
Kbi 2( ) Lt1 mm sin t1 34
0
0
0
0
0
Kbi 3( ) Lt2 mm sin t2 54
Kbi 4( ) Lt2 mm sin t2 51
Tc
Cb0
Ra0
0
0
0
Cb4
Ra0
Cb0
Ra1
Cb1
Ra1
0
0
0
0
Cb1
Ra2
Cb2
Ra2
0
0
0
0
Cb2
Ra3
Cb3
Ra3
0
0
0
0
Cb3
Ra4
Cb4
Ra4
0
Cb1
Lt1 mm sin t1 32
Cb2
Lt1 mm sin t1 34
0
0
0
0
0
Cb3
Lt2 mm sin t2 54
Cb4
Lt2 mm sin t2 51
C matrix
C Cr Rt Tc
K matrix
K Kr Rt Tk
New Equations of Motion for Dual Drive System
I K amp C matricies rearranged to place driving pulley in 1st row + 1st column and driven in 2nd row + 2nd column
IA augment I3
I0
I1
I2
I4
I5
I6
IC augment I0
I3
I1
I2
I4
I5
I6
I1kgmm2 1 106
kg m2
0 0 0 0 0 0
Ia stack I1kgmm2 IAT 0
T
IAT 1
T
IAT 2
T
IAT 4
T
IAT 5
T
IAT 6
T
Ic stack I1kgmm2 ICT 3
T
ICT 1
T
ICT 2
T
ICT 4
T
ICT 5
T
ICT 6
T
Appendix C 155
RtA augment Rt3
Rt0
Rt1
Rt2
Rt4
RtC augment Rt0
Rt3
Rt1
Rt2
Rt4
Rta stack RtAT 3
T
RtAT 0
T
RtAT 1
T
RtAT 2
T
RtAT 4
T
RtAT 5
T
RtAT 6
T
Rtc stack RtCT 0
T
RtCT 3
T
RtCT 1
T
RtCT 2
T
RtCT 4
T
RtCT 5
T
RtCT 6
T
TkA augment Tk3
Tk0
Tk1
Tk2
Tk4
Tk5
Tk6
Tka stack TkAT 3
T
TkAT 0
T
TkAT 1
T
TkAT 2
T
TkAT 4
T
TkC augment Tk0
Tk3
Tk1
Tk2
Tk4
Tk5
Tk6
Tkc stack TkCT 0
T
TkCT 3
T
TkCT 1
T
TkCT 2
T
TkCT 4
T
TcA augment Tc3
Tc0
Tc1
Tc2
Tc4
Tc5
Tc6
Tca stack TcAT 3
T
TcAT 0
T
TcAT 1
T
TcAT 2
T
TcAT 4
T
TcC augment Tc0
Tc3
Tc1
Tc2
Tc4
Tc5
Tc6
Tcc stack TcAT 0
T
TcAT 3
T
TcAT 1
T
TcAT 2
T
TcAT 4
T
Ka Kr Rta Tka Kc Kr Rtc Tkc Ca Cr Rta Tca Cc Cr Rtc Tcc
CHECK
KA augment K3
K0
K1
K2
K4
K5
K6
KC augment K0
K3
K1
K2
K4
K5
K6
CA augment C3
C0
C1
C2
C4
C5
C6
CC augment C0
C3
C1
C2
C4
C5
C6
Appendix C 156
Kacheck stack KAT 3
T
KAT 0
T
KAT 1
T
KAT 2
T
KAT 4
T
KAT 5
T
KAT 6
T
Kccheck stack KCT 0
T
KCT 3
T
KCT 1
T
KCT 2
T
KCT 4
T
KCT 5
T
KCT 6
T
Cacheck stack CAT 3
T
CAT 0
T
CAT 1
T
CAT 2
T
CAT 4
T
CAT 5
T
CAT 6
T
Cccheck stack CCT 0
T
CCT 3
T
CCT 1
T
CCT 2
T
CCT 4
T
CCT 5
T
CCT 6
T
Results for System switching from ISG as DRIVING pulley to Crankshaft as Drivi ng Pulley
Modified Submatricies for ISG Driving Phase --gt CS Driving Phase
Unit step function to provide shift from crankshaft DRIVING case (ie ISG driven case) to crankshaft DRIVEN
case (ie ISG driving case)
H n( ) 1 n 750if
0 n 750if
lt-- crankshaft DRIVING case (Phase change bw 2 cases occurs when n
reaches start speed)
I11mod n( ) Ic0 0
H n( ) 1if
Ia0 0
H n( ) 0if
I22mod n( )submatrix Ic 1 6 1 6( )
UnitsOf I( )H n( ) 1if
submatrix Ia 1 6 1 6( )
UnitsOf I( )H n( ) 0if
K11mod n( )
Kc0 0
UnitsOf K( )H n( ) 1if
Ka0 0
UnitsOf K( )H n( ) 0if
C11modn( )
Cc0 0
UnitsOf C( )H n( ) 1if
Ca0 0
UnitsOf C( )H n( ) 0if
K22mod n( )submatrix Kc 1 6 1 6( )
UnitsOf K( )H n( ) 1if
submatrix Ka 1 6 1 6( )
UnitsOf K( )H n( ) 0if
C22modn( )submatrix Cc 1 6 1 6( )
UnitsOf C( )H n( ) 1if
submatrix Ca 1 6 1 6( )
UnitsOf C( )H n( ) 0if
K21mod n( )submatrix Kc 1 6 0 0( )
UnitsOf K( )H n( ) 1if
submatrix Ka 1 6 0 0( )
UnitsOf K( )H n( ) 0if
C21modn( )submatrix Cc 1 6 0 0( )
UnitsOf C( )H n( ) 1if
submatrix Ca 1 6 0 0( )
UnitsOf C( )H n( ) 0if
K12mod n( )submatrix Kc 0 0 1 6( )
UnitsOf K( )H n( ) 1if
submatrix Ka 0 0 1 6( )
UnitsOf K( )H n( ) 0if
C12modn( )submatrix Cc 0 0 1 6( )
UnitsOf C( )H n( ) 1if
submatrix Ca 0 0 1 6( )
UnitsOf C( )H n( ) 0if
Appendix C 157
2mod n( ) I22mod n( )1
K22mod n( ) mod n( ) sort eigenvals 2mod n( ) nmod n( )mod n( )
2
EVmodn( ) augmenteigenvec 2mod n( ) mod n( )0
max eigenvec 2mod n( ) mod n( )0
eigenvec 2mod n( ) mod n( )1
max eigenvec 2mod n( ) mod n( )1
eigenvec 2mod n( ) mod n( )2
max eigenvec 2mod n( ) mod n( )2
eigenvec 2mod n( ) mod n( )3
max eigenvec 2mod n( ) mod n( )3
eigenvec 2mod n( ) mod n( )4
max eigenvec 2mod n( ) mod n( )4
eigenvec 2mod n( ) mod n( )5
max eigenvec 2mod n( ) mod n( )5
modeshapesmod n( ) stack nmod n( )T
EVmodn( )
t 0 0001 1
mode1a t( ) EVmod100( )0
sin nmod 100( )0 t mode2a t( ) EVmod100( )1
sin nmod 100( )1 t
mode1c t( ) EVmod800( )0
sin nmod 800( )0 t mode2c t( ) EVmod800( )1
sin nmod 800( )1 t
Pulley responses amp torque requirement for crankshaft amp alternator pulleys pulley1 and 4 respectively
The system equation becomes
I14q14 -double-dot + C1144 q14 -dot + K1144 q14 + C12qm-dot + K12qm = Qc
I2qm-double-dot + C22qm-dot + K22qm + C21q1-dot + K21q1 = 0
Pulley responses
Qm = - [(K22 - 2I2) + jC22 ]-1(K21 + jC21 )Q1
Torque requirement for crank shaft Pulley 1
qc = [(K11 -2I1) + jC11 ]Q1 + (K12 + jC12 )Qm
Torque requirement for alternator shaft Pulley 4
qa = [(K44 -2I4) + jC44 ]Q4 + (K12 + jC12 )Qm
Appendix C 158
Let DRIVING pulley have a unit amplitude 1 = 1 and let the system frequency be calculated based on
engine speed n
n 60 90 6000 n( )4n
60 a n( )
2n Ra0
60 Ra3
mod n( ) n( ) H n( ) 1if
a n( ) H n( ) 0if
Ymod n( ) K22mod n( ) mod n( ) 2 I22mod n( )
j mod n( ) C22modn( )
mmod n( ) Ymod n( )( )1
K21mod n( ) j mod n( ) C21modn( )
Crankshaft amp ISG required torques
Let input from DRIVING pulley be an angular displacement with constant amplitude of angular acceleration
Ac n( ) 650 1 n( )Ac n( )
n( ) 2
Let Qm = QmQ1(n) for n lt 750
and Qm = QmQ4(n) for n gt 750
Aa n( )42
I3 3
1a n( )Aa n( )
a n( ) 2
Qc0 4
qcmod n( ) K11mod n( ) mod n( ) 2
I11mod n( )
j mod n( ) C11modn( )
1 n( ) K12mod n( ) j mod n( ) C12modn( ) mmod n( ) 1 n( )
H n( ) 1if
Qc0 H n( ) 0if
qamod n( ) K11mod n( ) mod n( ) 2
I11mod n( )
j mod n( ) C11modn( )
1a n( ) K12mod n( ) jmod n( ) C12modn( ) mmod n( ) 1a n( ) Qc0
H n( ) 0if
0 H n( ) 1if
Q n( ) 48 n
Ra0
Ra3
48
18000
(ISG torque requirement alternate equation)
Appendix C 159
Dynamic tensioner arm torques
Qtt1mod n( )Kt Kt1
UnitsOf Kt( )j mod n( )
Ct Ct1
UnitsOf Cr( )
mmod n( )4 1 n( )
H n( ) 1if
Kt Kt1
UnitsOf Kt( )j mod n( )
Ct Ct1
UnitsOf Cr( )
mmod n( )4 1a n( )
H n( ) 0if
Qtt2mod n( )Kt Kt2
UnitsOf Kt( )j mod n( )
Ct Ct2
UnitsOf Cr( )
mmod n( )5 1 n( )
H n( ) 1if
Kt Kt2
UnitsOf Kt( )j mod n( )
Ct Ct2
UnitsOf Cr( )
mmod n( )5 1a n( )
H n( ) 0if
Appendix C 160
Dynamic belt span tensions
d n( ) 1 n( ) H n( ) 1if
1a n( ) H n( ) 0if
mod n( )
d n( )
mmod n( ) d n( ) 0 0
mmod n( ) d n( ) 1 0
mmod n( ) d n( ) 2 0
mmod n( ) d n( ) 3 0
mmod n( ) d n( ) 4 0
mmod n( ) d n( ) 5 0
Tm n( ) j n( )Tcc
UnitsOf Tcc( )
Tkc
UnitsOf Tkc( )
mod n( )
H n( ) 1if
j n( )Tca
UnitsOf Tca( )
Tka
UnitsOf Tka( )
mod n( )
H n( ) 0if
Tm n( ) j n( )Tcc
UnitsOf Tcc( )
Tkc
UnitsOf Tkc( )
mod n( )
H n( ) 1if
j n( )Tca
UnitsOf Tca( )
Tka
UnitsOf Tka( )
mod n( )
H n( ) 0if
(tensions for driving pulley belt spans)
Appendix C 161
C3 Static Analysis
Static Analysis using K Tk amp Q matricies amp Ts
For static case K = Q
Tension T = T0 + Tks
Thus T = K-1QTks + T0
Q1 68N m Qt1 0N m Qt2 0N m Ts 300N
Qc
Q4
Q2
Q3
Q5
Qt1
Qt2
Qc
5
2
0
0
0
0
J Qa
Q1
Q2
Q3
Q5
Qt1
Qt2
Qa
68
2
0
0
0
0
N m
cK22mod 900( )( )
1
N mQc A
K22mod 600( )1
N mQa
a
A0
A1
A2
0
A3
A4
A5
0
c1
c2
c0
c3
c4
c5
Tc Tk Ts Ta Tk a Ts
162
APPENDIX D
MATLAB Functions amp Scripts
D1 Parametric Analysis
D11 TwinMainm
The following function script performs the parametric analysis for the B-ISG system with a Twin
Tensioner It calls the function TwinTenStaticTensionm The parametric analysis perturbs a
single input parameter for the called function TwinTenStaticTensionm The main function takes
an initial input value for the Twin Tensioner‟s stiffness parameters Kto Kt1o Kt2o and
geometric parameters D3o D5o X3o Y3o X5o and Y5o An input parameter is allowed to
increment by six percent over a range from sixty percent below its initial value to sixty percent
above its initial value The coordinate parameters are incremented through a mesh of Cartesian
points with prescribed boundaries The TwinMainm function plots the parametric results
______________________________________________________________________________
clc
clear all
Static tension for single tensioner system for CS and Alt driving
Initial Conditions
Kto = 20626
Kt1o = 10314
Kt2o = 16502
D3o = 007240
D5o = 007240
X3o =0292761
Y3o =087
X5o =12057
Y5o =09193
Pertubations of initial parameters
Kt = (Kto-060Kto)006Kto(Kto+060Kto)
Kt1 = (Kt1o-060Kt1o)006Kt1o(Kt1o+060Kt1o)
Kt2 = (Kt2o-060Kt2o)006Kt2o(Kt2o+060Kt2o)
D3 = (D3o-060D3o)006D3o(D3o+060D3o)
D5 = (D5o-060D5o)006D5o(D5o+060D5o)
No of data points
s = 21
T = zeros(5s)
Ta = zeros(5s)
Parametric Plots
for i = 1s
Appendix D 163
[T(i)Ta(i)] = TwinTenStaticTension(Kt(i)Kt1oKt2oD3oD5oX3oY3oX5oY5o)
end
figure
[AXH1H2] = plotyy(Kt()T(1)Kt()Ta(4)plot) hold on
H3 = line(Kt()T(5)ParentAX(1)) hold on
H4 = line(Kt()Ta(3)Parent AX(2)) hold off
set(AXYLimModeautoYTickModeauto)
title(Static Tension vs Coupled Stiffness bw Arms 1 amp 2)
xlabel(Kt (Nmrad))
set(get(AX(1)Ylabel)StringCS Span Tension (N))
set(get(AX(2)Ylabel)StringISG Span Tension (N))
set(H1LineStyle-Colorb)
set(H2LineStyle-Colorg)
set(H3LineStyleColorget(H1Color))
set(H4LineStyleColorget(H2Color))
legend([H1H2H3H4]CS Tautest SpanISG Tautest SpanCS Slackest SpanISG
Slackest SpanLocationBest)
for i = 1s
[T(i)Ta(i)] = TwinTenStaticTension(KtoKt1(i)Kt2oD3oD5oX3oY3oX5oY5o)
end
figure
[AXH1H2] = plotyy(Kt1()T(1)Kt1()Ta(4)plot) hold on
H3 = line(Kt1()T(5)ParentAX(1)) hold on
H4 = line(Kt1()Ta(3)Parent AX(2)) hold off
set(AXYLimModeautoYTickModeauto)
title(Static Tension vs Tensioner Arm 1 Stiffness)
xlabel(Kt1 (Nmrad))
set(get(AX(1)Ylabel)StringCS Span Tension (N))
set(get(AX(2)Ylabel)StringISG Span Tension (N))
set(H1LineStyle-Colorb)
set(H2LineStyle-Colorg)
set(H3LineStyleColorget(H1Color))
set(H4LineStyleColorget(H2Color))
legend([H1H2H3H4]CS Tautest SpanISG Tautest SpanCS Slackest SpanISG
Slackest SpanLocationBest)
for i = 1s
[T(i)Ta(i)] = TwinTenStaticTension(KtoKt1oKt2(i)D3oD5oX3oY3oX5oY5o)
end
figure
[AXH1H2] = plotyy(Kt2()T(1)Kt2()Ta(4)plot) hold on
H3 = line(Kt2()T(5)ParentAX(1)) hold on
H4 = line(Kt2()Ta(3)Parent AX(2)) hold off
set(AXYLimModeautoYTickModeauto)
title(Static Tension vs Tensioner Arm 2 Stiffness)
xlabel(Kt2 (Nmrad))
set(get(AX(1)Ylabel)StringCS Span Tension (N))
Appendix D 164
set(get(AX(2)Ylabel)StringISG Span Tension (N))
set(H1LineStyle-Colorb)
set(H2LineStyle-Colorg)
set(H3LineStyleColorget(H1Color))
set(H4LineStyleColorget(H2Color))
legend([H1H2H3H4]CS Tautest SpanISG Tautest SpanCS Slackest SpanISG
Slackest SpanLocationBest)
for i = 1s
[T(i)Ta(i)] = TwinTenStaticTension(KtoKt1oKt2oD3(i)D5oX3oY3oX5oY5o)
end
figure
[AXH1H2] = plotyy(D3()T(1)D3()Ta(4)plot) hold on
H3 = line(D3()T(5)ParentAX(1)) hold on
H4 = line(D3()Ta(3)Parent AX(2)) hold off
set(AXYLimModeautoYTickModeauto)
title(Static Tension vs Tensioner Pulley 1 Diameter)
xlabel(D3 (m))
set(get(AX(1)Ylabel)StringCS Span Tension (N))
set(get(AX(2)Ylabel)StringISG Span Tension (N))
set(H1LineStyle-Colorb)
set(H2LineStyle-Colorg)
set(H3LineStyleColorget(H1Color))
set(H4LineStyleColorget(H2Color))
legend([H1H2H3H4]CS Tautest SpanISG Tautest SpanCS Slackest SpanISG
Slackest SpanLocationBest)
for i = 1s
[T(i)Ta(i)] = TwinTenStaticTension(KtoKt1oKt2oD3oD5(i)X3oY3oX5oY5o)
end
figure
[AXH1H2] = plotyy(D5()T(1)D5()Ta(4)plot) hold on
H3 = line(D5()T(5)ParentAX(1)) hold on
H4 = line(D5()Ta(3)Parent AX(2)) hold off
set(AXYLimModeautoYTickModeauto)
title(Static Tension vs Tensioner Pulley 2 Diameter)
xlabel(D5 (m))
set(get(AX(1)Ylabel)StringCS Span Tension (N))
set(get(AX(2)Ylabel)StringISG Span Tension (N))
set(H1LineStyle-Colorb)
set(H2LineStyle-Colorg)
set(H3LineStyleColorget(H1Color))
set(H4LineStyleColorget(H2Color))
legend([H1H2H3H4]CS Tautest SpanISG Tautest SpanCS Slackest SpanISG
Slackest SpanLocationBest)
Mesh points
m = 101
n = 101
Appendix D 165
T = zeros(5nm)
Ta = zeros(5nm)
[ixxiyy] = meshgrid(1m1n)
minX3 = 0260200
maxX3 = 0317677
minY3 = -0056640
maxY3 = 0228456
midX3 = 0311641
X3 = minX3 + (ixx-1)(maxX3-minX3)(m-1)
Y3 = minY3 + (iyy-1)(maxY3-minY3)(n-1)
for i = 1n
for j = 1m
if ((X3(ij)lt midX3)ampamp(Y3(ij)gt=(sqrt((0087945^2)-((X3(ij)-0224)^2)-
006395)))ampamp(Y3(ij)lt=(-1sqrt(((00703^2)-((X3(ij)-
024759)^2)))+016664)))||((X3(ij)gt=midX3)ampamp(Y3(ij)gt=(35548X3(ij)-
11134868))ampamp(Y3(ij)lt=(-1(sqrt(((00703^2)-((X3(ij)-024759)^2))))+016664))) mx+b
lt= y lt= circle4
[T(ij)Ta(ij)] =
TwinTenStaticTension(KtoKt1oKt2oD3oD5oX3(ij)Y3(ij)X5oY5o)
else
T(ij) = zeros(511)
Ta(ij) = zeros(511)
end
end
end
figure
Z1 = squeeze(T(1))
surf(X3Y3real(Z1))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(CS DRIVING Tautest Span Tension vs Tensioner Pulley 1 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(CS Span Tension (N))
figure
Z5 = squeeze(T(5))
surf(X3Y3real(Z5))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(CS DRIVING Slackest Span Tension vs Tensioner Pulley 1 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
Appendix D 166
zlabel(CS Span Tension (N))
figure
Za4 = squeeze(Ta(4))
surf(X3Y3real(Za4))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(ISG DRIVING Tautest Span Tension vs Tensioner Pulley 1 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(ISG Span Tension (N))
figure
Za3 = squeeze(Ta(3))
surf(X3Y3real(Za3))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(ISG DRIVING Slackest Span Tension vs Tensioner Pulley 1 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(ISG Span Tension (N))
minX5 = -0037093
maxX5 = 0212509
minY5 = 00362
maxY5 = 0228456
midX5a = 0131965
midX5b = 017729
X5 = minX5 + (ixx-1)(maxX5-minX5)(m-1)
Y5 = minY5 + (iyy-1)(maxY5-minY5)(n-1)
for i = 1n
for j = 1m
if
(X5(ij)ltmidX5a)ampamp(Y5(ij)lt=(0386X5(ij)+0146468))ampamp(Y5(ij)gt=(sqrt((0136525^2)-
(X5(ij)^2))))
[T(ij)Ta(ij)] =
TwinTenStaticTension(KtoKt1oKt2oD3oD5oX3oY3oX5(ij)Y5(ij))
elseif
((X5(ij)gt=midX5a)ampamp(X5(ij)ltmidX5b))ampamp(Y5(ij)gt=00362)ampamp(Y5(ij)lt=(0386X5(ij)+0
146468))
[T(ij)Ta(ij)] =
TwinTenStaticTension(KtoKt1oKt2oD3oD5oX3oY3oX5(ij)Y5(ij))
elseif (X5(ij)gt=midX5b)ampamp(Y5(ij)gt=(sqrt((00703^2)-(((X5(ij)-
024759)^2)))+016664))ampamp(Y5(ij)lt=(0386X5(ij)+0146468))
Appendix D 167
[T(ij)Ta(ij)] =
TwinTenStaticTension(KtoKt1oKt2oD3oD5oX3oY3oX5(ij)Y5(ij))
else
T(ij) = zeros(511)
Ta(ij) = zeros(511)
end
end
end
figure
Z1 = squeeze(T(1))
surf(X5Y5real(Z1))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(CS DRIVING Tautest Span Tension vs Tensioner Pulley 2 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(CS Span Tension (N))
figure
Z5 = squeeze(T(5))
surf(X5Y5real(Z5))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(CS DRIVING Slackest Span Tension vs Tensioner Pulley 2 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(CS Span Tension (N))
figure
Za4 = squeeze(Ta(4))
surf(X5Y5real(Za4))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(ISG DRIVING Tautest Span Tension vs Tensioner Pulley 2 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(ISG Span Tension (N))
figure
Za3 = squeeze(Ta(3))
surf(X5Y5real(Za3))
ZLim([50 500])
axis tight
Appendix D 168
colormap jet
colorbar
title(ISG DRIVING Slackest Span Tension vs Tensioner Pulley 2 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(ISG Span Tension (N))
D12 TwinTenStaticTensionm
The function TwinTenStaticTensionm simulates the static model of the B-ISG system with a
Twin Tensioner This function returns 3 vectors the static tension of each belt span in the
crankshaft- and ISG-driving phases of operation and the angle of displacement of each rigid
body in the ISG- driving phase It takes the input parameters kt kt1 kt2 for the tensioner arm
stiffness D3 and D5 for the tensioner pulley diameters and X3Y3 X5 and Y5 for the tensioner
arm pulley coordinates The function is called in the parametric analysis solution script
TwinMainm and in the optimization solution script OptimizationTwinm
D2 Optimization
D21 OptimizationTwinm
The following script is for the main function OptimizationTwinm It performs an optimization
search for the B-ISG system with a Twin Tensioner It takes an input for a parameter vector
containing values for the design variables The program calls the objective function
objfunTwinm and the constraint function confunTwinm The program can perform a genetic
algorithm (GA) optimization search or a hybrid GA optimization that includes a localized search
The optimal solution vector corresponding to the design variables and the optimal objective
function value is returned The program inputs the optimized values for the design variables into
TwinTenStaticTensionm This called function returns the optimized static state of tensions for
the crankshaft- and ISG- driving phases and for the angle of displacement of the rigid bodies in
the ISG driving phase
______________________________________________________________________________
clc
clear all
Initial values for variables
Kto = 20626
Kt1o = 10314
Kt2o = 16502
X3o = 0292761
Y3o = 0087
X5o = 012057
Appendix D 169
Y5o = 009193
w0 =[Kto Kt1o Kt2o X3o Y3o X5o Y5o] Start Point (row vector)
Variable ranges
minKt = Kto - 1Kto
maxKt = Kto + 1Kto
minKt1 = Kt1o - 1Kt1o
maxKt1 = Kt1o + 1Kt1o
minKt2 = Kt2o - 1Kt2o
maxKt2 = Kt2o + 1Kt2o
minX3 = 0260200
maxX3 = 0317677
minY3 = -0056640
maxY3 = 0228456
minX5 = -0037093
maxX5 = 0212509
minY5 = 00362
maxY5 = 0228456
ObjectiveFunction = objfunTwin
nvars = 7 Number of variables
ConstraintFunction = confunTwin
Uncomment next two lines (and comment the two functions after them) to use GA algorithm
for optimization
options = gaoptimset(InitialPopulationw0PopInitRange[minKt minKt1 minKt2 minX3
minY3 minX5 minY5 maxKt maxKt1 maxKt2 maxX3 maxY3 maxX5
maxY5]PopulationSize100Displayfinal)
[wfvalexitflagoutput] =
ga(ObjectiveFunctionnvars[][][][][][]ConstraintFunctionoptions)
fminconOptions = optimset(DisplayiterLargeScaleoff) Largescale off since gradient not
provided
options = gaoptimset(InitialPopulationw0PopInitRange[minKt minKt1 minKt2 minX3
minY3 minX5 minY5 maxKt maxKt1 maxKt2 maxX3 maxY3 maxX5
maxY5]PopulationSize100HybridFcnfmincon fminconOptions)
[wfvalexitflagoutput] = ga(ObjectiveFunctionnvars[][][][][][]ConstraintFunctionoptions)
[TTaThetaDegA] = TwinTenStaticTension(w(1)w(2)w(3)w(4)w(5)w(6)w(7))
D22 confunTwinm
The constraint function confunTwinm is used by the main optimization program to ensure
input values are constrained within the prescribed regions The function makes use of inequality
constraints for seven constrained variables corresponding to the design variables It takes an
input vector corresponding to the design variables and returns a data set of the vector values that
satisfy the prescribed constraints
Appendix D 170
D23 objfunTwinm
This function is the objective function for the main optimization program It outputs a value for
a weighted objective function or a non-weighted objective function relating the optimization of
the static tension The program takes an input vector containing a set of values for the design
variables that are within prescribed constraints The description of the function is similar to
TwinTenStaticTensionm but differs in the fact that it only returns a scalar value which is the
value of the objective function
171
VITA
ADEBUKOLA OLATUNDE
Email adebukolaolatundegmailcom
Adebukola Olatunde is a graduate research student at the University of Toronto in Toronto
Ontario Canada She obtained a Bachelor‟s Degree in Mechanical Engineering from McMaster
University in Hamilton Ontario Canada in 2002 Upon graduation she pursued a graduate
degree in mechanical engineering at the University of Toronto with a specialization in
mechanical systems dynamics and vibrations and environmental engineering In September
2008 she completed the requirements for the Master of Applied Science degree in Mechanical
Engineering She has held the position of teaching assistant for undergraduate courses in
dynamics and vibrations Adebukola has completed course work in professional education She
is a registered member of professional engineering organizations including the Professional
Engineer‟s of Ontario Engineer-in-Training program the Canadian Society of Mechanical
Engineers and the National Society of Black Engineers She intends to practice as a professional
engineering consultant in mechanical design
iv
ACKNOLOWEDGEMENTS
I would like to express deep gratitude to Dr Jean Zu for her guidance throughout the duration of
my studies and for providing me with the opportunity to conduct this thesis
I wish to thank the individuals of Litens Automotive who have provided guidance and data for
the research work Special thanks to Mike Clark Seeva Karuendiran and Dr Qiu for their time
and help
I thank my committee members Dr Naguib and Dr Sun for contributing their time to my
research work
My sincerest thanks to my research colleague David for his knowledge and support Many
thanks to my lab mates Qiming Hansong Ali Ming Andrew and Peyman for their guidance
I want to especially thank Dr Cleghorn Leslie Sinclair and Dr Zu for the opportunities to
teach These experiences have served to enrich my graduate studies As well thank you to Dr
Cleghorn for guidance in my research work
I am also in debt to my classmates and teaching colleagues throughout my time at the University
of Toronto especially Aaron and Mohammed for their support in my development as a graduate
researcher and teacher
v
CONTENTS
ABSTRACT ii
DEDICATION iii
ACKNOWLEDGEMENTS iv
CONTENTS v
LIST OF TABLES ix
LIST OF FIGURES xi
LIST OF SYMBOLS xvi
Chapter 1 INTRODUCTION 1
11 Background 1
12 Motivation 3
13 Thesis Objectives and Scope of Research 4
14 Organization and Content of Thesis 5
Chapter 2 LITERATURE REVIEW 7
21 Introduction 7
22 B-ISG System 8
221 ISG in Hybrids 8
2211 Full Hybrids 9
2212 Power Hybrids 10
2213 Mild Hybrids 11
2214 Micro Hybrids 11
222 B-ISG Structure Location and Function 13
2221 Structure and Location 13
2222 Functionalities 14
23 Belt Drive Modeling 15
24 Tensioners for B-ISG System 18
241 Tensioners Structures Function and Location 18
242 Systematic Review of Tensioner Designs for a B-ISG System 20
25 Summary 24
vi
Chapter 3 MODELING OF B-ISG SYSTEM 25
31 Overview 25
32 B-ISG Tensioner Design 25
33 Geometric Model of a B-ISG System with a Twin Tensioner 27
34 Equations of Motion for a B-ISG System with a Twin Tensioner 32
341 Dynamic Model of the B-ISG System 32
3411 Derivation of Equations of Motion 32
3412 Modeling of Phase Change 41
3413 Natural Frequencies Mode Shapes and Dynamic Responses 42
3414 Crankshaft Pulley Driving Torque Acceleration and Displacement 44
3415 ISG Pulley Driving Torque Acceleration and Displacement 46
3416 Tensioner Arms Dynamic Torques 48
3417 Dynamic Belt Span Tensions 49
342 Static Model of the B-ISG System 49
35 Simulations 50
351 Geometric Analysis 51
352 Dynamic Analysis 52
3521 Natural Frequency and Mode Shape 54
3522 Dynamic Response 58
3523 ISG Pulley and Crankshaft Pulley Torque Requirement 61
3524 Tensioner Arm Torque Requirement 62
3525 Dynamic Belt Span Tension 63
353 Static Analysis 66
36 Summary 69
Chapter 4 PARAMETRIC ANALYSIS OF A B-ISG TWIN TENSIONER 71
41 Introduction 71
42 Methodology 71
43 Results and Discussion 74
431 Influence of Tensioner Arm Stiffness on Static Tension 74
432 Influence of Tensioner Pulley Diameter on Static Tension 78
433 Influence of Tensioner Pulley 1 Coordinates on Static Tension 80
434 Influence of Tensioner Pulley 2 Coordinates on Static Tension 86
vii
44 Conclusion 92
Chapter 5 OPTIMIZATION OF A B-ISG TWIN TENSIONER 95
51 Optimization Problem 95
511 Selection of Design Variables 95
512 Objective Function amp Constraints 97
52 Optimization Method 100
521 Genetic Algorithm 100
522 Hybrid Optimization Algorithm 101
53 Results and Discussion 101
531 Parameter Settings amp Stopping Criteria for Simulations 101
532 Optimization Simulations 102
533 Discussion 106
54 Conclusion 109
Chapter 6 CONCLUSION AND RECOMMENDATIONS111
61 Summary 111
62 Conclusion 112
63 Recommendations for Future Work 113
REFERENCES 116
APPENDICIES 123
A Passive Dual Tensioner Designs from Patent Literature 123
B B-ISG Serpentine Belt Drive with Single Tensioner Equation of Motion 138
C MathCAD Scripts 145
C1 Geometric Analysis 145
C2 Dynamic Analysis 152
C3 Static Analysis 161
D MATLAB Functions amp Scripts 162
D1 Parametric Analysis 162
D11 TwinMainm 162
D12 TwinTenStaticTensionm 168
D2 Optimization 168
D21 OptimizationTwinm - Optimization Function 168
viii
D22 confunTwinm 169
D23 objfunTwinm 170
VITA 171
ix
LIST OF TABLES
21 Passive Dual Tensioner Designs from Patent Literature
31 Selected Contact Point Types on the ith Pulley and Types for the ith Belt Span
32 Coordinate Points for Pulley Centres and Twin Tensioner Pivot
33 Geometric Results of B-ISG System with Twin Tensioner
34 Data for Input Parameters used in Dynamic and Static Computations
35 Static Solution for Belt Span Tensions in Crankshaft and ISG Driving Cases for a B-ISG
Serpentine Belt Drive with a Single Tensioner
36 Static Solution for Belt Span Tensions in Crankshaft and ISG Driving Cases for a B-ISG
Serpentine Belt Drive with a Twin Tensioner
41 Initial Values Increments and Ranges for Parameters of Twin Tensioner
51 Summary of Parametric Analysis Data for Twin Tensioner Properties
52a GA Optimization Results for Twin Tensioner Parameters and Objective Function
52b Computations for Tensions and Angles from GA Optimization Results
53a Hybrid Optimization Results for Twin Tensioner Parameters and Objective Function
53b Computations for Tensions and Angles from Hybrid Optimization Results
54a Non-Weighted Optimization Results for Twin Tensioner Parameters and Objective
Function
54b Computations for Tensions and Angles from Non-Weighted Optimizations
x
55 Weighted Optimization Results for Static Tensions for Optimal B-ISG System with a
Twin Tensioner
56 Non-Weighted Optimization Results for Static Tensions for Optimal B-ISG System with a
Twin Tensioner
xi
LIST OF FIGURES
21 Hybrid Functions
31 Schematic of the Twin Tensioner
32 B-ISG Serpentine Belt Drive with Twin Tensioner
33 Angles Coordinates and Possible Belt Contact Points for the ith and i+1th Pulleys
34 Twin Tensioner Free Body Diagram of a Four-Degree of Freedom System
35 Free Body Diagram for Non-Tensioner Pulleys
36a ISG Driving Case Natural Frequency of System and Mode Shapes for Responsive Rigid
Bodies
36b ISG Driving Case First Mode Responses
36c ISG Driving Case Second Mode Responses
37a Crankshaft Driving Case Natural Frequency of System and Mode Shapes for Responsive
Rigid Bodies
37b Crankshaft Driving Case First Mode Responses
37c Crankshaft Driving Case Second Mode Responses
38 Crankshaft Pulley Dynamic Response (for crankshaft driven case)
39 ISG Pulley Dynamic Response (for ISG driven case)
310 Air Conditioner Pulley Dynamic Response
311 Tensioner Pulley 1 Dynamic Response
xii
312 Tensioner Pulley 2 Dynamic Response
313 Tensioner Arm 1 Dynamic Response
314 Tensioner Arm 2 Dynamic Response
315 Required Driving Torque for the ISG Pulley
316 Required Driving Torque for the Crankshaft Pulley
317 Dynamic Torque for Tensioner Arm 1
318 Dynamic Torque for Tensioner Arm 2
319 Span 1 (between crankshaft and air conditioner) Dynamic Belt Span Tension
320 Span 2 (between air conditioner and tensioner 1) Dynamic Belt Span Tension
321 Span 3 (between tensioner 1 and ISG) Dynamic Belt Span Tension
322 Span 4 (between ISG and tensioner 2) Dynamic Belt Span Tension
323 Span 5 (between tensioner 2 and crankshaft) Dynamic Belt Span Tension
324 B-ISG Serpentine Belt Drive with Single Tensioner
41a Regions 1 and 2 and their Associated Guidelines for Coordinates of Tensioner Pulleys 1
amp 2
41b Regions 1 and 2 in Cartesian Space
42 Parametric Analysis for Coupled Stiffness Arm Constant kt (N∙mrad)
43 Parametric Analysis for Stiffness of Arm 1 kt1 (N∙mrad)
44 Parametric Analysis for Stiffness of Arm 2 kt2 (N∙mrad)
xiii
45 Parametric Analysis for Pulley 1 Diameter D3 (m)
46 Parametric Analysis for Pulley 2 Diameter D5 (m)
47 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Tautest Span
Tension in Crankshaft Driving Case
48 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Slackest Span
Tension in Crankshaft Driving Case
49 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Tautest Span
Tension in ISG Driving Case
410 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Slackest Span
Tension in ISG Driving Case
411 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Tautest Span
Tension in Crankshaft Driving Case
412 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Slackest Span
Tension in Crankshaft Driving Case
413 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Tautest Span
Tension in ISG Driving Case
414 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Slackest Span
Tension in ISG Driving Case
51 Static Stability of the B-ISG Twin Tensioner Based on the Angular Displacement of
Tensioner Arms 1 and 2
A1 Proposed design by Bayerische Motoren Werke AG corresponding to patent nos
EP1420192-A2 and DE10253450-A1
A2a First of four proposed designs (and its various configurations) by Bosch GMBH
corresponding to patent no WO0026532-A1
A2b Second of four proposed designs by Bosch GMBH corresponding to patent no
WO0026532-A1
A2c Third of four proposed designs (and its various configurations) by Bosch GMBH
corresponding to patent no WO0026532-A1
A2d Fourth of four proposed designs (and its various configurations) by Bosch GMBH
corresponding to patent no WO0026532-A1
A3 Proposed design by Daimler Chrysler AG corresponding to patent no DE10324268-A1
A4 Proposed design by Dayco Products LLC corresponding to patent no US6942589-B2
xiv
A5 Proposed design by Gates Corp corresponding to patent no WO2003038309-A
A6 Proposed design by General Motors Corp corresponding to patent no US20060287146-
A1
A7 Proposed design by INA Schaeffler KG corresponding to patent no DE10044645-A1
A8a First of three proposed designs by INA Schaeffler KG corresponding to patent no
DE10159073-A1
A8b Second of three proposed designs by INA Schaeffler KG corresponding to patent no
DE10159073-A1
A8c Third of three proposed designs by INA Schaeffler KG corresponding to patent no
DE10159073-A1
A9 Proposed design by INA Schaeffler KG corresponding to patent no DE10359641-A1
A10 Proposed design by INA Schaeffler KG corresponding to patent no EP1723350-A1
A11 Proposed design by INA Schaeffler KG corresponding to patent no EP1738093-A1
A12 Proposed design by INA Schaeffler KG corresponding to patent no DE102004012395-
A1
A13 Proposed design by INA Schaeffler KG corresponding to patent nos DE102005017038-
A1and WO2006108461-A1
A14 Proposed design by Litens Automotive GMBH et al corresponding to patent no
US20010007839-A1
A15 Proposed design by Mitsubishi Jidosha Eng KK and Mitsubishi Motor Corp
corresponding to patent no JP2005083514-A
A16 Proposed design by Nissan Motor Co Ltd corresponding to patent no JP3565040-B2
A17 Proposed design by NTN Corp corresponding to patent no JP2006189073-A
A18 Proposed design by Valeo Equipment Electriques Moteur corresponding to patent nos
EP1658432 and WO2005015007
B1 Single Tensioner B-ISG System
B2 Free-body Diagram of ith Pulley
xv
B3 Free-body Diagram of Single Tensioner
C1 Schematic of B-ISG System with Twin Tensioner
C2 Possible Contact Points
xvi
LIST OF SYMBOLS
Latin Letters
A Belt cord cross-sectional area
C Damping matrix of the system
cb Belt damping
119888119894119887 Belt damping constant of the ith belt span
119914119946119946 Damping matrix element in the ith row and ith column
ct Damping acting between tensioner arms 1 and 2
cti Damping of the ith tensioner arm
DCS Diameter of crankshaft pulley
DISG Diameter of ISG pulley
ft Belt transition frequency
H(n) Phase change function
I Inertial matrix of the system
119920119938 Inertial matrix under ISG driving phase
119920119940 Inertial matrix under crankshaft driving phase
Ii Inertia of the ith pulley
Iti Inertia of the ith tensioner arm
119920120784120784 Submatrix of inertial matrix I
j Imaginary coordinate (ie (-1)12
)
K Stiffness matrix of the system
xvii
119896119887 Belt factor
119870119887 Belt cord stiffness
119896119894119887 Belt stiffness constant of the ith belt span
kt Spring stiffness acting between tensioner arms 1 and 2
kti Coil spring of the ith tensioner arm
119922120784120784 Submatrix of stiffness matrix K
Lfi Lbi Lengths of possible belt span connections from the ith pulley
Lti Length of the ith tensioner arm
Modeia Mode shape of the ith rigid body in the ISG driving phase
Modeic Mode shape of the ith rigid body in the crankshaft driving phase
n Engine speed
N Motor speed
nCS rpm of crankshaft pulley
NF Motor speed without load
nISG rpm of ISG pulley
Q Required torque matrix
qc Amplitude of the required crankshaft torque
QcsISG Required torque of the driving pulley (crankshaft or ISG)
Qm Required torque matrix of driven rigid bodies
Qti Dynamic torque of the ith tensioner arm
Ri Radius of the ith pulley
T Matrix of belt span static tensions
xviii
Trsquo Dynamic belt tension matrix
119931119940 Damping matrix due to the belt
119931119948 Stiffness matrix due to the belt
Ti Tension of the ith belt span
To Initial belt tension for the system
Ts Stall torque
Tti Tension for the neighbouring belt spans of the ith tensioner pulley
(XiYi) Coordinates of the ith pulley centre
XYfi XYbi XYfbi
XYbfi Possible connection points on the ith pulley leading to the ith belt span
XYf2i XYb2i
XYfb2i XYbf2i Possible connection points on the ith pulley leading to the (i-1)th belt span
Greek Letters
αi Angle between the datum and the line connecting the ith and (i+1)th pulley
centres
βji Angle of orientation for the ith belt span
120597θti(t) 120579 ti(t)
120579 ti(t)
Angular displacement velocity and acceleration (rotational coordinate) of the
ith tensioner arm
120637119938 General coordinate matrix under ISG driving phase
120637119940 General coordinate matrix under crankshaft driving phase
θfi θbi Angles between the datum and the belt connection spans with lengths Lfi and
Lbi respectively
Θi Amplitude of displacement of the ith pulley
xix
θi(t) 120579 i(t) 120579 i(t) Angular position velocity and acceleration (rotational coordinate) of the ith
pulley
θti Angle of the ith tensioner arm
θtoi Initial pivot angle of the ith tensioner arm
θm Angular displacement matrix of driven rigid bodies
Θm Amplitude of displacement of driven rigid bodies
ρ Belt cord density
120601119894 Belt wrap angle on the ith pulley
φmax Belt maximum phase angle
φ0deg Belt phase angle at zero frequency
ω Frequency of the system
ωcs Angular frequency of crankshaft pulley
ωISG Angular frequency of the ISG pulley
120654119951 Natural frequency of system
1
CHAPTER 1 INTRODUCTION
11 Background
Belt drive systems are the means of power transmission in conventional automobiles The
emergence of hybrid technologies specifically the Belt-driven Integrated Starter-generator (B-
ISG) has placed higher demands on belt drives than ever before The presence of an integrated
starter-generator (ISG) in a belt transmission places excessive strain on the belt leading to
premature belt failure This phenomenon has motivated automotive makers to design a tensioner
that is suitable for the B-ISG system
The belt drive is also known interchangeably as the front-end accessory drive-belt (FEAD) the
belt accessory-drive system (BAS) or the belt transmission system In a traditional setting the
role of this system is to transmit torque generated by an internal combustion engine (ICE) in
order to reliably drive multiple peripheral devices mounted on the engine block The high speed
torque is transmitted through a crankshaft pulley to a serpentine belt The serpentine belt is a
single continuous member that winds around the driving and driven accessory pulleys of the
drive system Serpentine belts used in automotive applications consist of several layers The
load-bearing layer is a flexible member consisting of high stiffness fibers [1] It is covered by a
protective layer to guard against mechanical damage and is bound below by a visco-elastic layer
that provides the required shock absorption and grip against the rigid pulleys [1] The accessory
devices may include an alternator power steering pump water pump and air conditioner
compressor among others
Introduction 2
The B-ISG system is a transmission system characteristic to micro-hybrid automobiles It is akin
to traditional belt drives differing in the fact that an electric motor called an integrated starter-
generator (ISG) replaces the original alternator re-starts the engine from idle speed and provides
braking regeneration [2] The re-start function of the micro-hybrid transmission is known as
stop-start In the B-ISG setting the ISG is mounted on the belt drive The ISG produces a speed
of approximately 2000 to 2500rpm in order to spin the engine at approximately 750rpm and
upwards to produce an instantaneous start in the start-stop process [3] The high rotations per
minute (rpm) produced by the ISG consistently places much higher tension requirements on the
belt than when the crankshaft is driving the belt It is preferable not to exceed a range of 600N to
800N of tension on the belt since this exceeds the safe operating conditions of belts used in most
traditional drive systems [4] The traditional belt drive system‟s tensioner a single-arm
tensioner does not suitably reduce the high belt tension nor provide enough tension in the slack
belts spans occurring in the ISG phase of operation for the B-ISG system
In order for the belt to transfer torque in a drive system its initial tension must be set to a value
that is sufficient to keep all spans rigid This value must not be too low as to allow any one span
to be slack during the drive‟s phases of operation Furthermore the belt must not be ldquoinstalled
with too high a tensionrdquo since this can lead to ldquopremature failure of the bearings supporting the
drive and driven pulleys and of the belt itselfrdquo [5] The presence of a tensioning mechanism in
an automotive belt drive allows for an enhanced belt life and performance since pre-tensioning
of the belt is normally not sufficient for all phases of belt drive operation A tensioner allows for
the system to cope with moderate to severe changes in belt span tensions
Introduction 3
Traditional automotive tensioners for belt drives of an ICE consist of a single spring-loaded
arm This type of tensioner is normally designed to provide a passive response to changes in belt
span tension The introduction of the ISG electric motor into the traditional belt drive with a
single-arm tensioner results in the presence of excessively slack spans and excessively tight
spans in the belt The tension requirements in the ISG-driving phase which differ from the
crankshaft-driving phase are poorly met by a traditional single-arm passive tensioner
Tensioners can be divided into two general classes passive and active In both classes the
single-arm tensioner design approach is the norm The passive class of tensioners employ purely
mechanical power to achieve tensioning of the belt while the active class also known as
automatic tensioners typically use some sort of electronic actuation Automatic tensioners have
been employed by various automotive manufacturers however ldquosuch devices add mass
complication and cost to each enginerdquo [5]
12 Motivation
The motivation for the research undertaken arises from the undesirable presence of high belt
tension in automotive belt drives Manufacturers of automotive belt drives have presented
numerous approaches for tension mechanism designs As mentioned in the preceding section
the automation of the traditional single-arm tensioner has disadvantages for manufacturers A
survey of the literature reveals that few quantitative investigations in comparison to the
qualitative investigations provided through patent literature have been conducted in the area of
passive and dual tensioner configurations As such the author of the research project has selected
to investigate the performance of a passive twin-arm tensioner design The theoretical tensioner
Introduction 4
configuration is motivated by research and developments of industry partner Litens
Automotivendash a manufacturer of automotive belt drive systems and components Litens‟
specialty in automotive tensioners has provided a basis for the research work conducted
13 Thesis Objectives and Scope of Research
The objective of this project is to model and investigate a system containing a passive twin-arm
tensioner in a B-ISG serpentine belt drive where the driving pulley alternates between a
crankshaft pulley and an ISG pulley The modeling of a serpentine belt drive system is in
continuation of the work done by post-doctoral fellow Zhen Mu in development of the priority
software known as FEAD at the University of Toronto Firstly for the B-ISG system with a
twin-arm tensioner the geometric state and its equations of motion (EOM) describing the
dynamic and static states are derived The modeling approach was verified by deriving the
geometric properties and the EOM of the system with a single tensioner arm and comparing its
crankshaft-phase‟s simulation results with FEAD software simulations This also provides
comparison of the new twin-arm tensioner belt drive model with the former single-arm tensioner
equipped belt drive model Secondly the model for the static system is investigated through
analysis of the tensioner parameters Thirdly the design variables selected from the parametric
analysis are used for optimization of the new system with respect to its criteria for desired
performance
Introduction 5
14 Organization and Content of the Thesis
This thesis presents the investigation of a passive twin-arm tensioner design in a B-ISG
serpentine belt drive system which is distinguished by having its driving pulley alternate
between a crankshaft pulley and an ISG pulley
Chapter 2 presents the literature reviewed relevant to the area of the thesis topic The context of
the research discusses the function and location of the ISG in hybrid technologies in order to
provide a background for the B-ISG system The attributes of the B-ISG are then discussed
Subsequently a description is given of the developments made in modeling belt drive systems
At the close of the chapter the prior art in tensioner designs and investigations are discussed
The third chapter describes the system models and theory for the B-ISG system with a twin-arm
tensioner Models for the geometric properties and the static and dynamic cases are derived The
simulation results of the system model are presented
Then the fourth chapter contains the parametric analysis The methodologies employed results
and a discussion are provided The design variables of the system to be considered in the
optimization are also discussed
The optimization of a B-ISG system with a passive twin-arm tensioner is presented in Chapter 5
The evaluation of optimization methods results of optimization and discussion of the results are
included Chapter 6 concludes the thesis work in summarizing the response to the thesis
Introduction 6
objectives and concluding the results of the investigation of the objectives Recommendations for
future work in the design and analysis of a B-ISG tensioner design are also described
7
CHAPTER 2 LITERATURE REVIEW
21 Introduction
This literature review justifies the study of the thesis research the significance of the topic and
provides the overall framework for the project The design of a tensioner for a Belt-driven
Integrated Starter-generator (B-ISG) system is a link in the chain of power transmission
developments in hybrid automobiles This chapter will begin with the context of the B-ISG
followed by a review of the hybrid classifications and the critical role of the ISG for each type
The function location and structure of the B-ISG system are then discussed Then a discussion
of the modeling of automotive belt transmissions is presented A systematic review of the prior
art and current state of tensioning mechanisms for B-ISG systems amalgamates the literature and
research evidence relevant to the thesis topic which is the design of a B-ISG tensioner
The Belt-driven Integrated Starter-generator (B-ISG) system is a part of a hybrid class that is
distinguished from other hybrid classes by the structure functions and location of its ISG The
B-ISG unit is a hybrid technology applied to traditional automotive belt drives The use of a B-
ISG system to achieve a start-stop function in the car engine is estimated to cut fuel consumption
in conventional automobiles by up to ten percent and thus reduce CO2 emissions [6]
Environmental and legislative standards for reducing CO2 emissions in vehicles have called for
carmakers to produce less polluting and more efficient vehicle powertrain systems [7] The
transition to bdquocleaner‟ cars makes room for the introduction of the ISG machine into conventional
automotive belt drives [8] The reduction of CO2 emissions and the similarity of the B-ISG
Literature Review 8
transmission to that of conventional cars provide the motivation for the thesis research
Consequently the micro-hybrid class of cars is especially discussed in the literature review since
it contains the B-ISG type of transmission system The micro-hybrid class is one of several
hybrid classes
A look at the performance of a belt-drive under the influence of an ISG is rooted in the
developments of hybrid technology The distinction of the ISG function and its location in each
hybrid class is discussed in the following section
22 B-ISG System
221 ISG in Hybrids
This section of the review discusses the standard classes of hybrid cars which are full power
mild and micro- hybrids Special attention is given to hybrid vehicle architectures involving
internal combustion engines (ICEs) as the main power source This is done for the sake of
comparison between hybrid classes since the ICE is the standard power source for B-ISG micro-
hybrids which is the focus of the research The term conventional car vehicle or automobile
henceforth refers to a vehicle powered solely by a gas or diesel ICE
A hybrid vehicle has a drive system that uses a combination of energy devices This may include
an ICE a battery and an electric motor typically an ISG Two systems exist in the classification
of hybrid vehicles The older system of classification separates hybrids into two classes series
hybrids and parallel hybrids In the older system many modern hybrid vehicles have modes of
operation matching both categories classifying them under either of the two classes [9] The
Literature Review 9
new system of classification has four classes full power mild and micro Under these classes
vehicles are more often under a sole category [9] In both systems an ICE may act as the primary
source of power otherwise it may be a fuel cell The fuel used by the ICE may be gas (petrol)
diesel or an alternative fuel such as ethanol bio-diesel or natural gas
2211 Full Hybrids
In a full hybrid car the ICE is used to power the integrated starter-generator (ISG) which stores
electrical energy in the batteries to be used to power an electric traction motor [8] The electric
traction motor is akin to a second ISG as it generates power and provides torque output It also
supplies an extra boost to the wheels during acceleration and drives up steep inclines A full
hybrid vehicle is able to move by electrical power only It can be driven by the ISG powering
the electric traction motor without the engine running This silent acceleration known as electric
launch is normally employed when accelerating from standstill [9] Full hybrids can generate
and consume energy at the same time Full hybrid vehicles also use regenerative braking [8]
The ISG allows this by converting from an electric traction motor to a generator when braking or
decelerating The kinetic energy from the car‟s motion is then turned into electricity and stored
in the batteries For full hybrids to achieve this they often use break-by-wire a form of
electronically controlled braking technology
A high-voltage (ie 36- or 42-volt) ISG is employed in full hybrids to start the ICE It spins the
engine more than 900 rpm whereas conventional 12-volt starter motors spin the engine at
approximately 250 rpm [9] Thus the full hybrid vehicle is able to have an instantaneous start In
full hybrids the ISG is placed in the position of the flywheel and can have its motion decoupled
Literature Review 10
from the engine [9] The ISG device also allows full hybrids to have engine start-stop also called
an idle-stop ability The idle-stop function refers to when the engine shuts down as soon as a
vehicle stops from its ICE driving mode which saves on the fuel it normally burns while idling
[8] The vehicle returns to the engine driving mode of operation by way of the ISG‟s start-up of
the crankshaft which restarts the engine in less than 300 milliseconds [9] In summary at
standstill the tachometer of the engine drops to 0 rpm since the engine has ceased the engine is
started only when needed which is often several seconds after acceleration has begun The
engine start-stop feature is achieved by way of an electronic control system that shuts off the ICE
when it is not needed to assist in driving the wheels or to produce electricity for recharging the
batteries The start-stop feature by itself is estimated to produce a ten percent fuel gain in hybrids
over conventional vehicles particularly in urban driving conditions [9] Since the ICE is
required to provide only the average horsepower used by the vehicle the engine is downsized in
comparison to a conventional automobile that obtains all its power from an ICE Frequently in
full hybrids the ICE uses an alternative operating strategy such as the Atkinson Cycle which has
a higher efficiency while having a lower power output Examples of full hybrids include the
Ford Escape and the Toyota Prius [9]
2212 Power Hybrids
Akin to the full hybrid the ISG of the power hybrid enables the same features electric launch
regenerative braking and engine idle-stop The distinguishing characteristic from full hybrids is
the ICE is not downsized to meet only the average power demand [9] Thus the engine of a
power hybrid is large and produces a high amount of horsepower compared to the former
Overall a power hybrid has the assist of a full size ICE and therefore has more torque and a
Literature Review 11
greater acceleration performance than a full hybrid or a conventional vehicle with the same size
ICE [9] The Lexus RX400h unit is an example of a power hybrid [9]
2213 Mild Hybrids
In the hybrid types discussed thus far the ISG is positioned between the engine and transmission
to provide traction for the wheels and for regenerative braking Often times the armature or rotor
of the electric motor-generator which is the ISG replaces the engine flywheel in full and power
hybrids [9] In the case of the mild hybrid the ISG is not decoupled from the ICE and hence it is
not able to drive the wheels apart from the engine It remains that the ISG shares the same shaft
with the ICE In this environment the electric launch feature does not exist since the ISG does
not turn the wheels independently of the engine and energy cannot be generated and consumed
at the same time However the ISG of the mild hybrid allows for the remaining features of the
full hybrid regenerative braking and engine idle-stop including the fact that the engine is
downsized to meet only the average demand for horsepower Mild hybrid vehicles include the
GMC Sierra pickup and 2003 to 2005 Honda Civic models [9]
2214 Micro Hybrids
Micro hybrid is the category of hybrids that can contain a B-ISG transmission and is also closest
to modern conventional vehicles This class normally features a gas or diesel ICE [9] The
conventional automobile is modified by installing an ISG unit on the mechanical drive in place
of or in addition to the starter motor The starter motor typically 12-volts is removed only in
the case that the ISG device passes cold start testing which is also dependent on the engine size
[10] Various mechanical drives that may be employed include chain gear or belt drives or a
Literature Review 12
clutchgear arrangement The majority of literature pertaining to mechanical driven ISG
applications does not pursue clutchgear arrangements since it is associated with greater costs
and increased speed issues Findings by Henry et al [11] show that the belt drive in
comparison to chain and gear drives has a decreased cost (especially if the ISG is mounted
directly to the accessory drive) has no need for lubrication has less restriction in the packaging
environment and produces very low noise Also mounting the ISG unit on a separate belt from
that linking the accessory pulleys is undesirable since applying the ISG directly to the accessory
belt drive requires less engine transmission or vehicle modifications
As with full power and mild hybrids the presence of the ISG allows for the start-stop feature
The automobile‟s electronic control unit (ECU) is calibrated or engine control circuitry (a
separate ECU) is added to the conventional car in order to shut down the engine when the
vehicle is stopped [12] The control system also controls the charge cycle of the ISG [9] This
entails that it dictates the field current by way of a microprocessor to allow the system to defer
battery charge cycles until the vehicle is decelerating [13] This produces electricity to recharge
the battery primarily during deceleration and braking The B-ISG transmission of a micro hybrid
and its various components are discussed in the subsequent section Examples of micro hybrid
vehicles are the PSA Group‟s Citroen C2 and C3 [14] Ford‟s Fiesta [14] and BMW‟s Mini
Cooper D and various others of BMW‟s European models [15]
Literature Review 13
Figure 21 Hybrid Functions
Source Dr Daniel Kok FFA July 2004 modified [16]
Figure 21 shows that the higher the voltage available to the ISG unit the more hybrid functions
it is capable of performing It is noted that B-ISG transmissions of the micro-hybrid class may
also exceed the typical functions of micro-hybrids For instance Ford‟s HyTrans van (developed
in partnership with Ricardo UK Ltd Valeo SA Gates Corporation and the UK Department for
Transport) uses a B-ISG system and a 42-volt battery The van is diesel-powered and has
characteristics of a mild hybrid such as cold cranks and engine assists [17]
222 B-ISG Structure Location and Function
2221 Structure and Location
The ISG is composed of an electrical machine normally of the inductive type which includes a
stator (stationary part of the ISG) and a rotor (non-stationary part of the ISG) and a converter
comprising of a regulator a modulator switches and filters There are various configurations to
integrate the ISG unit into an automobile power train One configuration situates the ISG
directly on the crankshaft in the place of the present flywheel [11] This set-up is more compact
however it results in a longer power train which becomes a potential concern for transverse-
Literature Review 14
mounted engines [18] An alternative set-up is to have a side-mounted ISG This term is used to
describe the configuration of mounting the electrical device on the side of the mechanical drive
[18] As mentioned in Section 2214 a belt drive is used as the mechanical drive for the thesis
research hence the ISG is belt-mounted and the transmission becomes a belt-driven ISG system
In this arrangement the ISG replaces the alternator [13] and in some cases the starter motor may
be removed This design allows for the functions of the ISG system mentioned in the description
of the ISG role in micro-hybrids [9] The side-mounted ISG specifically the belt-mounted ISG
is more evolutionary to the conventional car since it ldquoallows for a more traditional under-hood
layoutrdquo [11]
2222 Functionalities
The primary duty of the ISG in a micro hybrid specifically in a B-ISG setting is to bring the
engine from rest to normal operating speeds within a time span ranging from 250 to 400 ms [3]
and in some high voltage settings to provide cold starting
The cold starting operation of the ISG refers to starting the engine from its off mode rather than
idle mode andor when the engine is at a low temperature for example -29 to -50 degrees
Celsius [2] If the ISG is used for cold starting the peak torque is determined by the torque
requirement for the cold starting operation of the target vehicle since it is greater than the
nominal torque For this function the ldquomachine has to provide a breakaway torque about 15 [to]
18 times the nominal cranking torque to overcome static torque and rotate the engine from 0 to
[between] 10 [and] 20rpmrdquo [2] This remains to be a challenge for the ISG as the 12-volt
architecture most commonly found in vehicles does not supply sufficient voltage [2] The
introduction of the ISG machine and other electrical units in vehicles encourages a transition
Literature Review 15
from a 12-volt or 14-volt to a 42-volt electrical architecture [19] The transition to 42-volt
architecture brings ldquopotential higher-voltage functionalities that come with an ISG systemrdquo [20]
At present ldquowhen the [ISG] machine cannot provide enough torque for initial cold engine
cranking the conventional starter will [remain] in the system and perform only for the initial
cranking while the stop-start function is taken over by the [ISG] machinerdquo [2] The ISG‟s launch
assist torque the torque required to bring the engine from idle speed to the speed at which it can
develop a higher torque output is 2000 to 2500 rpm for most gas engines [3]
Delphi‟s Energen 5 High Output 12-volt Belt-alternator-starter (or B-ISG) was implemented by
researchers on a 53 L V-8 engine with an automatic transmission in a Chevrolet Silverado truck
[21] The ISG was applied in a belt-mounted configuration and was used only for warm engine
re-starts The results of Wezenbeek et al [21] showed that the starting torque for a re-start by the
12-Volt ISG was 42 Nm ISG‟s have also been used in 14V 36V and 42V architectures [13]
23 Belt Drive Modeling
The modeling of a serpentine belt drive and tensioning mechanism has typically involved the
application of Newtonian equilibrium equations to rigid bodies in order to derive the equations of
motion for the system There are two modes of motion in a serpentine belt drive transverse
motion and rotational motion The former can be viewed as the motion of the belt directed
normal to the direction of the beltpulley contact plane similar to the vibratory motion of a taut
string that is fixed at either end However the study of the rotational motion in a belt drive is the
focus of the thesis research
Literature Review 16
Much work on the mechanics of the belt drive was carried out by Firbank [22] Firbank‟s
models helped to understand belt performance and the influence of driving and driven pulleys on
the tension member The first description of a serpentine belt drive for automotive use was in
1979 by Cassidy et al [23] and since this time there has been an increasing body of knowledge
on the mathematical modeling of serpentine belt drives Ulsoy et al [24] presented a design
methodology to improve the dynamic performance of instability mechanisms for belt tensioner
systems The mathematical model developed by Ulsoy et al [24] coupled the equations of
motion that were obtained through a dynamic equilibrium of moments about a pivot point the
equations of motion for the transverse vibration of the belt and the equations of motion for the
belt tension variations appearing in the transverse vibrations This along with the boundary and
initial conditions were used to describe the vibration and stability of the coupled belt-tensioner
system Their system also considered the geometry of the belt drive and tensioner motion
Hereafter Beikmann et al [25] predicted the belt drive vibration for a system composed of a
driving pulley driven pulley and a dynamic tensioner The authors coupled the linear equations
of transverse motion for the respective belt spans with the equations of motion for pulleys and a
tensioner This was used to form the free response of the system and evaluate its response
through a closed-form solution of the system‟s natural frequencies and mode shapes
A complex modal analysis of a serpentine belt drive system was carried out by Kraver et al [26]
to determine the effect of damping on rotational vibration mode solutions The equations of
motion developed for a multi-pulley flat belt system with viscous damping and elastic
Literature Review 17
properties including the presence of a rotary tensioner were manipulated to carry out the modal
analysis
Beikmann et al [27] also derived a nonlinear model to predict the operating state of a belt-
tensioner system by way of nonlinear numerical methods and an approximated linear closed-
form method The authors used this strategy to develop a single design parameter referred to as
a tensioner constant to measure the effectiveness of the tensioning mechanism in relation to its
operating state from a reference state The authors considered the steady state tensions in belt
spans as a result of accessory loads belt drive geometry and tensioner properties
Zhang and Zu [28] conducted a modal analysis for the response of a linear serpentine belt drive
system A non-iterative approach was used to explicitly form the equations for the system‟s
natural frequencies An exact closed-form expression for the dynamic response of the system
using eigenfunction expansion was derived with the system under steady-state conditions and
subject to harmonic excitation
The work conducted by Balaji and Mockensturm [29] considered a front-end accessory drive
(FEAD) with a decoupler or isolator attached to a pulley The rotational response for the FEAD
was found analytically by considering the system to be piecewise linear about the equilibrium
angular deflections The effect of their nonlinear terms was considered through numerical
integration of the derived equations of motion by way of the iterative methodndash fourth order
Runge-Kutta The authors in this case considered the longitudinal (ie rotational) vibration of
the belt spans only
Literature Review 18
The first to carry out the analysis of a serpentine belt drive system containing a two-pulley
tensioner was Nouri in 2005 [30] Nouri found the closed-form analytical solution of a
serpentine belt drive with a two-pulley tensioner for the case of sinusoidal excitation He
employed Runge Kutta method as well to solve the equations of motion to find the response of
the system under a general input from the crankshaft The author‟s work also included the
optimization of the tensioner design in order to minimize belt span vibrations due to crankshaft
excitation Furthermore the author applied active control techniques to the tensioner in a belt
drive system
The works discussed have made significant contributions to the research and development into
tensioner systems for serpentine belt drives These lead into the requirements for the structure
function and location of tensioner systems particularly for B-ISG transmissions
24 Tensioners for B-ISG System
241 Tensioners Structure Function and Location
Literature shows that the improvement of a serpentine belt life in a B-ISG system centers on the
tensioning mechanism redesign This mechanism as shown by researchers including
Wezenbeek et al [21] and Henry et al [11] is crucial in establishing the least tension in the belt
(above a zero value) in order to guard against failure by way of slip due to slack spans in the belt
and oscillations during engine re-start It is noted by Firbank [22] that the mechanics of a belt-
drive ldquois based on the idea that belt behaviour is governed by the elastic extension or contraction
of the belt arising from tension variationsrdquo [22] these variations may be compensated for by an
adjustable tensioner
Literature Review 19
The two types of tensioners are passive and active tensioners The former permits an applied
initial tension and then acts as an idler and normally employs mechanical power and can include
passive hydraulic actuation This type is cheaper than the latter and easier to package The latter
type is capable of continually adjusting the belt tension since it permits a lower static tension
Active tensioners typically employ electric or magnetic-electric actuation andor a combination
of active and passive actuators such as electrical actuation of a hydraulic force
Conventional belt tensioners comprise of a single tensioner arm that is fitted with a sole idler
pulley to engage a serpentine belt [31] A radial bearing is used to rotatably connect the idler
pulley to the tensioner arm [31] The tensioner arm is mounted on a pivot pin that is wrapped by
a bushing and is free to rotate [31] The pin covered by the bushing is fixed to the engine
housing [31] A rotary spring is wrapped about the bearing pin and bushing to provide a pre-
tension force to the belt via the tensioner arm and idler pulley thus taking up the slack due to the
changes in belt length [31] When the belt undergoes stretch under a load the spring drives the
tensioner arm and idler pulley further into the belt [31] Belt tension changes under the modes of
operation which can include when the crankshaft (or driving pulley) abruptly decelerates from a
steady-state condition and auxiliary components continue to rotate still in their own inherent
inertia and thus become the primary drivers [31] These fluctuations in belt tension lead to belt
flutter or skip and slip that may damage other components present in the belt drive [31]
Locating the tensioner on the slack side of the belt is intended to lower the initial static tension
[11] In conventional vehicles the engine always drives the alternator so the tensioner is located
in the belt span that links the crankshaft and alternator pulleys In a B-ISG setting the slack span
Literature Review 20
of the belt alternates between the driving mode of the ISG and the driving mode of the crankshaft
[32] Research by Henry et al [11] and also the summary of prior art for tensioners in Table
21 show that placing the idlertensioner pulley in the slack span in the case that the ISG is
driving instead of in the slack span when the crankshaft is driving allows for easier packaging
and for the least static tension Designs shown in Table 21 place the tensioneridler pulley in the
same span as Henry et al [11] or in both the slack and taut spans if using a double
tensioneridler configuration
242 Systematic Review of Tensioner Designs for a B-ISG System
The proposals for belt tensioner devices to manage the issue of high peaks in belt tension for B-
ISG settings are largely in patent records as the re-design of a tensioner has been primarily a
concern of automotive makers thus far A systematic review of the patent literature has been
conducted in order to identify evaluate and collate relevant tensioning mechanism designs
applicable to a B-ISG setting Its research objective is to influence the selection of a tensioner
configuration for the thesis study
The predefined search strategy used by the researcher has been to consider patents dating only
post-2000 as many patents dating earlier are referred to in later patents as they are developed on
in most cases by the original inventor (eg an INA Schaeffler KG patent published in 2000 may
refer to its own earlier patent presented in 1999) Patents dating pre-2000 that do not have any
successor were also considered The inclusion and exclusion criteria and rationales that were
used to assess potential patents are as follows
Inclusion of
Literature Review 21
tensioner designs with two arms andor two pivots andor two pulleys
mechanical tensioners (ie exclusion of magnetic or electrical actuators or any
combination of active actuators) in order to minimize cost
tension devices that are an independent structure apart from the ISG structure in order to
reduce the required modification to the accessory belt drive of a conventional automobile
and
advanced designs that have not been further developed upon in a subsequent patent by the
inventor or an outside party
Table 21 provides a collation of the results for the systematic review based on the selection
criteria Illustrations of the collated patent designs may be seen in Appendix A It is noted that
the patent literature pertaining to these designs in most cases provides minimal numerical data
for belt tensions achieved by the tensioning mechanism In most cases only claims concerning
the outcome in belt performance achievable by the said tension device is stated in the patent
Table 21 Passive Dual Tensioner Designs from Patent Literature
Bayerische
Motoren Werke
AG
Patents EP1420192-A2 DE10253450-A1 [33]
Design Approach
2 tensioner pulleys (idlers) and 2 tension arms are mounted outside the periphery of the belt drive these form tiltable clamping arms around a common axis of rotation
A torsion spring is used at bearing bushings to mount tension arms at ISG shaft
Each tension arm cooperates with torsion spring mechanism to rotate through a damping
device in order to apply appropriate pressure to taut and slack spans of the belt in
different modes of operation
Bosch GMBH Patent WO0026532 et al [34]
Design Approach
2 tension pulleys each one is mounted on the return and load spans of the driven and
driving pulley respectively
Idlers (tension pulleys) each connect to a spring which is attached on one end to a fixed point
Literature Review 22
Idlers‟ motions are independent of each other and correspond to the tautness or
slackness in their respective spans
Or alternatively a spring connects the idler pulleys and one of the two idlers is fixed at
its axis of rotation
Daimler Chrysler
AG
Patents DE10324268-A1 [35]
Design Approach
2 idlers are given a working force by a self-aligning bearing
Bearing supports auxiliary unit (ISG) and is arranged concentrically with the axle
auxiliary unit pulley
Dayco Products
LLC
Patents US6942589-B2 et al [36]
Design Approach
2 tension arms are each rotatably coupled to an idler pulley
One idler pulley is on the tight belt span while the other idler pulley is on the slack belt
span
Tension arms maintain constant angle between one another
One arm forms a positive differential angle with the belt and the remaining arm forms a negative differential angle with the belt
Idler pulleys are on opposite sides of the ISG pulley
Gates Corporation Patents US20060249118-A1 WO2003038309-A [37]
Design Approach
A tensioner pulley contacts the belt at the slack span during start-up (ISG-driving mode)
A tensioner is asymmetrically biased in direction tending to cause power transmission
belt to be under tension
McVicar et al
(Firm General
Motors Corp)
Patent US20060287146-A1 [38]
Design Approach
2 tension pulleys and carrier arms with a central pivot are mounted to the engine
One tension arm and pulley moderately biases one side of belt run to take up slack
during engine start-up while other tension arm and pulley holds appropriate bias against
taut span of belt
A hydraulic strut is connected to one arm to provide moderate bias to belt during normal
engine operation and velocity sensitive resistance to increasing belt forces during engine
start-up
INA Schaeffler
KG et al
Patents DE10044645-A1 [39] DE10159073-A1 [40] EP1723350-A1 et al [41]
DE10359641-A1 et al [42] EP1738093-A1 et al [43] DE102004012395-A1 [44]
WO2006108461-A1 et al [45]
Design Approach
2 tension arms and 2 pulleys approach ndash o Mutually independent tensioning arms are supported for rotation in the same
plane of the housing part
o Idler pulley corresponding to each tensioning arm engages with different
sections of belt
o When high tension span alternates with slack span of belt drive one tension
arm will increase pressure on current slack span of belt and the other will
decrease pressure accordingly on taut span
o Or when the span under highest tension changes one tensioner arm moves out
of the belt drive periphery to a dead center due to a resulting force from the taut
span of the ISG starting mode
o Deflection of the taut span acts on associated pulley to apply a counter-moment to the other idler pulley on the slack span
Literature Review 23
o The 2 lever arms are of different lengths and each have an idler pulley of
different diameters and different wrap angles of belt (see DE10045143-A1 et
al)
1 tensioner arm and 2 pulleys approach ndash
o 2 idler pulleys are pinned to a beam arranged on a clamping arm that is tiltably
linked to the beam o The ISG machine is supported by a shock absorber
o During ISG start-up one idler pulley is induced to a dead center position while
it pulls the remaining idler pulley into a clamping position until force
equilibrium takes place
o A shock absorber is laid out such that its supporting spring action provides
necessary preloading at the idler pulley in the direction of the taut span during
ISG start-up mode
Litens Automotive
Group Ltd
Patents US6506137-B2 et al [46]
Design Approach
2 tension pulleys on opposite sides of the ISG pulley engage the belt
They are positioned such that their applied forces result in opposing directed moments with respect to the tension device‟s axis of pivot
The pivot axis varies relative to the force applied to each tension pulley
Diameters of the tensioner pulleys are approximately equal and belt wrap angles of the
tensioner pulleys are approximately equal
A limited swivel angle for the tensioner arms work cycle is permitted
Mitsubishi Jidosha
Eng KK
Mitsubishi Motor
Corp
Patents JP2005083514-A [47]
Design Approach
2 tensioners are used
1 tensioner is held on the slack span of the driving pulley in a locked condition and a
second tensioner is held on the slack side of the starting (driven) pulley in a free condition
Nissan Patents JP3565040-B2 et al [48]
Design Approach
A single tensioner is on the slack span once ISG pulley is in start-up mode
The tension device is comprised of a oil pressure tensioner and a half ratchet mechanism
(a plunger which performs retreat actuation according to the energizing force of the oil
pressure spring and load received from the ISG)
The tensioner is equipped with a relief valve to keep a predetermined load lower than the
maximum load added by the ISG device
NTN Corp Patent JP2006189073-A [49]
Design Approach
An automatic tensioner is equipped with a hydraulic damper mechanism comprised of a
screw bolt using saw-screwed teeth and a cylinder nut a return spring and a spring seat
in a pressure chamber (within the screw bolt) a rod seat (that is fitted to the lower end of
the cylinder nut) a spring support (arranged on varying diameter stepped recessed
sections of the rod seat) and a check valve with an openingclosing passage
The cylinder and screw bolt act as the rigidity buffer under excessive loads during ISG
start-up mode of operation
Valeo Equipment
Electriques
Moteur
Patents EP1658432 WO2005015007 [50]
Design Approach
ldquoThe invention relates to a system or a starter (10) in which a pulley (80) is rotationally mounted on a section (22) of a shaft which axially extends inside a pulley (80) and
Literature Review 24
forwards at least partially outside a support element (200) and is characterized in that
the free front end (23) of said shaft section (22) is carried by an arm (206) connected to
the support element (200)rdquo
The author notes that published patents and patent applications may retain patent numbers for multiple patent
offices (ie European Patent Office German Patent Office etc) In such cases the published patent number or in
the absence of such a number the published patent application number has been specified However published
patent documents in the above cases also served as the document (ie identical) to the published patent if available
Quoted from patent abstract as machine translation is poor
25 Summary
The research on tensioner designs from the patent literature demonstrates a lack of quantifiable
data for the performance of a twin tensioner particularly suited to a B-ISG system The review of
the literature for the modeling theory of serpentine belt drives and design of tensioners shows
few belt drive models that are specific to a B-ISG setting Hence the literature review supports
the thesis objective of modeling a B-ISG tensioner specifically one that has a passive twin
tensioner configuration and as well measuring the tensioner‟s performance The survey of
hybrid classes reveals that the micro-hybrid class is the only class employing a closely
conventional belt transmission and hence its B-ISG transmission is applicable for tensioner
investigation The patent designs for tensioners contribute to the development of the tensioner
design to be studied in the following chapter
25
CHAPTER 3 MODELING OF B-ISG SYSTEM
31 Overview
The derivation of a theoretical model for a B-ISG system uses real life data to explore the
conceptual system under realistic conditions The literature and prior art of tensioner designs
leads the researcher to make the following modeling contributions a proposed design for a
passive two-pulley tensioner computation of geometric attributes for a B-ISG system with the
proposed tensioner and derivation of the system‟s equations of motion (EOM) under dynamic
and static states as well as deriving the EOM for the B-ISG system with only a passive single-
pulley tensioner for comparison The principles of dynamic equilibrium are applied to the
conceptual system to derive the EOM
32 B-ISG Tensioner Design
The proposed design for a passive two pulley tensioner configures two tensioners about a single
fixed pivot point in the interior space of a serpentine belt drive One end of each tensioner arm
coincides with the centre point of a tensioner pulley and this point marks the axis of rotation of
the pulley The other end of each arm is pivoted about a point so that the arms share the same
axis of rotation This conceptual design henceforth is called a Twin Tensioner Figure 31 shows
a schematic for the proposed design
Modeling of B-ISG 26
Figure 31 Schematic of the Twin Tensioner
The tensioner pulley coordinates are described by (XiYi) their radii by Ri their arm lengths Lti
and their angles θti The rotation of the arms is resisted by stiffness kt of a coil spring acting
between the two arms and spring stiffness kti acting between each arm and the pivot point The
motion of each arm is dampened by dampers and akin to the springs a damper acts between the
two arms ct and a damper cti acts between each arm and the pivot point The result is a
tensioning mechanism with four degrees of freedom (DOF) that includes independent rotations
of the two pulleys and two arms
The following section relates the geometry of the rigid bodies in a B-ISG system equipped with a
Twin Tensioner to their respective motions
Modeling of B-ISG 27
33 Geometric Model of a B-ISG System with a Twin Tensioner
The B-ISG system with the Twin Tensioner is shown in Figure 32 The geometry of the drive
provides the lengths of the belt spans and angles of wrap for the belt and pulley contact surfaces
These variables are crucial to resolve the components of forces and moment arms acting on each
rigid body in the system and are used in the derivation of the EOM in section 34 Zhen Mu‟s
geometric modeling approach [51] used in the development of the software FEAD was applied
to the Twin Tensioner system to compute the system‟s unique geometric attributes
Figure 32 B-ISG Serpentine Belt Drive with Twin Tensioner
It is noted that in Figure 31 and Figure 32 showing the schematic of the Twin Tensioner and
the overall system respectively that for the purpose of the geometric computations the forward
direction follows the convention of the numbering order counterclockwise The numbering
order is in reverse to the actual direction of the belt motion which is in the clockwise direction in
this study The fourth pulley is identified as an ISG unit pulley However the properties used
for the ISG pulley‟s geometry inertia stiffness and damping is modeled as a conventional
Modeling of B-ISG 28
alternator pulley This pulley is conceptualized as an ISG when it is modeled as the driving
pulley at which point the requirements of the ISG are solved for and its non-inertia attributes
are not needed to be ascribed
Figure 33 shows the geometric attributes needed to resolve the wrap angle of the belt on each
pulley Variables (XiYi) and XYfi XYbi XYfbi and XYbfi are the ith pulley centre coordinates and
its possible belt connection points respectively Length Lfi is the length of the span connecting
the points XYfi and XYf(i+1) or XYbi and XYb(i+1) on the ith and (i+1)th pulleys respectively
Similarly Lbi is the length of the span between XYfbi and XYfb(i+1) or XYbfi and XYbf (i+1) on the
ith and (i+1)th pulleys respectively Angles αi θfi and θbi represent the angle between a line
connecting the ith and (i+1)th pulley centres and the angles of the belt connection spans with
lengths Lfi and Lbi respectively Ri is the radius of the ith pulley
Figure 33 Angles Coordinates and Possible Belt Contact Points for the ith and i+1th Pulleys
[modified] [51]
Modeling of B-ISG 29
The angle between the horizontal and the line connecting the ith and (i+1)th pulley centres αi is
calculated using Zhen‟s method [51] This method uses the pulley‟s coordinates and a cosine
trigonometric relation
i acos
Xi 1
Xi
Xi 1
Xi
2
Yi 1
Yi
2
Yi 1
Yi
if
(31a)
i 2 acos
Xi 1
Xi
Xi 1
Xi
2
Yi 1
Yi
2
Yi 1
Yi
if
(31b)
The lengths for connecting the possible belt spans are described by the variables Lfi and Lbi
The centre point coordinates and the radii of the pulleys are related through the solution of
triangles which they form to define values of the possible belt span lengths
Lfi
Xi 1
Xi
2
Yi 1
Yi
2
Ri 1
Ri
2
(32a)
Lbi
Xi 1
Xi
2
Yi 1
Yi
2
Ri 1
Ri
2
(32b)
The set of possible belt span lengths leads to the calculation of θfi and θbi the angles between the
line connecting the ith and (i+1)th pulley centres and the possible contact point on the pulley
perimeter
Modeling of B-ISG 30
(33a)
(33b)
The array of possible belt connection points comes about from the use of the pulley centre
coordinates and their radii and the sine of the sum or differences of αi and θfi or θbi The angle
αi is calculated in equations (31a) and (31b) and angles θfi and θbi are calculated in equations
(33a) and (33b) The formula to compute the array of points is shown in equations (34) and
(35) for the ith and (i+1)th pulleys Equation (34) describes the forward belt connection point
on the ith pulley which is in the span leading forward to the next (i+1)th pulley
(34a)
(34b)
(34c)
(34d)
bi atan
Lbi
Ri
Ri 1
Modeling of B-ISG 31
Equation (35) describes the backward belt connection point on the ith pulley This point sits on
the ith pulley in the contacting belt span which leads backward to connect with the (i-1)th
pulley
(35a)
(35b)
(35c)
(35d)
The selection of the coordinates from the array of possible connection points requires a graphic
user interface allowing for the points to be chosen based on observation This was achieved
using the MathCAD software package as demonstrated in the MathCAD scripts found in
Appendix C The belt connection points can be chosen so as to have a pulley on the interior or
exterior space of the serpentine belt drive The method used in the thesis research was to plot the
array of points in the MathCAD environment with distinct symbols used for each pair of points
and to select the belt connection points accordingly By observation of the selected point types
the type of belt span connection is also chosen Selected point and belt span types are shown in
Table 31
Modeling of B-ISG 32
Table 31 Selected Contact Point Types on the ith Pulley and Types for the ith Belt Span
Pulley Forward Contact
Point
Backwards Contact
Point
Belt Span
Connection
1 Crankshaft XYf1 XYbf21 Lf1
2 Air Conditioning XYfb2 XYf22 Lb2
3 Tensioner 1 XYbf3 XYfb23 Lb3
4 AlternatorISG XYfb4 XYbf24 Lb4
5 Tensioner 2 XYbf5 XYfb25 Lb5
The inscribed angles βji between the datum and the forward connection point on the ith pulley
and βji between the datum and its backward connection point are found through solving the
angle of the arc along the pulley circumference between the datum and specified point The
wrap angle ϕi is found as the difference between the two inscribed angles for each connection
point on the pulley The angle between each belt span and the horizontal as well as the initial
angle of the tensioner arms are found using arctangent relations Furthermore the total length of
the belt is determined by the sum of the lengths of the belt spans
34 Equations of Motion for a B-ISG System with a Twin Tensioner
341 Dynamic Model of the B-ISG System
3411 Derivation of Equations of Motion
This section derives the inertia damping stiffness and torque matrices for the entire system
Moment equilibrium equations are applied to each rigid body in the system and net force
equations are applied to each belt span From these two sets of equations the inertia damping
Modeling of B-ISG 33
and stiffness terms are grouped as factors against acceleration velocity and displacement
coordinates respectively and the torque matrix is resolved concurrently
A system whose motion can be described by n independent coordinates is called an n-DOF
system Consider the free body diagram of the Twin Tensioner in Figure 34 in which each
pulley of inertia Ii is supported on an arm of inertia Iti It is assumed that the pulleys are
constrained to rotate about their respective central axes and the arms are free to rotate about their
respective pivot points then at any time the position of each pulley can be described by a
rotational coordinate θi(t) and a coordinate θti(t) can denote the rotation of each arm Thus the
tensioner system comprises of four rigid bodies where each is described by one coordinate and
hence is a four-DOF system It is important to note that each rigid body is treated as a point
mass In addition inertial rotation in the positive direction is consistent with the direction of belt
motion The belt span tensions Ti and coupled radii Ri apply moments to the pulleys
Figure 34 Twin Tensioner Free Body Diagram of a Four-Degree of Freedom System
Modeling of B-ISG 34
For the serpentine belt system considered in the thesis research there are seven rigid bodies each
having a one-DOF of motion The EOM for a seven-DOF system form second-order coupled
differential equations meaning that each equation includes all of the general coordinates and
includes up to the second-order time derivatives of these coordinates The EOM can be
obtained by applying D‟Alembert‟s principle that the sum of the moments taken about any point
including the couples equals to zero Therefore the inertial couple the product of the inertia and
acceleration is equated to the moment sum as shown in equation (35)
I ∙ θ = ΣM (35)
The moment equilibrium equations for the Twin Tensioner in Figure 34 where the positive
direction is in the clockwise direction are shown in equations (36) through to (310) The
numbering convention used for each rigid body corresponds to the labeled serpentine belt drive
system shown in Figure 32 Qi represents the required torque of the ith rigid body ci is the
damping constant of the ith rigid body βji is the angle of orientation for the ith belt span and
120597120579119905119894 120579 119905119894 and 120579 119905119894 are the angular displacement angular velocity and angular acceleration of the ith
tensioner arm The initial angle of the ith tensioner arm is described by θtoi
minusI3 ∙ θ 3 = T3 ∙ R3 minus T2 ∙ R3 minus Q3 + c3 ∙ θ 3 (36)
minusI5 ∙ θ 5 = minusT4 ∙ R5 + T5 ∙ R5 minus Q5 + c5 ∙ θ 5 (37)
Modeling of B-ISG 35
It1 ∙ θ t1 = minusTt1 ∙ Lt1 ∙ sin θto 1 minus βj4 + sin θto 1 minus βj5 minus kt ∙ partθt1 minus partθt2 minus kt1 ∙
partθt1 minus ct ∙ partθ t1 minus partθ t2 minus ct1 ∙ partθ t1 (38)
It2 ∙ θ t2 = minusTt2 ∙ Lt2 ∙ sin θto 2 minus βj2 + sin θto 1 minus βj3 minus kt ∙ partθt2 minus partθt1 minus kt2 ∙ partθt2 minus
ct ∙ partθ t2 minus partθ t1 minus ct2 ∙ partθ t2 (39)
partθt1 = θt1 minus θto 1 (310a)
partθt2 = θt2 minus θto 2 (310b)
The free body diagrams for the remaining rigid bodies crankshaft pulley air conditioner pulley
and ISG pulley are in the general form of Figure 35 The sum of the moments about the axes of
rotation are taken for these structures in equations (311) through to (313)
Figure 35 Free Body Diagram for Non-Tensioner Pulleys
Modeling of B-ISG 36
I1 ∙ θ 1 = T5 ∙ R1 minus T1 ∙ R1 + Q1 minus c1 ∙ θ 1 (311)
I2 ∙ θ 2 = T1 ∙ R2 minus T2 ∙ R2 + Q2 minus c2 ∙ θ 2 (312)
I4 ∙ θ 4 = T3 ∙ R4 minus T4 ∙ R4 + Q4 minus c4 ∙ θ 4 (313)
The relationship between belt tensions and rigid body displacements is in the general form of
equation (314) where 119827119836 and 119827119844 are damping and stiffness matrices due to the belt respectively
with each factorized by a radial arm length This relationship is described for each span in
equations (315) through to (320) The belt damping constant for the ith belt span is cib
119827prime = 119827119836 ∙ 120521 + 119827119844 ∙ 120521 (314)
T1 = To + k1b ∙ R1 ∙ θ1 minus R2 ∙ θ2 + c1
b ∙ [R1 ∙ θ 1 minus R2 ∙ θ 2] (315)
T2 = To + k2b ∙ R2 ∙ θ2 minus R3 ∙ θ3 + Lt1 ∙ [sin θto 1 minus βj2 ] ∙ (θt1 minus θto 1) + c2
b ∙ [R2 ∙ θ 2 minus R3 ∙
θ 3 + Lt1 ∙ [sin θto 1 minus βj2 ] ∙ (θ t1)] (316)
T3 = To + k3b ∙ R3 ∙ θ3 minus R4 ∙ θ4 + Lt1 ∙ [sin θto 1 minus βj3 ] ∙ (θt1 minus θto 1) + c3
b ∙ [R3 ∙ θ 3 minus R4 ∙
θ 4 + Lt1 ∙ [sin θto 1 minus βj3 ] ∙ (θ t2)] (317)
Modeling of B-ISG 37
T4 = To + k4b ∙ R4 ∙ θ4 minus R5 ∙ θ5 + Lt2 ∙ [sin θto 2 minus βj4 ] ∙ (θt2 minus θto 2) + c4
b ∙ [R4 ∙ θ 4 minus R5 ∙
θ 5 + Lt2 ∙ [sin θto 2 minus βj4 ] ∙ (θ t1)] (318)
T5 = To + k5b ∙ R5 ∙ θ5 minus R1 ∙ θ1 + Lt2 ∙ [sin θto 2 minus βj5 ] ∙ (θt2 minus θto 2) + c5
b ∙ [R5 ∙ θ 5 minus R1 ∙
θ 1 + Lt2 ∙ [sin θto 2 minus βj5 ] ∙ (θ t2)] (319)
Tprime = Ti minus To (320)
Since the applied torques on the tensioner pulleys Q3 and Q4 are zero the static equilibrium
equation of the pulleys show that the adjacent spans of each tensioner pulley are equal to each
other Hence equations (321) and (322) are denoted as follows
Tt1 = T2 = T3 (321)
Tt2 = T4 = T5 (322)
Equations (310a) (310b) and (314) through to (322) are substituted into the EOMs described
in equations (36) to (39) and (311) to (313) The newly formed equations can be arranged
and written in matrix form as shown in equations (323) through to (328) The general
coordinate matrix 120521 and its first and second derivatives are shown in the EOM below
119816 ∙ 120521 + 119810 ∙ 120521 + 119818 ∙ 120521 = 119824 (323)
Modeling of B-ISG 38
The inertia matrix I includes the inertia of each rigid body in its diagonal elements The
damping matrix C includes variables 119888119894119887 the damping of the ith belt span 119877119894 its radius 120573119895119894 its
angle 119871119905119894 the ith tensioner arm‟s length 120579119905119900119894 its initial pivot angle and 119888119905 and 119888119905119894 the ith
tensioner arm viscous damping constants Stiffness matrix K contains 119896119894119887 the ith belt span
stiffness and 119896119905 and 119896119905119894 the ith tensioner arm stiffness constants and akin to the damping
matrix the variables 119877119894 119871119905119894 120579119905119900119894 and 120573119895119894 The belt span stiffness is computed in equation
(326b) where 119870119887 represents the belt cord stiffness 119896119887 is the belt factor obtained from
experimental data 120573119895119894 is the angle of orientation for the span between the jth and ith pulleys and
ϕi is the belt wrap angle on the ith pulley
Modeling of B-ISG 39
119816 =
I1 0 0 0 0 0 00 I2 0 0 0 0 00 0 I3 0 0 0 00 0 0 I4 0 0 00 0 0 0 I5 0 00 0 0 0 0 It1 00 0 0 0 0 0 It2
(324)
119810 =
c1
b ∙ R12 + c5
b ∙ R12 + c1 minusc1
b ∙ R1 ∙ R2 0 0 minusc5b ∙ R1 ∙ R5 0 c5
b ∙ R1 ∙ Lt2 ∙ sin θto 2 minus βj5
minusc1b ∙ R1 ∙ R2 c2
b ∙ R22 + c1
b ∙ R22 + c2 minusc2
b ∙ R2 ∙ R3 0 0 c2b ∙ R2 ∙ Lt1 ∙ sin θto 1 minus βj2 0
0 minusc2b ∙ R2 ∙ R3 c3
b ∙ R32 + c2
b ∙ R32 + c3 minusc3
b ∙ R3 ∙ R4 0 C36 0
0 0 minusc3b ∙ R3 ∙ R4 c4
b ∙ R42 + c3
b ∙ R42 + c4 minusc4
b ∙ R4 ∙ R5 minusc3b ∙ R4 ∙ Lt1 ∙ sin θto 1 minus βj3 c4
b ∙ R4 ∙ Lt2 ∙ sin θto 2 minus βj4
minusc5b ∙ R1 ∙ R5 0 0 minusc4
b ∙ R4 ∙ R5 c5b ∙ R5
2 + c4b ∙ R5
2 + c5 0 C57
0 0 0 0 0 ct +ct1 minusct
0 0 0 0 0 minusct ct +ct1
(325a)
C36 = 1198773 ∙ 1198711199051 ∙ [1198883119887 ∙ 119904119894119899 1205791199051199001 minus 1205731198953 minus 1198882
119887 ∙ 119904119894119899 1205791199051199001 minus 1205731198952 ] (325b)
C57 = 1198775 ∙ 1198711199052 ∙ [1198885119887 ∙ 119904119894119899 1205791199051199002 minus 1205731198955 minus 1198884
119887 ∙ 119904119894119899 1205791199051199002 minus 1205731198954 ] (325c)
Modeling of B-ISG 40
119818 =
k1
b ∙ R12 + k5
b ∙ R12 minusk1
b ∙ R1 ∙ R2 0 0 minusk5b ∙ R1 ∙ R5 0 k5
b ∙ R1 ∙ Lt2 ∙ sin θto 2 minus βj5
minusk1b ∙ R1 ∙ R2 k2
b ∙ R22 + k1
b ∙ R22 minusk2
b ∙ R2 ∙ R3 0 0 k2b ∙ R2 ∙ Lt1 ∙ sin θto 2 minus βj2 0
0 minusk2b ∙ R2 ∙ R3 k3
b ∙ R32 + k2
b ∙ R32 minusk3
b ∙ R3 ∙ R4 0 R3 ∙ Lt1 ∙ [k3b ∙ sin θto 1 minus βj3 minus k2
b ∙ sin θto 1 minus βj2 ] 0
0 0 minusk3b ∙ R3 ∙ R4 k4
b ∙ R42 + k3
b ∙ R42 minusk4
b ∙ R4 ∙ R5 minusk3b ∙ R4 ∙ Lt1 ∙ sin θto 1 minus βj3 k4
b ∙ R4 ∙ Lt2 ∙ sin θto 2 minus βj4
minusk5b ∙ R1 ∙ R5 0 0 minusk4
b ∙ R4 ∙ R5 k5b ∙ R5
2 + k4b ∙ R5
2 0 R5 ∙ Lt2 ∙ [k5b ∙ sin θto 2 minus βj5 minus k4
b ∙ sin θto 2 minus βj4 ]
0 0 0 0 0 kt +kt1 minuskt
0 0 0 0 0 minuskt kt +kt1
(326a)
k119894b =
Kb
Li + kb ∙ Ri ∙ϕi+1
2 + Ri ∙ϕi
2
(326b)
120521 =
θ1
θ2
θ3
θ4
θ5
partθt1
partθt2
(327)
119824 =
Q1
Q2
Q3
Q4
Q5
Qt1
Qt2
(328)
Modeling of B-ISG 41
3412 Modeling of Phase Change
The phase change from the crankshaft pulley being the driving pulley to the ISG pulley being the
driving pulley is described through a conditional equality based on a set of Boolean conditions
When the crankshaft is driving the rows and the columns of the EOM are swapped such that the
new order for rows and columns is 1 (crankshaft pulley) 4 (ISG pulley) 2 (air conditioner
pulley) 3 (tensioner 1 pulley) 5 (tensioner 2 pulley) 6 (tensioner arm 1) and 7 (tensioner arm 2)
When the ISG is driving the order is the same except that the second row and second column
terms relating to the ISG pulley become the first row and first column while the crankshaft
pulley terms (previously in the first row and first column) become the second row and second
column Hence the order for all rows and columns of the matrices making up the EOM in
equation (322) switches between 1423567 (when the crankshaft pulley is driving) and
4123567 (when the ISG pulley is driving) For example in the crankshaft driving and ISG
driving phases the general coordinate matrix and the inertia matrix become the following
120521119940 =
1205791
1205794
1205792
1205793
1205795
1205971205791199051
1205971205791199052
and 120521119938 =
1205794
1205791
1205792
1205793
1205795
1205971205791199051
1205971205791199052
(329a amp b)
119816119940 =
I1 0 0 0 0 0 00 I4 0 0 0 0 00 0 I2 0 0 0 00 0 0 I3 0 0 00 0 0 0 I5 0 00 0 0 0 0 It1 00 0 0 0 0 0 It2
and 119816119938 =
I4 0 0 0 0 0 00 I1 0 0 0 0 00 0 I2 0 0 0 00 0 0 I3 0 0 00 0 0 0 I5 0 00 0 0 0 0 It1 00 0 0 0 0 0 It2
(329c amp d)
Modeling of B-ISG 42
where subscripts c and a denote the crankshaft pulley driving phase and the ISG pulley driving
phase respectively
The condition for phase change is based on the engine speed n in units of rpm Equation (330)
demonstrates the phase change
H(n) = 1 119899 ge 750 (Crankshaft driving phase)0 119899 lt 750 (ISG driving phase)
(330)
When the crankshaft pulley is the driving pulley the ISG pulley becomes the driven pulley and
following suit when the ISG pulley is the driving pulley the crankshaft pulley becomes the
driven pulley These modes of operation mean that the system will predict two different sets of
natural frequencies and mode shapes Using a Boolean condition to allow for a swap between
the first and second rows as well as between the first and second columns of the EOM matrices
I C and K allows for a continuous plot of the dynamic response to be plotted for the ISG pulley
throughout its driving and driven phases as well as for that of the crankshaft pulley
3413 Natural Frequencies Mode Shapes and Dynamic Responses
Assuming the system undergoes simple harmonic motion its matrix of natural frequencies 120596119899
and modeshapes are found by solving the eigenvalue problem shown in equation (331a)
ωn ∙ 119816120784120784 minus 11981822 ∙ 120495m = 120782 (331a)
The displacement amplitude Θm is denoted implicitly in equation (331d)
Modeling of B-ISG 43
120521119846 = θ2 θ3 θ5 θ6 partθt1 partθt2 T for H n = 1 (331b)
120521119846 = θ1 θ3 θ5 θ6 partθt1 partθt2 T for H n = 0 (331c)
θm = 120495119846 ∙ sin(ω ∙ t) (331d)
I2 and K22 are submatrices of I and K respectively meaning the first row and column of each of
the original matrices are removed The eigenvalue problem is reached by considering the
undamped and unforced motion of the system Furthermore the dynamic responses are found by
knowing that the torque requirements in the matrixndash Qm for the driven pulleys and the tensioner
arms are zero in the dynamic case which signifies a response of the system to an input solely
from the driving pulley
I1 120782120782 119816120784120784
θ 1120521 119846
+ C11 119810120783120784119810120784120783 119810120784120784
θ 1120521 119846
+ K11 119818120783120784
119818120784120783 119818120784120784 θ1
120521119846 =
QCS ISG
119824119846 (332)
1
In the case of equation (331) θm is the submatrix identified in equations (331b) through to
(331d) Therein θ1 denotes the general coordinate for the driving pulley so that in the case the
phase change function H(n) is equal to zero θ1 becomes θ4 and the order of the rows and
columns for the remaining matrices correspond to the value of H(n) as mentioned earlier in
section 3412 For simple harmonic motion the motion of the driven pulleys are described as
1 The driving torque 119876119862119878119868119878119866 denotes the crankshaft torque 119876119862119878 when the crankshaft pulley is driving or the ISG
torque 119876119868119878119866 when the ISG pulley is in its driving function
Modeling of B-ISG 44
θm = 120495119846 ∙ sin(ω ∙ t) (333)
The dynamic response of the system to an input from the driving pulley under the assumption of
sinusoidal motion is expressed in equation (334)
120495119846 = [(119818120784120784 minusω2 ∙ 119816120784120784) + 119895ω ∙ 119810120784120784]minus1 ∙ (119818120784120783 + 119895ω ∙ 119810120784120783) ∙ Θ1 (334)
3414 Crankshaft Pulley Driving Torque Acceleration and Displacement
Subsequently the crankshaft pulley driving torque acceleration and displacement are firstly
discussed It is assumed in the thesis research for the purpose of modeling that the engine
serving the crankshaft is of the four cylinder type The input torque provided by a four-cylinder
engine is assumed to be dominated by two torque pulses per revolution of the crankshaft which
is represented by the factor of 2 on the steady component of the angular velocity in equation
(335) The torque requirement of the crankshaft pulley when it is the driving pulley is
Qc = qc ∙ sin(2 ∙ ωcs ∙ t) (335)
The amplitude of the required crankshaft torque qc is expressed in equation (336) and is
derived from equation (332)
qc = K11 minus ω2 ∙ I1 + 119895 ∙ ω ∙ C11 ∙ Θ1 + (119818120783120784 + 119895 ∙ ω ∙ 119810120783120784) ∙ 120495119846 (336)
Modeling of B-ISG 45
The angular frequency for the system in radians per second (rads) ω when the crankshaft
pulley is driving can be found as a function of the engine speed in rotations per minute (rpm) n
and by taking into account the double pulse per crankshaft revolution
ω = 2 ∙ ωcs = 4 ∙ π ∙ n
60
(337)
The system is considered when the amplitude of the crankshaft‟s angular acceleration is assumed
to be constant and equal to 650 rads2 during the crankshaft pulley driving phase The amplitude
of the excitation angular input from the engine is shown in equation (339b) and is found as a
result of (338)
θ 1CS = 650 ∙ sin(ω ∙ t) (338)
θ1CS = minus650
ω2sin(ω ∙ t)
(339a) where
Θ1CS = minus650
ω2
(339b)
Modeling of B-ISG 46
3415 ISG Pulley Driving Torque Acceleration and Displacement
Secondly the torque acceleration and the displacement of the ISG pulley in its driving phase is
discussed The torque for the ISG when it is in its driving function is assumed constant Ratings
for the ISG are taken from experiments performed by researchers Wezenbeek et al [21] on an
Energen 5 High Output Belt-alternator-starter (BAS) unit from Delphi The 12-Volt BAS which
can also be called a B-ISG was reported to have a maximum allowable speed of 18000 rpm [21]
As well it was noted that the ISG pulley was sized appropriately and the engine speed was
limited to ensure that an over-speed condition of the ISG pulley would not occur [21] The stall
torque rating for the Energen ISG was reported to be 48 Nm at the electric machine shaft [21]
The formula for the torque of a permanent magnet DC motor for any given speed (equation
(340)) is used to approximate the torque of the ISG in its driving mode[52]
QISG = Ts minus (N ∙ Ts divide NF) (340)2
Knowing the stall torque (the torque at 0 rpm) Ts and the maximum rpm of the motor when it is
not under load NF allows for the torque produced 119876119868119878119866 to be found for a given motor speed N
Experimental data from Litens Automotive Group [53] shows that for engine fire-up upon ISG
re-start the crankshaft must go from 0 rpm to an idle speed of approximately 750 rpm The
pulley installed on the ISG shaft in the case of the thesis research has a diameter of 6820 mm
(DISG) while that of the crankshaft has a diameter of 20065 mm (DCS) which makes the
2 The equation for the required driving torque for the ISG pulley may also be computed from the formula shown in
(336) Figure 315 for the driving torque of the ISG pulley shows that (336) and (340) produce similar results for
the required driving torque See Figure 315 for comparison of these results
Modeling of B-ISG 47
crankshaft to ISG pulley ratio approximately 2941 This ratio is used to determine the ISG
speed in equation (341)
nISG = nCS ∙DCS
DISG
(341)
For a crankshaft speed of 750 rpm the required ISG speed nISG is found from equation (341) to
be approximately 220656 rpm Thus the ISG torque during start-up is found from equation
(340) where N is equated to the value of nISG NF is assumed to be 18000 rpm and the stall
torque is allotted the value of 48 Nm The result is a required torque of approximately 42 Nm
for the ISG The acceleration of the ISG pulley is found by taking into account the torque
developed by the rotor and the polar moment of inertia of the pulley [54]
A1ISG = θ 1ISG = QISG IISG (342)
In torsional motion the function for angular displacement of input excitation is sinusoidal since
the electric motor is assumed to be resonating As a result of constant angular acceleration the
angular displacement of the ISG pulley in its driving mode is found in equation 343
θ1ISG = Θ1ISG ∙ sin(ωISG ∙ t) (343)
Knowing that acceleration is the second derivative of the displacement the amplitude of
displacement is solved subsequently [55]
Modeling of B-ISG 48
θ 1ISG = minusωISG2 ∙ Θ
1ISG ∙ sin(ωISG ∙ t) (344)
θ 1ISG = minusωISG2 ∙ Θ
1ISG
(345a)
Θ1ISG =minusQISG IISG
ωISG2
(345b)
In this case the angular frequency for the system 120596 is equivalent to 120596119868119878119866 that is the angular
frequency of the ISG pulley which can be expressed as a function of its speed in rpm
ω = ωISG =2 ∙ π ∙ nISG
60
(346a)
or in terms of the crankshaft rpm by substituting equation (341) into (346a)
ω =2 ∙ π
60∙ nCS ∙
DCS
DISG
(346b)
3416 Tensioner Arms Dynamic Torques
The dynamic torque for the tensioner arms are shown in equations (347) and (348)
Qt1 = kt + kt1 + 119895 ∙ ω ∙ (ct + ct1) ∙ (Θt1 ∙ Θ1) (347)
Modeling of B-ISG 49
Qt2 = kt + kt2 + 119895 ∙ ω ∙ (ct + ct2) ∙ (Θt2 ∙ Θ1) (348)
3417 Dynamic Belt Span Tensions
Furthermore the dynamic belt span tensions are derived from equation (314) and described in
matrix form in equations (349) and (350)
119827prime = 119895 ∙ ω ∙ 119827119836 + 119827119844 ∙ 120495119847 (349)
where
120495119847 = Θ1
120495119846 (350)
342 Static Model of the B-ISG System
It is fitting to pursue the derivation of the static model from the system using the dynamic EOM
For the system under static conditions equations (314) and (323) simplify to equations (351)
and (352) respectively
119827prime = 119827119844 ∙ 120521 (351)
119824 = 119818 ∙ 120521 (352)
Modeling of B-ISG 50
As noted in other chapters the focus of the B-ISG tensioner investigation especially for the
parametric and optimization studies in the subsequent chapters is to determine its effect on the
static belt span tensions Therein equations (351) and (352) are used to derive the expressions
for static tension in each belt span 119931prime is the tension solely due to deflection of the belt span
Equation (320) demonstrates the relationship between the tension due to belt response and the
initial tension also known as pre-tension The static tension 119931 is found by summing the initial
tension 1198790 with the expression for the dynamic tension shown in equations (315) through to
(319) and by substituting the expressions for the rigid bodies‟ displacements from equation
(352) and the relationship shown in equation (320) into equation (351)
119827 = 119827119844 ∙ (119818minus120783 ∙ 119824) + T0 (353)3
35 Simulations
The methods used to develop the geometric dynamic and static models of the Twin Tensioner B-
ISG system in the previous sections of this chapter were verified using the software FEAD The
input data for a single tensioner B-ISG system was entered into FEAD [51] to simulate the
crankshaft driving phase alone since the ISG phase is inapplicable in the FEAD [51] software
FEAD‟s [51] results agreed with those found in the simulation of the single tensioner system‟s
geometric model and EOMs in MathCAD software Furthermore the geometric simulation
3 For the purposes of the static tension the original order for the rows and columns of the stiffness matrix K and the
torque matrix Q are maintained as depicted in (326) and (328) In performing the inverse of K and its
multiplication with Q the first row and first column (in the case of the K matrix) are removed in the crankshaft
driving case whereas the fourth row and fourth column are removed in the ISG driving case Then the product for
the displacement120637 resulting from (119922minus120783 ∙ 119928) has a zero added to serve as the first element of the column matrix in
the crankshaft driving case or as the fourth element in the ISG driving case This is shown in detail in Appendix
C3 of MathCAD scripts
Modeling of B-ISG 51
results for both of the twin and single tensioner B-ISG systems were found to be in agreement as
well
351 Geometric Analysis
The initial coordinate inputs for the centre points of the five pulleys and the Twin Tensioner
pivot point are described as Cartesian coordinates and shown in Table 32 which also includes
the diameters for the pulleys
Table 32 Coordinate Points for Pulley Centres and Twin Tensioner Pivot [56]
Rigid Body Diameter [mm] Cartesian Coordinate [Xi Yi] [mm]
1Crankshaft Pulley 20065 [00]
2 Air Conditioner Pulley 10349 [224 -6395]
3 Tensioner Pulley 1 7240 [292761 87]
4 ISG Pulley 6820 [24759 16664]
5 Tensioner Pulley 2 7240 [12057 9193]
6 Tensioner Arm Pivot --- [201384 62516]
The geometric results for the B-ISG system are shown in Table 33
Table 33 Geometric Results of B-ISG System with Twin Tensioner
Pulley Forward
Connection Point
Backward
Connection Point
Wrap
Angle
ϕi (deg)
Angle of
Belt Span
βji (deg)
Length of
Belt Span
Li (mm)
1 Crankshaft [-6818-100093] [453889475] 202996 356103 227828
2 Air
Conditioning [275299-5717] [220484 -115575] 101425 277528 14064
3 Tensioner 1 [25887599735] [256873 82257] 28126 69403 58658
4 ISG [218374184225] [27951154644] 169554 58956 129513
5 Tensioner 2 [10419659645] [15158673262] 8585 333107 65949
Total Length of Belt (mm) 1243
Modeling of B-ISG 52
352 Dynamic Analysis
The dynamic results for the system include the natural frequencies mode shapes driven pulley
and tensioner arm responses the required torque for each driving pulley the dynamic torque for
each tensioner arm and the dynamic tension for each belt span These results for the model were
computed in equations (331a) through to (331d) for natural frequencies and mode shapes in
equation (334) for the driven pulley and tensioner arm responses in equation (336) for the
crankshaft pulley driving torque in equation (340) for the ISG pulley driving torque in
equations (347) and (348) for the tensioner arm torques and lastly in equation (349) for the
dynamic tension of each belt span Figures 36 through to 323 respectively display these
results The EOM simulations can also be contrasted with those of a similar system being a B-
ISG serpentine belt drive that is equipped with a single tensioner arm and single tensioner pulley
which interacts only in the span bridging the ISG and crankshaft pulleys The EOM for a B-ISG
with a single tensioner is presented in Appendix B
It is assumed for the sake of the dynamic and static computations that the system
does not have an isolator present on any pulley
has negligible rotational damping of the pulley shafts
has negligible belt span damping and that this damping does not differ amongst
spans (ie c1b = ∙∙∙ = ci
b = 0)
has quasi-static belt stretch where its belt experiences purely elastic deformation
has fixed axes for the pulley centres and tensioner pivot
has only one accessory pulley being modeled as an air conditioner pulley and
Modeling of B-ISG 53
has a rotational belt response that is decoupled from the transverse response of the
belt
The input parameter values of the dynamic (and static) computations as influenced by the above
assumptions for the present system equipped with a Twin Tensioner are shown in Table 34
Table 34 Data for Input Parameters used in Dynamic and Static Computations [56]
Rigid Body Data
Pulley Inertia
[kg∙mm2]
Damping
[N∙m∙srad]
Stiffness
[N∙mrad]
Required
Torque
[Nm]
Crankshaft 10 000 0 0 4
Air Conditioner 2 230 0 0 2
Tensioner 1 300 1x10-4
0 0
ISG 3000 0 0 5
Tensioner 2 300 1x10-4
0 0
Tensioner Arm 1 1500 1000 10314 0
Tensioner Arm 2 1500 1000 16502 0
Tensioner Arm
couple 1000 20626
Belt Data
Initial belt tension [N] To 300
Belt cord stiffness [Nmmmm] Kb 120 00000
Belt phase angle at zero frequency [deg] φ0deg 000
Belt transition frequency [Hz] ft 000
Belt maximum phase angle [deg] φmax 000
Belt factor [magnitude] kb 0500
Belt cord density [kgm3] ρ 1000
Belt cord cross-sectional area [mm2] A 693
Modeling of B-ISG 54
These values are for the driven cases for the ISG and crankshaft pulleys respectively In the
driving case for either pulley the inertia of the rigid body is defined as 1 kg∙mm2 and the driving
torque is determined in equations (335) and (340) for the crankshaft and ISG pulleys
respectively
It is noted that because of the belt data for the phase angle at zero frequency the transition
frequency and the maximum phase angle are all zero and hence the belt damping is assumed to
be constant between frequencies These three values are typically used to generate a phase angle
versus frequency curve for the belt where the phase angle is dependent on the frequency The
curve defined by equation (354) is normally symmetric with the lowest phase angle achieved at
0 Hz and the highest phase angle achieved at the prescribed transition frequency f The belt
damping would then be found by solving for cb in the following equation
tanφ = cb ∙ 2 ∙ π ∙ f (354)
Nevertheless the assumption for constant damping between frequencies is also in harmony with
the remaining assumptions which assume damping of the belt spans to be negligible and
constant between belt spans
3521 Natural Frequency and Mode Shape
The set of natural frequencies and mode shapes for the system are shown in Figures 36 and 37
under the cases of the ISG pulley driving and the crankshaft pulley driving The forcing
frequency for the system differs for each case due to the change in driving pulley Modeic and
Modeia denote the ith rigid body according to the numbering convention used in Figure 32 in
the crankshaft and ISG driving cases respectively
Modeling of B-ISG 55
Natural Frequency ωn [Hz]
Crankshaft Pulley ΔΘ4
Air Conditioner Pulley ΔΘ
Tensioner Pulley 1 ΔΘ
Tensioner Pulley 2 ΔΘ
Tensioner Arm 1 ΔΘ
Tensioner Arm 2 ΔΘ
Figure 36a ISG Driving Case Natural Frequency of System and Mode Shapes for Responsive
Rigid Bodies
Figure 36b ISG Driving Case First Mode Responses
4 ΔΘ signifies the term amplitude of angular displacement for the indicated rigid body
Modeling of B-ISG 56
Figure 36c ISG Driving Case Second Mode Responses
Natural Frequency ωn [Hz]
ISG Pulley ΔΘ5
Air Conditioner Pulley ΔΘ
Tensioner Pulley 1 ΔΘ
Tensioner Pulley 2 ΔΘ
Tensioner Arm 1 ΔΘ
Tensioner Arm 2 ΔΘ
Figure 37a Crankshaft Driving Case Natural Frequency of System and Mode Shapes for
Responsive Rigid Bodies
5 ΔΘ signifies the term amplitude of angular displacement for the indicated rigid body
Modeling of B-ISG 57
Figure 37b Crankshaft Driving Case First Mode Responses
Figure 37c Crankshaft Driving Case Second Mode Responses
Modeling of B-ISG 58
3522 Dynamic Response
The dynamic response specifically the magnitude of angular displacement for each rigid body is
plotted in Figures 38 through to 314 as a function of the crankshaft pulley speed n This is
fitting to the analysis since the crankshaft pulley‟s rpm decides the mode of operation for the
system in particular it determines whether the crankshaft pulley or ISG pulley is the driving
pulley
Figure 38 Crankshaft Pulley Dynamic Response (for crankshaft driven case)
Figure 39 ISG Pulley Dynamic Response (for ISG driven case)
Modeling of B-ISG 59
Figure 310 Air Conditioner Pulley Dynamic Response
Figure 311 Tensioner Pulley 1 Dynamic Response
Crankshaft Driving Phase ISG
Driving
Phase
Crankshaft Driving Phase ISG
Driving
Phase
Modeling of B-ISG 60
Figure 312 Tensioner Pulley 2 Dynamic Response
Figure 313 Tensioner Arm 1 Dynamic Response
Crankshaft Driving Phase ISG
Driving
Phase
Crankshaft Driving Phase ISG
Driving Phase
Modeling of B-ISG 61
Figure 314 Tensioner Arm 2 Dynamic Response
3523 ISG Pulley and Crankshaft Pulley Torque Requirement
Figures 315 and 316 respectively showcase the required torques for the ISG pulley in its driving
mode and the crankshaft pulley in its driving mode
Figure 315 Required Driving Torque for the ISG Pulley
Crankshaft Driving Phase ISG
Driving Phase
Modeling of B-ISG 62
Figure 315 shows two plots for the required driving torque of the ISG pulley The dashed line
labeled as Q(n) simulates the application of equation (340) which models the ISG torque as a
permanent magnet DC motor The additional solid line labeled as qamod uses the formula in
equation (336) which determines the load torque of the driving pulley based on the pulley
responses Figure 315 provides a comparison of the results
Figure 316 Required Driving Torque for the Crankshaft Pulley
3524 Tensioner Arms Torque Requirements
The torque for the tensioner arms are shown in Figures 317 and 318
Modeling of B-ISG 63
Figure 317 Dynamic Torque for Tensioner Arm 1
Figure 318 Dynamic Torque for Tensioner Arm 2
3525 Dynamic Belt Span Tension
The dynamic tensions for the belt spans are shown in Figures 319 through to 323 The values
plotted represent the magnitude of the dynamic tension
Crankshaft Driving Phase ISG
Driving Phase
Crankshaft Driving Phase ISG
Driving
Phase
Modeling of B-ISG 64
Figure 319 Span 1 (between crankshaft and air conditioner) Dynamic Belt Span Tension
Figure 320 Span 2 (between air conditioner and tensioner 1) Dynamic Belt Span Tension
Crankshaft Driving Phase ISG
Driving Phase
Crankshaft Driving Phase ISG
Driving
Phase
Modeling of B-ISG 65
Figure 321 Span 3 (between tensioner 1 and ISG) Dynamic Belt Span Tension
Figure 322 Span 4 (between ISG and tensioner 2) Dynamic Belt Span Tension
Crankshaft Driving Phase ISG
Driving
Phase
Crankshaft Driving Phase ISG
Driving Phase
Modeling of B-ISG 66
Figure 323 Span 5 (between tensioner 2 and crankshaft) Dynamic Belt Span Tension
The dynamic results for the system serve to show the conditions of the system for a set of input
parameters The following chapter targets the focus of the thesis research by analyzing the affect
of changing the input parameters on the static conditions of the system It is the static results that
are the focus of the thesis and is thus analyzed in Chapters 4 and 5 in the parametric and
optimization studies respectively The dynamic analysis has been used to complete the picture of
the system‟s state under set values for input parameters
353 Static Analysis
Before looking at the static results for the system under study in brevity the static results for a
B-ISG serpentine belt drive with a single tensioner are presented In this theoretical system the
tensioner arm and tensioner pulley that interacts with the span between the air conditioner and
ISG pulleys of the original system are removed as shown in Figure 324
Crankshaft Driving Phase ISG
Driving
Phase
Modeling of B-ISG 67
Figure 324 B-ISG Serpentine Belt Drive with Single Tensioner
The complete static model as well as the dynamic model for the system in Figure 324 is found
in Appendix B The results of the static tension for each belt span of the single tensioner system
when the crankshaft is driving and the ISG is driving are shown in Table 35
Table 35 Static Solution for Belt Span Tensions in Crankshaft and ISG Driving Cases for a B-
ISG Serpentine Belt Drive with a Single Tensioner
Belt Span between Pulleys Crankshaft Driving Case [N] ISG Driving Case [N]
Crankshaft ndash Air Conditioner 481239 -361076
Air Conditioner ndash ISG 442588 -399727
ISG ndash Tensioner 29596 316721
Tensioner ndash Crankshaft 29596 316721
The tensions in Table 35 are computed with an initial tension of 300N This value for pre-
tension allows the spans in the case that the crankshaft pulley is driving to be suitably tensioned
Modeling of B-ISG 68
Whereas in the case of the ISG pulley driving the first and second spans are excessively slack
Therein an additional pretension of approximately 400N would be required which would raise
the highest tension span to over 700N This leads to the motivation of the thesis researchndash to
reduce the static belt tensions when the ISG is driving As mentioned in Chapter 1 these
tensions should be minimized to prolong belt life preferably within the range of 600 to 800N
As well it is desirable to minimize the amount of pretension exerted on the belt The current
design uses a pre-tension of 300N The above results would lead to a required pre-tension of
more than 700N to keep all spans of the belt suitably in tension (well above 0N) in order to allow
the belt to exhibit high performance in power transmission and come near to the safe threshold
This is the rationale for investigating a Twin Tensioner configuration shown in Figure 32 for
the B-ISG serpentine belt drive under study For the theoretical system with a Twin Tensioner
the following static results in Table 36 are achieved
Table 36 Static Solution for Belt Span Tensions in Crankshaft and ISG Driving Cases for a B-
ISG Serpentine Belt Drive with a Twin Tensioner
Belt Span between Pulleys Crankshaft Driving Case [N] ISG Driving Case [N]
Crankshaft ndash Air Conditioner 465848 -284152
Air Conditioner ndash Tensioner 1 427197 -322803
Tensioner 1 ndash ISG 427197 -322803
ISG ndash Tensioner 2 28057 393645
Tensioner 2 ndash Crankshaft 28057 393645
The results in Table 36 show that the span following the ISG in the case between the Tensioner
1 and ISG pulleys is less slack than in the former single tensioner set-up However there
remains an excessive amount of pre-tension required to keep all spans suitably tensioned
Modeling of B-ISG 69
36 Summary
The simulation of the model for the B-ISG system with the Twin Tensioner shows that the mode
shapes of the rigid bodies within the system (Figures 36a to 37c) are greater in magnitude when
the ISG pulley is driving than when the crankshaft pulley is driving The dynamic responses of
the system as shown in Figures 38 and 310 to 314 is small for the crankshaft pulley and are
negligible for the remaining driven bodies when the ISG is driving For the crankshaft driving
phase there is greater dynamic response for the driven rigid bodies of the system including for
that of the ISG pulley
As the engine speed increases the torque requirement for the ISG was found to vary between
approximately 41Nm and 54Nm (before dropping steeply to approximately 3Nm at an engine
speed of about 720rpm) when modeled after equation (336) or between approximately 48Nm
and 34Nm when modeled after equation (340) In contrast the torque for the crankshaft peaks
at approximately 92Nm and 52Nm at an approximate engine speed of 1450rpm and 5000rpm
respectively The dynamic torque of the first tensioner arm was shown to peak at approximately
15Nm at the transition engine speed 750rpm and again at approximately 15Nm at an
approximate engine speed of about 1450rpm A small peak of about 3Nm was also seen at an
engine speed of 5000rpm Similarly for the second tensioner arm a torque peak of
approximately 20Nm was seen at 750rpm and 1450rpm and a smaller peak of about 8Nm was
seen at an engine speed of 5000rpm
The trend for the dynamic tensions is that the peaks are highest in the ISG driving portion of the
B-ISG operation in most cases and in a few cases they are seen to be close in magnitude to that
Modeling of B-ISG 70
of the highest peaks in the crankshaft driving portion The dynamic tension for the first belt span
peaked at approximately 780Nm 830Nm and 500Nm at engine speeds of 750rpm 1450rpm
5000rpm respectively For the dynamic tension of the second belt span peaks of approximately
1250Nm 675Nm and 760Nm were seen at the same respective engine speeds for the 3 peaks of
the former span At these same engine speeds the third belt span exhibited tension peaks at
approximately 1400Nm 650Nm and 890Nm The tension peaks of the fourth span were
approximately 165Nm 150Nm and 100Nm at engine speeds 750rpm 1450rpm and 5000rpm
The fifth span experienced peaks of approximately 165Nm 170Nm and 120Nm at the same
respective engine speeds of the fourth span
The simulation results for the static tension of the B-ISG system with the Twin Tensioner reveal
that taut spans of the crankshaft driving case are lower in the ISG driving case The largest
change is an approximate decrease of 750N in spans 1 through 3 while spans 4 and 5 increase
by approximately 113N It can be seen that the spans in highest tension (1 2 and 3) in the
crankshaft driving phase become excessively slack in the ISG driving phase There is a smaller
change between the tension values for the spans in the least tension in the crankshaft driving
phase and their corresponding span in the ISG driving phase
The summary of the simulation results are used as a benchmark for the optimized system shown
in Chapter 5 The static tension simulation results are investigated through a parametric study of
the Twin Tensioner system in Chapter 4 The optimization of the system is then based on the
selected design variables from the outcome of Chapter 4
71
CHAPTER 4 PARAMETRIC ANALYSIS OF A B-ISG
TWIN TENSIONER
41 Introduction
The parameters for the proposed Twin Tensioner for a Belt-driven Integrated Starter-generator
(B-ISG) system are investigated through a parametric analysis This analysis seeks to understand
how changing one parameter influences the static belt span tensions for the system Since the
thesis research focuses on the design of a tensioning mechanism to support static tension only
the parameters specific to the actual Twin Tensioner and applicable to the static case were
considered The parameters pertaining to accessory pulley properties such as radii or various
belt properties such as belt span stiffness are not considered In the analyses a single parameter
is varied over a prescribed range while all other parameters are held constant The pivot point
described by Cartesian Coordinates [X6Y6] for the tensioner arms is held constant in all cases
42 Methodology
The parametric study method applies to the general case of a function evaluated over changes in
one of its dependent variables The methodology is illustrated for the B-ISG system‟s function
for static tension which is evaluated for each change in one of its Twin Tensioner‟s parameters
The original data used for the system is based on sample vehicle data provided by Litens [56]
Table 41 provides the initial data for the parameters as well as the incremental change and
maxima and minima limits The increment Δi for the ith parameter is chosen arbitrarily Limits
for each parameter have been chosen to be plus or minus sixty percent of its initial value
Parametric Analysis 72
Table 41 Initial Values Increments and Ranges for Parameters of Twin Tensioner
Parameter Name Initial Value Increment (+- Δi) Minimum
value Maximum value
Coupled Spring
Stiffness kt
20626
N∙mrad 1238 N∙mrad 8250 N∙mrad 33002 N∙mrad
Tensioner Arm 1
Stiffness kt1
10314
N∙mrad 0619 N∙mrad 4126 N∙mrad 16502 N∙mrad
Tensioner Arm 2
Stiffness kt2
16502
N∙mrad 0990 N∙mrad 6601 N∙mrad 26403 N∙mrad
Tensioner Pulley 1
Diameter D3 007240 m 4344 ∙ 10
-3 m 00290 m 0116 m
Tensioner Pulley 2
Diameter D5 007240 m 4344 ∙ 10
-3 m 00290 m 0116 m
Tensioner Pulley 1
Initial Coordinates
[0292761
0087] m See Figure 41 for region of possible tensioner pulley
coordinates Tensioner Pulley 2
Initial Coordinates
[012057
009193] m
The mesh of possible points for the centre coordinates of tensioner pulley 1 and tensioner pulley
2 are designated as Region 1 and Region 2 respectively in Figures 41a and 41b
Figure 41a Regions 1 and 2 and their Associated Guidelines for Coordinates of Tensioner
Pulleys 1 amp 2
CS
AC
ISG
Ten 1
Ten 11
Region II
Region I
Parametric Analysis 73
Figure 41b Regions 1 and 2 in Cartesian Space
The selection for the minimum and maximum tensioner pulley centre coordinates and their
increments are not selected arbitrarily or without derivation as the other tensioner parameters
The coordinates for the pulley centres are identified using Intergraph‟s SmartSketch software a
graphing suite in MathCAD to model the regions of space The following descriptions are used
to describe the possible positions for the tensioner pulleys
Tensioner pulleys are situated such that they are exterior to the interior space created by
the serpentine belt thus they sit bdquooutside‟ the belt loop
The highest point on the tensioner pulley does not exceed the tangent line connecting the
upper hemispheres of the pulleys on either side of it
The tensioner pulleys may not overlap any other pulley
Parametric Analysis 74
Boundaries for regions described as Region 1 in span 2 and 3 and Region 2 in span 4
and 5 is selected based on the above criteria and their lower boundaries are selected
arbitrarily
These criteria were used to define the equation for each boundary line and leads to a set of
Boolean conditions that relate the x-coordinate and y-coordinate for each Cartesian pair The
density for the mesh of points in each region is arbitrarily selected as 101 x-points and 101 y-
points in each space for the purposes of the parametric analysis The outline of this method is
described in the MATLAB scripts contained in Appendix D
The results of the parametric analysis are shown for the slackest and tautest spans in each driving
case As was demonstrated in the literature review the tautest span immediately precedes the
driving pulley and the slackest span immediately follows the driving pulley in the direction of
the belt motion Thus in the case for the crankshaft driving the tautest span is in the first span
and the slackest span is in the fifth span Whereas in the ISG driving case the tautest span is in
the fourth span and the slackest span is in the third span Hence the parametric figures in this
chapter display only the tautest and slackest span values for both driving cases so as to describe
the maximum and minimum values for tension present in the given belt
43 Results amp Discussion
431 Influence of Tensioner Arm Stiffness on Static Tension
The parametric analysis begins with changing the stiffness value for the coil spring coupled
between tensioner arms 1 and 2 This stiffness value kt is changed over a range from sixty
percent less than its initial value kt0 to sixty percent more than its original value as shown in
Parametric Analysis 75
Table 41 The results of the static tension are shown in Figure 42 for the tautest and slackest
spans for both the crankshaft and ISG driving cases
Figure 42 Parametric Analysis for Coupled Stiffness Arm Constant kt (N∙mrad)
As kt increases in the crankshaft driving phase for the B-ISG system the highest tension
decreases from 4691N to 4646N while the lowest tension decreases from 2838N to 2793N
In the ISG driving phase the highest tension increases from 378N to 3998N and the lowest
tension increases from -3384N to -3167N Thus a change of approximately -45N is found in
the crankshaft driving case and approximately +22N is found in the ISG driving case for both the
tautest and slackest spans
Parametric Analysis 76
The second parameter analyzed is the stiffness value for tensioner arm 1 The results of this are
shown in Figure 43
Figure 43 Parametric Analysis for Stiffness of Arm 1 kt1 (N∙mrad)
In Figure 43 as kt1 increases an increase from 4628N to 4681N is observed for the tension of
the tautest span when the crankshaft is driving which is a change of +53N The same value for
net change is found in the slackest span for the same driving condition whose tension increases
from 2775N to 2828N For the case when the B-ISG system is in the ISG driving phase the
change is larger a value of -261N for the tautest span that changes from 4088N to 3827N and
for the slackest span that changes from -3077N to -3338N
Parametric Analysis 77
The change in static tension for the spans as the stiffness of arm 2 varies is demonstrated in
Figure 44
Figure 44 Parametric Analysis for Stiffness of Arm 2 kt2 (N∙mrad)
In this case it is observed that as kt2 increases the tautest span for the B-ISG system in the
crankshaft driving case decreases from 4675N to 4643N as well as the slackest span which
decreases from 2822N to 279N which is an overall change of -32N for both spans Whereas in
the ISG driving case a more noticeable change is once again found a difference of +144N
This is a result of the tautest span increasing from 3863N to 4007N and the slackest span
increasing from -3301N to -3157N
Parametric Analysis 78
432 Influence of Tensioner Pulley Diameter on Static Tension
The change in the diameter of tensioner pulley 1 D3 and its effect on static tension is shown in
Figure 45
Figure 45 Parametric Analysis for Pulley 1 Diameter D3 (m)
The change in the tautest and slackest spans for the B-ISG system‟s crankshaft driving case is
from 3248N to 425N and from 1395N to 240N respectively Peaks are seen at 4799N and
2946N for the respective spans This is a change of approximately +100N and a maximum
change of 1551N for both spans For the ISG driving case the tautest and slackest spans
decrease from 1083N to 6158N and 367N to -1006N Global minimums of 3246N and -391N
for the respective spans are seen This nets a change of approximately -467N and a maximum
change of approximately -759N
Parametric Analysis 79
The effect of changing the diameter of tensioner pulley 2 on the static tension is examined in
Figure 46
Figure 46 Parametric Analysis for Pulley 2 Diameter D5 (m)
The tautest and slackest spans in the crankshaft driving mode of the belt undergo a change from
4583N to 4721N and from 273N to 2869N respectively Therein as D5 increases the trend is
that for both spans there is an increase in tension of approximately 14N Contrastingly the spans
experience a decrease in the ISG driving case as D5 increases The tension of the tautest span
goes from 4296N to 3635N and that of the slackest span goes from -2866N to -3529N This
equals a decrease of approximately 66N for both spans
Parametric Analysis 80
433 Influence of Tensioner Pulley 1 Coordinates on Static Tension
The influence of the coordinates of tensioner pulley 1 on the value of tension in the tautest span
for the B-ISG system‟s crankshaft driving case is demonstrated in Figure 47
Figure 47 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Tautest Span
Tension in Crankshaft Driving Case
The region shown in Figure 47 corresponds to region 1 which is the realm of the positions for
tensioner pulley 1 The possible pulley coordinates in this case are represented by the non-blue
area reaching to the perimeter of the plot It is evident in the darkest red region of the plot
where the y-coordinate is between approximately 0m and 0075m and the x-coordinate is
(N)
Parametric Analysis 81
between approximately 026m and 031m that the highest value of tension is experienced in the
tautest span for the crankshaft driving case The range of tension for Region 1 in the tautest span
when the crankshaft is driving is between a maximum of approximately 500N and a minimum of
approximately 300N This equals an overall difference of 200N in tension for the tautest span by
moving the position of pulley 1 The lowest values for tension are obtained when the pulley
coordinates are approximately -0025m to 015m for the y-coordinate and approximately 031m
to 032m for the x-coordinate which corresponds to the yellow region An area of low tension is
also seen in the area where the y-coordinate is approximately 0m and the x-coordinate is
approximately between 026m and 027m
The changes in tension for the slackest span under the condition of the crankshaft pulley being
the driving pulley are shown in Figure 48
Parametric Analysis 82
Figure 48 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Slackest Span
Tension in Crankshaft Driving Case
Once again the possible coordinate points for tensioner pulley 1 in the B-ISG system are
represented by the non-blue region For the slackest span in the crankshaft driving case it is seen
that the lowest tension is approximately 125N while the highest tension is approximately 325N
This is an overall change of 200N that is achieved in the region The highest values are achieved
in the space where the y-coordinates are approximately 0m to 0075m and the x-coordinate
ranges from 026m to 031m which corresponds to the deep red region The lowest tension
values are achieved in the space where the y-coordinate ranges from approximately -0025m to
015m and the x-coordinate ranges from 031m to 032m which corresponds to the light blue-
green region of the plot The area containing a y-coordinate of approximately 0m and x-
(N)
Parametric Analysis 83
coordinates that are approximately between 026m and 027m also show minimum tension for
the slack span The regions of the x-y coordinates for the maximum and minimum tensions are
alike to the tautest span in Region 1 for the crankshaft driving case as well as was seen in Figure
47
The tension for the tautest span in the case that the ISG is driving in the B-ISG system is found
in Figure 49
Figure 49 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Tautest Span
Tension in ISG Driving Case
(N)
Parametric Analysis 84
Region 1 is represented by the coordinate values shown in the non-dark blue space of the plot in
Figure 49 The tautest span in the case of the ISG driving experiences a range of tension values
in Region 1 from 200N up to 1100N equaling a difference of 900N The minimum tension
values are achieved in the medium to light blue region This includes y-coordinates of
approximately 0m to 0075m and x-coordinates of approximately 026m to 03m The
maximum tension values are in the darkest red area inclusive of y-coordinates -0025m to 015m
and x-coordinates 031m to 032m in addition to y-coordinate of approximately 0m and x-
coordinates of approximately 026m to 027m It can be observed that aforementioned regions
for minimum and maximum tensions in Figure 49 are reverse to those seen in Figures 47 and
48 for the crankshaft driving case
The change in tension for the slackest span of the B-ISG system when it is driven by the ISG is
shown in Figure 410
Parametric Analysis 85
Figure 410 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Slackest Span
Tension in ISG Driving Case
Figure 410 exhibits the realm of possible points for tensioner pulley 1 for the case of the ISG
driving in the non-yellow-green area The minimum tension values are achieved in the darkest
blue area where the minimum tension is approximately -500N This area corresponds to y-
coordinates from approximately 0m to 005m and x-coordinates from approximately 026m to
03m The area of a maximum tension is approximately 400N and corresponds to the darkest red
area inclusive of y-coordinates -0025m to 015m and x-coordinates 031m to 032m as well as
the coordinates for y equaling approximately 0m and for x equaling approximately 026m to
027m The difference between maximum and minimum tensions in this case is approximately
900N It is noticed once again that the space of x- and y-coordinates containing the maximum
(N)
Parametric Analysis 86
tension is in the similar location to that of the described space for minimum tension in the
crankshaft driving case in Figure 47 and 48
434 Influence of Tensioner Pulley 2 Coordinates on Static Tension
The influence of pulley 2 coordinates on the tension value for the tautest span when the
crankshaft is driving the B-ISG system is shown in Figure 411 and is represented by the values
corresponding to the non-blue area
Figure 411 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Tautest Span
Tension in Crankshaft Driving Case
In Figure 411 the possible coordinates are contained within Region 2 The maximum tension
value is approximately 500N and is found in the darkest red space including approximately y-
(N)
Parametric Analysis 87
coordinates 004m to 014m and x-coordinates 0025m to 0175m and also y-coordinates 013m
to 02m corresponding to the x-coordinate at 0175m A minimum tension value of
approximately 350N is found in the yellow space and includes approximately y-coordinates
008m to 018m and x-coordinates 016m to 02m The difference in tension values is 150N
The analysis of the change in coordinates for tension pulley 2 on the value for tension in the
slackest span is shown in Figure 412 in the non-blue region
Figure 412 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Slackest Span
Tension in Crankshaft Driving Case
The value of 325N is the highest tension for the slack span in the crankshaft driving case of the
B-ISG system and is found in the deep-red region where the y-coordinates are between
(N)
Parametric Analysis 88
approximately 004m and 013m and the x-coordinates are approximately between 0025m and
016m as well as where y is between 013m and 02m and x is approximately 0175m The
lowest tension value for the slack span is approximately 150N and is found in the green-blue
space where y-coordinates are between approximately 01m and 022m and the x-coordinates
are between approximately 016m and 021m The overall difference in minimum and maximum
tension values is 175N The spaces for the maximum and minimum tension values are similar in
location to that found in Figure 411 for the tautest span in the crankshaft driving case
Figure 413 provides the theoretical data for the tension values of the tautest span as the position
of the B-ISG system‟s tensioner pulley 2 changes in the ISG driving case Possible points are in
the space of values which correspond to the non-dark-blue region in Figure 413
Parametric Analysis 89
Figure 413 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Tautest Span
Tension in ISG Driving Case
In Figure 413 the region for high tension reaches a value of approximately 950N and the region
for low tension reaches approximately 250N This equals a difference of 700N between
maximum and minimum tension values for the tautest span in the B-ISG system‟s ISG driving
case The coordinate points within the space that maximum tension is reached is in the dark red
region and includes y-coordinates from approximately 008m to 022m and x-coordinates from
approximately 016m to 021m The coordinate points within the space that minimum tension is
reached is in the blue-green region and includes y-coordinates from approximately 004m to
013m and the corresponding x-coordinates from approximately 0025m to 015m An additional
small region of minimum tension is seen in the area where the x-coordinate is approximately
(N)
Parametric Analysis 90
0175m and the y-coordinates are approximately between 013m and 02m The location for the
area of pulley centre points that achieve maximum and minimum tension values is approximately
located in the reverse positions on the plot when compared to that of the case for the crankshaft
driving in Figures 411 and 412 Therein the trend seen for pulley coordinates for the second
tensioner pulley follows suit with that of the first tensioner pulley which is that the area of points
for maximum tension in the crankshaft driving case becomes the approximate area of points for
minimum tension in the ISG driving case and vice versa
In Figure 414 the results of the parametric analysis on the coordinates of the second tensioner
pulley and its effect on the slackest span‟s tension in the ISG driving case is shown Similar to
earlier figures the non-dark yellow region represents Region 2 that contains the possible points
for the pulley‟s Cartesian coordinates
Parametric Analysis 91
Figure 414 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Slackest
Span Tension in ISG Driving Case
Figure 414 demonstrates a difference of approximately 725N between the highest and lowest
tension values for the slackest span of the B-ISG system in the ISG driving case The highest
tension values are approximately 225N The area of points that allow the second tension pulley
to achieve maximum tension in the belt span includes y-coordinates from approximately 01m to
022m and the corresponding x-coordinates from approximately 016m to 021m This
corresponds to the darkest red region in Figure 414 The coordinate values where the lowest
tension being approximately -500N is achieved include y-coordinate values from
approximately 004m to 013m and the corresponding x-coordinates from approximately 0025m
to 015m corresponding to the darkest blue region A dark blue region of lowest tension is also
(N)
Parametric Analysis 92
seen in the area where y is approximately between 013m and 02m and the x-coordinate is
approximately 0175m The regions for maximum and minimum tension values are observed to
be similar to those found in Figure 413 and alike to Figure 413 to be in reverse to those found
in Figure 411 and 412 for the tautest and slackest spans in the crankshaft driving case So as for
the changes in tensioner pulley 2 coordinates the areas for minimum tension in Region 2 of the
ISG driving case are similar to the areas for maximum tension in Region 2 of the crankshaft
driving case and vice versa for the maximum tension of the ISG driving case and the minimum
tension for the crankshaft driving case in Region 2
44 Conclusion
Overall the trend in the plots of Figures 47 48 411 and 412 indicate in the crankshaft driving
portion that the B-ISG system‟s belt span tensions experience the following effect
Minimum tension for the tautest span is achieved when tensioner pulley 1 centre
coordinates are located closer to the right side boundary and bottom left boundary of
Region 1 or when tensioner pulley 2 centre coordinates are within the upper right space
(near to the ISG pulley) and the space closer to the top boundary of Region 2
Maximum tension for the slackest span is achieved when the first tensioner pulley‟s
coordinates are located in the mid space and near to the bottom boundary of Region 1
and when the second tensioner pulley‟s coordinates are located near to the bottom left
boundary of Region 2 which is the boundary nearest to the crankshaft pulley
Parametric Analysis 93
The trend for minimizing the tautest span signifies that the tension for the slackest span is also
minimized at the same time As well maximizing the slackest span signifies that the tension for
the tautest span is also maximized at the same time too
The trend for the B-ISG system‟s ISG driving case as can be seen in Figures 49 410 413 and
414 is approximately in reverse to that of the crankshaft driving case for the system Wherein
points corresponding to minimum tension in Regions 1 and 2 in the ISG case are approximately
the same as points corresponding to maximum tension in the Regions for the crankshaft case and
vice versa for the ISG cases‟ areas of maximum tension
Minimum tension for the tautest span is present when the first tensioner pulley‟s
coordinates are near to mid to lower boundary of Region 1 and when the second
tensioner pulley‟s coordinates are close to the bottom left boundary of Region 2 which
is the furthest boundary from the ISG pulley and closest to the crankshaft pulley
Maximum tension for the slackest span is achieved when the first tensioner pulley is
located close to the right boundary of Region 1 and when the second tensioner pulley is
located near the right boundary and towards the top right boundary of Region 2
It is observed in Figures 47 to 414 and alike to Figures 42 to 46 the tautest and slackest
spans decrease or increase together Thus it can be assumed that the tension values in these
spans and likely the remaining spans outside of the tautest and slackest spans follow suit
Therein when parameters are changed to minimize one belt span‟s tension the remaining spans
will also have their tension values reduced Figures 42 through to 413 showed this clearly
where the overall change in the tension of the tautest and slackest spans changed by
Parametric Analysis 94
approximately the same values for each separate case of the crankshaft driving and the ISG
driving in the B-ISG system
Design variables are selected in the following chapter from the parameters that have been
analyzed in the present chapter The influence of changing parameters on the static tension
values for the various spans is further explored through an optimization study of the static belt
tension for the B-ISG system equipped with a Twin Tensioner in the following chapter Chapter
5
95
CHAPTER 5 OPTIMIZATION OF A B-ISG TENSIONER
The objective of the optimization analysis is to minimize the absolute magnitude of the static
tension in the ISG-operating mode of the serpentine belt drive The optimization seeks to
optimize the performance of the proposed Twin Tensioner design by using its properties as the
design variables for the objective function The optimization task begins with the selection of
these design variables for the objective function and then the selection of an optimization
method The results of the optimization will be compared with the results of the analytical
model for the static system and with the parametric analysis‟ data
51 Optimization Problem
511 Selection of Design Variables
The optimal system corresponds to the properties of the Twin Tensioner that result in minimized
magnitudes of static tension for the various belt spans Therein the design variables for the
optimization procedure are selected from amongst the Twin Tensioner‟s properties In the
parametric analysis of Chapter 4 the tensioner properties presented included
coupled stiffness kt
tensioner arm 1 stiffness kt1
tensioner arm 2 stiffness kt2
tensioner pulley 1 diameter D3
tensioner pulley 2 diameter D5
tensioner pulley 1 initial coordinates [X3Y3] and
Optimization 96
tensioner pulley 2 initial coordinates [X5Y5]
It was observed in the former chapter that perturbations of the stiffness and geometric parameters
caused a change between the lowest and highest values for the static tension especially in the
case of perturbations in the geometric parameters diameter and coordinates Table 51
summarizes the observed changes in the belt span tensions corresponding to the Twin Tensioner
parameters‟ maximum and minimum values
Table 51 Summary of Parametric Analysis Data for Twin Tensioner Properties
Parameter Symbol
Original Tensions in TautSlack Span (Crankshaft
Mode) [N]
Tension at
Min | Max Parameter6 for
Crankshaft Mode [N]
Percent Change from Original for
Min | Max Tensions []
Original Tension in TautSlack Span (ISG Mode)
[N]
Tension at
Min | Max Parameter Value in ISG Mode [N]
Percent Change from Original Tension for
Min | Max Tensions []
kt
465848 (taut) 4691 4646 07 -03 393645 (taut) 378 3998 -40 16
28057 (slack) 2838 2793 12 -05 -322803 (slack) -3384 -3167 -48 19
kt1
465848 (taut) 4628 4681 -07 05 393645 (taut) 4088 3827 38 -28
28057 (slack) 2775 2828 -11 08 -322803 (slack) -3077 -3338 47 -34
kt2
465848 (taut) 4675 4643 04 -03 393645 (taut) 3863 4007 -19 18
28057 (slack) 2822 279 06 -06 -322803 (slack) -3301 -3157 -23 22
D3 465848 (taut) 3248 425 -303 -88 393645 (taut) 1083 6158 1751 564
28057 (slack) 1395 240 -503 -145 -322803 (slack) 367 -1006 2137 688
D5 465848 (taut) 4583 4721 -16 13 393645 (taut) 4296 3635 91 -77
28057 (slack) 273 2869 -27 23 -322803 (slack) -2866 -3529 112 -93
[X3Y3] 465848 (taut) 300 500 -356 73 393645 (taut) 200 1100 -492 1794
28057 (slack) 125 325 -554 158 -322803 (slack) -500 400 -549 2239
6 The values for the tension for each of the taut and slack spans provided correspond to the minimum and maximum
values of the parameter listed in each case such that the columns of identical colour correspond to each other For
the coordinate parameters the minimum and maximum parameter value is inadmissible The tension values in these
cases are simply the minimum and maximum tension values achieved by the coordinate parameter listed
Optimization 97
[X5Y5] 465848 (taut) 350 500 -249 73 393645 (taut) 250 950 -365 1413
28057 (slack) 150 325 -465 158 -322803 (slack) -500 225 -549 1697
The results of the parametric analyses for the Twin Tensioner parameters show that there is a
noticeable percent change between the initial tensions and the tensions corresponding to each of
the minima and maxima parameter values or in the case of the coordinates between the
minimum and maximum tensions for the spans Thus the parametric data does not encourage
exclusion of any of the tensioner parameters from being selected as a design variable As a
theoretical experiment the optimization procedure seeks to find feasible physical solutions
Hence economic criteria are considered in the selection of the design variables from among the
Twin Tensioner‟s parameters Of the tensioner properties it is found that the diameter of the
tensioner pulleys has the largest impact on cost Adding mass to a tensioner pulley as a result of
increasing the diameter and consequently its inertia increases the cost of material Material cost
is most significant in the manufacture process of pulleys as their manufacturing is largely
automated [4] Furthermore varying the structure of a pulley requires retooling which also
increases the cost to manufacture As such the tensioner pulley diameters D3 and D5 are
excluded from being selected as design variables The remaining tensioner properties the
stiffness parameters and the initial coordinates of the pulley centres are selected as the design
variables for the objective function of the optimization process
512 Objective Function amp Constraints
In order to deal with two objective functions for a taut span and a slack span a weighted
approach was employed This emerges from the results of Chapter 3 for the static model and
Chapter 4 for the parametric study for the static system which show that a high tension span and
Optimization 98
a highly slack span exist in the ISG-driving phase of the B-ISG system Therein the first
objective function of equation (51a) is described as equaling fifty percent of the absolute tension
value of the tautest span and fifty percent of the absolute tension value of the slackest span for
the case of the ISG driving only The second objective function uses a non-weighted approach
and is described as the absolute tension of the slackest span when the ISG is driving A non-
weighted approach is motivated by the phenomenon of a fixed difference that is seen between
the slackest and tautest spans of the optimal designs found in the weighted optimization
simulations Equations (51a) through to (51c) display the objective functions
The limits for the design variables are expanded from those used in the parametric analysis for
the non-coordinate parameters kt kt1 and kt2 so that they are permitted to vary from
approximately 0 to approximately 200 of the initial value for each parameter kt0 kt10 and kt20
respectively In the case of the coordinate parameters [X3Y3] and [X5Y5] the x- and y-
coordinates are permitted to vary within the spaces Region 1 and Region 2 respectively which
were prescribed in Chapter 4 Figure 41a and 41b
Aside from the design variables design constraints on the system include the requirement for
static stability of the Twin Tensioner An optimal solution for the B-ISG system must achieve
the goal of the objective function which is to minimize the absolute tensions in the system
However for an optimal solution to be feasible the movement of the tensioner arm must remain
within an appropriate threshold In practice an automotive tensioner arm for the belt
transmission may be considered stable if its movement remains within a 10 degree range of
Optimization 99
motion [4] As such the angle of displacement for tensioner arms 1 and 2 are designated by θ t1
and θt2 respectively in the listed constraints
The optimization task is described in equations 51a to 52 Variables a through to g represent
scalar limits for the x-coordinate for corresponding ranges of the y-coordinate
Minimize 119879119908119890119894119892 119893119905119890119889 = 05 ∙ 119879119905119886119906119905 + 05 ∙ 119879119904119897119886119888119896
or119879119899119900119899 minus119908119890119894119892 119893119905119890119889 = 119879119904119897119886119888119896
(51a)
where
119879119905119886119906119905 = 119891119905119886119906119905 119896119905 1198961199051 1198961199052 1198833 1198843 1198835 1198845 (51b)
119879119904119897119886119888119896 = 119891119904119897119886119888119896 (119896119905 1198961199051 1198961199052 1198833 119884311988351198845) (51c)
Subject to
(1198961199050 minus 1 ∙ 1198961199050) le 119896119905 le (1198961199050 + 11198961199050)(11989611990510 minus 1 ∙ 11989611990510) le 1198961199051 le (11989611990510 + 111989611990510)(11989611990520 minus 1 ∙ 11989611990520) le 1198961199052 le (11989611990520 + 111989611990520)
119886 le 1198833 le 119888
1198931 1198833 le 1198843 le 1198933 1198833 119891119900119903 119886 le 1198833 lt 119887
1198932 1198833 le 1198843 le 1198933 1198833 119891119900119903 119887 le 1198833 le 119888119889 le 1198835 le 119892
1198934 1198835 le 1198845 le 1198937 1198835 for 119889 le 1198835 lt 1198901198935(1198835) le 1198845 le 1198937(1198835) for 119890 le 1198835 lt 119891
1198936 1198835 le 1198845 le 1198937 1198833 for 119891 le 1198833 le 119892 1205791199051 le 10deg 1205791199052 le 10deg
(52)
The functions for the taut and slack spans represent the fourth and third span respectively in the
case of the ISG driving The equations for the tensions of the aforementioned spans are shown
in equation 51a to 51c and are derived from equation 353 The constraints for the
optimization are described in equation 52
Optimization 100
52 Optimization Method
A twofold approach was used in the optimization method A global search alone and then a
hybrid search comprising of a global search and a local search The Genetic Algorithm is used
as the global search method and a Quadratic Sequential Programming algorithm is used for the
local search method
521 Genetic Algorithm
Genetic Algorithm (GA) is a part of the growing genre of evolutionary algorithms [57] The
optimization approach differs from classical search approaches by its ease of use and global
perspective [57] GA mimics biological evolution theory by using a ldquocross-over of heritable
information random mutation and selection on the basis of fitness between generationsrdquo [58] to
form a robust search algorithm that requires minimal problem information [57] The parameter
sets are represented as sample points modeled as bdquochromosomes‟ or data strings that are
measured against how well they allow the model to achieve the optimization task [58] GA is
stochastic which means that its algorithm uses random choices to generate subsequent sampling
points rather than using a set rule to generate the following sample This avoids the pitfall of
gradient-based techniques that may focus on local maxima or minima and end-up neglecting
regions containing higher peaks or lower valleys [57] Furthermore due to the randomness of
the GA‟s search strategy it is able to search a population (a region of possible parameter sets)
faster than other optimization techniques The GA approach is viewed as a universal
optimization approach while many classical methods viewed to be efficient for one optimization
problem may be seen as inefficient for others However because GA is a probabilistic algorithm
its solution for the objective function may only be near to a global optimum As such the current
Optimization 101
state of stochastic or global optimization methods has been to refine results of the GA with a
local search and optimization procedure
522 Hybrid Optimization Algorithm
In order to enhance the result of the objective function found by the GA a Hybrid optimization
function is implemented in MATLAB software The Hybrid optimization function combines a
global search GA with a local search Sequential Quadratic Programming (SQP) The hybrid
process refines the value of the objective function found through GA by using the final set of
points found by the algorithm as the initial point of the SQP algorithm The GA function
determines the region containing a global optimum and then the SQP algorithm uses a gradient
based technique to find a solution closer to the global optimum The MATLAB algorithm a
constrained minimization function known as fmincon uses an SQP method that approximates the
Hessian for the Lagrangian function (ie the second derivatives of the Lagrangian) by way of a
quasi-Newton approach to generate a quadratic program (QP) sub-problem [59] The solution
for the QP provides the search direction of the line search procedure used when each iteration is
performed [59]
53 Results and Discussion
531 Parameter Settings amp Stopping Criteria for Simulations
The parameter settings for the optimization procedure included setting the stall time limit to
200s This is the interval of time the GA is given to find an improvement in the value of the
objective function This is an increase from MATLAB‟s default of 20s Increasing the stall time
limit allows for the optimization search to consistently converge without being limited by time
Optimization 102
The population size used in finding the optimal solution is set at 100 This value was chosen
after varying the population size between 50 and 2000 showed no change in the value of the
objective function The max number of generations is set at 100 The time limit for the
algorithm is set at infinite The limiting factor serving as the stopping condition for the
optimization search was the function tolerance which is set at 1x10-6
This allows the program
to run until the ratio of the change in the objective function over the stall generations is less than
the value for function tolerance The stall generation setting is defined as the number of
generations since the last improvement of the objective function value and is limited to 50
532 Optimization Simulations
The results of the genetic algorithm optimization simulations performed in MATLAB are shown
in the following tables Table 52a and Table 52b
Table 52a GA Optimization Results for Twin Tensioner Parameters and Objective Function
Trial
No
Genetic Algorithm Optimization Method
Objective
Function
Value [N]
kt [N∙mrad]
kt1 [N∙mrad]
kt2 [N∙mrad]
[X3Y3]
[m] [X5Y5]
[m]
1 3582241 314069 204844 165020 [02928 00703] [01618 01036]
2 3582241 103646 205284 198901 [03009 00607] [01283 00809]
3 3582241 126431 204740 43549 [03010 00631] [01311 01147]
4 3582241 180285 206230 254870 [03095 00865] [01080 01675]
5 3582241 74757 204559 189077 [03084 00617] [01265 00718]
Original Design 20626 10314 16502 [02928 0087] [01206 00919]
Optimization 103
Table 52b Computations for Tensions and Angles from GA Optimization Results
Trial No
Genetic Algorithm Optimization Method
Slackest Tension [N] Tautest Tension [N] Tensioner Arm 1
Displacement [deg]
Tensioner Arm 2
Displacement [deg]
T3 T4 Θt1 Θt2
1 -1572307 5592176 -00025 -49748
2 -4054309 3110174 -00002 -20213
3 -3930858 3233624 -00004 -38370
4 -1309751 5854731 -00010 -49525
5 -4092446 3072036 -00000 -17703
Original Design -322803 393645 16410 -4571
For each trial above the GA function required 4 generations each consisting of 20 900 function
evaluations before finding no change in the optimal objective function value according to
stopping conditions
The results of the Hybrid function optimization are provided in Tables 53a and 53b below
Table 53a Hybrid Optimization Results for Twin Tensioner Parameters and Objective Function
Trial
No
Hybrid Optimization Method
Objective
Function
Value [N]
of
Function
Evals ( of
Iterations)
kt [N∙mrad]
kt1 [N∙mrad]
kt2 [N∙mrad]
[X3Y3]
[m] [X5Y5]
[m]
1 3582241 16 (1) 16065 205846 229494 [02780 00581] [01679 01288]
2 3582241 20 (1) 249227 205635 25218 [02901 00634] [01559 00870]
3 3582241 16 (1) 297295 204878 320479 [02962 00702] [01336 01447]
4 3582241 53 (1) 241433 204262 229683 [02912 00647] [00047 01465]
Optimization 104
5 3582241 21 (1) 379096 205548 188888 [02973 00703] [01206 01376]
Original Design 20626 10314 16502 [02928 0087] [01206 00919]
Table 53b Computations for Tensions and Angles from Hybrid Optimization Results
Trial No
Hybrid Algorithm Optimization Method
Slackest Tension [N] Tautest Tension [N] Tensioner Arm 1
Displacement [deg]
Tensioner Arm 2
Displacement [deg]
T3 T4 Θt1 Θt2
1 -2584641 4579841 -02430 67549
2 -3708747 3455736 -00023 -41068
3 -1707181 5457302 -00099 -43944
4 -269178 6895304 00006 -25366
5 -2982335 4182148 -00003 -41134
Original Design -322803 393645 16410 -4571
In Table 53a it can be seen that iterations of 16 20 21 or 53 were required for the local search
algorithm following the GA to find an optimal solution Once again the GA function
computed 4 generations which consisted of approximately 20 900 function evaluations before
securing an optimum solution
The simulation results of the non-weighted hybrid optimization approach are shown in tables
54a and 54b below
Optimization 105
Table 54a Non-Weighted Optimization Results for Twin Tensioner Parameters and Objective
Function
Trial
No
Objective
Function
Value [N]
of
Function
Evals ( of
Iterations)
kt [N∙mrad]
kt1 [N∙mrad]
kt2 [N∙mrad]
[X3Y3]
[m] [X5Y5]
[m]
Genetic Algorithm Optimization Method
1 33509e
-004 20900 (4) 321799 75530 212653 [02860 00602] [01082 01858]
Hybrid Optimization Method
1 73214e
-011 381 (13) 234881 14730 323358 [02952 00688] [00048 01466]
Original Design 20626 10314 16502 [02928 0087] [01206 00919]
Table 54b Computations for Tensions and Angles from Non-Weighted Optimizations
Trial No Slackest Tension [N] Tautest Tension [N]
Tensioner Arm 1
Displacement [deg]
Tensioner Arm 2
Displacement [deg]
T3 T4 Θt1 Θt2
Genetic Algorithm Optimization Method
1 -00003 7164479 -00588 -06213
Hybrid Optimization Method
1 -00000 7164482 15543 -16254
Original Design -322803 393645 16410 -4571
The weighted optimization data of Table 54a shows that the GA simulation again used 4
generations containing 20 900 function evaluations to conduct a global search for the optimal
system While the weighted Hybrid optimization used 13 iterations (consisting of 381 function
evaluations) after its GA run which used the same number of generations and function
evaluations as the GA run in the non-weighted simulations Tables 54a and 54b show the data
Optimization 106
for only one trial for each of the non-weighted GA and hybrid methods since only a single
optimal point exists in this case
533 Discussion
The optimal design from each search method can be selected based on the least amount of
additional pre-tension (corresponding to the largest magnitude of negative tension) that would
need to be added to the system This is in harmony with the goal of the optimization of the B-
ISG system as stated earlier to minimize the static tension for the tautest span and at the same
time minimize the absolute static tension of the slackest span for the ISG driving case As well
the angular displacements corresponding to each trial‟s results show that the Twin Tensioner is
under static stability Therein the optimal solution may be selected as the design parameters
corresponding to Trial 4 of the GA simulations to Trial 4 of the Hybrid simulations or to either
of the trials for the non-weighted optimization simulations
Given the ability of the Hybrid optimization to refine the results obtained in the GA
optimization the results of Trial 4 of the Hybrid simulations are selected as the most optimal
design from the weighted objective function approaches It is interesting to note that the Hybrid
case for the least slackness in belt span tension corresponds to a significantly larger number of
function evaluations than that of the remaining Hybrid cases This anomaly however does not
invalidate the other Hybrid cases since each still satisfy the design constraints Using the data
for the optimized system in Trial 4 (of the Hybrid optimization) the static tensions for the belt
spans in both of the B-ISG‟s phases of operation are as follows
Optimization 107
Table 55 Weighted Optimization Results for Static Tensions for Optimal B-ISG System with a
Twin Tensioner
Belt Span between Pulleys Crankshaft Driving Case [N] ISG Driving Case [N]
Optimized Original Optimized Original
Crankshaft ndash Air Conditioner 3926599 465848 117333 -284152
Air Conditioner ndash Tensioner 1 3540088 427197 -269178 -322803
Tensioner 1 ndash ISG 3540088 427197 -269178 -322803
ISG ndash Tensioner 2 2073813 28057 6895304 393645
Tensioner 2 ndash Crankshaft 2073813 28057 6895304 393645
Additional Pretension
Required (approximate) + 27000 +322803 + 27000 +322803
In Table 54b it is evident that the non-weighted class of optimization simulations achieves the
least amount of required pre-tension to be added to the system The computed tension results
corresponding to both of the non-weighted GA and Hybrid approaches are approximately
equivalent Therein either of their solution parameters may also be called the most optimal
design The Hybrid solution parameters are selected as the optimal design once again due to the
refinement of the GA output contained in the Hybrid approach and its corresponding belt
tensions are listed in Table 56 below
Optimization 108
Table 56 Non-Weighted Optimization Results for Static Tensions for Optimal B-ISG System
with a Twin Tensioner
Belt Span between Pulleys Crankshaft Driving Case [N] ISG Driving Case [N]
Optimized Original Optimized Original
Crankshaft ndash Air Conditioner 3891862 465848 386511 -284152
Air Conditioner ndash Tensioner 1 3505351 427197 -00000 -322803
Tensioner 1 ndash ISG 3505351 427197 -00000 -322803
ISG ndash Tensioner 2 2039076 28057 7164482 393645
Tensioner 2 ndash Crankshaft 2039076 28057 7164482 393645
Additional Pretension
Required (approximate) + 0000 +322803 + 00000 +322803
The results of the simulation experiments are limited by the following considerations
System equations are coupled so that a fixed difference remains between tautest and
slackest spans
A limited number of simulation trials have been performed
There are multiple optimal design points for the weighted optimization search methods
Remaining tensioner parameters tensioner pulley diameters and their stiffness have not
been included in the design variables for the experiments
The belt factor kb used in the modeling of the system‟s belt has been obtained
experimentally and may be open to further sources of error
Therein the conclusions obtained and interpretations of the simulation data can be limited by the
above noted comments on the optimization experiments
Optimization 109
54 Conclusion
The outcomes the trends in the experimental data and the optimal designs can be concluded
from the optimization simulations The simulation outcomes demonstrate that in all cases the
weighted optimization functions reached an identical value for the objective function whereas
the values reached for the parameters varied widely
The lowest tension values for the tautest and slackest span were achieved in Trial 5 of the GA
optimization approach In reiteration in the presence of slack spans the tension value of the
slackest span must be added to the initial static tension for the belt Therein for the former case
an amount of at least 409N would need to be added to the 300N of pre-tension already applied to
the system (see Table 34) The highest tension values for the spans were achieved in Trial 4 of
the weighted Hybrid optimization approach and in both trials of the non-weighted optimization
approaches In the former the weighted Hybrid trial the tension value achieved in the slackest
span was approximately -27N signifying that only at least 27N would need to be added to the
present pre-tension value for the system The tension of the slackest span in the non-weighted
approach was approximately 0N signifying that the minimum required additional tension is 0N
for the system
The optimized solutions for the tension values in each span show that there is consistently a fixed
difference of 716448N between the tautest and slackest span tension values as seen in Tables
52b 53b and 54b This difference is identical to the difference between the tautest and slackest
spans of the B-ISG system for the original values of the design parameters while in its ISG
mode As well the optimal stiffness parameters for the weighted Hybrid optimization case are
Optimization 110
greater than their original values except for that of the stiffness factor of tensioner arm 1
Likewise for the non-weighted Hybrid optimization case the stiffness parameters are above their
original values without exceptions The coordinates of the optimal solutions are within close
approximation to each other and also both match the regions for moderately low tension in
Regions 1 and 2 of the ISG driving case as is shown in Figures 49 410 413 and 414
The results of the non-weighted Hybrid optimization trial and Trial 4 of the weighted Hybrid
optimization simulations are selected as the most optimal designs for the B-ISG Twin Tensioner
In these designs the Twin Tensioner is shown in Table 53b and 54b to have static stability and
to maintain suitable tensions in the ISG driving phase The tensioner parameters for the optimal
designs allow for one of the lowest amounts of additional pre-tension to be added to the system
out of all the findings from the simulations which were conducted
111
CHAPTER 6 CONCLUSION
61 Summary
The primary aim of the thesis is to reduce the magnitude of static tension in the belt spans of a
Belt-driven Integrated Starter-generator (B-ISG) system by the design and investigation of a
Twin Tensioner It is established that the operating phases of the B-ISG system produced two
cases for static tension outcomes an ISG driving case and a crankshaft driving case The
approach taken in this thesis study includes the derivation of a system model for the geometric
properties as well as for the dynamic and static states of the B-ISG system The static state of a
B-ISG system with a single tensioner mechanism is highlighted for comparison with the static
state of the Twin Tensioner-equipped B-ISG system
It is observed that there is an overall reduction in the magnitudes of the static belt tensions with
the presence of a Twin Tensioner over that of a single tensioner The influences of the geometric
and stiffness properties of the Twin Tensioner affecting the static tensions in the system are
analyzed in a parametric study It is found that there is a notable change in the static tensions
produced as result of perturbations in each respective tensioner property This demonstrates
there are no reasons to not further consider a tensioner property based solely on its influence on
the B-ISG system‟s static tensions The phenomenon of higher magnitudes for static tensions in
the ISG mode of operation over that of the crankshaft mode of operation particularly in
excessively slack spans provides the motivation for optimizing the ISG case alone for static
tension The optimization method uses weighted and non-weighted approaches with genetic
algorithm (GA) and hybrid GA searches The most optimal design has been found to be one in
Conclusion 112
which the magnitude of tension in the excessively slack spans in the ISG driving case are
significantly lower than in that of the original B-ISG Twin Tensioner design
62 Conclusion
The conclusions that can be drawn from the study of a Twin Tensioner for a B-ISG system
include the following
1 The simulations of the dynamic model demonstrate that the mode shapes for the system
are greater in the ISG-phase of operation
2 It was observed in the output of the dynamic responses that the system‟s rigid bodies
experienced larger displacements when the crankshaft was driving over that of the ISG-
driving phase It was also noted that the transition speed marking the phase change from
the ISG driving to the crankshaft driving occurred before the system reached either of its
first natural frequencies
3 The magnitudes for static belt tensions as well as dynamic tensions for a B-ISG system
are consistently greater in its ISG operating phase than in its crankshaft operating phase
4 A Twin Tensioner is able to reduce the magnitudes of the static tension for the belt spans
of a B-ISG system in comparison to when only a single tensioner mechanism is present
5 The parametric study of the B-ISG system demonstrates that the slackest and tautest belt
spans decrease or increase together for either phase of operation
6 Perturbations in the Twin Tensioner‟s geometric and stiffness properties have a
significant influence on the magnitudes of the static tension of the slackest and tautest
belt spans The coordinate position of each pulley in the Twin Tensioner configuration
Conclusion 113
has the greatest influence on the belt span static tensions out of all the tensioner
properties considered
7 Optimization of the B-ISG system shows a fixed difference trend between the static
tension of the slackest and tautest belt spans for the B-ISG system
8 The values of the design variables for the most optimal system are found using a hybrid
algorithm approach The slackest span for the optimal system is significantly less slack
than that of the original design Therein less additional pretension is required to be added
to the system to compensate for slack spans in the ISG-driving phase of operation
63 Recommendation for Future Work
The investigation of the B-ISG Twin Tensioner encourages the following future work
1 The optimization of the B-ISG system with the inclusion of diametric Twin Tensioner
properties would provide a complete picture as to the highest possible performance
outcome that the Twin Tensioner is able to have with respect to the static tensions
achieved in the belt spans
2 A larger number of optimization trials using the genetic algorithm (GA) and hybrid GA
under weighted and other approaches would investigate the scope of optimal designs
available in the Twin Tensioner for the B-ISG system
3 A model of the system without the simplification of constant damping may produce
results that are more representative of realistic operating conditions of the serpentine belt
drive A finite element analysis of the Twin Tensioner B-ISG system may provide more
applicable findings
Conclusion 114
4 Investigation of the transverse motion coupled with the rotational belt motion in an
optimized B-ISG system equipped with a Twin Tensioner may also provide a closer look
at the system under realistic conditions In addition the affect of the Twin Tensioner on
transverse motion can determine whether significant improvements in the magnitudes of
static belt span tensions are still being achieved
5 The recommendation to conduct modal decoupling of the B-ISG system‟s static model is
motivated by the fixed difference trend between the slackest and tautest belt span
tensions shown in Chapter 5 The modal decoupling of the system would allow for its
matrices comprising the equations of motion to be diagonalized and therein to decouple
the system equations Modal analysis would transform the system from physical
coordinates into natural coordinates or modal coordinates which would lead to the
decoupling of system responses
6 An investigation and optimization of the dynamic belt span tensions for a B-ISG system
with a Twin Tensioner would increase understanding of the full impact of a Twin
Tensioner mechanism on the state of the B-ISG system It would be informative to
analyze the mode shapes of the first and second modes as well as the required torques of
the driving pulleys and the resulting torque of each of the tensioner arms The
observation of the dynamic belt span tensions would also give direction as to how
damping of the system may or may not be changed
7 Further comparison with the Twin Tensioner B-ISG system‟s dynamic and static states
including the Twin Tensioner‟s stability in each versus a B-ISG system with a single
tensioner would further demonstrate the improvements or dis-improvements in the Twin
Tensioner‟s performance on a B-ISG system
Conclusion 115
8 The influence of the belt properties on the dynamic and static tensions for a B-ISG
system with a Twin Tensioner can also be investigated This again will show the
evidence of improvements or dis-improvement in the Twin Tensioner‟s performance
within a B-ISG setting
9 Lastly an experimental apparatus of the B-ISG system with a Twin Tensioner can be
designed and constructed Suitable instrumentation can be set-up to measure belt span
tensions (both static and dynamic) belt motion and numerous other system qualities
This would provide extensive guidance as to finding the most appropriate theoretical
model for the system Experimental data would provide a bench mark for evaluating the
theoretical simulation results of the Twin Tensioner-equipped B-ISG system
116
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[14] National Alternative Fuels Training Consortium (NAFTC) (2005 Oct 3) Tech stuff
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[15] Green Car Congress BMW to Apply Start-Stop and Brake Regen to MINIs Up to 60
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[21] PJ Wezenbeek (Zytec Systems Ltd) D G Evans (General Motors Powertrain) D P
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(Delphi Corp) Combustion Assisted Belt-Cranking of a V-8 Engine at 12-Volts SAE
Technical Papers vol 113 pp 396-407 2004 Document no 2004-01-0569
[22] T C Firbank Mechanics of the Belt Drive International Journal of Mechanical
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[23] R L Cassidy S K Fan R S MacDonald and W F Samson Serpentine Extended Life
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[24] A G Ulsoy J E Whitesell and M D Hooven Design of Belt-Tensioner Systems for
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Transactions of the ASME vol 107 pp 282-290 July 1985
[25] R S Beikmann N C Perkins and A G Ulsoy Free Vibration of Serpentine Belt Drive
Systems Journal of Vibrations and Acoustics Transactions of the ASME vol 118 pp
406-413 1996
[26] T C Kraver G W Fan and J J Shah Complex Modal Analysis of a Flat Belt Pulley
System with Belt Damping and Coulomb-Damped Tensioner Journal of Mechanical
Design Transactions of the ASME vol 118 pp 306-311 Jun 1996
[27] R S Beikmann N C Perkins and A G Ulsoy Design and Analysis of Automotive
Serpentine Belt Drive Systems for Steady State Performance Journal of Mechanical
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[28] L Zhang and J W Zu Modal Analysis of Serpentine Belt Drive Systems Journal of
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[29] R Balaji and E M Mockensturm Dynamic analysis of a front-end accessory drive with a
decouplerisolator International Journal of Vehicle Design vol 39 pp 208-231 2005
[30] M Nouri Design Optimization and Active Control of Serpentine Belt Drive Systems with
Two-pulley Tensioners University of Toronto 2005
[31] G J Spicer (Litens Automotive Inc) Tensioner for use in eg belt drive system has
electronic actuator associated with clutch spring for engaging International
WO2005119089-A1 Jun 6 2005 2005
[32] Bando Chemical Industries Ltd and Litens Automotive GmbH About belt-type starter
system Feb 27 2002
[33] H Lemberger and R Jungjohann (Bayerische Motoren Werke AG) Tension device for an
envelope drive of a device especially a belt drive of a starter generator of an internal
combustion engine comprises a support part Europe EP1420192-A2 May 19 2004 2003
[34] P Ahner and M Ackermann (Bosch GMBH) Belt drive especially for internal
combustion engines to drive accessories in an automobile Germany DE19849886-A1
May 11 2000 1998
[35] N Freisinger K Hagemann J Sievert P Struebel and M Treusch (Daimler Chrysler AG)
Belt tensioning device for belt drive between engine and starter generator of motor
vehicle has self-aligning bearing that supports auxiliary unit and provides working force to
tensioners for tensioning belt Germany DE10324268 Dec 16 2004 2003
[36] C R Rogers (Dayco Products LLC) Offset starter generator drive system for a vehicle
engine has a dual arm pivoted tensioner United States US6942589-B2 Feb 8 2005 2002
[37] A Serkh and I Ali (Gates Corp) Internal combustion engine has belt drive system with
tensioner asymmetrically biased in direction tending to cause power transmission belt to be
under tension International WO2003038309-A1 May 8 2003 2002
References 120
[38] P J Mcvicar and C A Thurston (General Motors Corp) Belt alternator starter accessory
drive with dual tensioner United States US20060287146-A1 Dec 21 2006 2005
[39] W Petri and M Bogner (INA Schaeffler KG) Traction drive especially for driving
internal combustion engine units has arrangement for demand regulated setting of tension
consisting of unit with housing with limited rotation and pulley German DE10044645-
A1 Mar 21 2002 2000
[40] M Bogner (INA Schaeffler KG) Belt drive tensioner for a starter-generator of an IC
engine has locking system for locking tensioning element in an engine operating mode
locking system is directly connected to pivot arm follows arm control movements
German DE10159073-A1 Jun 12 2003 2001
[41] R Painta M Bogner and H Graf (INA Schaeffler KG) Traction mechanism drive esp
belt drive has belt tensioning pulley mounted on generator shaft and uncoupled from it via
freewheel to dampen load peaks Europe EP1723350-A1 Nov 22 2006 2005
[42] W Petri (INA Schaeffler KG) Drive unit for a combustion engine having a starter
generator and a belt drive has tensioner with spring and counter hydraulic force Germany
DE10359641-A1 Jul 28 2005 2003
[43] H Stief M Bogner B Hartmann T Kraft and M Schmid (INA Schaeffler KG) Traction
drive especially belt drive for short-duration driving of starter generator has tensioning
device with lever arm deflectable against restoring force and with end stop limiting
deflection travel Europe EP1738093-A1 Jan 3 2007 2005
[44] M Ulm (INA Schaeffler KG DE) Tension unit eg for drive in machine such as
combustion engine has belt or chain drive with wheels turning and connected with starter
generator and unit has two idlers arranged at clamping arm with machine stored by shock
absorber Germany DE102004012395-A1 Sep 29 2005 2004
[45] M Bogner (INA Schaeffler KG) Belt drive for starter motor-generator auxiliary assembly
has limited movement at the starter belt section tensioner roller bringing it into a dead point
position on starting the motor International WO2006108461-A1 Oct 19 2006 2006
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[46] W Guhr (Litens Automotive GMBH) Automotive motor and drive assembly includes
tension device positioned within belt drive system having combination starter United
States US2001007839-A1 Jul 12 2001 2001
[47] K Kuniaki K Masahiko H Kazuyuki I Shuichi and T Masaki (Mitsubishi Jidosha Eng
KK and Mitsubishi Motor Corp) Tension adjustment method of belt for starter generator
in vehicle involves shifting auto-tensioners between lock state and free state to adjust
tension of belt during driving of crank pulley Japan JP2005083514-A Mar 31 2005
2003
[48] Nissan Motor Co Ltd Winding gear for starting engine of hybrid motor vehicle has
tensioner tightening chain while cranking engine and slackens chain after start of engine
provided to span side of chain Japan JP3565040-B2 Sep 15 2004 1998
[49] S Sato and H Hayakawa (NTN Corp) Auto tensioner for ancillary drive belts has
cylinder nut and screw bolt in hydraulic damper mechanism provided in middle of cylinder
acting as start-up rigidity buffer component Japan JP2006189073-A Jul 20 2006 2005
[50] G Vadin-Michaud (Valeo Equip Electrique Moteur) Pulley and belt starting system for a
thermal engine for a motor vehicle Europe EP1658432 May 24 2006 2005
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2005
[52] W E Johns Notes on Motors [Electronic] 2003 [2008 June] Available at
httpwwwgizmologynetmotorshtm
[53] Litens Automotive Group Ltd DC BAS System - Conventional Start Input Profile Nov
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[54] Arnold Magnetic Technologies Corp General Motor Terminology [Electronic] pp 7
[2008 June] Available at httpwwwgrouparnoldcommtcpdfweb_motor_glossarypdf
[55] Douglas W Jones Stepping Motors University of Iowa - Department of Computer
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httpwwwcsuiowaedu~jonesstepphysicshtml
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[56] Litens Automotive Group Ltd (2004 Jan 31) FEAD software input data for test project
[57] K Deb Multi-Objective Optimization using Evolutionary Algorithms Toronto John Wiley
amp Sons Ltd 2001 pp 81-85
[58] P E McSharry (2004 May 11) Department of Engineering Science University of Oxford
[httpwwwengoxacuksamppubsgawbreppdf]
[59] The MathWorks Inc MATLAB vol 750342 (R2007b) Aug 15 2007
123
APPENDIX A
Passive Dual Tensioner Designs from Patent Literature
Figure A1 Proposed design by Bayerische Motoren Werke AG corresponding to patent nos EP1420192-A2 and DE10253450-A1
Source European Patent Office espcenet (publication nos EP1420192-A2 and DE10253450-A1 accessed May 2007) epespacenetcom [33]
Figure A1 label identification 1 ndash tightner 2 ndash belt drive
3 ndash starter generator
4 ndash internal-combustion engine
4‟ ndash crankshaft-lateral drive disk
5 ndash generator housing
6 ndash common axis of rotation
7 ndash featherspring of tiltable clamping arms
8 ndash clamping arm
9 ndash clamping arm
10 11 ndash idlers
12 12‟ ndash Zugtrum 13 13‟ ndash Leertrum
14 ndash carry-hurries 15 ndash generator wave
16 ndash bush
17 ndash absorption mechanism
18 18‟ ndash support arms
19 19‟ ndash auxiliary straight lines
20 ndash pipe
21 ndash torsion bar
22 ndash breaking through
23 ndash featherspring
24 ndash friction disk
25 ndash screw connection 26 ndash Wellscheibe
(European Patent Office May 2007) [33]
Appendix A 124
Figure A2a First of four proposed designs (and its various configurations) by Bosch GMBH
corresponding to patent no WO0026532-A1
Source European Patent Office espcenet (publication no WO0026532-A1 accessed Jun 2007)
epespacenetcom [34]
Figure A2b Second of four proposed designs by Bosch GMBH corresponding to patent no
WO0026532-A1
Source European Patent Office espcenet (publication no WO0026532-A1 accessed Jun 2007) epespacenetcom [34]
Figure A2c Third of four proposed designs (and its various configurations) by Bosch GMBH
corresponding to patent no WO0026532-A1
Source European Patent Office espcenet (publication no WO0026532-A1 accessed Jun 2007)
epespacenetcom [34]
Appendix A 125
Figure A2d Fourth of four proposed designs (and its various configurations) by Bosch GMBH
corresponding to patent no WO0026532-A1
Source European Patent Office espcenet (publication no WO0026532-A1 accessed Jun 2007)
epespacenetcom [34]
Figure A2a through to A2d label identification 10 ndash engine wheel
11 ndash [generator] 13 ndash spring
14 ndash belt
16 17 ndash tensioning pulleys
18 19 ndash springs
20 21 ndash fixed points
25ab ndash carriers of idlers
25c ndash gang bolt
(European Patent Office June 2007) [34]
Figure A3 Proposed design by Daimler Chrysler AG corresponding to patent no DE10324268-A1
Source European Patent Office espcenet (publication no DE10324268-A1 accessed May 2007)
epespacenetcom [35]
Figure A3 label identification
Appendix A 126
10 12 ndash belt pulleys
14 ndash auxiliary unit
16 ndash belt
22-1 22-2 ndash belt tensioners
(European Patent Office May 2007) [35]
Figure A4 Proposed design by Dayco Products LLC corresponding to patent no US6942589-B2
Source European Patent Office espcenet (publication no US6942589-B2 accessed Jun 2007)
epespacenetcom [36]
Figure A4 label identification 12 ndash belt
14 ndash tensioner
16 ndash generator pulley
18 ndash crankshaft pulley
22 ndash slack span 24 ndash tight span
32 34 ndash arms
33 35 ndash pulley
(European Patent Office June 2007) [36]
Appendix A 127
Figure A5 Proposed design by Gates Corp corresponding to patent no WO2003038309-A
Source European Patent Office espcenet (publication no WO2003038309-A accessed Jun 2007)
epespacenetcom [37]
Figure A5 label identification 12 ndash motorgenerator
14 ndash motorgenerator pulley 26 ndash belt tensioner
28 ndash belt tensioner pulley
30 ndash transmission belt
(European Patent Office June 2007) [37]
Figure A6 Proposed design by General Motors Corp corresponding to patent no US20060287146-A1
Source European Patent Office espcenet (publication no US20060287146-A1 accessed Jun 2007)
epespacenetcom [38]
Appendix A 128
Figure A6 label identification 28 ndash tensioner
32 ndash carrier arm
34 ndash secondary carrier arm
46 ndash tensioner pulley
58 ndash secondary tensioner pulley
(European Patent Office June 2007) [38]
Figure A7 Proposed design by INA Schaeffler KG corresponding to patent no DE10044645-A1
Source European Patent Office espcenet (publication no DE10044645-A1 accessed Jun 2007)
epespacenetcom [39]
Figure A7 label identification 2 ndash internal combustion engine
3 ndash traction element
11 ndash housing with limited rotation 12 13 ndash direction changing pulleys
(European Patent Office June 2007) [39]
Appendix A 129
Figure A8a First of three proposed designs by INA Schaeffler KG corresponding to patent no
DE10159073-A1
Source European Patent Office espcenet (publication no DE10159073-A1 accessed May 2007)
epespacenetcom [40]
Figure A8b Second of three proposed designs by INA Schaeffler KG corresponding to patent no
DE10159073-A1
Source European Patent Office espcenet (publication no DE10159073-A1 accessed May 2007)
epespacenetcom [40]
Appendix A 130
Figure A8c Third of three proposed designs by INA Schaeffler KG corresponding to patent no
DE10159073-A1
Source European Patent Office espcenet (publication no DE10159073-A1 accessed May 2007)
epespacenetcom [40]
Figure A8a A8b and A8c label identification 1 ndash tightener [tensioner]
2 ndash idler
3 ndash drawing means
4 ndash swivel arm
5 ndash axis of rotation
6 ndash drawing means impulse [belt]
7 ndash crankshaft
8 ndash starter generator
9 ndash bolting volume 10a ndash bolting device system
10b ndash bolting device system
10c ndash bolting device system
11 ndash friction body
12 ndash lateral surface
13 ndash bolting tape end
14 ndash bolting tape end
15 ndash control member
16 ndash torsion bar
17 ndash base
18 ndash pylon
19 ndash hub
20 ndash annular gap
21 ndash Gleitlagerbuchse
23 ndash [nil]
23 ndash friction disk
24 ndash turning camps 25 ndash teeth
26 ndash elbow levers
27 ndash clamping wedge
28 ndash internal contour
29 ndash longitudinal guidance
30 ndash system
31 ndash sensor
32 ndash clamping gap
(European Patent Office May 2007) [40]
Appendix A 131
Figure A9 Proposed design by INA Schaeffler KG corresponding to patent no DE10359641-A1
Source European Patent Office espcenet (publication no DE10359641-A1 accessed Jun 2007)
epespacenetcom [42]
Figure A9 label identification 8 ndash cylinder
10 ndash rod
12 ndash spring plate
13 ndash spring
14 ndash pressure lead
(European Patent Office June 2007) [42]
Appendix A 132
Figure A10 Proposed design by INA Schaeffler KG corresponding to patent no EP1723350-A1
Source European Patent Office espcenet (publication no EP1723350-A1 accessed Jun 2007) epespacenetcom [41]
Figure A10 label identification 4 ndash pulley
5 ndash hydraulic element 11 ndash freewheel
12 ndash shaft
(European Patent Office June 2007) [41]
Figure A11 Proposed design by INA Schaeffler KG corresponding to patent no EP1738093-A1
Source European Patent Office espcenet (publication no EP1738093-A1 accessed Jun 2007)
epespacenetcom [43]
Figure A11 label identification 1 ndash traction drive
2 ndash traction belt
3 ndash starter generator
Appendix A 133
7 ndash tension device
9 ndash lever arm
10 ndash guide roller
16 ndash end stop
(European Patent Office June 2007) [43]
Figure A12 Proposed design by INA Schaeffler KG corresponding to patent no DE102004012395-A1
Source European Patent Office espcenet (publication no DE102004012395-A1 accessed May 2007) epespacenetcom [44]
Figure A12 label identification 1 ndash belt drive
2 ndash belts
3 ndash wheel of the internal-combustion engine
4 ndash wheel of a Nebenaggregats
5 ndash wheel of the starter generator
6 ndash clamping unit
7 ndash idler
8 ndash idler
9 ndash scale beams
10 ndash drive place
11 ndash clamping arm
12 ndash camps
13 ndash coupling point
14 ndash shock absorber element
15 ndash arrow
(European Patent Office May 2007) [44]
Appendix A 134
Figure A13 Proposed design by INA Schaeffler KG corresponding to patent nos DE102005017038-A1and WO2006108461-A1
Source European Patent Office espcenet (publication nos DE102005017038-A1and WO2006108461-A1 accessed May 2007) epespacenetcom [45]
Figure A13 label identification 1 ndash belt
2 ndash wheel of the crankshaft KW
3 ndash wheel of a climatic compressor AC
4 ndash wheel of a starter generator SG
5 ndash wheel of a water pump WP
6 ndash first clamping system
7 ndash first tension adjuster lever arm
8 ndash first tension adjuster role
9 ndash second clamping system
10 ndash second tension adjuster lever arm
11 ndash second tension adjuster role 12 ndash guide roller
13 ndash drive-conditioned Zugtrum
(generatorischer enterprise (GE))
13 ndash starter-conditioned Leertrum
(starter enterprise (SE))
14 ndash drive-conditioned Leertrum (GE)
14 ndash starter-conditioned Zugtrum (SE)
14a ndash drive-conditioned Leertrumast (GE)
14a ndash starter-conditioned Zugtrumast (SE)
14b ndash drive-conditioned Leertrumast (GE)
14b ndash starter-conditioned Zugtrumast (SE)
(European Patent Office May 2007) [45]
Figure A14 Proposed design by Litens Automotive GMBH et al corresponding to patent no
US20010007839-A1
Appendix A 135
Source European Patent Office espcenet (publication no US20010007839-A1 accessed Jun 2007)
epespacenetcom [46]
Figure A14 label identification E - belt
K - crankshaft
R1 ndash first tension pulley
R2 ndash second tension pulley
S ndash tension device
T ndash drive system
1 ndash belt pulley
4 ndash belt pulley
(European Patent Office June 2007) [46]
Figure A15 Proposed design by Mitsubishi Jidosha Eng KK and Mitsubishi Motor Corp corresponding
to patent no JP2005083514-A
Source Industrial Property Digital Library and Japanese Patent Office Patent amp Utility Model Gazette DB (document no A 2005-083514 accessed May 2007) wwwipdlinpitgojp [47]
Figure A15 label identification 1 ndash Pulley for Starting
2 ndash Crank Pulley
3 ndash AC Pulley
4a ndash 1st roller
4b ndash 2nd roller
5 ndash Idler Pulley
6 ndash Belt
7c ndash Starter generator control section
7d ndash Idle stop control means
8 ndash WP Pulley
9 ndash IG Switch
10 ndash Engine
11 ndash Starter Generator
12 ndash Driving Shaft
Appendix A 136
7 ndash ECU
7a ndash 1st auto tensioner control section (the 1st auto
tensioner control means)
7b ndash 2nd auto tensioner control section (the 2nd auto
tensioner control means)
13 ndash Air-conditioner Compressor
14a ndash 1st auto tensioner
14b ndash 2nd auto tensioner
18 ndash Water Pump
(Industrial Property Digital Library May 2007) [47]
Figure A16 Proposed design by Nissan Motor Co Ltd corresponding to patent no JP3565040-B2
Source European Patent Office espcenet (publication no JP3565040-B2 accessed Jun 2007) epespacenetcom [48]
Figure A16 label identification 3 ndash chain [or belt]
5 ndash tensioner
4 ndash belt pulley
(European Patent Office June 2007) [48]
Figure A17 Proposed design by NTN Corp corresponding to patent no JP2006189073-A
Appendix A 137
Source European Patent Office espcenet (publication no JP2006189073-A accessed Jun 2007)
epespacenetcom [49]
Figure A17 label identification 5d - flange
6 ndash tensile strength spring
10 ndash actuator
10c ndash cylinder
12 ndash rod
20 ndash hydraulic damper mechanism
21 ndash cylinder nut
22 ndash screw bolt
(European Patent Office June 2007) [49]
Figure A18 Proposed design by Valeo Equipment Electriques Moteur corresponding to patent nos
EP1658432 and WO2005015007
Source European Patent Office espcenet (publication nos EP1658432 and WO2005015007
accessed May 2007) epespacenetcom [50]
Figure A18 abbreviated list of label identifications
10 ndash starter
22 ndash shaft section
23 ndash free front end
80 ndash pulley
200 ndash support element
206 - arm
(European Patent Office May 2007) [50]
The author notes that the list of labels corresponding to Figures A1a through to A7 are generated
from machine translations translated from the documentrsquos original language (ie German)
Consequently words may be translated inaccurately or not at all
138
APPENDIX B
B-ISG Serpentine Belt Drive with Single Tensioner
Equation of Motion
The equations of motion (EOM) for a B-ISG serpentine belt drive with a single tensioner are
shown The EOM has been derived similarly to that of the same system with a twin tensioner
that was provided in Chapter 3 The assumptions for the twin tensioner B-ISG system are
applicable for the single tensioner B-ISG system as well
Figure B1 shows the B-ISG system with a single tensioner pulley and arm The pulleys are
numbered 1 through 4 and their associated belt spans are numbered accordingly
Figure B1 Single Tensioner B-ISG System
Appendix B 139
The free-body diagram for the ith non-tensioner pulley in the system shown above is found in
Figure B2 The moment of inertia for the ith pulley is designated as Ii while the angular
displacement velocity and acceleration is designated as 120579119905119894 120579 119905119894 and 120579 119905119894 respectively The
required torque is Qi the angular damping is Ci and the tension of the ith span is Ti
Figure B2 Free-body Diagram of ith Pulley
The positive motion designated is assumed to be in the clockwise direction The radius for the
ith pulley is represented by Ri The equilibrium equations for the ith pulley are as follows
I1 ∙ θ 1 = T4 ∙ R1 minus T1 ∙ R1 + Q1 minus c1 ∙ θ 1 (B1)
I2 ∙ θ 2 = T1 ∙ R2 minus T2 ∙ R2 + Q2 minus c2 ∙ θ 2 (B2)
I3 ∙ θ 3 = T2 ∙ R3 minus T3 ∙ R3 + Q3 minus c3 ∙ θ 3 (B3)
Appendix B 140
A free-body diagram for the single tensioner pulley is shown in Figure B3 The rotational
stiffness and damping for the tensioner arm is designated as kt and ct respectively The angular
rotation and velocity for the arm is 120579119905119894 and 120579 119905119894 respectively
Figure B3 Free-body Diagram of Single Tensioner
From figure B2 the equations of equilibrium are resolved for the tensioner pulley The angle of
orientation for the ith belt span is designated by 120573119895119894
minusI4 ∙ θ 4 = minusT3 ∙ R4 + T4 ∙ R4 + Q4 + c4 ∙ θ 4 (B4)
It ∙ θ t = minusTt ∙ Lt ∙ sin θto minus βj4 + sin θto 1 minus βj5 minus kt ∙ partθt minus ct ∙ partθ t
(B5)
Appendix B 141
partθt = θt minus θto (B6)
The dynamic tension matrix Trsquo is proportional to the damping (Tc) and stiffness (Tk) matrices
that are due to belt damping (119888119894119887 ) and belt stiffness (119896119894
119887 ) respectively
119827prime = 119827119836 ∙ 120521 + 119827119844 ∙ 120521 (B7)
The initial tension is represented by To and the initial angle of the tensioner arm is represented
by 120579119905119900 The equation for the tension of the ith span is shown in the following equations
T1 = To + k1b ∙ R1 ∙ θ1 minus R2 ∙ θ2 + c1
b ∙ [R1 ∙ θ 1 minus R2 ∙ θ 2] (B8)
T2 = To + k2b ∙ R2 ∙ θ2 minus R3 ∙ θ3 + c2
b ∙ [R2 ∙ θ 2 minus R3 ∙ θ 3)] (B9)
T3 = To + k3b ∙ R3 ∙ θ3 minus R4 ∙ θ4 + Lt ∙ [sin θto minus βj3 ] ∙ (θt minus θto ) + c3
b ∙ [R3 ∙ θ 3 minus R4 ∙
θ 4 + Lt ∙ [sin θto minus βj3 ] ∙ (θ t)] (B10)
T4 = To + k4b ∙ R4 ∙ θ4 minus R1 ∙ θ1 + Lt ∙ [sin θto minus βj4 ] ∙ (θt minus θto ) + c4
b ∙ [R4 ∙ θ 4 minus R1 ∙
θ 1 + Lt ∙ [sin θto minus βj4 ] ∙ (θ t)] (B11)
Tprime = Ti minus To (B12)
Tt = T3 = T4 (B13)
Appendix B 142
The EOM for the single tensioner B-ISG system is found by substitution of equations B8 to
B13 into B1 to B5 The matrices in the EOM include the inertial matrix I damping matrix C
stiffness matrix K and the required torque matrix Q as well as the angular displacement
velocity and acceleration matrices 120521 120521 and 120521 respectively
119816 ∙ 120521 + 119810 ∙ 120521 + 119818 ∙ 120521 = 119824 (B14)
119816 =
I1 0 0 0 00 I2 0 0 00 0 I3 0 00 0 0 I4 00 0 0 0 It1
(B15)
The stiffness matrix includes kb the belt factor Kb the belt cord stiffness 120601119894 the wrap angle of
the belt on the ith pulley and Kbi the stiffness factor of the ith belt span Cb represents the belt
damping for each span and βji is the angle of orientation for the span between the jth and ith
pulleys It is noted in the terms of the stiffness and damping matrices below that the numerical
subscripts refer to the (i+1)th pulley The term Qt may be found within the required torque
matrix and represents the required torque for the tensioner arm As well the term It1 represents
the moment of inertia for the tensioner arm
Appendix B 143
K =
(B16)
Kbi =Kb
Li + kb ∙ Ri ∙ϕi+1
2 + Ri ∙ϕi
2
(B17)
C =
(B18)
Appendix B 144
Appendix B 144
120521 =
θ1
θ2
θ3
θ4
partθt
(B19)
119824 =
Q1
Q2
Q3
Q4
Qt
(B20)
Simulations of the EOM for the B-ISG system with a single tensioner were performed in FEAD
[51] software for dynamic and static cases This allowed for the methodology for deriving the
EOM to be verified and then applied to the B-ISG system with a twin tensioner The natural
frequencies modes shapes dynamic responses tensioner arm torques as well as the crankshaft
required torque only and the dynamic tensions were solved from the EOM as described in (331)
to (339) of Chapter 3 and as well as for the static tension from (351) to (353) of Chapter 3
This permitted verification of the complete derivation methodology and allowed for comparison
of the static tension of the B-ISG system belt spans in the case that a single tensioner is present
and in the case that a Twin Tensioner is present [51]
145
APPENDIX C
MathCAD Scripts
C1 Geometric Analysis
1 - CS
2 - AC
4 - Alt
3 - Ten1
5 - Ten 2
6 - Ten Pivot
1
2
3
4
5
Figure C1 Schematic of B-ISG
System with Twin Tensioner
Coordinate Input DataXY1 0 0( ) XY4 24759 16664( )
XY2 224 6395( ) XY5 12057 9193( )
XY3 292761 87( ) XY6 201384 62516( )
Computations
Lt1 XY30 0
XY60 0
2
XY30 1
XY60 1
2
Lt2 XY50 0
XY60 0
2
XY50 1
XY60 1
2
t1 atan2 XY30 0
XY60 0
XY30 1
XY60 1
t2 atan2 XY50 0
XY60 0
XY50 1
XY60 1
XY
XY10 0
XY20 0
XY30 0
XY40 0
XY50 0
XY60 0
XY10 1
XY20 1
XY30 1
XY40 1
XY50 1
XY60 1
x XY
0 y XY
1
Appendix C 146
i - angle bw horizontal and l ine from ith pulley center to (i+1)th pulley center
Adjust last number in range variable p to correspond to number of pulleys
p 0 1 4
k p( ) p 1( ) p 4if
0 otherwise
condition1 p( ) acos
XYk p( ) 0
XYp 0
XYk p( ) 0
XYp 0
2
XYk p( ) 1
XYp 1
2
condition2 p( ) 2 acos
XYk p( ) 0
XYp 0
XYk p( ) 0
XYp 0
2
XYk p( ) 1
XYp 1
2
p( ) if XYk p( ) 1
XYp 1
condition1 p( ) condition2 p( )
Lfi Lbi - connection belt span lengths
D1 20065mm D2 10349mm D3 7240mm D4 6820mm D5 7240mm
D
D1
D2
D3
D4
D5
Lf p( ) XYk p( ) 0
XYp 0
2
XYk p( ) 1
XYp 1
2
1
mm
Dk p( )
2
Dp
2
2
Lb p( ) XYk p( ) 0
XYp 0
2
XYk p( ) 1
XYp 1
2
1
mm
Dk p( )
2
Dp
2
2
fi bi - angle bw ith pulley center connection l ine and contact points Pbfi (or Pfbi) and Pbi
(or Pfi) respecti vely l
f p( ) atanLf p( ) mm
Dp
2
Dk p( )
2
Dp
Dk p( )
if
atanLf p( ) mm
Dk p( )
2
Dp
2
Dp
Dk p( )
if
2D
pD
k p( )if
b p( ) atan
mmLb p( )
Dp
2
Dk p( )
2
Appendix C 147
XYfi XYbi XYfbi XYbfi - 4 possible contact points for i th pulley
XYf p( ) XYp 0
Dp
2 mmcos p( ) f p( )
XYp 1
Dp
2 mmsin p( ) f p( )
XYb p( ) XYp 0
Dp
2 mmcos p( ) f p( )
XYp 1
Dp
2 mmsin p( ) f p( )
XYfb p( ) XYp 0
Dp
2 mmcos p( ) b p( )
XYp 1
Dp
2 mmsin p( ) b p( )
XYbf p( ) XYp 0
Dp
2 mmcos p( ) b p( )
XYp 1
Dp
2 mmsin p( ) b p( )
XYfi+1 XYbi+1 XYfbi+1 XYbfi+1 - 4 possible contact points for i+1th pulley
XYf2 p( ) XYk p( ) 0
Dk p( )
2 mmcos p( ) f p( )
XYk p( ) 1
Dk p( )
2 mmsin p( ) f p( )
XYb2 p( ) XYk p( ) 0
Dk p( )
2 mmcos p( ) f p( )
XYk p( ) 1
Dk p( )
2 mmsin p( ) f p( )
XYfb2 p( ) XYk p( ) 0
Dk p( )
2 mmcos p( ) b p( ) XY
k p( ) 1
Dk p( )
2 mmsin p( ) b p( )
XYbf2 p( ) XYk p( ) 0
Dk p( )
2 mmcos p( ) b p( ) XY
k p( ) 1
Dk p( )
2 mmsin p( ) b p( )
Row 1 --gt Pulley 1 Row i --gt Pulley i
XYfi
XYf 0( )0 0
XYf 1( )0 0
XYf 2( )0 0
XYf 3( )0 0
XYf 4( )0 0
XYf 0( )0 1
XYf 1( )0 1
XYf 2( )0 1
XYf 3( )0 1
XYf 4( )0 1
XYfi
6818
269222
335325
251552
108978
100093
89099
60875
200509
207158
x1 XYfi0
y1 XYfi1
Appendix C 148
XYbi
XYb 0( )0 0
XYb 1( )0 0
XYb 2( )0 0
XYb 3( )0 0
XYb 4( )0 0
XYb 0( )0 1
XYb 1( )0 1
XYb 2( )0 1
XYb 3( )0 1
XYb 4( )0 1
XYbi
47054
18575
269403
244841
164847
88606
291
30965
132651
166182
x2 XYbi0
y2 XYbi1
XYfbi
XYfb 0( )0 0
XYfb 1( )0 0
XYfb 2( )0 0
XYfb 3( )0 0
XYfb 4( )0 0
XYfb 0( )0 1
XYfb 1( )0 1
XYfb 2( )0 1
XYfb 3( )0 1
XYfb 4( )0 1
XYfbi
42113
275543
322697
229969
9452
91058
59383
75509
195834
177002
x3 XYfbi0
y3 XYfbi1
XYbfi
XYbf 0( )0 0
XYbf 1( )0 0
XYbf 2( )0 0
XYbf 3( )0 0
XYbf 4( )0 0
XYbf 0( )0 1
XYbf 1( )0 1
XYbf 2( )0 1
XYbf 3( )0 1
XYbf 4( )0 1
XYbfi
8384
211903
266707
224592
140427
551
13639
50105
141463
143331
x4 XYbfi0
y4 XYbfi1
Row 1 --gt Pulley 2 Row i --gt Pulley i+1
XYf2i
XYf2 0( )0 0
XYf2 1( )0 0
XYf2 2( )0 0
XYf2 3( )0 0
XYf2 4( )0 0
XYf2 0( )0 1
XYf2 1( )0 1
XYf2 2( )0 1
XYf2 3( )0 1
XYf2 4( )0 1
XYf2x XYf2i0
XYf2y XYf2i1
XYb2i
XYb2 0( )0 0
XYb2 1( )0 0
XYb2 2( )0 0
XYb2 3( )0 0
XYb2 4( )0 0
XYb2 0( )0 1
XYb2 1( )0 1
XYb2 2( )0 1
XYb2 3( )0 1
XYb2 4( )0 1
XYb2x XYb2i0
XYb2y XYb2i1
Appendix C 149
XYfb2i
XYfb2 0( )0 0
XYfb2 1( )0 0
XYfb2 2( )0 0
XYfb2 3( )0 0
XYfb2 4( )0 0
XYfb2 0( )0 1
XYfb2 1( )0 1
XYfb2 2( )0 1
XYfb2 3( )0 1
XYfb2 4( )0 1
XYfb2x XYfb2i
0
XYfb2y XYfb2i1
XYbf2i
XYbf2 0( )0 0
XYbf2 1( )0 0
XYbf2 2( )0 0
XYbf2 3( )0 0
XYbf2 4( )0 0
XYbf2 0( )0 1
XYbf2 1( )0 1
XYbf2 2( )0 1
XYbf2 3( )0 1
XYbf2 4( )0 1
XYbf2x XYbf2i0
XYbf2y XYbf2i1
100 40 20 80 140 200 260 320 380 440 500150
110
70
30
10
50
90
130
170
210
250Figure C2 Possible Contact Points
250
150
y1
y2
y3
y4
y
XYf2y
XYb2y
XYfb2y
XYbf2y
500100 x1 x2 x3 x4 x XYf2x XYb2x XYfb2x XYbf2x
Appendix C 150
Xij Yij - selected contact point on ith pulley for span from ith pulley to jth pulley
XY15 XYbf2iT 4
XY12 XYfiT 0
Pulley 1 contact pts
XY21 XYf2iT 0
XY23 XYfbiT 1
Pulley 2 contact pts
XY32 XYfb2iT 1
XY34 XYbfiT 2
Pulley 3 contact pts
XY43 XYbf2iT 2
XY45 XYfbiT 3
Pulley 4 contact pts
XY54 XYfb2iT 3
XY51 XYbfiT 4
Pulley 5 contact pts
By observation the lengths of span i is the following
L1 Lf 0( ) L2 Lb 1( ) L3 Lb 2( ) L4 Lb 3( ) L5 Lb 4( ) Li
L1
L2
L3
L4
L5
mm
i Angle between horizontal and span of ith pulley
i
atan
XY121
XY211
XY12
0XY21
0
atan
XY231
XY321
XY23
0XY32
0
atan
XY341
XY431
XY34
0XY43
0
atan
XY451
XY541
XY45
0XY54
0
atan
XY511
XY151
XY51
0XY15
0
Appendix C 151
Pulley 1 Pulley 2 Pulley 3 Pulley 4 Pulley 5
12 i0 2 21 i0 32 i1 2 43 i2 54 i3
15 i4 2 23 i1 34 i2 45 i3 51 i4
15
21
32
43
54
12
23
34
45
51
Wrap angle i for the ith pulley
1 2 atan2 XY150
XY151
atan2 XY120
XY121
2 atan2 XY210
XY1 0
XY211
XY1 1
atan2 XY230
XY1 0
XY231
XY1 1
3 2 atan2 XY320
XY2 0
XY321
XY2 1
atan2 XY340
XY2 0
XY341
XY2 1
4 atan2 XY430
XY3 0
XY431
XY3 1
atan2 XY450
XY3 0
XY451
XY3 1
5 atan2 XY540
XY4 0
XY541
XY4 1
atan2 XY510
XY4 0
XY511
XY4 1
1
2
3
4
5
Lb length of belt
Lbelt Li1
2
0
4
p
Dpp
Input Data for B-ISG System
Kt 20626Nm
rad (spring stiffness between tensioner arms 1
and 2)
Kt1 10314Nm
rad (stiffness for spring attached at arm 1 only)
Kt2 16502Nm
rad (stiffness for spring attached at arm 2 only)
Appendix C 152
C2 Dynamic Analysis
I C K moment of inertia damping and stiffness matrices respectively
u 0 1 4 v 0 1 4 (new counter variables where final value = no of pulleys + no of ten arms)
RaD
2
Appendix C 153
RaD
2
Ii =gt moment of inertia for ith pulley where i-1 and i represent ten arms
Ii0
0
1
2
3
4
5
6
10000
2230
300
3000
300
1500
1500
I diag Ii( ) kg mm2
Ci =gt Rotational damping and belt damping for the ith pulley where i-1 and i represent tensioner arms
1000kg
m3
CrossArea 693mm2
0 M CrossArea Lbelt M 0086kg
cb 2 KbM
Lbelt
Cb
cb
cb
cb
cb
cb
Cri
0
0
010
0
010
N mmsec
rad
Ct 1000N mmsec
rad Ct1 1000 N mm
sec
rad Ct2 1000N mm
sec
rad
Cr
Cri0
0
0
0
0
0
0
0
Cri1
0
0
0
0
0
0
0
Cri2
0
0
0
0
0
0
0
Cri3
0
0
0
0
0
0
0
Cri4
0
0
0
0
0
0
0
Ct Ct1
Ct
0
0
0
0
0
Ct
Ct Ct2
Rt
Ra0
Ra1
0
0
0
0
0
0
Ra1
Ra2
0
0
Lt1 mm sin t1 32
0
0
0
Ra2
Ra3
0
Lt1 mm sin t1 34
0
0
0
0
Ra3
Ra4
0
Lt2 mm sin t2 54
Ra0
0
0
0
Ra4
0
Lt2 mm sin t2 51
Appendix C 154
Kr
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Kt Kt1
Kt
0
0
0
0
0
Kt
Kt Kt2
Tk
Kbi 0( ) Ra0
0
0
0
Kbi 4( ) Ra0
Kbi 0( ) Ra1
Kbi 1( ) Ra1
0
0
0
0
Kbi 1( ) Ra2
Kbi 2( ) Ra2
0
0
0
0
Kbi 2( ) Ra3
Kbi 3( ) Ra3
0
0
0
0
Kbi 3( ) Ra4
Kbi 4( ) Ra4
0
Kbi 1( ) Lt1 mm sin t1 32
Kbi 2( ) Lt1 mm sin t1 34
0
0
0
0
0
Kbi 3( ) Lt2 mm sin t2 54
Kbi 4( ) Lt2 mm sin t2 51
Tc
Cb0
Ra0
0
0
0
Cb4
Ra0
Cb0
Ra1
Cb1
Ra1
0
0
0
0
Cb1
Ra2
Cb2
Ra2
0
0
0
0
Cb2
Ra3
Cb3
Ra3
0
0
0
0
Cb3
Ra4
Cb4
Ra4
0
Cb1
Lt1 mm sin t1 32
Cb2
Lt1 mm sin t1 34
0
0
0
0
0
Cb3
Lt2 mm sin t2 54
Cb4
Lt2 mm sin t2 51
C matrix
C Cr Rt Tc
K matrix
K Kr Rt Tk
New Equations of Motion for Dual Drive System
I K amp C matricies rearranged to place driving pulley in 1st row + 1st column and driven in 2nd row + 2nd column
IA augment I3
I0
I1
I2
I4
I5
I6
IC augment I0
I3
I1
I2
I4
I5
I6
I1kgmm2 1 106
kg m2
0 0 0 0 0 0
Ia stack I1kgmm2 IAT 0
T
IAT 1
T
IAT 2
T
IAT 4
T
IAT 5
T
IAT 6
T
Ic stack I1kgmm2 ICT 3
T
ICT 1
T
ICT 2
T
ICT 4
T
ICT 5
T
ICT 6
T
Appendix C 155
RtA augment Rt3
Rt0
Rt1
Rt2
Rt4
RtC augment Rt0
Rt3
Rt1
Rt2
Rt4
Rta stack RtAT 3
T
RtAT 0
T
RtAT 1
T
RtAT 2
T
RtAT 4
T
RtAT 5
T
RtAT 6
T
Rtc stack RtCT 0
T
RtCT 3
T
RtCT 1
T
RtCT 2
T
RtCT 4
T
RtCT 5
T
RtCT 6
T
TkA augment Tk3
Tk0
Tk1
Tk2
Tk4
Tk5
Tk6
Tka stack TkAT 3
T
TkAT 0
T
TkAT 1
T
TkAT 2
T
TkAT 4
T
TkC augment Tk0
Tk3
Tk1
Tk2
Tk4
Tk5
Tk6
Tkc stack TkCT 0
T
TkCT 3
T
TkCT 1
T
TkCT 2
T
TkCT 4
T
TcA augment Tc3
Tc0
Tc1
Tc2
Tc4
Tc5
Tc6
Tca stack TcAT 3
T
TcAT 0
T
TcAT 1
T
TcAT 2
T
TcAT 4
T
TcC augment Tc0
Tc3
Tc1
Tc2
Tc4
Tc5
Tc6
Tcc stack TcAT 0
T
TcAT 3
T
TcAT 1
T
TcAT 2
T
TcAT 4
T
Ka Kr Rta Tka Kc Kr Rtc Tkc Ca Cr Rta Tca Cc Cr Rtc Tcc
CHECK
KA augment K3
K0
K1
K2
K4
K5
K6
KC augment K0
K3
K1
K2
K4
K5
K6
CA augment C3
C0
C1
C2
C4
C5
C6
CC augment C0
C3
C1
C2
C4
C5
C6
Appendix C 156
Kacheck stack KAT 3
T
KAT 0
T
KAT 1
T
KAT 2
T
KAT 4
T
KAT 5
T
KAT 6
T
Kccheck stack KCT 0
T
KCT 3
T
KCT 1
T
KCT 2
T
KCT 4
T
KCT 5
T
KCT 6
T
Cacheck stack CAT 3
T
CAT 0
T
CAT 1
T
CAT 2
T
CAT 4
T
CAT 5
T
CAT 6
T
Cccheck stack CCT 0
T
CCT 3
T
CCT 1
T
CCT 2
T
CCT 4
T
CCT 5
T
CCT 6
T
Results for System switching from ISG as DRIVING pulley to Crankshaft as Drivi ng Pulley
Modified Submatricies for ISG Driving Phase --gt CS Driving Phase
Unit step function to provide shift from crankshaft DRIVING case (ie ISG driven case) to crankshaft DRIVEN
case (ie ISG driving case)
H n( ) 1 n 750if
0 n 750if
lt-- crankshaft DRIVING case (Phase change bw 2 cases occurs when n
reaches start speed)
I11mod n( ) Ic0 0
H n( ) 1if
Ia0 0
H n( ) 0if
I22mod n( )submatrix Ic 1 6 1 6( )
UnitsOf I( )H n( ) 1if
submatrix Ia 1 6 1 6( )
UnitsOf I( )H n( ) 0if
K11mod n( )
Kc0 0
UnitsOf K( )H n( ) 1if
Ka0 0
UnitsOf K( )H n( ) 0if
C11modn( )
Cc0 0
UnitsOf C( )H n( ) 1if
Ca0 0
UnitsOf C( )H n( ) 0if
K22mod n( )submatrix Kc 1 6 1 6( )
UnitsOf K( )H n( ) 1if
submatrix Ka 1 6 1 6( )
UnitsOf K( )H n( ) 0if
C22modn( )submatrix Cc 1 6 1 6( )
UnitsOf C( )H n( ) 1if
submatrix Ca 1 6 1 6( )
UnitsOf C( )H n( ) 0if
K21mod n( )submatrix Kc 1 6 0 0( )
UnitsOf K( )H n( ) 1if
submatrix Ka 1 6 0 0( )
UnitsOf K( )H n( ) 0if
C21modn( )submatrix Cc 1 6 0 0( )
UnitsOf C( )H n( ) 1if
submatrix Ca 1 6 0 0( )
UnitsOf C( )H n( ) 0if
K12mod n( )submatrix Kc 0 0 1 6( )
UnitsOf K( )H n( ) 1if
submatrix Ka 0 0 1 6( )
UnitsOf K( )H n( ) 0if
C12modn( )submatrix Cc 0 0 1 6( )
UnitsOf C( )H n( ) 1if
submatrix Ca 0 0 1 6( )
UnitsOf C( )H n( ) 0if
Appendix C 157
2mod n( ) I22mod n( )1
K22mod n( ) mod n( ) sort eigenvals 2mod n( ) nmod n( )mod n( )
2
EVmodn( ) augmenteigenvec 2mod n( ) mod n( )0
max eigenvec 2mod n( ) mod n( )0
eigenvec 2mod n( ) mod n( )1
max eigenvec 2mod n( ) mod n( )1
eigenvec 2mod n( ) mod n( )2
max eigenvec 2mod n( ) mod n( )2
eigenvec 2mod n( ) mod n( )3
max eigenvec 2mod n( ) mod n( )3
eigenvec 2mod n( ) mod n( )4
max eigenvec 2mod n( ) mod n( )4
eigenvec 2mod n( ) mod n( )5
max eigenvec 2mod n( ) mod n( )5
modeshapesmod n( ) stack nmod n( )T
EVmodn( )
t 0 0001 1
mode1a t( ) EVmod100( )0
sin nmod 100( )0 t mode2a t( ) EVmod100( )1
sin nmod 100( )1 t
mode1c t( ) EVmod800( )0
sin nmod 800( )0 t mode2c t( ) EVmod800( )1
sin nmod 800( )1 t
Pulley responses amp torque requirement for crankshaft amp alternator pulleys pulley1 and 4 respectively
The system equation becomes
I14q14 -double-dot + C1144 q14 -dot + K1144 q14 + C12qm-dot + K12qm = Qc
I2qm-double-dot + C22qm-dot + K22qm + C21q1-dot + K21q1 = 0
Pulley responses
Qm = - [(K22 - 2I2) + jC22 ]-1(K21 + jC21 )Q1
Torque requirement for crank shaft Pulley 1
qc = [(K11 -2I1) + jC11 ]Q1 + (K12 + jC12 )Qm
Torque requirement for alternator shaft Pulley 4
qa = [(K44 -2I4) + jC44 ]Q4 + (K12 + jC12 )Qm
Appendix C 158
Let DRIVING pulley have a unit amplitude 1 = 1 and let the system frequency be calculated based on
engine speed n
n 60 90 6000 n( )4n
60 a n( )
2n Ra0
60 Ra3
mod n( ) n( ) H n( ) 1if
a n( ) H n( ) 0if
Ymod n( ) K22mod n( ) mod n( ) 2 I22mod n( )
j mod n( ) C22modn( )
mmod n( ) Ymod n( )( )1
K21mod n( ) j mod n( ) C21modn( )
Crankshaft amp ISG required torques
Let input from DRIVING pulley be an angular displacement with constant amplitude of angular acceleration
Ac n( ) 650 1 n( )Ac n( )
n( ) 2
Let Qm = QmQ1(n) for n lt 750
and Qm = QmQ4(n) for n gt 750
Aa n( )42
I3 3
1a n( )Aa n( )
a n( ) 2
Qc0 4
qcmod n( ) K11mod n( ) mod n( ) 2
I11mod n( )
j mod n( ) C11modn( )
1 n( ) K12mod n( ) j mod n( ) C12modn( ) mmod n( ) 1 n( )
H n( ) 1if
Qc0 H n( ) 0if
qamod n( ) K11mod n( ) mod n( ) 2
I11mod n( )
j mod n( ) C11modn( )
1a n( ) K12mod n( ) jmod n( ) C12modn( ) mmod n( ) 1a n( ) Qc0
H n( ) 0if
0 H n( ) 1if
Q n( ) 48 n
Ra0
Ra3
48
18000
(ISG torque requirement alternate equation)
Appendix C 159
Dynamic tensioner arm torques
Qtt1mod n( )Kt Kt1
UnitsOf Kt( )j mod n( )
Ct Ct1
UnitsOf Cr( )
mmod n( )4 1 n( )
H n( ) 1if
Kt Kt1
UnitsOf Kt( )j mod n( )
Ct Ct1
UnitsOf Cr( )
mmod n( )4 1a n( )
H n( ) 0if
Qtt2mod n( )Kt Kt2
UnitsOf Kt( )j mod n( )
Ct Ct2
UnitsOf Cr( )
mmod n( )5 1 n( )
H n( ) 1if
Kt Kt2
UnitsOf Kt( )j mod n( )
Ct Ct2
UnitsOf Cr( )
mmod n( )5 1a n( )
H n( ) 0if
Appendix C 160
Dynamic belt span tensions
d n( ) 1 n( ) H n( ) 1if
1a n( ) H n( ) 0if
mod n( )
d n( )
mmod n( ) d n( ) 0 0
mmod n( ) d n( ) 1 0
mmod n( ) d n( ) 2 0
mmod n( ) d n( ) 3 0
mmod n( ) d n( ) 4 0
mmod n( ) d n( ) 5 0
Tm n( ) j n( )Tcc
UnitsOf Tcc( )
Tkc
UnitsOf Tkc( )
mod n( )
H n( ) 1if
j n( )Tca
UnitsOf Tca( )
Tka
UnitsOf Tka( )
mod n( )
H n( ) 0if
Tm n( ) j n( )Tcc
UnitsOf Tcc( )
Tkc
UnitsOf Tkc( )
mod n( )
H n( ) 1if
j n( )Tca
UnitsOf Tca( )
Tka
UnitsOf Tka( )
mod n( )
H n( ) 0if
(tensions for driving pulley belt spans)
Appendix C 161
C3 Static Analysis
Static Analysis using K Tk amp Q matricies amp Ts
For static case K = Q
Tension T = T0 + Tks
Thus T = K-1QTks + T0
Q1 68N m Qt1 0N m Qt2 0N m Ts 300N
Qc
Q4
Q2
Q3
Q5
Qt1
Qt2
Qc
5
2
0
0
0
0
J Qa
Q1
Q2
Q3
Q5
Qt1
Qt2
Qa
68
2
0
0
0
0
N m
cK22mod 900( )( )
1
N mQc A
K22mod 600( )1
N mQa
a
A0
A1
A2
0
A3
A4
A5
0
c1
c2
c0
c3
c4
c5
Tc Tk Ts Ta Tk a Ts
162
APPENDIX D
MATLAB Functions amp Scripts
D1 Parametric Analysis
D11 TwinMainm
The following function script performs the parametric analysis for the B-ISG system with a Twin
Tensioner It calls the function TwinTenStaticTensionm The parametric analysis perturbs a
single input parameter for the called function TwinTenStaticTensionm The main function takes
an initial input value for the Twin Tensioner‟s stiffness parameters Kto Kt1o Kt2o and
geometric parameters D3o D5o X3o Y3o X5o and Y5o An input parameter is allowed to
increment by six percent over a range from sixty percent below its initial value to sixty percent
above its initial value The coordinate parameters are incremented through a mesh of Cartesian
points with prescribed boundaries The TwinMainm function plots the parametric results
______________________________________________________________________________
clc
clear all
Static tension for single tensioner system for CS and Alt driving
Initial Conditions
Kto = 20626
Kt1o = 10314
Kt2o = 16502
D3o = 007240
D5o = 007240
X3o =0292761
Y3o =087
X5o =12057
Y5o =09193
Pertubations of initial parameters
Kt = (Kto-060Kto)006Kto(Kto+060Kto)
Kt1 = (Kt1o-060Kt1o)006Kt1o(Kt1o+060Kt1o)
Kt2 = (Kt2o-060Kt2o)006Kt2o(Kt2o+060Kt2o)
D3 = (D3o-060D3o)006D3o(D3o+060D3o)
D5 = (D5o-060D5o)006D5o(D5o+060D5o)
No of data points
s = 21
T = zeros(5s)
Ta = zeros(5s)
Parametric Plots
for i = 1s
Appendix D 163
[T(i)Ta(i)] = TwinTenStaticTension(Kt(i)Kt1oKt2oD3oD5oX3oY3oX5oY5o)
end
figure
[AXH1H2] = plotyy(Kt()T(1)Kt()Ta(4)plot) hold on
H3 = line(Kt()T(5)ParentAX(1)) hold on
H4 = line(Kt()Ta(3)Parent AX(2)) hold off
set(AXYLimModeautoYTickModeauto)
title(Static Tension vs Coupled Stiffness bw Arms 1 amp 2)
xlabel(Kt (Nmrad))
set(get(AX(1)Ylabel)StringCS Span Tension (N))
set(get(AX(2)Ylabel)StringISG Span Tension (N))
set(H1LineStyle-Colorb)
set(H2LineStyle-Colorg)
set(H3LineStyleColorget(H1Color))
set(H4LineStyleColorget(H2Color))
legend([H1H2H3H4]CS Tautest SpanISG Tautest SpanCS Slackest SpanISG
Slackest SpanLocationBest)
for i = 1s
[T(i)Ta(i)] = TwinTenStaticTension(KtoKt1(i)Kt2oD3oD5oX3oY3oX5oY5o)
end
figure
[AXH1H2] = plotyy(Kt1()T(1)Kt1()Ta(4)plot) hold on
H3 = line(Kt1()T(5)ParentAX(1)) hold on
H4 = line(Kt1()Ta(3)Parent AX(2)) hold off
set(AXYLimModeautoYTickModeauto)
title(Static Tension vs Tensioner Arm 1 Stiffness)
xlabel(Kt1 (Nmrad))
set(get(AX(1)Ylabel)StringCS Span Tension (N))
set(get(AX(2)Ylabel)StringISG Span Tension (N))
set(H1LineStyle-Colorb)
set(H2LineStyle-Colorg)
set(H3LineStyleColorget(H1Color))
set(H4LineStyleColorget(H2Color))
legend([H1H2H3H4]CS Tautest SpanISG Tautest SpanCS Slackest SpanISG
Slackest SpanLocationBest)
for i = 1s
[T(i)Ta(i)] = TwinTenStaticTension(KtoKt1oKt2(i)D3oD5oX3oY3oX5oY5o)
end
figure
[AXH1H2] = plotyy(Kt2()T(1)Kt2()Ta(4)plot) hold on
H3 = line(Kt2()T(5)ParentAX(1)) hold on
H4 = line(Kt2()Ta(3)Parent AX(2)) hold off
set(AXYLimModeautoYTickModeauto)
title(Static Tension vs Tensioner Arm 2 Stiffness)
xlabel(Kt2 (Nmrad))
set(get(AX(1)Ylabel)StringCS Span Tension (N))
Appendix D 164
set(get(AX(2)Ylabel)StringISG Span Tension (N))
set(H1LineStyle-Colorb)
set(H2LineStyle-Colorg)
set(H3LineStyleColorget(H1Color))
set(H4LineStyleColorget(H2Color))
legend([H1H2H3H4]CS Tautest SpanISG Tautest SpanCS Slackest SpanISG
Slackest SpanLocationBest)
for i = 1s
[T(i)Ta(i)] = TwinTenStaticTension(KtoKt1oKt2oD3(i)D5oX3oY3oX5oY5o)
end
figure
[AXH1H2] = plotyy(D3()T(1)D3()Ta(4)plot) hold on
H3 = line(D3()T(5)ParentAX(1)) hold on
H4 = line(D3()Ta(3)Parent AX(2)) hold off
set(AXYLimModeautoYTickModeauto)
title(Static Tension vs Tensioner Pulley 1 Diameter)
xlabel(D3 (m))
set(get(AX(1)Ylabel)StringCS Span Tension (N))
set(get(AX(2)Ylabel)StringISG Span Tension (N))
set(H1LineStyle-Colorb)
set(H2LineStyle-Colorg)
set(H3LineStyleColorget(H1Color))
set(H4LineStyleColorget(H2Color))
legend([H1H2H3H4]CS Tautest SpanISG Tautest SpanCS Slackest SpanISG
Slackest SpanLocationBest)
for i = 1s
[T(i)Ta(i)] = TwinTenStaticTension(KtoKt1oKt2oD3oD5(i)X3oY3oX5oY5o)
end
figure
[AXH1H2] = plotyy(D5()T(1)D5()Ta(4)plot) hold on
H3 = line(D5()T(5)ParentAX(1)) hold on
H4 = line(D5()Ta(3)Parent AX(2)) hold off
set(AXYLimModeautoYTickModeauto)
title(Static Tension vs Tensioner Pulley 2 Diameter)
xlabel(D5 (m))
set(get(AX(1)Ylabel)StringCS Span Tension (N))
set(get(AX(2)Ylabel)StringISG Span Tension (N))
set(H1LineStyle-Colorb)
set(H2LineStyle-Colorg)
set(H3LineStyleColorget(H1Color))
set(H4LineStyleColorget(H2Color))
legend([H1H2H3H4]CS Tautest SpanISG Tautest SpanCS Slackest SpanISG
Slackest SpanLocationBest)
Mesh points
m = 101
n = 101
Appendix D 165
T = zeros(5nm)
Ta = zeros(5nm)
[ixxiyy] = meshgrid(1m1n)
minX3 = 0260200
maxX3 = 0317677
minY3 = -0056640
maxY3 = 0228456
midX3 = 0311641
X3 = minX3 + (ixx-1)(maxX3-minX3)(m-1)
Y3 = minY3 + (iyy-1)(maxY3-minY3)(n-1)
for i = 1n
for j = 1m
if ((X3(ij)lt midX3)ampamp(Y3(ij)gt=(sqrt((0087945^2)-((X3(ij)-0224)^2)-
006395)))ampamp(Y3(ij)lt=(-1sqrt(((00703^2)-((X3(ij)-
024759)^2)))+016664)))||((X3(ij)gt=midX3)ampamp(Y3(ij)gt=(35548X3(ij)-
11134868))ampamp(Y3(ij)lt=(-1(sqrt(((00703^2)-((X3(ij)-024759)^2))))+016664))) mx+b
lt= y lt= circle4
[T(ij)Ta(ij)] =
TwinTenStaticTension(KtoKt1oKt2oD3oD5oX3(ij)Y3(ij)X5oY5o)
else
T(ij) = zeros(511)
Ta(ij) = zeros(511)
end
end
end
figure
Z1 = squeeze(T(1))
surf(X3Y3real(Z1))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(CS DRIVING Tautest Span Tension vs Tensioner Pulley 1 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(CS Span Tension (N))
figure
Z5 = squeeze(T(5))
surf(X3Y3real(Z5))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(CS DRIVING Slackest Span Tension vs Tensioner Pulley 1 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
Appendix D 166
zlabel(CS Span Tension (N))
figure
Za4 = squeeze(Ta(4))
surf(X3Y3real(Za4))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(ISG DRIVING Tautest Span Tension vs Tensioner Pulley 1 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(ISG Span Tension (N))
figure
Za3 = squeeze(Ta(3))
surf(X3Y3real(Za3))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(ISG DRIVING Slackest Span Tension vs Tensioner Pulley 1 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(ISG Span Tension (N))
minX5 = -0037093
maxX5 = 0212509
minY5 = 00362
maxY5 = 0228456
midX5a = 0131965
midX5b = 017729
X5 = minX5 + (ixx-1)(maxX5-minX5)(m-1)
Y5 = minY5 + (iyy-1)(maxY5-minY5)(n-1)
for i = 1n
for j = 1m
if
(X5(ij)ltmidX5a)ampamp(Y5(ij)lt=(0386X5(ij)+0146468))ampamp(Y5(ij)gt=(sqrt((0136525^2)-
(X5(ij)^2))))
[T(ij)Ta(ij)] =
TwinTenStaticTension(KtoKt1oKt2oD3oD5oX3oY3oX5(ij)Y5(ij))
elseif
((X5(ij)gt=midX5a)ampamp(X5(ij)ltmidX5b))ampamp(Y5(ij)gt=00362)ampamp(Y5(ij)lt=(0386X5(ij)+0
146468))
[T(ij)Ta(ij)] =
TwinTenStaticTension(KtoKt1oKt2oD3oD5oX3oY3oX5(ij)Y5(ij))
elseif (X5(ij)gt=midX5b)ampamp(Y5(ij)gt=(sqrt((00703^2)-(((X5(ij)-
024759)^2)))+016664))ampamp(Y5(ij)lt=(0386X5(ij)+0146468))
Appendix D 167
[T(ij)Ta(ij)] =
TwinTenStaticTension(KtoKt1oKt2oD3oD5oX3oY3oX5(ij)Y5(ij))
else
T(ij) = zeros(511)
Ta(ij) = zeros(511)
end
end
end
figure
Z1 = squeeze(T(1))
surf(X5Y5real(Z1))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(CS DRIVING Tautest Span Tension vs Tensioner Pulley 2 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(CS Span Tension (N))
figure
Z5 = squeeze(T(5))
surf(X5Y5real(Z5))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(CS DRIVING Slackest Span Tension vs Tensioner Pulley 2 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(CS Span Tension (N))
figure
Za4 = squeeze(Ta(4))
surf(X5Y5real(Za4))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(ISG DRIVING Tautest Span Tension vs Tensioner Pulley 2 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(ISG Span Tension (N))
figure
Za3 = squeeze(Ta(3))
surf(X5Y5real(Za3))
ZLim([50 500])
axis tight
Appendix D 168
colormap jet
colorbar
title(ISG DRIVING Slackest Span Tension vs Tensioner Pulley 2 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(ISG Span Tension (N))
D12 TwinTenStaticTensionm
The function TwinTenStaticTensionm simulates the static model of the B-ISG system with a
Twin Tensioner This function returns 3 vectors the static tension of each belt span in the
crankshaft- and ISG-driving phases of operation and the angle of displacement of each rigid
body in the ISG- driving phase It takes the input parameters kt kt1 kt2 for the tensioner arm
stiffness D3 and D5 for the tensioner pulley diameters and X3Y3 X5 and Y5 for the tensioner
arm pulley coordinates The function is called in the parametric analysis solution script
TwinMainm and in the optimization solution script OptimizationTwinm
D2 Optimization
D21 OptimizationTwinm
The following script is for the main function OptimizationTwinm It performs an optimization
search for the B-ISG system with a Twin Tensioner It takes an input for a parameter vector
containing values for the design variables The program calls the objective function
objfunTwinm and the constraint function confunTwinm The program can perform a genetic
algorithm (GA) optimization search or a hybrid GA optimization that includes a localized search
The optimal solution vector corresponding to the design variables and the optimal objective
function value is returned The program inputs the optimized values for the design variables into
TwinTenStaticTensionm This called function returns the optimized static state of tensions for
the crankshaft- and ISG- driving phases and for the angle of displacement of the rigid bodies in
the ISG driving phase
______________________________________________________________________________
clc
clear all
Initial values for variables
Kto = 20626
Kt1o = 10314
Kt2o = 16502
X3o = 0292761
Y3o = 0087
X5o = 012057
Appendix D 169
Y5o = 009193
w0 =[Kto Kt1o Kt2o X3o Y3o X5o Y5o] Start Point (row vector)
Variable ranges
minKt = Kto - 1Kto
maxKt = Kto + 1Kto
minKt1 = Kt1o - 1Kt1o
maxKt1 = Kt1o + 1Kt1o
minKt2 = Kt2o - 1Kt2o
maxKt2 = Kt2o + 1Kt2o
minX3 = 0260200
maxX3 = 0317677
minY3 = -0056640
maxY3 = 0228456
minX5 = -0037093
maxX5 = 0212509
minY5 = 00362
maxY5 = 0228456
ObjectiveFunction = objfunTwin
nvars = 7 Number of variables
ConstraintFunction = confunTwin
Uncomment next two lines (and comment the two functions after them) to use GA algorithm
for optimization
options = gaoptimset(InitialPopulationw0PopInitRange[minKt minKt1 minKt2 minX3
minY3 minX5 minY5 maxKt maxKt1 maxKt2 maxX3 maxY3 maxX5
maxY5]PopulationSize100Displayfinal)
[wfvalexitflagoutput] =
ga(ObjectiveFunctionnvars[][][][][][]ConstraintFunctionoptions)
fminconOptions = optimset(DisplayiterLargeScaleoff) Largescale off since gradient not
provided
options = gaoptimset(InitialPopulationw0PopInitRange[minKt minKt1 minKt2 minX3
minY3 minX5 minY5 maxKt maxKt1 maxKt2 maxX3 maxY3 maxX5
maxY5]PopulationSize100HybridFcnfmincon fminconOptions)
[wfvalexitflagoutput] = ga(ObjectiveFunctionnvars[][][][][][]ConstraintFunctionoptions)
[TTaThetaDegA] = TwinTenStaticTension(w(1)w(2)w(3)w(4)w(5)w(6)w(7))
D22 confunTwinm
The constraint function confunTwinm is used by the main optimization program to ensure
input values are constrained within the prescribed regions The function makes use of inequality
constraints for seven constrained variables corresponding to the design variables It takes an
input vector corresponding to the design variables and returns a data set of the vector values that
satisfy the prescribed constraints
Appendix D 170
D23 objfunTwinm
This function is the objective function for the main optimization program It outputs a value for
a weighted objective function or a non-weighted objective function relating the optimization of
the static tension The program takes an input vector containing a set of values for the design
variables that are within prescribed constraints The description of the function is similar to
TwinTenStaticTensionm but differs in the fact that it only returns a scalar value which is the
value of the objective function
171
VITA
ADEBUKOLA OLATUNDE
Email adebukolaolatundegmailcom
Adebukola Olatunde is a graduate research student at the University of Toronto in Toronto
Ontario Canada She obtained a Bachelor‟s Degree in Mechanical Engineering from McMaster
University in Hamilton Ontario Canada in 2002 Upon graduation she pursued a graduate
degree in mechanical engineering at the University of Toronto with a specialization in
mechanical systems dynamics and vibrations and environmental engineering In September
2008 she completed the requirements for the Master of Applied Science degree in Mechanical
Engineering She has held the position of teaching assistant for undergraduate courses in
dynamics and vibrations Adebukola has completed course work in professional education She
is a registered member of professional engineering organizations including the Professional
Engineer‟s of Ontario Engineer-in-Training program the Canadian Society of Mechanical
Engineers and the National Society of Black Engineers She intends to practice as a professional
engineering consultant in mechanical design
v
CONTENTS
ABSTRACT ii
DEDICATION iii
ACKNOWLEDGEMENTS iv
CONTENTS v
LIST OF TABLES ix
LIST OF FIGURES xi
LIST OF SYMBOLS xvi
Chapter 1 INTRODUCTION 1
11 Background 1
12 Motivation 3
13 Thesis Objectives and Scope of Research 4
14 Organization and Content of Thesis 5
Chapter 2 LITERATURE REVIEW 7
21 Introduction 7
22 B-ISG System 8
221 ISG in Hybrids 8
2211 Full Hybrids 9
2212 Power Hybrids 10
2213 Mild Hybrids 11
2214 Micro Hybrids 11
222 B-ISG Structure Location and Function 13
2221 Structure and Location 13
2222 Functionalities 14
23 Belt Drive Modeling 15
24 Tensioners for B-ISG System 18
241 Tensioners Structures Function and Location 18
242 Systematic Review of Tensioner Designs for a B-ISG System 20
25 Summary 24
vi
Chapter 3 MODELING OF B-ISG SYSTEM 25
31 Overview 25
32 B-ISG Tensioner Design 25
33 Geometric Model of a B-ISG System with a Twin Tensioner 27
34 Equations of Motion for a B-ISG System with a Twin Tensioner 32
341 Dynamic Model of the B-ISG System 32
3411 Derivation of Equations of Motion 32
3412 Modeling of Phase Change 41
3413 Natural Frequencies Mode Shapes and Dynamic Responses 42
3414 Crankshaft Pulley Driving Torque Acceleration and Displacement 44
3415 ISG Pulley Driving Torque Acceleration and Displacement 46
3416 Tensioner Arms Dynamic Torques 48
3417 Dynamic Belt Span Tensions 49
342 Static Model of the B-ISG System 49
35 Simulations 50
351 Geometric Analysis 51
352 Dynamic Analysis 52
3521 Natural Frequency and Mode Shape 54
3522 Dynamic Response 58
3523 ISG Pulley and Crankshaft Pulley Torque Requirement 61
3524 Tensioner Arm Torque Requirement 62
3525 Dynamic Belt Span Tension 63
353 Static Analysis 66
36 Summary 69
Chapter 4 PARAMETRIC ANALYSIS OF A B-ISG TWIN TENSIONER 71
41 Introduction 71
42 Methodology 71
43 Results and Discussion 74
431 Influence of Tensioner Arm Stiffness on Static Tension 74
432 Influence of Tensioner Pulley Diameter on Static Tension 78
433 Influence of Tensioner Pulley 1 Coordinates on Static Tension 80
434 Influence of Tensioner Pulley 2 Coordinates on Static Tension 86
vii
44 Conclusion 92
Chapter 5 OPTIMIZATION OF A B-ISG TWIN TENSIONER 95
51 Optimization Problem 95
511 Selection of Design Variables 95
512 Objective Function amp Constraints 97
52 Optimization Method 100
521 Genetic Algorithm 100
522 Hybrid Optimization Algorithm 101
53 Results and Discussion 101
531 Parameter Settings amp Stopping Criteria for Simulations 101
532 Optimization Simulations 102
533 Discussion 106
54 Conclusion 109
Chapter 6 CONCLUSION AND RECOMMENDATIONS111
61 Summary 111
62 Conclusion 112
63 Recommendations for Future Work 113
REFERENCES 116
APPENDICIES 123
A Passive Dual Tensioner Designs from Patent Literature 123
B B-ISG Serpentine Belt Drive with Single Tensioner Equation of Motion 138
C MathCAD Scripts 145
C1 Geometric Analysis 145
C2 Dynamic Analysis 152
C3 Static Analysis 161
D MATLAB Functions amp Scripts 162
D1 Parametric Analysis 162
D11 TwinMainm 162
D12 TwinTenStaticTensionm 168
D2 Optimization 168
D21 OptimizationTwinm - Optimization Function 168
viii
D22 confunTwinm 169
D23 objfunTwinm 170
VITA 171
ix
LIST OF TABLES
21 Passive Dual Tensioner Designs from Patent Literature
31 Selected Contact Point Types on the ith Pulley and Types for the ith Belt Span
32 Coordinate Points for Pulley Centres and Twin Tensioner Pivot
33 Geometric Results of B-ISG System with Twin Tensioner
34 Data for Input Parameters used in Dynamic and Static Computations
35 Static Solution for Belt Span Tensions in Crankshaft and ISG Driving Cases for a B-ISG
Serpentine Belt Drive with a Single Tensioner
36 Static Solution for Belt Span Tensions in Crankshaft and ISG Driving Cases for a B-ISG
Serpentine Belt Drive with a Twin Tensioner
41 Initial Values Increments and Ranges for Parameters of Twin Tensioner
51 Summary of Parametric Analysis Data for Twin Tensioner Properties
52a GA Optimization Results for Twin Tensioner Parameters and Objective Function
52b Computations for Tensions and Angles from GA Optimization Results
53a Hybrid Optimization Results for Twin Tensioner Parameters and Objective Function
53b Computations for Tensions and Angles from Hybrid Optimization Results
54a Non-Weighted Optimization Results for Twin Tensioner Parameters and Objective
Function
54b Computations for Tensions and Angles from Non-Weighted Optimizations
x
55 Weighted Optimization Results for Static Tensions for Optimal B-ISG System with a
Twin Tensioner
56 Non-Weighted Optimization Results for Static Tensions for Optimal B-ISG System with a
Twin Tensioner
xi
LIST OF FIGURES
21 Hybrid Functions
31 Schematic of the Twin Tensioner
32 B-ISG Serpentine Belt Drive with Twin Tensioner
33 Angles Coordinates and Possible Belt Contact Points for the ith and i+1th Pulleys
34 Twin Tensioner Free Body Diagram of a Four-Degree of Freedom System
35 Free Body Diagram for Non-Tensioner Pulleys
36a ISG Driving Case Natural Frequency of System and Mode Shapes for Responsive Rigid
Bodies
36b ISG Driving Case First Mode Responses
36c ISG Driving Case Second Mode Responses
37a Crankshaft Driving Case Natural Frequency of System and Mode Shapes for Responsive
Rigid Bodies
37b Crankshaft Driving Case First Mode Responses
37c Crankshaft Driving Case Second Mode Responses
38 Crankshaft Pulley Dynamic Response (for crankshaft driven case)
39 ISG Pulley Dynamic Response (for ISG driven case)
310 Air Conditioner Pulley Dynamic Response
311 Tensioner Pulley 1 Dynamic Response
xii
312 Tensioner Pulley 2 Dynamic Response
313 Tensioner Arm 1 Dynamic Response
314 Tensioner Arm 2 Dynamic Response
315 Required Driving Torque for the ISG Pulley
316 Required Driving Torque for the Crankshaft Pulley
317 Dynamic Torque for Tensioner Arm 1
318 Dynamic Torque for Tensioner Arm 2
319 Span 1 (between crankshaft and air conditioner) Dynamic Belt Span Tension
320 Span 2 (between air conditioner and tensioner 1) Dynamic Belt Span Tension
321 Span 3 (between tensioner 1 and ISG) Dynamic Belt Span Tension
322 Span 4 (between ISG and tensioner 2) Dynamic Belt Span Tension
323 Span 5 (between tensioner 2 and crankshaft) Dynamic Belt Span Tension
324 B-ISG Serpentine Belt Drive with Single Tensioner
41a Regions 1 and 2 and their Associated Guidelines for Coordinates of Tensioner Pulleys 1
amp 2
41b Regions 1 and 2 in Cartesian Space
42 Parametric Analysis for Coupled Stiffness Arm Constant kt (N∙mrad)
43 Parametric Analysis for Stiffness of Arm 1 kt1 (N∙mrad)
44 Parametric Analysis for Stiffness of Arm 2 kt2 (N∙mrad)
xiii
45 Parametric Analysis for Pulley 1 Diameter D3 (m)
46 Parametric Analysis for Pulley 2 Diameter D5 (m)
47 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Tautest Span
Tension in Crankshaft Driving Case
48 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Slackest Span
Tension in Crankshaft Driving Case
49 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Tautest Span
Tension in ISG Driving Case
410 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Slackest Span
Tension in ISG Driving Case
411 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Tautest Span
Tension in Crankshaft Driving Case
412 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Slackest Span
Tension in Crankshaft Driving Case
413 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Tautest Span
Tension in ISG Driving Case
414 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Slackest Span
Tension in ISG Driving Case
51 Static Stability of the B-ISG Twin Tensioner Based on the Angular Displacement of
Tensioner Arms 1 and 2
A1 Proposed design by Bayerische Motoren Werke AG corresponding to patent nos
EP1420192-A2 and DE10253450-A1
A2a First of four proposed designs (and its various configurations) by Bosch GMBH
corresponding to patent no WO0026532-A1
A2b Second of four proposed designs by Bosch GMBH corresponding to patent no
WO0026532-A1
A2c Third of four proposed designs (and its various configurations) by Bosch GMBH
corresponding to patent no WO0026532-A1
A2d Fourth of four proposed designs (and its various configurations) by Bosch GMBH
corresponding to patent no WO0026532-A1
A3 Proposed design by Daimler Chrysler AG corresponding to patent no DE10324268-A1
A4 Proposed design by Dayco Products LLC corresponding to patent no US6942589-B2
xiv
A5 Proposed design by Gates Corp corresponding to patent no WO2003038309-A
A6 Proposed design by General Motors Corp corresponding to patent no US20060287146-
A1
A7 Proposed design by INA Schaeffler KG corresponding to patent no DE10044645-A1
A8a First of three proposed designs by INA Schaeffler KG corresponding to patent no
DE10159073-A1
A8b Second of three proposed designs by INA Schaeffler KG corresponding to patent no
DE10159073-A1
A8c Third of three proposed designs by INA Schaeffler KG corresponding to patent no
DE10159073-A1
A9 Proposed design by INA Schaeffler KG corresponding to patent no DE10359641-A1
A10 Proposed design by INA Schaeffler KG corresponding to patent no EP1723350-A1
A11 Proposed design by INA Schaeffler KG corresponding to patent no EP1738093-A1
A12 Proposed design by INA Schaeffler KG corresponding to patent no DE102004012395-
A1
A13 Proposed design by INA Schaeffler KG corresponding to patent nos DE102005017038-
A1and WO2006108461-A1
A14 Proposed design by Litens Automotive GMBH et al corresponding to patent no
US20010007839-A1
A15 Proposed design by Mitsubishi Jidosha Eng KK and Mitsubishi Motor Corp
corresponding to patent no JP2005083514-A
A16 Proposed design by Nissan Motor Co Ltd corresponding to patent no JP3565040-B2
A17 Proposed design by NTN Corp corresponding to patent no JP2006189073-A
A18 Proposed design by Valeo Equipment Electriques Moteur corresponding to patent nos
EP1658432 and WO2005015007
B1 Single Tensioner B-ISG System
B2 Free-body Diagram of ith Pulley
xv
B3 Free-body Diagram of Single Tensioner
C1 Schematic of B-ISG System with Twin Tensioner
C2 Possible Contact Points
xvi
LIST OF SYMBOLS
Latin Letters
A Belt cord cross-sectional area
C Damping matrix of the system
cb Belt damping
119888119894119887 Belt damping constant of the ith belt span
119914119946119946 Damping matrix element in the ith row and ith column
ct Damping acting between tensioner arms 1 and 2
cti Damping of the ith tensioner arm
DCS Diameter of crankshaft pulley
DISG Diameter of ISG pulley
ft Belt transition frequency
H(n) Phase change function
I Inertial matrix of the system
119920119938 Inertial matrix under ISG driving phase
119920119940 Inertial matrix under crankshaft driving phase
Ii Inertia of the ith pulley
Iti Inertia of the ith tensioner arm
119920120784120784 Submatrix of inertial matrix I
j Imaginary coordinate (ie (-1)12
)
K Stiffness matrix of the system
xvii
119896119887 Belt factor
119870119887 Belt cord stiffness
119896119894119887 Belt stiffness constant of the ith belt span
kt Spring stiffness acting between tensioner arms 1 and 2
kti Coil spring of the ith tensioner arm
119922120784120784 Submatrix of stiffness matrix K
Lfi Lbi Lengths of possible belt span connections from the ith pulley
Lti Length of the ith tensioner arm
Modeia Mode shape of the ith rigid body in the ISG driving phase
Modeic Mode shape of the ith rigid body in the crankshaft driving phase
n Engine speed
N Motor speed
nCS rpm of crankshaft pulley
NF Motor speed without load
nISG rpm of ISG pulley
Q Required torque matrix
qc Amplitude of the required crankshaft torque
QcsISG Required torque of the driving pulley (crankshaft or ISG)
Qm Required torque matrix of driven rigid bodies
Qti Dynamic torque of the ith tensioner arm
Ri Radius of the ith pulley
T Matrix of belt span static tensions
xviii
Trsquo Dynamic belt tension matrix
119931119940 Damping matrix due to the belt
119931119948 Stiffness matrix due to the belt
Ti Tension of the ith belt span
To Initial belt tension for the system
Ts Stall torque
Tti Tension for the neighbouring belt spans of the ith tensioner pulley
(XiYi) Coordinates of the ith pulley centre
XYfi XYbi XYfbi
XYbfi Possible connection points on the ith pulley leading to the ith belt span
XYf2i XYb2i
XYfb2i XYbf2i Possible connection points on the ith pulley leading to the (i-1)th belt span
Greek Letters
αi Angle between the datum and the line connecting the ith and (i+1)th pulley
centres
βji Angle of orientation for the ith belt span
120597θti(t) 120579 ti(t)
120579 ti(t)
Angular displacement velocity and acceleration (rotational coordinate) of the
ith tensioner arm
120637119938 General coordinate matrix under ISG driving phase
120637119940 General coordinate matrix under crankshaft driving phase
θfi θbi Angles between the datum and the belt connection spans with lengths Lfi and
Lbi respectively
Θi Amplitude of displacement of the ith pulley
xix
θi(t) 120579 i(t) 120579 i(t) Angular position velocity and acceleration (rotational coordinate) of the ith
pulley
θti Angle of the ith tensioner arm
θtoi Initial pivot angle of the ith tensioner arm
θm Angular displacement matrix of driven rigid bodies
Θm Amplitude of displacement of driven rigid bodies
ρ Belt cord density
120601119894 Belt wrap angle on the ith pulley
φmax Belt maximum phase angle
φ0deg Belt phase angle at zero frequency
ω Frequency of the system
ωcs Angular frequency of crankshaft pulley
ωISG Angular frequency of the ISG pulley
120654119951 Natural frequency of system
1
CHAPTER 1 INTRODUCTION
11 Background
Belt drive systems are the means of power transmission in conventional automobiles The
emergence of hybrid technologies specifically the Belt-driven Integrated Starter-generator (B-
ISG) has placed higher demands on belt drives than ever before The presence of an integrated
starter-generator (ISG) in a belt transmission places excessive strain on the belt leading to
premature belt failure This phenomenon has motivated automotive makers to design a tensioner
that is suitable for the B-ISG system
The belt drive is also known interchangeably as the front-end accessory drive-belt (FEAD) the
belt accessory-drive system (BAS) or the belt transmission system In a traditional setting the
role of this system is to transmit torque generated by an internal combustion engine (ICE) in
order to reliably drive multiple peripheral devices mounted on the engine block The high speed
torque is transmitted through a crankshaft pulley to a serpentine belt The serpentine belt is a
single continuous member that winds around the driving and driven accessory pulleys of the
drive system Serpentine belts used in automotive applications consist of several layers The
load-bearing layer is a flexible member consisting of high stiffness fibers [1] It is covered by a
protective layer to guard against mechanical damage and is bound below by a visco-elastic layer
that provides the required shock absorption and grip against the rigid pulleys [1] The accessory
devices may include an alternator power steering pump water pump and air conditioner
compressor among others
Introduction 2
The B-ISG system is a transmission system characteristic to micro-hybrid automobiles It is akin
to traditional belt drives differing in the fact that an electric motor called an integrated starter-
generator (ISG) replaces the original alternator re-starts the engine from idle speed and provides
braking regeneration [2] The re-start function of the micro-hybrid transmission is known as
stop-start In the B-ISG setting the ISG is mounted on the belt drive The ISG produces a speed
of approximately 2000 to 2500rpm in order to spin the engine at approximately 750rpm and
upwards to produce an instantaneous start in the start-stop process [3] The high rotations per
minute (rpm) produced by the ISG consistently places much higher tension requirements on the
belt than when the crankshaft is driving the belt It is preferable not to exceed a range of 600N to
800N of tension on the belt since this exceeds the safe operating conditions of belts used in most
traditional drive systems [4] The traditional belt drive system‟s tensioner a single-arm
tensioner does not suitably reduce the high belt tension nor provide enough tension in the slack
belts spans occurring in the ISG phase of operation for the B-ISG system
In order for the belt to transfer torque in a drive system its initial tension must be set to a value
that is sufficient to keep all spans rigid This value must not be too low as to allow any one span
to be slack during the drive‟s phases of operation Furthermore the belt must not be ldquoinstalled
with too high a tensionrdquo since this can lead to ldquopremature failure of the bearings supporting the
drive and driven pulleys and of the belt itselfrdquo [5] The presence of a tensioning mechanism in
an automotive belt drive allows for an enhanced belt life and performance since pre-tensioning
of the belt is normally not sufficient for all phases of belt drive operation A tensioner allows for
the system to cope with moderate to severe changes in belt span tensions
Introduction 3
Traditional automotive tensioners for belt drives of an ICE consist of a single spring-loaded
arm This type of tensioner is normally designed to provide a passive response to changes in belt
span tension The introduction of the ISG electric motor into the traditional belt drive with a
single-arm tensioner results in the presence of excessively slack spans and excessively tight
spans in the belt The tension requirements in the ISG-driving phase which differ from the
crankshaft-driving phase are poorly met by a traditional single-arm passive tensioner
Tensioners can be divided into two general classes passive and active In both classes the
single-arm tensioner design approach is the norm The passive class of tensioners employ purely
mechanical power to achieve tensioning of the belt while the active class also known as
automatic tensioners typically use some sort of electronic actuation Automatic tensioners have
been employed by various automotive manufacturers however ldquosuch devices add mass
complication and cost to each enginerdquo [5]
12 Motivation
The motivation for the research undertaken arises from the undesirable presence of high belt
tension in automotive belt drives Manufacturers of automotive belt drives have presented
numerous approaches for tension mechanism designs As mentioned in the preceding section
the automation of the traditional single-arm tensioner has disadvantages for manufacturers A
survey of the literature reveals that few quantitative investigations in comparison to the
qualitative investigations provided through patent literature have been conducted in the area of
passive and dual tensioner configurations As such the author of the research project has selected
to investigate the performance of a passive twin-arm tensioner design The theoretical tensioner
Introduction 4
configuration is motivated by research and developments of industry partner Litens
Automotivendash a manufacturer of automotive belt drive systems and components Litens‟
specialty in automotive tensioners has provided a basis for the research work conducted
13 Thesis Objectives and Scope of Research
The objective of this project is to model and investigate a system containing a passive twin-arm
tensioner in a B-ISG serpentine belt drive where the driving pulley alternates between a
crankshaft pulley and an ISG pulley The modeling of a serpentine belt drive system is in
continuation of the work done by post-doctoral fellow Zhen Mu in development of the priority
software known as FEAD at the University of Toronto Firstly for the B-ISG system with a
twin-arm tensioner the geometric state and its equations of motion (EOM) describing the
dynamic and static states are derived The modeling approach was verified by deriving the
geometric properties and the EOM of the system with a single tensioner arm and comparing its
crankshaft-phase‟s simulation results with FEAD software simulations This also provides
comparison of the new twin-arm tensioner belt drive model with the former single-arm tensioner
equipped belt drive model Secondly the model for the static system is investigated through
analysis of the tensioner parameters Thirdly the design variables selected from the parametric
analysis are used for optimization of the new system with respect to its criteria for desired
performance
Introduction 5
14 Organization and Content of the Thesis
This thesis presents the investigation of a passive twin-arm tensioner design in a B-ISG
serpentine belt drive system which is distinguished by having its driving pulley alternate
between a crankshaft pulley and an ISG pulley
Chapter 2 presents the literature reviewed relevant to the area of the thesis topic The context of
the research discusses the function and location of the ISG in hybrid technologies in order to
provide a background for the B-ISG system The attributes of the B-ISG are then discussed
Subsequently a description is given of the developments made in modeling belt drive systems
At the close of the chapter the prior art in tensioner designs and investigations are discussed
The third chapter describes the system models and theory for the B-ISG system with a twin-arm
tensioner Models for the geometric properties and the static and dynamic cases are derived The
simulation results of the system model are presented
Then the fourth chapter contains the parametric analysis The methodologies employed results
and a discussion are provided The design variables of the system to be considered in the
optimization are also discussed
The optimization of a B-ISG system with a passive twin-arm tensioner is presented in Chapter 5
The evaluation of optimization methods results of optimization and discussion of the results are
included Chapter 6 concludes the thesis work in summarizing the response to the thesis
Introduction 6
objectives and concluding the results of the investigation of the objectives Recommendations for
future work in the design and analysis of a B-ISG tensioner design are also described
7
CHAPTER 2 LITERATURE REVIEW
21 Introduction
This literature review justifies the study of the thesis research the significance of the topic and
provides the overall framework for the project The design of a tensioner for a Belt-driven
Integrated Starter-generator (B-ISG) system is a link in the chain of power transmission
developments in hybrid automobiles This chapter will begin with the context of the B-ISG
followed by a review of the hybrid classifications and the critical role of the ISG for each type
The function location and structure of the B-ISG system are then discussed Then a discussion
of the modeling of automotive belt transmissions is presented A systematic review of the prior
art and current state of tensioning mechanisms for B-ISG systems amalgamates the literature and
research evidence relevant to the thesis topic which is the design of a B-ISG tensioner
The Belt-driven Integrated Starter-generator (B-ISG) system is a part of a hybrid class that is
distinguished from other hybrid classes by the structure functions and location of its ISG The
B-ISG unit is a hybrid technology applied to traditional automotive belt drives The use of a B-
ISG system to achieve a start-stop function in the car engine is estimated to cut fuel consumption
in conventional automobiles by up to ten percent and thus reduce CO2 emissions [6]
Environmental and legislative standards for reducing CO2 emissions in vehicles have called for
carmakers to produce less polluting and more efficient vehicle powertrain systems [7] The
transition to bdquocleaner‟ cars makes room for the introduction of the ISG machine into conventional
automotive belt drives [8] The reduction of CO2 emissions and the similarity of the B-ISG
Literature Review 8
transmission to that of conventional cars provide the motivation for the thesis research
Consequently the micro-hybrid class of cars is especially discussed in the literature review since
it contains the B-ISG type of transmission system The micro-hybrid class is one of several
hybrid classes
A look at the performance of a belt-drive under the influence of an ISG is rooted in the
developments of hybrid technology The distinction of the ISG function and its location in each
hybrid class is discussed in the following section
22 B-ISG System
221 ISG in Hybrids
This section of the review discusses the standard classes of hybrid cars which are full power
mild and micro- hybrids Special attention is given to hybrid vehicle architectures involving
internal combustion engines (ICEs) as the main power source This is done for the sake of
comparison between hybrid classes since the ICE is the standard power source for B-ISG micro-
hybrids which is the focus of the research The term conventional car vehicle or automobile
henceforth refers to a vehicle powered solely by a gas or diesel ICE
A hybrid vehicle has a drive system that uses a combination of energy devices This may include
an ICE a battery and an electric motor typically an ISG Two systems exist in the classification
of hybrid vehicles The older system of classification separates hybrids into two classes series
hybrids and parallel hybrids In the older system many modern hybrid vehicles have modes of
operation matching both categories classifying them under either of the two classes [9] The
Literature Review 9
new system of classification has four classes full power mild and micro Under these classes
vehicles are more often under a sole category [9] In both systems an ICE may act as the primary
source of power otherwise it may be a fuel cell The fuel used by the ICE may be gas (petrol)
diesel or an alternative fuel such as ethanol bio-diesel or natural gas
2211 Full Hybrids
In a full hybrid car the ICE is used to power the integrated starter-generator (ISG) which stores
electrical energy in the batteries to be used to power an electric traction motor [8] The electric
traction motor is akin to a second ISG as it generates power and provides torque output It also
supplies an extra boost to the wheels during acceleration and drives up steep inclines A full
hybrid vehicle is able to move by electrical power only It can be driven by the ISG powering
the electric traction motor without the engine running This silent acceleration known as electric
launch is normally employed when accelerating from standstill [9] Full hybrids can generate
and consume energy at the same time Full hybrid vehicles also use regenerative braking [8]
The ISG allows this by converting from an electric traction motor to a generator when braking or
decelerating The kinetic energy from the car‟s motion is then turned into electricity and stored
in the batteries For full hybrids to achieve this they often use break-by-wire a form of
electronically controlled braking technology
A high-voltage (ie 36- or 42-volt) ISG is employed in full hybrids to start the ICE It spins the
engine more than 900 rpm whereas conventional 12-volt starter motors spin the engine at
approximately 250 rpm [9] Thus the full hybrid vehicle is able to have an instantaneous start In
full hybrids the ISG is placed in the position of the flywheel and can have its motion decoupled
Literature Review 10
from the engine [9] The ISG device also allows full hybrids to have engine start-stop also called
an idle-stop ability The idle-stop function refers to when the engine shuts down as soon as a
vehicle stops from its ICE driving mode which saves on the fuel it normally burns while idling
[8] The vehicle returns to the engine driving mode of operation by way of the ISG‟s start-up of
the crankshaft which restarts the engine in less than 300 milliseconds [9] In summary at
standstill the tachometer of the engine drops to 0 rpm since the engine has ceased the engine is
started only when needed which is often several seconds after acceleration has begun The
engine start-stop feature is achieved by way of an electronic control system that shuts off the ICE
when it is not needed to assist in driving the wheels or to produce electricity for recharging the
batteries The start-stop feature by itself is estimated to produce a ten percent fuel gain in hybrids
over conventional vehicles particularly in urban driving conditions [9] Since the ICE is
required to provide only the average horsepower used by the vehicle the engine is downsized in
comparison to a conventional automobile that obtains all its power from an ICE Frequently in
full hybrids the ICE uses an alternative operating strategy such as the Atkinson Cycle which has
a higher efficiency while having a lower power output Examples of full hybrids include the
Ford Escape and the Toyota Prius [9]
2212 Power Hybrids
Akin to the full hybrid the ISG of the power hybrid enables the same features electric launch
regenerative braking and engine idle-stop The distinguishing characteristic from full hybrids is
the ICE is not downsized to meet only the average power demand [9] Thus the engine of a
power hybrid is large and produces a high amount of horsepower compared to the former
Overall a power hybrid has the assist of a full size ICE and therefore has more torque and a
Literature Review 11
greater acceleration performance than a full hybrid or a conventional vehicle with the same size
ICE [9] The Lexus RX400h unit is an example of a power hybrid [9]
2213 Mild Hybrids
In the hybrid types discussed thus far the ISG is positioned between the engine and transmission
to provide traction for the wheels and for regenerative braking Often times the armature or rotor
of the electric motor-generator which is the ISG replaces the engine flywheel in full and power
hybrids [9] In the case of the mild hybrid the ISG is not decoupled from the ICE and hence it is
not able to drive the wheels apart from the engine It remains that the ISG shares the same shaft
with the ICE In this environment the electric launch feature does not exist since the ISG does
not turn the wheels independently of the engine and energy cannot be generated and consumed
at the same time However the ISG of the mild hybrid allows for the remaining features of the
full hybrid regenerative braking and engine idle-stop including the fact that the engine is
downsized to meet only the average demand for horsepower Mild hybrid vehicles include the
GMC Sierra pickup and 2003 to 2005 Honda Civic models [9]
2214 Micro Hybrids
Micro hybrid is the category of hybrids that can contain a B-ISG transmission and is also closest
to modern conventional vehicles This class normally features a gas or diesel ICE [9] The
conventional automobile is modified by installing an ISG unit on the mechanical drive in place
of or in addition to the starter motor The starter motor typically 12-volts is removed only in
the case that the ISG device passes cold start testing which is also dependent on the engine size
[10] Various mechanical drives that may be employed include chain gear or belt drives or a
Literature Review 12
clutchgear arrangement The majority of literature pertaining to mechanical driven ISG
applications does not pursue clutchgear arrangements since it is associated with greater costs
and increased speed issues Findings by Henry et al [11] show that the belt drive in
comparison to chain and gear drives has a decreased cost (especially if the ISG is mounted
directly to the accessory drive) has no need for lubrication has less restriction in the packaging
environment and produces very low noise Also mounting the ISG unit on a separate belt from
that linking the accessory pulleys is undesirable since applying the ISG directly to the accessory
belt drive requires less engine transmission or vehicle modifications
As with full power and mild hybrids the presence of the ISG allows for the start-stop feature
The automobile‟s electronic control unit (ECU) is calibrated or engine control circuitry (a
separate ECU) is added to the conventional car in order to shut down the engine when the
vehicle is stopped [12] The control system also controls the charge cycle of the ISG [9] This
entails that it dictates the field current by way of a microprocessor to allow the system to defer
battery charge cycles until the vehicle is decelerating [13] This produces electricity to recharge
the battery primarily during deceleration and braking The B-ISG transmission of a micro hybrid
and its various components are discussed in the subsequent section Examples of micro hybrid
vehicles are the PSA Group‟s Citroen C2 and C3 [14] Ford‟s Fiesta [14] and BMW‟s Mini
Cooper D and various others of BMW‟s European models [15]
Literature Review 13
Figure 21 Hybrid Functions
Source Dr Daniel Kok FFA July 2004 modified [16]
Figure 21 shows that the higher the voltage available to the ISG unit the more hybrid functions
it is capable of performing It is noted that B-ISG transmissions of the micro-hybrid class may
also exceed the typical functions of micro-hybrids For instance Ford‟s HyTrans van (developed
in partnership with Ricardo UK Ltd Valeo SA Gates Corporation and the UK Department for
Transport) uses a B-ISG system and a 42-volt battery The van is diesel-powered and has
characteristics of a mild hybrid such as cold cranks and engine assists [17]
222 B-ISG Structure Location and Function
2221 Structure and Location
The ISG is composed of an electrical machine normally of the inductive type which includes a
stator (stationary part of the ISG) and a rotor (non-stationary part of the ISG) and a converter
comprising of a regulator a modulator switches and filters There are various configurations to
integrate the ISG unit into an automobile power train One configuration situates the ISG
directly on the crankshaft in the place of the present flywheel [11] This set-up is more compact
however it results in a longer power train which becomes a potential concern for transverse-
Literature Review 14
mounted engines [18] An alternative set-up is to have a side-mounted ISG This term is used to
describe the configuration of mounting the electrical device on the side of the mechanical drive
[18] As mentioned in Section 2214 a belt drive is used as the mechanical drive for the thesis
research hence the ISG is belt-mounted and the transmission becomes a belt-driven ISG system
In this arrangement the ISG replaces the alternator [13] and in some cases the starter motor may
be removed This design allows for the functions of the ISG system mentioned in the description
of the ISG role in micro-hybrids [9] The side-mounted ISG specifically the belt-mounted ISG
is more evolutionary to the conventional car since it ldquoallows for a more traditional under-hood
layoutrdquo [11]
2222 Functionalities
The primary duty of the ISG in a micro hybrid specifically in a B-ISG setting is to bring the
engine from rest to normal operating speeds within a time span ranging from 250 to 400 ms [3]
and in some high voltage settings to provide cold starting
The cold starting operation of the ISG refers to starting the engine from its off mode rather than
idle mode andor when the engine is at a low temperature for example -29 to -50 degrees
Celsius [2] If the ISG is used for cold starting the peak torque is determined by the torque
requirement for the cold starting operation of the target vehicle since it is greater than the
nominal torque For this function the ldquomachine has to provide a breakaway torque about 15 [to]
18 times the nominal cranking torque to overcome static torque and rotate the engine from 0 to
[between] 10 [and] 20rpmrdquo [2] This remains to be a challenge for the ISG as the 12-volt
architecture most commonly found in vehicles does not supply sufficient voltage [2] The
introduction of the ISG machine and other electrical units in vehicles encourages a transition
Literature Review 15
from a 12-volt or 14-volt to a 42-volt electrical architecture [19] The transition to 42-volt
architecture brings ldquopotential higher-voltage functionalities that come with an ISG systemrdquo [20]
At present ldquowhen the [ISG] machine cannot provide enough torque for initial cold engine
cranking the conventional starter will [remain] in the system and perform only for the initial
cranking while the stop-start function is taken over by the [ISG] machinerdquo [2] The ISG‟s launch
assist torque the torque required to bring the engine from idle speed to the speed at which it can
develop a higher torque output is 2000 to 2500 rpm for most gas engines [3]
Delphi‟s Energen 5 High Output 12-volt Belt-alternator-starter (or B-ISG) was implemented by
researchers on a 53 L V-8 engine with an automatic transmission in a Chevrolet Silverado truck
[21] The ISG was applied in a belt-mounted configuration and was used only for warm engine
re-starts The results of Wezenbeek et al [21] showed that the starting torque for a re-start by the
12-Volt ISG was 42 Nm ISG‟s have also been used in 14V 36V and 42V architectures [13]
23 Belt Drive Modeling
The modeling of a serpentine belt drive and tensioning mechanism has typically involved the
application of Newtonian equilibrium equations to rigid bodies in order to derive the equations of
motion for the system There are two modes of motion in a serpentine belt drive transverse
motion and rotational motion The former can be viewed as the motion of the belt directed
normal to the direction of the beltpulley contact plane similar to the vibratory motion of a taut
string that is fixed at either end However the study of the rotational motion in a belt drive is the
focus of the thesis research
Literature Review 16
Much work on the mechanics of the belt drive was carried out by Firbank [22] Firbank‟s
models helped to understand belt performance and the influence of driving and driven pulleys on
the tension member The first description of a serpentine belt drive for automotive use was in
1979 by Cassidy et al [23] and since this time there has been an increasing body of knowledge
on the mathematical modeling of serpentine belt drives Ulsoy et al [24] presented a design
methodology to improve the dynamic performance of instability mechanisms for belt tensioner
systems The mathematical model developed by Ulsoy et al [24] coupled the equations of
motion that were obtained through a dynamic equilibrium of moments about a pivot point the
equations of motion for the transverse vibration of the belt and the equations of motion for the
belt tension variations appearing in the transverse vibrations This along with the boundary and
initial conditions were used to describe the vibration and stability of the coupled belt-tensioner
system Their system also considered the geometry of the belt drive and tensioner motion
Hereafter Beikmann et al [25] predicted the belt drive vibration for a system composed of a
driving pulley driven pulley and a dynamic tensioner The authors coupled the linear equations
of transverse motion for the respective belt spans with the equations of motion for pulleys and a
tensioner This was used to form the free response of the system and evaluate its response
through a closed-form solution of the system‟s natural frequencies and mode shapes
A complex modal analysis of a serpentine belt drive system was carried out by Kraver et al [26]
to determine the effect of damping on rotational vibration mode solutions The equations of
motion developed for a multi-pulley flat belt system with viscous damping and elastic
Literature Review 17
properties including the presence of a rotary tensioner were manipulated to carry out the modal
analysis
Beikmann et al [27] also derived a nonlinear model to predict the operating state of a belt-
tensioner system by way of nonlinear numerical methods and an approximated linear closed-
form method The authors used this strategy to develop a single design parameter referred to as
a tensioner constant to measure the effectiveness of the tensioning mechanism in relation to its
operating state from a reference state The authors considered the steady state tensions in belt
spans as a result of accessory loads belt drive geometry and tensioner properties
Zhang and Zu [28] conducted a modal analysis for the response of a linear serpentine belt drive
system A non-iterative approach was used to explicitly form the equations for the system‟s
natural frequencies An exact closed-form expression for the dynamic response of the system
using eigenfunction expansion was derived with the system under steady-state conditions and
subject to harmonic excitation
The work conducted by Balaji and Mockensturm [29] considered a front-end accessory drive
(FEAD) with a decoupler or isolator attached to a pulley The rotational response for the FEAD
was found analytically by considering the system to be piecewise linear about the equilibrium
angular deflections The effect of their nonlinear terms was considered through numerical
integration of the derived equations of motion by way of the iterative methodndash fourth order
Runge-Kutta The authors in this case considered the longitudinal (ie rotational) vibration of
the belt spans only
Literature Review 18
The first to carry out the analysis of a serpentine belt drive system containing a two-pulley
tensioner was Nouri in 2005 [30] Nouri found the closed-form analytical solution of a
serpentine belt drive with a two-pulley tensioner for the case of sinusoidal excitation He
employed Runge Kutta method as well to solve the equations of motion to find the response of
the system under a general input from the crankshaft The author‟s work also included the
optimization of the tensioner design in order to minimize belt span vibrations due to crankshaft
excitation Furthermore the author applied active control techniques to the tensioner in a belt
drive system
The works discussed have made significant contributions to the research and development into
tensioner systems for serpentine belt drives These lead into the requirements for the structure
function and location of tensioner systems particularly for B-ISG transmissions
24 Tensioners for B-ISG System
241 Tensioners Structure Function and Location
Literature shows that the improvement of a serpentine belt life in a B-ISG system centers on the
tensioning mechanism redesign This mechanism as shown by researchers including
Wezenbeek et al [21] and Henry et al [11] is crucial in establishing the least tension in the belt
(above a zero value) in order to guard against failure by way of slip due to slack spans in the belt
and oscillations during engine re-start It is noted by Firbank [22] that the mechanics of a belt-
drive ldquois based on the idea that belt behaviour is governed by the elastic extension or contraction
of the belt arising from tension variationsrdquo [22] these variations may be compensated for by an
adjustable tensioner
Literature Review 19
The two types of tensioners are passive and active tensioners The former permits an applied
initial tension and then acts as an idler and normally employs mechanical power and can include
passive hydraulic actuation This type is cheaper than the latter and easier to package The latter
type is capable of continually adjusting the belt tension since it permits a lower static tension
Active tensioners typically employ electric or magnetic-electric actuation andor a combination
of active and passive actuators such as electrical actuation of a hydraulic force
Conventional belt tensioners comprise of a single tensioner arm that is fitted with a sole idler
pulley to engage a serpentine belt [31] A radial bearing is used to rotatably connect the idler
pulley to the tensioner arm [31] The tensioner arm is mounted on a pivot pin that is wrapped by
a bushing and is free to rotate [31] The pin covered by the bushing is fixed to the engine
housing [31] A rotary spring is wrapped about the bearing pin and bushing to provide a pre-
tension force to the belt via the tensioner arm and idler pulley thus taking up the slack due to the
changes in belt length [31] When the belt undergoes stretch under a load the spring drives the
tensioner arm and idler pulley further into the belt [31] Belt tension changes under the modes of
operation which can include when the crankshaft (or driving pulley) abruptly decelerates from a
steady-state condition and auxiliary components continue to rotate still in their own inherent
inertia and thus become the primary drivers [31] These fluctuations in belt tension lead to belt
flutter or skip and slip that may damage other components present in the belt drive [31]
Locating the tensioner on the slack side of the belt is intended to lower the initial static tension
[11] In conventional vehicles the engine always drives the alternator so the tensioner is located
in the belt span that links the crankshaft and alternator pulleys In a B-ISG setting the slack span
Literature Review 20
of the belt alternates between the driving mode of the ISG and the driving mode of the crankshaft
[32] Research by Henry et al [11] and also the summary of prior art for tensioners in Table
21 show that placing the idlertensioner pulley in the slack span in the case that the ISG is
driving instead of in the slack span when the crankshaft is driving allows for easier packaging
and for the least static tension Designs shown in Table 21 place the tensioneridler pulley in the
same span as Henry et al [11] or in both the slack and taut spans if using a double
tensioneridler configuration
242 Systematic Review of Tensioner Designs for a B-ISG System
The proposals for belt tensioner devices to manage the issue of high peaks in belt tension for B-
ISG settings are largely in patent records as the re-design of a tensioner has been primarily a
concern of automotive makers thus far A systematic review of the patent literature has been
conducted in order to identify evaluate and collate relevant tensioning mechanism designs
applicable to a B-ISG setting Its research objective is to influence the selection of a tensioner
configuration for the thesis study
The predefined search strategy used by the researcher has been to consider patents dating only
post-2000 as many patents dating earlier are referred to in later patents as they are developed on
in most cases by the original inventor (eg an INA Schaeffler KG patent published in 2000 may
refer to its own earlier patent presented in 1999) Patents dating pre-2000 that do not have any
successor were also considered The inclusion and exclusion criteria and rationales that were
used to assess potential patents are as follows
Inclusion of
Literature Review 21
tensioner designs with two arms andor two pivots andor two pulleys
mechanical tensioners (ie exclusion of magnetic or electrical actuators or any
combination of active actuators) in order to minimize cost
tension devices that are an independent structure apart from the ISG structure in order to
reduce the required modification to the accessory belt drive of a conventional automobile
and
advanced designs that have not been further developed upon in a subsequent patent by the
inventor or an outside party
Table 21 provides a collation of the results for the systematic review based on the selection
criteria Illustrations of the collated patent designs may be seen in Appendix A It is noted that
the patent literature pertaining to these designs in most cases provides minimal numerical data
for belt tensions achieved by the tensioning mechanism In most cases only claims concerning
the outcome in belt performance achievable by the said tension device is stated in the patent
Table 21 Passive Dual Tensioner Designs from Patent Literature
Bayerische
Motoren Werke
AG
Patents EP1420192-A2 DE10253450-A1 [33]
Design Approach
2 tensioner pulleys (idlers) and 2 tension arms are mounted outside the periphery of the belt drive these form tiltable clamping arms around a common axis of rotation
A torsion spring is used at bearing bushings to mount tension arms at ISG shaft
Each tension arm cooperates with torsion spring mechanism to rotate through a damping
device in order to apply appropriate pressure to taut and slack spans of the belt in
different modes of operation
Bosch GMBH Patent WO0026532 et al [34]
Design Approach
2 tension pulleys each one is mounted on the return and load spans of the driven and
driving pulley respectively
Idlers (tension pulleys) each connect to a spring which is attached on one end to a fixed point
Literature Review 22
Idlers‟ motions are independent of each other and correspond to the tautness or
slackness in their respective spans
Or alternatively a spring connects the idler pulleys and one of the two idlers is fixed at
its axis of rotation
Daimler Chrysler
AG
Patents DE10324268-A1 [35]
Design Approach
2 idlers are given a working force by a self-aligning bearing
Bearing supports auxiliary unit (ISG) and is arranged concentrically with the axle
auxiliary unit pulley
Dayco Products
LLC
Patents US6942589-B2 et al [36]
Design Approach
2 tension arms are each rotatably coupled to an idler pulley
One idler pulley is on the tight belt span while the other idler pulley is on the slack belt
span
Tension arms maintain constant angle between one another
One arm forms a positive differential angle with the belt and the remaining arm forms a negative differential angle with the belt
Idler pulleys are on opposite sides of the ISG pulley
Gates Corporation Patents US20060249118-A1 WO2003038309-A [37]
Design Approach
A tensioner pulley contacts the belt at the slack span during start-up (ISG-driving mode)
A tensioner is asymmetrically biased in direction tending to cause power transmission
belt to be under tension
McVicar et al
(Firm General
Motors Corp)
Patent US20060287146-A1 [38]
Design Approach
2 tension pulleys and carrier arms with a central pivot are mounted to the engine
One tension arm and pulley moderately biases one side of belt run to take up slack
during engine start-up while other tension arm and pulley holds appropriate bias against
taut span of belt
A hydraulic strut is connected to one arm to provide moderate bias to belt during normal
engine operation and velocity sensitive resistance to increasing belt forces during engine
start-up
INA Schaeffler
KG et al
Patents DE10044645-A1 [39] DE10159073-A1 [40] EP1723350-A1 et al [41]
DE10359641-A1 et al [42] EP1738093-A1 et al [43] DE102004012395-A1 [44]
WO2006108461-A1 et al [45]
Design Approach
2 tension arms and 2 pulleys approach ndash o Mutually independent tensioning arms are supported for rotation in the same
plane of the housing part
o Idler pulley corresponding to each tensioning arm engages with different
sections of belt
o When high tension span alternates with slack span of belt drive one tension
arm will increase pressure on current slack span of belt and the other will
decrease pressure accordingly on taut span
o Or when the span under highest tension changes one tensioner arm moves out
of the belt drive periphery to a dead center due to a resulting force from the taut
span of the ISG starting mode
o Deflection of the taut span acts on associated pulley to apply a counter-moment to the other idler pulley on the slack span
Literature Review 23
o The 2 lever arms are of different lengths and each have an idler pulley of
different diameters and different wrap angles of belt (see DE10045143-A1 et
al)
1 tensioner arm and 2 pulleys approach ndash
o 2 idler pulleys are pinned to a beam arranged on a clamping arm that is tiltably
linked to the beam o The ISG machine is supported by a shock absorber
o During ISG start-up one idler pulley is induced to a dead center position while
it pulls the remaining idler pulley into a clamping position until force
equilibrium takes place
o A shock absorber is laid out such that its supporting spring action provides
necessary preloading at the idler pulley in the direction of the taut span during
ISG start-up mode
Litens Automotive
Group Ltd
Patents US6506137-B2 et al [46]
Design Approach
2 tension pulleys on opposite sides of the ISG pulley engage the belt
They are positioned such that their applied forces result in opposing directed moments with respect to the tension device‟s axis of pivot
The pivot axis varies relative to the force applied to each tension pulley
Diameters of the tensioner pulleys are approximately equal and belt wrap angles of the
tensioner pulleys are approximately equal
A limited swivel angle for the tensioner arms work cycle is permitted
Mitsubishi Jidosha
Eng KK
Mitsubishi Motor
Corp
Patents JP2005083514-A [47]
Design Approach
2 tensioners are used
1 tensioner is held on the slack span of the driving pulley in a locked condition and a
second tensioner is held on the slack side of the starting (driven) pulley in a free condition
Nissan Patents JP3565040-B2 et al [48]
Design Approach
A single tensioner is on the slack span once ISG pulley is in start-up mode
The tension device is comprised of a oil pressure tensioner and a half ratchet mechanism
(a plunger which performs retreat actuation according to the energizing force of the oil
pressure spring and load received from the ISG)
The tensioner is equipped with a relief valve to keep a predetermined load lower than the
maximum load added by the ISG device
NTN Corp Patent JP2006189073-A [49]
Design Approach
An automatic tensioner is equipped with a hydraulic damper mechanism comprised of a
screw bolt using saw-screwed teeth and a cylinder nut a return spring and a spring seat
in a pressure chamber (within the screw bolt) a rod seat (that is fitted to the lower end of
the cylinder nut) a spring support (arranged on varying diameter stepped recessed
sections of the rod seat) and a check valve with an openingclosing passage
The cylinder and screw bolt act as the rigidity buffer under excessive loads during ISG
start-up mode of operation
Valeo Equipment
Electriques
Moteur
Patents EP1658432 WO2005015007 [50]
Design Approach
ldquoThe invention relates to a system or a starter (10) in which a pulley (80) is rotationally mounted on a section (22) of a shaft which axially extends inside a pulley (80) and
Literature Review 24
forwards at least partially outside a support element (200) and is characterized in that
the free front end (23) of said shaft section (22) is carried by an arm (206) connected to
the support element (200)rdquo
The author notes that published patents and patent applications may retain patent numbers for multiple patent
offices (ie European Patent Office German Patent Office etc) In such cases the published patent number or in
the absence of such a number the published patent application number has been specified However published
patent documents in the above cases also served as the document (ie identical) to the published patent if available
Quoted from patent abstract as machine translation is poor
25 Summary
The research on tensioner designs from the patent literature demonstrates a lack of quantifiable
data for the performance of a twin tensioner particularly suited to a B-ISG system The review of
the literature for the modeling theory of serpentine belt drives and design of tensioners shows
few belt drive models that are specific to a B-ISG setting Hence the literature review supports
the thesis objective of modeling a B-ISG tensioner specifically one that has a passive twin
tensioner configuration and as well measuring the tensioner‟s performance The survey of
hybrid classes reveals that the micro-hybrid class is the only class employing a closely
conventional belt transmission and hence its B-ISG transmission is applicable for tensioner
investigation The patent designs for tensioners contribute to the development of the tensioner
design to be studied in the following chapter
25
CHAPTER 3 MODELING OF B-ISG SYSTEM
31 Overview
The derivation of a theoretical model for a B-ISG system uses real life data to explore the
conceptual system under realistic conditions The literature and prior art of tensioner designs
leads the researcher to make the following modeling contributions a proposed design for a
passive two-pulley tensioner computation of geometric attributes for a B-ISG system with the
proposed tensioner and derivation of the system‟s equations of motion (EOM) under dynamic
and static states as well as deriving the EOM for the B-ISG system with only a passive single-
pulley tensioner for comparison The principles of dynamic equilibrium are applied to the
conceptual system to derive the EOM
32 B-ISG Tensioner Design
The proposed design for a passive two pulley tensioner configures two tensioners about a single
fixed pivot point in the interior space of a serpentine belt drive One end of each tensioner arm
coincides with the centre point of a tensioner pulley and this point marks the axis of rotation of
the pulley The other end of each arm is pivoted about a point so that the arms share the same
axis of rotation This conceptual design henceforth is called a Twin Tensioner Figure 31 shows
a schematic for the proposed design
Modeling of B-ISG 26
Figure 31 Schematic of the Twin Tensioner
The tensioner pulley coordinates are described by (XiYi) their radii by Ri their arm lengths Lti
and their angles θti The rotation of the arms is resisted by stiffness kt of a coil spring acting
between the two arms and spring stiffness kti acting between each arm and the pivot point The
motion of each arm is dampened by dampers and akin to the springs a damper acts between the
two arms ct and a damper cti acts between each arm and the pivot point The result is a
tensioning mechanism with four degrees of freedom (DOF) that includes independent rotations
of the two pulleys and two arms
The following section relates the geometry of the rigid bodies in a B-ISG system equipped with a
Twin Tensioner to their respective motions
Modeling of B-ISG 27
33 Geometric Model of a B-ISG System with a Twin Tensioner
The B-ISG system with the Twin Tensioner is shown in Figure 32 The geometry of the drive
provides the lengths of the belt spans and angles of wrap for the belt and pulley contact surfaces
These variables are crucial to resolve the components of forces and moment arms acting on each
rigid body in the system and are used in the derivation of the EOM in section 34 Zhen Mu‟s
geometric modeling approach [51] used in the development of the software FEAD was applied
to the Twin Tensioner system to compute the system‟s unique geometric attributes
Figure 32 B-ISG Serpentine Belt Drive with Twin Tensioner
It is noted that in Figure 31 and Figure 32 showing the schematic of the Twin Tensioner and
the overall system respectively that for the purpose of the geometric computations the forward
direction follows the convention of the numbering order counterclockwise The numbering
order is in reverse to the actual direction of the belt motion which is in the clockwise direction in
this study The fourth pulley is identified as an ISG unit pulley However the properties used
for the ISG pulley‟s geometry inertia stiffness and damping is modeled as a conventional
Modeling of B-ISG 28
alternator pulley This pulley is conceptualized as an ISG when it is modeled as the driving
pulley at which point the requirements of the ISG are solved for and its non-inertia attributes
are not needed to be ascribed
Figure 33 shows the geometric attributes needed to resolve the wrap angle of the belt on each
pulley Variables (XiYi) and XYfi XYbi XYfbi and XYbfi are the ith pulley centre coordinates and
its possible belt connection points respectively Length Lfi is the length of the span connecting
the points XYfi and XYf(i+1) or XYbi and XYb(i+1) on the ith and (i+1)th pulleys respectively
Similarly Lbi is the length of the span between XYfbi and XYfb(i+1) or XYbfi and XYbf (i+1) on the
ith and (i+1)th pulleys respectively Angles αi θfi and θbi represent the angle between a line
connecting the ith and (i+1)th pulley centres and the angles of the belt connection spans with
lengths Lfi and Lbi respectively Ri is the radius of the ith pulley
Figure 33 Angles Coordinates and Possible Belt Contact Points for the ith and i+1th Pulleys
[modified] [51]
Modeling of B-ISG 29
The angle between the horizontal and the line connecting the ith and (i+1)th pulley centres αi is
calculated using Zhen‟s method [51] This method uses the pulley‟s coordinates and a cosine
trigonometric relation
i acos
Xi 1
Xi
Xi 1
Xi
2
Yi 1
Yi
2
Yi 1
Yi
if
(31a)
i 2 acos
Xi 1
Xi
Xi 1
Xi
2
Yi 1
Yi
2
Yi 1
Yi
if
(31b)
The lengths for connecting the possible belt spans are described by the variables Lfi and Lbi
The centre point coordinates and the radii of the pulleys are related through the solution of
triangles which they form to define values of the possible belt span lengths
Lfi
Xi 1
Xi
2
Yi 1
Yi
2
Ri 1
Ri
2
(32a)
Lbi
Xi 1
Xi
2
Yi 1
Yi
2
Ri 1
Ri
2
(32b)
The set of possible belt span lengths leads to the calculation of θfi and θbi the angles between the
line connecting the ith and (i+1)th pulley centres and the possible contact point on the pulley
perimeter
Modeling of B-ISG 30
(33a)
(33b)
The array of possible belt connection points comes about from the use of the pulley centre
coordinates and their radii and the sine of the sum or differences of αi and θfi or θbi The angle
αi is calculated in equations (31a) and (31b) and angles θfi and θbi are calculated in equations
(33a) and (33b) The formula to compute the array of points is shown in equations (34) and
(35) for the ith and (i+1)th pulleys Equation (34) describes the forward belt connection point
on the ith pulley which is in the span leading forward to the next (i+1)th pulley
(34a)
(34b)
(34c)
(34d)
bi atan
Lbi
Ri
Ri 1
Modeling of B-ISG 31
Equation (35) describes the backward belt connection point on the ith pulley This point sits on
the ith pulley in the contacting belt span which leads backward to connect with the (i-1)th
pulley
(35a)
(35b)
(35c)
(35d)
The selection of the coordinates from the array of possible connection points requires a graphic
user interface allowing for the points to be chosen based on observation This was achieved
using the MathCAD software package as demonstrated in the MathCAD scripts found in
Appendix C The belt connection points can be chosen so as to have a pulley on the interior or
exterior space of the serpentine belt drive The method used in the thesis research was to plot the
array of points in the MathCAD environment with distinct symbols used for each pair of points
and to select the belt connection points accordingly By observation of the selected point types
the type of belt span connection is also chosen Selected point and belt span types are shown in
Table 31
Modeling of B-ISG 32
Table 31 Selected Contact Point Types on the ith Pulley and Types for the ith Belt Span
Pulley Forward Contact
Point
Backwards Contact
Point
Belt Span
Connection
1 Crankshaft XYf1 XYbf21 Lf1
2 Air Conditioning XYfb2 XYf22 Lb2
3 Tensioner 1 XYbf3 XYfb23 Lb3
4 AlternatorISG XYfb4 XYbf24 Lb4
5 Tensioner 2 XYbf5 XYfb25 Lb5
The inscribed angles βji between the datum and the forward connection point on the ith pulley
and βji between the datum and its backward connection point are found through solving the
angle of the arc along the pulley circumference between the datum and specified point The
wrap angle ϕi is found as the difference between the two inscribed angles for each connection
point on the pulley The angle between each belt span and the horizontal as well as the initial
angle of the tensioner arms are found using arctangent relations Furthermore the total length of
the belt is determined by the sum of the lengths of the belt spans
34 Equations of Motion for a B-ISG System with a Twin Tensioner
341 Dynamic Model of the B-ISG System
3411 Derivation of Equations of Motion
This section derives the inertia damping stiffness and torque matrices for the entire system
Moment equilibrium equations are applied to each rigid body in the system and net force
equations are applied to each belt span From these two sets of equations the inertia damping
Modeling of B-ISG 33
and stiffness terms are grouped as factors against acceleration velocity and displacement
coordinates respectively and the torque matrix is resolved concurrently
A system whose motion can be described by n independent coordinates is called an n-DOF
system Consider the free body diagram of the Twin Tensioner in Figure 34 in which each
pulley of inertia Ii is supported on an arm of inertia Iti It is assumed that the pulleys are
constrained to rotate about their respective central axes and the arms are free to rotate about their
respective pivot points then at any time the position of each pulley can be described by a
rotational coordinate θi(t) and a coordinate θti(t) can denote the rotation of each arm Thus the
tensioner system comprises of four rigid bodies where each is described by one coordinate and
hence is a four-DOF system It is important to note that each rigid body is treated as a point
mass In addition inertial rotation in the positive direction is consistent with the direction of belt
motion The belt span tensions Ti and coupled radii Ri apply moments to the pulleys
Figure 34 Twin Tensioner Free Body Diagram of a Four-Degree of Freedom System
Modeling of B-ISG 34
For the serpentine belt system considered in the thesis research there are seven rigid bodies each
having a one-DOF of motion The EOM for a seven-DOF system form second-order coupled
differential equations meaning that each equation includes all of the general coordinates and
includes up to the second-order time derivatives of these coordinates The EOM can be
obtained by applying D‟Alembert‟s principle that the sum of the moments taken about any point
including the couples equals to zero Therefore the inertial couple the product of the inertia and
acceleration is equated to the moment sum as shown in equation (35)
I ∙ θ = ΣM (35)
The moment equilibrium equations for the Twin Tensioner in Figure 34 where the positive
direction is in the clockwise direction are shown in equations (36) through to (310) The
numbering convention used for each rigid body corresponds to the labeled serpentine belt drive
system shown in Figure 32 Qi represents the required torque of the ith rigid body ci is the
damping constant of the ith rigid body βji is the angle of orientation for the ith belt span and
120597120579119905119894 120579 119905119894 and 120579 119905119894 are the angular displacement angular velocity and angular acceleration of the ith
tensioner arm The initial angle of the ith tensioner arm is described by θtoi
minusI3 ∙ θ 3 = T3 ∙ R3 minus T2 ∙ R3 minus Q3 + c3 ∙ θ 3 (36)
minusI5 ∙ θ 5 = minusT4 ∙ R5 + T5 ∙ R5 minus Q5 + c5 ∙ θ 5 (37)
Modeling of B-ISG 35
It1 ∙ θ t1 = minusTt1 ∙ Lt1 ∙ sin θto 1 minus βj4 + sin θto 1 minus βj5 minus kt ∙ partθt1 minus partθt2 minus kt1 ∙
partθt1 minus ct ∙ partθ t1 minus partθ t2 minus ct1 ∙ partθ t1 (38)
It2 ∙ θ t2 = minusTt2 ∙ Lt2 ∙ sin θto 2 minus βj2 + sin θto 1 minus βj3 minus kt ∙ partθt2 minus partθt1 minus kt2 ∙ partθt2 minus
ct ∙ partθ t2 minus partθ t1 minus ct2 ∙ partθ t2 (39)
partθt1 = θt1 minus θto 1 (310a)
partθt2 = θt2 minus θto 2 (310b)
The free body diagrams for the remaining rigid bodies crankshaft pulley air conditioner pulley
and ISG pulley are in the general form of Figure 35 The sum of the moments about the axes of
rotation are taken for these structures in equations (311) through to (313)
Figure 35 Free Body Diagram for Non-Tensioner Pulleys
Modeling of B-ISG 36
I1 ∙ θ 1 = T5 ∙ R1 minus T1 ∙ R1 + Q1 minus c1 ∙ θ 1 (311)
I2 ∙ θ 2 = T1 ∙ R2 minus T2 ∙ R2 + Q2 minus c2 ∙ θ 2 (312)
I4 ∙ θ 4 = T3 ∙ R4 minus T4 ∙ R4 + Q4 minus c4 ∙ θ 4 (313)
The relationship between belt tensions and rigid body displacements is in the general form of
equation (314) where 119827119836 and 119827119844 are damping and stiffness matrices due to the belt respectively
with each factorized by a radial arm length This relationship is described for each span in
equations (315) through to (320) The belt damping constant for the ith belt span is cib
119827prime = 119827119836 ∙ 120521 + 119827119844 ∙ 120521 (314)
T1 = To + k1b ∙ R1 ∙ θ1 minus R2 ∙ θ2 + c1
b ∙ [R1 ∙ θ 1 minus R2 ∙ θ 2] (315)
T2 = To + k2b ∙ R2 ∙ θ2 minus R3 ∙ θ3 + Lt1 ∙ [sin θto 1 minus βj2 ] ∙ (θt1 minus θto 1) + c2
b ∙ [R2 ∙ θ 2 minus R3 ∙
θ 3 + Lt1 ∙ [sin θto 1 minus βj2 ] ∙ (θ t1)] (316)
T3 = To + k3b ∙ R3 ∙ θ3 minus R4 ∙ θ4 + Lt1 ∙ [sin θto 1 minus βj3 ] ∙ (θt1 minus θto 1) + c3
b ∙ [R3 ∙ θ 3 minus R4 ∙
θ 4 + Lt1 ∙ [sin θto 1 minus βj3 ] ∙ (θ t2)] (317)
Modeling of B-ISG 37
T4 = To + k4b ∙ R4 ∙ θ4 minus R5 ∙ θ5 + Lt2 ∙ [sin θto 2 minus βj4 ] ∙ (θt2 minus θto 2) + c4
b ∙ [R4 ∙ θ 4 minus R5 ∙
θ 5 + Lt2 ∙ [sin θto 2 minus βj4 ] ∙ (θ t1)] (318)
T5 = To + k5b ∙ R5 ∙ θ5 minus R1 ∙ θ1 + Lt2 ∙ [sin θto 2 minus βj5 ] ∙ (θt2 minus θto 2) + c5
b ∙ [R5 ∙ θ 5 minus R1 ∙
θ 1 + Lt2 ∙ [sin θto 2 minus βj5 ] ∙ (θ t2)] (319)
Tprime = Ti minus To (320)
Since the applied torques on the tensioner pulleys Q3 and Q4 are zero the static equilibrium
equation of the pulleys show that the adjacent spans of each tensioner pulley are equal to each
other Hence equations (321) and (322) are denoted as follows
Tt1 = T2 = T3 (321)
Tt2 = T4 = T5 (322)
Equations (310a) (310b) and (314) through to (322) are substituted into the EOMs described
in equations (36) to (39) and (311) to (313) The newly formed equations can be arranged
and written in matrix form as shown in equations (323) through to (328) The general
coordinate matrix 120521 and its first and second derivatives are shown in the EOM below
119816 ∙ 120521 + 119810 ∙ 120521 + 119818 ∙ 120521 = 119824 (323)
Modeling of B-ISG 38
The inertia matrix I includes the inertia of each rigid body in its diagonal elements The
damping matrix C includes variables 119888119894119887 the damping of the ith belt span 119877119894 its radius 120573119895119894 its
angle 119871119905119894 the ith tensioner arm‟s length 120579119905119900119894 its initial pivot angle and 119888119905 and 119888119905119894 the ith
tensioner arm viscous damping constants Stiffness matrix K contains 119896119894119887 the ith belt span
stiffness and 119896119905 and 119896119905119894 the ith tensioner arm stiffness constants and akin to the damping
matrix the variables 119877119894 119871119905119894 120579119905119900119894 and 120573119895119894 The belt span stiffness is computed in equation
(326b) where 119870119887 represents the belt cord stiffness 119896119887 is the belt factor obtained from
experimental data 120573119895119894 is the angle of orientation for the span between the jth and ith pulleys and
ϕi is the belt wrap angle on the ith pulley
Modeling of B-ISG 39
119816 =
I1 0 0 0 0 0 00 I2 0 0 0 0 00 0 I3 0 0 0 00 0 0 I4 0 0 00 0 0 0 I5 0 00 0 0 0 0 It1 00 0 0 0 0 0 It2
(324)
119810 =
c1
b ∙ R12 + c5
b ∙ R12 + c1 minusc1
b ∙ R1 ∙ R2 0 0 minusc5b ∙ R1 ∙ R5 0 c5
b ∙ R1 ∙ Lt2 ∙ sin θto 2 minus βj5
minusc1b ∙ R1 ∙ R2 c2
b ∙ R22 + c1
b ∙ R22 + c2 minusc2
b ∙ R2 ∙ R3 0 0 c2b ∙ R2 ∙ Lt1 ∙ sin θto 1 minus βj2 0
0 minusc2b ∙ R2 ∙ R3 c3
b ∙ R32 + c2
b ∙ R32 + c3 minusc3
b ∙ R3 ∙ R4 0 C36 0
0 0 minusc3b ∙ R3 ∙ R4 c4
b ∙ R42 + c3
b ∙ R42 + c4 minusc4
b ∙ R4 ∙ R5 minusc3b ∙ R4 ∙ Lt1 ∙ sin θto 1 minus βj3 c4
b ∙ R4 ∙ Lt2 ∙ sin θto 2 minus βj4
minusc5b ∙ R1 ∙ R5 0 0 minusc4
b ∙ R4 ∙ R5 c5b ∙ R5
2 + c4b ∙ R5
2 + c5 0 C57
0 0 0 0 0 ct +ct1 minusct
0 0 0 0 0 minusct ct +ct1
(325a)
C36 = 1198773 ∙ 1198711199051 ∙ [1198883119887 ∙ 119904119894119899 1205791199051199001 minus 1205731198953 minus 1198882
119887 ∙ 119904119894119899 1205791199051199001 minus 1205731198952 ] (325b)
C57 = 1198775 ∙ 1198711199052 ∙ [1198885119887 ∙ 119904119894119899 1205791199051199002 minus 1205731198955 minus 1198884
119887 ∙ 119904119894119899 1205791199051199002 minus 1205731198954 ] (325c)
Modeling of B-ISG 40
119818 =
k1
b ∙ R12 + k5
b ∙ R12 minusk1
b ∙ R1 ∙ R2 0 0 minusk5b ∙ R1 ∙ R5 0 k5
b ∙ R1 ∙ Lt2 ∙ sin θto 2 minus βj5
minusk1b ∙ R1 ∙ R2 k2
b ∙ R22 + k1
b ∙ R22 minusk2
b ∙ R2 ∙ R3 0 0 k2b ∙ R2 ∙ Lt1 ∙ sin θto 2 minus βj2 0
0 minusk2b ∙ R2 ∙ R3 k3
b ∙ R32 + k2
b ∙ R32 minusk3
b ∙ R3 ∙ R4 0 R3 ∙ Lt1 ∙ [k3b ∙ sin θto 1 minus βj3 minus k2
b ∙ sin θto 1 minus βj2 ] 0
0 0 minusk3b ∙ R3 ∙ R4 k4
b ∙ R42 + k3
b ∙ R42 minusk4
b ∙ R4 ∙ R5 minusk3b ∙ R4 ∙ Lt1 ∙ sin θto 1 minus βj3 k4
b ∙ R4 ∙ Lt2 ∙ sin θto 2 minus βj4
minusk5b ∙ R1 ∙ R5 0 0 minusk4
b ∙ R4 ∙ R5 k5b ∙ R5
2 + k4b ∙ R5
2 0 R5 ∙ Lt2 ∙ [k5b ∙ sin θto 2 minus βj5 minus k4
b ∙ sin θto 2 minus βj4 ]
0 0 0 0 0 kt +kt1 minuskt
0 0 0 0 0 minuskt kt +kt1
(326a)
k119894b =
Kb
Li + kb ∙ Ri ∙ϕi+1
2 + Ri ∙ϕi
2
(326b)
120521 =
θ1
θ2
θ3
θ4
θ5
partθt1
partθt2
(327)
119824 =
Q1
Q2
Q3
Q4
Q5
Qt1
Qt2
(328)
Modeling of B-ISG 41
3412 Modeling of Phase Change
The phase change from the crankshaft pulley being the driving pulley to the ISG pulley being the
driving pulley is described through a conditional equality based on a set of Boolean conditions
When the crankshaft is driving the rows and the columns of the EOM are swapped such that the
new order for rows and columns is 1 (crankshaft pulley) 4 (ISG pulley) 2 (air conditioner
pulley) 3 (tensioner 1 pulley) 5 (tensioner 2 pulley) 6 (tensioner arm 1) and 7 (tensioner arm 2)
When the ISG is driving the order is the same except that the second row and second column
terms relating to the ISG pulley become the first row and first column while the crankshaft
pulley terms (previously in the first row and first column) become the second row and second
column Hence the order for all rows and columns of the matrices making up the EOM in
equation (322) switches between 1423567 (when the crankshaft pulley is driving) and
4123567 (when the ISG pulley is driving) For example in the crankshaft driving and ISG
driving phases the general coordinate matrix and the inertia matrix become the following
120521119940 =
1205791
1205794
1205792
1205793
1205795
1205971205791199051
1205971205791199052
and 120521119938 =
1205794
1205791
1205792
1205793
1205795
1205971205791199051
1205971205791199052
(329a amp b)
119816119940 =
I1 0 0 0 0 0 00 I4 0 0 0 0 00 0 I2 0 0 0 00 0 0 I3 0 0 00 0 0 0 I5 0 00 0 0 0 0 It1 00 0 0 0 0 0 It2
and 119816119938 =
I4 0 0 0 0 0 00 I1 0 0 0 0 00 0 I2 0 0 0 00 0 0 I3 0 0 00 0 0 0 I5 0 00 0 0 0 0 It1 00 0 0 0 0 0 It2
(329c amp d)
Modeling of B-ISG 42
where subscripts c and a denote the crankshaft pulley driving phase and the ISG pulley driving
phase respectively
The condition for phase change is based on the engine speed n in units of rpm Equation (330)
demonstrates the phase change
H(n) = 1 119899 ge 750 (Crankshaft driving phase)0 119899 lt 750 (ISG driving phase)
(330)
When the crankshaft pulley is the driving pulley the ISG pulley becomes the driven pulley and
following suit when the ISG pulley is the driving pulley the crankshaft pulley becomes the
driven pulley These modes of operation mean that the system will predict two different sets of
natural frequencies and mode shapes Using a Boolean condition to allow for a swap between
the first and second rows as well as between the first and second columns of the EOM matrices
I C and K allows for a continuous plot of the dynamic response to be plotted for the ISG pulley
throughout its driving and driven phases as well as for that of the crankshaft pulley
3413 Natural Frequencies Mode Shapes and Dynamic Responses
Assuming the system undergoes simple harmonic motion its matrix of natural frequencies 120596119899
and modeshapes are found by solving the eigenvalue problem shown in equation (331a)
ωn ∙ 119816120784120784 minus 11981822 ∙ 120495m = 120782 (331a)
The displacement amplitude Θm is denoted implicitly in equation (331d)
Modeling of B-ISG 43
120521119846 = θ2 θ3 θ5 θ6 partθt1 partθt2 T for H n = 1 (331b)
120521119846 = θ1 θ3 θ5 θ6 partθt1 partθt2 T for H n = 0 (331c)
θm = 120495119846 ∙ sin(ω ∙ t) (331d)
I2 and K22 are submatrices of I and K respectively meaning the first row and column of each of
the original matrices are removed The eigenvalue problem is reached by considering the
undamped and unforced motion of the system Furthermore the dynamic responses are found by
knowing that the torque requirements in the matrixndash Qm for the driven pulleys and the tensioner
arms are zero in the dynamic case which signifies a response of the system to an input solely
from the driving pulley
I1 120782120782 119816120784120784
θ 1120521 119846
+ C11 119810120783120784119810120784120783 119810120784120784
θ 1120521 119846
+ K11 119818120783120784
119818120784120783 119818120784120784 θ1
120521119846 =
QCS ISG
119824119846 (332)
1
In the case of equation (331) θm is the submatrix identified in equations (331b) through to
(331d) Therein θ1 denotes the general coordinate for the driving pulley so that in the case the
phase change function H(n) is equal to zero θ1 becomes θ4 and the order of the rows and
columns for the remaining matrices correspond to the value of H(n) as mentioned earlier in
section 3412 For simple harmonic motion the motion of the driven pulleys are described as
1 The driving torque 119876119862119878119868119878119866 denotes the crankshaft torque 119876119862119878 when the crankshaft pulley is driving or the ISG
torque 119876119868119878119866 when the ISG pulley is in its driving function
Modeling of B-ISG 44
θm = 120495119846 ∙ sin(ω ∙ t) (333)
The dynamic response of the system to an input from the driving pulley under the assumption of
sinusoidal motion is expressed in equation (334)
120495119846 = [(119818120784120784 minusω2 ∙ 119816120784120784) + 119895ω ∙ 119810120784120784]minus1 ∙ (119818120784120783 + 119895ω ∙ 119810120784120783) ∙ Θ1 (334)
3414 Crankshaft Pulley Driving Torque Acceleration and Displacement
Subsequently the crankshaft pulley driving torque acceleration and displacement are firstly
discussed It is assumed in the thesis research for the purpose of modeling that the engine
serving the crankshaft is of the four cylinder type The input torque provided by a four-cylinder
engine is assumed to be dominated by two torque pulses per revolution of the crankshaft which
is represented by the factor of 2 on the steady component of the angular velocity in equation
(335) The torque requirement of the crankshaft pulley when it is the driving pulley is
Qc = qc ∙ sin(2 ∙ ωcs ∙ t) (335)
The amplitude of the required crankshaft torque qc is expressed in equation (336) and is
derived from equation (332)
qc = K11 minus ω2 ∙ I1 + 119895 ∙ ω ∙ C11 ∙ Θ1 + (119818120783120784 + 119895 ∙ ω ∙ 119810120783120784) ∙ 120495119846 (336)
Modeling of B-ISG 45
The angular frequency for the system in radians per second (rads) ω when the crankshaft
pulley is driving can be found as a function of the engine speed in rotations per minute (rpm) n
and by taking into account the double pulse per crankshaft revolution
ω = 2 ∙ ωcs = 4 ∙ π ∙ n
60
(337)
The system is considered when the amplitude of the crankshaft‟s angular acceleration is assumed
to be constant and equal to 650 rads2 during the crankshaft pulley driving phase The amplitude
of the excitation angular input from the engine is shown in equation (339b) and is found as a
result of (338)
θ 1CS = 650 ∙ sin(ω ∙ t) (338)
θ1CS = minus650
ω2sin(ω ∙ t)
(339a) where
Θ1CS = minus650
ω2
(339b)
Modeling of B-ISG 46
3415 ISG Pulley Driving Torque Acceleration and Displacement
Secondly the torque acceleration and the displacement of the ISG pulley in its driving phase is
discussed The torque for the ISG when it is in its driving function is assumed constant Ratings
for the ISG are taken from experiments performed by researchers Wezenbeek et al [21] on an
Energen 5 High Output Belt-alternator-starter (BAS) unit from Delphi The 12-Volt BAS which
can also be called a B-ISG was reported to have a maximum allowable speed of 18000 rpm [21]
As well it was noted that the ISG pulley was sized appropriately and the engine speed was
limited to ensure that an over-speed condition of the ISG pulley would not occur [21] The stall
torque rating for the Energen ISG was reported to be 48 Nm at the electric machine shaft [21]
The formula for the torque of a permanent magnet DC motor for any given speed (equation
(340)) is used to approximate the torque of the ISG in its driving mode[52]
QISG = Ts minus (N ∙ Ts divide NF) (340)2
Knowing the stall torque (the torque at 0 rpm) Ts and the maximum rpm of the motor when it is
not under load NF allows for the torque produced 119876119868119878119866 to be found for a given motor speed N
Experimental data from Litens Automotive Group [53] shows that for engine fire-up upon ISG
re-start the crankshaft must go from 0 rpm to an idle speed of approximately 750 rpm The
pulley installed on the ISG shaft in the case of the thesis research has a diameter of 6820 mm
(DISG) while that of the crankshaft has a diameter of 20065 mm (DCS) which makes the
2 The equation for the required driving torque for the ISG pulley may also be computed from the formula shown in
(336) Figure 315 for the driving torque of the ISG pulley shows that (336) and (340) produce similar results for
the required driving torque See Figure 315 for comparison of these results
Modeling of B-ISG 47
crankshaft to ISG pulley ratio approximately 2941 This ratio is used to determine the ISG
speed in equation (341)
nISG = nCS ∙DCS
DISG
(341)
For a crankshaft speed of 750 rpm the required ISG speed nISG is found from equation (341) to
be approximately 220656 rpm Thus the ISG torque during start-up is found from equation
(340) where N is equated to the value of nISG NF is assumed to be 18000 rpm and the stall
torque is allotted the value of 48 Nm The result is a required torque of approximately 42 Nm
for the ISG The acceleration of the ISG pulley is found by taking into account the torque
developed by the rotor and the polar moment of inertia of the pulley [54]
A1ISG = θ 1ISG = QISG IISG (342)
In torsional motion the function for angular displacement of input excitation is sinusoidal since
the electric motor is assumed to be resonating As a result of constant angular acceleration the
angular displacement of the ISG pulley in its driving mode is found in equation 343
θ1ISG = Θ1ISG ∙ sin(ωISG ∙ t) (343)
Knowing that acceleration is the second derivative of the displacement the amplitude of
displacement is solved subsequently [55]
Modeling of B-ISG 48
θ 1ISG = minusωISG2 ∙ Θ
1ISG ∙ sin(ωISG ∙ t) (344)
θ 1ISG = minusωISG2 ∙ Θ
1ISG
(345a)
Θ1ISG =minusQISG IISG
ωISG2
(345b)
In this case the angular frequency for the system 120596 is equivalent to 120596119868119878119866 that is the angular
frequency of the ISG pulley which can be expressed as a function of its speed in rpm
ω = ωISG =2 ∙ π ∙ nISG
60
(346a)
or in terms of the crankshaft rpm by substituting equation (341) into (346a)
ω =2 ∙ π
60∙ nCS ∙
DCS
DISG
(346b)
3416 Tensioner Arms Dynamic Torques
The dynamic torque for the tensioner arms are shown in equations (347) and (348)
Qt1 = kt + kt1 + 119895 ∙ ω ∙ (ct + ct1) ∙ (Θt1 ∙ Θ1) (347)
Modeling of B-ISG 49
Qt2 = kt + kt2 + 119895 ∙ ω ∙ (ct + ct2) ∙ (Θt2 ∙ Θ1) (348)
3417 Dynamic Belt Span Tensions
Furthermore the dynamic belt span tensions are derived from equation (314) and described in
matrix form in equations (349) and (350)
119827prime = 119895 ∙ ω ∙ 119827119836 + 119827119844 ∙ 120495119847 (349)
where
120495119847 = Θ1
120495119846 (350)
342 Static Model of the B-ISG System
It is fitting to pursue the derivation of the static model from the system using the dynamic EOM
For the system under static conditions equations (314) and (323) simplify to equations (351)
and (352) respectively
119827prime = 119827119844 ∙ 120521 (351)
119824 = 119818 ∙ 120521 (352)
Modeling of B-ISG 50
As noted in other chapters the focus of the B-ISG tensioner investigation especially for the
parametric and optimization studies in the subsequent chapters is to determine its effect on the
static belt span tensions Therein equations (351) and (352) are used to derive the expressions
for static tension in each belt span 119931prime is the tension solely due to deflection of the belt span
Equation (320) demonstrates the relationship between the tension due to belt response and the
initial tension also known as pre-tension The static tension 119931 is found by summing the initial
tension 1198790 with the expression for the dynamic tension shown in equations (315) through to
(319) and by substituting the expressions for the rigid bodies‟ displacements from equation
(352) and the relationship shown in equation (320) into equation (351)
119827 = 119827119844 ∙ (119818minus120783 ∙ 119824) + T0 (353)3
35 Simulations
The methods used to develop the geometric dynamic and static models of the Twin Tensioner B-
ISG system in the previous sections of this chapter were verified using the software FEAD The
input data for a single tensioner B-ISG system was entered into FEAD [51] to simulate the
crankshaft driving phase alone since the ISG phase is inapplicable in the FEAD [51] software
FEAD‟s [51] results agreed with those found in the simulation of the single tensioner system‟s
geometric model and EOMs in MathCAD software Furthermore the geometric simulation
3 For the purposes of the static tension the original order for the rows and columns of the stiffness matrix K and the
torque matrix Q are maintained as depicted in (326) and (328) In performing the inverse of K and its
multiplication with Q the first row and first column (in the case of the K matrix) are removed in the crankshaft
driving case whereas the fourth row and fourth column are removed in the ISG driving case Then the product for
the displacement120637 resulting from (119922minus120783 ∙ 119928) has a zero added to serve as the first element of the column matrix in
the crankshaft driving case or as the fourth element in the ISG driving case This is shown in detail in Appendix
C3 of MathCAD scripts
Modeling of B-ISG 51
results for both of the twin and single tensioner B-ISG systems were found to be in agreement as
well
351 Geometric Analysis
The initial coordinate inputs for the centre points of the five pulleys and the Twin Tensioner
pivot point are described as Cartesian coordinates and shown in Table 32 which also includes
the diameters for the pulleys
Table 32 Coordinate Points for Pulley Centres and Twin Tensioner Pivot [56]
Rigid Body Diameter [mm] Cartesian Coordinate [Xi Yi] [mm]
1Crankshaft Pulley 20065 [00]
2 Air Conditioner Pulley 10349 [224 -6395]
3 Tensioner Pulley 1 7240 [292761 87]
4 ISG Pulley 6820 [24759 16664]
5 Tensioner Pulley 2 7240 [12057 9193]
6 Tensioner Arm Pivot --- [201384 62516]
The geometric results for the B-ISG system are shown in Table 33
Table 33 Geometric Results of B-ISG System with Twin Tensioner
Pulley Forward
Connection Point
Backward
Connection Point
Wrap
Angle
ϕi (deg)
Angle of
Belt Span
βji (deg)
Length of
Belt Span
Li (mm)
1 Crankshaft [-6818-100093] [453889475] 202996 356103 227828
2 Air
Conditioning [275299-5717] [220484 -115575] 101425 277528 14064
3 Tensioner 1 [25887599735] [256873 82257] 28126 69403 58658
4 ISG [218374184225] [27951154644] 169554 58956 129513
5 Tensioner 2 [10419659645] [15158673262] 8585 333107 65949
Total Length of Belt (mm) 1243
Modeling of B-ISG 52
352 Dynamic Analysis
The dynamic results for the system include the natural frequencies mode shapes driven pulley
and tensioner arm responses the required torque for each driving pulley the dynamic torque for
each tensioner arm and the dynamic tension for each belt span These results for the model were
computed in equations (331a) through to (331d) for natural frequencies and mode shapes in
equation (334) for the driven pulley and tensioner arm responses in equation (336) for the
crankshaft pulley driving torque in equation (340) for the ISG pulley driving torque in
equations (347) and (348) for the tensioner arm torques and lastly in equation (349) for the
dynamic tension of each belt span Figures 36 through to 323 respectively display these
results The EOM simulations can also be contrasted with those of a similar system being a B-
ISG serpentine belt drive that is equipped with a single tensioner arm and single tensioner pulley
which interacts only in the span bridging the ISG and crankshaft pulleys The EOM for a B-ISG
with a single tensioner is presented in Appendix B
It is assumed for the sake of the dynamic and static computations that the system
does not have an isolator present on any pulley
has negligible rotational damping of the pulley shafts
has negligible belt span damping and that this damping does not differ amongst
spans (ie c1b = ∙∙∙ = ci
b = 0)
has quasi-static belt stretch where its belt experiences purely elastic deformation
has fixed axes for the pulley centres and tensioner pivot
has only one accessory pulley being modeled as an air conditioner pulley and
Modeling of B-ISG 53
has a rotational belt response that is decoupled from the transverse response of the
belt
The input parameter values of the dynamic (and static) computations as influenced by the above
assumptions for the present system equipped with a Twin Tensioner are shown in Table 34
Table 34 Data for Input Parameters used in Dynamic and Static Computations [56]
Rigid Body Data
Pulley Inertia
[kg∙mm2]
Damping
[N∙m∙srad]
Stiffness
[N∙mrad]
Required
Torque
[Nm]
Crankshaft 10 000 0 0 4
Air Conditioner 2 230 0 0 2
Tensioner 1 300 1x10-4
0 0
ISG 3000 0 0 5
Tensioner 2 300 1x10-4
0 0
Tensioner Arm 1 1500 1000 10314 0
Tensioner Arm 2 1500 1000 16502 0
Tensioner Arm
couple 1000 20626
Belt Data
Initial belt tension [N] To 300
Belt cord stiffness [Nmmmm] Kb 120 00000
Belt phase angle at zero frequency [deg] φ0deg 000
Belt transition frequency [Hz] ft 000
Belt maximum phase angle [deg] φmax 000
Belt factor [magnitude] kb 0500
Belt cord density [kgm3] ρ 1000
Belt cord cross-sectional area [mm2] A 693
Modeling of B-ISG 54
These values are for the driven cases for the ISG and crankshaft pulleys respectively In the
driving case for either pulley the inertia of the rigid body is defined as 1 kg∙mm2 and the driving
torque is determined in equations (335) and (340) for the crankshaft and ISG pulleys
respectively
It is noted that because of the belt data for the phase angle at zero frequency the transition
frequency and the maximum phase angle are all zero and hence the belt damping is assumed to
be constant between frequencies These three values are typically used to generate a phase angle
versus frequency curve for the belt where the phase angle is dependent on the frequency The
curve defined by equation (354) is normally symmetric with the lowest phase angle achieved at
0 Hz and the highest phase angle achieved at the prescribed transition frequency f The belt
damping would then be found by solving for cb in the following equation
tanφ = cb ∙ 2 ∙ π ∙ f (354)
Nevertheless the assumption for constant damping between frequencies is also in harmony with
the remaining assumptions which assume damping of the belt spans to be negligible and
constant between belt spans
3521 Natural Frequency and Mode Shape
The set of natural frequencies and mode shapes for the system are shown in Figures 36 and 37
under the cases of the ISG pulley driving and the crankshaft pulley driving The forcing
frequency for the system differs for each case due to the change in driving pulley Modeic and
Modeia denote the ith rigid body according to the numbering convention used in Figure 32 in
the crankshaft and ISG driving cases respectively
Modeling of B-ISG 55
Natural Frequency ωn [Hz]
Crankshaft Pulley ΔΘ4
Air Conditioner Pulley ΔΘ
Tensioner Pulley 1 ΔΘ
Tensioner Pulley 2 ΔΘ
Tensioner Arm 1 ΔΘ
Tensioner Arm 2 ΔΘ
Figure 36a ISG Driving Case Natural Frequency of System and Mode Shapes for Responsive
Rigid Bodies
Figure 36b ISG Driving Case First Mode Responses
4 ΔΘ signifies the term amplitude of angular displacement for the indicated rigid body
Modeling of B-ISG 56
Figure 36c ISG Driving Case Second Mode Responses
Natural Frequency ωn [Hz]
ISG Pulley ΔΘ5
Air Conditioner Pulley ΔΘ
Tensioner Pulley 1 ΔΘ
Tensioner Pulley 2 ΔΘ
Tensioner Arm 1 ΔΘ
Tensioner Arm 2 ΔΘ
Figure 37a Crankshaft Driving Case Natural Frequency of System and Mode Shapes for
Responsive Rigid Bodies
5 ΔΘ signifies the term amplitude of angular displacement for the indicated rigid body
Modeling of B-ISG 57
Figure 37b Crankshaft Driving Case First Mode Responses
Figure 37c Crankshaft Driving Case Second Mode Responses
Modeling of B-ISG 58
3522 Dynamic Response
The dynamic response specifically the magnitude of angular displacement for each rigid body is
plotted in Figures 38 through to 314 as a function of the crankshaft pulley speed n This is
fitting to the analysis since the crankshaft pulley‟s rpm decides the mode of operation for the
system in particular it determines whether the crankshaft pulley or ISG pulley is the driving
pulley
Figure 38 Crankshaft Pulley Dynamic Response (for crankshaft driven case)
Figure 39 ISG Pulley Dynamic Response (for ISG driven case)
Modeling of B-ISG 59
Figure 310 Air Conditioner Pulley Dynamic Response
Figure 311 Tensioner Pulley 1 Dynamic Response
Crankshaft Driving Phase ISG
Driving
Phase
Crankshaft Driving Phase ISG
Driving
Phase
Modeling of B-ISG 60
Figure 312 Tensioner Pulley 2 Dynamic Response
Figure 313 Tensioner Arm 1 Dynamic Response
Crankshaft Driving Phase ISG
Driving
Phase
Crankshaft Driving Phase ISG
Driving Phase
Modeling of B-ISG 61
Figure 314 Tensioner Arm 2 Dynamic Response
3523 ISG Pulley and Crankshaft Pulley Torque Requirement
Figures 315 and 316 respectively showcase the required torques for the ISG pulley in its driving
mode and the crankshaft pulley in its driving mode
Figure 315 Required Driving Torque for the ISG Pulley
Crankshaft Driving Phase ISG
Driving Phase
Modeling of B-ISG 62
Figure 315 shows two plots for the required driving torque of the ISG pulley The dashed line
labeled as Q(n) simulates the application of equation (340) which models the ISG torque as a
permanent magnet DC motor The additional solid line labeled as qamod uses the formula in
equation (336) which determines the load torque of the driving pulley based on the pulley
responses Figure 315 provides a comparison of the results
Figure 316 Required Driving Torque for the Crankshaft Pulley
3524 Tensioner Arms Torque Requirements
The torque for the tensioner arms are shown in Figures 317 and 318
Modeling of B-ISG 63
Figure 317 Dynamic Torque for Tensioner Arm 1
Figure 318 Dynamic Torque for Tensioner Arm 2
3525 Dynamic Belt Span Tension
The dynamic tensions for the belt spans are shown in Figures 319 through to 323 The values
plotted represent the magnitude of the dynamic tension
Crankshaft Driving Phase ISG
Driving Phase
Crankshaft Driving Phase ISG
Driving
Phase
Modeling of B-ISG 64
Figure 319 Span 1 (between crankshaft and air conditioner) Dynamic Belt Span Tension
Figure 320 Span 2 (between air conditioner and tensioner 1) Dynamic Belt Span Tension
Crankshaft Driving Phase ISG
Driving Phase
Crankshaft Driving Phase ISG
Driving
Phase
Modeling of B-ISG 65
Figure 321 Span 3 (between tensioner 1 and ISG) Dynamic Belt Span Tension
Figure 322 Span 4 (between ISG and tensioner 2) Dynamic Belt Span Tension
Crankshaft Driving Phase ISG
Driving
Phase
Crankshaft Driving Phase ISG
Driving Phase
Modeling of B-ISG 66
Figure 323 Span 5 (between tensioner 2 and crankshaft) Dynamic Belt Span Tension
The dynamic results for the system serve to show the conditions of the system for a set of input
parameters The following chapter targets the focus of the thesis research by analyzing the affect
of changing the input parameters on the static conditions of the system It is the static results that
are the focus of the thesis and is thus analyzed in Chapters 4 and 5 in the parametric and
optimization studies respectively The dynamic analysis has been used to complete the picture of
the system‟s state under set values for input parameters
353 Static Analysis
Before looking at the static results for the system under study in brevity the static results for a
B-ISG serpentine belt drive with a single tensioner are presented In this theoretical system the
tensioner arm and tensioner pulley that interacts with the span between the air conditioner and
ISG pulleys of the original system are removed as shown in Figure 324
Crankshaft Driving Phase ISG
Driving
Phase
Modeling of B-ISG 67
Figure 324 B-ISG Serpentine Belt Drive with Single Tensioner
The complete static model as well as the dynamic model for the system in Figure 324 is found
in Appendix B The results of the static tension for each belt span of the single tensioner system
when the crankshaft is driving and the ISG is driving are shown in Table 35
Table 35 Static Solution for Belt Span Tensions in Crankshaft and ISG Driving Cases for a B-
ISG Serpentine Belt Drive with a Single Tensioner
Belt Span between Pulleys Crankshaft Driving Case [N] ISG Driving Case [N]
Crankshaft ndash Air Conditioner 481239 -361076
Air Conditioner ndash ISG 442588 -399727
ISG ndash Tensioner 29596 316721
Tensioner ndash Crankshaft 29596 316721
The tensions in Table 35 are computed with an initial tension of 300N This value for pre-
tension allows the spans in the case that the crankshaft pulley is driving to be suitably tensioned
Modeling of B-ISG 68
Whereas in the case of the ISG pulley driving the first and second spans are excessively slack
Therein an additional pretension of approximately 400N would be required which would raise
the highest tension span to over 700N This leads to the motivation of the thesis researchndash to
reduce the static belt tensions when the ISG is driving As mentioned in Chapter 1 these
tensions should be minimized to prolong belt life preferably within the range of 600 to 800N
As well it is desirable to minimize the amount of pretension exerted on the belt The current
design uses a pre-tension of 300N The above results would lead to a required pre-tension of
more than 700N to keep all spans of the belt suitably in tension (well above 0N) in order to allow
the belt to exhibit high performance in power transmission and come near to the safe threshold
This is the rationale for investigating a Twin Tensioner configuration shown in Figure 32 for
the B-ISG serpentine belt drive under study For the theoretical system with a Twin Tensioner
the following static results in Table 36 are achieved
Table 36 Static Solution for Belt Span Tensions in Crankshaft and ISG Driving Cases for a B-
ISG Serpentine Belt Drive with a Twin Tensioner
Belt Span between Pulleys Crankshaft Driving Case [N] ISG Driving Case [N]
Crankshaft ndash Air Conditioner 465848 -284152
Air Conditioner ndash Tensioner 1 427197 -322803
Tensioner 1 ndash ISG 427197 -322803
ISG ndash Tensioner 2 28057 393645
Tensioner 2 ndash Crankshaft 28057 393645
The results in Table 36 show that the span following the ISG in the case between the Tensioner
1 and ISG pulleys is less slack than in the former single tensioner set-up However there
remains an excessive amount of pre-tension required to keep all spans suitably tensioned
Modeling of B-ISG 69
36 Summary
The simulation of the model for the B-ISG system with the Twin Tensioner shows that the mode
shapes of the rigid bodies within the system (Figures 36a to 37c) are greater in magnitude when
the ISG pulley is driving than when the crankshaft pulley is driving The dynamic responses of
the system as shown in Figures 38 and 310 to 314 is small for the crankshaft pulley and are
negligible for the remaining driven bodies when the ISG is driving For the crankshaft driving
phase there is greater dynamic response for the driven rigid bodies of the system including for
that of the ISG pulley
As the engine speed increases the torque requirement for the ISG was found to vary between
approximately 41Nm and 54Nm (before dropping steeply to approximately 3Nm at an engine
speed of about 720rpm) when modeled after equation (336) or between approximately 48Nm
and 34Nm when modeled after equation (340) In contrast the torque for the crankshaft peaks
at approximately 92Nm and 52Nm at an approximate engine speed of 1450rpm and 5000rpm
respectively The dynamic torque of the first tensioner arm was shown to peak at approximately
15Nm at the transition engine speed 750rpm and again at approximately 15Nm at an
approximate engine speed of about 1450rpm A small peak of about 3Nm was also seen at an
engine speed of 5000rpm Similarly for the second tensioner arm a torque peak of
approximately 20Nm was seen at 750rpm and 1450rpm and a smaller peak of about 8Nm was
seen at an engine speed of 5000rpm
The trend for the dynamic tensions is that the peaks are highest in the ISG driving portion of the
B-ISG operation in most cases and in a few cases they are seen to be close in magnitude to that
Modeling of B-ISG 70
of the highest peaks in the crankshaft driving portion The dynamic tension for the first belt span
peaked at approximately 780Nm 830Nm and 500Nm at engine speeds of 750rpm 1450rpm
5000rpm respectively For the dynamic tension of the second belt span peaks of approximately
1250Nm 675Nm and 760Nm were seen at the same respective engine speeds for the 3 peaks of
the former span At these same engine speeds the third belt span exhibited tension peaks at
approximately 1400Nm 650Nm and 890Nm The tension peaks of the fourth span were
approximately 165Nm 150Nm and 100Nm at engine speeds 750rpm 1450rpm and 5000rpm
The fifth span experienced peaks of approximately 165Nm 170Nm and 120Nm at the same
respective engine speeds of the fourth span
The simulation results for the static tension of the B-ISG system with the Twin Tensioner reveal
that taut spans of the crankshaft driving case are lower in the ISG driving case The largest
change is an approximate decrease of 750N in spans 1 through 3 while spans 4 and 5 increase
by approximately 113N It can be seen that the spans in highest tension (1 2 and 3) in the
crankshaft driving phase become excessively slack in the ISG driving phase There is a smaller
change between the tension values for the spans in the least tension in the crankshaft driving
phase and their corresponding span in the ISG driving phase
The summary of the simulation results are used as a benchmark for the optimized system shown
in Chapter 5 The static tension simulation results are investigated through a parametric study of
the Twin Tensioner system in Chapter 4 The optimization of the system is then based on the
selected design variables from the outcome of Chapter 4
71
CHAPTER 4 PARAMETRIC ANALYSIS OF A B-ISG
TWIN TENSIONER
41 Introduction
The parameters for the proposed Twin Tensioner for a Belt-driven Integrated Starter-generator
(B-ISG) system are investigated through a parametric analysis This analysis seeks to understand
how changing one parameter influences the static belt span tensions for the system Since the
thesis research focuses on the design of a tensioning mechanism to support static tension only
the parameters specific to the actual Twin Tensioner and applicable to the static case were
considered The parameters pertaining to accessory pulley properties such as radii or various
belt properties such as belt span stiffness are not considered In the analyses a single parameter
is varied over a prescribed range while all other parameters are held constant The pivot point
described by Cartesian Coordinates [X6Y6] for the tensioner arms is held constant in all cases
42 Methodology
The parametric study method applies to the general case of a function evaluated over changes in
one of its dependent variables The methodology is illustrated for the B-ISG system‟s function
for static tension which is evaluated for each change in one of its Twin Tensioner‟s parameters
The original data used for the system is based on sample vehicle data provided by Litens [56]
Table 41 provides the initial data for the parameters as well as the incremental change and
maxima and minima limits The increment Δi for the ith parameter is chosen arbitrarily Limits
for each parameter have been chosen to be plus or minus sixty percent of its initial value
Parametric Analysis 72
Table 41 Initial Values Increments and Ranges for Parameters of Twin Tensioner
Parameter Name Initial Value Increment (+- Δi) Minimum
value Maximum value
Coupled Spring
Stiffness kt
20626
N∙mrad 1238 N∙mrad 8250 N∙mrad 33002 N∙mrad
Tensioner Arm 1
Stiffness kt1
10314
N∙mrad 0619 N∙mrad 4126 N∙mrad 16502 N∙mrad
Tensioner Arm 2
Stiffness kt2
16502
N∙mrad 0990 N∙mrad 6601 N∙mrad 26403 N∙mrad
Tensioner Pulley 1
Diameter D3 007240 m 4344 ∙ 10
-3 m 00290 m 0116 m
Tensioner Pulley 2
Diameter D5 007240 m 4344 ∙ 10
-3 m 00290 m 0116 m
Tensioner Pulley 1
Initial Coordinates
[0292761
0087] m See Figure 41 for region of possible tensioner pulley
coordinates Tensioner Pulley 2
Initial Coordinates
[012057
009193] m
The mesh of possible points for the centre coordinates of tensioner pulley 1 and tensioner pulley
2 are designated as Region 1 and Region 2 respectively in Figures 41a and 41b
Figure 41a Regions 1 and 2 and their Associated Guidelines for Coordinates of Tensioner
Pulleys 1 amp 2
CS
AC
ISG
Ten 1
Ten 11
Region II
Region I
Parametric Analysis 73
Figure 41b Regions 1 and 2 in Cartesian Space
The selection for the minimum and maximum tensioner pulley centre coordinates and their
increments are not selected arbitrarily or without derivation as the other tensioner parameters
The coordinates for the pulley centres are identified using Intergraph‟s SmartSketch software a
graphing suite in MathCAD to model the regions of space The following descriptions are used
to describe the possible positions for the tensioner pulleys
Tensioner pulleys are situated such that they are exterior to the interior space created by
the serpentine belt thus they sit bdquooutside‟ the belt loop
The highest point on the tensioner pulley does not exceed the tangent line connecting the
upper hemispheres of the pulleys on either side of it
The tensioner pulleys may not overlap any other pulley
Parametric Analysis 74
Boundaries for regions described as Region 1 in span 2 and 3 and Region 2 in span 4
and 5 is selected based on the above criteria and their lower boundaries are selected
arbitrarily
These criteria were used to define the equation for each boundary line and leads to a set of
Boolean conditions that relate the x-coordinate and y-coordinate for each Cartesian pair The
density for the mesh of points in each region is arbitrarily selected as 101 x-points and 101 y-
points in each space for the purposes of the parametric analysis The outline of this method is
described in the MATLAB scripts contained in Appendix D
The results of the parametric analysis are shown for the slackest and tautest spans in each driving
case As was demonstrated in the literature review the tautest span immediately precedes the
driving pulley and the slackest span immediately follows the driving pulley in the direction of
the belt motion Thus in the case for the crankshaft driving the tautest span is in the first span
and the slackest span is in the fifth span Whereas in the ISG driving case the tautest span is in
the fourth span and the slackest span is in the third span Hence the parametric figures in this
chapter display only the tautest and slackest span values for both driving cases so as to describe
the maximum and minimum values for tension present in the given belt
43 Results amp Discussion
431 Influence of Tensioner Arm Stiffness on Static Tension
The parametric analysis begins with changing the stiffness value for the coil spring coupled
between tensioner arms 1 and 2 This stiffness value kt is changed over a range from sixty
percent less than its initial value kt0 to sixty percent more than its original value as shown in
Parametric Analysis 75
Table 41 The results of the static tension are shown in Figure 42 for the tautest and slackest
spans for both the crankshaft and ISG driving cases
Figure 42 Parametric Analysis for Coupled Stiffness Arm Constant kt (N∙mrad)
As kt increases in the crankshaft driving phase for the B-ISG system the highest tension
decreases from 4691N to 4646N while the lowest tension decreases from 2838N to 2793N
In the ISG driving phase the highest tension increases from 378N to 3998N and the lowest
tension increases from -3384N to -3167N Thus a change of approximately -45N is found in
the crankshaft driving case and approximately +22N is found in the ISG driving case for both the
tautest and slackest spans
Parametric Analysis 76
The second parameter analyzed is the stiffness value for tensioner arm 1 The results of this are
shown in Figure 43
Figure 43 Parametric Analysis for Stiffness of Arm 1 kt1 (N∙mrad)
In Figure 43 as kt1 increases an increase from 4628N to 4681N is observed for the tension of
the tautest span when the crankshaft is driving which is a change of +53N The same value for
net change is found in the slackest span for the same driving condition whose tension increases
from 2775N to 2828N For the case when the B-ISG system is in the ISG driving phase the
change is larger a value of -261N for the tautest span that changes from 4088N to 3827N and
for the slackest span that changes from -3077N to -3338N
Parametric Analysis 77
The change in static tension for the spans as the stiffness of arm 2 varies is demonstrated in
Figure 44
Figure 44 Parametric Analysis for Stiffness of Arm 2 kt2 (N∙mrad)
In this case it is observed that as kt2 increases the tautest span for the B-ISG system in the
crankshaft driving case decreases from 4675N to 4643N as well as the slackest span which
decreases from 2822N to 279N which is an overall change of -32N for both spans Whereas in
the ISG driving case a more noticeable change is once again found a difference of +144N
This is a result of the tautest span increasing from 3863N to 4007N and the slackest span
increasing from -3301N to -3157N
Parametric Analysis 78
432 Influence of Tensioner Pulley Diameter on Static Tension
The change in the diameter of tensioner pulley 1 D3 and its effect on static tension is shown in
Figure 45
Figure 45 Parametric Analysis for Pulley 1 Diameter D3 (m)
The change in the tautest and slackest spans for the B-ISG system‟s crankshaft driving case is
from 3248N to 425N and from 1395N to 240N respectively Peaks are seen at 4799N and
2946N for the respective spans This is a change of approximately +100N and a maximum
change of 1551N for both spans For the ISG driving case the tautest and slackest spans
decrease from 1083N to 6158N and 367N to -1006N Global minimums of 3246N and -391N
for the respective spans are seen This nets a change of approximately -467N and a maximum
change of approximately -759N
Parametric Analysis 79
The effect of changing the diameter of tensioner pulley 2 on the static tension is examined in
Figure 46
Figure 46 Parametric Analysis for Pulley 2 Diameter D5 (m)
The tautest and slackest spans in the crankshaft driving mode of the belt undergo a change from
4583N to 4721N and from 273N to 2869N respectively Therein as D5 increases the trend is
that for both spans there is an increase in tension of approximately 14N Contrastingly the spans
experience a decrease in the ISG driving case as D5 increases The tension of the tautest span
goes from 4296N to 3635N and that of the slackest span goes from -2866N to -3529N This
equals a decrease of approximately 66N for both spans
Parametric Analysis 80
433 Influence of Tensioner Pulley 1 Coordinates on Static Tension
The influence of the coordinates of tensioner pulley 1 on the value of tension in the tautest span
for the B-ISG system‟s crankshaft driving case is demonstrated in Figure 47
Figure 47 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Tautest Span
Tension in Crankshaft Driving Case
The region shown in Figure 47 corresponds to region 1 which is the realm of the positions for
tensioner pulley 1 The possible pulley coordinates in this case are represented by the non-blue
area reaching to the perimeter of the plot It is evident in the darkest red region of the plot
where the y-coordinate is between approximately 0m and 0075m and the x-coordinate is
(N)
Parametric Analysis 81
between approximately 026m and 031m that the highest value of tension is experienced in the
tautest span for the crankshaft driving case The range of tension for Region 1 in the tautest span
when the crankshaft is driving is between a maximum of approximately 500N and a minimum of
approximately 300N This equals an overall difference of 200N in tension for the tautest span by
moving the position of pulley 1 The lowest values for tension are obtained when the pulley
coordinates are approximately -0025m to 015m for the y-coordinate and approximately 031m
to 032m for the x-coordinate which corresponds to the yellow region An area of low tension is
also seen in the area where the y-coordinate is approximately 0m and the x-coordinate is
approximately between 026m and 027m
The changes in tension for the slackest span under the condition of the crankshaft pulley being
the driving pulley are shown in Figure 48
Parametric Analysis 82
Figure 48 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Slackest Span
Tension in Crankshaft Driving Case
Once again the possible coordinate points for tensioner pulley 1 in the B-ISG system are
represented by the non-blue region For the slackest span in the crankshaft driving case it is seen
that the lowest tension is approximately 125N while the highest tension is approximately 325N
This is an overall change of 200N that is achieved in the region The highest values are achieved
in the space where the y-coordinates are approximately 0m to 0075m and the x-coordinate
ranges from 026m to 031m which corresponds to the deep red region The lowest tension
values are achieved in the space where the y-coordinate ranges from approximately -0025m to
015m and the x-coordinate ranges from 031m to 032m which corresponds to the light blue-
green region of the plot The area containing a y-coordinate of approximately 0m and x-
(N)
Parametric Analysis 83
coordinates that are approximately between 026m and 027m also show minimum tension for
the slack span The regions of the x-y coordinates for the maximum and minimum tensions are
alike to the tautest span in Region 1 for the crankshaft driving case as well as was seen in Figure
47
The tension for the tautest span in the case that the ISG is driving in the B-ISG system is found
in Figure 49
Figure 49 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Tautest Span
Tension in ISG Driving Case
(N)
Parametric Analysis 84
Region 1 is represented by the coordinate values shown in the non-dark blue space of the plot in
Figure 49 The tautest span in the case of the ISG driving experiences a range of tension values
in Region 1 from 200N up to 1100N equaling a difference of 900N The minimum tension
values are achieved in the medium to light blue region This includes y-coordinates of
approximately 0m to 0075m and x-coordinates of approximately 026m to 03m The
maximum tension values are in the darkest red area inclusive of y-coordinates -0025m to 015m
and x-coordinates 031m to 032m in addition to y-coordinate of approximately 0m and x-
coordinates of approximately 026m to 027m It can be observed that aforementioned regions
for minimum and maximum tensions in Figure 49 are reverse to those seen in Figures 47 and
48 for the crankshaft driving case
The change in tension for the slackest span of the B-ISG system when it is driven by the ISG is
shown in Figure 410
Parametric Analysis 85
Figure 410 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Slackest Span
Tension in ISG Driving Case
Figure 410 exhibits the realm of possible points for tensioner pulley 1 for the case of the ISG
driving in the non-yellow-green area The minimum tension values are achieved in the darkest
blue area where the minimum tension is approximately -500N This area corresponds to y-
coordinates from approximately 0m to 005m and x-coordinates from approximately 026m to
03m The area of a maximum tension is approximately 400N and corresponds to the darkest red
area inclusive of y-coordinates -0025m to 015m and x-coordinates 031m to 032m as well as
the coordinates for y equaling approximately 0m and for x equaling approximately 026m to
027m The difference between maximum and minimum tensions in this case is approximately
900N It is noticed once again that the space of x- and y-coordinates containing the maximum
(N)
Parametric Analysis 86
tension is in the similar location to that of the described space for minimum tension in the
crankshaft driving case in Figure 47 and 48
434 Influence of Tensioner Pulley 2 Coordinates on Static Tension
The influence of pulley 2 coordinates on the tension value for the tautest span when the
crankshaft is driving the B-ISG system is shown in Figure 411 and is represented by the values
corresponding to the non-blue area
Figure 411 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Tautest Span
Tension in Crankshaft Driving Case
In Figure 411 the possible coordinates are contained within Region 2 The maximum tension
value is approximately 500N and is found in the darkest red space including approximately y-
(N)
Parametric Analysis 87
coordinates 004m to 014m and x-coordinates 0025m to 0175m and also y-coordinates 013m
to 02m corresponding to the x-coordinate at 0175m A minimum tension value of
approximately 350N is found in the yellow space and includes approximately y-coordinates
008m to 018m and x-coordinates 016m to 02m The difference in tension values is 150N
The analysis of the change in coordinates for tension pulley 2 on the value for tension in the
slackest span is shown in Figure 412 in the non-blue region
Figure 412 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Slackest Span
Tension in Crankshaft Driving Case
The value of 325N is the highest tension for the slack span in the crankshaft driving case of the
B-ISG system and is found in the deep-red region where the y-coordinates are between
(N)
Parametric Analysis 88
approximately 004m and 013m and the x-coordinates are approximately between 0025m and
016m as well as where y is between 013m and 02m and x is approximately 0175m The
lowest tension value for the slack span is approximately 150N and is found in the green-blue
space where y-coordinates are between approximately 01m and 022m and the x-coordinates
are between approximately 016m and 021m The overall difference in minimum and maximum
tension values is 175N The spaces for the maximum and minimum tension values are similar in
location to that found in Figure 411 for the tautest span in the crankshaft driving case
Figure 413 provides the theoretical data for the tension values of the tautest span as the position
of the B-ISG system‟s tensioner pulley 2 changes in the ISG driving case Possible points are in
the space of values which correspond to the non-dark-blue region in Figure 413
Parametric Analysis 89
Figure 413 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Tautest Span
Tension in ISG Driving Case
In Figure 413 the region for high tension reaches a value of approximately 950N and the region
for low tension reaches approximately 250N This equals a difference of 700N between
maximum and minimum tension values for the tautest span in the B-ISG system‟s ISG driving
case The coordinate points within the space that maximum tension is reached is in the dark red
region and includes y-coordinates from approximately 008m to 022m and x-coordinates from
approximately 016m to 021m The coordinate points within the space that minimum tension is
reached is in the blue-green region and includes y-coordinates from approximately 004m to
013m and the corresponding x-coordinates from approximately 0025m to 015m An additional
small region of minimum tension is seen in the area where the x-coordinate is approximately
(N)
Parametric Analysis 90
0175m and the y-coordinates are approximately between 013m and 02m The location for the
area of pulley centre points that achieve maximum and minimum tension values is approximately
located in the reverse positions on the plot when compared to that of the case for the crankshaft
driving in Figures 411 and 412 Therein the trend seen for pulley coordinates for the second
tensioner pulley follows suit with that of the first tensioner pulley which is that the area of points
for maximum tension in the crankshaft driving case becomes the approximate area of points for
minimum tension in the ISG driving case and vice versa
In Figure 414 the results of the parametric analysis on the coordinates of the second tensioner
pulley and its effect on the slackest span‟s tension in the ISG driving case is shown Similar to
earlier figures the non-dark yellow region represents Region 2 that contains the possible points
for the pulley‟s Cartesian coordinates
Parametric Analysis 91
Figure 414 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Slackest
Span Tension in ISG Driving Case
Figure 414 demonstrates a difference of approximately 725N between the highest and lowest
tension values for the slackest span of the B-ISG system in the ISG driving case The highest
tension values are approximately 225N The area of points that allow the second tension pulley
to achieve maximum tension in the belt span includes y-coordinates from approximately 01m to
022m and the corresponding x-coordinates from approximately 016m to 021m This
corresponds to the darkest red region in Figure 414 The coordinate values where the lowest
tension being approximately -500N is achieved include y-coordinate values from
approximately 004m to 013m and the corresponding x-coordinates from approximately 0025m
to 015m corresponding to the darkest blue region A dark blue region of lowest tension is also
(N)
Parametric Analysis 92
seen in the area where y is approximately between 013m and 02m and the x-coordinate is
approximately 0175m The regions for maximum and minimum tension values are observed to
be similar to those found in Figure 413 and alike to Figure 413 to be in reverse to those found
in Figure 411 and 412 for the tautest and slackest spans in the crankshaft driving case So as for
the changes in tensioner pulley 2 coordinates the areas for minimum tension in Region 2 of the
ISG driving case are similar to the areas for maximum tension in Region 2 of the crankshaft
driving case and vice versa for the maximum tension of the ISG driving case and the minimum
tension for the crankshaft driving case in Region 2
44 Conclusion
Overall the trend in the plots of Figures 47 48 411 and 412 indicate in the crankshaft driving
portion that the B-ISG system‟s belt span tensions experience the following effect
Minimum tension for the tautest span is achieved when tensioner pulley 1 centre
coordinates are located closer to the right side boundary and bottom left boundary of
Region 1 or when tensioner pulley 2 centre coordinates are within the upper right space
(near to the ISG pulley) and the space closer to the top boundary of Region 2
Maximum tension for the slackest span is achieved when the first tensioner pulley‟s
coordinates are located in the mid space and near to the bottom boundary of Region 1
and when the second tensioner pulley‟s coordinates are located near to the bottom left
boundary of Region 2 which is the boundary nearest to the crankshaft pulley
Parametric Analysis 93
The trend for minimizing the tautest span signifies that the tension for the slackest span is also
minimized at the same time As well maximizing the slackest span signifies that the tension for
the tautest span is also maximized at the same time too
The trend for the B-ISG system‟s ISG driving case as can be seen in Figures 49 410 413 and
414 is approximately in reverse to that of the crankshaft driving case for the system Wherein
points corresponding to minimum tension in Regions 1 and 2 in the ISG case are approximately
the same as points corresponding to maximum tension in the Regions for the crankshaft case and
vice versa for the ISG cases‟ areas of maximum tension
Minimum tension for the tautest span is present when the first tensioner pulley‟s
coordinates are near to mid to lower boundary of Region 1 and when the second
tensioner pulley‟s coordinates are close to the bottom left boundary of Region 2 which
is the furthest boundary from the ISG pulley and closest to the crankshaft pulley
Maximum tension for the slackest span is achieved when the first tensioner pulley is
located close to the right boundary of Region 1 and when the second tensioner pulley is
located near the right boundary and towards the top right boundary of Region 2
It is observed in Figures 47 to 414 and alike to Figures 42 to 46 the tautest and slackest
spans decrease or increase together Thus it can be assumed that the tension values in these
spans and likely the remaining spans outside of the tautest and slackest spans follow suit
Therein when parameters are changed to minimize one belt span‟s tension the remaining spans
will also have their tension values reduced Figures 42 through to 413 showed this clearly
where the overall change in the tension of the tautest and slackest spans changed by
Parametric Analysis 94
approximately the same values for each separate case of the crankshaft driving and the ISG
driving in the B-ISG system
Design variables are selected in the following chapter from the parameters that have been
analyzed in the present chapter The influence of changing parameters on the static tension
values for the various spans is further explored through an optimization study of the static belt
tension for the B-ISG system equipped with a Twin Tensioner in the following chapter Chapter
5
95
CHAPTER 5 OPTIMIZATION OF A B-ISG TENSIONER
The objective of the optimization analysis is to minimize the absolute magnitude of the static
tension in the ISG-operating mode of the serpentine belt drive The optimization seeks to
optimize the performance of the proposed Twin Tensioner design by using its properties as the
design variables for the objective function The optimization task begins with the selection of
these design variables for the objective function and then the selection of an optimization
method The results of the optimization will be compared with the results of the analytical
model for the static system and with the parametric analysis‟ data
51 Optimization Problem
511 Selection of Design Variables
The optimal system corresponds to the properties of the Twin Tensioner that result in minimized
magnitudes of static tension for the various belt spans Therein the design variables for the
optimization procedure are selected from amongst the Twin Tensioner‟s properties In the
parametric analysis of Chapter 4 the tensioner properties presented included
coupled stiffness kt
tensioner arm 1 stiffness kt1
tensioner arm 2 stiffness kt2
tensioner pulley 1 diameter D3
tensioner pulley 2 diameter D5
tensioner pulley 1 initial coordinates [X3Y3] and
Optimization 96
tensioner pulley 2 initial coordinates [X5Y5]
It was observed in the former chapter that perturbations of the stiffness and geometric parameters
caused a change between the lowest and highest values for the static tension especially in the
case of perturbations in the geometric parameters diameter and coordinates Table 51
summarizes the observed changes in the belt span tensions corresponding to the Twin Tensioner
parameters‟ maximum and minimum values
Table 51 Summary of Parametric Analysis Data for Twin Tensioner Properties
Parameter Symbol
Original Tensions in TautSlack Span (Crankshaft
Mode) [N]
Tension at
Min | Max Parameter6 for
Crankshaft Mode [N]
Percent Change from Original for
Min | Max Tensions []
Original Tension in TautSlack Span (ISG Mode)
[N]
Tension at
Min | Max Parameter Value in ISG Mode [N]
Percent Change from Original Tension for
Min | Max Tensions []
kt
465848 (taut) 4691 4646 07 -03 393645 (taut) 378 3998 -40 16
28057 (slack) 2838 2793 12 -05 -322803 (slack) -3384 -3167 -48 19
kt1
465848 (taut) 4628 4681 -07 05 393645 (taut) 4088 3827 38 -28
28057 (slack) 2775 2828 -11 08 -322803 (slack) -3077 -3338 47 -34
kt2
465848 (taut) 4675 4643 04 -03 393645 (taut) 3863 4007 -19 18
28057 (slack) 2822 279 06 -06 -322803 (slack) -3301 -3157 -23 22
D3 465848 (taut) 3248 425 -303 -88 393645 (taut) 1083 6158 1751 564
28057 (slack) 1395 240 -503 -145 -322803 (slack) 367 -1006 2137 688
D5 465848 (taut) 4583 4721 -16 13 393645 (taut) 4296 3635 91 -77
28057 (slack) 273 2869 -27 23 -322803 (slack) -2866 -3529 112 -93
[X3Y3] 465848 (taut) 300 500 -356 73 393645 (taut) 200 1100 -492 1794
28057 (slack) 125 325 -554 158 -322803 (slack) -500 400 -549 2239
6 The values for the tension for each of the taut and slack spans provided correspond to the minimum and maximum
values of the parameter listed in each case such that the columns of identical colour correspond to each other For
the coordinate parameters the minimum and maximum parameter value is inadmissible The tension values in these
cases are simply the minimum and maximum tension values achieved by the coordinate parameter listed
Optimization 97
[X5Y5] 465848 (taut) 350 500 -249 73 393645 (taut) 250 950 -365 1413
28057 (slack) 150 325 -465 158 -322803 (slack) -500 225 -549 1697
The results of the parametric analyses for the Twin Tensioner parameters show that there is a
noticeable percent change between the initial tensions and the tensions corresponding to each of
the minima and maxima parameter values or in the case of the coordinates between the
minimum and maximum tensions for the spans Thus the parametric data does not encourage
exclusion of any of the tensioner parameters from being selected as a design variable As a
theoretical experiment the optimization procedure seeks to find feasible physical solutions
Hence economic criteria are considered in the selection of the design variables from among the
Twin Tensioner‟s parameters Of the tensioner properties it is found that the diameter of the
tensioner pulleys has the largest impact on cost Adding mass to a tensioner pulley as a result of
increasing the diameter and consequently its inertia increases the cost of material Material cost
is most significant in the manufacture process of pulleys as their manufacturing is largely
automated [4] Furthermore varying the structure of a pulley requires retooling which also
increases the cost to manufacture As such the tensioner pulley diameters D3 and D5 are
excluded from being selected as design variables The remaining tensioner properties the
stiffness parameters and the initial coordinates of the pulley centres are selected as the design
variables for the objective function of the optimization process
512 Objective Function amp Constraints
In order to deal with two objective functions for a taut span and a slack span a weighted
approach was employed This emerges from the results of Chapter 3 for the static model and
Chapter 4 for the parametric study for the static system which show that a high tension span and
Optimization 98
a highly slack span exist in the ISG-driving phase of the B-ISG system Therein the first
objective function of equation (51a) is described as equaling fifty percent of the absolute tension
value of the tautest span and fifty percent of the absolute tension value of the slackest span for
the case of the ISG driving only The second objective function uses a non-weighted approach
and is described as the absolute tension of the slackest span when the ISG is driving A non-
weighted approach is motivated by the phenomenon of a fixed difference that is seen between
the slackest and tautest spans of the optimal designs found in the weighted optimization
simulations Equations (51a) through to (51c) display the objective functions
The limits for the design variables are expanded from those used in the parametric analysis for
the non-coordinate parameters kt kt1 and kt2 so that they are permitted to vary from
approximately 0 to approximately 200 of the initial value for each parameter kt0 kt10 and kt20
respectively In the case of the coordinate parameters [X3Y3] and [X5Y5] the x- and y-
coordinates are permitted to vary within the spaces Region 1 and Region 2 respectively which
were prescribed in Chapter 4 Figure 41a and 41b
Aside from the design variables design constraints on the system include the requirement for
static stability of the Twin Tensioner An optimal solution for the B-ISG system must achieve
the goal of the objective function which is to minimize the absolute tensions in the system
However for an optimal solution to be feasible the movement of the tensioner arm must remain
within an appropriate threshold In practice an automotive tensioner arm for the belt
transmission may be considered stable if its movement remains within a 10 degree range of
Optimization 99
motion [4] As such the angle of displacement for tensioner arms 1 and 2 are designated by θ t1
and θt2 respectively in the listed constraints
The optimization task is described in equations 51a to 52 Variables a through to g represent
scalar limits for the x-coordinate for corresponding ranges of the y-coordinate
Minimize 119879119908119890119894119892 119893119905119890119889 = 05 ∙ 119879119905119886119906119905 + 05 ∙ 119879119904119897119886119888119896
or119879119899119900119899 minus119908119890119894119892 119893119905119890119889 = 119879119904119897119886119888119896
(51a)
where
119879119905119886119906119905 = 119891119905119886119906119905 119896119905 1198961199051 1198961199052 1198833 1198843 1198835 1198845 (51b)
119879119904119897119886119888119896 = 119891119904119897119886119888119896 (119896119905 1198961199051 1198961199052 1198833 119884311988351198845) (51c)
Subject to
(1198961199050 minus 1 ∙ 1198961199050) le 119896119905 le (1198961199050 + 11198961199050)(11989611990510 minus 1 ∙ 11989611990510) le 1198961199051 le (11989611990510 + 111989611990510)(11989611990520 minus 1 ∙ 11989611990520) le 1198961199052 le (11989611990520 + 111989611990520)
119886 le 1198833 le 119888
1198931 1198833 le 1198843 le 1198933 1198833 119891119900119903 119886 le 1198833 lt 119887
1198932 1198833 le 1198843 le 1198933 1198833 119891119900119903 119887 le 1198833 le 119888119889 le 1198835 le 119892
1198934 1198835 le 1198845 le 1198937 1198835 for 119889 le 1198835 lt 1198901198935(1198835) le 1198845 le 1198937(1198835) for 119890 le 1198835 lt 119891
1198936 1198835 le 1198845 le 1198937 1198833 for 119891 le 1198833 le 119892 1205791199051 le 10deg 1205791199052 le 10deg
(52)
The functions for the taut and slack spans represent the fourth and third span respectively in the
case of the ISG driving The equations for the tensions of the aforementioned spans are shown
in equation 51a to 51c and are derived from equation 353 The constraints for the
optimization are described in equation 52
Optimization 100
52 Optimization Method
A twofold approach was used in the optimization method A global search alone and then a
hybrid search comprising of a global search and a local search The Genetic Algorithm is used
as the global search method and a Quadratic Sequential Programming algorithm is used for the
local search method
521 Genetic Algorithm
Genetic Algorithm (GA) is a part of the growing genre of evolutionary algorithms [57] The
optimization approach differs from classical search approaches by its ease of use and global
perspective [57] GA mimics biological evolution theory by using a ldquocross-over of heritable
information random mutation and selection on the basis of fitness between generationsrdquo [58] to
form a robust search algorithm that requires minimal problem information [57] The parameter
sets are represented as sample points modeled as bdquochromosomes‟ or data strings that are
measured against how well they allow the model to achieve the optimization task [58] GA is
stochastic which means that its algorithm uses random choices to generate subsequent sampling
points rather than using a set rule to generate the following sample This avoids the pitfall of
gradient-based techniques that may focus on local maxima or minima and end-up neglecting
regions containing higher peaks or lower valleys [57] Furthermore due to the randomness of
the GA‟s search strategy it is able to search a population (a region of possible parameter sets)
faster than other optimization techniques The GA approach is viewed as a universal
optimization approach while many classical methods viewed to be efficient for one optimization
problem may be seen as inefficient for others However because GA is a probabilistic algorithm
its solution for the objective function may only be near to a global optimum As such the current
Optimization 101
state of stochastic or global optimization methods has been to refine results of the GA with a
local search and optimization procedure
522 Hybrid Optimization Algorithm
In order to enhance the result of the objective function found by the GA a Hybrid optimization
function is implemented in MATLAB software The Hybrid optimization function combines a
global search GA with a local search Sequential Quadratic Programming (SQP) The hybrid
process refines the value of the objective function found through GA by using the final set of
points found by the algorithm as the initial point of the SQP algorithm The GA function
determines the region containing a global optimum and then the SQP algorithm uses a gradient
based technique to find a solution closer to the global optimum The MATLAB algorithm a
constrained minimization function known as fmincon uses an SQP method that approximates the
Hessian for the Lagrangian function (ie the second derivatives of the Lagrangian) by way of a
quasi-Newton approach to generate a quadratic program (QP) sub-problem [59] The solution
for the QP provides the search direction of the line search procedure used when each iteration is
performed [59]
53 Results and Discussion
531 Parameter Settings amp Stopping Criteria for Simulations
The parameter settings for the optimization procedure included setting the stall time limit to
200s This is the interval of time the GA is given to find an improvement in the value of the
objective function This is an increase from MATLAB‟s default of 20s Increasing the stall time
limit allows for the optimization search to consistently converge without being limited by time
Optimization 102
The population size used in finding the optimal solution is set at 100 This value was chosen
after varying the population size between 50 and 2000 showed no change in the value of the
objective function The max number of generations is set at 100 The time limit for the
algorithm is set at infinite The limiting factor serving as the stopping condition for the
optimization search was the function tolerance which is set at 1x10-6
This allows the program
to run until the ratio of the change in the objective function over the stall generations is less than
the value for function tolerance The stall generation setting is defined as the number of
generations since the last improvement of the objective function value and is limited to 50
532 Optimization Simulations
The results of the genetic algorithm optimization simulations performed in MATLAB are shown
in the following tables Table 52a and Table 52b
Table 52a GA Optimization Results for Twin Tensioner Parameters and Objective Function
Trial
No
Genetic Algorithm Optimization Method
Objective
Function
Value [N]
kt [N∙mrad]
kt1 [N∙mrad]
kt2 [N∙mrad]
[X3Y3]
[m] [X5Y5]
[m]
1 3582241 314069 204844 165020 [02928 00703] [01618 01036]
2 3582241 103646 205284 198901 [03009 00607] [01283 00809]
3 3582241 126431 204740 43549 [03010 00631] [01311 01147]
4 3582241 180285 206230 254870 [03095 00865] [01080 01675]
5 3582241 74757 204559 189077 [03084 00617] [01265 00718]
Original Design 20626 10314 16502 [02928 0087] [01206 00919]
Optimization 103
Table 52b Computations for Tensions and Angles from GA Optimization Results
Trial No
Genetic Algorithm Optimization Method
Slackest Tension [N] Tautest Tension [N] Tensioner Arm 1
Displacement [deg]
Tensioner Arm 2
Displacement [deg]
T3 T4 Θt1 Θt2
1 -1572307 5592176 -00025 -49748
2 -4054309 3110174 -00002 -20213
3 -3930858 3233624 -00004 -38370
4 -1309751 5854731 -00010 -49525
5 -4092446 3072036 -00000 -17703
Original Design -322803 393645 16410 -4571
For each trial above the GA function required 4 generations each consisting of 20 900 function
evaluations before finding no change in the optimal objective function value according to
stopping conditions
The results of the Hybrid function optimization are provided in Tables 53a and 53b below
Table 53a Hybrid Optimization Results for Twin Tensioner Parameters and Objective Function
Trial
No
Hybrid Optimization Method
Objective
Function
Value [N]
of
Function
Evals ( of
Iterations)
kt [N∙mrad]
kt1 [N∙mrad]
kt2 [N∙mrad]
[X3Y3]
[m] [X5Y5]
[m]
1 3582241 16 (1) 16065 205846 229494 [02780 00581] [01679 01288]
2 3582241 20 (1) 249227 205635 25218 [02901 00634] [01559 00870]
3 3582241 16 (1) 297295 204878 320479 [02962 00702] [01336 01447]
4 3582241 53 (1) 241433 204262 229683 [02912 00647] [00047 01465]
Optimization 104
5 3582241 21 (1) 379096 205548 188888 [02973 00703] [01206 01376]
Original Design 20626 10314 16502 [02928 0087] [01206 00919]
Table 53b Computations for Tensions and Angles from Hybrid Optimization Results
Trial No
Hybrid Algorithm Optimization Method
Slackest Tension [N] Tautest Tension [N] Tensioner Arm 1
Displacement [deg]
Tensioner Arm 2
Displacement [deg]
T3 T4 Θt1 Θt2
1 -2584641 4579841 -02430 67549
2 -3708747 3455736 -00023 -41068
3 -1707181 5457302 -00099 -43944
4 -269178 6895304 00006 -25366
5 -2982335 4182148 -00003 -41134
Original Design -322803 393645 16410 -4571
In Table 53a it can be seen that iterations of 16 20 21 or 53 were required for the local search
algorithm following the GA to find an optimal solution Once again the GA function
computed 4 generations which consisted of approximately 20 900 function evaluations before
securing an optimum solution
The simulation results of the non-weighted hybrid optimization approach are shown in tables
54a and 54b below
Optimization 105
Table 54a Non-Weighted Optimization Results for Twin Tensioner Parameters and Objective
Function
Trial
No
Objective
Function
Value [N]
of
Function
Evals ( of
Iterations)
kt [N∙mrad]
kt1 [N∙mrad]
kt2 [N∙mrad]
[X3Y3]
[m] [X5Y5]
[m]
Genetic Algorithm Optimization Method
1 33509e
-004 20900 (4) 321799 75530 212653 [02860 00602] [01082 01858]
Hybrid Optimization Method
1 73214e
-011 381 (13) 234881 14730 323358 [02952 00688] [00048 01466]
Original Design 20626 10314 16502 [02928 0087] [01206 00919]
Table 54b Computations for Tensions and Angles from Non-Weighted Optimizations
Trial No Slackest Tension [N] Tautest Tension [N]
Tensioner Arm 1
Displacement [deg]
Tensioner Arm 2
Displacement [deg]
T3 T4 Θt1 Θt2
Genetic Algorithm Optimization Method
1 -00003 7164479 -00588 -06213
Hybrid Optimization Method
1 -00000 7164482 15543 -16254
Original Design -322803 393645 16410 -4571
The weighted optimization data of Table 54a shows that the GA simulation again used 4
generations containing 20 900 function evaluations to conduct a global search for the optimal
system While the weighted Hybrid optimization used 13 iterations (consisting of 381 function
evaluations) after its GA run which used the same number of generations and function
evaluations as the GA run in the non-weighted simulations Tables 54a and 54b show the data
Optimization 106
for only one trial for each of the non-weighted GA and hybrid methods since only a single
optimal point exists in this case
533 Discussion
The optimal design from each search method can be selected based on the least amount of
additional pre-tension (corresponding to the largest magnitude of negative tension) that would
need to be added to the system This is in harmony with the goal of the optimization of the B-
ISG system as stated earlier to minimize the static tension for the tautest span and at the same
time minimize the absolute static tension of the slackest span for the ISG driving case As well
the angular displacements corresponding to each trial‟s results show that the Twin Tensioner is
under static stability Therein the optimal solution may be selected as the design parameters
corresponding to Trial 4 of the GA simulations to Trial 4 of the Hybrid simulations or to either
of the trials for the non-weighted optimization simulations
Given the ability of the Hybrid optimization to refine the results obtained in the GA
optimization the results of Trial 4 of the Hybrid simulations are selected as the most optimal
design from the weighted objective function approaches It is interesting to note that the Hybrid
case for the least slackness in belt span tension corresponds to a significantly larger number of
function evaluations than that of the remaining Hybrid cases This anomaly however does not
invalidate the other Hybrid cases since each still satisfy the design constraints Using the data
for the optimized system in Trial 4 (of the Hybrid optimization) the static tensions for the belt
spans in both of the B-ISG‟s phases of operation are as follows
Optimization 107
Table 55 Weighted Optimization Results for Static Tensions for Optimal B-ISG System with a
Twin Tensioner
Belt Span between Pulleys Crankshaft Driving Case [N] ISG Driving Case [N]
Optimized Original Optimized Original
Crankshaft ndash Air Conditioner 3926599 465848 117333 -284152
Air Conditioner ndash Tensioner 1 3540088 427197 -269178 -322803
Tensioner 1 ndash ISG 3540088 427197 -269178 -322803
ISG ndash Tensioner 2 2073813 28057 6895304 393645
Tensioner 2 ndash Crankshaft 2073813 28057 6895304 393645
Additional Pretension
Required (approximate) + 27000 +322803 + 27000 +322803
In Table 54b it is evident that the non-weighted class of optimization simulations achieves the
least amount of required pre-tension to be added to the system The computed tension results
corresponding to both of the non-weighted GA and Hybrid approaches are approximately
equivalent Therein either of their solution parameters may also be called the most optimal
design The Hybrid solution parameters are selected as the optimal design once again due to the
refinement of the GA output contained in the Hybrid approach and its corresponding belt
tensions are listed in Table 56 below
Optimization 108
Table 56 Non-Weighted Optimization Results for Static Tensions for Optimal B-ISG System
with a Twin Tensioner
Belt Span between Pulleys Crankshaft Driving Case [N] ISG Driving Case [N]
Optimized Original Optimized Original
Crankshaft ndash Air Conditioner 3891862 465848 386511 -284152
Air Conditioner ndash Tensioner 1 3505351 427197 -00000 -322803
Tensioner 1 ndash ISG 3505351 427197 -00000 -322803
ISG ndash Tensioner 2 2039076 28057 7164482 393645
Tensioner 2 ndash Crankshaft 2039076 28057 7164482 393645
Additional Pretension
Required (approximate) + 0000 +322803 + 00000 +322803
The results of the simulation experiments are limited by the following considerations
System equations are coupled so that a fixed difference remains between tautest and
slackest spans
A limited number of simulation trials have been performed
There are multiple optimal design points for the weighted optimization search methods
Remaining tensioner parameters tensioner pulley diameters and their stiffness have not
been included in the design variables for the experiments
The belt factor kb used in the modeling of the system‟s belt has been obtained
experimentally and may be open to further sources of error
Therein the conclusions obtained and interpretations of the simulation data can be limited by the
above noted comments on the optimization experiments
Optimization 109
54 Conclusion
The outcomes the trends in the experimental data and the optimal designs can be concluded
from the optimization simulations The simulation outcomes demonstrate that in all cases the
weighted optimization functions reached an identical value for the objective function whereas
the values reached for the parameters varied widely
The lowest tension values for the tautest and slackest span were achieved in Trial 5 of the GA
optimization approach In reiteration in the presence of slack spans the tension value of the
slackest span must be added to the initial static tension for the belt Therein for the former case
an amount of at least 409N would need to be added to the 300N of pre-tension already applied to
the system (see Table 34) The highest tension values for the spans were achieved in Trial 4 of
the weighted Hybrid optimization approach and in both trials of the non-weighted optimization
approaches In the former the weighted Hybrid trial the tension value achieved in the slackest
span was approximately -27N signifying that only at least 27N would need to be added to the
present pre-tension value for the system The tension of the slackest span in the non-weighted
approach was approximately 0N signifying that the minimum required additional tension is 0N
for the system
The optimized solutions for the tension values in each span show that there is consistently a fixed
difference of 716448N between the tautest and slackest span tension values as seen in Tables
52b 53b and 54b This difference is identical to the difference between the tautest and slackest
spans of the B-ISG system for the original values of the design parameters while in its ISG
mode As well the optimal stiffness parameters for the weighted Hybrid optimization case are
Optimization 110
greater than their original values except for that of the stiffness factor of tensioner arm 1
Likewise for the non-weighted Hybrid optimization case the stiffness parameters are above their
original values without exceptions The coordinates of the optimal solutions are within close
approximation to each other and also both match the regions for moderately low tension in
Regions 1 and 2 of the ISG driving case as is shown in Figures 49 410 413 and 414
The results of the non-weighted Hybrid optimization trial and Trial 4 of the weighted Hybrid
optimization simulations are selected as the most optimal designs for the B-ISG Twin Tensioner
In these designs the Twin Tensioner is shown in Table 53b and 54b to have static stability and
to maintain suitable tensions in the ISG driving phase The tensioner parameters for the optimal
designs allow for one of the lowest amounts of additional pre-tension to be added to the system
out of all the findings from the simulations which were conducted
111
CHAPTER 6 CONCLUSION
61 Summary
The primary aim of the thesis is to reduce the magnitude of static tension in the belt spans of a
Belt-driven Integrated Starter-generator (B-ISG) system by the design and investigation of a
Twin Tensioner It is established that the operating phases of the B-ISG system produced two
cases for static tension outcomes an ISG driving case and a crankshaft driving case The
approach taken in this thesis study includes the derivation of a system model for the geometric
properties as well as for the dynamic and static states of the B-ISG system The static state of a
B-ISG system with a single tensioner mechanism is highlighted for comparison with the static
state of the Twin Tensioner-equipped B-ISG system
It is observed that there is an overall reduction in the magnitudes of the static belt tensions with
the presence of a Twin Tensioner over that of a single tensioner The influences of the geometric
and stiffness properties of the Twin Tensioner affecting the static tensions in the system are
analyzed in a parametric study It is found that there is a notable change in the static tensions
produced as result of perturbations in each respective tensioner property This demonstrates
there are no reasons to not further consider a tensioner property based solely on its influence on
the B-ISG system‟s static tensions The phenomenon of higher magnitudes for static tensions in
the ISG mode of operation over that of the crankshaft mode of operation particularly in
excessively slack spans provides the motivation for optimizing the ISG case alone for static
tension The optimization method uses weighted and non-weighted approaches with genetic
algorithm (GA) and hybrid GA searches The most optimal design has been found to be one in
Conclusion 112
which the magnitude of tension in the excessively slack spans in the ISG driving case are
significantly lower than in that of the original B-ISG Twin Tensioner design
62 Conclusion
The conclusions that can be drawn from the study of a Twin Tensioner for a B-ISG system
include the following
1 The simulations of the dynamic model demonstrate that the mode shapes for the system
are greater in the ISG-phase of operation
2 It was observed in the output of the dynamic responses that the system‟s rigid bodies
experienced larger displacements when the crankshaft was driving over that of the ISG-
driving phase It was also noted that the transition speed marking the phase change from
the ISG driving to the crankshaft driving occurred before the system reached either of its
first natural frequencies
3 The magnitudes for static belt tensions as well as dynamic tensions for a B-ISG system
are consistently greater in its ISG operating phase than in its crankshaft operating phase
4 A Twin Tensioner is able to reduce the magnitudes of the static tension for the belt spans
of a B-ISG system in comparison to when only a single tensioner mechanism is present
5 The parametric study of the B-ISG system demonstrates that the slackest and tautest belt
spans decrease or increase together for either phase of operation
6 Perturbations in the Twin Tensioner‟s geometric and stiffness properties have a
significant influence on the magnitudes of the static tension of the slackest and tautest
belt spans The coordinate position of each pulley in the Twin Tensioner configuration
Conclusion 113
has the greatest influence on the belt span static tensions out of all the tensioner
properties considered
7 Optimization of the B-ISG system shows a fixed difference trend between the static
tension of the slackest and tautest belt spans for the B-ISG system
8 The values of the design variables for the most optimal system are found using a hybrid
algorithm approach The slackest span for the optimal system is significantly less slack
than that of the original design Therein less additional pretension is required to be added
to the system to compensate for slack spans in the ISG-driving phase of operation
63 Recommendation for Future Work
The investigation of the B-ISG Twin Tensioner encourages the following future work
1 The optimization of the B-ISG system with the inclusion of diametric Twin Tensioner
properties would provide a complete picture as to the highest possible performance
outcome that the Twin Tensioner is able to have with respect to the static tensions
achieved in the belt spans
2 A larger number of optimization trials using the genetic algorithm (GA) and hybrid GA
under weighted and other approaches would investigate the scope of optimal designs
available in the Twin Tensioner for the B-ISG system
3 A model of the system without the simplification of constant damping may produce
results that are more representative of realistic operating conditions of the serpentine belt
drive A finite element analysis of the Twin Tensioner B-ISG system may provide more
applicable findings
Conclusion 114
4 Investigation of the transverse motion coupled with the rotational belt motion in an
optimized B-ISG system equipped with a Twin Tensioner may also provide a closer look
at the system under realistic conditions In addition the affect of the Twin Tensioner on
transverse motion can determine whether significant improvements in the magnitudes of
static belt span tensions are still being achieved
5 The recommendation to conduct modal decoupling of the B-ISG system‟s static model is
motivated by the fixed difference trend between the slackest and tautest belt span
tensions shown in Chapter 5 The modal decoupling of the system would allow for its
matrices comprising the equations of motion to be diagonalized and therein to decouple
the system equations Modal analysis would transform the system from physical
coordinates into natural coordinates or modal coordinates which would lead to the
decoupling of system responses
6 An investigation and optimization of the dynamic belt span tensions for a B-ISG system
with a Twin Tensioner would increase understanding of the full impact of a Twin
Tensioner mechanism on the state of the B-ISG system It would be informative to
analyze the mode shapes of the first and second modes as well as the required torques of
the driving pulleys and the resulting torque of each of the tensioner arms The
observation of the dynamic belt span tensions would also give direction as to how
damping of the system may or may not be changed
7 Further comparison with the Twin Tensioner B-ISG system‟s dynamic and static states
including the Twin Tensioner‟s stability in each versus a B-ISG system with a single
tensioner would further demonstrate the improvements or dis-improvements in the Twin
Tensioner‟s performance on a B-ISG system
Conclusion 115
8 The influence of the belt properties on the dynamic and static tensions for a B-ISG
system with a Twin Tensioner can also be investigated This again will show the
evidence of improvements or dis-improvement in the Twin Tensioner‟s performance
within a B-ISG setting
9 Lastly an experimental apparatus of the B-ISG system with a Twin Tensioner can be
designed and constructed Suitable instrumentation can be set-up to measure belt span
tensions (both static and dynamic) belt motion and numerous other system qualities
This would provide extensive guidance as to finding the most appropriate theoretical
model for the system Experimental data would provide a bench mark for evaluating the
theoretical simulation results of the Twin Tensioner-equipped B-ISG system
116
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[14] National Alternative Fuels Training Consortium (NAFTC) (2005 Oct 3) Tech stuff
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[15] Green Car Congress BMW to Apply Start-Stop and Brake Regen to MINIs Up to 60
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[21] PJ Wezenbeek (Zytec Systems Ltd) D G Evans (General Motors Powertrain) D P
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(Delphi Corp) Combustion Assisted Belt-Cranking of a V-8 Engine at 12-Volts SAE
Technical Papers vol 113 pp 396-407 2004 Document no 2004-01-0569
[22] T C Firbank Mechanics of the Belt Drive International Journal of Mechanical
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[23] R L Cassidy S K Fan R S MacDonald and W F Samson Serpentine Extended Life
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[24] A G Ulsoy J E Whitesell and M D Hooven Design of Belt-Tensioner Systems for
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Transactions of the ASME vol 107 pp 282-290 July 1985
[25] R S Beikmann N C Perkins and A G Ulsoy Free Vibration of Serpentine Belt Drive
Systems Journal of Vibrations and Acoustics Transactions of the ASME vol 118 pp
406-413 1996
[26] T C Kraver G W Fan and J J Shah Complex Modal Analysis of a Flat Belt Pulley
System with Belt Damping and Coulomb-Damped Tensioner Journal of Mechanical
Design Transactions of the ASME vol 118 pp 306-311 Jun 1996
[27] R S Beikmann N C Perkins and A G Ulsoy Design and Analysis of Automotive
Serpentine Belt Drive Systems for Steady State Performance Journal of Mechanical
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[28] L Zhang and J W Zu Modal Analysis of Serpentine Belt Drive Systems Journal of
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[29] R Balaji and E M Mockensturm Dynamic analysis of a front-end accessory drive with a
decouplerisolator International Journal of Vehicle Design vol 39 pp 208-231 2005
[30] M Nouri Design Optimization and Active Control of Serpentine Belt Drive Systems with
Two-pulley Tensioners University of Toronto 2005
[31] G J Spicer (Litens Automotive Inc) Tensioner for use in eg belt drive system has
electronic actuator associated with clutch spring for engaging International
WO2005119089-A1 Jun 6 2005 2005
[32] Bando Chemical Industries Ltd and Litens Automotive GmbH About belt-type starter
system Feb 27 2002
[33] H Lemberger and R Jungjohann (Bayerische Motoren Werke AG) Tension device for an
envelope drive of a device especially a belt drive of a starter generator of an internal
combustion engine comprises a support part Europe EP1420192-A2 May 19 2004 2003
[34] P Ahner and M Ackermann (Bosch GMBH) Belt drive especially for internal
combustion engines to drive accessories in an automobile Germany DE19849886-A1
May 11 2000 1998
[35] N Freisinger K Hagemann J Sievert P Struebel and M Treusch (Daimler Chrysler AG)
Belt tensioning device for belt drive between engine and starter generator of motor
vehicle has self-aligning bearing that supports auxiliary unit and provides working force to
tensioners for tensioning belt Germany DE10324268 Dec 16 2004 2003
[36] C R Rogers (Dayco Products LLC) Offset starter generator drive system for a vehicle
engine has a dual arm pivoted tensioner United States US6942589-B2 Feb 8 2005 2002
[37] A Serkh and I Ali (Gates Corp) Internal combustion engine has belt drive system with
tensioner asymmetrically biased in direction tending to cause power transmission belt to be
under tension International WO2003038309-A1 May 8 2003 2002
References 120
[38] P J Mcvicar and C A Thurston (General Motors Corp) Belt alternator starter accessory
drive with dual tensioner United States US20060287146-A1 Dec 21 2006 2005
[39] W Petri and M Bogner (INA Schaeffler KG) Traction drive especially for driving
internal combustion engine units has arrangement for demand regulated setting of tension
consisting of unit with housing with limited rotation and pulley German DE10044645-
A1 Mar 21 2002 2000
[40] M Bogner (INA Schaeffler KG) Belt drive tensioner for a starter-generator of an IC
engine has locking system for locking tensioning element in an engine operating mode
locking system is directly connected to pivot arm follows arm control movements
German DE10159073-A1 Jun 12 2003 2001
[41] R Painta M Bogner and H Graf (INA Schaeffler KG) Traction mechanism drive esp
belt drive has belt tensioning pulley mounted on generator shaft and uncoupled from it via
freewheel to dampen load peaks Europe EP1723350-A1 Nov 22 2006 2005
[42] W Petri (INA Schaeffler KG) Drive unit for a combustion engine having a starter
generator and a belt drive has tensioner with spring and counter hydraulic force Germany
DE10359641-A1 Jul 28 2005 2003
[43] H Stief M Bogner B Hartmann T Kraft and M Schmid (INA Schaeffler KG) Traction
drive especially belt drive for short-duration driving of starter generator has tensioning
device with lever arm deflectable against restoring force and with end stop limiting
deflection travel Europe EP1738093-A1 Jan 3 2007 2005
[44] M Ulm (INA Schaeffler KG DE) Tension unit eg for drive in machine such as
combustion engine has belt or chain drive with wheels turning and connected with starter
generator and unit has two idlers arranged at clamping arm with machine stored by shock
absorber Germany DE102004012395-A1 Sep 29 2005 2004
[45] M Bogner (INA Schaeffler KG) Belt drive for starter motor-generator auxiliary assembly
has limited movement at the starter belt section tensioner roller bringing it into a dead point
position on starting the motor International WO2006108461-A1 Oct 19 2006 2006
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[47] K Kuniaki K Masahiko H Kazuyuki I Shuichi and T Masaki (Mitsubishi Jidosha Eng
KK and Mitsubishi Motor Corp) Tension adjustment method of belt for starter generator
in vehicle involves shifting auto-tensioners between lock state and free state to adjust
tension of belt during driving of crank pulley Japan JP2005083514-A Mar 31 2005
2003
[48] Nissan Motor Co Ltd Winding gear for starting engine of hybrid motor vehicle has
tensioner tightening chain while cranking engine and slackens chain after start of engine
provided to span side of chain Japan JP3565040-B2 Sep 15 2004 1998
[49] S Sato and H Hayakawa (NTN Corp) Auto tensioner for ancillary drive belts has
cylinder nut and screw bolt in hydraulic damper mechanism provided in middle of cylinder
acting as start-up rigidity buffer component Japan JP2006189073-A Jul 20 2006 2005
[50] G Vadin-Michaud (Valeo Equip Electrique Moteur) Pulley and belt starting system for a
thermal engine for a motor vehicle Europe EP1658432 May 24 2006 2005
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2005
[52] W E Johns Notes on Motors [Electronic] 2003 [2008 June] Available at
httpwwwgizmologynetmotorshtm
[53] Litens Automotive Group Ltd DC BAS System - Conventional Start Input Profile Nov
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[2008 June] Available at httpwwwgrouparnoldcommtcpdfweb_motor_glossarypdf
[55] Douglas W Jones Stepping Motors University of Iowa - Department of Computer
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httpwwwcsuiowaedu~jonesstepphysicshtml
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[56] Litens Automotive Group Ltd (2004 Jan 31) FEAD software input data for test project
[57] K Deb Multi-Objective Optimization using Evolutionary Algorithms Toronto John Wiley
amp Sons Ltd 2001 pp 81-85
[58] P E McSharry (2004 May 11) Department of Engineering Science University of Oxford
[httpwwwengoxacuksamppubsgawbreppdf]
[59] The MathWorks Inc MATLAB vol 750342 (R2007b) Aug 15 2007
123
APPENDIX A
Passive Dual Tensioner Designs from Patent Literature
Figure A1 Proposed design by Bayerische Motoren Werke AG corresponding to patent nos EP1420192-A2 and DE10253450-A1
Source European Patent Office espcenet (publication nos EP1420192-A2 and DE10253450-A1 accessed May 2007) epespacenetcom [33]
Figure A1 label identification 1 ndash tightner 2 ndash belt drive
3 ndash starter generator
4 ndash internal-combustion engine
4‟ ndash crankshaft-lateral drive disk
5 ndash generator housing
6 ndash common axis of rotation
7 ndash featherspring of tiltable clamping arms
8 ndash clamping arm
9 ndash clamping arm
10 11 ndash idlers
12 12‟ ndash Zugtrum 13 13‟ ndash Leertrum
14 ndash carry-hurries 15 ndash generator wave
16 ndash bush
17 ndash absorption mechanism
18 18‟ ndash support arms
19 19‟ ndash auxiliary straight lines
20 ndash pipe
21 ndash torsion bar
22 ndash breaking through
23 ndash featherspring
24 ndash friction disk
25 ndash screw connection 26 ndash Wellscheibe
(European Patent Office May 2007) [33]
Appendix A 124
Figure A2a First of four proposed designs (and its various configurations) by Bosch GMBH
corresponding to patent no WO0026532-A1
Source European Patent Office espcenet (publication no WO0026532-A1 accessed Jun 2007)
epespacenetcom [34]
Figure A2b Second of four proposed designs by Bosch GMBH corresponding to patent no
WO0026532-A1
Source European Patent Office espcenet (publication no WO0026532-A1 accessed Jun 2007) epespacenetcom [34]
Figure A2c Third of four proposed designs (and its various configurations) by Bosch GMBH
corresponding to patent no WO0026532-A1
Source European Patent Office espcenet (publication no WO0026532-A1 accessed Jun 2007)
epespacenetcom [34]
Appendix A 125
Figure A2d Fourth of four proposed designs (and its various configurations) by Bosch GMBH
corresponding to patent no WO0026532-A1
Source European Patent Office espcenet (publication no WO0026532-A1 accessed Jun 2007)
epespacenetcom [34]
Figure A2a through to A2d label identification 10 ndash engine wheel
11 ndash [generator] 13 ndash spring
14 ndash belt
16 17 ndash tensioning pulleys
18 19 ndash springs
20 21 ndash fixed points
25ab ndash carriers of idlers
25c ndash gang bolt
(European Patent Office June 2007) [34]
Figure A3 Proposed design by Daimler Chrysler AG corresponding to patent no DE10324268-A1
Source European Patent Office espcenet (publication no DE10324268-A1 accessed May 2007)
epespacenetcom [35]
Figure A3 label identification
Appendix A 126
10 12 ndash belt pulleys
14 ndash auxiliary unit
16 ndash belt
22-1 22-2 ndash belt tensioners
(European Patent Office May 2007) [35]
Figure A4 Proposed design by Dayco Products LLC corresponding to patent no US6942589-B2
Source European Patent Office espcenet (publication no US6942589-B2 accessed Jun 2007)
epespacenetcom [36]
Figure A4 label identification 12 ndash belt
14 ndash tensioner
16 ndash generator pulley
18 ndash crankshaft pulley
22 ndash slack span 24 ndash tight span
32 34 ndash arms
33 35 ndash pulley
(European Patent Office June 2007) [36]
Appendix A 127
Figure A5 Proposed design by Gates Corp corresponding to patent no WO2003038309-A
Source European Patent Office espcenet (publication no WO2003038309-A accessed Jun 2007)
epespacenetcom [37]
Figure A5 label identification 12 ndash motorgenerator
14 ndash motorgenerator pulley 26 ndash belt tensioner
28 ndash belt tensioner pulley
30 ndash transmission belt
(European Patent Office June 2007) [37]
Figure A6 Proposed design by General Motors Corp corresponding to patent no US20060287146-A1
Source European Patent Office espcenet (publication no US20060287146-A1 accessed Jun 2007)
epespacenetcom [38]
Appendix A 128
Figure A6 label identification 28 ndash tensioner
32 ndash carrier arm
34 ndash secondary carrier arm
46 ndash tensioner pulley
58 ndash secondary tensioner pulley
(European Patent Office June 2007) [38]
Figure A7 Proposed design by INA Schaeffler KG corresponding to patent no DE10044645-A1
Source European Patent Office espcenet (publication no DE10044645-A1 accessed Jun 2007)
epespacenetcom [39]
Figure A7 label identification 2 ndash internal combustion engine
3 ndash traction element
11 ndash housing with limited rotation 12 13 ndash direction changing pulleys
(European Patent Office June 2007) [39]
Appendix A 129
Figure A8a First of three proposed designs by INA Schaeffler KG corresponding to patent no
DE10159073-A1
Source European Patent Office espcenet (publication no DE10159073-A1 accessed May 2007)
epespacenetcom [40]
Figure A8b Second of three proposed designs by INA Schaeffler KG corresponding to patent no
DE10159073-A1
Source European Patent Office espcenet (publication no DE10159073-A1 accessed May 2007)
epespacenetcom [40]
Appendix A 130
Figure A8c Third of three proposed designs by INA Schaeffler KG corresponding to patent no
DE10159073-A1
Source European Patent Office espcenet (publication no DE10159073-A1 accessed May 2007)
epespacenetcom [40]
Figure A8a A8b and A8c label identification 1 ndash tightener [tensioner]
2 ndash idler
3 ndash drawing means
4 ndash swivel arm
5 ndash axis of rotation
6 ndash drawing means impulse [belt]
7 ndash crankshaft
8 ndash starter generator
9 ndash bolting volume 10a ndash bolting device system
10b ndash bolting device system
10c ndash bolting device system
11 ndash friction body
12 ndash lateral surface
13 ndash bolting tape end
14 ndash bolting tape end
15 ndash control member
16 ndash torsion bar
17 ndash base
18 ndash pylon
19 ndash hub
20 ndash annular gap
21 ndash Gleitlagerbuchse
23 ndash [nil]
23 ndash friction disk
24 ndash turning camps 25 ndash teeth
26 ndash elbow levers
27 ndash clamping wedge
28 ndash internal contour
29 ndash longitudinal guidance
30 ndash system
31 ndash sensor
32 ndash clamping gap
(European Patent Office May 2007) [40]
Appendix A 131
Figure A9 Proposed design by INA Schaeffler KG corresponding to patent no DE10359641-A1
Source European Patent Office espcenet (publication no DE10359641-A1 accessed Jun 2007)
epespacenetcom [42]
Figure A9 label identification 8 ndash cylinder
10 ndash rod
12 ndash spring plate
13 ndash spring
14 ndash pressure lead
(European Patent Office June 2007) [42]
Appendix A 132
Figure A10 Proposed design by INA Schaeffler KG corresponding to patent no EP1723350-A1
Source European Patent Office espcenet (publication no EP1723350-A1 accessed Jun 2007) epespacenetcom [41]
Figure A10 label identification 4 ndash pulley
5 ndash hydraulic element 11 ndash freewheel
12 ndash shaft
(European Patent Office June 2007) [41]
Figure A11 Proposed design by INA Schaeffler KG corresponding to patent no EP1738093-A1
Source European Patent Office espcenet (publication no EP1738093-A1 accessed Jun 2007)
epespacenetcom [43]
Figure A11 label identification 1 ndash traction drive
2 ndash traction belt
3 ndash starter generator
Appendix A 133
7 ndash tension device
9 ndash lever arm
10 ndash guide roller
16 ndash end stop
(European Patent Office June 2007) [43]
Figure A12 Proposed design by INA Schaeffler KG corresponding to patent no DE102004012395-A1
Source European Patent Office espcenet (publication no DE102004012395-A1 accessed May 2007) epespacenetcom [44]
Figure A12 label identification 1 ndash belt drive
2 ndash belts
3 ndash wheel of the internal-combustion engine
4 ndash wheel of a Nebenaggregats
5 ndash wheel of the starter generator
6 ndash clamping unit
7 ndash idler
8 ndash idler
9 ndash scale beams
10 ndash drive place
11 ndash clamping arm
12 ndash camps
13 ndash coupling point
14 ndash shock absorber element
15 ndash arrow
(European Patent Office May 2007) [44]
Appendix A 134
Figure A13 Proposed design by INA Schaeffler KG corresponding to patent nos DE102005017038-A1and WO2006108461-A1
Source European Patent Office espcenet (publication nos DE102005017038-A1and WO2006108461-A1 accessed May 2007) epespacenetcom [45]
Figure A13 label identification 1 ndash belt
2 ndash wheel of the crankshaft KW
3 ndash wheel of a climatic compressor AC
4 ndash wheel of a starter generator SG
5 ndash wheel of a water pump WP
6 ndash first clamping system
7 ndash first tension adjuster lever arm
8 ndash first tension adjuster role
9 ndash second clamping system
10 ndash second tension adjuster lever arm
11 ndash second tension adjuster role 12 ndash guide roller
13 ndash drive-conditioned Zugtrum
(generatorischer enterprise (GE))
13 ndash starter-conditioned Leertrum
(starter enterprise (SE))
14 ndash drive-conditioned Leertrum (GE)
14 ndash starter-conditioned Zugtrum (SE)
14a ndash drive-conditioned Leertrumast (GE)
14a ndash starter-conditioned Zugtrumast (SE)
14b ndash drive-conditioned Leertrumast (GE)
14b ndash starter-conditioned Zugtrumast (SE)
(European Patent Office May 2007) [45]
Figure A14 Proposed design by Litens Automotive GMBH et al corresponding to patent no
US20010007839-A1
Appendix A 135
Source European Patent Office espcenet (publication no US20010007839-A1 accessed Jun 2007)
epespacenetcom [46]
Figure A14 label identification E - belt
K - crankshaft
R1 ndash first tension pulley
R2 ndash second tension pulley
S ndash tension device
T ndash drive system
1 ndash belt pulley
4 ndash belt pulley
(European Patent Office June 2007) [46]
Figure A15 Proposed design by Mitsubishi Jidosha Eng KK and Mitsubishi Motor Corp corresponding
to patent no JP2005083514-A
Source Industrial Property Digital Library and Japanese Patent Office Patent amp Utility Model Gazette DB (document no A 2005-083514 accessed May 2007) wwwipdlinpitgojp [47]
Figure A15 label identification 1 ndash Pulley for Starting
2 ndash Crank Pulley
3 ndash AC Pulley
4a ndash 1st roller
4b ndash 2nd roller
5 ndash Idler Pulley
6 ndash Belt
7c ndash Starter generator control section
7d ndash Idle stop control means
8 ndash WP Pulley
9 ndash IG Switch
10 ndash Engine
11 ndash Starter Generator
12 ndash Driving Shaft
Appendix A 136
7 ndash ECU
7a ndash 1st auto tensioner control section (the 1st auto
tensioner control means)
7b ndash 2nd auto tensioner control section (the 2nd auto
tensioner control means)
13 ndash Air-conditioner Compressor
14a ndash 1st auto tensioner
14b ndash 2nd auto tensioner
18 ndash Water Pump
(Industrial Property Digital Library May 2007) [47]
Figure A16 Proposed design by Nissan Motor Co Ltd corresponding to patent no JP3565040-B2
Source European Patent Office espcenet (publication no JP3565040-B2 accessed Jun 2007) epespacenetcom [48]
Figure A16 label identification 3 ndash chain [or belt]
5 ndash tensioner
4 ndash belt pulley
(European Patent Office June 2007) [48]
Figure A17 Proposed design by NTN Corp corresponding to patent no JP2006189073-A
Appendix A 137
Source European Patent Office espcenet (publication no JP2006189073-A accessed Jun 2007)
epespacenetcom [49]
Figure A17 label identification 5d - flange
6 ndash tensile strength spring
10 ndash actuator
10c ndash cylinder
12 ndash rod
20 ndash hydraulic damper mechanism
21 ndash cylinder nut
22 ndash screw bolt
(European Patent Office June 2007) [49]
Figure A18 Proposed design by Valeo Equipment Electriques Moteur corresponding to patent nos
EP1658432 and WO2005015007
Source European Patent Office espcenet (publication nos EP1658432 and WO2005015007
accessed May 2007) epespacenetcom [50]
Figure A18 abbreviated list of label identifications
10 ndash starter
22 ndash shaft section
23 ndash free front end
80 ndash pulley
200 ndash support element
206 - arm
(European Patent Office May 2007) [50]
The author notes that the list of labels corresponding to Figures A1a through to A7 are generated
from machine translations translated from the documentrsquos original language (ie German)
Consequently words may be translated inaccurately or not at all
138
APPENDIX B
B-ISG Serpentine Belt Drive with Single Tensioner
Equation of Motion
The equations of motion (EOM) for a B-ISG serpentine belt drive with a single tensioner are
shown The EOM has been derived similarly to that of the same system with a twin tensioner
that was provided in Chapter 3 The assumptions for the twin tensioner B-ISG system are
applicable for the single tensioner B-ISG system as well
Figure B1 shows the B-ISG system with a single tensioner pulley and arm The pulleys are
numbered 1 through 4 and their associated belt spans are numbered accordingly
Figure B1 Single Tensioner B-ISG System
Appendix B 139
The free-body diagram for the ith non-tensioner pulley in the system shown above is found in
Figure B2 The moment of inertia for the ith pulley is designated as Ii while the angular
displacement velocity and acceleration is designated as 120579119905119894 120579 119905119894 and 120579 119905119894 respectively The
required torque is Qi the angular damping is Ci and the tension of the ith span is Ti
Figure B2 Free-body Diagram of ith Pulley
The positive motion designated is assumed to be in the clockwise direction The radius for the
ith pulley is represented by Ri The equilibrium equations for the ith pulley are as follows
I1 ∙ θ 1 = T4 ∙ R1 minus T1 ∙ R1 + Q1 minus c1 ∙ θ 1 (B1)
I2 ∙ θ 2 = T1 ∙ R2 minus T2 ∙ R2 + Q2 minus c2 ∙ θ 2 (B2)
I3 ∙ θ 3 = T2 ∙ R3 minus T3 ∙ R3 + Q3 minus c3 ∙ θ 3 (B3)
Appendix B 140
A free-body diagram for the single tensioner pulley is shown in Figure B3 The rotational
stiffness and damping for the tensioner arm is designated as kt and ct respectively The angular
rotation and velocity for the arm is 120579119905119894 and 120579 119905119894 respectively
Figure B3 Free-body Diagram of Single Tensioner
From figure B2 the equations of equilibrium are resolved for the tensioner pulley The angle of
orientation for the ith belt span is designated by 120573119895119894
minusI4 ∙ θ 4 = minusT3 ∙ R4 + T4 ∙ R4 + Q4 + c4 ∙ θ 4 (B4)
It ∙ θ t = minusTt ∙ Lt ∙ sin θto minus βj4 + sin θto 1 minus βj5 minus kt ∙ partθt minus ct ∙ partθ t
(B5)
Appendix B 141
partθt = θt minus θto (B6)
The dynamic tension matrix Trsquo is proportional to the damping (Tc) and stiffness (Tk) matrices
that are due to belt damping (119888119894119887 ) and belt stiffness (119896119894
119887 ) respectively
119827prime = 119827119836 ∙ 120521 + 119827119844 ∙ 120521 (B7)
The initial tension is represented by To and the initial angle of the tensioner arm is represented
by 120579119905119900 The equation for the tension of the ith span is shown in the following equations
T1 = To + k1b ∙ R1 ∙ θ1 minus R2 ∙ θ2 + c1
b ∙ [R1 ∙ θ 1 minus R2 ∙ θ 2] (B8)
T2 = To + k2b ∙ R2 ∙ θ2 minus R3 ∙ θ3 + c2
b ∙ [R2 ∙ θ 2 minus R3 ∙ θ 3)] (B9)
T3 = To + k3b ∙ R3 ∙ θ3 minus R4 ∙ θ4 + Lt ∙ [sin θto minus βj3 ] ∙ (θt minus θto ) + c3
b ∙ [R3 ∙ θ 3 minus R4 ∙
θ 4 + Lt ∙ [sin θto minus βj3 ] ∙ (θ t)] (B10)
T4 = To + k4b ∙ R4 ∙ θ4 minus R1 ∙ θ1 + Lt ∙ [sin θto minus βj4 ] ∙ (θt minus θto ) + c4
b ∙ [R4 ∙ θ 4 minus R1 ∙
θ 1 + Lt ∙ [sin θto minus βj4 ] ∙ (θ t)] (B11)
Tprime = Ti minus To (B12)
Tt = T3 = T4 (B13)
Appendix B 142
The EOM for the single tensioner B-ISG system is found by substitution of equations B8 to
B13 into B1 to B5 The matrices in the EOM include the inertial matrix I damping matrix C
stiffness matrix K and the required torque matrix Q as well as the angular displacement
velocity and acceleration matrices 120521 120521 and 120521 respectively
119816 ∙ 120521 + 119810 ∙ 120521 + 119818 ∙ 120521 = 119824 (B14)
119816 =
I1 0 0 0 00 I2 0 0 00 0 I3 0 00 0 0 I4 00 0 0 0 It1
(B15)
The stiffness matrix includes kb the belt factor Kb the belt cord stiffness 120601119894 the wrap angle of
the belt on the ith pulley and Kbi the stiffness factor of the ith belt span Cb represents the belt
damping for each span and βji is the angle of orientation for the span between the jth and ith
pulleys It is noted in the terms of the stiffness and damping matrices below that the numerical
subscripts refer to the (i+1)th pulley The term Qt may be found within the required torque
matrix and represents the required torque for the tensioner arm As well the term It1 represents
the moment of inertia for the tensioner arm
Appendix B 143
K =
(B16)
Kbi =Kb
Li + kb ∙ Ri ∙ϕi+1
2 + Ri ∙ϕi
2
(B17)
C =
(B18)
Appendix B 144
Appendix B 144
120521 =
θ1
θ2
θ3
θ4
partθt
(B19)
119824 =
Q1
Q2
Q3
Q4
Qt
(B20)
Simulations of the EOM for the B-ISG system with a single tensioner were performed in FEAD
[51] software for dynamic and static cases This allowed for the methodology for deriving the
EOM to be verified and then applied to the B-ISG system with a twin tensioner The natural
frequencies modes shapes dynamic responses tensioner arm torques as well as the crankshaft
required torque only and the dynamic tensions were solved from the EOM as described in (331)
to (339) of Chapter 3 and as well as for the static tension from (351) to (353) of Chapter 3
This permitted verification of the complete derivation methodology and allowed for comparison
of the static tension of the B-ISG system belt spans in the case that a single tensioner is present
and in the case that a Twin Tensioner is present [51]
145
APPENDIX C
MathCAD Scripts
C1 Geometric Analysis
1 - CS
2 - AC
4 - Alt
3 - Ten1
5 - Ten 2
6 - Ten Pivot
1
2
3
4
5
Figure C1 Schematic of B-ISG
System with Twin Tensioner
Coordinate Input DataXY1 0 0( ) XY4 24759 16664( )
XY2 224 6395( ) XY5 12057 9193( )
XY3 292761 87( ) XY6 201384 62516( )
Computations
Lt1 XY30 0
XY60 0
2
XY30 1
XY60 1
2
Lt2 XY50 0
XY60 0
2
XY50 1
XY60 1
2
t1 atan2 XY30 0
XY60 0
XY30 1
XY60 1
t2 atan2 XY50 0
XY60 0
XY50 1
XY60 1
XY
XY10 0
XY20 0
XY30 0
XY40 0
XY50 0
XY60 0
XY10 1
XY20 1
XY30 1
XY40 1
XY50 1
XY60 1
x XY
0 y XY
1
Appendix C 146
i - angle bw horizontal and l ine from ith pulley center to (i+1)th pulley center
Adjust last number in range variable p to correspond to number of pulleys
p 0 1 4
k p( ) p 1( ) p 4if
0 otherwise
condition1 p( ) acos
XYk p( ) 0
XYp 0
XYk p( ) 0
XYp 0
2
XYk p( ) 1
XYp 1
2
condition2 p( ) 2 acos
XYk p( ) 0
XYp 0
XYk p( ) 0
XYp 0
2
XYk p( ) 1
XYp 1
2
p( ) if XYk p( ) 1
XYp 1
condition1 p( ) condition2 p( )
Lfi Lbi - connection belt span lengths
D1 20065mm D2 10349mm D3 7240mm D4 6820mm D5 7240mm
D
D1
D2
D3
D4
D5
Lf p( ) XYk p( ) 0
XYp 0
2
XYk p( ) 1
XYp 1
2
1
mm
Dk p( )
2
Dp
2
2
Lb p( ) XYk p( ) 0
XYp 0
2
XYk p( ) 1
XYp 1
2
1
mm
Dk p( )
2
Dp
2
2
fi bi - angle bw ith pulley center connection l ine and contact points Pbfi (or Pfbi) and Pbi
(or Pfi) respecti vely l
f p( ) atanLf p( ) mm
Dp
2
Dk p( )
2
Dp
Dk p( )
if
atanLf p( ) mm
Dk p( )
2
Dp
2
Dp
Dk p( )
if
2D
pD
k p( )if
b p( ) atan
mmLb p( )
Dp
2
Dk p( )
2
Appendix C 147
XYfi XYbi XYfbi XYbfi - 4 possible contact points for i th pulley
XYf p( ) XYp 0
Dp
2 mmcos p( ) f p( )
XYp 1
Dp
2 mmsin p( ) f p( )
XYb p( ) XYp 0
Dp
2 mmcos p( ) f p( )
XYp 1
Dp
2 mmsin p( ) f p( )
XYfb p( ) XYp 0
Dp
2 mmcos p( ) b p( )
XYp 1
Dp
2 mmsin p( ) b p( )
XYbf p( ) XYp 0
Dp
2 mmcos p( ) b p( )
XYp 1
Dp
2 mmsin p( ) b p( )
XYfi+1 XYbi+1 XYfbi+1 XYbfi+1 - 4 possible contact points for i+1th pulley
XYf2 p( ) XYk p( ) 0
Dk p( )
2 mmcos p( ) f p( )
XYk p( ) 1
Dk p( )
2 mmsin p( ) f p( )
XYb2 p( ) XYk p( ) 0
Dk p( )
2 mmcos p( ) f p( )
XYk p( ) 1
Dk p( )
2 mmsin p( ) f p( )
XYfb2 p( ) XYk p( ) 0
Dk p( )
2 mmcos p( ) b p( ) XY
k p( ) 1
Dk p( )
2 mmsin p( ) b p( )
XYbf2 p( ) XYk p( ) 0
Dk p( )
2 mmcos p( ) b p( ) XY
k p( ) 1
Dk p( )
2 mmsin p( ) b p( )
Row 1 --gt Pulley 1 Row i --gt Pulley i
XYfi
XYf 0( )0 0
XYf 1( )0 0
XYf 2( )0 0
XYf 3( )0 0
XYf 4( )0 0
XYf 0( )0 1
XYf 1( )0 1
XYf 2( )0 1
XYf 3( )0 1
XYf 4( )0 1
XYfi
6818
269222
335325
251552
108978
100093
89099
60875
200509
207158
x1 XYfi0
y1 XYfi1
Appendix C 148
XYbi
XYb 0( )0 0
XYb 1( )0 0
XYb 2( )0 0
XYb 3( )0 0
XYb 4( )0 0
XYb 0( )0 1
XYb 1( )0 1
XYb 2( )0 1
XYb 3( )0 1
XYb 4( )0 1
XYbi
47054
18575
269403
244841
164847
88606
291
30965
132651
166182
x2 XYbi0
y2 XYbi1
XYfbi
XYfb 0( )0 0
XYfb 1( )0 0
XYfb 2( )0 0
XYfb 3( )0 0
XYfb 4( )0 0
XYfb 0( )0 1
XYfb 1( )0 1
XYfb 2( )0 1
XYfb 3( )0 1
XYfb 4( )0 1
XYfbi
42113
275543
322697
229969
9452
91058
59383
75509
195834
177002
x3 XYfbi0
y3 XYfbi1
XYbfi
XYbf 0( )0 0
XYbf 1( )0 0
XYbf 2( )0 0
XYbf 3( )0 0
XYbf 4( )0 0
XYbf 0( )0 1
XYbf 1( )0 1
XYbf 2( )0 1
XYbf 3( )0 1
XYbf 4( )0 1
XYbfi
8384
211903
266707
224592
140427
551
13639
50105
141463
143331
x4 XYbfi0
y4 XYbfi1
Row 1 --gt Pulley 2 Row i --gt Pulley i+1
XYf2i
XYf2 0( )0 0
XYf2 1( )0 0
XYf2 2( )0 0
XYf2 3( )0 0
XYf2 4( )0 0
XYf2 0( )0 1
XYf2 1( )0 1
XYf2 2( )0 1
XYf2 3( )0 1
XYf2 4( )0 1
XYf2x XYf2i0
XYf2y XYf2i1
XYb2i
XYb2 0( )0 0
XYb2 1( )0 0
XYb2 2( )0 0
XYb2 3( )0 0
XYb2 4( )0 0
XYb2 0( )0 1
XYb2 1( )0 1
XYb2 2( )0 1
XYb2 3( )0 1
XYb2 4( )0 1
XYb2x XYb2i0
XYb2y XYb2i1
Appendix C 149
XYfb2i
XYfb2 0( )0 0
XYfb2 1( )0 0
XYfb2 2( )0 0
XYfb2 3( )0 0
XYfb2 4( )0 0
XYfb2 0( )0 1
XYfb2 1( )0 1
XYfb2 2( )0 1
XYfb2 3( )0 1
XYfb2 4( )0 1
XYfb2x XYfb2i
0
XYfb2y XYfb2i1
XYbf2i
XYbf2 0( )0 0
XYbf2 1( )0 0
XYbf2 2( )0 0
XYbf2 3( )0 0
XYbf2 4( )0 0
XYbf2 0( )0 1
XYbf2 1( )0 1
XYbf2 2( )0 1
XYbf2 3( )0 1
XYbf2 4( )0 1
XYbf2x XYbf2i0
XYbf2y XYbf2i1
100 40 20 80 140 200 260 320 380 440 500150
110
70
30
10
50
90
130
170
210
250Figure C2 Possible Contact Points
250
150
y1
y2
y3
y4
y
XYf2y
XYb2y
XYfb2y
XYbf2y
500100 x1 x2 x3 x4 x XYf2x XYb2x XYfb2x XYbf2x
Appendix C 150
Xij Yij - selected contact point on ith pulley for span from ith pulley to jth pulley
XY15 XYbf2iT 4
XY12 XYfiT 0
Pulley 1 contact pts
XY21 XYf2iT 0
XY23 XYfbiT 1
Pulley 2 contact pts
XY32 XYfb2iT 1
XY34 XYbfiT 2
Pulley 3 contact pts
XY43 XYbf2iT 2
XY45 XYfbiT 3
Pulley 4 contact pts
XY54 XYfb2iT 3
XY51 XYbfiT 4
Pulley 5 contact pts
By observation the lengths of span i is the following
L1 Lf 0( ) L2 Lb 1( ) L3 Lb 2( ) L4 Lb 3( ) L5 Lb 4( ) Li
L1
L2
L3
L4
L5
mm
i Angle between horizontal and span of ith pulley
i
atan
XY121
XY211
XY12
0XY21
0
atan
XY231
XY321
XY23
0XY32
0
atan
XY341
XY431
XY34
0XY43
0
atan
XY451
XY541
XY45
0XY54
0
atan
XY511
XY151
XY51
0XY15
0
Appendix C 151
Pulley 1 Pulley 2 Pulley 3 Pulley 4 Pulley 5
12 i0 2 21 i0 32 i1 2 43 i2 54 i3
15 i4 2 23 i1 34 i2 45 i3 51 i4
15
21
32
43
54
12
23
34
45
51
Wrap angle i for the ith pulley
1 2 atan2 XY150
XY151
atan2 XY120
XY121
2 atan2 XY210
XY1 0
XY211
XY1 1
atan2 XY230
XY1 0
XY231
XY1 1
3 2 atan2 XY320
XY2 0
XY321
XY2 1
atan2 XY340
XY2 0
XY341
XY2 1
4 atan2 XY430
XY3 0
XY431
XY3 1
atan2 XY450
XY3 0
XY451
XY3 1
5 atan2 XY540
XY4 0
XY541
XY4 1
atan2 XY510
XY4 0
XY511
XY4 1
1
2
3
4
5
Lb length of belt
Lbelt Li1
2
0
4
p
Dpp
Input Data for B-ISG System
Kt 20626Nm
rad (spring stiffness between tensioner arms 1
and 2)
Kt1 10314Nm
rad (stiffness for spring attached at arm 1 only)
Kt2 16502Nm
rad (stiffness for spring attached at arm 2 only)
Appendix C 152
C2 Dynamic Analysis
I C K moment of inertia damping and stiffness matrices respectively
u 0 1 4 v 0 1 4 (new counter variables where final value = no of pulleys + no of ten arms)
RaD
2
Appendix C 153
RaD
2
Ii =gt moment of inertia for ith pulley where i-1 and i represent ten arms
Ii0
0
1
2
3
4
5
6
10000
2230
300
3000
300
1500
1500
I diag Ii( ) kg mm2
Ci =gt Rotational damping and belt damping for the ith pulley where i-1 and i represent tensioner arms
1000kg
m3
CrossArea 693mm2
0 M CrossArea Lbelt M 0086kg
cb 2 KbM
Lbelt
Cb
cb
cb
cb
cb
cb
Cri
0
0
010
0
010
N mmsec
rad
Ct 1000N mmsec
rad Ct1 1000 N mm
sec
rad Ct2 1000N mm
sec
rad
Cr
Cri0
0
0
0
0
0
0
0
Cri1
0
0
0
0
0
0
0
Cri2
0
0
0
0
0
0
0
Cri3
0
0
0
0
0
0
0
Cri4
0
0
0
0
0
0
0
Ct Ct1
Ct
0
0
0
0
0
Ct
Ct Ct2
Rt
Ra0
Ra1
0
0
0
0
0
0
Ra1
Ra2
0
0
Lt1 mm sin t1 32
0
0
0
Ra2
Ra3
0
Lt1 mm sin t1 34
0
0
0
0
Ra3
Ra4
0
Lt2 mm sin t2 54
Ra0
0
0
0
Ra4
0
Lt2 mm sin t2 51
Appendix C 154
Kr
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Kt Kt1
Kt
0
0
0
0
0
Kt
Kt Kt2
Tk
Kbi 0( ) Ra0
0
0
0
Kbi 4( ) Ra0
Kbi 0( ) Ra1
Kbi 1( ) Ra1
0
0
0
0
Kbi 1( ) Ra2
Kbi 2( ) Ra2
0
0
0
0
Kbi 2( ) Ra3
Kbi 3( ) Ra3
0
0
0
0
Kbi 3( ) Ra4
Kbi 4( ) Ra4
0
Kbi 1( ) Lt1 mm sin t1 32
Kbi 2( ) Lt1 mm sin t1 34
0
0
0
0
0
Kbi 3( ) Lt2 mm sin t2 54
Kbi 4( ) Lt2 mm sin t2 51
Tc
Cb0
Ra0
0
0
0
Cb4
Ra0
Cb0
Ra1
Cb1
Ra1
0
0
0
0
Cb1
Ra2
Cb2
Ra2
0
0
0
0
Cb2
Ra3
Cb3
Ra3
0
0
0
0
Cb3
Ra4
Cb4
Ra4
0
Cb1
Lt1 mm sin t1 32
Cb2
Lt1 mm sin t1 34
0
0
0
0
0
Cb3
Lt2 mm sin t2 54
Cb4
Lt2 mm sin t2 51
C matrix
C Cr Rt Tc
K matrix
K Kr Rt Tk
New Equations of Motion for Dual Drive System
I K amp C matricies rearranged to place driving pulley in 1st row + 1st column and driven in 2nd row + 2nd column
IA augment I3
I0
I1
I2
I4
I5
I6
IC augment I0
I3
I1
I2
I4
I5
I6
I1kgmm2 1 106
kg m2
0 0 0 0 0 0
Ia stack I1kgmm2 IAT 0
T
IAT 1
T
IAT 2
T
IAT 4
T
IAT 5
T
IAT 6
T
Ic stack I1kgmm2 ICT 3
T
ICT 1
T
ICT 2
T
ICT 4
T
ICT 5
T
ICT 6
T
Appendix C 155
RtA augment Rt3
Rt0
Rt1
Rt2
Rt4
RtC augment Rt0
Rt3
Rt1
Rt2
Rt4
Rta stack RtAT 3
T
RtAT 0
T
RtAT 1
T
RtAT 2
T
RtAT 4
T
RtAT 5
T
RtAT 6
T
Rtc stack RtCT 0
T
RtCT 3
T
RtCT 1
T
RtCT 2
T
RtCT 4
T
RtCT 5
T
RtCT 6
T
TkA augment Tk3
Tk0
Tk1
Tk2
Tk4
Tk5
Tk6
Tka stack TkAT 3
T
TkAT 0
T
TkAT 1
T
TkAT 2
T
TkAT 4
T
TkC augment Tk0
Tk3
Tk1
Tk2
Tk4
Tk5
Tk6
Tkc stack TkCT 0
T
TkCT 3
T
TkCT 1
T
TkCT 2
T
TkCT 4
T
TcA augment Tc3
Tc0
Tc1
Tc2
Tc4
Tc5
Tc6
Tca stack TcAT 3
T
TcAT 0
T
TcAT 1
T
TcAT 2
T
TcAT 4
T
TcC augment Tc0
Tc3
Tc1
Tc2
Tc4
Tc5
Tc6
Tcc stack TcAT 0
T
TcAT 3
T
TcAT 1
T
TcAT 2
T
TcAT 4
T
Ka Kr Rta Tka Kc Kr Rtc Tkc Ca Cr Rta Tca Cc Cr Rtc Tcc
CHECK
KA augment K3
K0
K1
K2
K4
K5
K6
KC augment K0
K3
K1
K2
K4
K5
K6
CA augment C3
C0
C1
C2
C4
C5
C6
CC augment C0
C3
C1
C2
C4
C5
C6
Appendix C 156
Kacheck stack KAT 3
T
KAT 0
T
KAT 1
T
KAT 2
T
KAT 4
T
KAT 5
T
KAT 6
T
Kccheck stack KCT 0
T
KCT 3
T
KCT 1
T
KCT 2
T
KCT 4
T
KCT 5
T
KCT 6
T
Cacheck stack CAT 3
T
CAT 0
T
CAT 1
T
CAT 2
T
CAT 4
T
CAT 5
T
CAT 6
T
Cccheck stack CCT 0
T
CCT 3
T
CCT 1
T
CCT 2
T
CCT 4
T
CCT 5
T
CCT 6
T
Results for System switching from ISG as DRIVING pulley to Crankshaft as Drivi ng Pulley
Modified Submatricies for ISG Driving Phase --gt CS Driving Phase
Unit step function to provide shift from crankshaft DRIVING case (ie ISG driven case) to crankshaft DRIVEN
case (ie ISG driving case)
H n( ) 1 n 750if
0 n 750if
lt-- crankshaft DRIVING case (Phase change bw 2 cases occurs when n
reaches start speed)
I11mod n( ) Ic0 0
H n( ) 1if
Ia0 0
H n( ) 0if
I22mod n( )submatrix Ic 1 6 1 6( )
UnitsOf I( )H n( ) 1if
submatrix Ia 1 6 1 6( )
UnitsOf I( )H n( ) 0if
K11mod n( )
Kc0 0
UnitsOf K( )H n( ) 1if
Ka0 0
UnitsOf K( )H n( ) 0if
C11modn( )
Cc0 0
UnitsOf C( )H n( ) 1if
Ca0 0
UnitsOf C( )H n( ) 0if
K22mod n( )submatrix Kc 1 6 1 6( )
UnitsOf K( )H n( ) 1if
submatrix Ka 1 6 1 6( )
UnitsOf K( )H n( ) 0if
C22modn( )submatrix Cc 1 6 1 6( )
UnitsOf C( )H n( ) 1if
submatrix Ca 1 6 1 6( )
UnitsOf C( )H n( ) 0if
K21mod n( )submatrix Kc 1 6 0 0( )
UnitsOf K( )H n( ) 1if
submatrix Ka 1 6 0 0( )
UnitsOf K( )H n( ) 0if
C21modn( )submatrix Cc 1 6 0 0( )
UnitsOf C( )H n( ) 1if
submatrix Ca 1 6 0 0( )
UnitsOf C( )H n( ) 0if
K12mod n( )submatrix Kc 0 0 1 6( )
UnitsOf K( )H n( ) 1if
submatrix Ka 0 0 1 6( )
UnitsOf K( )H n( ) 0if
C12modn( )submatrix Cc 0 0 1 6( )
UnitsOf C( )H n( ) 1if
submatrix Ca 0 0 1 6( )
UnitsOf C( )H n( ) 0if
Appendix C 157
2mod n( ) I22mod n( )1
K22mod n( ) mod n( ) sort eigenvals 2mod n( ) nmod n( )mod n( )
2
EVmodn( ) augmenteigenvec 2mod n( ) mod n( )0
max eigenvec 2mod n( ) mod n( )0
eigenvec 2mod n( ) mod n( )1
max eigenvec 2mod n( ) mod n( )1
eigenvec 2mod n( ) mod n( )2
max eigenvec 2mod n( ) mod n( )2
eigenvec 2mod n( ) mod n( )3
max eigenvec 2mod n( ) mod n( )3
eigenvec 2mod n( ) mod n( )4
max eigenvec 2mod n( ) mod n( )4
eigenvec 2mod n( ) mod n( )5
max eigenvec 2mod n( ) mod n( )5
modeshapesmod n( ) stack nmod n( )T
EVmodn( )
t 0 0001 1
mode1a t( ) EVmod100( )0
sin nmod 100( )0 t mode2a t( ) EVmod100( )1
sin nmod 100( )1 t
mode1c t( ) EVmod800( )0
sin nmod 800( )0 t mode2c t( ) EVmod800( )1
sin nmod 800( )1 t
Pulley responses amp torque requirement for crankshaft amp alternator pulleys pulley1 and 4 respectively
The system equation becomes
I14q14 -double-dot + C1144 q14 -dot + K1144 q14 + C12qm-dot + K12qm = Qc
I2qm-double-dot + C22qm-dot + K22qm + C21q1-dot + K21q1 = 0
Pulley responses
Qm = - [(K22 - 2I2) + jC22 ]-1(K21 + jC21 )Q1
Torque requirement for crank shaft Pulley 1
qc = [(K11 -2I1) + jC11 ]Q1 + (K12 + jC12 )Qm
Torque requirement for alternator shaft Pulley 4
qa = [(K44 -2I4) + jC44 ]Q4 + (K12 + jC12 )Qm
Appendix C 158
Let DRIVING pulley have a unit amplitude 1 = 1 and let the system frequency be calculated based on
engine speed n
n 60 90 6000 n( )4n
60 a n( )
2n Ra0
60 Ra3
mod n( ) n( ) H n( ) 1if
a n( ) H n( ) 0if
Ymod n( ) K22mod n( ) mod n( ) 2 I22mod n( )
j mod n( ) C22modn( )
mmod n( ) Ymod n( )( )1
K21mod n( ) j mod n( ) C21modn( )
Crankshaft amp ISG required torques
Let input from DRIVING pulley be an angular displacement with constant amplitude of angular acceleration
Ac n( ) 650 1 n( )Ac n( )
n( ) 2
Let Qm = QmQ1(n) for n lt 750
and Qm = QmQ4(n) for n gt 750
Aa n( )42
I3 3
1a n( )Aa n( )
a n( ) 2
Qc0 4
qcmod n( ) K11mod n( ) mod n( ) 2
I11mod n( )
j mod n( ) C11modn( )
1 n( ) K12mod n( ) j mod n( ) C12modn( ) mmod n( ) 1 n( )
H n( ) 1if
Qc0 H n( ) 0if
qamod n( ) K11mod n( ) mod n( ) 2
I11mod n( )
j mod n( ) C11modn( )
1a n( ) K12mod n( ) jmod n( ) C12modn( ) mmod n( ) 1a n( ) Qc0
H n( ) 0if
0 H n( ) 1if
Q n( ) 48 n
Ra0
Ra3
48
18000
(ISG torque requirement alternate equation)
Appendix C 159
Dynamic tensioner arm torques
Qtt1mod n( )Kt Kt1
UnitsOf Kt( )j mod n( )
Ct Ct1
UnitsOf Cr( )
mmod n( )4 1 n( )
H n( ) 1if
Kt Kt1
UnitsOf Kt( )j mod n( )
Ct Ct1
UnitsOf Cr( )
mmod n( )4 1a n( )
H n( ) 0if
Qtt2mod n( )Kt Kt2
UnitsOf Kt( )j mod n( )
Ct Ct2
UnitsOf Cr( )
mmod n( )5 1 n( )
H n( ) 1if
Kt Kt2
UnitsOf Kt( )j mod n( )
Ct Ct2
UnitsOf Cr( )
mmod n( )5 1a n( )
H n( ) 0if
Appendix C 160
Dynamic belt span tensions
d n( ) 1 n( ) H n( ) 1if
1a n( ) H n( ) 0if
mod n( )
d n( )
mmod n( ) d n( ) 0 0
mmod n( ) d n( ) 1 0
mmod n( ) d n( ) 2 0
mmod n( ) d n( ) 3 0
mmod n( ) d n( ) 4 0
mmod n( ) d n( ) 5 0
Tm n( ) j n( )Tcc
UnitsOf Tcc( )
Tkc
UnitsOf Tkc( )
mod n( )
H n( ) 1if
j n( )Tca
UnitsOf Tca( )
Tka
UnitsOf Tka( )
mod n( )
H n( ) 0if
Tm n( ) j n( )Tcc
UnitsOf Tcc( )
Tkc
UnitsOf Tkc( )
mod n( )
H n( ) 1if
j n( )Tca
UnitsOf Tca( )
Tka
UnitsOf Tka( )
mod n( )
H n( ) 0if
(tensions for driving pulley belt spans)
Appendix C 161
C3 Static Analysis
Static Analysis using K Tk amp Q matricies amp Ts
For static case K = Q
Tension T = T0 + Tks
Thus T = K-1QTks + T0
Q1 68N m Qt1 0N m Qt2 0N m Ts 300N
Qc
Q4
Q2
Q3
Q5
Qt1
Qt2
Qc
5
2
0
0
0
0
J Qa
Q1
Q2
Q3
Q5
Qt1
Qt2
Qa
68
2
0
0
0
0
N m
cK22mod 900( )( )
1
N mQc A
K22mod 600( )1
N mQa
a
A0
A1
A2
0
A3
A4
A5
0
c1
c2
c0
c3
c4
c5
Tc Tk Ts Ta Tk a Ts
162
APPENDIX D
MATLAB Functions amp Scripts
D1 Parametric Analysis
D11 TwinMainm
The following function script performs the parametric analysis for the B-ISG system with a Twin
Tensioner It calls the function TwinTenStaticTensionm The parametric analysis perturbs a
single input parameter for the called function TwinTenStaticTensionm The main function takes
an initial input value for the Twin Tensioner‟s stiffness parameters Kto Kt1o Kt2o and
geometric parameters D3o D5o X3o Y3o X5o and Y5o An input parameter is allowed to
increment by six percent over a range from sixty percent below its initial value to sixty percent
above its initial value The coordinate parameters are incremented through a mesh of Cartesian
points with prescribed boundaries The TwinMainm function plots the parametric results
______________________________________________________________________________
clc
clear all
Static tension for single tensioner system for CS and Alt driving
Initial Conditions
Kto = 20626
Kt1o = 10314
Kt2o = 16502
D3o = 007240
D5o = 007240
X3o =0292761
Y3o =087
X5o =12057
Y5o =09193
Pertubations of initial parameters
Kt = (Kto-060Kto)006Kto(Kto+060Kto)
Kt1 = (Kt1o-060Kt1o)006Kt1o(Kt1o+060Kt1o)
Kt2 = (Kt2o-060Kt2o)006Kt2o(Kt2o+060Kt2o)
D3 = (D3o-060D3o)006D3o(D3o+060D3o)
D5 = (D5o-060D5o)006D5o(D5o+060D5o)
No of data points
s = 21
T = zeros(5s)
Ta = zeros(5s)
Parametric Plots
for i = 1s
Appendix D 163
[T(i)Ta(i)] = TwinTenStaticTension(Kt(i)Kt1oKt2oD3oD5oX3oY3oX5oY5o)
end
figure
[AXH1H2] = plotyy(Kt()T(1)Kt()Ta(4)plot) hold on
H3 = line(Kt()T(5)ParentAX(1)) hold on
H4 = line(Kt()Ta(3)Parent AX(2)) hold off
set(AXYLimModeautoYTickModeauto)
title(Static Tension vs Coupled Stiffness bw Arms 1 amp 2)
xlabel(Kt (Nmrad))
set(get(AX(1)Ylabel)StringCS Span Tension (N))
set(get(AX(2)Ylabel)StringISG Span Tension (N))
set(H1LineStyle-Colorb)
set(H2LineStyle-Colorg)
set(H3LineStyleColorget(H1Color))
set(H4LineStyleColorget(H2Color))
legend([H1H2H3H4]CS Tautest SpanISG Tautest SpanCS Slackest SpanISG
Slackest SpanLocationBest)
for i = 1s
[T(i)Ta(i)] = TwinTenStaticTension(KtoKt1(i)Kt2oD3oD5oX3oY3oX5oY5o)
end
figure
[AXH1H2] = plotyy(Kt1()T(1)Kt1()Ta(4)plot) hold on
H3 = line(Kt1()T(5)ParentAX(1)) hold on
H4 = line(Kt1()Ta(3)Parent AX(2)) hold off
set(AXYLimModeautoYTickModeauto)
title(Static Tension vs Tensioner Arm 1 Stiffness)
xlabel(Kt1 (Nmrad))
set(get(AX(1)Ylabel)StringCS Span Tension (N))
set(get(AX(2)Ylabel)StringISG Span Tension (N))
set(H1LineStyle-Colorb)
set(H2LineStyle-Colorg)
set(H3LineStyleColorget(H1Color))
set(H4LineStyleColorget(H2Color))
legend([H1H2H3H4]CS Tautest SpanISG Tautest SpanCS Slackest SpanISG
Slackest SpanLocationBest)
for i = 1s
[T(i)Ta(i)] = TwinTenStaticTension(KtoKt1oKt2(i)D3oD5oX3oY3oX5oY5o)
end
figure
[AXH1H2] = plotyy(Kt2()T(1)Kt2()Ta(4)plot) hold on
H3 = line(Kt2()T(5)ParentAX(1)) hold on
H4 = line(Kt2()Ta(3)Parent AX(2)) hold off
set(AXYLimModeautoYTickModeauto)
title(Static Tension vs Tensioner Arm 2 Stiffness)
xlabel(Kt2 (Nmrad))
set(get(AX(1)Ylabel)StringCS Span Tension (N))
Appendix D 164
set(get(AX(2)Ylabel)StringISG Span Tension (N))
set(H1LineStyle-Colorb)
set(H2LineStyle-Colorg)
set(H3LineStyleColorget(H1Color))
set(H4LineStyleColorget(H2Color))
legend([H1H2H3H4]CS Tautest SpanISG Tautest SpanCS Slackest SpanISG
Slackest SpanLocationBest)
for i = 1s
[T(i)Ta(i)] = TwinTenStaticTension(KtoKt1oKt2oD3(i)D5oX3oY3oX5oY5o)
end
figure
[AXH1H2] = plotyy(D3()T(1)D3()Ta(4)plot) hold on
H3 = line(D3()T(5)ParentAX(1)) hold on
H4 = line(D3()Ta(3)Parent AX(2)) hold off
set(AXYLimModeautoYTickModeauto)
title(Static Tension vs Tensioner Pulley 1 Diameter)
xlabel(D3 (m))
set(get(AX(1)Ylabel)StringCS Span Tension (N))
set(get(AX(2)Ylabel)StringISG Span Tension (N))
set(H1LineStyle-Colorb)
set(H2LineStyle-Colorg)
set(H3LineStyleColorget(H1Color))
set(H4LineStyleColorget(H2Color))
legend([H1H2H3H4]CS Tautest SpanISG Tautest SpanCS Slackest SpanISG
Slackest SpanLocationBest)
for i = 1s
[T(i)Ta(i)] = TwinTenStaticTension(KtoKt1oKt2oD3oD5(i)X3oY3oX5oY5o)
end
figure
[AXH1H2] = plotyy(D5()T(1)D5()Ta(4)plot) hold on
H3 = line(D5()T(5)ParentAX(1)) hold on
H4 = line(D5()Ta(3)Parent AX(2)) hold off
set(AXYLimModeautoYTickModeauto)
title(Static Tension vs Tensioner Pulley 2 Diameter)
xlabel(D5 (m))
set(get(AX(1)Ylabel)StringCS Span Tension (N))
set(get(AX(2)Ylabel)StringISG Span Tension (N))
set(H1LineStyle-Colorb)
set(H2LineStyle-Colorg)
set(H3LineStyleColorget(H1Color))
set(H4LineStyleColorget(H2Color))
legend([H1H2H3H4]CS Tautest SpanISG Tautest SpanCS Slackest SpanISG
Slackest SpanLocationBest)
Mesh points
m = 101
n = 101
Appendix D 165
T = zeros(5nm)
Ta = zeros(5nm)
[ixxiyy] = meshgrid(1m1n)
minX3 = 0260200
maxX3 = 0317677
minY3 = -0056640
maxY3 = 0228456
midX3 = 0311641
X3 = minX3 + (ixx-1)(maxX3-minX3)(m-1)
Y3 = minY3 + (iyy-1)(maxY3-minY3)(n-1)
for i = 1n
for j = 1m
if ((X3(ij)lt midX3)ampamp(Y3(ij)gt=(sqrt((0087945^2)-((X3(ij)-0224)^2)-
006395)))ampamp(Y3(ij)lt=(-1sqrt(((00703^2)-((X3(ij)-
024759)^2)))+016664)))||((X3(ij)gt=midX3)ampamp(Y3(ij)gt=(35548X3(ij)-
11134868))ampamp(Y3(ij)lt=(-1(sqrt(((00703^2)-((X3(ij)-024759)^2))))+016664))) mx+b
lt= y lt= circle4
[T(ij)Ta(ij)] =
TwinTenStaticTension(KtoKt1oKt2oD3oD5oX3(ij)Y3(ij)X5oY5o)
else
T(ij) = zeros(511)
Ta(ij) = zeros(511)
end
end
end
figure
Z1 = squeeze(T(1))
surf(X3Y3real(Z1))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(CS DRIVING Tautest Span Tension vs Tensioner Pulley 1 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(CS Span Tension (N))
figure
Z5 = squeeze(T(5))
surf(X3Y3real(Z5))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(CS DRIVING Slackest Span Tension vs Tensioner Pulley 1 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
Appendix D 166
zlabel(CS Span Tension (N))
figure
Za4 = squeeze(Ta(4))
surf(X3Y3real(Za4))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(ISG DRIVING Tautest Span Tension vs Tensioner Pulley 1 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(ISG Span Tension (N))
figure
Za3 = squeeze(Ta(3))
surf(X3Y3real(Za3))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(ISG DRIVING Slackest Span Tension vs Tensioner Pulley 1 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(ISG Span Tension (N))
minX5 = -0037093
maxX5 = 0212509
minY5 = 00362
maxY5 = 0228456
midX5a = 0131965
midX5b = 017729
X5 = minX5 + (ixx-1)(maxX5-minX5)(m-1)
Y5 = minY5 + (iyy-1)(maxY5-minY5)(n-1)
for i = 1n
for j = 1m
if
(X5(ij)ltmidX5a)ampamp(Y5(ij)lt=(0386X5(ij)+0146468))ampamp(Y5(ij)gt=(sqrt((0136525^2)-
(X5(ij)^2))))
[T(ij)Ta(ij)] =
TwinTenStaticTension(KtoKt1oKt2oD3oD5oX3oY3oX5(ij)Y5(ij))
elseif
((X5(ij)gt=midX5a)ampamp(X5(ij)ltmidX5b))ampamp(Y5(ij)gt=00362)ampamp(Y5(ij)lt=(0386X5(ij)+0
146468))
[T(ij)Ta(ij)] =
TwinTenStaticTension(KtoKt1oKt2oD3oD5oX3oY3oX5(ij)Y5(ij))
elseif (X5(ij)gt=midX5b)ampamp(Y5(ij)gt=(sqrt((00703^2)-(((X5(ij)-
024759)^2)))+016664))ampamp(Y5(ij)lt=(0386X5(ij)+0146468))
Appendix D 167
[T(ij)Ta(ij)] =
TwinTenStaticTension(KtoKt1oKt2oD3oD5oX3oY3oX5(ij)Y5(ij))
else
T(ij) = zeros(511)
Ta(ij) = zeros(511)
end
end
end
figure
Z1 = squeeze(T(1))
surf(X5Y5real(Z1))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(CS DRIVING Tautest Span Tension vs Tensioner Pulley 2 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(CS Span Tension (N))
figure
Z5 = squeeze(T(5))
surf(X5Y5real(Z5))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(CS DRIVING Slackest Span Tension vs Tensioner Pulley 2 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(CS Span Tension (N))
figure
Za4 = squeeze(Ta(4))
surf(X5Y5real(Za4))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(ISG DRIVING Tautest Span Tension vs Tensioner Pulley 2 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(ISG Span Tension (N))
figure
Za3 = squeeze(Ta(3))
surf(X5Y5real(Za3))
ZLim([50 500])
axis tight
Appendix D 168
colormap jet
colorbar
title(ISG DRIVING Slackest Span Tension vs Tensioner Pulley 2 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(ISG Span Tension (N))
D12 TwinTenStaticTensionm
The function TwinTenStaticTensionm simulates the static model of the B-ISG system with a
Twin Tensioner This function returns 3 vectors the static tension of each belt span in the
crankshaft- and ISG-driving phases of operation and the angle of displacement of each rigid
body in the ISG- driving phase It takes the input parameters kt kt1 kt2 for the tensioner arm
stiffness D3 and D5 for the tensioner pulley diameters and X3Y3 X5 and Y5 for the tensioner
arm pulley coordinates The function is called in the parametric analysis solution script
TwinMainm and in the optimization solution script OptimizationTwinm
D2 Optimization
D21 OptimizationTwinm
The following script is for the main function OptimizationTwinm It performs an optimization
search for the B-ISG system with a Twin Tensioner It takes an input for a parameter vector
containing values for the design variables The program calls the objective function
objfunTwinm and the constraint function confunTwinm The program can perform a genetic
algorithm (GA) optimization search or a hybrid GA optimization that includes a localized search
The optimal solution vector corresponding to the design variables and the optimal objective
function value is returned The program inputs the optimized values for the design variables into
TwinTenStaticTensionm This called function returns the optimized static state of tensions for
the crankshaft- and ISG- driving phases and for the angle of displacement of the rigid bodies in
the ISG driving phase
______________________________________________________________________________
clc
clear all
Initial values for variables
Kto = 20626
Kt1o = 10314
Kt2o = 16502
X3o = 0292761
Y3o = 0087
X5o = 012057
Appendix D 169
Y5o = 009193
w0 =[Kto Kt1o Kt2o X3o Y3o X5o Y5o] Start Point (row vector)
Variable ranges
minKt = Kto - 1Kto
maxKt = Kto + 1Kto
minKt1 = Kt1o - 1Kt1o
maxKt1 = Kt1o + 1Kt1o
minKt2 = Kt2o - 1Kt2o
maxKt2 = Kt2o + 1Kt2o
minX3 = 0260200
maxX3 = 0317677
minY3 = -0056640
maxY3 = 0228456
minX5 = -0037093
maxX5 = 0212509
minY5 = 00362
maxY5 = 0228456
ObjectiveFunction = objfunTwin
nvars = 7 Number of variables
ConstraintFunction = confunTwin
Uncomment next two lines (and comment the two functions after them) to use GA algorithm
for optimization
options = gaoptimset(InitialPopulationw0PopInitRange[minKt minKt1 minKt2 minX3
minY3 minX5 minY5 maxKt maxKt1 maxKt2 maxX3 maxY3 maxX5
maxY5]PopulationSize100Displayfinal)
[wfvalexitflagoutput] =
ga(ObjectiveFunctionnvars[][][][][][]ConstraintFunctionoptions)
fminconOptions = optimset(DisplayiterLargeScaleoff) Largescale off since gradient not
provided
options = gaoptimset(InitialPopulationw0PopInitRange[minKt minKt1 minKt2 minX3
minY3 minX5 minY5 maxKt maxKt1 maxKt2 maxX3 maxY3 maxX5
maxY5]PopulationSize100HybridFcnfmincon fminconOptions)
[wfvalexitflagoutput] = ga(ObjectiveFunctionnvars[][][][][][]ConstraintFunctionoptions)
[TTaThetaDegA] = TwinTenStaticTension(w(1)w(2)w(3)w(4)w(5)w(6)w(7))
D22 confunTwinm
The constraint function confunTwinm is used by the main optimization program to ensure
input values are constrained within the prescribed regions The function makes use of inequality
constraints for seven constrained variables corresponding to the design variables It takes an
input vector corresponding to the design variables and returns a data set of the vector values that
satisfy the prescribed constraints
Appendix D 170
D23 objfunTwinm
This function is the objective function for the main optimization program It outputs a value for
a weighted objective function or a non-weighted objective function relating the optimization of
the static tension The program takes an input vector containing a set of values for the design
variables that are within prescribed constraints The description of the function is similar to
TwinTenStaticTensionm but differs in the fact that it only returns a scalar value which is the
value of the objective function
171
VITA
ADEBUKOLA OLATUNDE
Email adebukolaolatundegmailcom
Adebukola Olatunde is a graduate research student at the University of Toronto in Toronto
Ontario Canada She obtained a Bachelor‟s Degree in Mechanical Engineering from McMaster
University in Hamilton Ontario Canada in 2002 Upon graduation she pursued a graduate
degree in mechanical engineering at the University of Toronto with a specialization in
mechanical systems dynamics and vibrations and environmental engineering In September
2008 she completed the requirements for the Master of Applied Science degree in Mechanical
Engineering She has held the position of teaching assistant for undergraduate courses in
dynamics and vibrations Adebukola has completed course work in professional education She
is a registered member of professional engineering organizations including the Professional
Engineer‟s of Ontario Engineer-in-Training program the Canadian Society of Mechanical
Engineers and the National Society of Black Engineers She intends to practice as a professional
engineering consultant in mechanical design
vi
Chapter 3 MODELING OF B-ISG SYSTEM 25
31 Overview 25
32 B-ISG Tensioner Design 25
33 Geometric Model of a B-ISG System with a Twin Tensioner 27
34 Equations of Motion for a B-ISG System with a Twin Tensioner 32
341 Dynamic Model of the B-ISG System 32
3411 Derivation of Equations of Motion 32
3412 Modeling of Phase Change 41
3413 Natural Frequencies Mode Shapes and Dynamic Responses 42
3414 Crankshaft Pulley Driving Torque Acceleration and Displacement 44
3415 ISG Pulley Driving Torque Acceleration and Displacement 46
3416 Tensioner Arms Dynamic Torques 48
3417 Dynamic Belt Span Tensions 49
342 Static Model of the B-ISG System 49
35 Simulations 50
351 Geometric Analysis 51
352 Dynamic Analysis 52
3521 Natural Frequency and Mode Shape 54
3522 Dynamic Response 58
3523 ISG Pulley and Crankshaft Pulley Torque Requirement 61
3524 Tensioner Arm Torque Requirement 62
3525 Dynamic Belt Span Tension 63
353 Static Analysis 66
36 Summary 69
Chapter 4 PARAMETRIC ANALYSIS OF A B-ISG TWIN TENSIONER 71
41 Introduction 71
42 Methodology 71
43 Results and Discussion 74
431 Influence of Tensioner Arm Stiffness on Static Tension 74
432 Influence of Tensioner Pulley Diameter on Static Tension 78
433 Influence of Tensioner Pulley 1 Coordinates on Static Tension 80
434 Influence of Tensioner Pulley 2 Coordinates on Static Tension 86
vii
44 Conclusion 92
Chapter 5 OPTIMIZATION OF A B-ISG TWIN TENSIONER 95
51 Optimization Problem 95
511 Selection of Design Variables 95
512 Objective Function amp Constraints 97
52 Optimization Method 100
521 Genetic Algorithm 100
522 Hybrid Optimization Algorithm 101
53 Results and Discussion 101
531 Parameter Settings amp Stopping Criteria for Simulations 101
532 Optimization Simulations 102
533 Discussion 106
54 Conclusion 109
Chapter 6 CONCLUSION AND RECOMMENDATIONS111
61 Summary 111
62 Conclusion 112
63 Recommendations for Future Work 113
REFERENCES 116
APPENDICIES 123
A Passive Dual Tensioner Designs from Patent Literature 123
B B-ISG Serpentine Belt Drive with Single Tensioner Equation of Motion 138
C MathCAD Scripts 145
C1 Geometric Analysis 145
C2 Dynamic Analysis 152
C3 Static Analysis 161
D MATLAB Functions amp Scripts 162
D1 Parametric Analysis 162
D11 TwinMainm 162
D12 TwinTenStaticTensionm 168
D2 Optimization 168
D21 OptimizationTwinm - Optimization Function 168
viii
D22 confunTwinm 169
D23 objfunTwinm 170
VITA 171
ix
LIST OF TABLES
21 Passive Dual Tensioner Designs from Patent Literature
31 Selected Contact Point Types on the ith Pulley and Types for the ith Belt Span
32 Coordinate Points for Pulley Centres and Twin Tensioner Pivot
33 Geometric Results of B-ISG System with Twin Tensioner
34 Data for Input Parameters used in Dynamic and Static Computations
35 Static Solution for Belt Span Tensions in Crankshaft and ISG Driving Cases for a B-ISG
Serpentine Belt Drive with a Single Tensioner
36 Static Solution for Belt Span Tensions in Crankshaft and ISG Driving Cases for a B-ISG
Serpentine Belt Drive with a Twin Tensioner
41 Initial Values Increments and Ranges for Parameters of Twin Tensioner
51 Summary of Parametric Analysis Data for Twin Tensioner Properties
52a GA Optimization Results for Twin Tensioner Parameters and Objective Function
52b Computations for Tensions and Angles from GA Optimization Results
53a Hybrid Optimization Results for Twin Tensioner Parameters and Objective Function
53b Computations for Tensions and Angles from Hybrid Optimization Results
54a Non-Weighted Optimization Results for Twin Tensioner Parameters and Objective
Function
54b Computations for Tensions and Angles from Non-Weighted Optimizations
x
55 Weighted Optimization Results for Static Tensions for Optimal B-ISG System with a
Twin Tensioner
56 Non-Weighted Optimization Results for Static Tensions for Optimal B-ISG System with a
Twin Tensioner
xi
LIST OF FIGURES
21 Hybrid Functions
31 Schematic of the Twin Tensioner
32 B-ISG Serpentine Belt Drive with Twin Tensioner
33 Angles Coordinates and Possible Belt Contact Points for the ith and i+1th Pulleys
34 Twin Tensioner Free Body Diagram of a Four-Degree of Freedom System
35 Free Body Diagram for Non-Tensioner Pulleys
36a ISG Driving Case Natural Frequency of System and Mode Shapes for Responsive Rigid
Bodies
36b ISG Driving Case First Mode Responses
36c ISG Driving Case Second Mode Responses
37a Crankshaft Driving Case Natural Frequency of System and Mode Shapes for Responsive
Rigid Bodies
37b Crankshaft Driving Case First Mode Responses
37c Crankshaft Driving Case Second Mode Responses
38 Crankshaft Pulley Dynamic Response (for crankshaft driven case)
39 ISG Pulley Dynamic Response (for ISG driven case)
310 Air Conditioner Pulley Dynamic Response
311 Tensioner Pulley 1 Dynamic Response
xii
312 Tensioner Pulley 2 Dynamic Response
313 Tensioner Arm 1 Dynamic Response
314 Tensioner Arm 2 Dynamic Response
315 Required Driving Torque for the ISG Pulley
316 Required Driving Torque for the Crankshaft Pulley
317 Dynamic Torque for Tensioner Arm 1
318 Dynamic Torque for Tensioner Arm 2
319 Span 1 (between crankshaft and air conditioner) Dynamic Belt Span Tension
320 Span 2 (between air conditioner and tensioner 1) Dynamic Belt Span Tension
321 Span 3 (between tensioner 1 and ISG) Dynamic Belt Span Tension
322 Span 4 (between ISG and tensioner 2) Dynamic Belt Span Tension
323 Span 5 (between tensioner 2 and crankshaft) Dynamic Belt Span Tension
324 B-ISG Serpentine Belt Drive with Single Tensioner
41a Regions 1 and 2 and their Associated Guidelines for Coordinates of Tensioner Pulleys 1
amp 2
41b Regions 1 and 2 in Cartesian Space
42 Parametric Analysis for Coupled Stiffness Arm Constant kt (N∙mrad)
43 Parametric Analysis for Stiffness of Arm 1 kt1 (N∙mrad)
44 Parametric Analysis for Stiffness of Arm 2 kt2 (N∙mrad)
xiii
45 Parametric Analysis for Pulley 1 Diameter D3 (m)
46 Parametric Analysis for Pulley 2 Diameter D5 (m)
47 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Tautest Span
Tension in Crankshaft Driving Case
48 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Slackest Span
Tension in Crankshaft Driving Case
49 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Tautest Span
Tension in ISG Driving Case
410 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Slackest Span
Tension in ISG Driving Case
411 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Tautest Span
Tension in Crankshaft Driving Case
412 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Slackest Span
Tension in Crankshaft Driving Case
413 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Tautest Span
Tension in ISG Driving Case
414 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Slackest Span
Tension in ISG Driving Case
51 Static Stability of the B-ISG Twin Tensioner Based on the Angular Displacement of
Tensioner Arms 1 and 2
A1 Proposed design by Bayerische Motoren Werke AG corresponding to patent nos
EP1420192-A2 and DE10253450-A1
A2a First of four proposed designs (and its various configurations) by Bosch GMBH
corresponding to patent no WO0026532-A1
A2b Second of four proposed designs by Bosch GMBH corresponding to patent no
WO0026532-A1
A2c Third of four proposed designs (and its various configurations) by Bosch GMBH
corresponding to patent no WO0026532-A1
A2d Fourth of four proposed designs (and its various configurations) by Bosch GMBH
corresponding to patent no WO0026532-A1
A3 Proposed design by Daimler Chrysler AG corresponding to patent no DE10324268-A1
A4 Proposed design by Dayco Products LLC corresponding to patent no US6942589-B2
xiv
A5 Proposed design by Gates Corp corresponding to patent no WO2003038309-A
A6 Proposed design by General Motors Corp corresponding to patent no US20060287146-
A1
A7 Proposed design by INA Schaeffler KG corresponding to patent no DE10044645-A1
A8a First of three proposed designs by INA Schaeffler KG corresponding to patent no
DE10159073-A1
A8b Second of three proposed designs by INA Schaeffler KG corresponding to patent no
DE10159073-A1
A8c Third of three proposed designs by INA Schaeffler KG corresponding to patent no
DE10159073-A1
A9 Proposed design by INA Schaeffler KG corresponding to patent no DE10359641-A1
A10 Proposed design by INA Schaeffler KG corresponding to patent no EP1723350-A1
A11 Proposed design by INA Schaeffler KG corresponding to patent no EP1738093-A1
A12 Proposed design by INA Schaeffler KG corresponding to patent no DE102004012395-
A1
A13 Proposed design by INA Schaeffler KG corresponding to patent nos DE102005017038-
A1and WO2006108461-A1
A14 Proposed design by Litens Automotive GMBH et al corresponding to patent no
US20010007839-A1
A15 Proposed design by Mitsubishi Jidosha Eng KK and Mitsubishi Motor Corp
corresponding to patent no JP2005083514-A
A16 Proposed design by Nissan Motor Co Ltd corresponding to patent no JP3565040-B2
A17 Proposed design by NTN Corp corresponding to patent no JP2006189073-A
A18 Proposed design by Valeo Equipment Electriques Moteur corresponding to patent nos
EP1658432 and WO2005015007
B1 Single Tensioner B-ISG System
B2 Free-body Diagram of ith Pulley
xv
B3 Free-body Diagram of Single Tensioner
C1 Schematic of B-ISG System with Twin Tensioner
C2 Possible Contact Points
xvi
LIST OF SYMBOLS
Latin Letters
A Belt cord cross-sectional area
C Damping matrix of the system
cb Belt damping
119888119894119887 Belt damping constant of the ith belt span
119914119946119946 Damping matrix element in the ith row and ith column
ct Damping acting between tensioner arms 1 and 2
cti Damping of the ith tensioner arm
DCS Diameter of crankshaft pulley
DISG Diameter of ISG pulley
ft Belt transition frequency
H(n) Phase change function
I Inertial matrix of the system
119920119938 Inertial matrix under ISG driving phase
119920119940 Inertial matrix under crankshaft driving phase
Ii Inertia of the ith pulley
Iti Inertia of the ith tensioner arm
119920120784120784 Submatrix of inertial matrix I
j Imaginary coordinate (ie (-1)12
)
K Stiffness matrix of the system
xvii
119896119887 Belt factor
119870119887 Belt cord stiffness
119896119894119887 Belt stiffness constant of the ith belt span
kt Spring stiffness acting between tensioner arms 1 and 2
kti Coil spring of the ith tensioner arm
119922120784120784 Submatrix of stiffness matrix K
Lfi Lbi Lengths of possible belt span connections from the ith pulley
Lti Length of the ith tensioner arm
Modeia Mode shape of the ith rigid body in the ISG driving phase
Modeic Mode shape of the ith rigid body in the crankshaft driving phase
n Engine speed
N Motor speed
nCS rpm of crankshaft pulley
NF Motor speed without load
nISG rpm of ISG pulley
Q Required torque matrix
qc Amplitude of the required crankshaft torque
QcsISG Required torque of the driving pulley (crankshaft or ISG)
Qm Required torque matrix of driven rigid bodies
Qti Dynamic torque of the ith tensioner arm
Ri Radius of the ith pulley
T Matrix of belt span static tensions
xviii
Trsquo Dynamic belt tension matrix
119931119940 Damping matrix due to the belt
119931119948 Stiffness matrix due to the belt
Ti Tension of the ith belt span
To Initial belt tension for the system
Ts Stall torque
Tti Tension for the neighbouring belt spans of the ith tensioner pulley
(XiYi) Coordinates of the ith pulley centre
XYfi XYbi XYfbi
XYbfi Possible connection points on the ith pulley leading to the ith belt span
XYf2i XYb2i
XYfb2i XYbf2i Possible connection points on the ith pulley leading to the (i-1)th belt span
Greek Letters
αi Angle between the datum and the line connecting the ith and (i+1)th pulley
centres
βji Angle of orientation for the ith belt span
120597θti(t) 120579 ti(t)
120579 ti(t)
Angular displacement velocity and acceleration (rotational coordinate) of the
ith tensioner arm
120637119938 General coordinate matrix under ISG driving phase
120637119940 General coordinate matrix under crankshaft driving phase
θfi θbi Angles between the datum and the belt connection spans with lengths Lfi and
Lbi respectively
Θi Amplitude of displacement of the ith pulley
xix
θi(t) 120579 i(t) 120579 i(t) Angular position velocity and acceleration (rotational coordinate) of the ith
pulley
θti Angle of the ith tensioner arm
θtoi Initial pivot angle of the ith tensioner arm
θm Angular displacement matrix of driven rigid bodies
Θm Amplitude of displacement of driven rigid bodies
ρ Belt cord density
120601119894 Belt wrap angle on the ith pulley
φmax Belt maximum phase angle
φ0deg Belt phase angle at zero frequency
ω Frequency of the system
ωcs Angular frequency of crankshaft pulley
ωISG Angular frequency of the ISG pulley
120654119951 Natural frequency of system
1
CHAPTER 1 INTRODUCTION
11 Background
Belt drive systems are the means of power transmission in conventional automobiles The
emergence of hybrid technologies specifically the Belt-driven Integrated Starter-generator (B-
ISG) has placed higher demands on belt drives than ever before The presence of an integrated
starter-generator (ISG) in a belt transmission places excessive strain on the belt leading to
premature belt failure This phenomenon has motivated automotive makers to design a tensioner
that is suitable for the B-ISG system
The belt drive is also known interchangeably as the front-end accessory drive-belt (FEAD) the
belt accessory-drive system (BAS) or the belt transmission system In a traditional setting the
role of this system is to transmit torque generated by an internal combustion engine (ICE) in
order to reliably drive multiple peripheral devices mounted on the engine block The high speed
torque is transmitted through a crankshaft pulley to a serpentine belt The serpentine belt is a
single continuous member that winds around the driving and driven accessory pulleys of the
drive system Serpentine belts used in automotive applications consist of several layers The
load-bearing layer is a flexible member consisting of high stiffness fibers [1] It is covered by a
protective layer to guard against mechanical damage and is bound below by a visco-elastic layer
that provides the required shock absorption and grip against the rigid pulleys [1] The accessory
devices may include an alternator power steering pump water pump and air conditioner
compressor among others
Introduction 2
The B-ISG system is a transmission system characteristic to micro-hybrid automobiles It is akin
to traditional belt drives differing in the fact that an electric motor called an integrated starter-
generator (ISG) replaces the original alternator re-starts the engine from idle speed and provides
braking regeneration [2] The re-start function of the micro-hybrid transmission is known as
stop-start In the B-ISG setting the ISG is mounted on the belt drive The ISG produces a speed
of approximately 2000 to 2500rpm in order to spin the engine at approximately 750rpm and
upwards to produce an instantaneous start in the start-stop process [3] The high rotations per
minute (rpm) produced by the ISG consistently places much higher tension requirements on the
belt than when the crankshaft is driving the belt It is preferable not to exceed a range of 600N to
800N of tension on the belt since this exceeds the safe operating conditions of belts used in most
traditional drive systems [4] The traditional belt drive system‟s tensioner a single-arm
tensioner does not suitably reduce the high belt tension nor provide enough tension in the slack
belts spans occurring in the ISG phase of operation for the B-ISG system
In order for the belt to transfer torque in a drive system its initial tension must be set to a value
that is sufficient to keep all spans rigid This value must not be too low as to allow any one span
to be slack during the drive‟s phases of operation Furthermore the belt must not be ldquoinstalled
with too high a tensionrdquo since this can lead to ldquopremature failure of the bearings supporting the
drive and driven pulleys and of the belt itselfrdquo [5] The presence of a tensioning mechanism in
an automotive belt drive allows for an enhanced belt life and performance since pre-tensioning
of the belt is normally not sufficient for all phases of belt drive operation A tensioner allows for
the system to cope with moderate to severe changes in belt span tensions
Introduction 3
Traditional automotive tensioners for belt drives of an ICE consist of a single spring-loaded
arm This type of tensioner is normally designed to provide a passive response to changes in belt
span tension The introduction of the ISG electric motor into the traditional belt drive with a
single-arm tensioner results in the presence of excessively slack spans and excessively tight
spans in the belt The tension requirements in the ISG-driving phase which differ from the
crankshaft-driving phase are poorly met by a traditional single-arm passive tensioner
Tensioners can be divided into two general classes passive and active In both classes the
single-arm tensioner design approach is the norm The passive class of tensioners employ purely
mechanical power to achieve tensioning of the belt while the active class also known as
automatic tensioners typically use some sort of electronic actuation Automatic tensioners have
been employed by various automotive manufacturers however ldquosuch devices add mass
complication and cost to each enginerdquo [5]
12 Motivation
The motivation for the research undertaken arises from the undesirable presence of high belt
tension in automotive belt drives Manufacturers of automotive belt drives have presented
numerous approaches for tension mechanism designs As mentioned in the preceding section
the automation of the traditional single-arm tensioner has disadvantages for manufacturers A
survey of the literature reveals that few quantitative investigations in comparison to the
qualitative investigations provided through patent literature have been conducted in the area of
passive and dual tensioner configurations As such the author of the research project has selected
to investigate the performance of a passive twin-arm tensioner design The theoretical tensioner
Introduction 4
configuration is motivated by research and developments of industry partner Litens
Automotivendash a manufacturer of automotive belt drive systems and components Litens‟
specialty in automotive tensioners has provided a basis for the research work conducted
13 Thesis Objectives and Scope of Research
The objective of this project is to model and investigate a system containing a passive twin-arm
tensioner in a B-ISG serpentine belt drive where the driving pulley alternates between a
crankshaft pulley and an ISG pulley The modeling of a serpentine belt drive system is in
continuation of the work done by post-doctoral fellow Zhen Mu in development of the priority
software known as FEAD at the University of Toronto Firstly for the B-ISG system with a
twin-arm tensioner the geometric state and its equations of motion (EOM) describing the
dynamic and static states are derived The modeling approach was verified by deriving the
geometric properties and the EOM of the system with a single tensioner arm and comparing its
crankshaft-phase‟s simulation results with FEAD software simulations This also provides
comparison of the new twin-arm tensioner belt drive model with the former single-arm tensioner
equipped belt drive model Secondly the model for the static system is investigated through
analysis of the tensioner parameters Thirdly the design variables selected from the parametric
analysis are used for optimization of the new system with respect to its criteria for desired
performance
Introduction 5
14 Organization and Content of the Thesis
This thesis presents the investigation of a passive twin-arm tensioner design in a B-ISG
serpentine belt drive system which is distinguished by having its driving pulley alternate
between a crankshaft pulley and an ISG pulley
Chapter 2 presents the literature reviewed relevant to the area of the thesis topic The context of
the research discusses the function and location of the ISG in hybrid technologies in order to
provide a background for the B-ISG system The attributes of the B-ISG are then discussed
Subsequently a description is given of the developments made in modeling belt drive systems
At the close of the chapter the prior art in tensioner designs and investigations are discussed
The third chapter describes the system models and theory for the B-ISG system with a twin-arm
tensioner Models for the geometric properties and the static and dynamic cases are derived The
simulation results of the system model are presented
Then the fourth chapter contains the parametric analysis The methodologies employed results
and a discussion are provided The design variables of the system to be considered in the
optimization are also discussed
The optimization of a B-ISG system with a passive twin-arm tensioner is presented in Chapter 5
The evaluation of optimization methods results of optimization and discussion of the results are
included Chapter 6 concludes the thesis work in summarizing the response to the thesis
Introduction 6
objectives and concluding the results of the investigation of the objectives Recommendations for
future work in the design and analysis of a B-ISG tensioner design are also described
7
CHAPTER 2 LITERATURE REVIEW
21 Introduction
This literature review justifies the study of the thesis research the significance of the topic and
provides the overall framework for the project The design of a tensioner for a Belt-driven
Integrated Starter-generator (B-ISG) system is a link in the chain of power transmission
developments in hybrid automobiles This chapter will begin with the context of the B-ISG
followed by a review of the hybrid classifications and the critical role of the ISG for each type
The function location and structure of the B-ISG system are then discussed Then a discussion
of the modeling of automotive belt transmissions is presented A systematic review of the prior
art and current state of tensioning mechanisms for B-ISG systems amalgamates the literature and
research evidence relevant to the thesis topic which is the design of a B-ISG tensioner
The Belt-driven Integrated Starter-generator (B-ISG) system is a part of a hybrid class that is
distinguished from other hybrid classes by the structure functions and location of its ISG The
B-ISG unit is a hybrid technology applied to traditional automotive belt drives The use of a B-
ISG system to achieve a start-stop function in the car engine is estimated to cut fuel consumption
in conventional automobiles by up to ten percent and thus reduce CO2 emissions [6]
Environmental and legislative standards for reducing CO2 emissions in vehicles have called for
carmakers to produce less polluting and more efficient vehicle powertrain systems [7] The
transition to bdquocleaner‟ cars makes room for the introduction of the ISG machine into conventional
automotive belt drives [8] The reduction of CO2 emissions and the similarity of the B-ISG
Literature Review 8
transmission to that of conventional cars provide the motivation for the thesis research
Consequently the micro-hybrid class of cars is especially discussed in the literature review since
it contains the B-ISG type of transmission system The micro-hybrid class is one of several
hybrid classes
A look at the performance of a belt-drive under the influence of an ISG is rooted in the
developments of hybrid technology The distinction of the ISG function and its location in each
hybrid class is discussed in the following section
22 B-ISG System
221 ISG in Hybrids
This section of the review discusses the standard classes of hybrid cars which are full power
mild and micro- hybrids Special attention is given to hybrid vehicle architectures involving
internal combustion engines (ICEs) as the main power source This is done for the sake of
comparison between hybrid classes since the ICE is the standard power source for B-ISG micro-
hybrids which is the focus of the research The term conventional car vehicle or automobile
henceforth refers to a vehicle powered solely by a gas or diesel ICE
A hybrid vehicle has a drive system that uses a combination of energy devices This may include
an ICE a battery and an electric motor typically an ISG Two systems exist in the classification
of hybrid vehicles The older system of classification separates hybrids into two classes series
hybrids and parallel hybrids In the older system many modern hybrid vehicles have modes of
operation matching both categories classifying them under either of the two classes [9] The
Literature Review 9
new system of classification has four classes full power mild and micro Under these classes
vehicles are more often under a sole category [9] In both systems an ICE may act as the primary
source of power otherwise it may be a fuel cell The fuel used by the ICE may be gas (petrol)
diesel or an alternative fuel such as ethanol bio-diesel or natural gas
2211 Full Hybrids
In a full hybrid car the ICE is used to power the integrated starter-generator (ISG) which stores
electrical energy in the batteries to be used to power an electric traction motor [8] The electric
traction motor is akin to a second ISG as it generates power and provides torque output It also
supplies an extra boost to the wheels during acceleration and drives up steep inclines A full
hybrid vehicle is able to move by electrical power only It can be driven by the ISG powering
the electric traction motor without the engine running This silent acceleration known as electric
launch is normally employed when accelerating from standstill [9] Full hybrids can generate
and consume energy at the same time Full hybrid vehicles also use regenerative braking [8]
The ISG allows this by converting from an electric traction motor to a generator when braking or
decelerating The kinetic energy from the car‟s motion is then turned into electricity and stored
in the batteries For full hybrids to achieve this they often use break-by-wire a form of
electronically controlled braking technology
A high-voltage (ie 36- or 42-volt) ISG is employed in full hybrids to start the ICE It spins the
engine more than 900 rpm whereas conventional 12-volt starter motors spin the engine at
approximately 250 rpm [9] Thus the full hybrid vehicle is able to have an instantaneous start In
full hybrids the ISG is placed in the position of the flywheel and can have its motion decoupled
Literature Review 10
from the engine [9] The ISG device also allows full hybrids to have engine start-stop also called
an idle-stop ability The idle-stop function refers to when the engine shuts down as soon as a
vehicle stops from its ICE driving mode which saves on the fuel it normally burns while idling
[8] The vehicle returns to the engine driving mode of operation by way of the ISG‟s start-up of
the crankshaft which restarts the engine in less than 300 milliseconds [9] In summary at
standstill the tachometer of the engine drops to 0 rpm since the engine has ceased the engine is
started only when needed which is often several seconds after acceleration has begun The
engine start-stop feature is achieved by way of an electronic control system that shuts off the ICE
when it is not needed to assist in driving the wheels or to produce electricity for recharging the
batteries The start-stop feature by itself is estimated to produce a ten percent fuel gain in hybrids
over conventional vehicles particularly in urban driving conditions [9] Since the ICE is
required to provide only the average horsepower used by the vehicle the engine is downsized in
comparison to a conventional automobile that obtains all its power from an ICE Frequently in
full hybrids the ICE uses an alternative operating strategy such as the Atkinson Cycle which has
a higher efficiency while having a lower power output Examples of full hybrids include the
Ford Escape and the Toyota Prius [9]
2212 Power Hybrids
Akin to the full hybrid the ISG of the power hybrid enables the same features electric launch
regenerative braking and engine idle-stop The distinguishing characteristic from full hybrids is
the ICE is not downsized to meet only the average power demand [9] Thus the engine of a
power hybrid is large and produces a high amount of horsepower compared to the former
Overall a power hybrid has the assist of a full size ICE and therefore has more torque and a
Literature Review 11
greater acceleration performance than a full hybrid or a conventional vehicle with the same size
ICE [9] The Lexus RX400h unit is an example of a power hybrid [9]
2213 Mild Hybrids
In the hybrid types discussed thus far the ISG is positioned between the engine and transmission
to provide traction for the wheels and for regenerative braking Often times the armature or rotor
of the electric motor-generator which is the ISG replaces the engine flywheel in full and power
hybrids [9] In the case of the mild hybrid the ISG is not decoupled from the ICE and hence it is
not able to drive the wheels apart from the engine It remains that the ISG shares the same shaft
with the ICE In this environment the electric launch feature does not exist since the ISG does
not turn the wheels independently of the engine and energy cannot be generated and consumed
at the same time However the ISG of the mild hybrid allows for the remaining features of the
full hybrid regenerative braking and engine idle-stop including the fact that the engine is
downsized to meet only the average demand for horsepower Mild hybrid vehicles include the
GMC Sierra pickup and 2003 to 2005 Honda Civic models [9]
2214 Micro Hybrids
Micro hybrid is the category of hybrids that can contain a B-ISG transmission and is also closest
to modern conventional vehicles This class normally features a gas or diesel ICE [9] The
conventional automobile is modified by installing an ISG unit on the mechanical drive in place
of or in addition to the starter motor The starter motor typically 12-volts is removed only in
the case that the ISG device passes cold start testing which is also dependent on the engine size
[10] Various mechanical drives that may be employed include chain gear or belt drives or a
Literature Review 12
clutchgear arrangement The majority of literature pertaining to mechanical driven ISG
applications does not pursue clutchgear arrangements since it is associated with greater costs
and increased speed issues Findings by Henry et al [11] show that the belt drive in
comparison to chain and gear drives has a decreased cost (especially if the ISG is mounted
directly to the accessory drive) has no need for lubrication has less restriction in the packaging
environment and produces very low noise Also mounting the ISG unit on a separate belt from
that linking the accessory pulleys is undesirable since applying the ISG directly to the accessory
belt drive requires less engine transmission or vehicle modifications
As with full power and mild hybrids the presence of the ISG allows for the start-stop feature
The automobile‟s electronic control unit (ECU) is calibrated or engine control circuitry (a
separate ECU) is added to the conventional car in order to shut down the engine when the
vehicle is stopped [12] The control system also controls the charge cycle of the ISG [9] This
entails that it dictates the field current by way of a microprocessor to allow the system to defer
battery charge cycles until the vehicle is decelerating [13] This produces electricity to recharge
the battery primarily during deceleration and braking The B-ISG transmission of a micro hybrid
and its various components are discussed in the subsequent section Examples of micro hybrid
vehicles are the PSA Group‟s Citroen C2 and C3 [14] Ford‟s Fiesta [14] and BMW‟s Mini
Cooper D and various others of BMW‟s European models [15]
Literature Review 13
Figure 21 Hybrid Functions
Source Dr Daniel Kok FFA July 2004 modified [16]
Figure 21 shows that the higher the voltage available to the ISG unit the more hybrid functions
it is capable of performing It is noted that B-ISG transmissions of the micro-hybrid class may
also exceed the typical functions of micro-hybrids For instance Ford‟s HyTrans van (developed
in partnership with Ricardo UK Ltd Valeo SA Gates Corporation and the UK Department for
Transport) uses a B-ISG system and a 42-volt battery The van is diesel-powered and has
characteristics of a mild hybrid such as cold cranks and engine assists [17]
222 B-ISG Structure Location and Function
2221 Structure and Location
The ISG is composed of an electrical machine normally of the inductive type which includes a
stator (stationary part of the ISG) and a rotor (non-stationary part of the ISG) and a converter
comprising of a regulator a modulator switches and filters There are various configurations to
integrate the ISG unit into an automobile power train One configuration situates the ISG
directly on the crankshaft in the place of the present flywheel [11] This set-up is more compact
however it results in a longer power train which becomes a potential concern for transverse-
Literature Review 14
mounted engines [18] An alternative set-up is to have a side-mounted ISG This term is used to
describe the configuration of mounting the electrical device on the side of the mechanical drive
[18] As mentioned in Section 2214 a belt drive is used as the mechanical drive for the thesis
research hence the ISG is belt-mounted and the transmission becomes a belt-driven ISG system
In this arrangement the ISG replaces the alternator [13] and in some cases the starter motor may
be removed This design allows for the functions of the ISG system mentioned in the description
of the ISG role in micro-hybrids [9] The side-mounted ISG specifically the belt-mounted ISG
is more evolutionary to the conventional car since it ldquoallows for a more traditional under-hood
layoutrdquo [11]
2222 Functionalities
The primary duty of the ISG in a micro hybrid specifically in a B-ISG setting is to bring the
engine from rest to normal operating speeds within a time span ranging from 250 to 400 ms [3]
and in some high voltage settings to provide cold starting
The cold starting operation of the ISG refers to starting the engine from its off mode rather than
idle mode andor when the engine is at a low temperature for example -29 to -50 degrees
Celsius [2] If the ISG is used for cold starting the peak torque is determined by the torque
requirement for the cold starting operation of the target vehicle since it is greater than the
nominal torque For this function the ldquomachine has to provide a breakaway torque about 15 [to]
18 times the nominal cranking torque to overcome static torque and rotate the engine from 0 to
[between] 10 [and] 20rpmrdquo [2] This remains to be a challenge for the ISG as the 12-volt
architecture most commonly found in vehicles does not supply sufficient voltage [2] The
introduction of the ISG machine and other electrical units in vehicles encourages a transition
Literature Review 15
from a 12-volt or 14-volt to a 42-volt electrical architecture [19] The transition to 42-volt
architecture brings ldquopotential higher-voltage functionalities that come with an ISG systemrdquo [20]
At present ldquowhen the [ISG] machine cannot provide enough torque for initial cold engine
cranking the conventional starter will [remain] in the system and perform only for the initial
cranking while the stop-start function is taken over by the [ISG] machinerdquo [2] The ISG‟s launch
assist torque the torque required to bring the engine from idle speed to the speed at which it can
develop a higher torque output is 2000 to 2500 rpm for most gas engines [3]
Delphi‟s Energen 5 High Output 12-volt Belt-alternator-starter (or B-ISG) was implemented by
researchers on a 53 L V-8 engine with an automatic transmission in a Chevrolet Silverado truck
[21] The ISG was applied in a belt-mounted configuration and was used only for warm engine
re-starts The results of Wezenbeek et al [21] showed that the starting torque for a re-start by the
12-Volt ISG was 42 Nm ISG‟s have also been used in 14V 36V and 42V architectures [13]
23 Belt Drive Modeling
The modeling of a serpentine belt drive and tensioning mechanism has typically involved the
application of Newtonian equilibrium equations to rigid bodies in order to derive the equations of
motion for the system There are two modes of motion in a serpentine belt drive transverse
motion and rotational motion The former can be viewed as the motion of the belt directed
normal to the direction of the beltpulley contact plane similar to the vibratory motion of a taut
string that is fixed at either end However the study of the rotational motion in a belt drive is the
focus of the thesis research
Literature Review 16
Much work on the mechanics of the belt drive was carried out by Firbank [22] Firbank‟s
models helped to understand belt performance and the influence of driving and driven pulleys on
the tension member The first description of a serpentine belt drive for automotive use was in
1979 by Cassidy et al [23] and since this time there has been an increasing body of knowledge
on the mathematical modeling of serpentine belt drives Ulsoy et al [24] presented a design
methodology to improve the dynamic performance of instability mechanisms for belt tensioner
systems The mathematical model developed by Ulsoy et al [24] coupled the equations of
motion that were obtained through a dynamic equilibrium of moments about a pivot point the
equations of motion for the transverse vibration of the belt and the equations of motion for the
belt tension variations appearing in the transverse vibrations This along with the boundary and
initial conditions were used to describe the vibration and stability of the coupled belt-tensioner
system Their system also considered the geometry of the belt drive and tensioner motion
Hereafter Beikmann et al [25] predicted the belt drive vibration for a system composed of a
driving pulley driven pulley and a dynamic tensioner The authors coupled the linear equations
of transverse motion for the respective belt spans with the equations of motion for pulleys and a
tensioner This was used to form the free response of the system and evaluate its response
through a closed-form solution of the system‟s natural frequencies and mode shapes
A complex modal analysis of a serpentine belt drive system was carried out by Kraver et al [26]
to determine the effect of damping on rotational vibration mode solutions The equations of
motion developed for a multi-pulley flat belt system with viscous damping and elastic
Literature Review 17
properties including the presence of a rotary tensioner were manipulated to carry out the modal
analysis
Beikmann et al [27] also derived a nonlinear model to predict the operating state of a belt-
tensioner system by way of nonlinear numerical methods and an approximated linear closed-
form method The authors used this strategy to develop a single design parameter referred to as
a tensioner constant to measure the effectiveness of the tensioning mechanism in relation to its
operating state from a reference state The authors considered the steady state tensions in belt
spans as a result of accessory loads belt drive geometry and tensioner properties
Zhang and Zu [28] conducted a modal analysis for the response of a linear serpentine belt drive
system A non-iterative approach was used to explicitly form the equations for the system‟s
natural frequencies An exact closed-form expression for the dynamic response of the system
using eigenfunction expansion was derived with the system under steady-state conditions and
subject to harmonic excitation
The work conducted by Balaji and Mockensturm [29] considered a front-end accessory drive
(FEAD) with a decoupler or isolator attached to a pulley The rotational response for the FEAD
was found analytically by considering the system to be piecewise linear about the equilibrium
angular deflections The effect of their nonlinear terms was considered through numerical
integration of the derived equations of motion by way of the iterative methodndash fourth order
Runge-Kutta The authors in this case considered the longitudinal (ie rotational) vibration of
the belt spans only
Literature Review 18
The first to carry out the analysis of a serpentine belt drive system containing a two-pulley
tensioner was Nouri in 2005 [30] Nouri found the closed-form analytical solution of a
serpentine belt drive with a two-pulley tensioner for the case of sinusoidal excitation He
employed Runge Kutta method as well to solve the equations of motion to find the response of
the system under a general input from the crankshaft The author‟s work also included the
optimization of the tensioner design in order to minimize belt span vibrations due to crankshaft
excitation Furthermore the author applied active control techniques to the tensioner in a belt
drive system
The works discussed have made significant contributions to the research and development into
tensioner systems for serpentine belt drives These lead into the requirements for the structure
function and location of tensioner systems particularly for B-ISG transmissions
24 Tensioners for B-ISG System
241 Tensioners Structure Function and Location
Literature shows that the improvement of a serpentine belt life in a B-ISG system centers on the
tensioning mechanism redesign This mechanism as shown by researchers including
Wezenbeek et al [21] and Henry et al [11] is crucial in establishing the least tension in the belt
(above a zero value) in order to guard against failure by way of slip due to slack spans in the belt
and oscillations during engine re-start It is noted by Firbank [22] that the mechanics of a belt-
drive ldquois based on the idea that belt behaviour is governed by the elastic extension or contraction
of the belt arising from tension variationsrdquo [22] these variations may be compensated for by an
adjustable tensioner
Literature Review 19
The two types of tensioners are passive and active tensioners The former permits an applied
initial tension and then acts as an idler and normally employs mechanical power and can include
passive hydraulic actuation This type is cheaper than the latter and easier to package The latter
type is capable of continually adjusting the belt tension since it permits a lower static tension
Active tensioners typically employ electric or magnetic-electric actuation andor a combination
of active and passive actuators such as electrical actuation of a hydraulic force
Conventional belt tensioners comprise of a single tensioner arm that is fitted with a sole idler
pulley to engage a serpentine belt [31] A radial bearing is used to rotatably connect the idler
pulley to the tensioner arm [31] The tensioner arm is mounted on a pivot pin that is wrapped by
a bushing and is free to rotate [31] The pin covered by the bushing is fixed to the engine
housing [31] A rotary spring is wrapped about the bearing pin and bushing to provide a pre-
tension force to the belt via the tensioner arm and idler pulley thus taking up the slack due to the
changes in belt length [31] When the belt undergoes stretch under a load the spring drives the
tensioner arm and idler pulley further into the belt [31] Belt tension changes under the modes of
operation which can include when the crankshaft (or driving pulley) abruptly decelerates from a
steady-state condition and auxiliary components continue to rotate still in their own inherent
inertia and thus become the primary drivers [31] These fluctuations in belt tension lead to belt
flutter or skip and slip that may damage other components present in the belt drive [31]
Locating the tensioner on the slack side of the belt is intended to lower the initial static tension
[11] In conventional vehicles the engine always drives the alternator so the tensioner is located
in the belt span that links the crankshaft and alternator pulleys In a B-ISG setting the slack span
Literature Review 20
of the belt alternates between the driving mode of the ISG and the driving mode of the crankshaft
[32] Research by Henry et al [11] and also the summary of prior art for tensioners in Table
21 show that placing the idlertensioner pulley in the slack span in the case that the ISG is
driving instead of in the slack span when the crankshaft is driving allows for easier packaging
and for the least static tension Designs shown in Table 21 place the tensioneridler pulley in the
same span as Henry et al [11] or in both the slack and taut spans if using a double
tensioneridler configuration
242 Systematic Review of Tensioner Designs for a B-ISG System
The proposals for belt tensioner devices to manage the issue of high peaks in belt tension for B-
ISG settings are largely in patent records as the re-design of a tensioner has been primarily a
concern of automotive makers thus far A systematic review of the patent literature has been
conducted in order to identify evaluate and collate relevant tensioning mechanism designs
applicable to a B-ISG setting Its research objective is to influence the selection of a tensioner
configuration for the thesis study
The predefined search strategy used by the researcher has been to consider patents dating only
post-2000 as many patents dating earlier are referred to in later patents as they are developed on
in most cases by the original inventor (eg an INA Schaeffler KG patent published in 2000 may
refer to its own earlier patent presented in 1999) Patents dating pre-2000 that do not have any
successor were also considered The inclusion and exclusion criteria and rationales that were
used to assess potential patents are as follows
Inclusion of
Literature Review 21
tensioner designs with two arms andor two pivots andor two pulleys
mechanical tensioners (ie exclusion of magnetic or electrical actuators or any
combination of active actuators) in order to minimize cost
tension devices that are an independent structure apart from the ISG structure in order to
reduce the required modification to the accessory belt drive of a conventional automobile
and
advanced designs that have not been further developed upon in a subsequent patent by the
inventor or an outside party
Table 21 provides a collation of the results for the systematic review based on the selection
criteria Illustrations of the collated patent designs may be seen in Appendix A It is noted that
the patent literature pertaining to these designs in most cases provides minimal numerical data
for belt tensions achieved by the tensioning mechanism In most cases only claims concerning
the outcome in belt performance achievable by the said tension device is stated in the patent
Table 21 Passive Dual Tensioner Designs from Patent Literature
Bayerische
Motoren Werke
AG
Patents EP1420192-A2 DE10253450-A1 [33]
Design Approach
2 tensioner pulleys (idlers) and 2 tension arms are mounted outside the periphery of the belt drive these form tiltable clamping arms around a common axis of rotation
A torsion spring is used at bearing bushings to mount tension arms at ISG shaft
Each tension arm cooperates with torsion spring mechanism to rotate through a damping
device in order to apply appropriate pressure to taut and slack spans of the belt in
different modes of operation
Bosch GMBH Patent WO0026532 et al [34]
Design Approach
2 tension pulleys each one is mounted on the return and load spans of the driven and
driving pulley respectively
Idlers (tension pulleys) each connect to a spring which is attached on one end to a fixed point
Literature Review 22
Idlers‟ motions are independent of each other and correspond to the tautness or
slackness in their respective spans
Or alternatively a spring connects the idler pulleys and one of the two idlers is fixed at
its axis of rotation
Daimler Chrysler
AG
Patents DE10324268-A1 [35]
Design Approach
2 idlers are given a working force by a self-aligning bearing
Bearing supports auxiliary unit (ISG) and is arranged concentrically with the axle
auxiliary unit pulley
Dayco Products
LLC
Patents US6942589-B2 et al [36]
Design Approach
2 tension arms are each rotatably coupled to an idler pulley
One idler pulley is on the tight belt span while the other idler pulley is on the slack belt
span
Tension arms maintain constant angle between one another
One arm forms a positive differential angle with the belt and the remaining arm forms a negative differential angle with the belt
Idler pulleys are on opposite sides of the ISG pulley
Gates Corporation Patents US20060249118-A1 WO2003038309-A [37]
Design Approach
A tensioner pulley contacts the belt at the slack span during start-up (ISG-driving mode)
A tensioner is asymmetrically biased in direction tending to cause power transmission
belt to be under tension
McVicar et al
(Firm General
Motors Corp)
Patent US20060287146-A1 [38]
Design Approach
2 tension pulleys and carrier arms with a central pivot are mounted to the engine
One tension arm and pulley moderately biases one side of belt run to take up slack
during engine start-up while other tension arm and pulley holds appropriate bias against
taut span of belt
A hydraulic strut is connected to one arm to provide moderate bias to belt during normal
engine operation and velocity sensitive resistance to increasing belt forces during engine
start-up
INA Schaeffler
KG et al
Patents DE10044645-A1 [39] DE10159073-A1 [40] EP1723350-A1 et al [41]
DE10359641-A1 et al [42] EP1738093-A1 et al [43] DE102004012395-A1 [44]
WO2006108461-A1 et al [45]
Design Approach
2 tension arms and 2 pulleys approach ndash o Mutually independent tensioning arms are supported for rotation in the same
plane of the housing part
o Idler pulley corresponding to each tensioning arm engages with different
sections of belt
o When high tension span alternates with slack span of belt drive one tension
arm will increase pressure on current slack span of belt and the other will
decrease pressure accordingly on taut span
o Or when the span under highest tension changes one tensioner arm moves out
of the belt drive periphery to a dead center due to a resulting force from the taut
span of the ISG starting mode
o Deflection of the taut span acts on associated pulley to apply a counter-moment to the other idler pulley on the slack span
Literature Review 23
o The 2 lever arms are of different lengths and each have an idler pulley of
different diameters and different wrap angles of belt (see DE10045143-A1 et
al)
1 tensioner arm and 2 pulleys approach ndash
o 2 idler pulleys are pinned to a beam arranged on a clamping arm that is tiltably
linked to the beam o The ISG machine is supported by a shock absorber
o During ISG start-up one idler pulley is induced to a dead center position while
it pulls the remaining idler pulley into a clamping position until force
equilibrium takes place
o A shock absorber is laid out such that its supporting spring action provides
necessary preloading at the idler pulley in the direction of the taut span during
ISG start-up mode
Litens Automotive
Group Ltd
Patents US6506137-B2 et al [46]
Design Approach
2 tension pulleys on opposite sides of the ISG pulley engage the belt
They are positioned such that their applied forces result in opposing directed moments with respect to the tension device‟s axis of pivot
The pivot axis varies relative to the force applied to each tension pulley
Diameters of the tensioner pulleys are approximately equal and belt wrap angles of the
tensioner pulleys are approximately equal
A limited swivel angle for the tensioner arms work cycle is permitted
Mitsubishi Jidosha
Eng KK
Mitsubishi Motor
Corp
Patents JP2005083514-A [47]
Design Approach
2 tensioners are used
1 tensioner is held on the slack span of the driving pulley in a locked condition and a
second tensioner is held on the slack side of the starting (driven) pulley in a free condition
Nissan Patents JP3565040-B2 et al [48]
Design Approach
A single tensioner is on the slack span once ISG pulley is in start-up mode
The tension device is comprised of a oil pressure tensioner and a half ratchet mechanism
(a plunger which performs retreat actuation according to the energizing force of the oil
pressure spring and load received from the ISG)
The tensioner is equipped with a relief valve to keep a predetermined load lower than the
maximum load added by the ISG device
NTN Corp Patent JP2006189073-A [49]
Design Approach
An automatic tensioner is equipped with a hydraulic damper mechanism comprised of a
screw bolt using saw-screwed teeth and a cylinder nut a return spring and a spring seat
in a pressure chamber (within the screw bolt) a rod seat (that is fitted to the lower end of
the cylinder nut) a spring support (arranged on varying diameter stepped recessed
sections of the rod seat) and a check valve with an openingclosing passage
The cylinder and screw bolt act as the rigidity buffer under excessive loads during ISG
start-up mode of operation
Valeo Equipment
Electriques
Moteur
Patents EP1658432 WO2005015007 [50]
Design Approach
ldquoThe invention relates to a system or a starter (10) in which a pulley (80) is rotationally mounted on a section (22) of a shaft which axially extends inside a pulley (80) and
Literature Review 24
forwards at least partially outside a support element (200) and is characterized in that
the free front end (23) of said shaft section (22) is carried by an arm (206) connected to
the support element (200)rdquo
The author notes that published patents and patent applications may retain patent numbers for multiple patent
offices (ie European Patent Office German Patent Office etc) In such cases the published patent number or in
the absence of such a number the published patent application number has been specified However published
patent documents in the above cases also served as the document (ie identical) to the published patent if available
Quoted from patent abstract as machine translation is poor
25 Summary
The research on tensioner designs from the patent literature demonstrates a lack of quantifiable
data for the performance of a twin tensioner particularly suited to a B-ISG system The review of
the literature for the modeling theory of serpentine belt drives and design of tensioners shows
few belt drive models that are specific to a B-ISG setting Hence the literature review supports
the thesis objective of modeling a B-ISG tensioner specifically one that has a passive twin
tensioner configuration and as well measuring the tensioner‟s performance The survey of
hybrid classes reveals that the micro-hybrid class is the only class employing a closely
conventional belt transmission and hence its B-ISG transmission is applicable for tensioner
investigation The patent designs for tensioners contribute to the development of the tensioner
design to be studied in the following chapter
25
CHAPTER 3 MODELING OF B-ISG SYSTEM
31 Overview
The derivation of a theoretical model for a B-ISG system uses real life data to explore the
conceptual system under realistic conditions The literature and prior art of tensioner designs
leads the researcher to make the following modeling contributions a proposed design for a
passive two-pulley tensioner computation of geometric attributes for a B-ISG system with the
proposed tensioner and derivation of the system‟s equations of motion (EOM) under dynamic
and static states as well as deriving the EOM for the B-ISG system with only a passive single-
pulley tensioner for comparison The principles of dynamic equilibrium are applied to the
conceptual system to derive the EOM
32 B-ISG Tensioner Design
The proposed design for a passive two pulley tensioner configures two tensioners about a single
fixed pivot point in the interior space of a serpentine belt drive One end of each tensioner arm
coincides with the centre point of a tensioner pulley and this point marks the axis of rotation of
the pulley The other end of each arm is pivoted about a point so that the arms share the same
axis of rotation This conceptual design henceforth is called a Twin Tensioner Figure 31 shows
a schematic for the proposed design
Modeling of B-ISG 26
Figure 31 Schematic of the Twin Tensioner
The tensioner pulley coordinates are described by (XiYi) their radii by Ri their arm lengths Lti
and their angles θti The rotation of the arms is resisted by stiffness kt of a coil spring acting
between the two arms and spring stiffness kti acting between each arm and the pivot point The
motion of each arm is dampened by dampers and akin to the springs a damper acts between the
two arms ct and a damper cti acts between each arm and the pivot point The result is a
tensioning mechanism with four degrees of freedom (DOF) that includes independent rotations
of the two pulleys and two arms
The following section relates the geometry of the rigid bodies in a B-ISG system equipped with a
Twin Tensioner to their respective motions
Modeling of B-ISG 27
33 Geometric Model of a B-ISG System with a Twin Tensioner
The B-ISG system with the Twin Tensioner is shown in Figure 32 The geometry of the drive
provides the lengths of the belt spans and angles of wrap for the belt and pulley contact surfaces
These variables are crucial to resolve the components of forces and moment arms acting on each
rigid body in the system and are used in the derivation of the EOM in section 34 Zhen Mu‟s
geometric modeling approach [51] used in the development of the software FEAD was applied
to the Twin Tensioner system to compute the system‟s unique geometric attributes
Figure 32 B-ISG Serpentine Belt Drive with Twin Tensioner
It is noted that in Figure 31 and Figure 32 showing the schematic of the Twin Tensioner and
the overall system respectively that for the purpose of the geometric computations the forward
direction follows the convention of the numbering order counterclockwise The numbering
order is in reverse to the actual direction of the belt motion which is in the clockwise direction in
this study The fourth pulley is identified as an ISG unit pulley However the properties used
for the ISG pulley‟s geometry inertia stiffness and damping is modeled as a conventional
Modeling of B-ISG 28
alternator pulley This pulley is conceptualized as an ISG when it is modeled as the driving
pulley at which point the requirements of the ISG are solved for and its non-inertia attributes
are not needed to be ascribed
Figure 33 shows the geometric attributes needed to resolve the wrap angle of the belt on each
pulley Variables (XiYi) and XYfi XYbi XYfbi and XYbfi are the ith pulley centre coordinates and
its possible belt connection points respectively Length Lfi is the length of the span connecting
the points XYfi and XYf(i+1) or XYbi and XYb(i+1) on the ith and (i+1)th pulleys respectively
Similarly Lbi is the length of the span between XYfbi and XYfb(i+1) or XYbfi and XYbf (i+1) on the
ith and (i+1)th pulleys respectively Angles αi θfi and θbi represent the angle between a line
connecting the ith and (i+1)th pulley centres and the angles of the belt connection spans with
lengths Lfi and Lbi respectively Ri is the radius of the ith pulley
Figure 33 Angles Coordinates and Possible Belt Contact Points for the ith and i+1th Pulleys
[modified] [51]
Modeling of B-ISG 29
The angle between the horizontal and the line connecting the ith and (i+1)th pulley centres αi is
calculated using Zhen‟s method [51] This method uses the pulley‟s coordinates and a cosine
trigonometric relation
i acos
Xi 1
Xi
Xi 1
Xi
2
Yi 1
Yi
2
Yi 1
Yi
if
(31a)
i 2 acos
Xi 1
Xi
Xi 1
Xi
2
Yi 1
Yi
2
Yi 1
Yi
if
(31b)
The lengths for connecting the possible belt spans are described by the variables Lfi and Lbi
The centre point coordinates and the radii of the pulleys are related through the solution of
triangles which they form to define values of the possible belt span lengths
Lfi
Xi 1
Xi
2
Yi 1
Yi
2
Ri 1
Ri
2
(32a)
Lbi
Xi 1
Xi
2
Yi 1
Yi
2
Ri 1
Ri
2
(32b)
The set of possible belt span lengths leads to the calculation of θfi and θbi the angles between the
line connecting the ith and (i+1)th pulley centres and the possible contact point on the pulley
perimeter
Modeling of B-ISG 30
(33a)
(33b)
The array of possible belt connection points comes about from the use of the pulley centre
coordinates and their radii and the sine of the sum or differences of αi and θfi or θbi The angle
αi is calculated in equations (31a) and (31b) and angles θfi and θbi are calculated in equations
(33a) and (33b) The formula to compute the array of points is shown in equations (34) and
(35) for the ith and (i+1)th pulleys Equation (34) describes the forward belt connection point
on the ith pulley which is in the span leading forward to the next (i+1)th pulley
(34a)
(34b)
(34c)
(34d)
bi atan
Lbi
Ri
Ri 1
Modeling of B-ISG 31
Equation (35) describes the backward belt connection point on the ith pulley This point sits on
the ith pulley in the contacting belt span which leads backward to connect with the (i-1)th
pulley
(35a)
(35b)
(35c)
(35d)
The selection of the coordinates from the array of possible connection points requires a graphic
user interface allowing for the points to be chosen based on observation This was achieved
using the MathCAD software package as demonstrated in the MathCAD scripts found in
Appendix C The belt connection points can be chosen so as to have a pulley on the interior or
exterior space of the serpentine belt drive The method used in the thesis research was to plot the
array of points in the MathCAD environment with distinct symbols used for each pair of points
and to select the belt connection points accordingly By observation of the selected point types
the type of belt span connection is also chosen Selected point and belt span types are shown in
Table 31
Modeling of B-ISG 32
Table 31 Selected Contact Point Types on the ith Pulley and Types for the ith Belt Span
Pulley Forward Contact
Point
Backwards Contact
Point
Belt Span
Connection
1 Crankshaft XYf1 XYbf21 Lf1
2 Air Conditioning XYfb2 XYf22 Lb2
3 Tensioner 1 XYbf3 XYfb23 Lb3
4 AlternatorISG XYfb4 XYbf24 Lb4
5 Tensioner 2 XYbf5 XYfb25 Lb5
The inscribed angles βji between the datum and the forward connection point on the ith pulley
and βji between the datum and its backward connection point are found through solving the
angle of the arc along the pulley circumference between the datum and specified point The
wrap angle ϕi is found as the difference between the two inscribed angles for each connection
point on the pulley The angle between each belt span and the horizontal as well as the initial
angle of the tensioner arms are found using arctangent relations Furthermore the total length of
the belt is determined by the sum of the lengths of the belt spans
34 Equations of Motion for a B-ISG System with a Twin Tensioner
341 Dynamic Model of the B-ISG System
3411 Derivation of Equations of Motion
This section derives the inertia damping stiffness and torque matrices for the entire system
Moment equilibrium equations are applied to each rigid body in the system and net force
equations are applied to each belt span From these two sets of equations the inertia damping
Modeling of B-ISG 33
and stiffness terms are grouped as factors against acceleration velocity and displacement
coordinates respectively and the torque matrix is resolved concurrently
A system whose motion can be described by n independent coordinates is called an n-DOF
system Consider the free body diagram of the Twin Tensioner in Figure 34 in which each
pulley of inertia Ii is supported on an arm of inertia Iti It is assumed that the pulleys are
constrained to rotate about their respective central axes and the arms are free to rotate about their
respective pivot points then at any time the position of each pulley can be described by a
rotational coordinate θi(t) and a coordinate θti(t) can denote the rotation of each arm Thus the
tensioner system comprises of four rigid bodies where each is described by one coordinate and
hence is a four-DOF system It is important to note that each rigid body is treated as a point
mass In addition inertial rotation in the positive direction is consistent with the direction of belt
motion The belt span tensions Ti and coupled radii Ri apply moments to the pulleys
Figure 34 Twin Tensioner Free Body Diagram of a Four-Degree of Freedom System
Modeling of B-ISG 34
For the serpentine belt system considered in the thesis research there are seven rigid bodies each
having a one-DOF of motion The EOM for a seven-DOF system form second-order coupled
differential equations meaning that each equation includes all of the general coordinates and
includes up to the second-order time derivatives of these coordinates The EOM can be
obtained by applying D‟Alembert‟s principle that the sum of the moments taken about any point
including the couples equals to zero Therefore the inertial couple the product of the inertia and
acceleration is equated to the moment sum as shown in equation (35)
I ∙ θ = ΣM (35)
The moment equilibrium equations for the Twin Tensioner in Figure 34 where the positive
direction is in the clockwise direction are shown in equations (36) through to (310) The
numbering convention used for each rigid body corresponds to the labeled serpentine belt drive
system shown in Figure 32 Qi represents the required torque of the ith rigid body ci is the
damping constant of the ith rigid body βji is the angle of orientation for the ith belt span and
120597120579119905119894 120579 119905119894 and 120579 119905119894 are the angular displacement angular velocity and angular acceleration of the ith
tensioner arm The initial angle of the ith tensioner arm is described by θtoi
minusI3 ∙ θ 3 = T3 ∙ R3 minus T2 ∙ R3 minus Q3 + c3 ∙ θ 3 (36)
minusI5 ∙ θ 5 = minusT4 ∙ R5 + T5 ∙ R5 minus Q5 + c5 ∙ θ 5 (37)
Modeling of B-ISG 35
It1 ∙ θ t1 = minusTt1 ∙ Lt1 ∙ sin θto 1 minus βj4 + sin θto 1 minus βj5 minus kt ∙ partθt1 minus partθt2 minus kt1 ∙
partθt1 minus ct ∙ partθ t1 minus partθ t2 minus ct1 ∙ partθ t1 (38)
It2 ∙ θ t2 = minusTt2 ∙ Lt2 ∙ sin θto 2 minus βj2 + sin θto 1 minus βj3 minus kt ∙ partθt2 minus partθt1 minus kt2 ∙ partθt2 minus
ct ∙ partθ t2 minus partθ t1 minus ct2 ∙ partθ t2 (39)
partθt1 = θt1 minus θto 1 (310a)
partθt2 = θt2 minus θto 2 (310b)
The free body diagrams for the remaining rigid bodies crankshaft pulley air conditioner pulley
and ISG pulley are in the general form of Figure 35 The sum of the moments about the axes of
rotation are taken for these structures in equations (311) through to (313)
Figure 35 Free Body Diagram for Non-Tensioner Pulleys
Modeling of B-ISG 36
I1 ∙ θ 1 = T5 ∙ R1 minus T1 ∙ R1 + Q1 minus c1 ∙ θ 1 (311)
I2 ∙ θ 2 = T1 ∙ R2 minus T2 ∙ R2 + Q2 minus c2 ∙ θ 2 (312)
I4 ∙ θ 4 = T3 ∙ R4 minus T4 ∙ R4 + Q4 minus c4 ∙ θ 4 (313)
The relationship between belt tensions and rigid body displacements is in the general form of
equation (314) where 119827119836 and 119827119844 are damping and stiffness matrices due to the belt respectively
with each factorized by a radial arm length This relationship is described for each span in
equations (315) through to (320) The belt damping constant for the ith belt span is cib
119827prime = 119827119836 ∙ 120521 + 119827119844 ∙ 120521 (314)
T1 = To + k1b ∙ R1 ∙ θ1 minus R2 ∙ θ2 + c1
b ∙ [R1 ∙ θ 1 minus R2 ∙ θ 2] (315)
T2 = To + k2b ∙ R2 ∙ θ2 minus R3 ∙ θ3 + Lt1 ∙ [sin θto 1 minus βj2 ] ∙ (θt1 minus θto 1) + c2
b ∙ [R2 ∙ θ 2 minus R3 ∙
θ 3 + Lt1 ∙ [sin θto 1 minus βj2 ] ∙ (θ t1)] (316)
T3 = To + k3b ∙ R3 ∙ θ3 minus R4 ∙ θ4 + Lt1 ∙ [sin θto 1 minus βj3 ] ∙ (θt1 minus θto 1) + c3
b ∙ [R3 ∙ θ 3 minus R4 ∙
θ 4 + Lt1 ∙ [sin θto 1 minus βj3 ] ∙ (θ t2)] (317)
Modeling of B-ISG 37
T4 = To + k4b ∙ R4 ∙ θ4 minus R5 ∙ θ5 + Lt2 ∙ [sin θto 2 minus βj4 ] ∙ (θt2 minus θto 2) + c4
b ∙ [R4 ∙ θ 4 minus R5 ∙
θ 5 + Lt2 ∙ [sin θto 2 minus βj4 ] ∙ (θ t1)] (318)
T5 = To + k5b ∙ R5 ∙ θ5 minus R1 ∙ θ1 + Lt2 ∙ [sin θto 2 minus βj5 ] ∙ (θt2 minus θto 2) + c5
b ∙ [R5 ∙ θ 5 minus R1 ∙
θ 1 + Lt2 ∙ [sin θto 2 minus βj5 ] ∙ (θ t2)] (319)
Tprime = Ti minus To (320)
Since the applied torques on the tensioner pulleys Q3 and Q4 are zero the static equilibrium
equation of the pulleys show that the adjacent spans of each tensioner pulley are equal to each
other Hence equations (321) and (322) are denoted as follows
Tt1 = T2 = T3 (321)
Tt2 = T4 = T5 (322)
Equations (310a) (310b) and (314) through to (322) are substituted into the EOMs described
in equations (36) to (39) and (311) to (313) The newly formed equations can be arranged
and written in matrix form as shown in equations (323) through to (328) The general
coordinate matrix 120521 and its first and second derivatives are shown in the EOM below
119816 ∙ 120521 + 119810 ∙ 120521 + 119818 ∙ 120521 = 119824 (323)
Modeling of B-ISG 38
The inertia matrix I includes the inertia of each rigid body in its diagonal elements The
damping matrix C includes variables 119888119894119887 the damping of the ith belt span 119877119894 its radius 120573119895119894 its
angle 119871119905119894 the ith tensioner arm‟s length 120579119905119900119894 its initial pivot angle and 119888119905 and 119888119905119894 the ith
tensioner arm viscous damping constants Stiffness matrix K contains 119896119894119887 the ith belt span
stiffness and 119896119905 and 119896119905119894 the ith tensioner arm stiffness constants and akin to the damping
matrix the variables 119877119894 119871119905119894 120579119905119900119894 and 120573119895119894 The belt span stiffness is computed in equation
(326b) where 119870119887 represents the belt cord stiffness 119896119887 is the belt factor obtained from
experimental data 120573119895119894 is the angle of orientation for the span between the jth and ith pulleys and
ϕi is the belt wrap angle on the ith pulley
Modeling of B-ISG 39
119816 =
I1 0 0 0 0 0 00 I2 0 0 0 0 00 0 I3 0 0 0 00 0 0 I4 0 0 00 0 0 0 I5 0 00 0 0 0 0 It1 00 0 0 0 0 0 It2
(324)
119810 =
c1
b ∙ R12 + c5
b ∙ R12 + c1 minusc1
b ∙ R1 ∙ R2 0 0 minusc5b ∙ R1 ∙ R5 0 c5
b ∙ R1 ∙ Lt2 ∙ sin θto 2 minus βj5
minusc1b ∙ R1 ∙ R2 c2
b ∙ R22 + c1
b ∙ R22 + c2 minusc2
b ∙ R2 ∙ R3 0 0 c2b ∙ R2 ∙ Lt1 ∙ sin θto 1 minus βj2 0
0 minusc2b ∙ R2 ∙ R3 c3
b ∙ R32 + c2
b ∙ R32 + c3 minusc3
b ∙ R3 ∙ R4 0 C36 0
0 0 minusc3b ∙ R3 ∙ R4 c4
b ∙ R42 + c3
b ∙ R42 + c4 minusc4
b ∙ R4 ∙ R5 minusc3b ∙ R4 ∙ Lt1 ∙ sin θto 1 minus βj3 c4
b ∙ R4 ∙ Lt2 ∙ sin θto 2 minus βj4
minusc5b ∙ R1 ∙ R5 0 0 minusc4
b ∙ R4 ∙ R5 c5b ∙ R5
2 + c4b ∙ R5
2 + c5 0 C57
0 0 0 0 0 ct +ct1 minusct
0 0 0 0 0 minusct ct +ct1
(325a)
C36 = 1198773 ∙ 1198711199051 ∙ [1198883119887 ∙ 119904119894119899 1205791199051199001 minus 1205731198953 minus 1198882
119887 ∙ 119904119894119899 1205791199051199001 minus 1205731198952 ] (325b)
C57 = 1198775 ∙ 1198711199052 ∙ [1198885119887 ∙ 119904119894119899 1205791199051199002 minus 1205731198955 minus 1198884
119887 ∙ 119904119894119899 1205791199051199002 minus 1205731198954 ] (325c)
Modeling of B-ISG 40
119818 =
k1
b ∙ R12 + k5
b ∙ R12 minusk1
b ∙ R1 ∙ R2 0 0 minusk5b ∙ R1 ∙ R5 0 k5
b ∙ R1 ∙ Lt2 ∙ sin θto 2 minus βj5
minusk1b ∙ R1 ∙ R2 k2
b ∙ R22 + k1
b ∙ R22 minusk2
b ∙ R2 ∙ R3 0 0 k2b ∙ R2 ∙ Lt1 ∙ sin θto 2 minus βj2 0
0 minusk2b ∙ R2 ∙ R3 k3
b ∙ R32 + k2
b ∙ R32 minusk3
b ∙ R3 ∙ R4 0 R3 ∙ Lt1 ∙ [k3b ∙ sin θto 1 minus βj3 minus k2
b ∙ sin θto 1 minus βj2 ] 0
0 0 minusk3b ∙ R3 ∙ R4 k4
b ∙ R42 + k3
b ∙ R42 minusk4
b ∙ R4 ∙ R5 minusk3b ∙ R4 ∙ Lt1 ∙ sin θto 1 minus βj3 k4
b ∙ R4 ∙ Lt2 ∙ sin θto 2 minus βj4
minusk5b ∙ R1 ∙ R5 0 0 minusk4
b ∙ R4 ∙ R5 k5b ∙ R5
2 + k4b ∙ R5
2 0 R5 ∙ Lt2 ∙ [k5b ∙ sin θto 2 minus βj5 minus k4
b ∙ sin θto 2 minus βj4 ]
0 0 0 0 0 kt +kt1 minuskt
0 0 0 0 0 minuskt kt +kt1
(326a)
k119894b =
Kb
Li + kb ∙ Ri ∙ϕi+1
2 + Ri ∙ϕi
2
(326b)
120521 =
θ1
θ2
θ3
θ4
θ5
partθt1
partθt2
(327)
119824 =
Q1
Q2
Q3
Q4
Q5
Qt1
Qt2
(328)
Modeling of B-ISG 41
3412 Modeling of Phase Change
The phase change from the crankshaft pulley being the driving pulley to the ISG pulley being the
driving pulley is described through a conditional equality based on a set of Boolean conditions
When the crankshaft is driving the rows and the columns of the EOM are swapped such that the
new order for rows and columns is 1 (crankshaft pulley) 4 (ISG pulley) 2 (air conditioner
pulley) 3 (tensioner 1 pulley) 5 (tensioner 2 pulley) 6 (tensioner arm 1) and 7 (tensioner arm 2)
When the ISG is driving the order is the same except that the second row and second column
terms relating to the ISG pulley become the first row and first column while the crankshaft
pulley terms (previously in the first row and first column) become the second row and second
column Hence the order for all rows and columns of the matrices making up the EOM in
equation (322) switches between 1423567 (when the crankshaft pulley is driving) and
4123567 (when the ISG pulley is driving) For example in the crankshaft driving and ISG
driving phases the general coordinate matrix and the inertia matrix become the following
120521119940 =
1205791
1205794
1205792
1205793
1205795
1205971205791199051
1205971205791199052
and 120521119938 =
1205794
1205791
1205792
1205793
1205795
1205971205791199051
1205971205791199052
(329a amp b)
119816119940 =
I1 0 0 0 0 0 00 I4 0 0 0 0 00 0 I2 0 0 0 00 0 0 I3 0 0 00 0 0 0 I5 0 00 0 0 0 0 It1 00 0 0 0 0 0 It2
and 119816119938 =
I4 0 0 0 0 0 00 I1 0 0 0 0 00 0 I2 0 0 0 00 0 0 I3 0 0 00 0 0 0 I5 0 00 0 0 0 0 It1 00 0 0 0 0 0 It2
(329c amp d)
Modeling of B-ISG 42
where subscripts c and a denote the crankshaft pulley driving phase and the ISG pulley driving
phase respectively
The condition for phase change is based on the engine speed n in units of rpm Equation (330)
demonstrates the phase change
H(n) = 1 119899 ge 750 (Crankshaft driving phase)0 119899 lt 750 (ISG driving phase)
(330)
When the crankshaft pulley is the driving pulley the ISG pulley becomes the driven pulley and
following suit when the ISG pulley is the driving pulley the crankshaft pulley becomes the
driven pulley These modes of operation mean that the system will predict two different sets of
natural frequencies and mode shapes Using a Boolean condition to allow for a swap between
the first and second rows as well as between the first and second columns of the EOM matrices
I C and K allows for a continuous plot of the dynamic response to be plotted for the ISG pulley
throughout its driving and driven phases as well as for that of the crankshaft pulley
3413 Natural Frequencies Mode Shapes and Dynamic Responses
Assuming the system undergoes simple harmonic motion its matrix of natural frequencies 120596119899
and modeshapes are found by solving the eigenvalue problem shown in equation (331a)
ωn ∙ 119816120784120784 minus 11981822 ∙ 120495m = 120782 (331a)
The displacement amplitude Θm is denoted implicitly in equation (331d)
Modeling of B-ISG 43
120521119846 = θ2 θ3 θ5 θ6 partθt1 partθt2 T for H n = 1 (331b)
120521119846 = θ1 θ3 θ5 θ6 partθt1 partθt2 T for H n = 0 (331c)
θm = 120495119846 ∙ sin(ω ∙ t) (331d)
I2 and K22 are submatrices of I and K respectively meaning the first row and column of each of
the original matrices are removed The eigenvalue problem is reached by considering the
undamped and unforced motion of the system Furthermore the dynamic responses are found by
knowing that the torque requirements in the matrixndash Qm for the driven pulleys and the tensioner
arms are zero in the dynamic case which signifies a response of the system to an input solely
from the driving pulley
I1 120782120782 119816120784120784
θ 1120521 119846
+ C11 119810120783120784119810120784120783 119810120784120784
θ 1120521 119846
+ K11 119818120783120784
119818120784120783 119818120784120784 θ1
120521119846 =
QCS ISG
119824119846 (332)
1
In the case of equation (331) θm is the submatrix identified in equations (331b) through to
(331d) Therein θ1 denotes the general coordinate for the driving pulley so that in the case the
phase change function H(n) is equal to zero θ1 becomes θ4 and the order of the rows and
columns for the remaining matrices correspond to the value of H(n) as mentioned earlier in
section 3412 For simple harmonic motion the motion of the driven pulleys are described as
1 The driving torque 119876119862119878119868119878119866 denotes the crankshaft torque 119876119862119878 when the crankshaft pulley is driving or the ISG
torque 119876119868119878119866 when the ISG pulley is in its driving function
Modeling of B-ISG 44
θm = 120495119846 ∙ sin(ω ∙ t) (333)
The dynamic response of the system to an input from the driving pulley under the assumption of
sinusoidal motion is expressed in equation (334)
120495119846 = [(119818120784120784 minusω2 ∙ 119816120784120784) + 119895ω ∙ 119810120784120784]minus1 ∙ (119818120784120783 + 119895ω ∙ 119810120784120783) ∙ Θ1 (334)
3414 Crankshaft Pulley Driving Torque Acceleration and Displacement
Subsequently the crankshaft pulley driving torque acceleration and displacement are firstly
discussed It is assumed in the thesis research for the purpose of modeling that the engine
serving the crankshaft is of the four cylinder type The input torque provided by a four-cylinder
engine is assumed to be dominated by two torque pulses per revolution of the crankshaft which
is represented by the factor of 2 on the steady component of the angular velocity in equation
(335) The torque requirement of the crankshaft pulley when it is the driving pulley is
Qc = qc ∙ sin(2 ∙ ωcs ∙ t) (335)
The amplitude of the required crankshaft torque qc is expressed in equation (336) and is
derived from equation (332)
qc = K11 minus ω2 ∙ I1 + 119895 ∙ ω ∙ C11 ∙ Θ1 + (119818120783120784 + 119895 ∙ ω ∙ 119810120783120784) ∙ 120495119846 (336)
Modeling of B-ISG 45
The angular frequency for the system in radians per second (rads) ω when the crankshaft
pulley is driving can be found as a function of the engine speed in rotations per minute (rpm) n
and by taking into account the double pulse per crankshaft revolution
ω = 2 ∙ ωcs = 4 ∙ π ∙ n
60
(337)
The system is considered when the amplitude of the crankshaft‟s angular acceleration is assumed
to be constant and equal to 650 rads2 during the crankshaft pulley driving phase The amplitude
of the excitation angular input from the engine is shown in equation (339b) and is found as a
result of (338)
θ 1CS = 650 ∙ sin(ω ∙ t) (338)
θ1CS = minus650
ω2sin(ω ∙ t)
(339a) where
Θ1CS = minus650
ω2
(339b)
Modeling of B-ISG 46
3415 ISG Pulley Driving Torque Acceleration and Displacement
Secondly the torque acceleration and the displacement of the ISG pulley in its driving phase is
discussed The torque for the ISG when it is in its driving function is assumed constant Ratings
for the ISG are taken from experiments performed by researchers Wezenbeek et al [21] on an
Energen 5 High Output Belt-alternator-starter (BAS) unit from Delphi The 12-Volt BAS which
can also be called a B-ISG was reported to have a maximum allowable speed of 18000 rpm [21]
As well it was noted that the ISG pulley was sized appropriately and the engine speed was
limited to ensure that an over-speed condition of the ISG pulley would not occur [21] The stall
torque rating for the Energen ISG was reported to be 48 Nm at the electric machine shaft [21]
The formula for the torque of a permanent magnet DC motor for any given speed (equation
(340)) is used to approximate the torque of the ISG in its driving mode[52]
QISG = Ts minus (N ∙ Ts divide NF) (340)2
Knowing the stall torque (the torque at 0 rpm) Ts and the maximum rpm of the motor when it is
not under load NF allows for the torque produced 119876119868119878119866 to be found for a given motor speed N
Experimental data from Litens Automotive Group [53] shows that for engine fire-up upon ISG
re-start the crankshaft must go from 0 rpm to an idle speed of approximately 750 rpm The
pulley installed on the ISG shaft in the case of the thesis research has a diameter of 6820 mm
(DISG) while that of the crankshaft has a diameter of 20065 mm (DCS) which makes the
2 The equation for the required driving torque for the ISG pulley may also be computed from the formula shown in
(336) Figure 315 for the driving torque of the ISG pulley shows that (336) and (340) produce similar results for
the required driving torque See Figure 315 for comparison of these results
Modeling of B-ISG 47
crankshaft to ISG pulley ratio approximately 2941 This ratio is used to determine the ISG
speed in equation (341)
nISG = nCS ∙DCS
DISG
(341)
For a crankshaft speed of 750 rpm the required ISG speed nISG is found from equation (341) to
be approximately 220656 rpm Thus the ISG torque during start-up is found from equation
(340) where N is equated to the value of nISG NF is assumed to be 18000 rpm and the stall
torque is allotted the value of 48 Nm The result is a required torque of approximately 42 Nm
for the ISG The acceleration of the ISG pulley is found by taking into account the torque
developed by the rotor and the polar moment of inertia of the pulley [54]
A1ISG = θ 1ISG = QISG IISG (342)
In torsional motion the function for angular displacement of input excitation is sinusoidal since
the electric motor is assumed to be resonating As a result of constant angular acceleration the
angular displacement of the ISG pulley in its driving mode is found in equation 343
θ1ISG = Θ1ISG ∙ sin(ωISG ∙ t) (343)
Knowing that acceleration is the second derivative of the displacement the amplitude of
displacement is solved subsequently [55]
Modeling of B-ISG 48
θ 1ISG = minusωISG2 ∙ Θ
1ISG ∙ sin(ωISG ∙ t) (344)
θ 1ISG = minusωISG2 ∙ Θ
1ISG
(345a)
Θ1ISG =minusQISG IISG
ωISG2
(345b)
In this case the angular frequency for the system 120596 is equivalent to 120596119868119878119866 that is the angular
frequency of the ISG pulley which can be expressed as a function of its speed in rpm
ω = ωISG =2 ∙ π ∙ nISG
60
(346a)
or in terms of the crankshaft rpm by substituting equation (341) into (346a)
ω =2 ∙ π
60∙ nCS ∙
DCS
DISG
(346b)
3416 Tensioner Arms Dynamic Torques
The dynamic torque for the tensioner arms are shown in equations (347) and (348)
Qt1 = kt + kt1 + 119895 ∙ ω ∙ (ct + ct1) ∙ (Θt1 ∙ Θ1) (347)
Modeling of B-ISG 49
Qt2 = kt + kt2 + 119895 ∙ ω ∙ (ct + ct2) ∙ (Θt2 ∙ Θ1) (348)
3417 Dynamic Belt Span Tensions
Furthermore the dynamic belt span tensions are derived from equation (314) and described in
matrix form in equations (349) and (350)
119827prime = 119895 ∙ ω ∙ 119827119836 + 119827119844 ∙ 120495119847 (349)
where
120495119847 = Θ1
120495119846 (350)
342 Static Model of the B-ISG System
It is fitting to pursue the derivation of the static model from the system using the dynamic EOM
For the system under static conditions equations (314) and (323) simplify to equations (351)
and (352) respectively
119827prime = 119827119844 ∙ 120521 (351)
119824 = 119818 ∙ 120521 (352)
Modeling of B-ISG 50
As noted in other chapters the focus of the B-ISG tensioner investigation especially for the
parametric and optimization studies in the subsequent chapters is to determine its effect on the
static belt span tensions Therein equations (351) and (352) are used to derive the expressions
for static tension in each belt span 119931prime is the tension solely due to deflection of the belt span
Equation (320) demonstrates the relationship between the tension due to belt response and the
initial tension also known as pre-tension The static tension 119931 is found by summing the initial
tension 1198790 with the expression for the dynamic tension shown in equations (315) through to
(319) and by substituting the expressions for the rigid bodies‟ displacements from equation
(352) and the relationship shown in equation (320) into equation (351)
119827 = 119827119844 ∙ (119818minus120783 ∙ 119824) + T0 (353)3
35 Simulations
The methods used to develop the geometric dynamic and static models of the Twin Tensioner B-
ISG system in the previous sections of this chapter were verified using the software FEAD The
input data for a single tensioner B-ISG system was entered into FEAD [51] to simulate the
crankshaft driving phase alone since the ISG phase is inapplicable in the FEAD [51] software
FEAD‟s [51] results agreed with those found in the simulation of the single tensioner system‟s
geometric model and EOMs in MathCAD software Furthermore the geometric simulation
3 For the purposes of the static tension the original order for the rows and columns of the stiffness matrix K and the
torque matrix Q are maintained as depicted in (326) and (328) In performing the inverse of K and its
multiplication with Q the first row and first column (in the case of the K matrix) are removed in the crankshaft
driving case whereas the fourth row and fourth column are removed in the ISG driving case Then the product for
the displacement120637 resulting from (119922minus120783 ∙ 119928) has a zero added to serve as the first element of the column matrix in
the crankshaft driving case or as the fourth element in the ISG driving case This is shown in detail in Appendix
C3 of MathCAD scripts
Modeling of B-ISG 51
results for both of the twin and single tensioner B-ISG systems were found to be in agreement as
well
351 Geometric Analysis
The initial coordinate inputs for the centre points of the five pulleys and the Twin Tensioner
pivot point are described as Cartesian coordinates and shown in Table 32 which also includes
the diameters for the pulleys
Table 32 Coordinate Points for Pulley Centres and Twin Tensioner Pivot [56]
Rigid Body Diameter [mm] Cartesian Coordinate [Xi Yi] [mm]
1Crankshaft Pulley 20065 [00]
2 Air Conditioner Pulley 10349 [224 -6395]
3 Tensioner Pulley 1 7240 [292761 87]
4 ISG Pulley 6820 [24759 16664]
5 Tensioner Pulley 2 7240 [12057 9193]
6 Tensioner Arm Pivot --- [201384 62516]
The geometric results for the B-ISG system are shown in Table 33
Table 33 Geometric Results of B-ISG System with Twin Tensioner
Pulley Forward
Connection Point
Backward
Connection Point
Wrap
Angle
ϕi (deg)
Angle of
Belt Span
βji (deg)
Length of
Belt Span
Li (mm)
1 Crankshaft [-6818-100093] [453889475] 202996 356103 227828
2 Air
Conditioning [275299-5717] [220484 -115575] 101425 277528 14064
3 Tensioner 1 [25887599735] [256873 82257] 28126 69403 58658
4 ISG [218374184225] [27951154644] 169554 58956 129513
5 Tensioner 2 [10419659645] [15158673262] 8585 333107 65949
Total Length of Belt (mm) 1243
Modeling of B-ISG 52
352 Dynamic Analysis
The dynamic results for the system include the natural frequencies mode shapes driven pulley
and tensioner arm responses the required torque for each driving pulley the dynamic torque for
each tensioner arm and the dynamic tension for each belt span These results for the model were
computed in equations (331a) through to (331d) for natural frequencies and mode shapes in
equation (334) for the driven pulley and tensioner arm responses in equation (336) for the
crankshaft pulley driving torque in equation (340) for the ISG pulley driving torque in
equations (347) and (348) for the tensioner arm torques and lastly in equation (349) for the
dynamic tension of each belt span Figures 36 through to 323 respectively display these
results The EOM simulations can also be contrasted with those of a similar system being a B-
ISG serpentine belt drive that is equipped with a single tensioner arm and single tensioner pulley
which interacts only in the span bridging the ISG and crankshaft pulleys The EOM for a B-ISG
with a single tensioner is presented in Appendix B
It is assumed for the sake of the dynamic and static computations that the system
does not have an isolator present on any pulley
has negligible rotational damping of the pulley shafts
has negligible belt span damping and that this damping does not differ amongst
spans (ie c1b = ∙∙∙ = ci
b = 0)
has quasi-static belt stretch where its belt experiences purely elastic deformation
has fixed axes for the pulley centres and tensioner pivot
has only one accessory pulley being modeled as an air conditioner pulley and
Modeling of B-ISG 53
has a rotational belt response that is decoupled from the transverse response of the
belt
The input parameter values of the dynamic (and static) computations as influenced by the above
assumptions for the present system equipped with a Twin Tensioner are shown in Table 34
Table 34 Data for Input Parameters used in Dynamic and Static Computations [56]
Rigid Body Data
Pulley Inertia
[kg∙mm2]
Damping
[N∙m∙srad]
Stiffness
[N∙mrad]
Required
Torque
[Nm]
Crankshaft 10 000 0 0 4
Air Conditioner 2 230 0 0 2
Tensioner 1 300 1x10-4
0 0
ISG 3000 0 0 5
Tensioner 2 300 1x10-4
0 0
Tensioner Arm 1 1500 1000 10314 0
Tensioner Arm 2 1500 1000 16502 0
Tensioner Arm
couple 1000 20626
Belt Data
Initial belt tension [N] To 300
Belt cord stiffness [Nmmmm] Kb 120 00000
Belt phase angle at zero frequency [deg] φ0deg 000
Belt transition frequency [Hz] ft 000
Belt maximum phase angle [deg] φmax 000
Belt factor [magnitude] kb 0500
Belt cord density [kgm3] ρ 1000
Belt cord cross-sectional area [mm2] A 693
Modeling of B-ISG 54
These values are for the driven cases for the ISG and crankshaft pulleys respectively In the
driving case for either pulley the inertia of the rigid body is defined as 1 kg∙mm2 and the driving
torque is determined in equations (335) and (340) for the crankshaft and ISG pulleys
respectively
It is noted that because of the belt data for the phase angle at zero frequency the transition
frequency and the maximum phase angle are all zero and hence the belt damping is assumed to
be constant between frequencies These three values are typically used to generate a phase angle
versus frequency curve for the belt where the phase angle is dependent on the frequency The
curve defined by equation (354) is normally symmetric with the lowest phase angle achieved at
0 Hz and the highest phase angle achieved at the prescribed transition frequency f The belt
damping would then be found by solving for cb in the following equation
tanφ = cb ∙ 2 ∙ π ∙ f (354)
Nevertheless the assumption for constant damping between frequencies is also in harmony with
the remaining assumptions which assume damping of the belt spans to be negligible and
constant between belt spans
3521 Natural Frequency and Mode Shape
The set of natural frequencies and mode shapes for the system are shown in Figures 36 and 37
under the cases of the ISG pulley driving and the crankshaft pulley driving The forcing
frequency for the system differs for each case due to the change in driving pulley Modeic and
Modeia denote the ith rigid body according to the numbering convention used in Figure 32 in
the crankshaft and ISG driving cases respectively
Modeling of B-ISG 55
Natural Frequency ωn [Hz]
Crankshaft Pulley ΔΘ4
Air Conditioner Pulley ΔΘ
Tensioner Pulley 1 ΔΘ
Tensioner Pulley 2 ΔΘ
Tensioner Arm 1 ΔΘ
Tensioner Arm 2 ΔΘ
Figure 36a ISG Driving Case Natural Frequency of System and Mode Shapes for Responsive
Rigid Bodies
Figure 36b ISG Driving Case First Mode Responses
4 ΔΘ signifies the term amplitude of angular displacement for the indicated rigid body
Modeling of B-ISG 56
Figure 36c ISG Driving Case Second Mode Responses
Natural Frequency ωn [Hz]
ISG Pulley ΔΘ5
Air Conditioner Pulley ΔΘ
Tensioner Pulley 1 ΔΘ
Tensioner Pulley 2 ΔΘ
Tensioner Arm 1 ΔΘ
Tensioner Arm 2 ΔΘ
Figure 37a Crankshaft Driving Case Natural Frequency of System and Mode Shapes for
Responsive Rigid Bodies
5 ΔΘ signifies the term amplitude of angular displacement for the indicated rigid body
Modeling of B-ISG 57
Figure 37b Crankshaft Driving Case First Mode Responses
Figure 37c Crankshaft Driving Case Second Mode Responses
Modeling of B-ISG 58
3522 Dynamic Response
The dynamic response specifically the magnitude of angular displacement for each rigid body is
plotted in Figures 38 through to 314 as a function of the crankshaft pulley speed n This is
fitting to the analysis since the crankshaft pulley‟s rpm decides the mode of operation for the
system in particular it determines whether the crankshaft pulley or ISG pulley is the driving
pulley
Figure 38 Crankshaft Pulley Dynamic Response (for crankshaft driven case)
Figure 39 ISG Pulley Dynamic Response (for ISG driven case)
Modeling of B-ISG 59
Figure 310 Air Conditioner Pulley Dynamic Response
Figure 311 Tensioner Pulley 1 Dynamic Response
Crankshaft Driving Phase ISG
Driving
Phase
Crankshaft Driving Phase ISG
Driving
Phase
Modeling of B-ISG 60
Figure 312 Tensioner Pulley 2 Dynamic Response
Figure 313 Tensioner Arm 1 Dynamic Response
Crankshaft Driving Phase ISG
Driving
Phase
Crankshaft Driving Phase ISG
Driving Phase
Modeling of B-ISG 61
Figure 314 Tensioner Arm 2 Dynamic Response
3523 ISG Pulley and Crankshaft Pulley Torque Requirement
Figures 315 and 316 respectively showcase the required torques for the ISG pulley in its driving
mode and the crankshaft pulley in its driving mode
Figure 315 Required Driving Torque for the ISG Pulley
Crankshaft Driving Phase ISG
Driving Phase
Modeling of B-ISG 62
Figure 315 shows two plots for the required driving torque of the ISG pulley The dashed line
labeled as Q(n) simulates the application of equation (340) which models the ISG torque as a
permanent magnet DC motor The additional solid line labeled as qamod uses the formula in
equation (336) which determines the load torque of the driving pulley based on the pulley
responses Figure 315 provides a comparison of the results
Figure 316 Required Driving Torque for the Crankshaft Pulley
3524 Tensioner Arms Torque Requirements
The torque for the tensioner arms are shown in Figures 317 and 318
Modeling of B-ISG 63
Figure 317 Dynamic Torque for Tensioner Arm 1
Figure 318 Dynamic Torque for Tensioner Arm 2
3525 Dynamic Belt Span Tension
The dynamic tensions for the belt spans are shown in Figures 319 through to 323 The values
plotted represent the magnitude of the dynamic tension
Crankshaft Driving Phase ISG
Driving Phase
Crankshaft Driving Phase ISG
Driving
Phase
Modeling of B-ISG 64
Figure 319 Span 1 (between crankshaft and air conditioner) Dynamic Belt Span Tension
Figure 320 Span 2 (between air conditioner and tensioner 1) Dynamic Belt Span Tension
Crankshaft Driving Phase ISG
Driving Phase
Crankshaft Driving Phase ISG
Driving
Phase
Modeling of B-ISG 65
Figure 321 Span 3 (between tensioner 1 and ISG) Dynamic Belt Span Tension
Figure 322 Span 4 (between ISG and tensioner 2) Dynamic Belt Span Tension
Crankshaft Driving Phase ISG
Driving
Phase
Crankshaft Driving Phase ISG
Driving Phase
Modeling of B-ISG 66
Figure 323 Span 5 (between tensioner 2 and crankshaft) Dynamic Belt Span Tension
The dynamic results for the system serve to show the conditions of the system for a set of input
parameters The following chapter targets the focus of the thesis research by analyzing the affect
of changing the input parameters on the static conditions of the system It is the static results that
are the focus of the thesis and is thus analyzed in Chapters 4 and 5 in the parametric and
optimization studies respectively The dynamic analysis has been used to complete the picture of
the system‟s state under set values for input parameters
353 Static Analysis
Before looking at the static results for the system under study in brevity the static results for a
B-ISG serpentine belt drive with a single tensioner are presented In this theoretical system the
tensioner arm and tensioner pulley that interacts with the span between the air conditioner and
ISG pulleys of the original system are removed as shown in Figure 324
Crankshaft Driving Phase ISG
Driving
Phase
Modeling of B-ISG 67
Figure 324 B-ISG Serpentine Belt Drive with Single Tensioner
The complete static model as well as the dynamic model for the system in Figure 324 is found
in Appendix B The results of the static tension for each belt span of the single tensioner system
when the crankshaft is driving and the ISG is driving are shown in Table 35
Table 35 Static Solution for Belt Span Tensions in Crankshaft and ISG Driving Cases for a B-
ISG Serpentine Belt Drive with a Single Tensioner
Belt Span between Pulleys Crankshaft Driving Case [N] ISG Driving Case [N]
Crankshaft ndash Air Conditioner 481239 -361076
Air Conditioner ndash ISG 442588 -399727
ISG ndash Tensioner 29596 316721
Tensioner ndash Crankshaft 29596 316721
The tensions in Table 35 are computed with an initial tension of 300N This value for pre-
tension allows the spans in the case that the crankshaft pulley is driving to be suitably tensioned
Modeling of B-ISG 68
Whereas in the case of the ISG pulley driving the first and second spans are excessively slack
Therein an additional pretension of approximately 400N would be required which would raise
the highest tension span to over 700N This leads to the motivation of the thesis researchndash to
reduce the static belt tensions when the ISG is driving As mentioned in Chapter 1 these
tensions should be minimized to prolong belt life preferably within the range of 600 to 800N
As well it is desirable to minimize the amount of pretension exerted on the belt The current
design uses a pre-tension of 300N The above results would lead to a required pre-tension of
more than 700N to keep all spans of the belt suitably in tension (well above 0N) in order to allow
the belt to exhibit high performance in power transmission and come near to the safe threshold
This is the rationale for investigating a Twin Tensioner configuration shown in Figure 32 for
the B-ISG serpentine belt drive under study For the theoretical system with a Twin Tensioner
the following static results in Table 36 are achieved
Table 36 Static Solution for Belt Span Tensions in Crankshaft and ISG Driving Cases for a B-
ISG Serpentine Belt Drive with a Twin Tensioner
Belt Span between Pulleys Crankshaft Driving Case [N] ISG Driving Case [N]
Crankshaft ndash Air Conditioner 465848 -284152
Air Conditioner ndash Tensioner 1 427197 -322803
Tensioner 1 ndash ISG 427197 -322803
ISG ndash Tensioner 2 28057 393645
Tensioner 2 ndash Crankshaft 28057 393645
The results in Table 36 show that the span following the ISG in the case between the Tensioner
1 and ISG pulleys is less slack than in the former single tensioner set-up However there
remains an excessive amount of pre-tension required to keep all spans suitably tensioned
Modeling of B-ISG 69
36 Summary
The simulation of the model for the B-ISG system with the Twin Tensioner shows that the mode
shapes of the rigid bodies within the system (Figures 36a to 37c) are greater in magnitude when
the ISG pulley is driving than when the crankshaft pulley is driving The dynamic responses of
the system as shown in Figures 38 and 310 to 314 is small for the crankshaft pulley and are
negligible for the remaining driven bodies when the ISG is driving For the crankshaft driving
phase there is greater dynamic response for the driven rigid bodies of the system including for
that of the ISG pulley
As the engine speed increases the torque requirement for the ISG was found to vary between
approximately 41Nm and 54Nm (before dropping steeply to approximately 3Nm at an engine
speed of about 720rpm) when modeled after equation (336) or between approximately 48Nm
and 34Nm when modeled after equation (340) In contrast the torque for the crankshaft peaks
at approximately 92Nm and 52Nm at an approximate engine speed of 1450rpm and 5000rpm
respectively The dynamic torque of the first tensioner arm was shown to peak at approximately
15Nm at the transition engine speed 750rpm and again at approximately 15Nm at an
approximate engine speed of about 1450rpm A small peak of about 3Nm was also seen at an
engine speed of 5000rpm Similarly for the second tensioner arm a torque peak of
approximately 20Nm was seen at 750rpm and 1450rpm and a smaller peak of about 8Nm was
seen at an engine speed of 5000rpm
The trend for the dynamic tensions is that the peaks are highest in the ISG driving portion of the
B-ISG operation in most cases and in a few cases they are seen to be close in magnitude to that
Modeling of B-ISG 70
of the highest peaks in the crankshaft driving portion The dynamic tension for the first belt span
peaked at approximately 780Nm 830Nm and 500Nm at engine speeds of 750rpm 1450rpm
5000rpm respectively For the dynamic tension of the second belt span peaks of approximately
1250Nm 675Nm and 760Nm were seen at the same respective engine speeds for the 3 peaks of
the former span At these same engine speeds the third belt span exhibited tension peaks at
approximately 1400Nm 650Nm and 890Nm The tension peaks of the fourth span were
approximately 165Nm 150Nm and 100Nm at engine speeds 750rpm 1450rpm and 5000rpm
The fifth span experienced peaks of approximately 165Nm 170Nm and 120Nm at the same
respective engine speeds of the fourth span
The simulation results for the static tension of the B-ISG system with the Twin Tensioner reveal
that taut spans of the crankshaft driving case are lower in the ISG driving case The largest
change is an approximate decrease of 750N in spans 1 through 3 while spans 4 and 5 increase
by approximately 113N It can be seen that the spans in highest tension (1 2 and 3) in the
crankshaft driving phase become excessively slack in the ISG driving phase There is a smaller
change between the tension values for the spans in the least tension in the crankshaft driving
phase and their corresponding span in the ISG driving phase
The summary of the simulation results are used as a benchmark for the optimized system shown
in Chapter 5 The static tension simulation results are investigated through a parametric study of
the Twin Tensioner system in Chapter 4 The optimization of the system is then based on the
selected design variables from the outcome of Chapter 4
71
CHAPTER 4 PARAMETRIC ANALYSIS OF A B-ISG
TWIN TENSIONER
41 Introduction
The parameters for the proposed Twin Tensioner for a Belt-driven Integrated Starter-generator
(B-ISG) system are investigated through a parametric analysis This analysis seeks to understand
how changing one parameter influences the static belt span tensions for the system Since the
thesis research focuses on the design of a tensioning mechanism to support static tension only
the parameters specific to the actual Twin Tensioner and applicable to the static case were
considered The parameters pertaining to accessory pulley properties such as radii or various
belt properties such as belt span stiffness are not considered In the analyses a single parameter
is varied over a prescribed range while all other parameters are held constant The pivot point
described by Cartesian Coordinates [X6Y6] for the tensioner arms is held constant in all cases
42 Methodology
The parametric study method applies to the general case of a function evaluated over changes in
one of its dependent variables The methodology is illustrated for the B-ISG system‟s function
for static tension which is evaluated for each change in one of its Twin Tensioner‟s parameters
The original data used for the system is based on sample vehicle data provided by Litens [56]
Table 41 provides the initial data for the parameters as well as the incremental change and
maxima and minima limits The increment Δi for the ith parameter is chosen arbitrarily Limits
for each parameter have been chosen to be plus or minus sixty percent of its initial value
Parametric Analysis 72
Table 41 Initial Values Increments and Ranges for Parameters of Twin Tensioner
Parameter Name Initial Value Increment (+- Δi) Minimum
value Maximum value
Coupled Spring
Stiffness kt
20626
N∙mrad 1238 N∙mrad 8250 N∙mrad 33002 N∙mrad
Tensioner Arm 1
Stiffness kt1
10314
N∙mrad 0619 N∙mrad 4126 N∙mrad 16502 N∙mrad
Tensioner Arm 2
Stiffness kt2
16502
N∙mrad 0990 N∙mrad 6601 N∙mrad 26403 N∙mrad
Tensioner Pulley 1
Diameter D3 007240 m 4344 ∙ 10
-3 m 00290 m 0116 m
Tensioner Pulley 2
Diameter D5 007240 m 4344 ∙ 10
-3 m 00290 m 0116 m
Tensioner Pulley 1
Initial Coordinates
[0292761
0087] m See Figure 41 for region of possible tensioner pulley
coordinates Tensioner Pulley 2
Initial Coordinates
[012057
009193] m
The mesh of possible points for the centre coordinates of tensioner pulley 1 and tensioner pulley
2 are designated as Region 1 and Region 2 respectively in Figures 41a and 41b
Figure 41a Regions 1 and 2 and their Associated Guidelines for Coordinates of Tensioner
Pulleys 1 amp 2
CS
AC
ISG
Ten 1
Ten 11
Region II
Region I
Parametric Analysis 73
Figure 41b Regions 1 and 2 in Cartesian Space
The selection for the minimum and maximum tensioner pulley centre coordinates and their
increments are not selected arbitrarily or without derivation as the other tensioner parameters
The coordinates for the pulley centres are identified using Intergraph‟s SmartSketch software a
graphing suite in MathCAD to model the regions of space The following descriptions are used
to describe the possible positions for the tensioner pulleys
Tensioner pulleys are situated such that they are exterior to the interior space created by
the serpentine belt thus they sit bdquooutside‟ the belt loop
The highest point on the tensioner pulley does not exceed the tangent line connecting the
upper hemispheres of the pulleys on either side of it
The tensioner pulleys may not overlap any other pulley
Parametric Analysis 74
Boundaries for regions described as Region 1 in span 2 and 3 and Region 2 in span 4
and 5 is selected based on the above criteria and their lower boundaries are selected
arbitrarily
These criteria were used to define the equation for each boundary line and leads to a set of
Boolean conditions that relate the x-coordinate and y-coordinate for each Cartesian pair The
density for the mesh of points in each region is arbitrarily selected as 101 x-points and 101 y-
points in each space for the purposes of the parametric analysis The outline of this method is
described in the MATLAB scripts contained in Appendix D
The results of the parametric analysis are shown for the slackest and tautest spans in each driving
case As was demonstrated in the literature review the tautest span immediately precedes the
driving pulley and the slackest span immediately follows the driving pulley in the direction of
the belt motion Thus in the case for the crankshaft driving the tautest span is in the first span
and the slackest span is in the fifth span Whereas in the ISG driving case the tautest span is in
the fourth span and the slackest span is in the third span Hence the parametric figures in this
chapter display only the tautest and slackest span values for both driving cases so as to describe
the maximum and minimum values for tension present in the given belt
43 Results amp Discussion
431 Influence of Tensioner Arm Stiffness on Static Tension
The parametric analysis begins with changing the stiffness value for the coil spring coupled
between tensioner arms 1 and 2 This stiffness value kt is changed over a range from sixty
percent less than its initial value kt0 to sixty percent more than its original value as shown in
Parametric Analysis 75
Table 41 The results of the static tension are shown in Figure 42 for the tautest and slackest
spans for both the crankshaft and ISG driving cases
Figure 42 Parametric Analysis for Coupled Stiffness Arm Constant kt (N∙mrad)
As kt increases in the crankshaft driving phase for the B-ISG system the highest tension
decreases from 4691N to 4646N while the lowest tension decreases from 2838N to 2793N
In the ISG driving phase the highest tension increases from 378N to 3998N and the lowest
tension increases from -3384N to -3167N Thus a change of approximately -45N is found in
the crankshaft driving case and approximately +22N is found in the ISG driving case for both the
tautest and slackest spans
Parametric Analysis 76
The second parameter analyzed is the stiffness value for tensioner arm 1 The results of this are
shown in Figure 43
Figure 43 Parametric Analysis for Stiffness of Arm 1 kt1 (N∙mrad)
In Figure 43 as kt1 increases an increase from 4628N to 4681N is observed for the tension of
the tautest span when the crankshaft is driving which is a change of +53N The same value for
net change is found in the slackest span for the same driving condition whose tension increases
from 2775N to 2828N For the case when the B-ISG system is in the ISG driving phase the
change is larger a value of -261N for the tautest span that changes from 4088N to 3827N and
for the slackest span that changes from -3077N to -3338N
Parametric Analysis 77
The change in static tension for the spans as the stiffness of arm 2 varies is demonstrated in
Figure 44
Figure 44 Parametric Analysis for Stiffness of Arm 2 kt2 (N∙mrad)
In this case it is observed that as kt2 increases the tautest span for the B-ISG system in the
crankshaft driving case decreases from 4675N to 4643N as well as the slackest span which
decreases from 2822N to 279N which is an overall change of -32N for both spans Whereas in
the ISG driving case a more noticeable change is once again found a difference of +144N
This is a result of the tautest span increasing from 3863N to 4007N and the slackest span
increasing from -3301N to -3157N
Parametric Analysis 78
432 Influence of Tensioner Pulley Diameter on Static Tension
The change in the diameter of tensioner pulley 1 D3 and its effect on static tension is shown in
Figure 45
Figure 45 Parametric Analysis for Pulley 1 Diameter D3 (m)
The change in the tautest and slackest spans for the B-ISG system‟s crankshaft driving case is
from 3248N to 425N and from 1395N to 240N respectively Peaks are seen at 4799N and
2946N for the respective spans This is a change of approximately +100N and a maximum
change of 1551N for both spans For the ISG driving case the tautest and slackest spans
decrease from 1083N to 6158N and 367N to -1006N Global minimums of 3246N and -391N
for the respective spans are seen This nets a change of approximately -467N and a maximum
change of approximately -759N
Parametric Analysis 79
The effect of changing the diameter of tensioner pulley 2 on the static tension is examined in
Figure 46
Figure 46 Parametric Analysis for Pulley 2 Diameter D5 (m)
The tautest and slackest spans in the crankshaft driving mode of the belt undergo a change from
4583N to 4721N and from 273N to 2869N respectively Therein as D5 increases the trend is
that for both spans there is an increase in tension of approximately 14N Contrastingly the spans
experience a decrease in the ISG driving case as D5 increases The tension of the tautest span
goes from 4296N to 3635N and that of the slackest span goes from -2866N to -3529N This
equals a decrease of approximately 66N for both spans
Parametric Analysis 80
433 Influence of Tensioner Pulley 1 Coordinates on Static Tension
The influence of the coordinates of tensioner pulley 1 on the value of tension in the tautest span
for the B-ISG system‟s crankshaft driving case is demonstrated in Figure 47
Figure 47 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Tautest Span
Tension in Crankshaft Driving Case
The region shown in Figure 47 corresponds to region 1 which is the realm of the positions for
tensioner pulley 1 The possible pulley coordinates in this case are represented by the non-blue
area reaching to the perimeter of the plot It is evident in the darkest red region of the plot
where the y-coordinate is between approximately 0m and 0075m and the x-coordinate is
(N)
Parametric Analysis 81
between approximately 026m and 031m that the highest value of tension is experienced in the
tautest span for the crankshaft driving case The range of tension for Region 1 in the tautest span
when the crankshaft is driving is between a maximum of approximately 500N and a minimum of
approximately 300N This equals an overall difference of 200N in tension for the tautest span by
moving the position of pulley 1 The lowest values for tension are obtained when the pulley
coordinates are approximately -0025m to 015m for the y-coordinate and approximately 031m
to 032m for the x-coordinate which corresponds to the yellow region An area of low tension is
also seen in the area where the y-coordinate is approximately 0m and the x-coordinate is
approximately between 026m and 027m
The changes in tension for the slackest span under the condition of the crankshaft pulley being
the driving pulley are shown in Figure 48
Parametric Analysis 82
Figure 48 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Slackest Span
Tension in Crankshaft Driving Case
Once again the possible coordinate points for tensioner pulley 1 in the B-ISG system are
represented by the non-blue region For the slackest span in the crankshaft driving case it is seen
that the lowest tension is approximately 125N while the highest tension is approximately 325N
This is an overall change of 200N that is achieved in the region The highest values are achieved
in the space where the y-coordinates are approximately 0m to 0075m and the x-coordinate
ranges from 026m to 031m which corresponds to the deep red region The lowest tension
values are achieved in the space where the y-coordinate ranges from approximately -0025m to
015m and the x-coordinate ranges from 031m to 032m which corresponds to the light blue-
green region of the plot The area containing a y-coordinate of approximately 0m and x-
(N)
Parametric Analysis 83
coordinates that are approximately between 026m and 027m also show minimum tension for
the slack span The regions of the x-y coordinates for the maximum and minimum tensions are
alike to the tautest span in Region 1 for the crankshaft driving case as well as was seen in Figure
47
The tension for the tautest span in the case that the ISG is driving in the B-ISG system is found
in Figure 49
Figure 49 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Tautest Span
Tension in ISG Driving Case
(N)
Parametric Analysis 84
Region 1 is represented by the coordinate values shown in the non-dark blue space of the plot in
Figure 49 The tautest span in the case of the ISG driving experiences a range of tension values
in Region 1 from 200N up to 1100N equaling a difference of 900N The minimum tension
values are achieved in the medium to light blue region This includes y-coordinates of
approximately 0m to 0075m and x-coordinates of approximately 026m to 03m The
maximum tension values are in the darkest red area inclusive of y-coordinates -0025m to 015m
and x-coordinates 031m to 032m in addition to y-coordinate of approximately 0m and x-
coordinates of approximately 026m to 027m It can be observed that aforementioned regions
for minimum and maximum tensions in Figure 49 are reverse to those seen in Figures 47 and
48 for the crankshaft driving case
The change in tension for the slackest span of the B-ISG system when it is driven by the ISG is
shown in Figure 410
Parametric Analysis 85
Figure 410 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Slackest Span
Tension in ISG Driving Case
Figure 410 exhibits the realm of possible points for tensioner pulley 1 for the case of the ISG
driving in the non-yellow-green area The minimum tension values are achieved in the darkest
blue area where the minimum tension is approximately -500N This area corresponds to y-
coordinates from approximately 0m to 005m and x-coordinates from approximately 026m to
03m The area of a maximum tension is approximately 400N and corresponds to the darkest red
area inclusive of y-coordinates -0025m to 015m and x-coordinates 031m to 032m as well as
the coordinates for y equaling approximately 0m and for x equaling approximately 026m to
027m The difference between maximum and minimum tensions in this case is approximately
900N It is noticed once again that the space of x- and y-coordinates containing the maximum
(N)
Parametric Analysis 86
tension is in the similar location to that of the described space for minimum tension in the
crankshaft driving case in Figure 47 and 48
434 Influence of Tensioner Pulley 2 Coordinates on Static Tension
The influence of pulley 2 coordinates on the tension value for the tautest span when the
crankshaft is driving the B-ISG system is shown in Figure 411 and is represented by the values
corresponding to the non-blue area
Figure 411 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Tautest Span
Tension in Crankshaft Driving Case
In Figure 411 the possible coordinates are contained within Region 2 The maximum tension
value is approximately 500N and is found in the darkest red space including approximately y-
(N)
Parametric Analysis 87
coordinates 004m to 014m and x-coordinates 0025m to 0175m and also y-coordinates 013m
to 02m corresponding to the x-coordinate at 0175m A minimum tension value of
approximately 350N is found in the yellow space and includes approximately y-coordinates
008m to 018m and x-coordinates 016m to 02m The difference in tension values is 150N
The analysis of the change in coordinates for tension pulley 2 on the value for tension in the
slackest span is shown in Figure 412 in the non-blue region
Figure 412 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Slackest Span
Tension in Crankshaft Driving Case
The value of 325N is the highest tension for the slack span in the crankshaft driving case of the
B-ISG system and is found in the deep-red region where the y-coordinates are between
(N)
Parametric Analysis 88
approximately 004m and 013m and the x-coordinates are approximately between 0025m and
016m as well as where y is between 013m and 02m and x is approximately 0175m The
lowest tension value for the slack span is approximately 150N and is found in the green-blue
space where y-coordinates are between approximately 01m and 022m and the x-coordinates
are between approximately 016m and 021m The overall difference in minimum and maximum
tension values is 175N The spaces for the maximum and minimum tension values are similar in
location to that found in Figure 411 for the tautest span in the crankshaft driving case
Figure 413 provides the theoretical data for the tension values of the tautest span as the position
of the B-ISG system‟s tensioner pulley 2 changes in the ISG driving case Possible points are in
the space of values which correspond to the non-dark-blue region in Figure 413
Parametric Analysis 89
Figure 413 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Tautest Span
Tension in ISG Driving Case
In Figure 413 the region for high tension reaches a value of approximately 950N and the region
for low tension reaches approximately 250N This equals a difference of 700N between
maximum and minimum tension values for the tautest span in the B-ISG system‟s ISG driving
case The coordinate points within the space that maximum tension is reached is in the dark red
region and includes y-coordinates from approximately 008m to 022m and x-coordinates from
approximately 016m to 021m The coordinate points within the space that minimum tension is
reached is in the blue-green region and includes y-coordinates from approximately 004m to
013m and the corresponding x-coordinates from approximately 0025m to 015m An additional
small region of minimum tension is seen in the area where the x-coordinate is approximately
(N)
Parametric Analysis 90
0175m and the y-coordinates are approximately between 013m and 02m The location for the
area of pulley centre points that achieve maximum and minimum tension values is approximately
located in the reverse positions on the plot when compared to that of the case for the crankshaft
driving in Figures 411 and 412 Therein the trend seen for pulley coordinates for the second
tensioner pulley follows suit with that of the first tensioner pulley which is that the area of points
for maximum tension in the crankshaft driving case becomes the approximate area of points for
minimum tension in the ISG driving case and vice versa
In Figure 414 the results of the parametric analysis on the coordinates of the second tensioner
pulley and its effect on the slackest span‟s tension in the ISG driving case is shown Similar to
earlier figures the non-dark yellow region represents Region 2 that contains the possible points
for the pulley‟s Cartesian coordinates
Parametric Analysis 91
Figure 414 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Slackest
Span Tension in ISG Driving Case
Figure 414 demonstrates a difference of approximately 725N between the highest and lowest
tension values for the slackest span of the B-ISG system in the ISG driving case The highest
tension values are approximately 225N The area of points that allow the second tension pulley
to achieve maximum tension in the belt span includes y-coordinates from approximately 01m to
022m and the corresponding x-coordinates from approximately 016m to 021m This
corresponds to the darkest red region in Figure 414 The coordinate values where the lowest
tension being approximately -500N is achieved include y-coordinate values from
approximately 004m to 013m and the corresponding x-coordinates from approximately 0025m
to 015m corresponding to the darkest blue region A dark blue region of lowest tension is also
(N)
Parametric Analysis 92
seen in the area where y is approximately between 013m and 02m and the x-coordinate is
approximately 0175m The regions for maximum and minimum tension values are observed to
be similar to those found in Figure 413 and alike to Figure 413 to be in reverse to those found
in Figure 411 and 412 for the tautest and slackest spans in the crankshaft driving case So as for
the changes in tensioner pulley 2 coordinates the areas for minimum tension in Region 2 of the
ISG driving case are similar to the areas for maximum tension in Region 2 of the crankshaft
driving case and vice versa for the maximum tension of the ISG driving case and the minimum
tension for the crankshaft driving case in Region 2
44 Conclusion
Overall the trend in the plots of Figures 47 48 411 and 412 indicate in the crankshaft driving
portion that the B-ISG system‟s belt span tensions experience the following effect
Minimum tension for the tautest span is achieved when tensioner pulley 1 centre
coordinates are located closer to the right side boundary and bottom left boundary of
Region 1 or when tensioner pulley 2 centre coordinates are within the upper right space
(near to the ISG pulley) and the space closer to the top boundary of Region 2
Maximum tension for the slackest span is achieved when the first tensioner pulley‟s
coordinates are located in the mid space and near to the bottom boundary of Region 1
and when the second tensioner pulley‟s coordinates are located near to the bottom left
boundary of Region 2 which is the boundary nearest to the crankshaft pulley
Parametric Analysis 93
The trend for minimizing the tautest span signifies that the tension for the slackest span is also
minimized at the same time As well maximizing the slackest span signifies that the tension for
the tautest span is also maximized at the same time too
The trend for the B-ISG system‟s ISG driving case as can be seen in Figures 49 410 413 and
414 is approximately in reverse to that of the crankshaft driving case for the system Wherein
points corresponding to minimum tension in Regions 1 and 2 in the ISG case are approximately
the same as points corresponding to maximum tension in the Regions for the crankshaft case and
vice versa for the ISG cases‟ areas of maximum tension
Minimum tension for the tautest span is present when the first tensioner pulley‟s
coordinates are near to mid to lower boundary of Region 1 and when the second
tensioner pulley‟s coordinates are close to the bottom left boundary of Region 2 which
is the furthest boundary from the ISG pulley and closest to the crankshaft pulley
Maximum tension for the slackest span is achieved when the first tensioner pulley is
located close to the right boundary of Region 1 and when the second tensioner pulley is
located near the right boundary and towards the top right boundary of Region 2
It is observed in Figures 47 to 414 and alike to Figures 42 to 46 the tautest and slackest
spans decrease or increase together Thus it can be assumed that the tension values in these
spans and likely the remaining spans outside of the tautest and slackest spans follow suit
Therein when parameters are changed to minimize one belt span‟s tension the remaining spans
will also have their tension values reduced Figures 42 through to 413 showed this clearly
where the overall change in the tension of the tautest and slackest spans changed by
Parametric Analysis 94
approximately the same values for each separate case of the crankshaft driving and the ISG
driving in the B-ISG system
Design variables are selected in the following chapter from the parameters that have been
analyzed in the present chapter The influence of changing parameters on the static tension
values for the various spans is further explored through an optimization study of the static belt
tension for the B-ISG system equipped with a Twin Tensioner in the following chapter Chapter
5
95
CHAPTER 5 OPTIMIZATION OF A B-ISG TENSIONER
The objective of the optimization analysis is to minimize the absolute magnitude of the static
tension in the ISG-operating mode of the serpentine belt drive The optimization seeks to
optimize the performance of the proposed Twin Tensioner design by using its properties as the
design variables for the objective function The optimization task begins with the selection of
these design variables for the objective function and then the selection of an optimization
method The results of the optimization will be compared with the results of the analytical
model for the static system and with the parametric analysis‟ data
51 Optimization Problem
511 Selection of Design Variables
The optimal system corresponds to the properties of the Twin Tensioner that result in minimized
magnitudes of static tension for the various belt spans Therein the design variables for the
optimization procedure are selected from amongst the Twin Tensioner‟s properties In the
parametric analysis of Chapter 4 the tensioner properties presented included
coupled stiffness kt
tensioner arm 1 stiffness kt1
tensioner arm 2 stiffness kt2
tensioner pulley 1 diameter D3
tensioner pulley 2 diameter D5
tensioner pulley 1 initial coordinates [X3Y3] and
Optimization 96
tensioner pulley 2 initial coordinates [X5Y5]
It was observed in the former chapter that perturbations of the stiffness and geometric parameters
caused a change between the lowest and highest values for the static tension especially in the
case of perturbations in the geometric parameters diameter and coordinates Table 51
summarizes the observed changes in the belt span tensions corresponding to the Twin Tensioner
parameters‟ maximum and minimum values
Table 51 Summary of Parametric Analysis Data for Twin Tensioner Properties
Parameter Symbol
Original Tensions in TautSlack Span (Crankshaft
Mode) [N]
Tension at
Min | Max Parameter6 for
Crankshaft Mode [N]
Percent Change from Original for
Min | Max Tensions []
Original Tension in TautSlack Span (ISG Mode)
[N]
Tension at
Min | Max Parameter Value in ISG Mode [N]
Percent Change from Original Tension for
Min | Max Tensions []
kt
465848 (taut) 4691 4646 07 -03 393645 (taut) 378 3998 -40 16
28057 (slack) 2838 2793 12 -05 -322803 (slack) -3384 -3167 -48 19
kt1
465848 (taut) 4628 4681 -07 05 393645 (taut) 4088 3827 38 -28
28057 (slack) 2775 2828 -11 08 -322803 (slack) -3077 -3338 47 -34
kt2
465848 (taut) 4675 4643 04 -03 393645 (taut) 3863 4007 -19 18
28057 (slack) 2822 279 06 -06 -322803 (slack) -3301 -3157 -23 22
D3 465848 (taut) 3248 425 -303 -88 393645 (taut) 1083 6158 1751 564
28057 (slack) 1395 240 -503 -145 -322803 (slack) 367 -1006 2137 688
D5 465848 (taut) 4583 4721 -16 13 393645 (taut) 4296 3635 91 -77
28057 (slack) 273 2869 -27 23 -322803 (slack) -2866 -3529 112 -93
[X3Y3] 465848 (taut) 300 500 -356 73 393645 (taut) 200 1100 -492 1794
28057 (slack) 125 325 -554 158 -322803 (slack) -500 400 -549 2239
6 The values for the tension for each of the taut and slack spans provided correspond to the minimum and maximum
values of the parameter listed in each case such that the columns of identical colour correspond to each other For
the coordinate parameters the minimum and maximum parameter value is inadmissible The tension values in these
cases are simply the minimum and maximum tension values achieved by the coordinate parameter listed
Optimization 97
[X5Y5] 465848 (taut) 350 500 -249 73 393645 (taut) 250 950 -365 1413
28057 (slack) 150 325 -465 158 -322803 (slack) -500 225 -549 1697
The results of the parametric analyses for the Twin Tensioner parameters show that there is a
noticeable percent change between the initial tensions and the tensions corresponding to each of
the minima and maxima parameter values or in the case of the coordinates between the
minimum and maximum tensions for the spans Thus the parametric data does not encourage
exclusion of any of the tensioner parameters from being selected as a design variable As a
theoretical experiment the optimization procedure seeks to find feasible physical solutions
Hence economic criteria are considered in the selection of the design variables from among the
Twin Tensioner‟s parameters Of the tensioner properties it is found that the diameter of the
tensioner pulleys has the largest impact on cost Adding mass to a tensioner pulley as a result of
increasing the diameter and consequently its inertia increases the cost of material Material cost
is most significant in the manufacture process of pulleys as their manufacturing is largely
automated [4] Furthermore varying the structure of a pulley requires retooling which also
increases the cost to manufacture As such the tensioner pulley diameters D3 and D5 are
excluded from being selected as design variables The remaining tensioner properties the
stiffness parameters and the initial coordinates of the pulley centres are selected as the design
variables for the objective function of the optimization process
512 Objective Function amp Constraints
In order to deal with two objective functions for a taut span and a slack span a weighted
approach was employed This emerges from the results of Chapter 3 for the static model and
Chapter 4 for the parametric study for the static system which show that a high tension span and
Optimization 98
a highly slack span exist in the ISG-driving phase of the B-ISG system Therein the first
objective function of equation (51a) is described as equaling fifty percent of the absolute tension
value of the tautest span and fifty percent of the absolute tension value of the slackest span for
the case of the ISG driving only The second objective function uses a non-weighted approach
and is described as the absolute tension of the slackest span when the ISG is driving A non-
weighted approach is motivated by the phenomenon of a fixed difference that is seen between
the slackest and tautest spans of the optimal designs found in the weighted optimization
simulations Equations (51a) through to (51c) display the objective functions
The limits for the design variables are expanded from those used in the parametric analysis for
the non-coordinate parameters kt kt1 and kt2 so that they are permitted to vary from
approximately 0 to approximately 200 of the initial value for each parameter kt0 kt10 and kt20
respectively In the case of the coordinate parameters [X3Y3] and [X5Y5] the x- and y-
coordinates are permitted to vary within the spaces Region 1 and Region 2 respectively which
were prescribed in Chapter 4 Figure 41a and 41b
Aside from the design variables design constraints on the system include the requirement for
static stability of the Twin Tensioner An optimal solution for the B-ISG system must achieve
the goal of the objective function which is to minimize the absolute tensions in the system
However for an optimal solution to be feasible the movement of the tensioner arm must remain
within an appropriate threshold In practice an automotive tensioner arm for the belt
transmission may be considered stable if its movement remains within a 10 degree range of
Optimization 99
motion [4] As such the angle of displacement for tensioner arms 1 and 2 are designated by θ t1
and θt2 respectively in the listed constraints
The optimization task is described in equations 51a to 52 Variables a through to g represent
scalar limits for the x-coordinate for corresponding ranges of the y-coordinate
Minimize 119879119908119890119894119892 119893119905119890119889 = 05 ∙ 119879119905119886119906119905 + 05 ∙ 119879119904119897119886119888119896
or119879119899119900119899 minus119908119890119894119892 119893119905119890119889 = 119879119904119897119886119888119896
(51a)
where
119879119905119886119906119905 = 119891119905119886119906119905 119896119905 1198961199051 1198961199052 1198833 1198843 1198835 1198845 (51b)
119879119904119897119886119888119896 = 119891119904119897119886119888119896 (119896119905 1198961199051 1198961199052 1198833 119884311988351198845) (51c)
Subject to
(1198961199050 minus 1 ∙ 1198961199050) le 119896119905 le (1198961199050 + 11198961199050)(11989611990510 minus 1 ∙ 11989611990510) le 1198961199051 le (11989611990510 + 111989611990510)(11989611990520 minus 1 ∙ 11989611990520) le 1198961199052 le (11989611990520 + 111989611990520)
119886 le 1198833 le 119888
1198931 1198833 le 1198843 le 1198933 1198833 119891119900119903 119886 le 1198833 lt 119887
1198932 1198833 le 1198843 le 1198933 1198833 119891119900119903 119887 le 1198833 le 119888119889 le 1198835 le 119892
1198934 1198835 le 1198845 le 1198937 1198835 for 119889 le 1198835 lt 1198901198935(1198835) le 1198845 le 1198937(1198835) for 119890 le 1198835 lt 119891
1198936 1198835 le 1198845 le 1198937 1198833 for 119891 le 1198833 le 119892 1205791199051 le 10deg 1205791199052 le 10deg
(52)
The functions for the taut and slack spans represent the fourth and third span respectively in the
case of the ISG driving The equations for the tensions of the aforementioned spans are shown
in equation 51a to 51c and are derived from equation 353 The constraints for the
optimization are described in equation 52
Optimization 100
52 Optimization Method
A twofold approach was used in the optimization method A global search alone and then a
hybrid search comprising of a global search and a local search The Genetic Algorithm is used
as the global search method and a Quadratic Sequential Programming algorithm is used for the
local search method
521 Genetic Algorithm
Genetic Algorithm (GA) is a part of the growing genre of evolutionary algorithms [57] The
optimization approach differs from classical search approaches by its ease of use and global
perspective [57] GA mimics biological evolution theory by using a ldquocross-over of heritable
information random mutation and selection on the basis of fitness between generationsrdquo [58] to
form a robust search algorithm that requires minimal problem information [57] The parameter
sets are represented as sample points modeled as bdquochromosomes‟ or data strings that are
measured against how well they allow the model to achieve the optimization task [58] GA is
stochastic which means that its algorithm uses random choices to generate subsequent sampling
points rather than using a set rule to generate the following sample This avoids the pitfall of
gradient-based techniques that may focus on local maxima or minima and end-up neglecting
regions containing higher peaks or lower valleys [57] Furthermore due to the randomness of
the GA‟s search strategy it is able to search a population (a region of possible parameter sets)
faster than other optimization techniques The GA approach is viewed as a universal
optimization approach while many classical methods viewed to be efficient for one optimization
problem may be seen as inefficient for others However because GA is a probabilistic algorithm
its solution for the objective function may only be near to a global optimum As such the current
Optimization 101
state of stochastic or global optimization methods has been to refine results of the GA with a
local search and optimization procedure
522 Hybrid Optimization Algorithm
In order to enhance the result of the objective function found by the GA a Hybrid optimization
function is implemented in MATLAB software The Hybrid optimization function combines a
global search GA with a local search Sequential Quadratic Programming (SQP) The hybrid
process refines the value of the objective function found through GA by using the final set of
points found by the algorithm as the initial point of the SQP algorithm The GA function
determines the region containing a global optimum and then the SQP algorithm uses a gradient
based technique to find a solution closer to the global optimum The MATLAB algorithm a
constrained minimization function known as fmincon uses an SQP method that approximates the
Hessian for the Lagrangian function (ie the second derivatives of the Lagrangian) by way of a
quasi-Newton approach to generate a quadratic program (QP) sub-problem [59] The solution
for the QP provides the search direction of the line search procedure used when each iteration is
performed [59]
53 Results and Discussion
531 Parameter Settings amp Stopping Criteria for Simulations
The parameter settings for the optimization procedure included setting the stall time limit to
200s This is the interval of time the GA is given to find an improvement in the value of the
objective function This is an increase from MATLAB‟s default of 20s Increasing the stall time
limit allows for the optimization search to consistently converge without being limited by time
Optimization 102
The population size used in finding the optimal solution is set at 100 This value was chosen
after varying the population size between 50 and 2000 showed no change in the value of the
objective function The max number of generations is set at 100 The time limit for the
algorithm is set at infinite The limiting factor serving as the stopping condition for the
optimization search was the function tolerance which is set at 1x10-6
This allows the program
to run until the ratio of the change in the objective function over the stall generations is less than
the value for function tolerance The stall generation setting is defined as the number of
generations since the last improvement of the objective function value and is limited to 50
532 Optimization Simulations
The results of the genetic algorithm optimization simulations performed in MATLAB are shown
in the following tables Table 52a and Table 52b
Table 52a GA Optimization Results for Twin Tensioner Parameters and Objective Function
Trial
No
Genetic Algorithm Optimization Method
Objective
Function
Value [N]
kt [N∙mrad]
kt1 [N∙mrad]
kt2 [N∙mrad]
[X3Y3]
[m] [X5Y5]
[m]
1 3582241 314069 204844 165020 [02928 00703] [01618 01036]
2 3582241 103646 205284 198901 [03009 00607] [01283 00809]
3 3582241 126431 204740 43549 [03010 00631] [01311 01147]
4 3582241 180285 206230 254870 [03095 00865] [01080 01675]
5 3582241 74757 204559 189077 [03084 00617] [01265 00718]
Original Design 20626 10314 16502 [02928 0087] [01206 00919]
Optimization 103
Table 52b Computations for Tensions and Angles from GA Optimization Results
Trial No
Genetic Algorithm Optimization Method
Slackest Tension [N] Tautest Tension [N] Tensioner Arm 1
Displacement [deg]
Tensioner Arm 2
Displacement [deg]
T3 T4 Θt1 Θt2
1 -1572307 5592176 -00025 -49748
2 -4054309 3110174 -00002 -20213
3 -3930858 3233624 -00004 -38370
4 -1309751 5854731 -00010 -49525
5 -4092446 3072036 -00000 -17703
Original Design -322803 393645 16410 -4571
For each trial above the GA function required 4 generations each consisting of 20 900 function
evaluations before finding no change in the optimal objective function value according to
stopping conditions
The results of the Hybrid function optimization are provided in Tables 53a and 53b below
Table 53a Hybrid Optimization Results for Twin Tensioner Parameters and Objective Function
Trial
No
Hybrid Optimization Method
Objective
Function
Value [N]
of
Function
Evals ( of
Iterations)
kt [N∙mrad]
kt1 [N∙mrad]
kt2 [N∙mrad]
[X3Y3]
[m] [X5Y5]
[m]
1 3582241 16 (1) 16065 205846 229494 [02780 00581] [01679 01288]
2 3582241 20 (1) 249227 205635 25218 [02901 00634] [01559 00870]
3 3582241 16 (1) 297295 204878 320479 [02962 00702] [01336 01447]
4 3582241 53 (1) 241433 204262 229683 [02912 00647] [00047 01465]
Optimization 104
5 3582241 21 (1) 379096 205548 188888 [02973 00703] [01206 01376]
Original Design 20626 10314 16502 [02928 0087] [01206 00919]
Table 53b Computations for Tensions and Angles from Hybrid Optimization Results
Trial No
Hybrid Algorithm Optimization Method
Slackest Tension [N] Tautest Tension [N] Tensioner Arm 1
Displacement [deg]
Tensioner Arm 2
Displacement [deg]
T3 T4 Θt1 Θt2
1 -2584641 4579841 -02430 67549
2 -3708747 3455736 -00023 -41068
3 -1707181 5457302 -00099 -43944
4 -269178 6895304 00006 -25366
5 -2982335 4182148 -00003 -41134
Original Design -322803 393645 16410 -4571
In Table 53a it can be seen that iterations of 16 20 21 or 53 were required for the local search
algorithm following the GA to find an optimal solution Once again the GA function
computed 4 generations which consisted of approximately 20 900 function evaluations before
securing an optimum solution
The simulation results of the non-weighted hybrid optimization approach are shown in tables
54a and 54b below
Optimization 105
Table 54a Non-Weighted Optimization Results for Twin Tensioner Parameters and Objective
Function
Trial
No
Objective
Function
Value [N]
of
Function
Evals ( of
Iterations)
kt [N∙mrad]
kt1 [N∙mrad]
kt2 [N∙mrad]
[X3Y3]
[m] [X5Y5]
[m]
Genetic Algorithm Optimization Method
1 33509e
-004 20900 (4) 321799 75530 212653 [02860 00602] [01082 01858]
Hybrid Optimization Method
1 73214e
-011 381 (13) 234881 14730 323358 [02952 00688] [00048 01466]
Original Design 20626 10314 16502 [02928 0087] [01206 00919]
Table 54b Computations for Tensions and Angles from Non-Weighted Optimizations
Trial No Slackest Tension [N] Tautest Tension [N]
Tensioner Arm 1
Displacement [deg]
Tensioner Arm 2
Displacement [deg]
T3 T4 Θt1 Θt2
Genetic Algorithm Optimization Method
1 -00003 7164479 -00588 -06213
Hybrid Optimization Method
1 -00000 7164482 15543 -16254
Original Design -322803 393645 16410 -4571
The weighted optimization data of Table 54a shows that the GA simulation again used 4
generations containing 20 900 function evaluations to conduct a global search for the optimal
system While the weighted Hybrid optimization used 13 iterations (consisting of 381 function
evaluations) after its GA run which used the same number of generations and function
evaluations as the GA run in the non-weighted simulations Tables 54a and 54b show the data
Optimization 106
for only one trial for each of the non-weighted GA and hybrid methods since only a single
optimal point exists in this case
533 Discussion
The optimal design from each search method can be selected based on the least amount of
additional pre-tension (corresponding to the largest magnitude of negative tension) that would
need to be added to the system This is in harmony with the goal of the optimization of the B-
ISG system as stated earlier to minimize the static tension for the tautest span and at the same
time minimize the absolute static tension of the slackest span for the ISG driving case As well
the angular displacements corresponding to each trial‟s results show that the Twin Tensioner is
under static stability Therein the optimal solution may be selected as the design parameters
corresponding to Trial 4 of the GA simulations to Trial 4 of the Hybrid simulations or to either
of the trials for the non-weighted optimization simulations
Given the ability of the Hybrid optimization to refine the results obtained in the GA
optimization the results of Trial 4 of the Hybrid simulations are selected as the most optimal
design from the weighted objective function approaches It is interesting to note that the Hybrid
case for the least slackness in belt span tension corresponds to a significantly larger number of
function evaluations than that of the remaining Hybrid cases This anomaly however does not
invalidate the other Hybrid cases since each still satisfy the design constraints Using the data
for the optimized system in Trial 4 (of the Hybrid optimization) the static tensions for the belt
spans in both of the B-ISG‟s phases of operation are as follows
Optimization 107
Table 55 Weighted Optimization Results for Static Tensions for Optimal B-ISG System with a
Twin Tensioner
Belt Span between Pulleys Crankshaft Driving Case [N] ISG Driving Case [N]
Optimized Original Optimized Original
Crankshaft ndash Air Conditioner 3926599 465848 117333 -284152
Air Conditioner ndash Tensioner 1 3540088 427197 -269178 -322803
Tensioner 1 ndash ISG 3540088 427197 -269178 -322803
ISG ndash Tensioner 2 2073813 28057 6895304 393645
Tensioner 2 ndash Crankshaft 2073813 28057 6895304 393645
Additional Pretension
Required (approximate) + 27000 +322803 + 27000 +322803
In Table 54b it is evident that the non-weighted class of optimization simulations achieves the
least amount of required pre-tension to be added to the system The computed tension results
corresponding to both of the non-weighted GA and Hybrid approaches are approximately
equivalent Therein either of their solution parameters may also be called the most optimal
design The Hybrid solution parameters are selected as the optimal design once again due to the
refinement of the GA output contained in the Hybrid approach and its corresponding belt
tensions are listed in Table 56 below
Optimization 108
Table 56 Non-Weighted Optimization Results for Static Tensions for Optimal B-ISG System
with a Twin Tensioner
Belt Span between Pulleys Crankshaft Driving Case [N] ISG Driving Case [N]
Optimized Original Optimized Original
Crankshaft ndash Air Conditioner 3891862 465848 386511 -284152
Air Conditioner ndash Tensioner 1 3505351 427197 -00000 -322803
Tensioner 1 ndash ISG 3505351 427197 -00000 -322803
ISG ndash Tensioner 2 2039076 28057 7164482 393645
Tensioner 2 ndash Crankshaft 2039076 28057 7164482 393645
Additional Pretension
Required (approximate) + 0000 +322803 + 00000 +322803
The results of the simulation experiments are limited by the following considerations
System equations are coupled so that a fixed difference remains between tautest and
slackest spans
A limited number of simulation trials have been performed
There are multiple optimal design points for the weighted optimization search methods
Remaining tensioner parameters tensioner pulley diameters and their stiffness have not
been included in the design variables for the experiments
The belt factor kb used in the modeling of the system‟s belt has been obtained
experimentally and may be open to further sources of error
Therein the conclusions obtained and interpretations of the simulation data can be limited by the
above noted comments on the optimization experiments
Optimization 109
54 Conclusion
The outcomes the trends in the experimental data and the optimal designs can be concluded
from the optimization simulations The simulation outcomes demonstrate that in all cases the
weighted optimization functions reached an identical value for the objective function whereas
the values reached for the parameters varied widely
The lowest tension values for the tautest and slackest span were achieved in Trial 5 of the GA
optimization approach In reiteration in the presence of slack spans the tension value of the
slackest span must be added to the initial static tension for the belt Therein for the former case
an amount of at least 409N would need to be added to the 300N of pre-tension already applied to
the system (see Table 34) The highest tension values for the spans were achieved in Trial 4 of
the weighted Hybrid optimization approach and in both trials of the non-weighted optimization
approaches In the former the weighted Hybrid trial the tension value achieved in the slackest
span was approximately -27N signifying that only at least 27N would need to be added to the
present pre-tension value for the system The tension of the slackest span in the non-weighted
approach was approximately 0N signifying that the minimum required additional tension is 0N
for the system
The optimized solutions for the tension values in each span show that there is consistently a fixed
difference of 716448N between the tautest and slackest span tension values as seen in Tables
52b 53b and 54b This difference is identical to the difference between the tautest and slackest
spans of the B-ISG system for the original values of the design parameters while in its ISG
mode As well the optimal stiffness parameters for the weighted Hybrid optimization case are
Optimization 110
greater than their original values except for that of the stiffness factor of tensioner arm 1
Likewise for the non-weighted Hybrid optimization case the stiffness parameters are above their
original values without exceptions The coordinates of the optimal solutions are within close
approximation to each other and also both match the regions for moderately low tension in
Regions 1 and 2 of the ISG driving case as is shown in Figures 49 410 413 and 414
The results of the non-weighted Hybrid optimization trial and Trial 4 of the weighted Hybrid
optimization simulations are selected as the most optimal designs for the B-ISG Twin Tensioner
In these designs the Twin Tensioner is shown in Table 53b and 54b to have static stability and
to maintain suitable tensions in the ISG driving phase The tensioner parameters for the optimal
designs allow for one of the lowest amounts of additional pre-tension to be added to the system
out of all the findings from the simulations which were conducted
111
CHAPTER 6 CONCLUSION
61 Summary
The primary aim of the thesis is to reduce the magnitude of static tension in the belt spans of a
Belt-driven Integrated Starter-generator (B-ISG) system by the design and investigation of a
Twin Tensioner It is established that the operating phases of the B-ISG system produced two
cases for static tension outcomes an ISG driving case and a crankshaft driving case The
approach taken in this thesis study includes the derivation of a system model for the geometric
properties as well as for the dynamic and static states of the B-ISG system The static state of a
B-ISG system with a single tensioner mechanism is highlighted for comparison with the static
state of the Twin Tensioner-equipped B-ISG system
It is observed that there is an overall reduction in the magnitudes of the static belt tensions with
the presence of a Twin Tensioner over that of a single tensioner The influences of the geometric
and stiffness properties of the Twin Tensioner affecting the static tensions in the system are
analyzed in a parametric study It is found that there is a notable change in the static tensions
produced as result of perturbations in each respective tensioner property This demonstrates
there are no reasons to not further consider a tensioner property based solely on its influence on
the B-ISG system‟s static tensions The phenomenon of higher magnitudes for static tensions in
the ISG mode of operation over that of the crankshaft mode of operation particularly in
excessively slack spans provides the motivation for optimizing the ISG case alone for static
tension The optimization method uses weighted and non-weighted approaches with genetic
algorithm (GA) and hybrid GA searches The most optimal design has been found to be one in
Conclusion 112
which the magnitude of tension in the excessively slack spans in the ISG driving case are
significantly lower than in that of the original B-ISG Twin Tensioner design
62 Conclusion
The conclusions that can be drawn from the study of a Twin Tensioner for a B-ISG system
include the following
1 The simulations of the dynamic model demonstrate that the mode shapes for the system
are greater in the ISG-phase of operation
2 It was observed in the output of the dynamic responses that the system‟s rigid bodies
experienced larger displacements when the crankshaft was driving over that of the ISG-
driving phase It was also noted that the transition speed marking the phase change from
the ISG driving to the crankshaft driving occurred before the system reached either of its
first natural frequencies
3 The magnitudes for static belt tensions as well as dynamic tensions for a B-ISG system
are consistently greater in its ISG operating phase than in its crankshaft operating phase
4 A Twin Tensioner is able to reduce the magnitudes of the static tension for the belt spans
of a B-ISG system in comparison to when only a single tensioner mechanism is present
5 The parametric study of the B-ISG system demonstrates that the slackest and tautest belt
spans decrease or increase together for either phase of operation
6 Perturbations in the Twin Tensioner‟s geometric and stiffness properties have a
significant influence on the magnitudes of the static tension of the slackest and tautest
belt spans The coordinate position of each pulley in the Twin Tensioner configuration
Conclusion 113
has the greatest influence on the belt span static tensions out of all the tensioner
properties considered
7 Optimization of the B-ISG system shows a fixed difference trend between the static
tension of the slackest and tautest belt spans for the B-ISG system
8 The values of the design variables for the most optimal system are found using a hybrid
algorithm approach The slackest span for the optimal system is significantly less slack
than that of the original design Therein less additional pretension is required to be added
to the system to compensate for slack spans in the ISG-driving phase of operation
63 Recommendation for Future Work
The investigation of the B-ISG Twin Tensioner encourages the following future work
1 The optimization of the B-ISG system with the inclusion of diametric Twin Tensioner
properties would provide a complete picture as to the highest possible performance
outcome that the Twin Tensioner is able to have with respect to the static tensions
achieved in the belt spans
2 A larger number of optimization trials using the genetic algorithm (GA) and hybrid GA
under weighted and other approaches would investigate the scope of optimal designs
available in the Twin Tensioner for the B-ISG system
3 A model of the system without the simplification of constant damping may produce
results that are more representative of realistic operating conditions of the serpentine belt
drive A finite element analysis of the Twin Tensioner B-ISG system may provide more
applicable findings
Conclusion 114
4 Investigation of the transverse motion coupled with the rotational belt motion in an
optimized B-ISG system equipped with a Twin Tensioner may also provide a closer look
at the system under realistic conditions In addition the affect of the Twin Tensioner on
transverse motion can determine whether significant improvements in the magnitudes of
static belt span tensions are still being achieved
5 The recommendation to conduct modal decoupling of the B-ISG system‟s static model is
motivated by the fixed difference trend between the slackest and tautest belt span
tensions shown in Chapter 5 The modal decoupling of the system would allow for its
matrices comprising the equations of motion to be diagonalized and therein to decouple
the system equations Modal analysis would transform the system from physical
coordinates into natural coordinates or modal coordinates which would lead to the
decoupling of system responses
6 An investigation and optimization of the dynamic belt span tensions for a B-ISG system
with a Twin Tensioner would increase understanding of the full impact of a Twin
Tensioner mechanism on the state of the B-ISG system It would be informative to
analyze the mode shapes of the first and second modes as well as the required torques of
the driving pulleys and the resulting torque of each of the tensioner arms The
observation of the dynamic belt span tensions would also give direction as to how
damping of the system may or may not be changed
7 Further comparison with the Twin Tensioner B-ISG system‟s dynamic and static states
including the Twin Tensioner‟s stability in each versus a B-ISG system with a single
tensioner would further demonstrate the improvements or dis-improvements in the Twin
Tensioner‟s performance on a B-ISG system
Conclusion 115
8 The influence of the belt properties on the dynamic and static tensions for a B-ISG
system with a Twin Tensioner can also be investigated This again will show the
evidence of improvements or dis-improvement in the Twin Tensioner‟s performance
within a B-ISG setting
9 Lastly an experimental apparatus of the B-ISG system with a Twin Tensioner can be
designed and constructed Suitable instrumentation can be set-up to measure belt span
tensions (both static and dynamic) belt motion and numerous other system qualities
This would provide extensive guidance as to finding the most appropriate theoretical
model for the system Experimental data would provide a bench mark for evaluating the
theoretical simulation results of the Twin Tensioner-equipped B-ISG system
116
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[14] National Alternative Fuels Training Consortium (NAFTC) (2005 Oct 3) Tech stuff
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[21] PJ Wezenbeek (Zytec Systems Ltd) D G Evans (General Motors Powertrain) D P
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(Delphi Corp) Combustion Assisted Belt-Cranking of a V-8 Engine at 12-Volts SAE
Technical Papers vol 113 pp 396-407 2004 Document no 2004-01-0569
[22] T C Firbank Mechanics of the Belt Drive International Journal of Mechanical
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[23] R L Cassidy S K Fan R S MacDonald and W F Samson Serpentine Extended Life
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[24] A G Ulsoy J E Whitesell and M D Hooven Design of Belt-Tensioner Systems for
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Transactions of the ASME vol 107 pp 282-290 July 1985
[25] R S Beikmann N C Perkins and A G Ulsoy Free Vibration of Serpentine Belt Drive
Systems Journal of Vibrations and Acoustics Transactions of the ASME vol 118 pp
406-413 1996
[26] T C Kraver G W Fan and J J Shah Complex Modal Analysis of a Flat Belt Pulley
System with Belt Damping and Coulomb-Damped Tensioner Journal of Mechanical
Design Transactions of the ASME vol 118 pp 306-311 Jun 1996
[27] R S Beikmann N C Perkins and A G Ulsoy Design and Analysis of Automotive
Serpentine Belt Drive Systems for Steady State Performance Journal of Mechanical
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[28] L Zhang and J W Zu Modal Analysis of Serpentine Belt Drive Systems Journal of
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[29] R Balaji and E M Mockensturm Dynamic analysis of a front-end accessory drive with a
decouplerisolator International Journal of Vehicle Design vol 39 pp 208-231 2005
[30] M Nouri Design Optimization and Active Control of Serpentine Belt Drive Systems with
Two-pulley Tensioners University of Toronto 2005
[31] G J Spicer (Litens Automotive Inc) Tensioner for use in eg belt drive system has
electronic actuator associated with clutch spring for engaging International
WO2005119089-A1 Jun 6 2005 2005
[32] Bando Chemical Industries Ltd and Litens Automotive GmbH About belt-type starter
system Feb 27 2002
[33] H Lemberger and R Jungjohann (Bayerische Motoren Werke AG) Tension device for an
envelope drive of a device especially a belt drive of a starter generator of an internal
combustion engine comprises a support part Europe EP1420192-A2 May 19 2004 2003
[34] P Ahner and M Ackermann (Bosch GMBH) Belt drive especially for internal
combustion engines to drive accessories in an automobile Germany DE19849886-A1
May 11 2000 1998
[35] N Freisinger K Hagemann J Sievert P Struebel and M Treusch (Daimler Chrysler AG)
Belt tensioning device for belt drive between engine and starter generator of motor
vehicle has self-aligning bearing that supports auxiliary unit and provides working force to
tensioners for tensioning belt Germany DE10324268 Dec 16 2004 2003
[36] C R Rogers (Dayco Products LLC) Offset starter generator drive system for a vehicle
engine has a dual arm pivoted tensioner United States US6942589-B2 Feb 8 2005 2002
[37] A Serkh and I Ali (Gates Corp) Internal combustion engine has belt drive system with
tensioner asymmetrically biased in direction tending to cause power transmission belt to be
under tension International WO2003038309-A1 May 8 2003 2002
References 120
[38] P J Mcvicar and C A Thurston (General Motors Corp) Belt alternator starter accessory
drive with dual tensioner United States US20060287146-A1 Dec 21 2006 2005
[39] W Petri and M Bogner (INA Schaeffler KG) Traction drive especially for driving
internal combustion engine units has arrangement for demand regulated setting of tension
consisting of unit with housing with limited rotation and pulley German DE10044645-
A1 Mar 21 2002 2000
[40] M Bogner (INA Schaeffler KG) Belt drive tensioner for a starter-generator of an IC
engine has locking system for locking tensioning element in an engine operating mode
locking system is directly connected to pivot arm follows arm control movements
German DE10159073-A1 Jun 12 2003 2001
[41] R Painta M Bogner and H Graf (INA Schaeffler KG) Traction mechanism drive esp
belt drive has belt tensioning pulley mounted on generator shaft and uncoupled from it via
freewheel to dampen load peaks Europe EP1723350-A1 Nov 22 2006 2005
[42] W Petri (INA Schaeffler KG) Drive unit for a combustion engine having a starter
generator and a belt drive has tensioner with spring and counter hydraulic force Germany
DE10359641-A1 Jul 28 2005 2003
[43] H Stief M Bogner B Hartmann T Kraft and M Schmid (INA Schaeffler KG) Traction
drive especially belt drive for short-duration driving of starter generator has tensioning
device with lever arm deflectable against restoring force and with end stop limiting
deflection travel Europe EP1738093-A1 Jan 3 2007 2005
[44] M Ulm (INA Schaeffler KG DE) Tension unit eg for drive in machine such as
combustion engine has belt or chain drive with wheels turning and connected with starter
generator and unit has two idlers arranged at clamping arm with machine stored by shock
absorber Germany DE102004012395-A1 Sep 29 2005 2004
[45] M Bogner (INA Schaeffler KG) Belt drive for starter motor-generator auxiliary assembly
has limited movement at the starter belt section tensioner roller bringing it into a dead point
position on starting the motor International WO2006108461-A1 Oct 19 2006 2006
References 121
[46] W Guhr (Litens Automotive GMBH) Automotive motor and drive assembly includes
tension device positioned within belt drive system having combination starter United
States US2001007839-A1 Jul 12 2001 2001
[47] K Kuniaki K Masahiko H Kazuyuki I Shuichi and T Masaki (Mitsubishi Jidosha Eng
KK and Mitsubishi Motor Corp) Tension adjustment method of belt for starter generator
in vehicle involves shifting auto-tensioners between lock state and free state to adjust
tension of belt during driving of crank pulley Japan JP2005083514-A Mar 31 2005
2003
[48] Nissan Motor Co Ltd Winding gear for starting engine of hybrid motor vehicle has
tensioner tightening chain while cranking engine and slackens chain after start of engine
provided to span side of chain Japan JP3565040-B2 Sep 15 2004 1998
[49] S Sato and H Hayakawa (NTN Corp) Auto tensioner for ancillary drive belts has
cylinder nut and screw bolt in hydraulic damper mechanism provided in middle of cylinder
acting as start-up rigidity buffer component Japan JP2006189073-A Jul 20 2006 2005
[50] G Vadin-Michaud (Valeo Equip Electrique Moteur) Pulley and belt starting system for a
thermal engine for a motor vehicle Europe EP1658432 May 24 2006 2005
[51] M Zhen University of Toronto and Litens Automotive Group Ltd FEAD vol 50
2005
[52] W E Johns Notes on Motors [Electronic] 2003 [2008 June] Available at
httpwwwgizmologynetmotorshtm
[53] Litens Automotive Group Ltd DC BAS System - Conventional Start Input Profile Nov
23 2007
[54] Arnold Magnetic Technologies Corp General Motor Terminology [Electronic] pp 7
[2008 June] Available at httpwwwgrouparnoldcommtcpdfweb_motor_glossarypdf
[55] Douglas W Jones Stepping Motors University of Iowa - Department of Computer
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httpwwwcsuiowaedu~jonesstepphysicshtml
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[56] Litens Automotive Group Ltd (2004 Jan 31) FEAD software input data for test project
[57] K Deb Multi-Objective Optimization using Evolutionary Algorithms Toronto John Wiley
amp Sons Ltd 2001 pp 81-85
[58] P E McSharry (2004 May 11) Department of Engineering Science University of Oxford
[httpwwwengoxacuksamppubsgawbreppdf]
[59] The MathWorks Inc MATLAB vol 750342 (R2007b) Aug 15 2007
123
APPENDIX A
Passive Dual Tensioner Designs from Patent Literature
Figure A1 Proposed design by Bayerische Motoren Werke AG corresponding to patent nos EP1420192-A2 and DE10253450-A1
Source European Patent Office espcenet (publication nos EP1420192-A2 and DE10253450-A1 accessed May 2007) epespacenetcom [33]
Figure A1 label identification 1 ndash tightner 2 ndash belt drive
3 ndash starter generator
4 ndash internal-combustion engine
4‟ ndash crankshaft-lateral drive disk
5 ndash generator housing
6 ndash common axis of rotation
7 ndash featherspring of tiltable clamping arms
8 ndash clamping arm
9 ndash clamping arm
10 11 ndash idlers
12 12‟ ndash Zugtrum 13 13‟ ndash Leertrum
14 ndash carry-hurries 15 ndash generator wave
16 ndash bush
17 ndash absorption mechanism
18 18‟ ndash support arms
19 19‟ ndash auxiliary straight lines
20 ndash pipe
21 ndash torsion bar
22 ndash breaking through
23 ndash featherspring
24 ndash friction disk
25 ndash screw connection 26 ndash Wellscheibe
(European Patent Office May 2007) [33]
Appendix A 124
Figure A2a First of four proposed designs (and its various configurations) by Bosch GMBH
corresponding to patent no WO0026532-A1
Source European Patent Office espcenet (publication no WO0026532-A1 accessed Jun 2007)
epespacenetcom [34]
Figure A2b Second of four proposed designs by Bosch GMBH corresponding to patent no
WO0026532-A1
Source European Patent Office espcenet (publication no WO0026532-A1 accessed Jun 2007) epespacenetcom [34]
Figure A2c Third of four proposed designs (and its various configurations) by Bosch GMBH
corresponding to patent no WO0026532-A1
Source European Patent Office espcenet (publication no WO0026532-A1 accessed Jun 2007)
epespacenetcom [34]
Appendix A 125
Figure A2d Fourth of four proposed designs (and its various configurations) by Bosch GMBH
corresponding to patent no WO0026532-A1
Source European Patent Office espcenet (publication no WO0026532-A1 accessed Jun 2007)
epespacenetcom [34]
Figure A2a through to A2d label identification 10 ndash engine wheel
11 ndash [generator] 13 ndash spring
14 ndash belt
16 17 ndash tensioning pulleys
18 19 ndash springs
20 21 ndash fixed points
25ab ndash carriers of idlers
25c ndash gang bolt
(European Patent Office June 2007) [34]
Figure A3 Proposed design by Daimler Chrysler AG corresponding to patent no DE10324268-A1
Source European Patent Office espcenet (publication no DE10324268-A1 accessed May 2007)
epespacenetcom [35]
Figure A3 label identification
Appendix A 126
10 12 ndash belt pulleys
14 ndash auxiliary unit
16 ndash belt
22-1 22-2 ndash belt tensioners
(European Patent Office May 2007) [35]
Figure A4 Proposed design by Dayco Products LLC corresponding to patent no US6942589-B2
Source European Patent Office espcenet (publication no US6942589-B2 accessed Jun 2007)
epespacenetcom [36]
Figure A4 label identification 12 ndash belt
14 ndash tensioner
16 ndash generator pulley
18 ndash crankshaft pulley
22 ndash slack span 24 ndash tight span
32 34 ndash arms
33 35 ndash pulley
(European Patent Office June 2007) [36]
Appendix A 127
Figure A5 Proposed design by Gates Corp corresponding to patent no WO2003038309-A
Source European Patent Office espcenet (publication no WO2003038309-A accessed Jun 2007)
epespacenetcom [37]
Figure A5 label identification 12 ndash motorgenerator
14 ndash motorgenerator pulley 26 ndash belt tensioner
28 ndash belt tensioner pulley
30 ndash transmission belt
(European Patent Office June 2007) [37]
Figure A6 Proposed design by General Motors Corp corresponding to patent no US20060287146-A1
Source European Patent Office espcenet (publication no US20060287146-A1 accessed Jun 2007)
epespacenetcom [38]
Appendix A 128
Figure A6 label identification 28 ndash tensioner
32 ndash carrier arm
34 ndash secondary carrier arm
46 ndash tensioner pulley
58 ndash secondary tensioner pulley
(European Patent Office June 2007) [38]
Figure A7 Proposed design by INA Schaeffler KG corresponding to patent no DE10044645-A1
Source European Patent Office espcenet (publication no DE10044645-A1 accessed Jun 2007)
epespacenetcom [39]
Figure A7 label identification 2 ndash internal combustion engine
3 ndash traction element
11 ndash housing with limited rotation 12 13 ndash direction changing pulleys
(European Patent Office June 2007) [39]
Appendix A 129
Figure A8a First of three proposed designs by INA Schaeffler KG corresponding to patent no
DE10159073-A1
Source European Patent Office espcenet (publication no DE10159073-A1 accessed May 2007)
epespacenetcom [40]
Figure A8b Second of three proposed designs by INA Schaeffler KG corresponding to patent no
DE10159073-A1
Source European Patent Office espcenet (publication no DE10159073-A1 accessed May 2007)
epespacenetcom [40]
Appendix A 130
Figure A8c Third of three proposed designs by INA Schaeffler KG corresponding to patent no
DE10159073-A1
Source European Patent Office espcenet (publication no DE10159073-A1 accessed May 2007)
epespacenetcom [40]
Figure A8a A8b and A8c label identification 1 ndash tightener [tensioner]
2 ndash idler
3 ndash drawing means
4 ndash swivel arm
5 ndash axis of rotation
6 ndash drawing means impulse [belt]
7 ndash crankshaft
8 ndash starter generator
9 ndash bolting volume 10a ndash bolting device system
10b ndash bolting device system
10c ndash bolting device system
11 ndash friction body
12 ndash lateral surface
13 ndash bolting tape end
14 ndash bolting tape end
15 ndash control member
16 ndash torsion bar
17 ndash base
18 ndash pylon
19 ndash hub
20 ndash annular gap
21 ndash Gleitlagerbuchse
23 ndash [nil]
23 ndash friction disk
24 ndash turning camps 25 ndash teeth
26 ndash elbow levers
27 ndash clamping wedge
28 ndash internal contour
29 ndash longitudinal guidance
30 ndash system
31 ndash sensor
32 ndash clamping gap
(European Patent Office May 2007) [40]
Appendix A 131
Figure A9 Proposed design by INA Schaeffler KG corresponding to patent no DE10359641-A1
Source European Patent Office espcenet (publication no DE10359641-A1 accessed Jun 2007)
epespacenetcom [42]
Figure A9 label identification 8 ndash cylinder
10 ndash rod
12 ndash spring plate
13 ndash spring
14 ndash pressure lead
(European Patent Office June 2007) [42]
Appendix A 132
Figure A10 Proposed design by INA Schaeffler KG corresponding to patent no EP1723350-A1
Source European Patent Office espcenet (publication no EP1723350-A1 accessed Jun 2007) epespacenetcom [41]
Figure A10 label identification 4 ndash pulley
5 ndash hydraulic element 11 ndash freewheel
12 ndash shaft
(European Patent Office June 2007) [41]
Figure A11 Proposed design by INA Schaeffler KG corresponding to patent no EP1738093-A1
Source European Patent Office espcenet (publication no EP1738093-A1 accessed Jun 2007)
epespacenetcom [43]
Figure A11 label identification 1 ndash traction drive
2 ndash traction belt
3 ndash starter generator
Appendix A 133
7 ndash tension device
9 ndash lever arm
10 ndash guide roller
16 ndash end stop
(European Patent Office June 2007) [43]
Figure A12 Proposed design by INA Schaeffler KG corresponding to patent no DE102004012395-A1
Source European Patent Office espcenet (publication no DE102004012395-A1 accessed May 2007) epespacenetcom [44]
Figure A12 label identification 1 ndash belt drive
2 ndash belts
3 ndash wheel of the internal-combustion engine
4 ndash wheel of a Nebenaggregats
5 ndash wheel of the starter generator
6 ndash clamping unit
7 ndash idler
8 ndash idler
9 ndash scale beams
10 ndash drive place
11 ndash clamping arm
12 ndash camps
13 ndash coupling point
14 ndash shock absorber element
15 ndash arrow
(European Patent Office May 2007) [44]
Appendix A 134
Figure A13 Proposed design by INA Schaeffler KG corresponding to patent nos DE102005017038-A1and WO2006108461-A1
Source European Patent Office espcenet (publication nos DE102005017038-A1and WO2006108461-A1 accessed May 2007) epespacenetcom [45]
Figure A13 label identification 1 ndash belt
2 ndash wheel of the crankshaft KW
3 ndash wheel of a climatic compressor AC
4 ndash wheel of a starter generator SG
5 ndash wheel of a water pump WP
6 ndash first clamping system
7 ndash first tension adjuster lever arm
8 ndash first tension adjuster role
9 ndash second clamping system
10 ndash second tension adjuster lever arm
11 ndash second tension adjuster role 12 ndash guide roller
13 ndash drive-conditioned Zugtrum
(generatorischer enterprise (GE))
13 ndash starter-conditioned Leertrum
(starter enterprise (SE))
14 ndash drive-conditioned Leertrum (GE)
14 ndash starter-conditioned Zugtrum (SE)
14a ndash drive-conditioned Leertrumast (GE)
14a ndash starter-conditioned Zugtrumast (SE)
14b ndash drive-conditioned Leertrumast (GE)
14b ndash starter-conditioned Zugtrumast (SE)
(European Patent Office May 2007) [45]
Figure A14 Proposed design by Litens Automotive GMBH et al corresponding to patent no
US20010007839-A1
Appendix A 135
Source European Patent Office espcenet (publication no US20010007839-A1 accessed Jun 2007)
epespacenetcom [46]
Figure A14 label identification E - belt
K - crankshaft
R1 ndash first tension pulley
R2 ndash second tension pulley
S ndash tension device
T ndash drive system
1 ndash belt pulley
4 ndash belt pulley
(European Patent Office June 2007) [46]
Figure A15 Proposed design by Mitsubishi Jidosha Eng KK and Mitsubishi Motor Corp corresponding
to patent no JP2005083514-A
Source Industrial Property Digital Library and Japanese Patent Office Patent amp Utility Model Gazette DB (document no A 2005-083514 accessed May 2007) wwwipdlinpitgojp [47]
Figure A15 label identification 1 ndash Pulley for Starting
2 ndash Crank Pulley
3 ndash AC Pulley
4a ndash 1st roller
4b ndash 2nd roller
5 ndash Idler Pulley
6 ndash Belt
7c ndash Starter generator control section
7d ndash Idle stop control means
8 ndash WP Pulley
9 ndash IG Switch
10 ndash Engine
11 ndash Starter Generator
12 ndash Driving Shaft
Appendix A 136
7 ndash ECU
7a ndash 1st auto tensioner control section (the 1st auto
tensioner control means)
7b ndash 2nd auto tensioner control section (the 2nd auto
tensioner control means)
13 ndash Air-conditioner Compressor
14a ndash 1st auto tensioner
14b ndash 2nd auto tensioner
18 ndash Water Pump
(Industrial Property Digital Library May 2007) [47]
Figure A16 Proposed design by Nissan Motor Co Ltd corresponding to patent no JP3565040-B2
Source European Patent Office espcenet (publication no JP3565040-B2 accessed Jun 2007) epespacenetcom [48]
Figure A16 label identification 3 ndash chain [or belt]
5 ndash tensioner
4 ndash belt pulley
(European Patent Office June 2007) [48]
Figure A17 Proposed design by NTN Corp corresponding to patent no JP2006189073-A
Appendix A 137
Source European Patent Office espcenet (publication no JP2006189073-A accessed Jun 2007)
epespacenetcom [49]
Figure A17 label identification 5d - flange
6 ndash tensile strength spring
10 ndash actuator
10c ndash cylinder
12 ndash rod
20 ndash hydraulic damper mechanism
21 ndash cylinder nut
22 ndash screw bolt
(European Patent Office June 2007) [49]
Figure A18 Proposed design by Valeo Equipment Electriques Moteur corresponding to patent nos
EP1658432 and WO2005015007
Source European Patent Office espcenet (publication nos EP1658432 and WO2005015007
accessed May 2007) epespacenetcom [50]
Figure A18 abbreviated list of label identifications
10 ndash starter
22 ndash shaft section
23 ndash free front end
80 ndash pulley
200 ndash support element
206 - arm
(European Patent Office May 2007) [50]
The author notes that the list of labels corresponding to Figures A1a through to A7 are generated
from machine translations translated from the documentrsquos original language (ie German)
Consequently words may be translated inaccurately or not at all
138
APPENDIX B
B-ISG Serpentine Belt Drive with Single Tensioner
Equation of Motion
The equations of motion (EOM) for a B-ISG serpentine belt drive with a single tensioner are
shown The EOM has been derived similarly to that of the same system with a twin tensioner
that was provided in Chapter 3 The assumptions for the twin tensioner B-ISG system are
applicable for the single tensioner B-ISG system as well
Figure B1 shows the B-ISG system with a single tensioner pulley and arm The pulleys are
numbered 1 through 4 and their associated belt spans are numbered accordingly
Figure B1 Single Tensioner B-ISG System
Appendix B 139
The free-body diagram for the ith non-tensioner pulley in the system shown above is found in
Figure B2 The moment of inertia for the ith pulley is designated as Ii while the angular
displacement velocity and acceleration is designated as 120579119905119894 120579 119905119894 and 120579 119905119894 respectively The
required torque is Qi the angular damping is Ci and the tension of the ith span is Ti
Figure B2 Free-body Diagram of ith Pulley
The positive motion designated is assumed to be in the clockwise direction The radius for the
ith pulley is represented by Ri The equilibrium equations for the ith pulley are as follows
I1 ∙ θ 1 = T4 ∙ R1 minus T1 ∙ R1 + Q1 minus c1 ∙ θ 1 (B1)
I2 ∙ θ 2 = T1 ∙ R2 minus T2 ∙ R2 + Q2 minus c2 ∙ θ 2 (B2)
I3 ∙ θ 3 = T2 ∙ R3 minus T3 ∙ R3 + Q3 minus c3 ∙ θ 3 (B3)
Appendix B 140
A free-body diagram for the single tensioner pulley is shown in Figure B3 The rotational
stiffness and damping for the tensioner arm is designated as kt and ct respectively The angular
rotation and velocity for the arm is 120579119905119894 and 120579 119905119894 respectively
Figure B3 Free-body Diagram of Single Tensioner
From figure B2 the equations of equilibrium are resolved for the tensioner pulley The angle of
orientation for the ith belt span is designated by 120573119895119894
minusI4 ∙ θ 4 = minusT3 ∙ R4 + T4 ∙ R4 + Q4 + c4 ∙ θ 4 (B4)
It ∙ θ t = minusTt ∙ Lt ∙ sin θto minus βj4 + sin θto 1 minus βj5 minus kt ∙ partθt minus ct ∙ partθ t
(B5)
Appendix B 141
partθt = θt minus θto (B6)
The dynamic tension matrix Trsquo is proportional to the damping (Tc) and stiffness (Tk) matrices
that are due to belt damping (119888119894119887 ) and belt stiffness (119896119894
119887 ) respectively
119827prime = 119827119836 ∙ 120521 + 119827119844 ∙ 120521 (B7)
The initial tension is represented by To and the initial angle of the tensioner arm is represented
by 120579119905119900 The equation for the tension of the ith span is shown in the following equations
T1 = To + k1b ∙ R1 ∙ θ1 minus R2 ∙ θ2 + c1
b ∙ [R1 ∙ θ 1 minus R2 ∙ θ 2] (B8)
T2 = To + k2b ∙ R2 ∙ θ2 minus R3 ∙ θ3 + c2
b ∙ [R2 ∙ θ 2 minus R3 ∙ θ 3)] (B9)
T3 = To + k3b ∙ R3 ∙ θ3 minus R4 ∙ θ4 + Lt ∙ [sin θto minus βj3 ] ∙ (θt minus θto ) + c3
b ∙ [R3 ∙ θ 3 minus R4 ∙
θ 4 + Lt ∙ [sin θto minus βj3 ] ∙ (θ t)] (B10)
T4 = To + k4b ∙ R4 ∙ θ4 minus R1 ∙ θ1 + Lt ∙ [sin θto minus βj4 ] ∙ (θt minus θto ) + c4
b ∙ [R4 ∙ θ 4 minus R1 ∙
θ 1 + Lt ∙ [sin θto minus βj4 ] ∙ (θ t)] (B11)
Tprime = Ti minus To (B12)
Tt = T3 = T4 (B13)
Appendix B 142
The EOM for the single tensioner B-ISG system is found by substitution of equations B8 to
B13 into B1 to B5 The matrices in the EOM include the inertial matrix I damping matrix C
stiffness matrix K and the required torque matrix Q as well as the angular displacement
velocity and acceleration matrices 120521 120521 and 120521 respectively
119816 ∙ 120521 + 119810 ∙ 120521 + 119818 ∙ 120521 = 119824 (B14)
119816 =
I1 0 0 0 00 I2 0 0 00 0 I3 0 00 0 0 I4 00 0 0 0 It1
(B15)
The stiffness matrix includes kb the belt factor Kb the belt cord stiffness 120601119894 the wrap angle of
the belt on the ith pulley and Kbi the stiffness factor of the ith belt span Cb represents the belt
damping for each span and βji is the angle of orientation for the span between the jth and ith
pulleys It is noted in the terms of the stiffness and damping matrices below that the numerical
subscripts refer to the (i+1)th pulley The term Qt may be found within the required torque
matrix and represents the required torque for the tensioner arm As well the term It1 represents
the moment of inertia for the tensioner arm
Appendix B 143
K =
(B16)
Kbi =Kb
Li + kb ∙ Ri ∙ϕi+1
2 + Ri ∙ϕi
2
(B17)
C =
(B18)
Appendix B 144
Appendix B 144
120521 =
θ1
θ2
θ3
θ4
partθt
(B19)
119824 =
Q1
Q2
Q3
Q4
Qt
(B20)
Simulations of the EOM for the B-ISG system with a single tensioner were performed in FEAD
[51] software for dynamic and static cases This allowed for the methodology for deriving the
EOM to be verified and then applied to the B-ISG system with a twin tensioner The natural
frequencies modes shapes dynamic responses tensioner arm torques as well as the crankshaft
required torque only and the dynamic tensions were solved from the EOM as described in (331)
to (339) of Chapter 3 and as well as for the static tension from (351) to (353) of Chapter 3
This permitted verification of the complete derivation methodology and allowed for comparison
of the static tension of the B-ISG system belt spans in the case that a single tensioner is present
and in the case that a Twin Tensioner is present [51]
145
APPENDIX C
MathCAD Scripts
C1 Geometric Analysis
1 - CS
2 - AC
4 - Alt
3 - Ten1
5 - Ten 2
6 - Ten Pivot
1
2
3
4
5
Figure C1 Schematic of B-ISG
System with Twin Tensioner
Coordinate Input DataXY1 0 0( ) XY4 24759 16664( )
XY2 224 6395( ) XY5 12057 9193( )
XY3 292761 87( ) XY6 201384 62516( )
Computations
Lt1 XY30 0
XY60 0
2
XY30 1
XY60 1
2
Lt2 XY50 0
XY60 0
2
XY50 1
XY60 1
2
t1 atan2 XY30 0
XY60 0
XY30 1
XY60 1
t2 atan2 XY50 0
XY60 0
XY50 1
XY60 1
XY
XY10 0
XY20 0
XY30 0
XY40 0
XY50 0
XY60 0
XY10 1
XY20 1
XY30 1
XY40 1
XY50 1
XY60 1
x XY
0 y XY
1
Appendix C 146
i - angle bw horizontal and l ine from ith pulley center to (i+1)th pulley center
Adjust last number in range variable p to correspond to number of pulleys
p 0 1 4
k p( ) p 1( ) p 4if
0 otherwise
condition1 p( ) acos
XYk p( ) 0
XYp 0
XYk p( ) 0
XYp 0
2
XYk p( ) 1
XYp 1
2
condition2 p( ) 2 acos
XYk p( ) 0
XYp 0
XYk p( ) 0
XYp 0
2
XYk p( ) 1
XYp 1
2
p( ) if XYk p( ) 1
XYp 1
condition1 p( ) condition2 p( )
Lfi Lbi - connection belt span lengths
D1 20065mm D2 10349mm D3 7240mm D4 6820mm D5 7240mm
D
D1
D2
D3
D4
D5
Lf p( ) XYk p( ) 0
XYp 0
2
XYk p( ) 1
XYp 1
2
1
mm
Dk p( )
2
Dp
2
2
Lb p( ) XYk p( ) 0
XYp 0
2
XYk p( ) 1
XYp 1
2
1
mm
Dk p( )
2
Dp
2
2
fi bi - angle bw ith pulley center connection l ine and contact points Pbfi (or Pfbi) and Pbi
(or Pfi) respecti vely l
f p( ) atanLf p( ) mm
Dp
2
Dk p( )
2
Dp
Dk p( )
if
atanLf p( ) mm
Dk p( )
2
Dp
2
Dp
Dk p( )
if
2D
pD
k p( )if
b p( ) atan
mmLb p( )
Dp
2
Dk p( )
2
Appendix C 147
XYfi XYbi XYfbi XYbfi - 4 possible contact points for i th pulley
XYf p( ) XYp 0
Dp
2 mmcos p( ) f p( )
XYp 1
Dp
2 mmsin p( ) f p( )
XYb p( ) XYp 0
Dp
2 mmcos p( ) f p( )
XYp 1
Dp
2 mmsin p( ) f p( )
XYfb p( ) XYp 0
Dp
2 mmcos p( ) b p( )
XYp 1
Dp
2 mmsin p( ) b p( )
XYbf p( ) XYp 0
Dp
2 mmcos p( ) b p( )
XYp 1
Dp
2 mmsin p( ) b p( )
XYfi+1 XYbi+1 XYfbi+1 XYbfi+1 - 4 possible contact points for i+1th pulley
XYf2 p( ) XYk p( ) 0
Dk p( )
2 mmcos p( ) f p( )
XYk p( ) 1
Dk p( )
2 mmsin p( ) f p( )
XYb2 p( ) XYk p( ) 0
Dk p( )
2 mmcos p( ) f p( )
XYk p( ) 1
Dk p( )
2 mmsin p( ) f p( )
XYfb2 p( ) XYk p( ) 0
Dk p( )
2 mmcos p( ) b p( ) XY
k p( ) 1
Dk p( )
2 mmsin p( ) b p( )
XYbf2 p( ) XYk p( ) 0
Dk p( )
2 mmcos p( ) b p( ) XY
k p( ) 1
Dk p( )
2 mmsin p( ) b p( )
Row 1 --gt Pulley 1 Row i --gt Pulley i
XYfi
XYf 0( )0 0
XYf 1( )0 0
XYf 2( )0 0
XYf 3( )0 0
XYf 4( )0 0
XYf 0( )0 1
XYf 1( )0 1
XYf 2( )0 1
XYf 3( )0 1
XYf 4( )0 1
XYfi
6818
269222
335325
251552
108978
100093
89099
60875
200509
207158
x1 XYfi0
y1 XYfi1
Appendix C 148
XYbi
XYb 0( )0 0
XYb 1( )0 0
XYb 2( )0 0
XYb 3( )0 0
XYb 4( )0 0
XYb 0( )0 1
XYb 1( )0 1
XYb 2( )0 1
XYb 3( )0 1
XYb 4( )0 1
XYbi
47054
18575
269403
244841
164847
88606
291
30965
132651
166182
x2 XYbi0
y2 XYbi1
XYfbi
XYfb 0( )0 0
XYfb 1( )0 0
XYfb 2( )0 0
XYfb 3( )0 0
XYfb 4( )0 0
XYfb 0( )0 1
XYfb 1( )0 1
XYfb 2( )0 1
XYfb 3( )0 1
XYfb 4( )0 1
XYfbi
42113
275543
322697
229969
9452
91058
59383
75509
195834
177002
x3 XYfbi0
y3 XYfbi1
XYbfi
XYbf 0( )0 0
XYbf 1( )0 0
XYbf 2( )0 0
XYbf 3( )0 0
XYbf 4( )0 0
XYbf 0( )0 1
XYbf 1( )0 1
XYbf 2( )0 1
XYbf 3( )0 1
XYbf 4( )0 1
XYbfi
8384
211903
266707
224592
140427
551
13639
50105
141463
143331
x4 XYbfi0
y4 XYbfi1
Row 1 --gt Pulley 2 Row i --gt Pulley i+1
XYf2i
XYf2 0( )0 0
XYf2 1( )0 0
XYf2 2( )0 0
XYf2 3( )0 0
XYf2 4( )0 0
XYf2 0( )0 1
XYf2 1( )0 1
XYf2 2( )0 1
XYf2 3( )0 1
XYf2 4( )0 1
XYf2x XYf2i0
XYf2y XYf2i1
XYb2i
XYb2 0( )0 0
XYb2 1( )0 0
XYb2 2( )0 0
XYb2 3( )0 0
XYb2 4( )0 0
XYb2 0( )0 1
XYb2 1( )0 1
XYb2 2( )0 1
XYb2 3( )0 1
XYb2 4( )0 1
XYb2x XYb2i0
XYb2y XYb2i1
Appendix C 149
XYfb2i
XYfb2 0( )0 0
XYfb2 1( )0 0
XYfb2 2( )0 0
XYfb2 3( )0 0
XYfb2 4( )0 0
XYfb2 0( )0 1
XYfb2 1( )0 1
XYfb2 2( )0 1
XYfb2 3( )0 1
XYfb2 4( )0 1
XYfb2x XYfb2i
0
XYfb2y XYfb2i1
XYbf2i
XYbf2 0( )0 0
XYbf2 1( )0 0
XYbf2 2( )0 0
XYbf2 3( )0 0
XYbf2 4( )0 0
XYbf2 0( )0 1
XYbf2 1( )0 1
XYbf2 2( )0 1
XYbf2 3( )0 1
XYbf2 4( )0 1
XYbf2x XYbf2i0
XYbf2y XYbf2i1
100 40 20 80 140 200 260 320 380 440 500150
110
70
30
10
50
90
130
170
210
250Figure C2 Possible Contact Points
250
150
y1
y2
y3
y4
y
XYf2y
XYb2y
XYfb2y
XYbf2y
500100 x1 x2 x3 x4 x XYf2x XYb2x XYfb2x XYbf2x
Appendix C 150
Xij Yij - selected contact point on ith pulley for span from ith pulley to jth pulley
XY15 XYbf2iT 4
XY12 XYfiT 0
Pulley 1 contact pts
XY21 XYf2iT 0
XY23 XYfbiT 1
Pulley 2 contact pts
XY32 XYfb2iT 1
XY34 XYbfiT 2
Pulley 3 contact pts
XY43 XYbf2iT 2
XY45 XYfbiT 3
Pulley 4 contact pts
XY54 XYfb2iT 3
XY51 XYbfiT 4
Pulley 5 contact pts
By observation the lengths of span i is the following
L1 Lf 0( ) L2 Lb 1( ) L3 Lb 2( ) L4 Lb 3( ) L5 Lb 4( ) Li
L1
L2
L3
L4
L5
mm
i Angle between horizontal and span of ith pulley
i
atan
XY121
XY211
XY12
0XY21
0
atan
XY231
XY321
XY23
0XY32
0
atan
XY341
XY431
XY34
0XY43
0
atan
XY451
XY541
XY45
0XY54
0
atan
XY511
XY151
XY51
0XY15
0
Appendix C 151
Pulley 1 Pulley 2 Pulley 3 Pulley 4 Pulley 5
12 i0 2 21 i0 32 i1 2 43 i2 54 i3
15 i4 2 23 i1 34 i2 45 i3 51 i4
15
21
32
43
54
12
23
34
45
51
Wrap angle i for the ith pulley
1 2 atan2 XY150
XY151
atan2 XY120
XY121
2 atan2 XY210
XY1 0
XY211
XY1 1
atan2 XY230
XY1 0
XY231
XY1 1
3 2 atan2 XY320
XY2 0
XY321
XY2 1
atan2 XY340
XY2 0
XY341
XY2 1
4 atan2 XY430
XY3 0
XY431
XY3 1
atan2 XY450
XY3 0
XY451
XY3 1
5 atan2 XY540
XY4 0
XY541
XY4 1
atan2 XY510
XY4 0
XY511
XY4 1
1
2
3
4
5
Lb length of belt
Lbelt Li1
2
0
4
p
Dpp
Input Data for B-ISG System
Kt 20626Nm
rad (spring stiffness between tensioner arms 1
and 2)
Kt1 10314Nm
rad (stiffness for spring attached at arm 1 only)
Kt2 16502Nm
rad (stiffness for spring attached at arm 2 only)
Appendix C 152
C2 Dynamic Analysis
I C K moment of inertia damping and stiffness matrices respectively
u 0 1 4 v 0 1 4 (new counter variables where final value = no of pulleys + no of ten arms)
RaD
2
Appendix C 153
RaD
2
Ii =gt moment of inertia for ith pulley where i-1 and i represent ten arms
Ii0
0
1
2
3
4
5
6
10000
2230
300
3000
300
1500
1500
I diag Ii( ) kg mm2
Ci =gt Rotational damping and belt damping for the ith pulley where i-1 and i represent tensioner arms
1000kg
m3
CrossArea 693mm2
0 M CrossArea Lbelt M 0086kg
cb 2 KbM
Lbelt
Cb
cb
cb
cb
cb
cb
Cri
0
0
010
0
010
N mmsec
rad
Ct 1000N mmsec
rad Ct1 1000 N mm
sec
rad Ct2 1000N mm
sec
rad
Cr
Cri0
0
0
0
0
0
0
0
Cri1
0
0
0
0
0
0
0
Cri2
0
0
0
0
0
0
0
Cri3
0
0
0
0
0
0
0
Cri4
0
0
0
0
0
0
0
Ct Ct1
Ct
0
0
0
0
0
Ct
Ct Ct2
Rt
Ra0
Ra1
0
0
0
0
0
0
Ra1
Ra2
0
0
Lt1 mm sin t1 32
0
0
0
Ra2
Ra3
0
Lt1 mm sin t1 34
0
0
0
0
Ra3
Ra4
0
Lt2 mm sin t2 54
Ra0
0
0
0
Ra4
0
Lt2 mm sin t2 51
Appendix C 154
Kr
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Kt Kt1
Kt
0
0
0
0
0
Kt
Kt Kt2
Tk
Kbi 0( ) Ra0
0
0
0
Kbi 4( ) Ra0
Kbi 0( ) Ra1
Kbi 1( ) Ra1
0
0
0
0
Kbi 1( ) Ra2
Kbi 2( ) Ra2
0
0
0
0
Kbi 2( ) Ra3
Kbi 3( ) Ra3
0
0
0
0
Kbi 3( ) Ra4
Kbi 4( ) Ra4
0
Kbi 1( ) Lt1 mm sin t1 32
Kbi 2( ) Lt1 mm sin t1 34
0
0
0
0
0
Kbi 3( ) Lt2 mm sin t2 54
Kbi 4( ) Lt2 mm sin t2 51
Tc
Cb0
Ra0
0
0
0
Cb4
Ra0
Cb0
Ra1
Cb1
Ra1
0
0
0
0
Cb1
Ra2
Cb2
Ra2
0
0
0
0
Cb2
Ra3
Cb3
Ra3
0
0
0
0
Cb3
Ra4
Cb4
Ra4
0
Cb1
Lt1 mm sin t1 32
Cb2
Lt1 mm sin t1 34
0
0
0
0
0
Cb3
Lt2 mm sin t2 54
Cb4
Lt2 mm sin t2 51
C matrix
C Cr Rt Tc
K matrix
K Kr Rt Tk
New Equations of Motion for Dual Drive System
I K amp C matricies rearranged to place driving pulley in 1st row + 1st column and driven in 2nd row + 2nd column
IA augment I3
I0
I1
I2
I4
I5
I6
IC augment I0
I3
I1
I2
I4
I5
I6
I1kgmm2 1 106
kg m2
0 0 0 0 0 0
Ia stack I1kgmm2 IAT 0
T
IAT 1
T
IAT 2
T
IAT 4
T
IAT 5
T
IAT 6
T
Ic stack I1kgmm2 ICT 3
T
ICT 1
T
ICT 2
T
ICT 4
T
ICT 5
T
ICT 6
T
Appendix C 155
RtA augment Rt3
Rt0
Rt1
Rt2
Rt4
RtC augment Rt0
Rt3
Rt1
Rt2
Rt4
Rta stack RtAT 3
T
RtAT 0
T
RtAT 1
T
RtAT 2
T
RtAT 4
T
RtAT 5
T
RtAT 6
T
Rtc stack RtCT 0
T
RtCT 3
T
RtCT 1
T
RtCT 2
T
RtCT 4
T
RtCT 5
T
RtCT 6
T
TkA augment Tk3
Tk0
Tk1
Tk2
Tk4
Tk5
Tk6
Tka stack TkAT 3
T
TkAT 0
T
TkAT 1
T
TkAT 2
T
TkAT 4
T
TkC augment Tk0
Tk3
Tk1
Tk2
Tk4
Tk5
Tk6
Tkc stack TkCT 0
T
TkCT 3
T
TkCT 1
T
TkCT 2
T
TkCT 4
T
TcA augment Tc3
Tc0
Tc1
Tc2
Tc4
Tc5
Tc6
Tca stack TcAT 3
T
TcAT 0
T
TcAT 1
T
TcAT 2
T
TcAT 4
T
TcC augment Tc0
Tc3
Tc1
Tc2
Tc4
Tc5
Tc6
Tcc stack TcAT 0
T
TcAT 3
T
TcAT 1
T
TcAT 2
T
TcAT 4
T
Ka Kr Rta Tka Kc Kr Rtc Tkc Ca Cr Rta Tca Cc Cr Rtc Tcc
CHECK
KA augment K3
K0
K1
K2
K4
K5
K6
KC augment K0
K3
K1
K2
K4
K5
K6
CA augment C3
C0
C1
C2
C4
C5
C6
CC augment C0
C3
C1
C2
C4
C5
C6
Appendix C 156
Kacheck stack KAT 3
T
KAT 0
T
KAT 1
T
KAT 2
T
KAT 4
T
KAT 5
T
KAT 6
T
Kccheck stack KCT 0
T
KCT 3
T
KCT 1
T
KCT 2
T
KCT 4
T
KCT 5
T
KCT 6
T
Cacheck stack CAT 3
T
CAT 0
T
CAT 1
T
CAT 2
T
CAT 4
T
CAT 5
T
CAT 6
T
Cccheck stack CCT 0
T
CCT 3
T
CCT 1
T
CCT 2
T
CCT 4
T
CCT 5
T
CCT 6
T
Results for System switching from ISG as DRIVING pulley to Crankshaft as Drivi ng Pulley
Modified Submatricies for ISG Driving Phase --gt CS Driving Phase
Unit step function to provide shift from crankshaft DRIVING case (ie ISG driven case) to crankshaft DRIVEN
case (ie ISG driving case)
H n( ) 1 n 750if
0 n 750if
lt-- crankshaft DRIVING case (Phase change bw 2 cases occurs when n
reaches start speed)
I11mod n( ) Ic0 0
H n( ) 1if
Ia0 0
H n( ) 0if
I22mod n( )submatrix Ic 1 6 1 6( )
UnitsOf I( )H n( ) 1if
submatrix Ia 1 6 1 6( )
UnitsOf I( )H n( ) 0if
K11mod n( )
Kc0 0
UnitsOf K( )H n( ) 1if
Ka0 0
UnitsOf K( )H n( ) 0if
C11modn( )
Cc0 0
UnitsOf C( )H n( ) 1if
Ca0 0
UnitsOf C( )H n( ) 0if
K22mod n( )submatrix Kc 1 6 1 6( )
UnitsOf K( )H n( ) 1if
submatrix Ka 1 6 1 6( )
UnitsOf K( )H n( ) 0if
C22modn( )submatrix Cc 1 6 1 6( )
UnitsOf C( )H n( ) 1if
submatrix Ca 1 6 1 6( )
UnitsOf C( )H n( ) 0if
K21mod n( )submatrix Kc 1 6 0 0( )
UnitsOf K( )H n( ) 1if
submatrix Ka 1 6 0 0( )
UnitsOf K( )H n( ) 0if
C21modn( )submatrix Cc 1 6 0 0( )
UnitsOf C( )H n( ) 1if
submatrix Ca 1 6 0 0( )
UnitsOf C( )H n( ) 0if
K12mod n( )submatrix Kc 0 0 1 6( )
UnitsOf K( )H n( ) 1if
submatrix Ka 0 0 1 6( )
UnitsOf K( )H n( ) 0if
C12modn( )submatrix Cc 0 0 1 6( )
UnitsOf C( )H n( ) 1if
submatrix Ca 0 0 1 6( )
UnitsOf C( )H n( ) 0if
Appendix C 157
2mod n( ) I22mod n( )1
K22mod n( ) mod n( ) sort eigenvals 2mod n( ) nmod n( )mod n( )
2
EVmodn( ) augmenteigenvec 2mod n( ) mod n( )0
max eigenvec 2mod n( ) mod n( )0
eigenvec 2mod n( ) mod n( )1
max eigenvec 2mod n( ) mod n( )1
eigenvec 2mod n( ) mod n( )2
max eigenvec 2mod n( ) mod n( )2
eigenvec 2mod n( ) mod n( )3
max eigenvec 2mod n( ) mod n( )3
eigenvec 2mod n( ) mod n( )4
max eigenvec 2mod n( ) mod n( )4
eigenvec 2mod n( ) mod n( )5
max eigenvec 2mod n( ) mod n( )5
modeshapesmod n( ) stack nmod n( )T
EVmodn( )
t 0 0001 1
mode1a t( ) EVmod100( )0
sin nmod 100( )0 t mode2a t( ) EVmod100( )1
sin nmod 100( )1 t
mode1c t( ) EVmod800( )0
sin nmod 800( )0 t mode2c t( ) EVmod800( )1
sin nmod 800( )1 t
Pulley responses amp torque requirement for crankshaft amp alternator pulleys pulley1 and 4 respectively
The system equation becomes
I14q14 -double-dot + C1144 q14 -dot + K1144 q14 + C12qm-dot + K12qm = Qc
I2qm-double-dot + C22qm-dot + K22qm + C21q1-dot + K21q1 = 0
Pulley responses
Qm = - [(K22 - 2I2) + jC22 ]-1(K21 + jC21 )Q1
Torque requirement for crank shaft Pulley 1
qc = [(K11 -2I1) + jC11 ]Q1 + (K12 + jC12 )Qm
Torque requirement for alternator shaft Pulley 4
qa = [(K44 -2I4) + jC44 ]Q4 + (K12 + jC12 )Qm
Appendix C 158
Let DRIVING pulley have a unit amplitude 1 = 1 and let the system frequency be calculated based on
engine speed n
n 60 90 6000 n( )4n
60 a n( )
2n Ra0
60 Ra3
mod n( ) n( ) H n( ) 1if
a n( ) H n( ) 0if
Ymod n( ) K22mod n( ) mod n( ) 2 I22mod n( )
j mod n( ) C22modn( )
mmod n( ) Ymod n( )( )1
K21mod n( ) j mod n( ) C21modn( )
Crankshaft amp ISG required torques
Let input from DRIVING pulley be an angular displacement with constant amplitude of angular acceleration
Ac n( ) 650 1 n( )Ac n( )
n( ) 2
Let Qm = QmQ1(n) for n lt 750
and Qm = QmQ4(n) for n gt 750
Aa n( )42
I3 3
1a n( )Aa n( )
a n( ) 2
Qc0 4
qcmod n( ) K11mod n( ) mod n( ) 2
I11mod n( )
j mod n( ) C11modn( )
1 n( ) K12mod n( ) j mod n( ) C12modn( ) mmod n( ) 1 n( )
H n( ) 1if
Qc0 H n( ) 0if
qamod n( ) K11mod n( ) mod n( ) 2
I11mod n( )
j mod n( ) C11modn( )
1a n( ) K12mod n( ) jmod n( ) C12modn( ) mmod n( ) 1a n( ) Qc0
H n( ) 0if
0 H n( ) 1if
Q n( ) 48 n
Ra0
Ra3
48
18000
(ISG torque requirement alternate equation)
Appendix C 159
Dynamic tensioner arm torques
Qtt1mod n( )Kt Kt1
UnitsOf Kt( )j mod n( )
Ct Ct1
UnitsOf Cr( )
mmod n( )4 1 n( )
H n( ) 1if
Kt Kt1
UnitsOf Kt( )j mod n( )
Ct Ct1
UnitsOf Cr( )
mmod n( )4 1a n( )
H n( ) 0if
Qtt2mod n( )Kt Kt2
UnitsOf Kt( )j mod n( )
Ct Ct2
UnitsOf Cr( )
mmod n( )5 1 n( )
H n( ) 1if
Kt Kt2
UnitsOf Kt( )j mod n( )
Ct Ct2
UnitsOf Cr( )
mmod n( )5 1a n( )
H n( ) 0if
Appendix C 160
Dynamic belt span tensions
d n( ) 1 n( ) H n( ) 1if
1a n( ) H n( ) 0if
mod n( )
d n( )
mmod n( ) d n( ) 0 0
mmod n( ) d n( ) 1 0
mmod n( ) d n( ) 2 0
mmod n( ) d n( ) 3 0
mmod n( ) d n( ) 4 0
mmod n( ) d n( ) 5 0
Tm n( ) j n( )Tcc
UnitsOf Tcc( )
Tkc
UnitsOf Tkc( )
mod n( )
H n( ) 1if
j n( )Tca
UnitsOf Tca( )
Tka
UnitsOf Tka( )
mod n( )
H n( ) 0if
Tm n( ) j n( )Tcc
UnitsOf Tcc( )
Tkc
UnitsOf Tkc( )
mod n( )
H n( ) 1if
j n( )Tca
UnitsOf Tca( )
Tka
UnitsOf Tka( )
mod n( )
H n( ) 0if
(tensions for driving pulley belt spans)
Appendix C 161
C3 Static Analysis
Static Analysis using K Tk amp Q matricies amp Ts
For static case K = Q
Tension T = T0 + Tks
Thus T = K-1QTks + T0
Q1 68N m Qt1 0N m Qt2 0N m Ts 300N
Qc
Q4
Q2
Q3
Q5
Qt1
Qt2
Qc
5
2
0
0
0
0
J Qa
Q1
Q2
Q3
Q5
Qt1
Qt2
Qa
68
2
0
0
0
0
N m
cK22mod 900( )( )
1
N mQc A
K22mod 600( )1
N mQa
a
A0
A1
A2
0
A3
A4
A5
0
c1
c2
c0
c3
c4
c5
Tc Tk Ts Ta Tk a Ts
162
APPENDIX D
MATLAB Functions amp Scripts
D1 Parametric Analysis
D11 TwinMainm
The following function script performs the parametric analysis for the B-ISG system with a Twin
Tensioner It calls the function TwinTenStaticTensionm The parametric analysis perturbs a
single input parameter for the called function TwinTenStaticTensionm The main function takes
an initial input value for the Twin Tensioner‟s stiffness parameters Kto Kt1o Kt2o and
geometric parameters D3o D5o X3o Y3o X5o and Y5o An input parameter is allowed to
increment by six percent over a range from sixty percent below its initial value to sixty percent
above its initial value The coordinate parameters are incremented through a mesh of Cartesian
points with prescribed boundaries The TwinMainm function plots the parametric results
______________________________________________________________________________
clc
clear all
Static tension for single tensioner system for CS and Alt driving
Initial Conditions
Kto = 20626
Kt1o = 10314
Kt2o = 16502
D3o = 007240
D5o = 007240
X3o =0292761
Y3o =087
X5o =12057
Y5o =09193
Pertubations of initial parameters
Kt = (Kto-060Kto)006Kto(Kto+060Kto)
Kt1 = (Kt1o-060Kt1o)006Kt1o(Kt1o+060Kt1o)
Kt2 = (Kt2o-060Kt2o)006Kt2o(Kt2o+060Kt2o)
D3 = (D3o-060D3o)006D3o(D3o+060D3o)
D5 = (D5o-060D5o)006D5o(D5o+060D5o)
No of data points
s = 21
T = zeros(5s)
Ta = zeros(5s)
Parametric Plots
for i = 1s
Appendix D 163
[T(i)Ta(i)] = TwinTenStaticTension(Kt(i)Kt1oKt2oD3oD5oX3oY3oX5oY5o)
end
figure
[AXH1H2] = plotyy(Kt()T(1)Kt()Ta(4)plot) hold on
H3 = line(Kt()T(5)ParentAX(1)) hold on
H4 = line(Kt()Ta(3)Parent AX(2)) hold off
set(AXYLimModeautoYTickModeauto)
title(Static Tension vs Coupled Stiffness bw Arms 1 amp 2)
xlabel(Kt (Nmrad))
set(get(AX(1)Ylabel)StringCS Span Tension (N))
set(get(AX(2)Ylabel)StringISG Span Tension (N))
set(H1LineStyle-Colorb)
set(H2LineStyle-Colorg)
set(H3LineStyleColorget(H1Color))
set(H4LineStyleColorget(H2Color))
legend([H1H2H3H4]CS Tautest SpanISG Tautest SpanCS Slackest SpanISG
Slackest SpanLocationBest)
for i = 1s
[T(i)Ta(i)] = TwinTenStaticTension(KtoKt1(i)Kt2oD3oD5oX3oY3oX5oY5o)
end
figure
[AXH1H2] = plotyy(Kt1()T(1)Kt1()Ta(4)plot) hold on
H3 = line(Kt1()T(5)ParentAX(1)) hold on
H4 = line(Kt1()Ta(3)Parent AX(2)) hold off
set(AXYLimModeautoYTickModeauto)
title(Static Tension vs Tensioner Arm 1 Stiffness)
xlabel(Kt1 (Nmrad))
set(get(AX(1)Ylabel)StringCS Span Tension (N))
set(get(AX(2)Ylabel)StringISG Span Tension (N))
set(H1LineStyle-Colorb)
set(H2LineStyle-Colorg)
set(H3LineStyleColorget(H1Color))
set(H4LineStyleColorget(H2Color))
legend([H1H2H3H4]CS Tautest SpanISG Tautest SpanCS Slackest SpanISG
Slackest SpanLocationBest)
for i = 1s
[T(i)Ta(i)] = TwinTenStaticTension(KtoKt1oKt2(i)D3oD5oX3oY3oX5oY5o)
end
figure
[AXH1H2] = plotyy(Kt2()T(1)Kt2()Ta(4)plot) hold on
H3 = line(Kt2()T(5)ParentAX(1)) hold on
H4 = line(Kt2()Ta(3)Parent AX(2)) hold off
set(AXYLimModeautoYTickModeauto)
title(Static Tension vs Tensioner Arm 2 Stiffness)
xlabel(Kt2 (Nmrad))
set(get(AX(1)Ylabel)StringCS Span Tension (N))
Appendix D 164
set(get(AX(2)Ylabel)StringISG Span Tension (N))
set(H1LineStyle-Colorb)
set(H2LineStyle-Colorg)
set(H3LineStyleColorget(H1Color))
set(H4LineStyleColorget(H2Color))
legend([H1H2H3H4]CS Tautest SpanISG Tautest SpanCS Slackest SpanISG
Slackest SpanLocationBest)
for i = 1s
[T(i)Ta(i)] = TwinTenStaticTension(KtoKt1oKt2oD3(i)D5oX3oY3oX5oY5o)
end
figure
[AXH1H2] = plotyy(D3()T(1)D3()Ta(4)plot) hold on
H3 = line(D3()T(5)ParentAX(1)) hold on
H4 = line(D3()Ta(3)Parent AX(2)) hold off
set(AXYLimModeautoYTickModeauto)
title(Static Tension vs Tensioner Pulley 1 Diameter)
xlabel(D3 (m))
set(get(AX(1)Ylabel)StringCS Span Tension (N))
set(get(AX(2)Ylabel)StringISG Span Tension (N))
set(H1LineStyle-Colorb)
set(H2LineStyle-Colorg)
set(H3LineStyleColorget(H1Color))
set(H4LineStyleColorget(H2Color))
legend([H1H2H3H4]CS Tautest SpanISG Tautest SpanCS Slackest SpanISG
Slackest SpanLocationBest)
for i = 1s
[T(i)Ta(i)] = TwinTenStaticTension(KtoKt1oKt2oD3oD5(i)X3oY3oX5oY5o)
end
figure
[AXH1H2] = plotyy(D5()T(1)D5()Ta(4)plot) hold on
H3 = line(D5()T(5)ParentAX(1)) hold on
H4 = line(D5()Ta(3)Parent AX(2)) hold off
set(AXYLimModeautoYTickModeauto)
title(Static Tension vs Tensioner Pulley 2 Diameter)
xlabel(D5 (m))
set(get(AX(1)Ylabel)StringCS Span Tension (N))
set(get(AX(2)Ylabel)StringISG Span Tension (N))
set(H1LineStyle-Colorb)
set(H2LineStyle-Colorg)
set(H3LineStyleColorget(H1Color))
set(H4LineStyleColorget(H2Color))
legend([H1H2H3H4]CS Tautest SpanISG Tautest SpanCS Slackest SpanISG
Slackest SpanLocationBest)
Mesh points
m = 101
n = 101
Appendix D 165
T = zeros(5nm)
Ta = zeros(5nm)
[ixxiyy] = meshgrid(1m1n)
minX3 = 0260200
maxX3 = 0317677
minY3 = -0056640
maxY3 = 0228456
midX3 = 0311641
X3 = minX3 + (ixx-1)(maxX3-minX3)(m-1)
Y3 = minY3 + (iyy-1)(maxY3-minY3)(n-1)
for i = 1n
for j = 1m
if ((X3(ij)lt midX3)ampamp(Y3(ij)gt=(sqrt((0087945^2)-((X3(ij)-0224)^2)-
006395)))ampamp(Y3(ij)lt=(-1sqrt(((00703^2)-((X3(ij)-
024759)^2)))+016664)))||((X3(ij)gt=midX3)ampamp(Y3(ij)gt=(35548X3(ij)-
11134868))ampamp(Y3(ij)lt=(-1(sqrt(((00703^2)-((X3(ij)-024759)^2))))+016664))) mx+b
lt= y lt= circle4
[T(ij)Ta(ij)] =
TwinTenStaticTension(KtoKt1oKt2oD3oD5oX3(ij)Y3(ij)X5oY5o)
else
T(ij) = zeros(511)
Ta(ij) = zeros(511)
end
end
end
figure
Z1 = squeeze(T(1))
surf(X3Y3real(Z1))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(CS DRIVING Tautest Span Tension vs Tensioner Pulley 1 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(CS Span Tension (N))
figure
Z5 = squeeze(T(5))
surf(X3Y3real(Z5))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(CS DRIVING Slackest Span Tension vs Tensioner Pulley 1 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
Appendix D 166
zlabel(CS Span Tension (N))
figure
Za4 = squeeze(Ta(4))
surf(X3Y3real(Za4))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(ISG DRIVING Tautest Span Tension vs Tensioner Pulley 1 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(ISG Span Tension (N))
figure
Za3 = squeeze(Ta(3))
surf(X3Y3real(Za3))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(ISG DRIVING Slackest Span Tension vs Tensioner Pulley 1 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(ISG Span Tension (N))
minX5 = -0037093
maxX5 = 0212509
minY5 = 00362
maxY5 = 0228456
midX5a = 0131965
midX5b = 017729
X5 = minX5 + (ixx-1)(maxX5-minX5)(m-1)
Y5 = minY5 + (iyy-1)(maxY5-minY5)(n-1)
for i = 1n
for j = 1m
if
(X5(ij)ltmidX5a)ampamp(Y5(ij)lt=(0386X5(ij)+0146468))ampamp(Y5(ij)gt=(sqrt((0136525^2)-
(X5(ij)^2))))
[T(ij)Ta(ij)] =
TwinTenStaticTension(KtoKt1oKt2oD3oD5oX3oY3oX5(ij)Y5(ij))
elseif
((X5(ij)gt=midX5a)ampamp(X5(ij)ltmidX5b))ampamp(Y5(ij)gt=00362)ampamp(Y5(ij)lt=(0386X5(ij)+0
146468))
[T(ij)Ta(ij)] =
TwinTenStaticTension(KtoKt1oKt2oD3oD5oX3oY3oX5(ij)Y5(ij))
elseif (X5(ij)gt=midX5b)ampamp(Y5(ij)gt=(sqrt((00703^2)-(((X5(ij)-
024759)^2)))+016664))ampamp(Y5(ij)lt=(0386X5(ij)+0146468))
Appendix D 167
[T(ij)Ta(ij)] =
TwinTenStaticTension(KtoKt1oKt2oD3oD5oX3oY3oX5(ij)Y5(ij))
else
T(ij) = zeros(511)
Ta(ij) = zeros(511)
end
end
end
figure
Z1 = squeeze(T(1))
surf(X5Y5real(Z1))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(CS DRIVING Tautest Span Tension vs Tensioner Pulley 2 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(CS Span Tension (N))
figure
Z5 = squeeze(T(5))
surf(X5Y5real(Z5))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(CS DRIVING Slackest Span Tension vs Tensioner Pulley 2 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(CS Span Tension (N))
figure
Za4 = squeeze(Ta(4))
surf(X5Y5real(Za4))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(ISG DRIVING Tautest Span Tension vs Tensioner Pulley 2 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(ISG Span Tension (N))
figure
Za3 = squeeze(Ta(3))
surf(X5Y5real(Za3))
ZLim([50 500])
axis tight
Appendix D 168
colormap jet
colorbar
title(ISG DRIVING Slackest Span Tension vs Tensioner Pulley 2 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(ISG Span Tension (N))
D12 TwinTenStaticTensionm
The function TwinTenStaticTensionm simulates the static model of the B-ISG system with a
Twin Tensioner This function returns 3 vectors the static tension of each belt span in the
crankshaft- and ISG-driving phases of operation and the angle of displacement of each rigid
body in the ISG- driving phase It takes the input parameters kt kt1 kt2 for the tensioner arm
stiffness D3 and D5 for the tensioner pulley diameters and X3Y3 X5 and Y5 for the tensioner
arm pulley coordinates The function is called in the parametric analysis solution script
TwinMainm and in the optimization solution script OptimizationTwinm
D2 Optimization
D21 OptimizationTwinm
The following script is for the main function OptimizationTwinm It performs an optimization
search for the B-ISG system with a Twin Tensioner It takes an input for a parameter vector
containing values for the design variables The program calls the objective function
objfunTwinm and the constraint function confunTwinm The program can perform a genetic
algorithm (GA) optimization search or a hybrid GA optimization that includes a localized search
The optimal solution vector corresponding to the design variables and the optimal objective
function value is returned The program inputs the optimized values for the design variables into
TwinTenStaticTensionm This called function returns the optimized static state of tensions for
the crankshaft- and ISG- driving phases and for the angle of displacement of the rigid bodies in
the ISG driving phase
______________________________________________________________________________
clc
clear all
Initial values for variables
Kto = 20626
Kt1o = 10314
Kt2o = 16502
X3o = 0292761
Y3o = 0087
X5o = 012057
Appendix D 169
Y5o = 009193
w0 =[Kto Kt1o Kt2o X3o Y3o X5o Y5o] Start Point (row vector)
Variable ranges
minKt = Kto - 1Kto
maxKt = Kto + 1Kto
minKt1 = Kt1o - 1Kt1o
maxKt1 = Kt1o + 1Kt1o
minKt2 = Kt2o - 1Kt2o
maxKt2 = Kt2o + 1Kt2o
minX3 = 0260200
maxX3 = 0317677
minY3 = -0056640
maxY3 = 0228456
minX5 = -0037093
maxX5 = 0212509
minY5 = 00362
maxY5 = 0228456
ObjectiveFunction = objfunTwin
nvars = 7 Number of variables
ConstraintFunction = confunTwin
Uncomment next two lines (and comment the two functions after them) to use GA algorithm
for optimization
options = gaoptimset(InitialPopulationw0PopInitRange[minKt minKt1 minKt2 minX3
minY3 minX5 minY5 maxKt maxKt1 maxKt2 maxX3 maxY3 maxX5
maxY5]PopulationSize100Displayfinal)
[wfvalexitflagoutput] =
ga(ObjectiveFunctionnvars[][][][][][]ConstraintFunctionoptions)
fminconOptions = optimset(DisplayiterLargeScaleoff) Largescale off since gradient not
provided
options = gaoptimset(InitialPopulationw0PopInitRange[minKt minKt1 minKt2 minX3
minY3 minX5 minY5 maxKt maxKt1 maxKt2 maxX3 maxY3 maxX5
maxY5]PopulationSize100HybridFcnfmincon fminconOptions)
[wfvalexitflagoutput] = ga(ObjectiveFunctionnvars[][][][][][]ConstraintFunctionoptions)
[TTaThetaDegA] = TwinTenStaticTension(w(1)w(2)w(3)w(4)w(5)w(6)w(7))
D22 confunTwinm
The constraint function confunTwinm is used by the main optimization program to ensure
input values are constrained within the prescribed regions The function makes use of inequality
constraints for seven constrained variables corresponding to the design variables It takes an
input vector corresponding to the design variables and returns a data set of the vector values that
satisfy the prescribed constraints
Appendix D 170
D23 objfunTwinm
This function is the objective function for the main optimization program It outputs a value for
a weighted objective function or a non-weighted objective function relating the optimization of
the static tension The program takes an input vector containing a set of values for the design
variables that are within prescribed constraints The description of the function is similar to
TwinTenStaticTensionm but differs in the fact that it only returns a scalar value which is the
value of the objective function
171
VITA
ADEBUKOLA OLATUNDE
Email adebukolaolatundegmailcom
Adebukola Olatunde is a graduate research student at the University of Toronto in Toronto
Ontario Canada She obtained a Bachelor‟s Degree in Mechanical Engineering from McMaster
University in Hamilton Ontario Canada in 2002 Upon graduation she pursued a graduate
degree in mechanical engineering at the University of Toronto with a specialization in
mechanical systems dynamics and vibrations and environmental engineering In September
2008 she completed the requirements for the Master of Applied Science degree in Mechanical
Engineering She has held the position of teaching assistant for undergraduate courses in
dynamics and vibrations Adebukola has completed course work in professional education She
is a registered member of professional engineering organizations including the Professional
Engineer‟s of Ontario Engineer-in-Training program the Canadian Society of Mechanical
Engineers and the National Society of Black Engineers She intends to practice as a professional
engineering consultant in mechanical design
vii
44 Conclusion 92
Chapter 5 OPTIMIZATION OF A B-ISG TWIN TENSIONER 95
51 Optimization Problem 95
511 Selection of Design Variables 95
512 Objective Function amp Constraints 97
52 Optimization Method 100
521 Genetic Algorithm 100
522 Hybrid Optimization Algorithm 101
53 Results and Discussion 101
531 Parameter Settings amp Stopping Criteria for Simulations 101
532 Optimization Simulations 102
533 Discussion 106
54 Conclusion 109
Chapter 6 CONCLUSION AND RECOMMENDATIONS111
61 Summary 111
62 Conclusion 112
63 Recommendations for Future Work 113
REFERENCES 116
APPENDICIES 123
A Passive Dual Tensioner Designs from Patent Literature 123
B B-ISG Serpentine Belt Drive with Single Tensioner Equation of Motion 138
C MathCAD Scripts 145
C1 Geometric Analysis 145
C2 Dynamic Analysis 152
C3 Static Analysis 161
D MATLAB Functions amp Scripts 162
D1 Parametric Analysis 162
D11 TwinMainm 162
D12 TwinTenStaticTensionm 168
D2 Optimization 168
D21 OptimizationTwinm - Optimization Function 168
viii
D22 confunTwinm 169
D23 objfunTwinm 170
VITA 171
ix
LIST OF TABLES
21 Passive Dual Tensioner Designs from Patent Literature
31 Selected Contact Point Types on the ith Pulley and Types for the ith Belt Span
32 Coordinate Points for Pulley Centres and Twin Tensioner Pivot
33 Geometric Results of B-ISG System with Twin Tensioner
34 Data for Input Parameters used in Dynamic and Static Computations
35 Static Solution for Belt Span Tensions in Crankshaft and ISG Driving Cases for a B-ISG
Serpentine Belt Drive with a Single Tensioner
36 Static Solution for Belt Span Tensions in Crankshaft and ISG Driving Cases for a B-ISG
Serpentine Belt Drive with a Twin Tensioner
41 Initial Values Increments and Ranges for Parameters of Twin Tensioner
51 Summary of Parametric Analysis Data for Twin Tensioner Properties
52a GA Optimization Results for Twin Tensioner Parameters and Objective Function
52b Computations for Tensions and Angles from GA Optimization Results
53a Hybrid Optimization Results for Twin Tensioner Parameters and Objective Function
53b Computations for Tensions and Angles from Hybrid Optimization Results
54a Non-Weighted Optimization Results for Twin Tensioner Parameters and Objective
Function
54b Computations for Tensions and Angles from Non-Weighted Optimizations
x
55 Weighted Optimization Results for Static Tensions for Optimal B-ISG System with a
Twin Tensioner
56 Non-Weighted Optimization Results for Static Tensions for Optimal B-ISG System with a
Twin Tensioner
xi
LIST OF FIGURES
21 Hybrid Functions
31 Schematic of the Twin Tensioner
32 B-ISG Serpentine Belt Drive with Twin Tensioner
33 Angles Coordinates and Possible Belt Contact Points for the ith and i+1th Pulleys
34 Twin Tensioner Free Body Diagram of a Four-Degree of Freedom System
35 Free Body Diagram for Non-Tensioner Pulleys
36a ISG Driving Case Natural Frequency of System and Mode Shapes for Responsive Rigid
Bodies
36b ISG Driving Case First Mode Responses
36c ISG Driving Case Second Mode Responses
37a Crankshaft Driving Case Natural Frequency of System and Mode Shapes for Responsive
Rigid Bodies
37b Crankshaft Driving Case First Mode Responses
37c Crankshaft Driving Case Second Mode Responses
38 Crankshaft Pulley Dynamic Response (for crankshaft driven case)
39 ISG Pulley Dynamic Response (for ISG driven case)
310 Air Conditioner Pulley Dynamic Response
311 Tensioner Pulley 1 Dynamic Response
xii
312 Tensioner Pulley 2 Dynamic Response
313 Tensioner Arm 1 Dynamic Response
314 Tensioner Arm 2 Dynamic Response
315 Required Driving Torque for the ISG Pulley
316 Required Driving Torque for the Crankshaft Pulley
317 Dynamic Torque for Tensioner Arm 1
318 Dynamic Torque for Tensioner Arm 2
319 Span 1 (between crankshaft and air conditioner) Dynamic Belt Span Tension
320 Span 2 (between air conditioner and tensioner 1) Dynamic Belt Span Tension
321 Span 3 (between tensioner 1 and ISG) Dynamic Belt Span Tension
322 Span 4 (between ISG and tensioner 2) Dynamic Belt Span Tension
323 Span 5 (between tensioner 2 and crankshaft) Dynamic Belt Span Tension
324 B-ISG Serpentine Belt Drive with Single Tensioner
41a Regions 1 and 2 and their Associated Guidelines for Coordinates of Tensioner Pulleys 1
amp 2
41b Regions 1 and 2 in Cartesian Space
42 Parametric Analysis for Coupled Stiffness Arm Constant kt (N∙mrad)
43 Parametric Analysis for Stiffness of Arm 1 kt1 (N∙mrad)
44 Parametric Analysis for Stiffness of Arm 2 kt2 (N∙mrad)
xiii
45 Parametric Analysis for Pulley 1 Diameter D3 (m)
46 Parametric Analysis for Pulley 2 Diameter D5 (m)
47 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Tautest Span
Tension in Crankshaft Driving Case
48 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Slackest Span
Tension in Crankshaft Driving Case
49 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Tautest Span
Tension in ISG Driving Case
410 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Slackest Span
Tension in ISG Driving Case
411 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Tautest Span
Tension in Crankshaft Driving Case
412 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Slackest Span
Tension in Crankshaft Driving Case
413 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Tautest Span
Tension in ISG Driving Case
414 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Slackest Span
Tension in ISG Driving Case
51 Static Stability of the B-ISG Twin Tensioner Based on the Angular Displacement of
Tensioner Arms 1 and 2
A1 Proposed design by Bayerische Motoren Werke AG corresponding to patent nos
EP1420192-A2 and DE10253450-A1
A2a First of four proposed designs (and its various configurations) by Bosch GMBH
corresponding to patent no WO0026532-A1
A2b Second of four proposed designs by Bosch GMBH corresponding to patent no
WO0026532-A1
A2c Third of four proposed designs (and its various configurations) by Bosch GMBH
corresponding to patent no WO0026532-A1
A2d Fourth of four proposed designs (and its various configurations) by Bosch GMBH
corresponding to patent no WO0026532-A1
A3 Proposed design by Daimler Chrysler AG corresponding to patent no DE10324268-A1
A4 Proposed design by Dayco Products LLC corresponding to patent no US6942589-B2
xiv
A5 Proposed design by Gates Corp corresponding to patent no WO2003038309-A
A6 Proposed design by General Motors Corp corresponding to patent no US20060287146-
A1
A7 Proposed design by INA Schaeffler KG corresponding to patent no DE10044645-A1
A8a First of three proposed designs by INA Schaeffler KG corresponding to patent no
DE10159073-A1
A8b Second of three proposed designs by INA Schaeffler KG corresponding to patent no
DE10159073-A1
A8c Third of three proposed designs by INA Schaeffler KG corresponding to patent no
DE10159073-A1
A9 Proposed design by INA Schaeffler KG corresponding to patent no DE10359641-A1
A10 Proposed design by INA Schaeffler KG corresponding to patent no EP1723350-A1
A11 Proposed design by INA Schaeffler KG corresponding to patent no EP1738093-A1
A12 Proposed design by INA Schaeffler KG corresponding to patent no DE102004012395-
A1
A13 Proposed design by INA Schaeffler KG corresponding to patent nos DE102005017038-
A1and WO2006108461-A1
A14 Proposed design by Litens Automotive GMBH et al corresponding to patent no
US20010007839-A1
A15 Proposed design by Mitsubishi Jidosha Eng KK and Mitsubishi Motor Corp
corresponding to patent no JP2005083514-A
A16 Proposed design by Nissan Motor Co Ltd corresponding to patent no JP3565040-B2
A17 Proposed design by NTN Corp corresponding to patent no JP2006189073-A
A18 Proposed design by Valeo Equipment Electriques Moteur corresponding to patent nos
EP1658432 and WO2005015007
B1 Single Tensioner B-ISG System
B2 Free-body Diagram of ith Pulley
xv
B3 Free-body Diagram of Single Tensioner
C1 Schematic of B-ISG System with Twin Tensioner
C2 Possible Contact Points
xvi
LIST OF SYMBOLS
Latin Letters
A Belt cord cross-sectional area
C Damping matrix of the system
cb Belt damping
119888119894119887 Belt damping constant of the ith belt span
119914119946119946 Damping matrix element in the ith row and ith column
ct Damping acting between tensioner arms 1 and 2
cti Damping of the ith tensioner arm
DCS Diameter of crankshaft pulley
DISG Diameter of ISG pulley
ft Belt transition frequency
H(n) Phase change function
I Inertial matrix of the system
119920119938 Inertial matrix under ISG driving phase
119920119940 Inertial matrix under crankshaft driving phase
Ii Inertia of the ith pulley
Iti Inertia of the ith tensioner arm
119920120784120784 Submatrix of inertial matrix I
j Imaginary coordinate (ie (-1)12
)
K Stiffness matrix of the system
xvii
119896119887 Belt factor
119870119887 Belt cord stiffness
119896119894119887 Belt stiffness constant of the ith belt span
kt Spring stiffness acting between tensioner arms 1 and 2
kti Coil spring of the ith tensioner arm
119922120784120784 Submatrix of stiffness matrix K
Lfi Lbi Lengths of possible belt span connections from the ith pulley
Lti Length of the ith tensioner arm
Modeia Mode shape of the ith rigid body in the ISG driving phase
Modeic Mode shape of the ith rigid body in the crankshaft driving phase
n Engine speed
N Motor speed
nCS rpm of crankshaft pulley
NF Motor speed without load
nISG rpm of ISG pulley
Q Required torque matrix
qc Amplitude of the required crankshaft torque
QcsISG Required torque of the driving pulley (crankshaft or ISG)
Qm Required torque matrix of driven rigid bodies
Qti Dynamic torque of the ith tensioner arm
Ri Radius of the ith pulley
T Matrix of belt span static tensions
xviii
Trsquo Dynamic belt tension matrix
119931119940 Damping matrix due to the belt
119931119948 Stiffness matrix due to the belt
Ti Tension of the ith belt span
To Initial belt tension for the system
Ts Stall torque
Tti Tension for the neighbouring belt spans of the ith tensioner pulley
(XiYi) Coordinates of the ith pulley centre
XYfi XYbi XYfbi
XYbfi Possible connection points on the ith pulley leading to the ith belt span
XYf2i XYb2i
XYfb2i XYbf2i Possible connection points on the ith pulley leading to the (i-1)th belt span
Greek Letters
αi Angle between the datum and the line connecting the ith and (i+1)th pulley
centres
βji Angle of orientation for the ith belt span
120597θti(t) 120579 ti(t)
120579 ti(t)
Angular displacement velocity and acceleration (rotational coordinate) of the
ith tensioner arm
120637119938 General coordinate matrix under ISG driving phase
120637119940 General coordinate matrix under crankshaft driving phase
θfi θbi Angles between the datum and the belt connection spans with lengths Lfi and
Lbi respectively
Θi Amplitude of displacement of the ith pulley
xix
θi(t) 120579 i(t) 120579 i(t) Angular position velocity and acceleration (rotational coordinate) of the ith
pulley
θti Angle of the ith tensioner arm
θtoi Initial pivot angle of the ith tensioner arm
θm Angular displacement matrix of driven rigid bodies
Θm Amplitude of displacement of driven rigid bodies
ρ Belt cord density
120601119894 Belt wrap angle on the ith pulley
φmax Belt maximum phase angle
φ0deg Belt phase angle at zero frequency
ω Frequency of the system
ωcs Angular frequency of crankshaft pulley
ωISG Angular frequency of the ISG pulley
120654119951 Natural frequency of system
1
CHAPTER 1 INTRODUCTION
11 Background
Belt drive systems are the means of power transmission in conventional automobiles The
emergence of hybrid technologies specifically the Belt-driven Integrated Starter-generator (B-
ISG) has placed higher demands on belt drives than ever before The presence of an integrated
starter-generator (ISG) in a belt transmission places excessive strain on the belt leading to
premature belt failure This phenomenon has motivated automotive makers to design a tensioner
that is suitable for the B-ISG system
The belt drive is also known interchangeably as the front-end accessory drive-belt (FEAD) the
belt accessory-drive system (BAS) or the belt transmission system In a traditional setting the
role of this system is to transmit torque generated by an internal combustion engine (ICE) in
order to reliably drive multiple peripheral devices mounted on the engine block The high speed
torque is transmitted through a crankshaft pulley to a serpentine belt The serpentine belt is a
single continuous member that winds around the driving and driven accessory pulleys of the
drive system Serpentine belts used in automotive applications consist of several layers The
load-bearing layer is a flexible member consisting of high stiffness fibers [1] It is covered by a
protective layer to guard against mechanical damage and is bound below by a visco-elastic layer
that provides the required shock absorption and grip against the rigid pulleys [1] The accessory
devices may include an alternator power steering pump water pump and air conditioner
compressor among others
Introduction 2
The B-ISG system is a transmission system characteristic to micro-hybrid automobiles It is akin
to traditional belt drives differing in the fact that an electric motor called an integrated starter-
generator (ISG) replaces the original alternator re-starts the engine from idle speed and provides
braking regeneration [2] The re-start function of the micro-hybrid transmission is known as
stop-start In the B-ISG setting the ISG is mounted on the belt drive The ISG produces a speed
of approximately 2000 to 2500rpm in order to spin the engine at approximately 750rpm and
upwards to produce an instantaneous start in the start-stop process [3] The high rotations per
minute (rpm) produced by the ISG consistently places much higher tension requirements on the
belt than when the crankshaft is driving the belt It is preferable not to exceed a range of 600N to
800N of tension on the belt since this exceeds the safe operating conditions of belts used in most
traditional drive systems [4] The traditional belt drive system‟s tensioner a single-arm
tensioner does not suitably reduce the high belt tension nor provide enough tension in the slack
belts spans occurring in the ISG phase of operation for the B-ISG system
In order for the belt to transfer torque in a drive system its initial tension must be set to a value
that is sufficient to keep all spans rigid This value must not be too low as to allow any one span
to be slack during the drive‟s phases of operation Furthermore the belt must not be ldquoinstalled
with too high a tensionrdquo since this can lead to ldquopremature failure of the bearings supporting the
drive and driven pulleys and of the belt itselfrdquo [5] The presence of a tensioning mechanism in
an automotive belt drive allows for an enhanced belt life and performance since pre-tensioning
of the belt is normally not sufficient for all phases of belt drive operation A tensioner allows for
the system to cope with moderate to severe changes in belt span tensions
Introduction 3
Traditional automotive tensioners for belt drives of an ICE consist of a single spring-loaded
arm This type of tensioner is normally designed to provide a passive response to changes in belt
span tension The introduction of the ISG electric motor into the traditional belt drive with a
single-arm tensioner results in the presence of excessively slack spans and excessively tight
spans in the belt The tension requirements in the ISG-driving phase which differ from the
crankshaft-driving phase are poorly met by a traditional single-arm passive tensioner
Tensioners can be divided into two general classes passive and active In both classes the
single-arm tensioner design approach is the norm The passive class of tensioners employ purely
mechanical power to achieve tensioning of the belt while the active class also known as
automatic tensioners typically use some sort of electronic actuation Automatic tensioners have
been employed by various automotive manufacturers however ldquosuch devices add mass
complication and cost to each enginerdquo [5]
12 Motivation
The motivation for the research undertaken arises from the undesirable presence of high belt
tension in automotive belt drives Manufacturers of automotive belt drives have presented
numerous approaches for tension mechanism designs As mentioned in the preceding section
the automation of the traditional single-arm tensioner has disadvantages for manufacturers A
survey of the literature reveals that few quantitative investigations in comparison to the
qualitative investigations provided through patent literature have been conducted in the area of
passive and dual tensioner configurations As such the author of the research project has selected
to investigate the performance of a passive twin-arm tensioner design The theoretical tensioner
Introduction 4
configuration is motivated by research and developments of industry partner Litens
Automotivendash a manufacturer of automotive belt drive systems and components Litens‟
specialty in automotive tensioners has provided a basis for the research work conducted
13 Thesis Objectives and Scope of Research
The objective of this project is to model and investigate a system containing a passive twin-arm
tensioner in a B-ISG serpentine belt drive where the driving pulley alternates between a
crankshaft pulley and an ISG pulley The modeling of a serpentine belt drive system is in
continuation of the work done by post-doctoral fellow Zhen Mu in development of the priority
software known as FEAD at the University of Toronto Firstly for the B-ISG system with a
twin-arm tensioner the geometric state and its equations of motion (EOM) describing the
dynamic and static states are derived The modeling approach was verified by deriving the
geometric properties and the EOM of the system with a single tensioner arm and comparing its
crankshaft-phase‟s simulation results with FEAD software simulations This also provides
comparison of the new twin-arm tensioner belt drive model with the former single-arm tensioner
equipped belt drive model Secondly the model for the static system is investigated through
analysis of the tensioner parameters Thirdly the design variables selected from the parametric
analysis are used for optimization of the new system with respect to its criteria for desired
performance
Introduction 5
14 Organization and Content of the Thesis
This thesis presents the investigation of a passive twin-arm tensioner design in a B-ISG
serpentine belt drive system which is distinguished by having its driving pulley alternate
between a crankshaft pulley and an ISG pulley
Chapter 2 presents the literature reviewed relevant to the area of the thesis topic The context of
the research discusses the function and location of the ISG in hybrid technologies in order to
provide a background for the B-ISG system The attributes of the B-ISG are then discussed
Subsequently a description is given of the developments made in modeling belt drive systems
At the close of the chapter the prior art in tensioner designs and investigations are discussed
The third chapter describes the system models and theory for the B-ISG system with a twin-arm
tensioner Models for the geometric properties and the static and dynamic cases are derived The
simulation results of the system model are presented
Then the fourth chapter contains the parametric analysis The methodologies employed results
and a discussion are provided The design variables of the system to be considered in the
optimization are also discussed
The optimization of a B-ISG system with a passive twin-arm tensioner is presented in Chapter 5
The evaluation of optimization methods results of optimization and discussion of the results are
included Chapter 6 concludes the thesis work in summarizing the response to the thesis
Introduction 6
objectives and concluding the results of the investigation of the objectives Recommendations for
future work in the design and analysis of a B-ISG tensioner design are also described
7
CHAPTER 2 LITERATURE REVIEW
21 Introduction
This literature review justifies the study of the thesis research the significance of the topic and
provides the overall framework for the project The design of a tensioner for a Belt-driven
Integrated Starter-generator (B-ISG) system is a link in the chain of power transmission
developments in hybrid automobiles This chapter will begin with the context of the B-ISG
followed by a review of the hybrid classifications and the critical role of the ISG for each type
The function location and structure of the B-ISG system are then discussed Then a discussion
of the modeling of automotive belt transmissions is presented A systematic review of the prior
art and current state of tensioning mechanisms for B-ISG systems amalgamates the literature and
research evidence relevant to the thesis topic which is the design of a B-ISG tensioner
The Belt-driven Integrated Starter-generator (B-ISG) system is a part of a hybrid class that is
distinguished from other hybrid classes by the structure functions and location of its ISG The
B-ISG unit is a hybrid technology applied to traditional automotive belt drives The use of a B-
ISG system to achieve a start-stop function in the car engine is estimated to cut fuel consumption
in conventional automobiles by up to ten percent and thus reduce CO2 emissions [6]
Environmental and legislative standards for reducing CO2 emissions in vehicles have called for
carmakers to produce less polluting and more efficient vehicle powertrain systems [7] The
transition to bdquocleaner‟ cars makes room for the introduction of the ISG machine into conventional
automotive belt drives [8] The reduction of CO2 emissions and the similarity of the B-ISG
Literature Review 8
transmission to that of conventional cars provide the motivation for the thesis research
Consequently the micro-hybrid class of cars is especially discussed in the literature review since
it contains the B-ISG type of transmission system The micro-hybrid class is one of several
hybrid classes
A look at the performance of a belt-drive under the influence of an ISG is rooted in the
developments of hybrid technology The distinction of the ISG function and its location in each
hybrid class is discussed in the following section
22 B-ISG System
221 ISG in Hybrids
This section of the review discusses the standard classes of hybrid cars which are full power
mild and micro- hybrids Special attention is given to hybrid vehicle architectures involving
internal combustion engines (ICEs) as the main power source This is done for the sake of
comparison between hybrid classes since the ICE is the standard power source for B-ISG micro-
hybrids which is the focus of the research The term conventional car vehicle or automobile
henceforth refers to a vehicle powered solely by a gas or diesel ICE
A hybrid vehicle has a drive system that uses a combination of energy devices This may include
an ICE a battery and an electric motor typically an ISG Two systems exist in the classification
of hybrid vehicles The older system of classification separates hybrids into two classes series
hybrids and parallel hybrids In the older system many modern hybrid vehicles have modes of
operation matching both categories classifying them under either of the two classes [9] The
Literature Review 9
new system of classification has four classes full power mild and micro Under these classes
vehicles are more often under a sole category [9] In both systems an ICE may act as the primary
source of power otherwise it may be a fuel cell The fuel used by the ICE may be gas (petrol)
diesel or an alternative fuel such as ethanol bio-diesel or natural gas
2211 Full Hybrids
In a full hybrid car the ICE is used to power the integrated starter-generator (ISG) which stores
electrical energy in the batteries to be used to power an electric traction motor [8] The electric
traction motor is akin to a second ISG as it generates power and provides torque output It also
supplies an extra boost to the wheels during acceleration and drives up steep inclines A full
hybrid vehicle is able to move by electrical power only It can be driven by the ISG powering
the electric traction motor without the engine running This silent acceleration known as electric
launch is normally employed when accelerating from standstill [9] Full hybrids can generate
and consume energy at the same time Full hybrid vehicles also use regenerative braking [8]
The ISG allows this by converting from an electric traction motor to a generator when braking or
decelerating The kinetic energy from the car‟s motion is then turned into electricity and stored
in the batteries For full hybrids to achieve this they often use break-by-wire a form of
electronically controlled braking technology
A high-voltage (ie 36- or 42-volt) ISG is employed in full hybrids to start the ICE It spins the
engine more than 900 rpm whereas conventional 12-volt starter motors spin the engine at
approximately 250 rpm [9] Thus the full hybrid vehicle is able to have an instantaneous start In
full hybrids the ISG is placed in the position of the flywheel and can have its motion decoupled
Literature Review 10
from the engine [9] The ISG device also allows full hybrids to have engine start-stop also called
an idle-stop ability The idle-stop function refers to when the engine shuts down as soon as a
vehicle stops from its ICE driving mode which saves on the fuel it normally burns while idling
[8] The vehicle returns to the engine driving mode of operation by way of the ISG‟s start-up of
the crankshaft which restarts the engine in less than 300 milliseconds [9] In summary at
standstill the tachometer of the engine drops to 0 rpm since the engine has ceased the engine is
started only when needed which is often several seconds after acceleration has begun The
engine start-stop feature is achieved by way of an electronic control system that shuts off the ICE
when it is not needed to assist in driving the wheels or to produce electricity for recharging the
batteries The start-stop feature by itself is estimated to produce a ten percent fuel gain in hybrids
over conventional vehicles particularly in urban driving conditions [9] Since the ICE is
required to provide only the average horsepower used by the vehicle the engine is downsized in
comparison to a conventional automobile that obtains all its power from an ICE Frequently in
full hybrids the ICE uses an alternative operating strategy such as the Atkinson Cycle which has
a higher efficiency while having a lower power output Examples of full hybrids include the
Ford Escape and the Toyota Prius [9]
2212 Power Hybrids
Akin to the full hybrid the ISG of the power hybrid enables the same features electric launch
regenerative braking and engine idle-stop The distinguishing characteristic from full hybrids is
the ICE is not downsized to meet only the average power demand [9] Thus the engine of a
power hybrid is large and produces a high amount of horsepower compared to the former
Overall a power hybrid has the assist of a full size ICE and therefore has more torque and a
Literature Review 11
greater acceleration performance than a full hybrid or a conventional vehicle with the same size
ICE [9] The Lexus RX400h unit is an example of a power hybrid [9]
2213 Mild Hybrids
In the hybrid types discussed thus far the ISG is positioned between the engine and transmission
to provide traction for the wheels and for regenerative braking Often times the armature or rotor
of the electric motor-generator which is the ISG replaces the engine flywheel in full and power
hybrids [9] In the case of the mild hybrid the ISG is not decoupled from the ICE and hence it is
not able to drive the wheels apart from the engine It remains that the ISG shares the same shaft
with the ICE In this environment the electric launch feature does not exist since the ISG does
not turn the wheels independently of the engine and energy cannot be generated and consumed
at the same time However the ISG of the mild hybrid allows for the remaining features of the
full hybrid regenerative braking and engine idle-stop including the fact that the engine is
downsized to meet only the average demand for horsepower Mild hybrid vehicles include the
GMC Sierra pickup and 2003 to 2005 Honda Civic models [9]
2214 Micro Hybrids
Micro hybrid is the category of hybrids that can contain a B-ISG transmission and is also closest
to modern conventional vehicles This class normally features a gas or diesel ICE [9] The
conventional automobile is modified by installing an ISG unit on the mechanical drive in place
of or in addition to the starter motor The starter motor typically 12-volts is removed only in
the case that the ISG device passes cold start testing which is also dependent on the engine size
[10] Various mechanical drives that may be employed include chain gear or belt drives or a
Literature Review 12
clutchgear arrangement The majority of literature pertaining to mechanical driven ISG
applications does not pursue clutchgear arrangements since it is associated with greater costs
and increased speed issues Findings by Henry et al [11] show that the belt drive in
comparison to chain and gear drives has a decreased cost (especially if the ISG is mounted
directly to the accessory drive) has no need for lubrication has less restriction in the packaging
environment and produces very low noise Also mounting the ISG unit on a separate belt from
that linking the accessory pulleys is undesirable since applying the ISG directly to the accessory
belt drive requires less engine transmission or vehicle modifications
As with full power and mild hybrids the presence of the ISG allows for the start-stop feature
The automobile‟s electronic control unit (ECU) is calibrated or engine control circuitry (a
separate ECU) is added to the conventional car in order to shut down the engine when the
vehicle is stopped [12] The control system also controls the charge cycle of the ISG [9] This
entails that it dictates the field current by way of a microprocessor to allow the system to defer
battery charge cycles until the vehicle is decelerating [13] This produces electricity to recharge
the battery primarily during deceleration and braking The B-ISG transmission of a micro hybrid
and its various components are discussed in the subsequent section Examples of micro hybrid
vehicles are the PSA Group‟s Citroen C2 and C3 [14] Ford‟s Fiesta [14] and BMW‟s Mini
Cooper D and various others of BMW‟s European models [15]
Literature Review 13
Figure 21 Hybrid Functions
Source Dr Daniel Kok FFA July 2004 modified [16]
Figure 21 shows that the higher the voltage available to the ISG unit the more hybrid functions
it is capable of performing It is noted that B-ISG transmissions of the micro-hybrid class may
also exceed the typical functions of micro-hybrids For instance Ford‟s HyTrans van (developed
in partnership with Ricardo UK Ltd Valeo SA Gates Corporation and the UK Department for
Transport) uses a B-ISG system and a 42-volt battery The van is diesel-powered and has
characteristics of a mild hybrid such as cold cranks and engine assists [17]
222 B-ISG Structure Location and Function
2221 Structure and Location
The ISG is composed of an electrical machine normally of the inductive type which includes a
stator (stationary part of the ISG) and a rotor (non-stationary part of the ISG) and a converter
comprising of a regulator a modulator switches and filters There are various configurations to
integrate the ISG unit into an automobile power train One configuration situates the ISG
directly on the crankshaft in the place of the present flywheel [11] This set-up is more compact
however it results in a longer power train which becomes a potential concern for transverse-
Literature Review 14
mounted engines [18] An alternative set-up is to have a side-mounted ISG This term is used to
describe the configuration of mounting the electrical device on the side of the mechanical drive
[18] As mentioned in Section 2214 a belt drive is used as the mechanical drive for the thesis
research hence the ISG is belt-mounted and the transmission becomes a belt-driven ISG system
In this arrangement the ISG replaces the alternator [13] and in some cases the starter motor may
be removed This design allows for the functions of the ISG system mentioned in the description
of the ISG role in micro-hybrids [9] The side-mounted ISG specifically the belt-mounted ISG
is more evolutionary to the conventional car since it ldquoallows for a more traditional under-hood
layoutrdquo [11]
2222 Functionalities
The primary duty of the ISG in a micro hybrid specifically in a B-ISG setting is to bring the
engine from rest to normal operating speeds within a time span ranging from 250 to 400 ms [3]
and in some high voltage settings to provide cold starting
The cold starting operation of the ISG refers to starting the engine from its off mode rather than
idle mode andor when the engine is at a low temperature for example -29 to -50 degrees
Celsius [2] If the ISG is used for cold starting the peak torque is determined by the torque
requirement for the cold starting operation of the target vehicle since it is greater than the
nominal torque For this function the ldquomachine has to provide a breakaway torque about 15 [to]
18 times the nominal cranking torque to overcome static torque and rotate the engine from 0 to
[between] 10 [and] 20rpmrdquo [2] This remains to be a challenge for the ISG as the 12-volt
architecture most commonly found in vehicles does not supply sufficient voltage [2] The
introduction of the ISG machine and other electrical units in vehicles encourages a transition
Literature Review 15
from a 12-volt or 14-volt to a 42-volt electrical architecture [19] The transition to 42-volt
architecture brings ldquopotential higher-voltage functionalities that come with an ISG systemrdquo [20]
At present ldquowhen the [ISG] machine cannot provide enough torque for initial cold engine
cranking the conventional starter will [remain] in the system and perform only for the initial
cranking while the stop-start function is taken over by the [ISG] machinerdquo [2] The ISG‟s launch
assist torque the torque required to bring the engine from idle speed to the speed at which it can
develop a higher torque output is 2000 to 2500 rpm for most gas engines [3]
Delphi‟s Energen 5 High Output 12-volt Belt-alternator-starter (or B-ISG) was implemented by
researchers on a 53 L V-8 engine with an automatic transmission in a Chevrolet Silverado truck
[21] The ISG was applied in a belt-mounted configuration and was used only for warm engine
re-starts The results of Wezenbeek et al [21] showed that the starting torque for a re-start by the
12-Volt ISG was 42 Nm ISG‟s have also been used in 14V 36V and 42V architectures [13]
23 Belt Drive Modeling
The modeling of a serpentine belt drive and tensioning mechanism has typically involved the
application of Newtonian equilibrium equations to rigid bodies in order to derive the equations of
motion for the system There are two modes of motion in a serpentine belt drive transverse
motion and rotational motion The former can be viewed as the motion of the belt directed
normal to the direction of the beltpulley contact plane similar to the vibratory motion of a taut
string that is fixed at either end However the study of the rotational motion in a belt drive is the
focus of the thesis research
Literature Review 16
Much work on the mechanics of the belt drive was carried out by Firbank [22] Firbank‟s
models helped to understand belt performance and the influence of driving and driven pulleys on
the tension member The first description of a serpentine belt drive for automotive use was in
1979 by Cassidy et al [23] and since this time there has been an increasing body of knowledge
on the mathematical modeling of serpentine belt drives Ulsoy et al [24] presented a design
methodology to improve the dynamic performance of instability mechanisms for belt tensioner
systems The mathematical model developed by Ulsoy et al [24] coupled the equations of
motion that were obtained through a dynamic equilibrium of moments about a pivot point the
equations of motion for the transverse vibration of the belt and the equations of motion for the
belt tension variations appearing in the transverse vibrations This along with the boundary and
initial conditions were used to describe the vibration and stability of the coupled belt-tensioner
system Their system also considered the geometry of the belt drive and tensioner motion
Hereafter Beikmann et al [25] predicted the belt drive vibration for a system composed of a
driving pulley driven pulley and a dynamic tensioner The authors coupled the linear equations
of transverse motion for the respective belt spans with the equations of motion for pulleys and a
tensioner This was used to form the free response of the system and evaluate its response
through a closed-form solution of the system‟s natural frequencies and mode shapes
A complex modal analysis of a serpentine belt drive system was carried out by Kraver et al [26]
to determine the effect of damping on rotational vibration mode solutions The equations of
motion developed for a multi-pulley flat belt system with viscous damping and elastic
Literature Review 17
properties including the presence of a rotary tensioner were manipulated to carry out the modal
analysis
Beikmann et al [27] also derived a nonlinear model to predict the operating state of a belt-
tensioner system by way of nonlinear numerical methods and an approximated linear closed-
form method The authors used this strategy to develop a single design parameter referred to as
a tensioner constant to measure the effectiveness of the tensioning mechanism in relation to its
operating state from a reference state The authors considered the steady state tensions in belt
spans as a result of accessory loads belt drive geometry and tensioner properties
Zhang and Zu [28] conducted a modal analysis for the response of a linear serpentine belt drive
system A non-iterative approach was used to explicitly form the equations for the system‟s
natural frequencies An exact closed-form expression for the dynamic response of the system
using eigenfunction expansion was derived with the system under steady-state conditions and
subject to harmonic excitation
The work conducted by Balaji and Mockensturm [29] considered a front-end accessory drive
(FEAD) with a decoupler or isolator attached to a pulley The rotational response for the FEAD
was found analytically by considering the system to be piecewise linear about the equilibrium
angular deflections The effect of their nonlinear terms was considered through numerical
integration of the derived equations of motion by way of the iterative methodndash fourth order
Runge-Kutta The authors in this case considered the longitudinal (ie rotational) vibration of
the belt spans only
Literature Review 18
The first to carry out the analysis of a serpentine belt drive system containing a two-pulley
tensioner was Nouri in 2005 [30] Nouri found the closed-form analytical solution of a
serpentine belt drive with a two-pulley tensioner for the case of sinusoidal excitation He
employed Runge Kutta method as well to solve the equations of motion to find the response of
the system under a general input from the crankshaft The author‟s work also included the
optimization of the tensioner design in order to minimize belt span vibrations due to crankshaft
excitation Furthermore the author applied active control techniques to the tensioner in a belt
drive system
The works discussed have made significant contributions to the research and development into
tensioner systems for serpentine belt drives These lead into the requirements for the structure
function and location of tensioner systems particularly for B-ISG transmissions
24 Tensioners for B-ISG System
241 Tensioners Structure Function and Location
Literature shows that the improvement of a serpentine belt life in a B-ISG system centers on the
tensioning mechanism redesign This mechanism as shown by researchers including
Wezenbeek et al [21] and Henry et al [11] is crucial in establishing the least tension in the belt
(above a zero value) in order to guard against failure by way of slip due to slack spans in the belt
and oscillations during engine re-start It is noted by Firbank [22] that the mechanics of a belt-
drive ldquois based on the idea that belt behaviour is governed by the elastic extension or contraction
of the belt arising from tension variationsrdquo [22] these variations may be compensated for by an
adjustable tensioner
Literature Review 19
The two types of tensioners are passive and active tensioners The former permits an applied
initial tension and then acts as an idler and normally employs mechanical power and can include
passive hydraulic actuation This type is cheaper than the latter and easier to package The latter
type is capable of continually adjusting the belt tension since it permits a lower static tension
Active tensioners typically employ electric or magnetic-electric actuation andor a combination
of active and passive actuators such as electrical actuation of a hydraulic force
Conventional belt tensioners comprise of a single tensioner arm that is fitted with a sole idler
pulley to engage a serpentine belt [31] A radial bearing is used to rotatably connect the idler
pulley to the tensioner arm [31] The tensioner arm is mounted on a pivot pin that is wrapped by
a bushing and is free to rotate [31] The pin covered by the bushing is fixed to the engine
housing [31] A rotary spring is wrapped about the bearing pin and bushing to provide a pre-
tension force to the belt via the tensioner arm and idler pulley thus taking up the slack due to the
changes in belt length [31] When the belt undergoes stretch under a load the spring drives the
tensioner arm and idler pulley further into the belt [31] Belt tension changes under the modes of
operation which can include when the crankshaft (or driving pulley) abruptly decelerates from a
steady-state condition and auxiliary components continue to rotate still in their own inherent
inertia and thus become the primary drivers [31] These fluctuations in belt tension lead to belt
flutter or skip and slip that may damage other components present in the belt drive [31]
Locating the tensioner on the slack side of the belt is intended to lower the initial static tension
[11] In conventional vehicles the engine always drives the alternator so the tensioner is located
in the belt span that links the crankshaft and alternator pulleys In a B-ISG setting the slack span
Literature Review 20
of the belt alternates between the driving mode of the ISG and the driving mode of the crankshaft
[32] Research by Henry et al [11] and also the summary of prior art for tensioners in Table
21 show that placing the idlertensioner pulley in the slack span in the case that the ISG is
driving instead of in the slack span when the crankshaft is driving allows for easier packaging
and for the least static tension Designs shown in Table 21 place the tensioneridler pulley in the
same span as Henry et al [11] or in both the slack and taut spans if using a double
tensioneridler configuration
242 Systematic Review of Tensioner Designs for a B-ISG System
The proposals for belt tensioner devices to manage the issue of high peaks in belt tension for B-
ISG settings are largely in patent records as the re-design of a tensioner has been primarily a
concern of automotive makers thus far A systematic review of the patent literature has been
conducted in order to identify evaluate and collate relevant tensioning mechanism designs
applicable to a B-ISG setting Its research objective is to influence the selection of a tensioner
configuration for the thesis study
The predefined search strategy used by the researcher has been to consider patents dating only
post-2000 as many patents dating earlier are referred to in later patents as they are developed on
in most cases by the original inventor (eg an INA Schaeffler KG patent published in 2000 may
refer to its own earlier patent presented in 1999) Patents dating pre-2000 that do not have any
successor were also considered The inclusion and exclusion criteria and rationales that were
used to assess potential patents are as follows
Inclusion of
Literature Review 21
tensioner designs with two arms andor two pivots andor two pulleys
mechanical tensioners (ie exclusion of magnetic or electrical actuators or any
combination of active actuators) in order to minimize cost
tension devices that are an independent structure apart from the ISG structure in order to
reduce the required modification to the accessory belt drive of a conventional automobile
and
advanced designs that have not been further developed upon in a subsequent patent by the
inventor or an outside party
Table 21 provides a collation of the results for the systematic review based on the selection
criteria Illustrations of the collated patent designs may be seen in Appendix A It is noted that
the patent literature pertaining to these designs in most cases provides minimal numerical data
for belt tensions achieved by the tensioning mechanism In most cases only claims concerning
the outcome in belt performance achievable by the said tension device is stated in the patent
Table 21 Passive Dual Tensioner Designs from Patent Literature
Bayerische
Motoren Werke
AG
Patents EP1420192-A2 DE10253450-A1 [33]
Design Approach
2 tensioner pulleys (idlers) and 2 tension arms are mounted outside the periphery of the belt drive these form tiltable clamping arms around a common axis of rotation
A torsion spring is used at bearing bushings to mount tension arms at ISG shaft
Each tension arm cooperates with torsion spring mechanism to rotate through a damping
device in order to apply appropriate pressure to taut and slack spans of the belt in
different modes of operation
Bosch GMBH Patent WO0026532 et al [34]
Design Approach
2 tension pulleys each one is mounted on the return and load spans of the driven and
driving pulley respectively
Idlers (tension pulleys) each connect to a spring which is attached on one end to a fixed point
Literature Review 22
Idlers‟ motions are independent of each other and correspond to the tautness or
slackness in their respective spans
Or alternatively a spring connects the idler pulleys and one of the two idlers is fixed at
its axis of rotation
Daimler Chrysler
AG
Patents DE10324268-A1 [35]
Design Approach
2 idlers are given a working force by a self-aligning bearing
Bearing supports auxiliary unit (ISG) and is arranged concentrically with the axle
auxiliary unit pulley
Dayco Products
LLC
Patents US6942589-B2 et al [36]
Design Approach
2 tension arms are each rotatably coupled to an idler pulley
One idler pulley is on the tight belt span while the other idler pulley is on the slack belt
span
Tension arms maintain constant angle between one another
One arm forms a positive differential angle with the belt and the remaining arm forms a negative differential angle with the belt
Idler pulleys are on opposite sides of the ISG pulley
Gates Corporation Patents US20060249118-A1 WO2003038309-A [37]
Design Approach
A tensioner pulley contacts the belt at the slack span during start-up (ISG-driving mode)
A tensioner is asymmetrically biased in direction tending to cause power transmission
belt to be under tension
McVicar et al
(Firm General
Motors Corp)
Patent US20060287146-A1 [38]
Design Approach
2 tension pulleys and carrier arms with a central pivot are mounted to the engine
One tension arm and pulley moderately biases one side of belt run to take up slack
during engine start-up while other tension arm and pulley holds appropriate bias against
taut span of belt
A hydraulic strut is connected to one arm to provide moderate bias to belt during normal
engine operation and velocity sensitive resistance to increasing belt forces during engine
start-up
INA Schaeffler
KG et al
Patents DE10044645-A1 [39] DE10159073-A1 [40] EP1723350-A1 et al [41]
DE10359641-A1 et al [42] EP1738093-A1 et al [43] DE102004012395-A1 [44]
WO2006108461-A1 et al [45]
Design Approach
2 tension arms and 2 pulleys approach ndash o Mutually independent tensioning arms are supported for rotation in the same
plane of the housing part
o Idler pulley corresponding to each tensioning arm engages with different
sections of belt
o When high tension span alternates with slack span of belt drive one tension
arm will increase pressure on current slack span of belt and the other will
decrease pressure accordingly on taut span
o Or when the span under highest tension changes one tensioner arm moves out
of the belt drive periphery to a dead center due to a resulting force from the taut
span of the ISG starting mode
o Deflection of the taut span acts on associated pulley to apply a counter-moment to the other idler pulley on the slack span
Literature Review 23
o The 2 lever arms are of different lengths and each have an idler pulley of
different diameters and different wrap angles of belt (see DE10045143-A1 et
al)
1 tensioner arm and 2 pulleys approach ndash
o 2 idler pulleys are pinned to a beam arranged on a clamping arm that is tiltably
linked to the beam o The ISG machine is supported by a shock absorber
o During ISG start-up one idler pulley is induced to a dead center position while
it pulls the remaining idler pulley into a clamping position until force
equilibrium takes place
o A shock absorber is laid out such that its supporting spring action provides
necessary preloading at the idler pulley in the direction of the taut span during
ISG start-up mode
Litens Automotive
Group Ltd
Patents US6506137-B2 et al [46]
Design Approach
2 tension pulleys on opposite sides of the ISG pulley engage the belt
They are positioned such that their applied forces result in opposing directed moments with respect to the tension device‟s axis of pivot
The pivot axis varies relative to the force applied to each tension pulley
Diameters of the tensioner pulleys are approximately equal and belt wrap angles of the
tensioner pulleys are approximately equal
A limited swivel angle for the tensioner arms work cycle is permitted
Mitsubishi Jidosha
Eng KK
Mitsubishi Motor
Corp
Patents JP2005083514-A [47]
Design Approach
2 tensioners are used
1 tensioner is held on the slack span of the driving pulley in a locked condition and a
second tensioner is held on the slack side of the starting (driven) pulley in a free condition
Nissan Patents JP3565040-B2 et al [48]
Design Approach
A single tensioner is on the slack span once ISG pulley is in start-up mode
The tension device is comprised of a oil pressure tensioner and a half ratchet mechanism
(a plunger which performs retreat actuation according to the energizing force of the oil
pressure spring and load received from the ISG)
The tensioner is equipped with a relief valve to keep a predetermined load lower than the
maximum load added by the ISG device
NTN Corp Patent JP2006189073-A [49]
Design Approach
An automatic tensioner is equipped with a hydraulic damper mechanism comprised of a
screw bolt using saw-screwed teeth and a cylinder nut a return spring and a spring seat
in a pressure chamber (within the screw bolt) a rod seat (that is fitted to the lower end of
the cylinder nut) a spring support (arranged on varying diameter stepped recessed
sections of the rod seat) and a check valve with an openingclosing passage
The cylinder and screw bolt act as the rigidity buffer under excessive loads during ISG
start-up mode of operation
Valeo Equipment
Electriques
Moteur
Patents EP1658432 WO2005015007 [50]
Design Approach
ldquoThe invention relates to a system or a starter (10) in which a pulley (80) is rotationally mounted on a section (22) of a shaft which axially extends inside a pulley (80) and
Literature Review 24
forwards at least partially outside a support element (200) and is characterized in that
the free front end (23) of said shaft section (22) is carried by an arm (206) connected to
the support element (200)rdquo
The author notes that published patents and patent applications may retain patent numbers for multiple patent
offices (ie European Patent Office German Patent Office etc) In such cases the published patent number or in
the absence of such a number the published patent application number has been specified However published
patent documents in the above cases also served as the document (ie identical) to the published patent if available
Quoted from patent abstract as machine translation is poor
25 Summary
The research on tensioner designs from the patent literature demonstrates a lack of quantifiable
data for the performance of a twin tensioner particularly suited to a B-ISG system The review of
the literature for the modeling theory of serpentine belt drives and design of tensioners shows
few belt drive models that are specific to a B-ISG setting Hence the literature review supports
the thesis objective of modeling a B-ISG tensioner specifically one that has a passive twin
tensioner configuration and as well measuring the tensioner‟s performance The survey of
hybrid classes reveals that the micro-hybrid class is the only class employing a closely
conventional belt transmission and hence its B-ISG transmission is applicable for tensioner
investigation The patent designs for tensioners contribute to the development of the tensioner
design to be studied in the following chapter
25
CHAPTER 3 MODELING OF B-ISG SYSTEM
31 Overview
The derivation of a theoretical model for a B-ISG system uses real life data to explore the
conceptual system under realistic conditions The literature and prior art of tensioner designs
leads the researcher to make the following modeling contributions a proposed design for a
passive two-pulley tensioner computation of geometric attributes for a B-ISG system with the
proposed tensioner and derivation of the system‟s equations of motion (EOM) under dynamic
and static states as well as deriving the EOM for the B-ISG system with only a passive single-
pulley tensioner for comparison The principles of dynamic equilibrium are applied to the
conceptual system to derive the EOM
32 B-ISG Tensioner Design
The proposed design for a passive two pulley tensioner configures two tensioners about a single
fixed pivot point in the interior space of a serpentine belt drive One end of each tensioner arm
coincides with the centre point of a tensioner pulley and this point marks the axis of rotation of
the pulley The other end of each arm is pivoted about a point so that the arms share the same
axis of rotation This conceptual design henceforth is called a Twin Tensioner Figure 31 shows
a schematic for the proposed design
Modeling of B-ISG 26
Figure 31 Schematic of the Twin Tensioner
The tensioner pulley coordinates are described by (XiYi) their radii by Ri their arm lengths Lti
and their angles θti The rotation of the arms is resisted by stiffness kt of a coil spring acting
between the two arms and spring stiffness kti acting between each arm and the pivot point The
motion of each arm is dampened by dampers and akin to the springs a damper acts between the
two arms ct and a damper cti acts between each arm and the pivot point The result is a
tensioning mechanism with four degrees of freedom (DOF) that includes independent rotations
of the two pulleys and two arms
The following section relates the geometry of the rigid bodies in a B-ISG system equipped with a
Twin Tensioner to their respective motions
Modeling of B-ISG 27
33 Geometric Model of a B-ISG System with a Twin Tensioner
The B-ISG system with the Twin Tensioner is shown in Figure 32 The geometry of the drive
provides the lengths of the belt spans and angles of wrap for the belt and pulley contact surfaces
These variables are crucial to resolve the components of forces and moment arms acting on each
rigid body in the system and are used in the derivation of the EOM in section 34 Zhen Mu‟s
geometric modeling approach [51] used in the development of the software FEAD was applied
to the Twin Tensioner system to compute the system‟s unique geometric attributes
Figure 32 B-ISG Serpentine Belt Drive with Twin Tensioner
It is noted that in Figure 31 and Figure 32 showing the schematic of the Twin Tensioner and
the overall system respectively that for the purpose of the geometric computations the forward
direction follows the convention of the numbering order counterclockwise The numbering
order is in reverse to the actual direction of the belt motion which is in the clockwise direction in
this study The fourth pulley is identified as an ISG unit pulley However the properties used
for the ISG pulley‟s geometry inertia stiffness and damping is modeled as a conventional
Modeling of B-ISG 28
alternator pulley This pulley is conceptualized as an ISG when it is modeled as the driving
pulley at which point the requirements of the ISG are solved for and its non-inertia attributes
are not needed to be ascribed
Figure 33 shows the geometric attributes needed to resolve the wrap angle of the belt on each
pulley Variables (XiYi) and XYfi XYbi XYfbi and XYbfi are the ith pulley centre coordinates and
its possible belt connection points respectively Length Lfi is the length of the span connecting
the points XYfi and XYf(i+1) or XYbi and XYb(i+1) on the ith and (i+1)th pulleys respectively
Similarly Lbi is the length of the span between XYfbi and XYfb(i+1) or XYbfi and XYbf (i+1) on the
ith and (i+1)th pulleys respectively Angles αi θfi and θbi represent the angle between a line
connecting the ith and (i+1)th pulley centres and the angles of the belt connection spans with
lengths Lfi and Lbi respectively Ri is the radius of the ith pulley
Figure 33 Angles Coordinates and Possible Belt Contact Points for the ith and i+1th Pulleys
[modified] [51]
Modeling of B-ISG 29
The angle between the horizontal and the line connecting the ith and (i+1)th pulley centres αi is
calculated using Zhen‟s method [51] This method uses the pulley‟s coordinates and a cosine
trigonometric relation
i acos
Xi 1
Xi
Xi 1
Xi
2
Yi 1
Yi
2
Yi 1
Yi
if
(31a)
i 2 acos
Xi 1
Xi
Xi 1
Xi
2
Yi 1
Yi
2
Yi 1
Yi
if
(31b)
The lengths for connecting the possible belt spans are described by the variables Lfi and Lbi
The centre point coordinates and the radii of the pulleys are related through the solution of
triangles which they form to define values of the possible belt span lengths
Lfi
Xi 1
Xi
2
Yi 1
Yi
2
Ri 1
Ri
2
(32a)
Lbi
Xi 1
Xi
2
Yi 1
Yi
2
Ri 1
Ri
2
(32b)
The set of possible belt span lengths leads to the calculation of θfi and θbi the angles between the
line connecting the ith and (i+1)th pulley centres and the possible contact point on the pulley
perimeter
Modeling of B-ISG 30
(33a)
(33b)
The array of possible belt connection points comes about from the use of the pulley centre
coordinates and their radii and the sine of the sum or differences of αi and θfi or θbi The angle
αi is calculated in equations (31a) and (31b) and angles θfi and θbi are calculated in equations
(33a) and (33b) The formula to compute the array of points is shown in equations (34) and
(35) for the ith and (i+1)th pulleys Equation (34) describes the forward belt connection point
on the ith pulley which is in the span leading forward to the next (i+1)th pulley
(34a)
(34b)
(34c)
(34d)
bi atan
Lbi
Ri
Ri 1
Modeling of B-ISG 31
Equation (35) describes the backward belt connection point on the ith pulley This point sits on
the ith pulley in the contacting belt span which leads backward to connect with the (i-1)th
pulley
(35a)
(35b)
(35c)
(35d)
The selection of the coordinates from the array of possible connection points requires a graphic
user interface allowing for the points to be chosen based on observation This was achieved
using the MathCAD software package as demonstrated in the MathCAD scripts found in
Appendix C The belt connection points can be chosen so as to have a pulley on the interior or
exterior space of the serpentine belt drive The method used in the thesis research was to plot the
array of points in the MathCAD environment with distinct symbols used for each pair of points
and to select the belt connection points accordingly By observation of the selected point types
the type of belt span connection is also chosen Selected point and belt span types are shown in
Table 31
Modeling of B-ISG 32
Table 31 Selected Contact Point Types on the ith Pulley and Types for the ith Belt Span
Pulley Forward Contact
Point
Backwards Contact
Point
Belt Span
Connection
1 Crankshaft XYf1 XYbf21 Lf1
2 Air Conditioning XYfb2 XYf22 Lb2
3 Tensioner 1 XYbf3 XYfb23 Lb3
4 AlternatorISG XYfb4 XYbf24 Lb4
5 Tensioner 2 XYbf5 XYfb25 Lb5
The inscribed angles βji between the datum and the forward connection point on the ith pulley
and βji between the datum and its backward connection point are found through solving the
angle of the arc along the pulley circumference between the datum and specified point The
wrap angle ϕi is found as the difference between the two inscribed angles for each connection
point on the pulley The angle between each belt span and the horizontal as well as the initial
angle of the tensioner arms are found using arctangent relations Furthermore the total length of
the belt is determined by the sum of the lengths of the belt spans
34 Equations of Motion for a B-ISG System with a Twin Tensioner
341 Dynamic Model of the B-ISG System
3411 Derivation of Equations of Motion
This section derives the inertia damping stiffness and torque matrices for the entire system
Moment equilibrium equations are applied to each rigid body in the system and net force
equations are applied to each belt span From these two sets of equations the inertia damping
Modeling of B-ISG 33
and stiffness terms are grouped as factors against acceleration velocity and displacement
coordinates respectively and the torque matrix is resolved concurrently
A system whose motion can be described by n independent coordinates is called an n-DOF
system Consider the free body diagram of the Twin Tensioner in Figure 34 in which each
pulley of inertia Ii is supported on an arm of inertia Iti It is assumed that the pulleys are
constrained to rotate about their respective central axes and the arms are free to rotate about their
respective pivot points then at any time the position of each pulley can be described by a
rotational coordinate θi(t) and a coordinate θti(t) can denote the rotation of each arm Thus the
tensioner system comprises of four rigid bodies where each is described by one coordinate and
hence is a four-DOF system It is important to note that each rigid body is treated as a point
mass In addition inertial rotation in the positive direction is consistent with the direction of belt
motion The belt span tensions Ti and coupled radii Ri apply moments to the pulleys
Figure 34 Twin Tensioner Free Body Diagram of a Four-Degree of Freedom System
Modeling of B-ISG 34
For the serpentine belt system considered in the thesis research there are seven rigid bodies each
having a one-DOF of motion The EOM for a seven-DOF system form second-order coupled
differential equations meaning that each equation includes all of the general coordinates and
includes up to the second-order time derivatives of these coordinates The EOM can be
obtained by applying D‟Alembert‟s principle that the sum of the moments taken about any point
including the couples equals to zero Therefore the inertial couple the product of the inertia and
acceleration is equated to the moment sum as shown in equation (35)
I ∙ θ = ΣM (35)
The moment equilibrium equations for the Twin Tensioner in Figure 34 where the positive
direction is in the clockwise direction are shown in equations (36) through to (310) The
numbering convention used for each rigid body corresponds to the labeled serpentine belt drive
system shown in Figure 32 Qi represents the required torque of the ith rigid body ci is the
damping constant of the ith rigid body βji is the angle of orientation for the ith belt span and
120597120579119905119894 120579 119905119894 and 120579 119905119894 are the angular displacement angular velocity and angular acceleration of the ith
tensioner arm The initial angle of the ith tensioner arm is described by θtoi
minusI3 ∙ θ 3 = T3 ∙ R3 minus T2 ∙ R3 minus Q3 + c3 ∙ θ 3 (36)
minusI5 ∙ θ 5 = minusT4 ∙ R5 + T5 ∙ R5 minus Q5 + c5 ∙ θ 5 (37)
Modeling of B-ISG 35
It1 ∙ θ t1 = minusTt1 ∙ Lt1 ∙ sin θto 1 minus βj4 + sin θto 1 minus βj5 minus kt ∙ partθt1 minus partθt2 minus kt1 ∙
partθt1 minus ct ∙ partθ t1 minus partθ t2 minus ct1 ∙ partθ t1 (38)
It2 ∙ θ t2 = minusTt2 ∙ Lt2 ∙ sin θto 2 minus βj2 + sin θto 1 minus βj3 minus kt ∙ partθt2 minus partθt1 minus kt2 ∙ partθt2 minus
ct ∙ partθ t2 minus partθ t1 minus ct2 ∙ partθ t2 (39)
partθt1 = θt1 minus θto 1 (310a)
partθt2 = θt2 minus θto 2 (310b)
The free body diagrams for the remaining rigid bodies crankshaft pulley air conditioner pulley
and ISG pulley are in the general form of Figure 35 The sum of the moments about the axes of
rotation are taken for these structures in equations (311) through to (313)
Figure 35 Free Body Diagram for Non-Tensioner Pulleys
Modeling of B-ISG 36
I1 ∙ θ 1 = T5 ∙ R1 minus T1 ∙ R1 + Q1 minus c1 ∙ θ 1 (311)
I2 ∙ θ 2 = T1 ∙ R2 minus T2 ∙ R2 + Q2 minus c2 ∙ θ 2 (312)
I4 ∙ θ 4 = T3 ∙ R4 minus T4 ∙ R4 + Q4 minus c4 ∙ θ 4 (313)
The relationship between belt tensions and rigid body displacements is in the general form of
equation (314) where 119827119836 and 119827119844 are damping and stiffness matrices due to the belt respectively
with each factorized by a radial arm length This relationship is described for each span in
equations (315) through to (320) The belt damping constant for the ith belt span is cib
119827prime = 119827119836 ∙ 120521 + 119827119844 ∙ 120521 (314)
T1 = To + k1b ∙ R1 ∙ θ1 minus R2 ∙ θ2 + c1
b ∙ [R1 ∙ θ 1 minus R2 ∙ θ 2] (315)
T2 = To + k2b ∙ R2 ∙ θ2 minus R3 ∙ θ3 + Lt1 ∙ [sin θto 1 minus βj2 ] ∙ (θt1 minus θto 1) + c2
b ∙ [R2 ∙ θ 2 minus R3 ∙
θ 3 + Lt1 ∙ [sin θto 1 minus βj2 ] ∙ (θ t1)] (316)
T3 = To + k3b ∙ R3 ∙ θ3 minus R4 ∙ θ4 + Lt1 ∙ [sin θto 1 minus βj3 ] ∙ (θt1 minus θto 1) + c3
b ∙ [R3 ∙ θ 3 minus R4 ∙
θ 4 + Lt1 ∙ [sin θto 1 minus βj3 ] ∙ (θ t2)] (317)
Modeling of B-ISG 37
T4 = To + k4b ∙ R4 ∙ θ4 minus R5 ∙ θ5 + Lt2 ∙ [sin θto 2 minus βj4 ] ∙ (θt2 minus θto 2) + c4
b ∙ [R4 ∙ θ 4 minus R5 ∙
θ 5 + Lt2 ∙ [sin θto 2 minus βj4 ] ∙ (θ t1)] (318)
T5 = To + k5b ∙ R5 ∙ θ5 minus R1 ∙ θ1 + Lt2 ∙ [sin θto 2 minus βj5 ] ∙ (θt2 minus θto 2) + c5
b ∙ [R5 ∙ θ 5 minus R1 ∙
θ 1 + Lt2 ∙ [sin θto 2 minus βj5 ] ∙ (θ t2)] (319)
Tprime = Ti minus To (320)
Since the applied torques on the tensioner pulleys Q3 and Q4 are zero the static equilibrium
equation of the pulleys show that the adjacent spans of each tensioner pulley are equal to each
other Hence equations (321) and (322) are denoted as follows
Tt1 = T2 = T3 (321)
Tt2 = T4 = T5 (322)
Equations (310a) (310b) and (314) through to (322) are substituted into the EOMs described
in equations (36) to (39) and (311) to (313) The newly formed equations can be arranged
and written in matrix form as shown in equations (323) through to (328) The general
coordinate matrix 120521 and its first and second derivatives are shown in the EOM below
119816 ∙ 120521 + 119810 ∙ 120521 + 119818 ∙ 120521 = 119824 (323)
Modeling of B-ISG 38
The inertia matrix I includes the inertia of each rigid body in its diagonal elements The
damping matrix C includes variables 119888119894119887 the damping of the ith belt span 119877119894 its radius 120573119895119894 its
angle 119871119905119894 the ith tensioner arm‟s length 120579119905119900119894 its initial pivot angle and 119888119905 and 119888119905119894 the ith
tensioner arm viscous damping constants Stiffness matrix K contains 119896119894119887 the ith belt span
stiffness and 119896119905 and 119896119905119894 the ith tensioner arm stiffness constants and akin to the damping
matrix the variables 119877119894 119871119905119894 120579119905119900119894 and 120573119895119894 The belt span stiffness is computed in equation
(326b) where 119870119887 represents the belt cord stiffness 119896119887 is the belt factor obtained from
experimental data 120573119895119894 is the angle of orientation for the span between the jth and ith pulleys and
ϕi is the belt wrap angle on the ith pulley
Modeling of B-ISG 39
119816 =
I1 0 0 0 0 0 00 I2 0 0 0 0 00 0 I3 0 0 0 00 0 0 I4 0 0 00 0 0 0 I5 0 00 0 0 0 0 It1 00 0 0 0 0 0 It2
(324)
119810 =
c1
b ∙ R12 + c5
b ∙ R12 + c1 minusc1
b ∙ R1 ∙ R2 0 0 minusc5b ∙ R1 ∙ R5 0 c5
b ∙ R1 ∙ Lt2 ∙ sin θto 2 minus βj5
minusc1b ∙ R1 ∙ R2 c2
b ∙ R22 + c1
b ∙ R22 + c2 minusc2
b ∙ R2 ∙ R3 0 0 c2b ∙ R2 ∙ Lt1 ∙ sin θto 1 minus βj2 0
0 minusc2b ∙ R2 ∙ R3 c3
b ∙ R32 + c2
b ∙ R32 + c3 minusc3
b ∙ R3 ∙ R4 0 C36 0
0 0 minusc3b ∙ R3 ∙ R4 c4
b ∙ R42 + c3
b ∙ R42 + c4 minusc4
b ∙ R4 ∙ R5 minusc3b ∙ R4 ∙ Lt1 ∙ sin θto 1 minus βj3 c4
b ∙ R4 ∙ Lt2 ∙ sin θto 2 minus βj4
minusc5b ∙ R1 ∙ R5 0 0 minusc4
b ∙ R4 ∙ R5 c5b ∙ R5
2 + c4b ∙ R5
2 + c5 0 C57
0 0 0 0 0 ct +ct1 minusct
0 0 0 0 0 minusct ct +ct1
(325a)
C36 = 1198773 ∙ 1198711199051 ∙ [1198883119887 ∙ 119904119894119899 1205791199051199001 minus 1205731198953 minus 1198882
119887 ∙ 119904119894119899 1205791199051199001 minus 1205731198952 ] (325b)
C57 = 1198775 ∙ 1198711199052 ∙ [1198885119887 ∙ 119904119894119899 1205791199051199002 minus 1205731198955 minus 1198884
119887 ∙ 119904119894119899 1205791199051199002 minus 1205731198954 ] (325c)
Modeling of B-ISG 40
119818 =
k1
b ∙ R12 + k5
b ∙ R12 minusk1
b ∙ R1 ∙ R2 0 0 minusk5b ∙ R1 ∙ R5 0 k5
b ∙ R1 ∙ Lt2 ∙ sin θto 2 minus βj5
minusk1b ∙ R1 ∙ R2 k2
b ∙ R22 + k1
b ∙ R22 minusk2
b ∙ R2 ∙ R3 0 0 k2b ∙ R2 ∙ Lt1 ∙ sin θto 2 minus βj2 0
0 minusk2b ∙ R2 ∙ R3 k3
b ∙ R32 + k2
b ∙ R32 minusk3
b ∙ R3 ∙ R4 0 R3 ∙ Lt1 ∙ [k3b ∙ sin θto 1 minus βj3 minus k2
b ∙ sin θto 1 minus βj2 ] 0
0 0 minusk3b ∙ R3 ∙ R4 k4
b ∙ R42 + k3
b ∙ R42 minusk4
b ∙ R4 ∙ R5 minusk3b ∙ R4 ∙ Lt1 ∙ sin θto 1 minus βj3 k4
b ∙ R4 ∙ Lt2 ∙ sin θto 2 minus βj4
minusk5b ∙ R1 ∙ R5 0 0 minusk4
b ∙ R4 ∙ R5 k5b ∙ R5
2 + k4b ∙ R5
2 0 R5 ∙ Lt2 ∙ [k5b ∙ sin θto 2 minus βj5 minus k4
b ∙ sin θto 2 minus βj4 ]
0 0 0 0 0 kt +kt1 minuskt
0 0 0 0 0 minuskt kt +kt1
(326a)
k119894b =
Kb
Li + kb ∙ Ri ∙ϕi+1
2 + Ri ∙ϕi
2
(326b)
120521 =
θ1
θ2
θ3
θ4
θ5
partθt1
partθt2
(327)
119824 =
Q1
Q2
Q3
Q4
Q5
Qt1
Qt2
(328)
Modeling of B-ISG 41
3412 Modeling of Phase Change
The phase change from the crankshaft pulley being the driving pulley to the ISG pulley being the
driving pulley is described through a conditional equality based on a set of Boolean conditions
When the crankshaft is driving the rows and the columns of the EOM are swapped such that the
new order for rows and columns is 1 (crankshaft pulley) 4 (ISG pulley) 2 (air conditioner
pulley) 3 (tensioner 1 pulley) 5 (tensioner 2 pulley) 6 (tensioner arm 1) and 7 (tensioner arm 2)
When the ISG is driving the order is the same except that the second row and second column
terms relating to the ISG pulley become the first row and first column while the crankshaft
pulley terms (previously in the first row and first column) become the second row and second
column Hence the order for all rows and columns of the matrices making up the EOM in
equation (322) switches between 1423567 (when the crankshaft pulley is driving) and
4123567 (when the ISG pulley is driving) For example in the crankshaft driving and ISG
driving phases the general coordinate matrix and the inertia matrix become the following
120521119940 =
1205791
1205794
1205792
1205793
1205795
1205971205791199051
1205971205791199052
and 120521119938 =
1205794
1205791
1205792
1205793
1205795
1205971205791199051
1205971205791199052
(329a amp b)
119816119940 =
I1 0 0 0 0 0 00 I4 0 0 0 0 00 0 I2 0 0 0 00 0 0 I3 0 0 00 0 0 0 I5 0 00 0 0 0 0 It1 00 0 0 0 0 0 It2
and 119816119938 =
I4 0 0 0 0 0 00 I1 0 0 0 0 00 0 I2 0 0 0 00 0 0 I3 0 0 00 0 0 0 I5 0 00 0 0 0 0 It1 00 0 0 0 0 0 It2
(329c amp d)
Modeling of B-ISG 42
where subscripts c and a denote the crankshaft pulley driving phase and the ISG pulley driving
phase respectively
The condition for phase change is based on the engine speed n in units of rpm Equation (330)
demonstrates the phase change
H(n) = 1 119899 ge 750 (Crankshaft driving phase)0 119899 lt 750 (ISG driving phase)
(330)
When the crankshaft pulley is the driving pulley the ISG pulley becomes the driven pulley and
following suit when the ISG pulley is the driving pulley the crankshaft pulley becomes the
driven pulley These modes of operation mean that the system will predict two different sets of
natural frequencies and mode shapes Using a Boolean condition to allow for a swap between
the first and second rows as well as between the first and second columns of the EOM matrices
I C and K allows for a continuous plot of the dynamic response to be plotted for the ISG pulley
throughout its driving and driven phases as well as for that of the crankshaft pulley
3413 Natural Frequencies Mode Shapes and Dynamic Responses
Assuming the system undergoes simple harmonic motion its matrix of natural frequencies 120596119899
and modeshapes are found by solving the eigenvalue problem shown in equation (331a)
ωn ∙ 119816120784120784 minus 11981822 ∙ 120495m = 120782 (331a)
The displacement amplitude Θm is denoted implicitly in equation (331d)
Modeling of B-ISG 43
120521119846 = θ2 θ3 θ5 θ6 partθt1 partθt2 T for H n = 1 (331b)
120521119846 = θ1 θ3 θ5 θ6 partθt1 partθt2 T for H n = 0 (331c)
θm = 120495119846 ∙ sin(ω ∙ t) (331d)
I2 and K22 are submatrices of I and K respectively meaning the first row and column of each of
the original matrices are removed The eigenvalue problem is reached by considering the
undamped and unforced motion of the system Furthermore the dynamic responses are found by
knowing that the torque requirements in the matrixndash Qm for the driven pulleys and the tensioner
arms are zero in the dynamic case which signifies a response of the system to an input solely
from the driving pulley
I1 120782120782 119816120784120784
θ 1120521 119846
+ C11 119810120783120784119810120784120783 119810120784120784
θ 1120521 119846
+ K11 119818120783120784
119818120784120783 119818120784120784 θ1
120521119846 =
QCS ISG
119824119846 (332)
1
In the case of equation (331) θm is the submatrix identified in equations (331b) through to
(331d) Therein θ1 denotes the general coordinate for the driving pulley so that in the case the
phase change function H(n) is equal to zero θ1 becomes θ4 and the order of the rows and
columns for the remaining matrices correspond to the value of H(n) as mentioned earlier in
section 3412 For simple harmonic motion the motion of the driven pulleys are described as
1 The driving torque 119876119862119878119868119878119866 denotes the crankshaft torque 119876119862119878 when the crankshaft pulley is driving or the ISG
torque 119876119868119878119866 when the ISG pulley is in its driving function
Modeling of B-ISG 44
θm = 120495119846 ∙ sin(ω ∙ t) (333)
The dynamic response of the system to an input from the driving pulley under the assumption of
sinusoidal motion is expressed in equation (334)
120495119846 = [(119818120784120784 minusω2 ∙ 119816120784120784) + 119895ω ∙ 119810120784120784]minus1 ∙ (119818120784120783 + 119895ω ∙ 119810120784120783) ∙ Θ1 (334)
3414 Crankshaft Pulley Driving Torque Acceleration and Displacement
Subsequently the crankshaft pulley driving torque acceleration and displacement are firstly
discussed It is assumed in the thesis research for the purpose of modeling that the engine
serving the crankshaft is of the four cylinder type The input torque provided by a four-cylinder
engine is assumed to be dominated by two torque pulses per revolution of the crankshaft which
is represented by the factor of 2 on the steady component of the angular velocity in equation
(335) The torque requirement of the crankshaft pulley when it is the driving pulley is
Qc = qc ∙ sin(2 ∙ ωcs ∙ t) (335)
The amplitude of the required crankshaft torque qc is expressed in equation (336) and is
derived from equation (332)
qc = K11 minus ω2 ∙ I1 + 119895 ∙ ω ∙ C11 ∙ Θ1 + (119818120783120784 + 119895 ∙ ω ∙ 119810120783120784) ∙ 120495119846 (336)
Modeling of B-ISG 45
The angular frequency for the system in radians per second (rads) ω when the crankshaft
pulley is driving can be found as a function of the engine speed in rotations per minute (rpm) n
and by taking into account the double pulse per crankshaft revolution
ω = 2 ∙ ωcs = 4 ∙ π ∙ n
60
(337)
The system is considered when the amplitude of the crankshaft‟s angular acceleration is assumed
to be constant and equal to 650 rads2 during the crankshaft pulley driving phase The amplitude
of the excitation angular input from the engine is shown in equation (339b) and is found as a
result of (338)
θ 1CS = 650 ∙ sin(ω ∙ t) (338)
θ1CS = minus650
ω2sin(ω ∙ t)
(339a) where
Θ1CS = minus650
ω2
(339b)
Modeling of B-ISG 46
3415 ISG Pulley Driving Torque Acceleration and Displacement
Secondly the torque acceleration and the displacement of the ISG pulley in its driving phase is
discussed The torque for the ISG when it is in its driving function is assumed constant Ratings
for the ISG are taken from experiments performed by researchers Wezenbeek et al [21] on an
Energen 5 High Output Belt-alternator-starter (BAS) unit from Delphi The 12-Volt BAS which
can also be called a B-ISG was reported to have a maximum allowable speed of 18000 rpm [21]
As well it was noted that the ISG pulley was sized appropriately and the engine speed was
limited to ensure that an over-speed condition of the ISG pulley would not occur [21] The stall
torque rating for the Energen ISG was reported to be 48 Nm at the electric machine shaft [21]
The formula for the torque of a permanent magnet DC motor for any given speed (equation
(340)) is used to approximate the torque of the ISG in its driving mode[52]
QISG = Ts minus (N ∙ Ts divide NF) (340)2
Knowing the stall torque (the torque at 0 rpm) Ts and the maximum rpm of the motor when it is
not under load NF allows for the torque produced 119876119868119878119866 to be found for a given motor speed N
Experimental data from Litens Automotive Group [53] shows that for engine fire-up upon ISG
re-start the crankshaft must go from 0 rpm to an idle speed of approximately 750 rpm The
pulley installed on the ISG shaft in the case of the thesis research has a diameter of 6820 mm
(DISG) while that of the crankshaft has a diameter of 20065 mm (DCS) which makes the
2 The equation for the required driving torque for the ISG pulley may also be computed from the formula shown in
(336) Figure 315 for the driving torque of the ISG pulley shows that (336) and (340) produce similar results for
the required driving torque See Figure 315 for comparison of these results
Modeling of B-ISG 47
crankshaft to ISG pulley ratio approximately 2941 This ratio is used to determine the ISG
speed in equation (341)
nISG = nCS ∙DCS
DISG
(341)
For a crankshaft speed of 750 rpm the required ISG speed nISG is found from equation (341) to
be approximately 220656 rpm Thus the ISG torque during start-up is found from equation
(340) where N is equated to the value of nISG NF is assumed to be 18000 rpm and the stall
torque is allotted the value of 48 Nm The result is a required torque of approximately 42 Nm
for the ISG The acceleration of the ISG pulley is found by taking into account the torque
developed by the rotor and the polar moment of inertia of the pulley [54]
A1ISG = θ 1ISG = QISG IISG (342)
In torsional motion the function for angular displacement of input excitation is sinusoidal since
the electric motor is assumed to be resonating As a result of constant angular acceleration the
angular displacement of the ISG pulley in its driving mode is found in equation 343
θ1ISG = Θ1ISG ∙ sin(ωISG ∙ t) (343)
Knowing that acceleration is the second derivative of the displacement the amplitude of
displacement is solved subsequently [55]
Modeling of B-ISG 48
θ 1ISG = minusωISG2 ∙ Θ
1ISG ∙ sin(ωISG ∙ t) (344)
θ 1ISG = minusωISG2 ∙ Θ
1ISG
(345a)
Θ1ISG =minusQISG IISG
ωISG2
(345b)
In this case the angular frequency for the system 120596 is equivalent to 120596119868119878119866 that is the angular
frequency of the ISG pulley which can be expressed as a function of its speed in rpm
ω = ωISG =2 ∙ π ∙ nISG
60
(346a)
or in terms of the crankshaft rpm by substituting equation (341) into (346a)
ω =2 ∙ π
60∙ nCS ∙
DCS
DISG
(346b)
3416 Tensioner Arms Dynamic Torques
The dynamic torque for the tensioner arms are shown in equations (347) and (348)
Qt1 = kt + kt1 + 119895 ∙ ω ∙ (ct + ct1) ∙ (Θt1 ∙ Θ1) (347)
Modeling of B-ISG 49
Qt2 = kt + kt2 + 119895 ∙ ω ∙ (ct + ct2) ∙ (Θt2 ∙ Θ1) (348)
3417 Dynamic Belt Span Tensions
Furthermore the dynamic belt span tensions are derived from equation (314) and described in
matrix form in equations (349) and (350)
119827prime = 119895 ∙ ω ∙ 119827119836 + 119827119844 ∙ 120495119847 (349)
where
120495119847 = Θ1
120495119846 (350)
342 Static Model of the B-ISG System
It is fitting to pursue the derivation of the static model from the system using the dynamic EOM
For the system under static conditions equations (314) and (323) simplify to equations (351)
and (352) respectively
119827prime = 119827119844 ∙ 120521 (351)
119824 = 119818 ∙ 120521 (352)
Modeling of B-ISG 50
As noted in other chapters the focus of the B-ISG tensioner investigation especially for the
parametric and optimization studies in the subsequent chapters is to determine its effect on the
static belt span tensions Therein equations (351) and (352) are used to derive the expressions
for static tension in each belt span 119931prime is the tension solely due to deflection of the belt span
Equation (320) demonstrates the relationship between the tension due to belt response and the
initial tension also known as pre-tension The static tension 119931 is found by summing the initial
tension 1198790 with the expression for the dynamic tension shown in equations (315) through to
(319) and by substituting the expressions for the rigid bodies‟ displacements from equation
(352) and the relationship shown in equation (320) into equation (351)
119827 = 119827119844 ∙ (119818minus120783 ∙ 119824) + T0 (353)3
35 Simulations
The methods used to develop the geometric dynamic and static models of the Twin Tensioner B-
ISG system in the previous sections of this chapter were verified using the software FEAD The
input data for a single tensioner B-ISG system was entered into FEAD [51] to simulate the
crankshaft driving phase alone since the ISG phase is inapplicable in the FEAD [51] software
FEAD‟s [51] results agreed with those found in the simulation of the single tensioner system‟s
geometric model and EOMs in MathCAD software Furthermore the geometric simulation
3 For the purposes of the static tension the original order for the rows and columns of the stiffness matrix K and the
torque matrix Q are maintained as depicted in (326) and (328) In performing the inverse of K and its
multiplication with Q the first row and first column (in the case of the K matrix) are removed in the crankshaft
driving case whereas the fourth row and fourth column are removed in the ISG driving case Then the product for
the displacement120637 resulting from (119922minus120783 ∙ 119928) has a zero added to serve as the first element of the column matrix in
the crankshaft driving case or as the fourth element in the ISG driving case This is shown in detail in Appendix
C3 of MathCAD scripts
Modeling of B-ISG 51
results for both of the twin and single tensioner B-ISG systems were found to be in agreement as
well
351 Geometric Analysis
The initial coordinate inputs for the centre points of the five pulleys and the Twin Tensioner
pivot point are described as Cartesian coordinates and shown in Table 32 which also includes
the diameters for the pulleys
Table 32 Coordinate Points for Pulley Centres and Twin Tensioner Pivot [56]
Rigid Body Diameter [mm] Cartesian Coordinate [Xi Yi] [mm]
1Crankshaft Pulley 20065 [00]
2 Air Conditioner Pulley 10349 [224 -6395]
3 Tensioner Pulley 1 7240 [292761 87]
4 ISG Pulley 6820 [24759 16664]
5 Tensioner Pulley 2 7240 [12057 9193]
6 Tensioner Arm Pivot --- [201384 62516]
The geometric results for the B-ISG system are shown in Table 33
Table 33 Geometric Results of B-ISG System with Twin Tensioner
Pulley Forward
Connection Point
Backward
Connection Point
Wrap
Angle
ϕi (deg)
Angle of
Belt Span
βji (deg)
Length of
Belt Span
Li (mm)
1 Crankshaft [-6818-100093] [453889475] 202996 356103 227828
2 Air
Conditioning [275299-5717] [220484 -115575] 101425 277528 14064
3 Tensioner 1 [25887599735] [256873 82257] 28126 69403 58658
4 ISG [218374184225] [27951154644] 169554 58956 129513
5 Tensioner 2 [10419659645] [15158673262] 8585 333107 65949
Total Length of Belt (mm) 1243
Modeling of B-ISG 52
352 Dynamic Analysis
The dynamic results for the system include the natural frequencies mode shapes driven pulley
and tensioner arm responses the required torque for each driving pulley the dynamic torque for
each tensioner arm and the dynamic tension for each belt span These results for the model were
computed in equations (331a) through to (331d) for natural frequencies and mode shapes in
equation (334) for the driven pulley and tensioner arm responses in equation (336) for the
crankshaft pulley driving torque in equation (340) for the ISG pulley driving torque in
equations (347) and (348) for the tensioner arm torques and lastly in equation (349) for the
dynamic tension of each belt span Figures 36 through to 323 respectively display these
results The EOM simulations can also be contrasted with those of a similar system being a B-
ISG serpentine belt drive that is equipped with a single tensioner arm and single tensioner pulley
which interacts only in the span bridging the ISG and crankshaft pulleys The EOM for a B-ISG
with a single tensioner is presented in Appendix B
It is assumed for the sake of the dynamic and static computations that the system
does not have an isolator present on any pulley
has negligible rotational damping of the pulley shafts
has negligible belt span damping and that this damping does not differ amongst
spans (ie c1b = ∙∙∙ = ci
b = 0)
has quasi-static belt stretch where its belt experiences purely elastic deformation
has fixed axes for the pulley centres and tensioner pivot
has only one accessory pulley being modeled as an air conditioner pulley and
Modeling of B-ISG 53
has a rotational belt response that is decoupled from the transverse response of the
belt
The input parameter values of the dynamic (and static) computations as influenced by the above
assumptions for the present system equipped with a Twin Tensioner are shown in Table 34
Table 34 Data for Input Parameters used in Dynamic and Static Computations [56]
Rigid Body Data
Pulley Inertia
[kg∙mm2]
Damping
[N∙m∙srad]
Stiffness
[N∙mrad]
Required
Torque
[Nm]
Crankshaft 10 000 0 0 4
Air Conditioner 2 230 0 0 2
Tensioner 1 300 1x10-4
0 0
ISG 3000 0 0 5
Tensioner 2 300 1x10-4
0 0
Tensioner Arm 1 1500 1000 10314 0
Tensioner Arm 2 1500 1000 16502 0
Tensioner Arm
couple 1000 20626
Belt Data
Initial belt tension [N] To 300
Belt cord stiffness [Nmmmm] Kb 120 00000
Belt phase angle at zero frequency [deg] φ0deg 000
Belt transition frequency [Hz] ft 000
Belt maximum phase angle [deg] φmax 000
Belt factor [magnitude] kb 0500
Belt cord density [kgm3] ρ 1000
Belt cord cross-sectional area [mm2] A 693
Modeling of B-ISG 54
These values are for the driven cases for the ISG and crankshaft pulleys respectively In the
driving case for either pulley the inertia of the rigid body is defined as 1 kg∙mm2 and the driving
torque is determined in equations (335) and (340) for the crankshaft and ISG pulleys
respectively
It is noted that because of the belt data for the phase angle at zero frequency the transition
frequency and the maximum phase angle are all zero and hence the belt damping is assumed to
be constant between frequencies These three values are typically used to generate a phase angle
versus frequency curve for the belt where the phase angle is dependent on the frequency The
curve defined by equation (354) is normally symmetric with the lowest phase angle achieved at
0 Hz and the highest phase angle achieved at the prescribed transition frequency f The belt
damping would then be found by solving for cb in the following equation
tanφ = cb ∙ 2 ∙ π ∙ f (354)
Nevertheless the assumption for constant damping between frequencies is also in harmony with
the remaining assumptions which assume damping of the belt spans to be negligible and
constant between belt spans
3521 Natural Frequency and Mode Shape
The set of natural frequencies and mode shapes for the system are shown in Figures 36 and 37
under the cases of the ISG pulley driving and the crankshaft pulley driving The forcing
frequency for the system differs for each case due to the change in driving pulley Modeic and
Modeia denote the ith rigid body according to the numbering convention used in Figure 32 in
the crankshaft and ISG driving cases respectively
Modeling of B-ISG 55
Natural Frequency ωn [Hz]
Crankshaft Pulley ΔΘ4
Air Conditioner Pulley ΔΘ
Tensioner Pulley 1 ΔΘ
Tensioner Pulley 2 ΔΘ
Tensioner Arm 1 ΔΘ
Tensioner Arm 2 ΔΘ
Figure 36a ISG Driving Case Natural Frequency of System and Mode Shapes for Responsive
Rigid Bodies
Figure 36b ISG Driving Case First Mode Responses
4 ΔΘ signifies the term amplitude of angular displacement for the indicated rigid body
Modeling of B-ISG 56
Figure 36c ISG Driving Case Second Mode Responses
Natural Frequency ωn [Hz]
ISG Pulley ΔΘ5
Air Conditioner Pulley ΔΘ
Tensioner Pulley 1 ΔΘ
Tensioner Pulley 2 ΔΘ
Tensioner Arm 1 ΔΘ
Tensioner Arm 2 ΔΘ
Figure 37a Crankshaft Driving Case Natural Frequency of System and Mode Shapes for
Responsive Rigid Bodies
5 ΔΘ signifies the term amplitude of angular displacement for the indicated rigid body
Modeling of B-ISG 57
Figure 37b Crankshaft Driving Case First Mode Responses
Figure 37c Crankshaft Driving Case Second Mode Responses
Modeling of B-ISG 58
3522 Dynamic Response
The dynamic response specifically the magnitude of angular displacement for each rigid body is
plotted in Figures 38 through to 314 as a function of the crankshaft pulley speed n This is
fitting to the analysis since the crankshaft pulley‟s rpm decides the mode of operation for the
system in particular it determines whether the crankshaft pulley or ISG pulley is the driving
pulley
Figure 38 Crankshaft Pulley Dynamic Response (for crankshaft driven case)
Figure 39 ISG Pulley Dynamic Response (for ISG driven case)
Modeling of B-ISG 59
Figure 310 Air Conditioner Pulley Dynamic Response
Figure 311 Tensioner Pulley 1 Dynamic Response
Crankshaft Driving Phase ISG
Driving
Phase
Crankshaft Driving Phase ISG
Driving
Phase
Modeling of B-ISG 60
Figure 312 Tensioner Pulley 2 Dynamic Response
Figure 313 Tensioner Arm 1 Dynamic Response
Crankshaft Driving Phase ISG
Driving
Phase
Crankshaft Driving Phase ISG
Driving Phase
Modeling of B-ISG 61
Figure 314 Tensioner Arm 2 Dynamic Response
3523 ISG Pulley and Crankshaft Pulley Torque Requirement
Figures 315 and 316 respectively showcase the required torques for the ISG pulley in its driving
mode and the crankshaft pulley in its driving mode
Figure 315 Required Driving Torque for the ISG Pulley
Crankshaft Driving Phase ISG
Driving Phase
Modeling of B-ISG 62
Figure 315 shows two plots for the required driving torque of the ISG pulley The dashed line
labeled as Q(n) simulates the application of equation (340) which models the ISG torque as a
permanent magnet DC motor The additional solid line labeled as qamod uses the formula in
equation (336) which determines the load torque of the driving pulley based on the pulley
responses Figure 315 provides a comparison of the results
Figure 316 Required Driving Torque for the Crankshaft Pulley
3524 Tensioner Arms Torque Requirements
The torque for the tensioner arms are shown in Figures 317 and 318
Modeling of B-ISG 63
Figure 317 Dynamic Torque for Tensioner Arm 1
Figure 318 Dynamic Torque for Tensioner Arm 2
3525 Dynamic Belt Span Tension
The dynamic tensions for the belt spans are shown in Figures 319 through to 323 The values
plotted represent the magnitude of the dynamic tension
Crankshaft Driving Phase ISG
Driving Phase
Crankshaft Driving Phase ISG
Driving
Phase
Modeling of B-ISG 64
Figure 319 Span 1 (between crankshaft and air conditioner) Dynamic Belt Span Tension
Figure 320 Span 2 (between air conditioner and tensioner 1) Dynamic Belt Span Tension
Crankshaft Driving Phase ISG
Driving Phase
Crankshaft Driving Phase ISG
Driving
Phase
Modeling of B-ISG 65
Figure 321 Span 3 (between tensioner 1 and ISG) Dynamic Belt Span Tension
Figure 322 Span 4 (between ISG and tensioner 2) Dynamic Belt Span Tension
Crankshaft Driving Phase ISG
Driving
Phase
Crankshaft Driving Phase ISG
Driving Phase
Modeling of B-ISG 66
Figure 323 Span 5 (between tensioner 2 and crankshaft) Dynamic Belt Span Tension
The dynamic results for the system serve to show the conditions of the system for a set of input
parameters The following chapter targets the focus of the thesis research by analyzing the affect
of changing the input parameters on the static conditions of the system It is the static results that
are the focus of the thesis and is thus analyzed in Chapters 4 and 5 in the parametric and
optimization studies respectively The dynamic analysis has been used to complete the picture of
the system‟s state under set values for input parameters
353 Static Analysis
Before looking at the static results for the system under study in brevity the static results for a
B-ISG serpentine belt drive with a single tensioner are presented In this theoretical system the
tensioner arm and tensioner pulley that interacts with the span between the air conditioner and
ISG pulleys of the original system are removed as shown in Figure 324
Crankshaft Driving Phase ISG
Driving
Phase
Modeling of B-ISG 67
Figure 324 B-ISG Serpentine Belt Drive with Single Tensioner
The complete static model as well as the dynamic model for the system in Figure 324 is found
in Appendix B The results of the static tension for each belt span of the single tensioner system
when the crankshaft is driving and the ISG is driving are shown in Table 35
Table 35 Static Solution for Belt Span Tensions in Crankshaft and ISG Driving Cases for a B-
ISG Serpentine Belt Drive with a Single Tensioner
Belt Span between Pulleys Crankshaft Driving Case [N] ISG Driving Case [N]
Crankshaft ndash Air Conditioner 481239 -361076
Air Conditioner ndash ISG 442588 -399727
ISG ndash Tensioner 29596 316721
Tensioner ndash Crankshaft 29596 316721
The tensions in Table 35 are computed with an initial tension of 300N This value for pre-
tension allows the spans in the case that the crankshaft pulley is driving to be suitably tensioned
Modeling of B-ISG 68
Whereas in the case of the ISG pulley driving the first and second spans are excessively slack
Therein an additional pretension of approximately 400N would be required which would raise
the highest tension span to over 700N This leads to the motivation of the thesis researchndash to
reduce the static belt tensions when the ISG is driving As mentioned in Chapter 1 these
tensions should be minimized to prolong belt life preferably within the range of 600 to 800N
As well it is desirable to minimize the amount of pretension exerted on the belt The current
design uses a pre-tension of 300N The above results would lead to a required pre-tension of
more than 700N to keep all spans of the belt suitably in tension (well above 0N) in order to allow
the belt to exhibit high performance in power transmission and come near to the safe threshold
This is the rationale for investigating a Twin Tensioner configuration shown in Figure 32 for
the B-ISG serpentine belt drive under study For the theoretical system with a Twin Tensioner
the following static results in Table 36 are achieved
Table 36 Static Solution for Belt Span Tensions in Crankshaft and ISG Driving Cases for a B-
ISG Serpentine Belt Drive with a Twin Tensioner
Belt Span between Pulleys Crankshaft Driving Case [N] ISG Driving Case [N]
Crankshaft ndash Air Conditioner 465848 -284152
Air Conditioner ndash Tensioner 1 427197 -322803
Tensioner 1 ndash ISG 427197 -322803
ISG ndash Tensioner 2 28057 393645
Tensioner 2 ndash Crankshaft 28057 393645
The results in Table 36 show that the span following the ISG in the case between the Tensioner
1 and ISG pulleys is less slack than in the former single tensioner set-up However there
remains an excessive amount of pre-tension required to keep all spans suitably tensioned
Modeling of B-ISG 69
36 Summary
The simulation of the model for the B-ISG system with the Twin Tensioner shows that the mode
shapes of the rigid bodies within the system (Figures 36a to 37c) are greater in magnitude when
the ISG pulley is driving than when the crankshaft pulley is driving The dynamic responses of
the system as shown in Figures 38 and 310 to 314 is small for the crankshaft pulley and are
negligible for the remaining driven bodies when the ISG is driving For the crankshaft driving
phase there is greater dynamic response for the driven rigid bodies of the system including for
that of the ISG pulley
As the engine speed increases the torque requirement for the ISG was found to vary between
approximately 41Nm and 54Nm (before dropping steeply to approximately 3Nm at an engine
speed of about 720rpm) when modeled after equation (336) or between approximately 48Nm
and 34Nm when modeled after equation (340) In contrast the torque for the crankshaft peaks
at approximately 92Nm and 52Nm at an approximate engine speed of 1450rpm and 5000rpm
respectively The dynamic torque of the first tensioner arm was shown to peak at approximately
15Nm at the transition engine speed 750rpm and again at approximately 15Nm at an
approximate engine speed of about 1450rpm A small peak of about 3Nm was also seen at an
engine speed of 5000rpm Similarly for the second tensioner arm a torque peak of
approximately 20Nm was seen at 750rpm and 1450rpm and a smaller peak of about 8Nm was
seen at an engine speed of 5000rpm
The trend for the dynamic tensions is that the peaks are highest in the ISG driving portion of the
B-ISG operation in most cases and in a few cases they are seen to be close in magnitude to that
Modeling of B-ISG 70
of the highest peaks in the crankshaft driving portion The dynamic tension for the first belt span
peaked at approximately 780Nm 830Nm and 500Nm at engine speeds of 750rpm 1450rpm
5000rpm respectively For the dynamic tension of the second belt span peaks of approximately
1250Nm 675Nm and 760Nm were seen at the same respective engine speeds for the 3 peaks of
the former span At these same engine speeds the third belt span exhibited tension peaks at
approximately 1400Nm 650Nm and 890Nm The tension peaks of the fourth span were
approximately 165Nm 150Nm and 100Nm at engine speeds 750rpm 1450rpm and 5000rpm
The fifth span experienced peaks of approximately 165Nm 170Nm and 120Nm at the same
respective engine speeds of the fourth span
The simulation results for the static tension of the B-ISG system with the Twin Tensioner reveal
that taut spans of the crankshaft driving case are lower in the ISG driving case The largest
change is an approximate decrease of 750N in spans 1 through 3 while spans 4 and 5 increase
by approximately 113N It can be seen that the spans in highest tension (1 2 and 3) in the
crankshaft driving phase become excessively slack in the ISG driving phase There is a smaller
change between the tension values for the spans in the least tension in the crankshaft driving
phase and their corresponding span in the ISG driving phase
The summary of the simulation results are used as a benchmark for the optimized system shown
in Chapter 5 The static tension simulation results are investigated through a parametric study of
the Twin Tensioner system in Chapter 4 The optimization of the system is then based on the
selected design variables from the outcome of Chapter 4
71
CHAPTER 4 PARAMETRIC ANALYSIS OF A B-ISG
TWIN TENSIONER
41 Introduction
The parameters for the proposed Twin Tensioner for a Belt-driven Integrated Starter-generator
(B-ISG) system are investigated through a parametric analysis This analysis seeks to understand
how changing one parameter influences the static belt span tensions for the system Since the
thesis research focuses on the design of a tensioning mechanism to support static tension only
the parameters specific to the actual Twin Tensioner and applicable to the static case were
considered The parameters pertaining to accessory pulley properties such as radii or various
belt properties such as belt span stiffness are not considered In the analyses a single parameter
is varied over a prescribed range while all other parameters are held constant The pivot point
described by Cartesian Coordinates [X6Y6] for the tensioner arms is held constant in all cases
42 Methodology
The parametric study method applies to the general case of a function evaluated over changes in
one of its dependent variables The methodology is illustrated for the B-ISG system‟s function
for static tension which is evaluated for each change in one of its Twin Tensioner‟s parameters
The original data used for the system is based on sample vehicle data provided by Litens [56]
Table 41 provides the initial data for the parameters as well as the incremental change and
maxima and minima limits The increment Δi for the ith parameter is chosen arbitrarily Limits
for each parameter have been chosen to be plus or minus sixty percent of its initial value
Parametric Analysis 72
Table 41 Initial Values Increments and Ranges for Parameters of Twin Tensioner
Parameter Name Initial Value Increment (+- Δi) Minimum
value Maximum value
Coupled Spring
Stiffness kt
20626
N∙mrad 1238 N∙mrad 8250 N∙mrad 33002 N∙mrad
Tensioner Arm 1
Stiffness kt1
10314
N∙mrad 0619 N∙mrad 4126 N∙mrad 16502 N∙mrad
Tensioner Arm 2
Stiffness kt2
16502
N∙mrad 0990 N∙mrad 6601 N∙mrad 26403 N∙mrad
Tensioner Pulley 1
Diameter D3 007240 m 4344 ∙ 10
-3 m 00290 m 0116 m
Tensioner Pulley 2
Diameter D5 007240 m 4344 ∙ 10
-3 m 00290 m 0116 m
Tensioner Pulley 1
Initial Coordinates
[0292761
0087] m See Figure 41 for region of possible tensioner pulley
coordinates Tensioner Pulley 2
Initial Coordinates
[012057
009193] m
The mesh of possible points for the centre coordinates of tensioner pulley 1 and tensioner pulley
2 are designated as Region 1 and Region 2 respectively in Figures 41a and 41b
Figure 41a Regions 1 and 2 and their Associated Guidelines for Coordinates of Tensioner
Pulleys 1 amp 2
CS
AC
ISG
Ten 1
Ten 11
Region II
Region I
Parametric Analysis 73
Figure 41b Regions 1 and 2 in Cartesian Space
The selection for the minimum and maximum tensioner pulley centre coordinates and their
increments are not selected arbitrarily or without derivation as the other tensioner parameters
The coordinates for the pulley centres are identified using Intergraph‟s SmartSketch software a
graphing suite in MathCAD to model the regions of space The following descriptions are used
to describe the possible positions for the tensioner pulleys
Tensioner pulleys are situated such that they are exterior to the interior space created by
the serpentine belt thus they sit bdquooutside‟ the belt loop
The highest point on the tensioner pulley does not exceed the tangent line connecting the
upper hemispheres of the pulleys on either side of it
The tensioner pulleys may not overlap any other pulley
Parametric Analysis 74
Boundaries for regions described as Region 1 in span 2 and 3 and Region 2 in span 4
and 5 is selected based on the above criteria and their lower boundaries are selected
arbitrarily
These criteria were used to define the equation for each boundary line and leads to a set of
Boolean conditions that relate the x-coordinate and y-coordinate for each Cartesian pair The
density for the mesh of points in each region is arbitrarily selected as 101 x-points and 101 y-
points in each space for the purposes of the parametric analysis The outline of this method is
described in the MATLAB scripts contained in Appendix D
The results of the parametric analysis are shown for the slackest and tautest spans in each driving
case As was demonstrated in the literature review the tautest span immediately precedes the
driving pulley and the slackest span immediately follows the driving pulley in the direction of
the belt motion Thus in the case for the crankshaft driving the tautest span is in the first span
and the slackest span is in the fifth span Whereas in the ISG driving case the tautest span is in
the fourth span and the slackest span is in the third span Hence the parametric figures in this
chapter display only the tautest and slackest span values for both driving cases so as to describe
the maximum and minimum values for tension present in the given belt
43 Results amp Discussion
431 Influence of Tensioner Arm Stiffness on Static Tension
The parametric analysis begins with changing the stiffness value for the coil spring coupled
between tensioner arms 1 and 2 This stiffness value kt is changed over a range from sixty
percent less than its initial value kt0 to sixty percent more than its original value as shown in
Parametric Analysis 75
Table 41 The results of the static tension are shown in Figure 42 for the tautest and slackest
spans for both the crankshaft and ISG driving cases
Figure 42 Parametric Analysis for Coupled Stiffness Arm Constant kt (N∙mrad)
As kt increases in the crankshaft driving phase for the B-ISG system the highest tension
decreases from 4691N to 4646N while the lowest tension decreases from 2838N to 2793N
In the ISG driving phase the highest tension increases from 378N to 3998N and the lowest
tension increases from -3384N to -3167N Thus a change of approximately -45N is found in
the crankshaft driving case and approximately +22N is found in the ISG driving case for both the
tautest and slackest spans
Parametric Analysis 76
The second parameter analyzed is the stiffness value for tensioner arm 1 The results of this are
shown in Figure 43
Figure 43 Parametric Analysis for Stiffness of Arm 1 kt1 (N∙mrad)
In Figure 43 as kt1 increases an increase from 4628N to 4681N is observed for the tension of
the tautest span when the crankshaft is driving which is a change of +53N The same value for
net change is found in the slackest span for the same driving condition whose tension increases
from 2775N to 2828N For the case when the B-ISG system is in the ISG driving phase the
change is larger a value of -261N for the tautest span that changes from 4088N to 3827N and
for the slackest span that changes from -3077N to -3338N
Parametric Analysis 77
The change in static tension for the spans as the stiffness of arm 2 varies is demonstrated in
Figure 44
Figure 44 Parametric Analysis for Stiffness of Arm 2 kt2 (N∙mrad)
In this case it is observed that as kt2 increases the tautest span for the B-ISG system in the
crankshaft driving case decreases from 4675N to 4643N as well as the slackest span which
decreases from 2822N to 279N which is an overall change of -32N for both spans Whereas in
the ISG driving case a more noticeable change is once again found a difference of +144N
This is a result of the tautest span increasing from 3863N to 4007N and the slackest span
increasing from -3301N to -3157N
Parametric Analysis 78
432 Influence of Tensioner Pulley Diameter on Static Tension
The change in the diameter of tensioner pulley 1 D3 and its effect on static tension is shown in
Figure 45
Figure 45 Parametric Analysis for Pulley 1 Diameter D3 (m)
The change in the tautest and slackest spans for the B-ISG system‟s crankshaft driving case is
from 3248N to 425N and from 1395N to 240N respectively Peaks are seen at 4799N and
2946N for the respective spans This is a change of approximately +100N and a maximum
change of 1551N for both spans For the ISG driving case the tautest and slackest spans
decrease from 1083N to 6158N and 367N to -1006N Global minimums of 3246N and -391N
for the respective spans are seen This nets a change of approximately -467N and a maximum
change of approximately -759N
Parametric Analysis 79
The effect of changing the diameter of tensioner pulley 2 on the static tension is examined in
Figure 46
Figure 46 Parametric Analysis for Pulley 2 Diameter D5 (m)
The tautest and slackest spans in the crankshaft driving mode of the belt undergo a change from
4583N to 4721N and from 273N to 2869N respectively Therein as D5 increases the trend is
that for both spans there is an increase in tension of approximately 14N Contrastingly the spans
experience a decrease in the ISG driving case as D5 increases The tension of the tautest span
goes from 4296N to 3635N and that of the slackest span goes from -2866N to -3529N This
equals a decrease of approximately 66N for both spans
Parametric Analysis 80
433 Influence of Tensioner Pulley 1 Coordinates on Static Tension
The influence of the coordinates of tensioner pulley 1 on the value of tension in the tautest span
for the B-ISG system‟s crankshaft driving case is demonstrated in Figure 47
Figure 47 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Tautest Span
Tension in Crankshaft Driving Case
The region shown in Figure 47 corresponds to region 1 which is the realm of the positions for
tensioner pulley 1 The possible pulley coordinates in this case are represented by the non-blue
area reaching to the perimeter of the plot It is evident in the darkest red region of the plot
where the y-coordinate is between approximately 0m and 0075m and the x-coordinate is
(N)
Parametric Analysis 81
between approximately 026m and 031m that the highest value of tension is experienced in the
tautest span for the crankshaft driving case The range of tension for Region 1 in the tautest span
when the crankshaft is driving is between a maximum of approximately 500N and a minimum of
approximately 300N This equals an overall difference of 200N in tension for the tautest span by
moving the position of pulley 1 The lowest values for tension are obtained when the pulley
coordinates are approximately -0025m to 015m for the y-coordinate and approximately 031m
to 032m for the x-coordinate which corresponds to the yellow region An area of low tension is
also seen in the area where the y-coordinate is approximately 0m and the x-coordinate is
approximately between 026m and 027m
The changes in tension for the slackest span under the condition of the crankshaft pulley being
the driving pulley are shown in Figure 48
Parametric Analysis 82
Figure 48 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Slackest Span
Tension in Crankshaft Driving Case
Once again the possible coordinate points for tensioner pulley 1 in the B-ISG system are
represented by the non-blue region For the slackest span in the crankshaft driving case it is seen
that the lowest tension is approximately 125N while the highest tension is approximately 325N
This is an overall change of 200N that is achieved in the region The highest values are achieved
in the space where the y-coordinates are approximately 0m to 0075m and the x-coordinate
ranges from 026m to 031m which corresponds to the deep red region The lowest tension
values are achieved in the space where the y-coordinate ranges from approximately -0025m to
015m and the x-coordinate ranges from 031m to 032m which corresponds to the light blue-
green region of the plot The area containing a y-coordinate of approximately 0m and x-
(N)
Parametric Analysis 83
coordinates that are approximately between 026m and 027m also show minimum tension for
the slack span The regions of the x-y coordinates for the maximum and minimum tensions are
alike to the tautest span in Region 1 for the crankshaft driving case as well as was seen in Figure
47
The tension for the tautest span in the case that the ISG is driving in the B-ISG system is found
in Figure 49
Figure 49 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Tautest Span
Tension in ISG Driving Case
(N)
Parametric Analysis 84
Region 1 is represented by the coordinate values shown in the non-dark blue space of the plot in
Figure 49 The tautest span in the case of the ISG driving experiences a range of tension values
in Region 1 from 200N up to 1100N equaling a difference of 900N The minimum tension
values are achieved in the medium to light blue region This includes y-coordinates of
approximately 0m to 0075m and x-coordinates of approximately 026m to 03m The
maximum tension values are in the darkest red area inclusive of y-coordinates -0025m to 015m
and x-coordinates 031m to 032m in addition to y-coordinate of approximately 0m and x-
coordinates of approximately 026m to 027m It can be observed that aforementioned regions
for minimum and maximum tensions in Figure 49 are reverse to those seen in Figures 47 and
48 for the crankshaft driving case
The change in tension for the slackest span of the B-ISG system when it is driven by the ISG is
shown in Figure 410
Parametric Analysis 85
Figure 410 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Slackest Span
Tension in ISG Driving Case
Figure 410 exhibits the realm of possible points for tensioner pulley 1 for the case of the ISG
driving in the non-yellow-green area The minimum tension values are achieved in the darkest
blue area where the minimum tension is approximately -500N This area corresponds to y-
coordinates from approximately 0m to 005m and x-coordinates from approximately 026m to
03m The area of a maximum tension is approximately 400N and corresponds to the darkest red
area inclusive of y-coordinates -0025m to 015m and x-coordinates 031m to 032m as well as
the coordinates for y equaling approximately 0m and for x equaling approximately 026m to
027m The difference between maximum and minimum tensions in this case is approximately
900N It is noticed once again that the space of x- and y-coordinates containing the maximum
(N)
Parametric Analysis 86
tension is in the similar location to that of the described space for minimum tension in the
crankshaft driving case in Figure 47 and 48
434 Influence of Tensioner Pulley 2 Coordinates on Static Tension
The influence of pulley 2 coordinates on the tension value for the tautest span when the
crankshaft is driving the B-ISG system is shown in Figure 411 and is represented by the values
corresponding to the non-blue area
Figure 411 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Tautest Span
Tension in Crankshaft Driving Case
In Figure 411 the possible coordinates are contained within Region 2 The maximum tension
value is approximately 500N and is found in the darkest red space including approximately y-
(N)
Parametric Analysis 87
coordinates 004m to 014m and x-coordinates 0025m to 0175m and also y-coordinates 013m
to 02m corresponding to the x-coordinate at 0175m A minimum tension value of
approximately 350N is found in the yellow space and includes approximately y-coordinates
008m to 018m and x-coordinates 016m to 02m The difference in tension values is 150N
The analysis of the change in coordinates for tension pulley 2 on the value for tension in the
slackest span is shown in Figure 412 in the non-blue region
Figure 412 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Slackest Span
Tension in Crankshaft Driving Case
The value of 325N is the highest tension for the slack span in the crankshaft driving case of the
B-ISG system and is found in the deep-red region where the y-coordinates are between
(N)
Parametric Analysis 88
approximately 004m and 013m and the x-coordinates are approximately between 0025m and
016m as well as where y is between 013m and 02m and x is approximately 0175m The
lowest tension value for the slack span is approximately 150N and is found in the green-blue
space where y-coordinates are between approximately 01m and 022m and the x-coordinates
are between approximately 016m and 021m The overall difference in minimum and maximum
tension values is 175N The spaces for the maximum and minimum tension values are similar in
location to that found in Figure 411 for the tautest span in the crankshaft driving case
Figure 413 provides the theoretical data for the tension values of the tautest span as the position
of the B-ISG system‟s tensioner pulley 2 changes in the ISG driving case Possible points are in
the space of values which correspond to the non-dark-blue region in Figure 413
Parametric Analysis 89
Figure 413 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Tautest Span
Tension in ISG Driving Case
In Figure 413 the region for high tension reaches a value of approximately 950N and the region
for low tension reaches approximately 250N This equals a difference of 700N between
maximum and minimum tension values for the tautest span in the B-ISG system‟s ISG driving
case The coordinate points within the space that maximum tension is reached is in the dark red
region and includes y-coordinates from approximately 008m to 022m and x-coordinates from
approximately 016m to 021m The coordinate points within the space that minimum tension is
reached is in the blue-green region and includes y-coordinates from approximately 004m to
013m and the corresponding x-coordinates from approximately 0025m to 015m An additional
small region of minimum tension is seen in the area where the x-coordinate is approximately
(N)
Parametric Analysis 90
0175m and the y-coordinates are approximately between 013m and 02m The location for the
area of pulley centre points that achieve maximum and minimum tension values is approximately
located in the reverse positions on the plot when compared to that of the case for the crankshaft
driving in Figures 411 and 412 Therein the trend seen for pulley coordinates for the second
tensioner pulley follows suit with that of the first tensioner pulley which is that the area of points
for maximum tension in the crankshaft driving case becomes the approximate area of points for
minimum tension in the ISG driving case and vice versa
In Figure 414 the results of the parametric analysis on the coordinates of the second tensioner
pulley and its effect on the slackest span‟s tension in the ISG driving case is shown Similar to
earlier figures the non-dark yellow region represents Region 2 that contains the possible points
for the pulley‟s Cartesian coordinates
Parametric Analysis 91
Figure 414 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Slackest
Span Tension in ISG Driving Case
Figure 414 demonstrates a difference of approximately 725N between the highest and lowest
tension values for the slackest span of the B-ISG system in the ISG driving case The highest
tension values are approximately 225N The area of points that allow the second tension pulley
to achieve maximum tension in the belt span includes y-coordinates from approximately 01m to
022m and the corresponding x-coordinates from approximately 016m to 021m This
corresponds to the darkest red region in Figure 414 The coordinate values where the lowest
tension being approximately -500N is achieved include y-coordinate values from
approximately 004m to 013m and the corresponding x-coordinates from approximately 0025m
to 015m corresponding to the darkest blue region A dark blue region of lowest tension is also
(N)
Parametric Analysis 92
seen in the area where y is approximately between 013m and 02m and the x-coordinate is
approximately 0175m The regions for maximum and minimum tension values are observed to
be similar to those found in Figure 413 and alike to Figure 413 to be in reverse to those found
in Figure 411 and 412 for the tautest and slackest spans in the crankshaft driving case So as for
the changes in tensioner pulley 2 coordinates the areas for minimum tension in Region 2 of the
ISG driving case are similar to the areas for maximum tension in Region 2 of the crankshaft
driving case and vice versa for the maximum tension of the ISG driving case and the minimum
tension for the crankshaft driving case in Region 2
44 Conclusion
Overall the trend in the plots of Figures 47 48 411 and 412 indicate in the crankshaft driving
portion that the B-ISG system‟s belt span tensions experience the following effect
Minimum tension for the tautest span is achieved when tensioner pulley 1 centre
coordinates are located closer to the right side boundary and bottom left boundary of
Region 1 or when tensioner pulley 2 centre coordinates are within the upper right space
(near to the ISG pulley) and the space closer to the top boundary of Region 2
Maximum tension for the slackest span is achieved when the first tensioner pulley‟s
coordinates are located in the mid space and near to the bottom boundary of Region 1
and when the second tensioner pulley‟s coordinates are located near to the bottom left
boundary of Region 2 which is the boundary nearest to the crankshaft pulley
Parametric Analysis 93
The trend for minimizing the tautest span signifies that the tension for the slackest span is also
minimized at the same time As well maximizing the slackest span signifies that the tension for
the tautest span is also maximized at the same time too
The trend for the B-ISG system‟s ISG driving case as can be seen in Figures 49 410 413 and
414 is approximately in reverse to that of the crankshaft driving case for the system Wherein
points corresponding to minimum tension in Regions 1 and 2 in the ISG case are approximately
the same as points corresponding to maximum tension in the Regions for the crankshaft case and
vice versa for the ISG cases‟ areas of maximum tension
Minimum tension for the tautest span is present when the first tensioner pulley‟s
coordinates are near to mid to lower boundary of Region 1 and when the second
tensioner pulley‟s coordinates are close to the bottom left boundary of Region 2 which
is the furthest boundary from the ISG pulley and closest to the crankshaft pulley
Maximum tension for the slackest span is achieved when the first tensioner pulley is
located close to the right boundary of Region 1 and when the second tensioner pulley is
located near the right boundary and towards the top right boundary of Region 2
It is observed in Figures 47 to 414 and alike to Figures 42 to 46 the tautest and slackest
spans decrease or increase together Thus it can be assumed that the tension values in these
spans and likely the remaining spans outside of the tautest and slackest spans follow suit
Therein when parameters are changed to minimize one belt span‟s tension the remaining spans
will also have their tension values reduced Figures 42 through to 413 showed this clearly
where the overall change in the tension of the tautest and slackest spans changed by
Parametric Analysis 94
approximately the same values for each separate case of the crankshaft driving and the ISG
driving in the B-ISG system
Design variables are selected in the following chapter from the parameters that have been
analyzed in the present chapter The influence of changing parameters on the static tension
values for the various spans is further explored through an optimization study of the static belt
tension for the B-ISG system equipped with a Twin Tensioner in the following chapter Chapter
5
95
CHAPTER 5 OPTIMIZATION OF A B-ISG TENSIONER
The objective of the optimization analysis is to minimize the absolute magnitude of the static
tension in the ISG-operating mode of the serpentine belt drive The optimization seeks to
optimize the performance of the proposed Twin Tensioner design by using its properties as the
design variables for the objective function The optimization task begins with the selection of
these design variables for the objective function and then the selection of an optimization
method The results of the optimization will be compared with the results of the analytical
model for the static system and with the parametric analysis‟ data
51 Optimization Problem
511 Selection of Design Variables
The optimal system corresponds to the properties of the Twin Tensioner that result in minimized
magnitudes of static tension for the various belt spans Therein the design variables for the
optimization procedure are selected from amongst the Twin Tensioner‟s properties In the
parametric analysis of Chapter 4 the tensioner properties presented included
coupled stiffness kt
tensioner arm 1 stiffness kt1
tensioner arm 2 stiffness kt2
tensioner pulley 1 diameter D3
tensioner pulley 2 diameter D5
tensioner pulley 1 initial coordinates [X3Y3] and
Optimization 96
tensioner pulley 2 initial coordinates [X5Y5]
It was observed in the former chapter that perturbations of the stiffness and geometric parameters
caused a change between the lowest and highest values for the static tension especially in the
case of perturbations in the geometric parameters diameter and coordinates Table 51
summarizes the observed changes in the belt span tensions corresponding to the Twin Tensioner
parameters‟ maximum and minimum values
Table 51 Summary of Parametric Analysis Data for Twin Tensioner Properties
Parameter Symbol
Original Tensions in TautSlack Span (Crankshaft
Mode) [N]
Tension at
Min | Max Parameter6 for
Crankshaft Mode [N]
Percent Change from Original for
Min | Max Tensions []
Original Tension in TautSlack Span (ISG Mode)
[N]
Tension at
Min | Max Parameter Value in ISG Mode [N]
Percent Change from Original Tension for
Min | Max Tensions []
kt
465848 (taut) 4691 4646 07 -03 393645 (taut) 378 3998 -40 16
28057 (slack) 2838 2793 12 -05 -322803 (slack) -3384 -3167 -48 19
kt1
465848 (taut) 4628 4681 -07 05 393645 (taut) 4088 3827 38 -28
28057 (slack) 2775 2828 -11 08 -322803 (slack) -3077 -3338 47 -34
kt2
465848 (taut) 4675 4643 04 -03 393645 (taut) 3863 4007 -19 18
28057 (slack) 2822 279 06 -06 -322803 (slack) -3301 -3157 -23 22
D3 465848 (taut) 3248 425 -303 -88 393645 (taut) 1083 6158 1751 564
28057 (slack) 1395 240 -503 -145 -322803 (slack) 367 -1006 2137 688
D5 465848 (taut) 4583 4721 -16 13 393645 (taut) 4296 3635 91 -77
28057 (slack) 273 2869 -27 23 -322803 (slack) -2866 -3529 112 -93
[X3Y3] 465848 (taut) 300 500 -356 73 393645 (taut) 200 1100 -492 1794
28057 (slack) 125 325 -554 158 -322803 (slack) -500 400 -549 2239
6 The values for the tension for each of the taut and slack spans provided correspond to the minimum and maximum
values of the parameter listed in each case such that the columns of identical colour correspond to each other For
the coordinate parameters the minimum and maximum parameter value is inadmissible The tension values in these
cases are simply the minimum and maximum tension values achieved by the coordinate parameter listed
Optimization 97
[X5Y5] 465848 (taut) 350 500 -249 73 393645 (taut) 250 950 -365 1413
28057 (slack) 150 325 -465 158 -322803 (slack) -500 225 -549 1697
The results of the parametric analyses for the Twin Tensioner parameters show that there is a
noticeable percent change between the initial tensions and the tensions corresponding to each of
the minima and maxima parameter values or in the case of the coordinates between the
minimum and maximum tensions for the spans Thus the parametric data does not encourage
exclusion of any of the tensioner parameters from being selected as a design variable As a
theoretical experiment the optimization procedure seeks to find feasible physical solutions
Hence economic criteria are considered in the selection of the design variables from among the
Twin Tensioner‟s parameters Of the tensioner properties it is found that the diameter of the
tensioner pulleys has the largest impact on cost Adding mass to a tensioner pulley as a result of
increasing the diameter and consequently its inertia increases the cost of material Material cost
is most significant in the manufacture process of pulleys as their manufacturing is largely
automated [4] Furthermore varying the structure of a pulley requires retooling which also
increases the cost to manufacture As such the tensioner pulley diameters D3 and D5 are
excluded from being selected as design variables The remaining tensioner properties the
stiffness parameters and the initial coordinates of the pulley centres are selected as the design
variables for the objective function of the optimization process
512 Objective Function amp Constraints
In order to deal with two objective functions for a taut span and a slack span a weighted
approach was employed This emerges from the results of Chapter 3 for the static model and
Chapter 4 for the parametric study for the static system which show that a high tension span and
Optimization 98
a highly slack span exist in the ISG-driving phase of the B-ISG system Therein the first
objective function of equation (51a) is described as equaling fifty percent of the absolute tension
value of the tautest span and fifty percent of the absolute tension value of the slackest span for
the case of the ISG driving only The second objective function uses a non-weighted approach
and is described as the absolute tension of the slackest span when the ISG is driving A non-
weighted approach is motivated by the phenomenon of a fixed difference that is seen between
the slackest and tautest spans of the optimal designs found in the weighted optimization
simulations Equations (51a) through to (51c) display the objective functions
The limits for the design variables are expanded from those used in the parametric analysis for
the non-coordinate parameters kt kt1 and kt2 so that they are permitted to vary from
approximately 0 to approximately 200 of the initial value for each parameter kt0 kt10 and kt20
respectively In the case of the coordinate parameters [X3Y3] and [X5Y5] the x- and y-
coordinates are permitted to vary within the spaces Region 1 and Region 2 respectively which
were prescribed in Chapter 4 Figure 41a and 41b
Aside from the design variables design constraints on the system include the requirement for
static stability of the Twin Tensioner An optimal solution for the B-ISG system must achieve
the goal of the objective function which is to minimize the absolute tensions in the system
However for an optimal solution to be feasible the movement of the tensioner arm must remain
within an appropriate threshold In practice an automotive tensioner arm for the belt
transmission may be considered stable if its movement remains within a 10 degree range of
Optimization 99
motion [4] As such the angle of displacement for tensioner arms 1 and 2 are designated by θ t1
and θt2 respectively in the listed constraints
The optimization task is described in equations 51a to 52 Variables a through to g represent
scalar limits for the x-coordinate for corresponding ranges of the y-coordinate
Minimize 119879119908119890119894119892 119893119905119890119889 = 05 ∙ 119879119905119886119906119905 + 05 ∙ 119879119904119897119886119888119896
or119879119899119900119899 minus119908119890119894119892 119893119905119890119889 = 119879119904119897119886119888119896
(51a)
where
119879119905119886119906119905 = 119891119905119886119906119905 119896119905 1198961199051 1198961199052 1198833 1198843 1198835 1198845 (51b)
119879119904119897119886119888119896 = 119891119904119897119886119888119896 (119896119905 1198961199051 1198961199052 1198833 119884311988351198845) (51c)
Subject to
(1198961199050 minus 1 ∙ 1198961199050) le 119896119905 le (1198961199050 + 11198961199050)(11989611990510 minus 1 ∙ 11989611990510) le 1198961199051 le (11989611990510 + 111989611990510)(11989611990520 minus 1 ∙ 11989611990520) le 1198961199052 le (11989611990520 + 111989611990520)
119886 le 1198833 le 119888
1198931 1198833 le 1198843 le 1198933 1198833 119891119900119903 119886 le 1198833 lt 119887
1198932 1198833 le 1198843 le 1198933 1198833 119891119900119903 119887 le 1198833 le 119888119889 le 1198835 le 119892
1198934 1198835 le 1198845 le 1198937 1198835 for 119889 le 1198835 lt 1198901198935(1198835) le 1198845 le 1198937(1198835) for 119890 le 1198835 lt 119891
1198936 1198835 le 1198845 le 1198937 1198833 for 119891 le 1198833 le 119892 1205791199051 le 10deg 1205791199052 le 10deg
(52)
The functions for the taut and slack spans represent the fourth and third span respectively in the
case of the ISG driving The equations for the tensions of the aforementioned spans are shown
in equation 51a to 51c and are derived from equation 353 The constraints for the
optimization are described in equation 52
Optimization 100
52 Optimization Method
A twofold approach was used in the optimization method A global search alone and then a
hybrid search comprising of a global search and a local search The Genetic Algorithm is used
as the global search method and a Quadratic Sequential Programming algorithm is used for the
local search method
521 Genetic Algorithm
Genetic Algorithm (GA) is a part of the growing genre of evolutionary algorithms [57] The
optimization approach differs from classical search approaches by its ease of use and global
perspective [57] GA mimics biological evolution theory by using a ldquocross-over of heritable
information random mutation and selection on the basis of fitness between generationsrdquo [58] to
form a robust search algorithm that requires minimal problem information [57] The parameter
sets are represented as sample points modeled as bdquochromosomes‟ or data strings that are
measured against how well they allow the model to achieve the optimization task [58] GA is
stochastic which means that its algorithm uses random choices to generate subsequent sampling
points rather than using a set rule to generate the following sample This avoids the pitfall of
gradient-based techniques that may focus on local maxima or minima and end-up neglecting
regions containing higher peaks or lower valleys [57] Furthermore due to the randomness of
the GA‟s search strategy it is able to search a population (a region of possible parameter sets)
faster than other optimization techniques The GA approach is viewed as a universal
optimization approach while many classical methods viewed to be efficient for one optimization
problem may be seen as inefficient for others However because GA is a probabilistic algorithm
its solution for the objective function may only be near to a global optimum As such the current
Optimization 101
state of stochastic or global optimization methods has been to refine results of the GA with a
local search and optimization procedure
522 Hybrid Optimization Algorithm
In order to enhance the result of the objective function found by the GA a Hybrid optimization
function is implemented in MATLAB software The Hybrid optimization function combines a
global search GA with a local search Sequential Quadratic Programming (SQP) The hybrid
process refines the value of the objective function found through GA by using the final set of
points found by the algorithm as the initial point of the SQP algorithm The GA function
determines the region containing a global optimum and then the SQP algorithm uses a gradient
based technique to find a solution closer to the global optimum The MATLAB algorithm a
constrained minimization function known as fmincon uses an SQP method that approximates the
Hessian for the Lagrangian function (ie the second derivatives of the Lagrangian) by way of a
quasi-Newton approach to generate a quadratic program (QP) sub-problem [59] The solution
for the QP provides the search direction of the line search procedure used when each iteration is
performed [59]
53 Results and Discussion
531 Parameter Settings amp Stopping Criteria for Simulations
The parameter settings for the optimization procedure included setting the stall time limit to
200s This is the interval of time the GA is given to find an improvement in the value of the
objective function This is an increase from MATLAB‟s default of 20s Increasing the stall time
limit allows for the optimization search to consistently converge without being limited by time
Optimization 102
The population size used in finding the optimal solution is set at 100 This value was chosen
after varying the population size between 50 and 2000 showed no change in the value of the
objective function The max number of generations is set at 100 The time limit for the
algorithm is set at infinite The limiting factor serving as the stopping condition for the
optimization search was the function tolerance which is set at 1x10-6
This allows the program
to run until the ratio of the change in the objective function over the stall generations is less than
the value for function tolerance The stall generation setting is defined as the number of
generations since the last improvement of the objective function value and is limited to 50
532 Optimization Simulations
The results of the genetic algorithm optimization simulations performed in MATLAB are shown
in the following tables Table 52a and Table 52b
Table 52a GA Optimization Results for Twin Tensioner Parameters and Objective Function
Trial
No
Genetic Algorithm Optimization Method
Objective
Function
Value [N]
kt [N∙mrad]
kt1 [N∙mrad]
kt2 [N∙mrad]
[X3Y3]
[m] [X5Y5]
[m]
1 3582241 314069 204844 165020 [02928 00703] [01618 01036]
2 3582241 103646 205284 198901 [03009 00607] [01283 00809]
3 3582241 126431 204740 43549 [03010 00631] [01311 01147]
4 3582241 180285 206230 254870 [03095 00865] [01080 01675]
5 3582241 74757 204559 189077 [03084 00617] [01265 00718]
Original Design 20626 10314 16502 [02928 0087] [01206 00919]
Optimization 103
Table 52b Computations for Tensions and Angles from GA Optimization Results
Trial No
Genetic Algorithm Optimization Method
Slackest Tension [N] Tautest Tension [N] Tensioner Arm 1
Displacement [deg]
Tensioner Arm 2
Displacement [deg]
T3 T4 Θt1 Θt2
1 -1572307 5592176 -00025 -49748
2 -4054309 3110174 -00002 -20213
3 -3930858 3233624 -00004 -38370
4 -1309751 5854731 -00010 -49525
5 -4092446 3072036 -00000 -17703
Original Design -322803 393645 16410 -4571
For each trial above the GA function required 4 generations each consisting of 20 900 function
evaluations before finding no change in the optimal objective function value according to
stopping conditions
The results of the Hybrid function optimization are provided in Tables 53a and 53b below
Table 53a Hybrid Optimization Results for Twin Tensioner Parameters and Objective Function
Trial
No
Hybrid Optimization Method
Objective
Function
Value [N]
of
Function
Evals ( of
Iterations)
kt [N∙mrad]
kt1 [N∙mrad]
kt2 [N∙mrad]
[X3Y3]
[m] [X5Y5]
[m]
1 3582241 16 (1) 16065 205846 229494 [02780 00581] [01679 01288]
2 3582241 20 (1) 249227 205635 25218 [02901 00634] [01559 00870]
3 3582241 16 (1) 297295 204878 320479 [02962 00702] [01336 01447]
4 3582241 53 (1) 241433 204262 229683 [02912 00647] [00047 01465]
Optimization 104
5 3582241 21 (1) 379096 205548 188888 [02973 00703] [01206 01376]
Original Design 20626 10314 16502 [02928 0087] [01206 00919]
Table 53b Computations for Tensions and Angles from Hybrid Optimization Results
Trial No
Hybrid Algorithm Optimization Method
Slackest Tension [N] Tautest Tension [N] Tensioner Arm 1
Displacement [deg]
Tensioner Arm 2
Displacement [deg]
T3 T4 Θt1 Θt2
1 -2584641 4579841 -02430 67549
2 -3708747 3455736 -00023 -41068
3 -1707181 5457302 -00099 -43944
4 -269178 6895304 00006 -25366
5 -2982335 4182148 -00003 -41134
Original Design -322803 393645 16410 -4571
In Table 53a it can be seen that iterations of 16 20 21 or 53 were required for the local search
algorithm following the GA to find an optimal solution Once again the GA function
computed 4 generations which consisted of approximately 20 900 function evaluations before
securing an optimum solution
The simulation results of the non-weighted hybrid optimization approach are shown in tables
54a and 54b below
Optimization 105
Table 54a Non-Weighted Optimization Results for Twin Tensioner Parameters and Objective
Function
Trial
No
Objective
Function
Value [N]
of
Function
Evals ( of
Iterations)
kt [N∙mrad]
kt1 [N∙mrad]
kt2 [N∙mrad]
[X3Y3]
[m] [X5Y5]
[m]
Genetic Algorithm Optimization Method
1 33509e
-004 20900 (4) 321799 75530 212653 [02860 00602] [01082 01858]
Hybrid Optimization Method
1 73214e
-011 381 (13) 234881 14730 323358 [02952 00688] [00048 01466]
Original Design 20626 10314 16502 [02928 0087] [01206 00919]
Table 54b Computations for Tensions and Angles from Non-Weighted Optimizations
Trial No Slackest Tension [N] Tautest Tension [N]
Tensioner Arm 1
Displacement [deg]
Tensioner Arm 2
Displacement [deg]
T3 T4 Θt1 Θt2
Genetic Algorithm Optimization Method
1 -00003 7164479 -00588 -06213
Hybrid Optimization Method
1 -00000 7164482 15543 -16254
Original Design -322803 393645 16410 -4571
The weighted optimization data of Table 54a shows that the GA simulation again used 4
generations containing 20 900 function evaluations to conduct a global search for the optimal
system While the weighted Hybrid optimization used 13 iterations (consisting of 381 function
evaluations) after its GA run which used the same number of generations and function
evaluations as the GA run in the non-weighted simulations Tables 54a and 54b show the data
Optimization 106
for only one trial for each of the non-weighted GA and hybrid methods since only a single
optimal point exists in this case
533 Discussion
The optimal design from each search method can be selected based on the least amount of
additional pre-tension (corresponding to the largest magnitude of negative tension) that would
need to be added to the system This is in harmony with the goal of the optimization of the B-
ISG system as stated earlier to minimize the static tension for the tautest span and at the same
time minimize the absolute static tension of the slackest span for the ISG driving case As well
the angular displacements corresponding to each trial‟s results show that the Twin Tensioner is
under static stability Therein the optimal solution may be selected as the design parameters
corresponding to Trial 4 of the GA simulations to Trial 4 of the Hybrid simulations or to either
of the trials for the non-weighted optimization simulations
Given the ability of the Hybrid optimization to refine the results obtained in the GA
optimization the results of Trial 4 of the Hybrid simulations are selected as the most optimal
design from the weighted objective function approaches It is interesting to note that the Hybrid
case for the least slackness in belt span tension corresponds to a significantly larger number of
function evaluations than that of the remaining Hybrid cases This anomaly however does not
invalidate the other Hybrid cases since each still satisfy the design constraints Using the data
for the optimized system in Trial 4 (of the Hybrid optimization) the static tensions for the belt
spans in both of the B-ISG‟s phases of operation are as follows
Optimization 107
Table 55 Weighted Optimization Results for Static Tensions for Optimal B-ISG System with a
Twin Tensioner
Belt Span between Pulleys Crankshaft Driving Case [N] ISG Driving Case [N]
Optimized Original Optimized Original
Crankshaft ndash Air Conditioner 3926599 465848 117333 -284152
Air Conditioner ndash Tensioner 1 3540088 427197 -269178 -322803
Tensioner 1 ndash ISG 3540088 427197 -269178 -322803
ISG ndash Tensioner 2 2073813 28057 6895304 393645
Tensioner 2 ndash Crankshaft 2073813 28057 6895304 393645
Additional Pretension
Required (approximate) + 27000 +322803 + 27000 +322803
In Table 54b it is evident that the non-weighted class of optimization simulations achieves the
least amount of required pre-tension to be added to the system The computed tension results
corresponding to both of the non-weighted GA and Hybrid approaches are approximately
equivalent Therein either of their solution parameters may also be called the most optimal
design The Hybrid solution parameters are selected as the optimal design once again due to the
refinement of the GA output contained in the Hybrid approach and its corresponding belt
tensions are listed in Table 56 below
Optimization 108
Table 56 Non-Weighted Optimization Results for Static Tensions for Optimal B-ISG System
with a Twin Tensioner
Belt Span between Pulleys Crankshaft Driving Case [N] ISG Driving Case [N]
Optimized Original Optimized Original
Crankshaft ndash Air Conditioner 3891862 465848 386511 -284152
Air Conditioner ndash Tensioner 1 3505351 427197 -00000 -322803
Tensioner 1 ndash ISG 3505351 427197 -00000 -322803
ISG ndash Tensioner 2 2039076 28057 7164482 393645
Tensioner 2 ndash Crankshaft 2039076 28057 7164482 393645
Additional Pretension
Required (approximate) + 0000 +322803 + 00000 +322803
The results of the simulation experiments are limited by the following considerations
System equations are coupled so that a fixed difference remains between tautest and
slackest spans
A limited number of simulation trials have been performed
There are multiple optimal design points for the weighted optimization search methods
Remaining tensioner parameters tensioner pulley diameters and their stiffness have not
been included in the design variables for the experiments
The belt factor kb used in the modeling of the system‟s belt has been obtained
experimentally and may be open to further sources of error
Therein the conclusions obtained and interpretations of the simulation data can be limited by the
above noted comments on the optimization experiments
Optimization 109
54 Conclusion
The outcomes the trends in the experimental data and the optimal designs can be concluded
from the optimization simulations The simulation outcomes demonstrate that in all cases the
weighted optimization functions reached an identical value for the objective function whereas
the values reached for the parameters varied widely
The lowest tension values for the tautest and slackest span were achieved in Trial 5 of the GA
optimization approach In reiteration in the presence of slack spans the tension value of the
slackest span must be added to the initial static tension for the belt Therein for the former case
an amount of at least 409N would need to be added to the 300N of pre-tension already applied to
the system (see Table 34) The highest tension values for the spans were achieved in Trial 4 of
the weighted Hybrid optimization approach and in both trials of the non-weighted optimization
approaches In the former the weighted Hybrid trial the tension value achieved in the slackest
span was approximately -27N signifying that only at least 27N would need to be added to the
present pre-tension value for the system The tension of the slackest span in the non-weighted
approach was approximately 0N signifying that the minimum required additional tension is 0N
for the system
The optimized solutions for the tension values in each span show that there is consistently a fixed
difference of 716448N between the tautest and slackest span tension values as seen in Tables
52b 53b and 54b This difference is identical to the difference between the tautest and slackest
spans of the B-ISG system for the original values of the design parameters while in its ISG
mode As well the optimal stiffness parameters for the weighted Hybrid optimization case are
Optimization 110
greater than their original values except for that of the stiffness factor of tensioner arm 1
Likewise for the non-weighted Hybrid optimization case the stiffness parameters are above their
original values without exceptions The coordinates of the optimal solutions are within close
approximation to each other and also both match the regions for moderately low tension in
Regions 1 and 2 of the ISG driving case as is shown in Figures 49 410 413 and 414
The results of the non-weighted Hybrid optimization trial and Trial 4 of the weighted Hybrid
optimization simulations are selected as the most optimal designs for the B-ISG Twin Tensioner
In these designs the Twin Tensioner is shown in Table 53b and 54b to have static stability and
to maintain suitable tensions in the ISG driving phase The tensioner parameters for the optimal
designs allow for one of the lowest amounts of additional pre-tension to be added to the system
out of all the findings from the simulations which were conducted
111
CHAPTER 6 CONCLUSION
61 Summary
The primary aim of the thesis is to reduce the magnitude of static tension in the belt spans of a
Belt-driven Integrated Starter-generator (B-ISG) system by the design and investigation of a
Twin Tensioner It is established that the operating phases of the B-ISG system produced two
cases for static tension outcomes an ISG driving case and a crankshaft driving case The
approach taken in this thesis study includes the derivation of a system model for the geometric
properties as well as for the dynamic and static states of the B-ISG system The static state of a
B-ISG system with a single tensioner mechanism is highlighted for comparison with the static
state of the Twin Tensioner-equipped B-ISG system
It is observed that there is an overall reduction in the magnitudes of the static belt tensions with
the presence of a Twin Tensioner over that of a single tensioner The influences of the geometric
and stiffness properties of the Twin Tensioner affecting the static tensions in the system are
analyzed in a parametric study It is found that there is a notable change in the static tensions
produced as result of perturbations in each respective tensioner property This demonstrates
there are no reasons to not further consider a tensioner property based solely on its influence on
the B-ISG system‟s static tensions The phenomenon of higher magnitudes for static tensions in
the ISG mode of operation over that of the crankshaft mode of operation particularly in
excessively slack spans provides the motivation for optimizing the ISG case alone for static
tension The optimization method uses weighted and non-weighted approaches with genetic
algorithm (GA) and hybrid GA searches The most optimal design has been found to be one in
Conclusion 112
which the magnitude of tension in the excessively slack spans in the ISG driving case are
significantly lower than in that of the original B-ISG Twin Tensioner design
62 Conclusion
The conclusions that can be drawn from the study of a Twin Tensioner for a B-ISG system
include the following
1 The simulations of the dynamic model demonstrate that the mode shapes for the system
are greater in the ISG-phase of operation
2 It was observed in the output of the dynamic responses that the system‟s rigid bodies
experienced larger displacements when the crankshaft was driving over that of the ISG-
driving phase It was also noted that the transition speed marking the phase change from
the ISG driving to the crankshaft driving occurred before the system reached either of its
first natural frequencies
3 The magnitudes for static belt tensions as well as dynamic tensions for a B-ISG system
are consistently greater in its ISG operating phase than in its crankshaft operating phase
4 A Twin Tensioner is able to reduce the magnitudes of the static tension for the belt spans
of a B-ISG system in comparison to when only a single tensioner mechanism is present
5 The parametric study of the B-ISG system demonstrates that the slackest and tautest belt
spans decrease or increase together for either phase of operation
6 Perturbations in the Twin Tensioner‟s geometric and stiffness properties have a
significant influence on the magnitudes of the static tension of the slackest and tautest
belt spans The coordinate position of each pulley in the Twin Tensioner configuration
Conclusion 113
has the greatest influence on the belt span static tensions out of all the tensioner
properties considered
7 Optimization of the B-ISG system shows a fixed difference trend between the static
tension of the slackest and tautest belt spans for the B-ISG system
8 The values of the design variables for the most optimal system are found using a hybrid
algorithm approach The slackest span for the optimal system is significantly less slack
than that of the original design Therein less additional pretension is required to be added
to the system to compensate for slack spans in the ISG-driving phase of operation
63 Recommendation for Future Work
The investigation of the B-ISG Twin Tensioner encourages the following future work
1 The optimization of the B-ISG system with the inclusion of diametric Twin Tensioner
properties would provide a complete picture as to the highest possible performance
outcome that the Twin Tensioner is able to have with respect to the static tensions
achieved in the belt spans
2 A larger number of optimization trials using the genetic algorithm (GA) and hybrid GA
under weighted and other approaches would investigate the scope of optimal designs
available in the Twin Tensioner for the B-ISG system
3 A model of the system without the simplification of constant damping may produce
results that are more representative of realistic operating conditions of the serpentine belt
drive A finite element analysis of the Twin Tensioner B-ISG system may provide more
applicable findings
Conclusion 114
4 Investigation of the transverse motion coupled with the rotational belt motion in an
optimized B-ISG system equipped with a Twin Tensioner may also provide a closer look
at the system under realistic conditions In addition the affect of the Twin Tensioner on
transverse motion can determine whether significant improvements in the magnitudes of
static belt span tensions are still being achieved
5 The recommendation to conduct modal decoupling of the B-ISG system‟s static model is
motivated by the fixed difference trend between the slackest and tautest belt span
tensions shown in Chapter 5 The modal decoupling of the system would allow for its
matrices comprising the equations of motion to be diagonalized and therein to decouple
the system equations Modal analysis would transform the system from physical
coordinates into natural coordinates or modal coordinates which would lead to the
decoupling of system responses
6 An investigation and optimization of the dynamic belt span tensions for a B-ISG system
with a Twin Tensioner would increase understanding of the full impact of a Twin
Tensioner mechanism on the state of the B-ISG system It would be informative to
analyze the mode shapes of the first and second modes as well as the required torques of
the driving pulleys and the resulting torque of each of the tensioner arms The
observation of the dynamic belt span tensions would also give direction as to how
damping of the system may or may not be changed
7 Further comparison with the Twin Tensioner B-ISG system‟s dynamic and static states
including the Twin Tensioner‟s stability in each versus a B-ISG system with a single
tensioner would further demonstrate the improvements or dis-improvements in the Twin
Tensioner‟s performance on a B-ISG system
Conclusion 115
8 The influence of the belt properties on the dynamic and static tensions for a B-ISG
system with a Twin Tensioner can also be investigated This again will show the
evidence of improvements or dis-improvement in the Twin Tensioner‟s performance
within a B-ISG setting
9 Lastly an experimental apparatus of the B-ISG system with a Twin Tensioner can be
designed and constructed Suitable instrumentation can be set-up to measure belt span
tensions (both static and dynamic) belt motion and numerous other system qualities
This would provide extensive guidance as to finding the most appropriate theoretical
model for the system Experimental data would provide a bench mark for evaluating the
theoretical simulation results of the Twin Tensioner-equipped B-ISG system
116
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[14] National Alternative Fuels Training Consortium (NAFTC) (2005 Oct 3) Tech stuff
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[15] Green Car Congress BMW to Apply Start-Stop and Brake Regen to MINIs Up to 60
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[21] PJ Wezenbeek (Zytec Systems Ltd) D G Evans (General Motors Powertrain) D P
Sczomak (General Motors Powertrain) J P Absmeier (Delphi Corp) and G T Fattic
(Delphi Corp) Combustion Assisted Belt-Cranking of a V-8 Engine at 12-Volts SAE
Technical Papers vol 113 pp 396-407 2004 Document no 2004-01-0569
[22] T C Firbank Mechanics of the Belt Drive International Journal of Mechanical
Sciences vol 12 pp 1053-1063 1970
[23] R L Cassidy S K Fan R S MacDonald and W F Samson Serpentine Extended Life
Accessory Drive SAE Papers 1979 Document no 760699
[24] A G Ulsoy J E Whitesell and M D Hooven Design of Belt-Tensioner Systems for
Dynamic Stability Journal of Vibration Acoustics Stress and Reliability in Design
Transactions of the ASME vol 107 pp 282-290 July 1985
[25] R S Beikmann N C Perkins and A G Ulsoy Free Vibration of Serpentine Belt Drive
Systems Journal of Vibrations and Acoustics Transactions of the ASME vol 118 pp
406-413 1996
[26] T C Kraver G W Fan and J J Shah Complex Modal Analysis of a Flat Belt Pulley
System with Belt Damping and Coulomb-Damped Tensioner Journal of Mechanical
Design Transactions of the ASME vol 118 pp 306-311 Jun 1996
[27] R S Beikmann N C Perkins and A G Ulsoy Design and Analysis of Automotive
Serpentine Belt Drive Systems for Steady State Performance Journal of Mechanical
Design Transactions of the ASME vol 119 pp 162-168 Jun 1997
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[28] L Zhang and J W Zu Modal Analysis of Serpentine Belt Drive Systems Journal of
Sound and Vibration vol 222 pp 259-279 1999
[29] R Balaji and E M Mockensturm Dynamic analysis of a front-end accessory drive with a
decouplerisolator International Journal of Vehicle Design vol 39 pp 208-231 2005
[30] M Nouri Design Optimization and Active Control of Serpentine Belt Drive Systems with
Two-pulley Tensioners University of Toronto 2005
[31] G J Spicer (Litens Automotive Inc) Tensioner for use in eg belt drive system has
electronic actuator associated with clutch spring for engaging International
WO2005119089-A1 Jun 6 2005 2005
[32] Bando Chemical Industries Ltd and Litens Automotive GmbH About belt-type starter
system Feb 27 2002
[33] H Lemberger and R Jungjohann (Bayerische Motoren Werke AG) Tension device for an
envelope drive of a device especially a belt drive of a starter generator of an internal
combustion engine comprises a support part Europe EP1420192-A2 May 19 2004 2003
[34] P Ahner and M Ackermann (Bosch GMBH) Belt drive especially for internal
combustion engines to drive accessories in an automobile Germany DE19849886-A1
May 11 2000 1998
[35] N Freisinger K Hagemann J Sievert P Struebel and M Treusch (Daimler Chrysler AG)
Belt tensioning device for belt drive between engine and starter generator of motor
vehicle has self-aligning bearing that supports auxiliary unit and provides working force to
tensioners for tensioning belt Germany DE10324268 Dec 16 2004 2003
[36] C R Rogers (Dayco Products LLC) Offset starter generator drive system for a vehicle
engine has a dual arm pivoted tensioner United States US6942589-B2 Feb 8 2005 2002
[37] A Serkh and I Ali (Gates Corp) Internal combustion engine has belt drive system with
tensioner asymmetrically biased in direction tending to cause power transmission belt to be
under tension International WO2003038309-A1 May 8 2003 2002
References 120
[38] P J Mcvicar and C A Thurston (General Motors Corp) Belt alternator starter accessory
drive with dual tensioner United States US20060287146-A1 Dec 21 2006 2005
[39] W Petri and M Bogner (INA Schaeffler KG) Traction drive especially for driving
internal combustion engine units has arrangement for demand regulated setting of tension
consisting of unit with housing with limited rotation and pulley German DE10044645-
A1 Mar 21 2002 2000
[40] M Bogner (INA Schaeffler KG) Belt drive tensioner for a starter-generator of an IC
engine has locking system for locking tensioning element in an engine operating mode
locking system is directly connected to pivot arm follows arm control movements
German DE10159073-A1 Jun 12 2003 2001
[41] R Painta M Bogner and H Graf (INA Schaeffler KG) Traction mechanism drive esp
belt drive has belt tensioning pulley mounted on generator shaft and uncoupled from it via
freewheel to dampen load peaks Europe EP1723350-A1 Nov 22 2006 2005
[42] W Petri (INA Schaeffler KG) Drive unit for a combustion engine having a starter
generator and a belt drive has tensioner with spring and counter hydraulic force Germany
DE10359641-A1 Jul 28 2005 2003
[43] H Stief M Bogner B Hartmann T Kraft and M Schmid (INA Schaeffler KG) Traction
drive especially belt drive for short-duration driving of starter generator has tensioning
device with lever arm deflectable against restoring force and with end stop limiting
deflection travel Europe EP1738093-A1 Jan 3 2007 2005
[44] M Ulm (INA Schaeffler KG DE) Tension unit eg for drive in machine such as
combustion engine has belt or chain drive with wheels turning and connected with starter
generator and unit has two idlers arranged at clamping arm with machine stored by shock
absorber Germany DE102004012395-A1 Sep 29 2005 2004
[45] M Bogner (INA Schaeffler KG) Belt drive for starter motor-generator auxiliary assembly
has limited movement at the starter belt section tensioner roller bringing it into a dead point
position on starting the motor International WO2006108461-A1 Oct 19 2006 2006
References 121
[46] W Guhr (Litens Automotive GMBH) Automotive motor and drive assembly includes
tension device positioned within belt drive system having combination starter United
States US2001007839-A1 Jul 12 2001 2001
[47] K Kuniaki K Masahiko H Kazuyuki I Shuichi and T Masaki (Mitsubishi Jidosha Eng
KK and Mitsubishi Motor Corp) Tension adjustment method of belt for starter generator
in vehicle involves shifting auto-tensioners between lock state and free state to adjust
tension of belt during driving of crank pulley Japan JP2005083514-A Mar 31 2005
2003
[48] Nissan Motor Co Ltd Winding gear for starting engine of hybrid motor vehicle has
tensioner tightening chain while cranking engine and slackens chain after start of engine
provided to span side of chain Japan JP3565040-B2 Sep 15 2004 1998
[49] S Sato and H Hayakawa (NTN Corp) Auto tensioner for ancillary drive belts has
cylinder nut and screw bolt in hydraulic damper mechanism provided in middle of cylinder
acting as start-up rigidity buffer component Japan JP2006189073-A Jul 20 2006 2005
[50] G Vadin-Michaud (Valeo Equip Electrique Moteur) Pulley and belt starting system for a
thermal engine for a motor vehicle Europe EP1658432 May 24 2006 2005
[51] M Zhen University of Toronto and Litens Automotive Group Ltd FEAD vol 50
2005
[52] W E Johns Notes on Motors [Electronic] 2003 [2008 June] Available at
httpwwwgizmologynetmotorshtm
[53] Litens Automotive Group Ltd DC BAS System - Conventional Start Input Profile Nov
23 2007
[54] Arnold Magnetic Technologies Corp General Motor Terminology [Electronic] pp 7
[2008 June] Available at httpwwwgrouparnoldcommtcpdfweb_motor_glossarypdf
[55] Douglas W Jones Stepping Motors University of Iowa - Department of Computer
Science [Electronic] Feb 14 2008 [2008 June] Available at
httpwwwcsuiowaedu~jonesstepphysicshtml
References 122
[56] Litens Automotive Group Ltd (2004 Jan 31) FEAD software input data for test project
[57] K Deb Multi-Objective Optimization using Evolutionary Algorithms Toronto John Wiley
amp Sons Ltd 2001 pp 81-85
[58] P E McSharry (2004 May 11) Department of Engineering Science University of Oxford
[httpwwwengoxacuksamppubsgawbreppdf]
[59] The MathWorks Inc MATLAB vol 750342 (R2007b) Aug 15 2007
123
APPENDIX A
Passive Dual Tensioner Designs from Patent Literature
Figure A1 Proposed design by Bayerische Motoren Werke AG corresponding to patent nos EP1420192-A2 and DE10253450-A1
Source European Patent Office espcenet (publication nos EP1420192-A2 and DE10253450-A1 accessed May 2007) epespacenetcom [33]
Figure A1 label identification 1 ndash tightner 2 ndash belt drive
3 ndash starter generator
4 ndash internal-combustion engine
4‟ ndash crankshaft-lateral drive disk
5 ndash generator housing
6 ndash common axis of rotation
7 ndash featherspring of tiltable clamping arms
8 ndash clamping arm
9 ndash clamping arm
10 11 ndash idlers
12 12‟ ndash Zugtrum 13 13‟ ndash Leertrum
14 ndash carry-hurries 15 ndash generator wave
16 ndash bush
17 ndash absorption mechanism
18 18‟ ndash support arms
19 19‟ ndash auxiliary straight lines
20 ndash pipe
21 ndash torsion bar
22 ndash breaking through
23 ndash featherspring
24 ndash friction disk
25 ndash screw connection 26 ndash Wellscheibe
(European Patent Office May 2007) [33]
Appendix A 124
Figure A2a First of four proposed designs (and its various configurations) by Bosch GMBH
corresponding to patent no WO0026532-A1
Source European Patent Office espcenet (publication no WO0026532-A1 accessed Jun 2007)
epespacenetcom [34]
Figure A2b Second of four proposed designs by Bosch GMBH corresponding to patent no
WO0026532-A1
Source European Patent Office espcenet (publication no WO0026532-A1 accessed Jun 2007) epespacenetcom [34]
Figure A2c Third of four proposed designs (and its various configurations) by Bosch GMBH
corresponding to patent no WO0026532-A1
Source European Patent Office espcenet (publication no WO0026532-A1 accessed Jun 2007)
epespacenetcom [34]
Appendix A 125
Figure A2d Fourth of four proposed designs (and its various configurations) by Bosch GMBH
corresponding to patent no WO0026532-A1
Source European Patent Office espcenet (publication no WO0026532-A1 accessed Jun 2007)
epespacenetcom [34]
Figure A2a through to A2d label identification 10 ndash engine wheel
11 ndash [generator] 13 ndash spring
14 ndash belt
16 17 ndash tensioning pulleys
18 19 ndash springs
20 21 ndash fixed points
25ab ndash carriers of idlers
25c ndash gang bolt
(European Patent Office June 2007) [34]
Figure A3 Proposed design by Daimler Chrysler AG corresponding to patent no DE10324268-A1
Source European Patent Office espcenet (publication no DE10324268-A1 accessed May 2007)
epespacenetcom [35]
Figure A3 label identification
Appendix A 126
10 12 ndash belt pulleys
14 ndash auxiliary unit
16 ndash belt
22-1 22-2 ndash belt tensioners
(European Patent Office May 2007) [35]
Figure A4 Proposed design by Dayco Products LLC corresponding to patent no US6942589-B2
Source European Patent Office espcenet (publication no US6942589-B2 accessed Jun 2007)
epespacenetcom [36]
Figure A4 label identification 12 ndash belt
14 ndash tensioner
16 ndash generator pulley
18 ndash crankshaft pulley
22 ndash slack span 24 ndash tight span
32 34 ndash arms
33 35 ndash pulley
(European Patent Office June 2007) [36]
Appendix A 127
Figure A5 Proposed design by Gates Corp corresponding to patent no WO2003038309-A
Source European Patent Office espcenet (publication no WO2003038309-A accessed Jun 2007)
epespacenetcom [37]
Figure A5 label identification 12 ndash motorgenerator
14 ndash motorgenerator pulley 26 ndash belt tensioner
28 ndash belt tensioner pulley
30 ndash transmission belt
(European Patent Office June 2007) [37]
Figure A6 Proposed design by General Motors Corp corresponding to patent no US20060287146-A1
Source European Patent Office espcenet (publication no US20060287146-A1 accessed Jun 2007)
epespacenetcom [38]
Appendix A 128
Figure A6 label identification 28 ndash tensioner
32 ndash carrier arm
34 ndash secondary carrier arm
46 ndash tensioner pulley
58 ndash secondary tensioner pulley
(European Patent Office June 2007) [38]
Figure A7 Proposed design by INA Schaeffler KG corresponding to patent no DE10044645-A1
Source European Patent Office espcenet (publication no DE10044645-A1 accessed Jun 2007)
epespacenetcom [39]
Figure A7 label identification 2 ndash internal combustion engine
3 ndash traction element
11 ndash housing with limited rotation 12 13 ndash direction changing pulleys
(European Patent Office June 2007) [39]
Appendix A 129
Figure A8a First of three proposed designs by INA Schaeffler KG corresponding to patent no
DE10159073-A1
Source European Patent Office espcenet (publication no DE10159073-A1 accessed May 2007)
epespacenetcom [40]
Figure A8b Second of three proposed designs by INA Schaeffler KG corresponding to patent no
DE10159073-A1
Source European Patent Office espcenet (publication no DE10159073-A1 accessed May 2007)
epespacenetcom [40]
Appendix A 130
Figure A8c Third of three proposed designs by INA Schaeffler KG corresponding to patent no
DE10159073-A1
Source European Patent Office espcenet (publication no DE10159073-A1 accessed May 2007)
epespacenetcom [40]
Figure A8a A8b and A8c label identification 1 ndash tightener [tensioner]
2 ndash idler
3 ndash drawing means
4 ndash swivel arm
5 ndash axis of rotation
6 ndash drawing means impulse [belt]
7 ndash crankshaft
8 ndash starter generator
9 ndash bolting volume 10a ndash bolting device system
10b ndash bolting device system
10c ndash bolting device system
11 ndash friction body
12 ndash lateral surface
13 ndash bolting tape end
14 ndash bolting tape end
15 ndash control member
16 ndash torsion bar
17 ndash base
18 ndash pylon
19 ndash hub
20 ndash annular gap
21 ndash Gleitlagerbuchse
23 ndash [nil]
23 ndash friction disk
24 ndash turning camps 25 ndash teeth
26 ndash elbow levers
27 ndash clamping wedge
28 ndash internal contour
29 ndash longitudinal guidance
30 ndash system
31 ndash sensor
32 ndash clamping gap
(European Patent Office May 2007) [40]
Appendix A 131
Figure A9 Proposed design by INA Schaeffler KG corresponding to patent no DE10359641-A1
Source European Patent Office espcenet (publication no DE10359641-A1 accessed Jun 2007)
epespacenetcom [42]
Figure A9 label identification 8 ndash cylinder
10 ndash rod
12 ndash spring plate
13 ndash spring
14 ndash pressure lead
(European Patent Office June 2007) [42]
Appendix A 132
Figure A10 Proposed design by INA Schaeffler KG corresponding to patent no EP1723350-A1
Source European Patent Office espcenet (publication no EP1723350-A1 accessed Jun 2007) epespacenetcom [41]
Figure A10 label identification 4 ndash pulley
5 ndash hydraulic element 11 ndash freewheel
12 ndash shaft
(European Patent Office June 2007) [41]
Figure A11 Proposed design by INA Schaeffler KG corresponding to patent no EP1738093-A1
Source European Patent Office espcenet (publication no EP1738093-A1 accessed Jun 2007)
epespacenetcom [43]
Figure A11 label identification 1 ndash traction drive
2 ndash traction belt
3 ndash starter generator
Appendix A 133
7 ndash tension device
9 ndash lever arm
10 ndash guide roller
16 ndash end stop
(European Patent Office June 2007) [43]
Figure A12 Proposed design by INA Schaeffler KG corresponding to patent no DE102004012395-A1
Source European Patent Office espcenet (publication no DE102004012395-A1 accessed May 2007) epespacenetcom [44]
Figure A12 label identification 1 ndash belt drive
2 ndash belts
3 ndash wheel of the internal-combustion engine
4 ndash wheel of a Nebenaggregats
5 ndash wheel of the starter generator
6 ndash clamping unit
7 ndash idler
8 ndash idler
9 ndash scale beams
10 ndash drive place
11 ndash clamping arm
12 ndash camps
13 ndash coupling point
14 ndash shock absorber element
15 ndash arrow
(European Patent Office May 2007) [44]
Appendix A 134
Figure A13 Proposed design by INA Schaeffler KG corresponding to patent nos DE102005017038-A1and WO2006108461-A1
Source European Patent Office espcenet (publication nos DE102005017038-A1and WO2006108461-A1 accessed May 2007) epespacenetcom [45]
Figure A13 label identification 1 ndash belt
2 ndash wheel of the crankshaft KW
3 ndash wheel of a climatic compressor AC
4 ndash wheel of a starter generator SG
5 ndash wheel of a water pump WP
6 ndash first clamping system
7 ndash first tension adjuster lever arm
8 ndash first tension adjuster role
9 ndash second clamping system
10 ndash second tension adjuster lever arm
11 ndash second tension adjuster role 12 ndash guide roller
13 ndash drive-conditioned Zugtrum
(generatorischer enterprise (GE))
13 ndash starter-conditioned Leertrum
(starter enterprise (SE))
14 ndash drive-conditioned Leertrum (GE)
14 ndash starter-conditioned Zugtrum (SE)
14a ndash drive-conditioned Leertrumast (GE)
14a ndash starter-conditioned Zugtrumast (SE)
14b ndash drive-conditioned Leertrumast (GE)
14b ndash starter-conditioned Zugtrumast (SE)
(European Patent Office May 2007) [45]
Figure A14 Proposed design by Litens Automotive GMBH et al corresponding to patent no
US20010007839-A1
Appendix A 135
Source European Patent Office espcenet (publication no US20010007839-A1 accessed Jun 2007)
epespacenetcom [46]
Figure A14 label identification E - belt
K - crankshaft
R1 ndash first tension pulley
R2 ndash second tension pulley
S ndash tension device
T ndash drive system
1 ndash belt pulley
4 ndash belt pulley
(European Patent Office June 2007) [46]
Figure A15 Proposed design by Mitsubishi Jidosha Eng KK and Mitsubishi Motor Corp corresponding
to patent no JP2005083514-A
Source Industrial Property Digital Library and Japanese Patent Office Patent amp Utility Model Gazette DB (document no A 2005-083514 accessed May 2007) wwwipdlinpitgojp [47]
Figure A15 label identification 1 ndash Pulley for Starting
2 ndash Crank Pulley
3 ndash AC Pulley
4a ndash 1st roller
4b ndash 2nd roller
5 ndash Idler Pulley
6 ndash Belt
7c ndash Starter generator control section
7d ndash Idle stop control means
8 ndash WP Pulley
9 ndash IG Switch
10 ndash Engine
11 ndash Starter Generator
12 ndash Driving Shaft
Appendix A 136
7 ndash ECU
7a ndash 1st auto tensioner control section (the 1st auto
tensioner control means)
7b ndash 2nd auto tensioner control section (the 2nd auto
tensioner control means)
13 ndash Air-conditioner Compressor
14a ndash 1st auto tensioner
14b ndash 2nd auto tensioner
18 ndash Water Pump
(Industrial Property Digital Library May 2007) [47]
Figure A16 Proposed design by Nissan Motor Co Ltd corresponding to patent no JP3565040-B2
Source European Patent Office espcenet (publication no JP3565040-B2 accessed Jun 2007) epespacenetcom [48]
Figure A16 label identification 3 ndash chain [or belt]
5 ndash tensioner
4 ndash belt pulley
(European Patent Office June 2007) [48]
Figure A17 Proposed design by NTN Corp corresponding to patent no JP2006189073-A
Appendix A 137
Source European Patent Office espcenet (publication no JP2006189073-A accessed Jun 2007)
epespacenetcom [49]
Figure A17 label identification 5d - flange
6 ndash tensile strength spring
10 ndash actuator
10c ndash cylinder
12 ndash rod
20 ndash hydraulic damper mechanism
21 ndash cylinder nut
22 ndash screw bolt
(European Patent Office June 2007) [49]
Figure A18 Proposed design by Valeo Equipment Electriques Moteur corresponding to patent nos
EP1658432 and WO2005015007
Source European Patent Office espcenet (publication nos EP1658432 and WO2005015007
accessed May 2007) epespacenetcom [50]
Figure A18 abbreviated list of label identifications
10 ndash starter
22 ndash shaft section
23 ndash free front end
80 ndash pulley
200 ndash support element
206 - arm
(European Patent Office May 2007) [50]
The author notes that the list of labels corresponding to Figures A1a through to A7 are generated
from machine translations translated from the documentrsquos original language (ie German)
Consequently words may be translated inaccurately or not at all
138
APPENDIX B
B-ISG Serpentine Belt Drive with Single Tensioner
Equation of Motion
The equations of motion (EOM) for a B-ISG serpentine belt drive with a single tensioner are
shown The EOM has been derived similarly to that of the same system with a twin tensioner
that was provided in Chapter 3 The assumptions for the twin tensioner B-ISG system are
applicable for the single tensioner B-ISG system as well
Figure B1 shows the B-ISG system with a single tensioner pulley and arm The pulleys are
numbered 1 through 4 and their associated belt spans are numbered accordingly
Figure B1 Single Tensioner B-ISG System
Appendix B 139
The free-body diagram for the ith non-tensioner pulley in the system shown above is found in
Figure B2 The moment of inertia for the ith pulley is designated as Ii while the angular
displacement velocity and acceleration is designated as 120579119905119894 120579 119905119894 and 120579 119905119894 respectively The
required torque is Qi the angular damping is Ci and the tension of the ith span is Ti
Figure B2 Free-body Diagram of ith Pulley
The positive motion designated is assumed to be in the clockwise direction The radius for the
ith pulley is represented by Ri The equilibrium equations for the ith pulley are as follows
I1 ∙ θ 1 = T4 ∙ R1 minus T1 ∙ R1 + Q1 minus c1 ∙ θ 1 (B1)
I2 ∙ θ 2 = T1 ∙ R2 minus T2 ∙ R2 + Q2 minus c2 ∙ θ 2 (B2)
I3 ∙ θ 3 = T2 ∙ R3 minus T3 ∙ R3 + Q3 minus c3 ∙ θ 3 (B3)
Appendix B 140
A free-body diagram for the single tensioner pulley is shown in Figure B3 The rotational
stiffness and damping for the tensioner arm is designated as kt and ct respectively The angular
rotation and velocity for the arm is 120579119905119894 and 120579 119905119894 respectively
Figure B3 Free-body Diagram of Single Tensioner
From figure B2 the equations of equilibrium are resolved for the tensioner pulley The angle of
orientation for the ith belt span is designated by 120573119895119894
minusI4 ∙ θ 4 = minusT3 ∙ R4 + T4 ∙ R4 + Q4 + c4 ∙ θ 4 (B4)
It ∙ θ t = minusTt ∙ Lt ∙ sin θto minus βj4 + sin θto 1 minus βj5 minus kt ∙ partθt minus ct ∙ partθ t
(B5)
Appendix B 141
partθt = θt minus θto (B6)
The dynamic tension matrix Trsquo is proportional to the damping (Tc) and stiffness (Tk) matrices
that are due to belt damping (119888119894119887 ) and belt stiffness (119896119894
119887 ) respectively
119827prime = 119827119836 ∙ 120521 + 119827119844 ∙ 120521 (B7)
The initial tension is represented by To and the initial angle of the tensioner arm is represented
by 120579119905119900 The equation for the tension of the ith span is shown in the following equations
T1 = To + k1b ∙ R1 ∙ θ1 minus R2 ∙ θ2 + c1
b ∙ [R1 ∙ θ 1 minus R2 ∙ θ 2] (B8)
T2 = To + k2b ∙ R2 ∙ θ2 minus R3 ∙ θ3 + c2
b ∙ [R2 ∙ θ 2 minus R3 ∙ θ 3)] (B9)
T3 = To + k3b ∙ R3 ∙ θ3 minus R4 ∙ θ4 + Lt ∙ [sin θto minus βj3 ] ∙ (θt minus θto ) + c3
b ∙ [R3 ∙ θ 3 minus R4 ∙
θ 4 + Lt ∙ [sin θto minus βj3 ] ∙ (θ t)] (B10)
T4 = To + k4b ∙ R4 ∙ θ4 minus R1 ∙ θ1 + Lt ∙ [sin θto minus βj4 ] ∙ (θt minus θto ) + c4
b ∙ [R4 ∙ θ 4 minus R1 ∙
θ 1 + Lt ∙ [sin θto minus βj4 ] ∙ (θ t)] (B11)
Tprime = Ti minus To (B12)
Tt = T3 = T4 (B13)
Appendix B 142
The EOM for the single tensioner B-ISG system is found by substitution of equations B8 to
B13 into B1 to B5 The matrices in the EOM include the inertial matrix I damping matrix C
stiffness matrix K and the required torque matrix Q as well as the angular displacement
velocity and acceleration matrices 120521 120521 and 120521 respectively
119816 ∙ 120521 + 119810 ∙ 120521 + 119818 ∙ 120521 = 119824 (B14)
119816 =
I1 0 0 0 00 I2 0 0 00 0 I3 0 00 0 0 I4 00 0 0 0 It1
(B15)
The stiffness matrix includes kb the belt factor Kb the belt cord stiffness 120601119894 the wrap angle of
the belt on the ith pulley and Kbi the stiffness factor of the ith belt span Cb represents the belt
damping for each span and βji is the angle of orientation for the span between the jth and ith
pulleys It is noted in the terms of the stiffness and damping matrices below that the numerical
subscripts refer to the (i+1)th pulley The term Qt may be found within the required torque
matrix and represents the required torque for the tensioner arm As well the term It1 represents
the moment of inertia for the tensioner arm
Appendix B 143
K =
(B16)
Kbi =Kb
Li + kb ∙ Ri ∙ϕi+1
2 + Ri ∙ϕi
2
(B17)
C =
(B18)
Appendix B 144
Appendix B 144
120521 =
θ1
θ2
θ3
θ4
partθt
(B19)
119824 =
Q1
Q2
Q3
Q4
Qt
(B20)
Simulations of the EOM for the B-ISG system with a single tensioner were performed in FEAD
[51] software for dynamic and static cases This allowed for the methodology for deriving the
EOM to be verified and then applied to the B-ISG system with a twin tensioner The natural
frequencies modes shapes dynamic responses tensioner arm torques as well as the crankshaft
required torque only and the dynamic tensions were solved from the EOM as described in (331)
to (339) of Chapter 3 and as well as for the static tension from (351) to (353) of Chapter 3
This permitted verification of the complete derivation methodology and allowed for comparison
of the static tension of the B-ISG system belt spans in the case that a single tensioner is present
and in the case that a Twin Tensioner is present [51]
145
APPENDIX C
MathCAD Scripts
C1 Geometric Analysis
1 - CS
2 - AC
4 - Alt
3 - Ten1
5 - Ten 2
6 - Ten Pivot
1
2
3
4
5
Figure C1 Schematic of B-ISG
System with Twin Tensioner
Coordinate Input DataXY1 0 0( ) XY4 24759 16664( )
XY2 224 6395( ) XY5 12057 9193( )
XY3 292761 87( ) XY6 201384 62516( )
Computations
Lt1 XY30 0
XY60 0
2
XY30 1
XY60 1
2
Lt2 XY50 0
XY60 0
2
XY50 1
XY60 1
2
t1 atan2 XY30 0
XY60 0
XY30 1
XY60 1
t2 atan2 XY50 0
XY60 0
XY50 1
XY60 1
XY
XY10 0
XY20 0
XY30 0
XY40 0
XY50 0
XY60 0
XY10 1
XY20 1
XY30 1
XY40 1
XY50 1
XY60 1
x XY
0 y XY
1
Appendix C 146
i - angle bw horizontal and l ine from ith pulley center to (i+1)th pulley center
Adjust last number in range variable p to correspond to number of pulleys
p 0 1 4
k p( ) p 1( ) p 4if
0 otherwise
condition1 p( ) acos
XYk p( ) 0
XYp 0
XYk p( ) 0
XYp 0
2
XYk p( ) 1
XYp 1
2
condition2 p( ) 2 acos
XYk p( ) 0
XYp 0
XYk p( ) 0
XYp 0
2
XYk p( ) 1
XYp 1
2
p( ) if XYk p( ) 1
XYp 1
condition1 p( ) condition2 p( )
Lfi Lbi - connection belt span lengths
D1 20065mm D2 10349mm D3 7240mm D4 6820mm D5 7240mm
D
D1
D2
D3
D4
D5
Lf p( ) XYk p( ) 0
XYp 0
2
XYk p( ) 1
XYp 1
2
1
mm
Dk p( )
2
Dp
2
2
Lb p( ) XYk p( ) 0
XYp 0
2
XYk p( ) 1
XYp 1
2
1
mm
Dk p( )
2
Dp
2
2
fi bi - angle bw ith pulley center connection l ine and contact points Pbfi (or Pfbi) and Pbi
(or Pfi) respecti vely l
f p( ) atanLf p( ) mm
Dp
2
Dk p( )
2
Dp
Dk p( )
if
atanLf p( ) mm
Dk p( )
2
Dp
2
Dp
Dk p( )
if
2D
pD
k p( )if
b p( ) atan
mmLb p( )
Dp
2
Dk p( )
2
Appendix C 147
XYfi XYbi XYfbi XYbfi - 4 possible contact points for i th pulley
XYf p( ) XYp 0
Dp
2 mmcos p( ) f p( )
XYp 1
Dp
2 mmsin p( ) f p( )
XYb p( ) XYp 0
Dp
2 mmcos p( ) f p( )
XYp 1
Dp
2 mmsin p( ) f p( )
XYfb p( ) XYp 0
Dp
2 mmcos p( ) b p( )
XYp 1
Dp
2 mmsin p( ) b p( )
XYbf p( ) XYp 0
Dp
2 mmcos p( ) b p( )
XYp 1
Dp
2 mmsin p( ) b p( )
XYfi+1 XYbi+1 XYfbi+1 XYbfi+1 - 4 possible contact points for i+1th pulley
XYf2 p( ) XYk p( ) 0
Dk p( )
2 mmcos p( ) f p( )
XYk p( ) 1
Dk p( )
2 mmsin p( ) f p( )
XYb2 p( ) XYk p( ) 0
Dk p( )
2 mmcos p( ) f p( )
XYk p( ) 1
Dk p( )
2 mmsin p( ) f p( )
XYfb2 p( ) XYk p( ) 0
Dk p( )
2 mmcos p( ) b p( ) XY
k p( ) 1
Dk p( )
2 mmsin p( ) b p( )
XYbf2 p( ) XYk p( ) 0
Dk p( )
2 mmcos p( ) b p( ) XY
k p( ) 1
Dk p( )
2 mmsin p( ) b p( )
Row 1 --gt Pulley 1 Row i --gt Pulley i
XYfi
XYf 0( )0 0
XYf 1( )0 0
XYf 2( )0 0
XYf 3( )0 0
XYf 4( )0 0
XYf 0( )0 1
XYf 1( )0 1
XYf 2( )0 1
XYf 3( )0 1
XYf 4( )0 1
XYfi
6818
269222
335325
251552
108978
100093
89099
60875
200509
207158
x1 XYfi0
y1 XYfi1
Appendix C 148
XYbi
XYb 0( )0 0
XYb 1( )0 0
XYb 2( )0 0
XYb 3( )0 0
XYb 4( )0 0
XYb 0( )0 1
XYb 1( )0 1
XYb 2( )0 1
XYb 3( )0 1
XYb 4( )0 1
XYbi
47054
18575
269403
244841
164847
88606
291
30965
132651
166182
x2 XYbi0
y2 XYbi1
XYfbi
XYfb 0( )0 0
XYfb 1( )0 0
XYfb 2( )0 0
XYfb 3( )0 0
XYfb 4( )0 0
XYfb 0( )0 1
XYfb 1( )0 1
XYfb 2( )0 1
XYfb 3( )0 1
XYfb 4( )0 1
XYfbi
42113
275543
322697
229969
9452
91058
59383
75509
195834
177002
x3 XYfbi0
y3 XYfbi1
XYbfi
XYbf 0( )0 0
XYbf 1( )0 0
XYbf 2( )0 0
XYbf 3( )0 0
XYbf 4( )0 0
XYbf 0( )0 1
XYbf 1( )0 1
XYbf 2( )0 1
XYbf 3( )0 1
XYbf 4( )0 1
XYbfi
8384
211903
266707
224592
140427
551
13639
50105
141463
143331
x4 XYbfi0
y4 XYbfi1
Row 1 --gt Pulley 2 Row i --gt Pulley i+1
XYf2i
XYf2 0( )0 0
XYf2 1( )0 0
XYf2 2( )0 0
XYf2 3( )0 0
XYf2 4( )0 0
XYf2 0( )0 1
XYf2 1( )0 1
XYf2 2( )0 1
XYf2 3( )0 1
XYf2 4( )0 1
XYf2x XYf2i0
XYf2y XYf2i1
XYb2i
XYb2 0( )0 0
XYb2 1( )0 0
XYb2 2( )0 0
XYb2 3( )0 0
XYb2 4( )0 0
XYb2 0( )0 1
XYb2 1( )0 1
XYb2 2( )0 1
XYb2 3( )0 1
XYb2 4( )0 1
XYb2x XYb2i0
XYb2y XYb2i1
Appendix C 149
XYfb2i
XYfb2 0( )0 0
XYfb2 1( )0 0
XYfb2 2( )0 0
XYfb2 3( )0 0
XYfb2 4( )0 0
XYfb2 0( )0 1
XYfb2 1( )0 1
XYfb2 2( )0 1
XYfb2 3( )0 1
XYfb2 4( )0 1
XYfb2x XYfb2i
0
XYfb2y XYfb2i1
XYbf2i
XYbf2 0( )0 0
XYbf2 1( )0 0
XYbf2 2( )0 0
XYbf2 3( )0 0
XYbf2 4( )0 0
XYbf2 0( )0 1
XYbf2 1( )0 1
XYbf2 2( )0 1
XYbf2 3( )0 1
XYbf2 4( )0 1
XYbf2x XYbf2i0
XYbf2y XYbf2i1
100 40 20 80 140 200 260 320 380 440 500150
110
70
30
10
50
90
130
170
210
250Figure C2 Possible Contact Points
250
150
y1
y2
y3
y4
y
XYf2y
XYb2y
XYfb2y
XYbf2y
500100 x1 x2 x3 x4 x XYf2x XYb2x XYfb2x XYbf2x
Appendix C 150
Xij Yij - selected contact point on ith pulley for span from ith pulley to jth pulley
XY15 XYbf2iT 4
XY12 XYfiT 0
Pulley 1 contact pts
XY21 XYf2iT 0
XY23 XYfbiT 1
Pulley 2 contact pts
XY32 XYfb2iT 1
XY34 XYbfiT 2
Pulley 3 contact pts
XY43 XYbf2iT 2
XY45 XYfbiT 3
Pulley 4 contact pts
XY54 XYfb2iT 3
XY51 XYbfiT 4
Pulley 5 contact pts
By observation the lengths of span i is the following
L1 Lf 0( ) L2 Lb 1( ) L3 Lb 2( ) L4 Lb 3( ) L5 Lb 4( ) Li
L1
L2
L3
L4
L5
mm
i Angle between horizontal and span of ith pulley
i
atan
XY121
XY211
XY12
0XY21
0
atan
XY231
XY321
XY23
0XY32
0
atan
XY341
XY431
XY34
0XY43
0
atan
XY451
XY541
XY45
0XY54
0
atan
XY511
XY151
XY51
0XY15
0
Appendix C 151
Pulley 1 Pulley 2 Pulley 3 Pulley 4 Pulley 5
12 i0 2 21 i0 32 i1 2 43 i2 54 i3
15 i4 2 23 i1 34 i2 45 i3 51 i4
15
21
32
43
54
12
23
34
45
51
Wrap angle i for the ith pulley
1 2 atan2 XY150
XY151
atan2 XY120
XY121
2 atan2 XY210
XY1 0
XY211
XY1 1
atan2 XY230
XY1 0
XY231
XY1 1
3 2 atan2 XY320
XY2 0
XY321
XY2 1
atan2 XY340
XY2 0
XY341
XY2 1
4 atan2 XY430
XY3 0
XY431
XY3 1
atan2 XY450
XY3 0
XY451
XY3 1
5 atan2 XY540
XY4 0
XY541
XY4 1
atan2 XY510
XY4 0
XY511
XY4 1
1
2
3
4
5
Lb length of belt
Lbelt Li1
2
0
4
p
Dpp
Input Data for B-ISG System
Kt 20626Nm
rad (spring stiffness between tensioner arms 1
and 2)
Kt1 10314Nm
rad (stiffness for spring attached at arm 1 only)
Kt2 16502Nm
rad (stiffness for spring attached at arm 2 only)
Appendix C 152
C2 Dynamic Analysis
I C K moment of inertia damping and stiffness matrices respectively
u 0 1 4 v 0 1 4 (new counter variables where final value = no of pulleys + no of ten arms)
RaD
2
Appendix C 153
RaD
2
Ii =gt moment of inertia for ith pulley where i-1 and i represent ten arms
Ii0
0
1
2
3
4
5
6
10000
2230
300
3000
300
1500
1500
I diag Ii( ) kg mm2
Ci =gt Rotational damping and belt damping for the ith pulley where i-1 and i represent tensioner arms
1000kg
m3
CrossArea 693mm2
0 M CrossArea Lbelt M 0086kg
cb 2 KbM
Lbelt
Cb
cb
cb
cb
cb
cb
Cri
0
0
010
0
010
N mmsec
rad
Ct 1000N mmsec
rad Ct1 1000 N mm
sec
rad Ct2 1000N mm
sec
rad
Cr
Cri0
0
0
0
0
0
0
0
Cri1
0
0
0
0
0
0
0
Cri2
0
0
0
0
0
0
0
Cri3
0
0
0
0
0
0
0
Cri4
0
0
0
0
0
0
0
Ct Ct1
Ct
0
0
0
0
0
Ct
Ct Ct2
Rt
Ra0
Ra1
0
0
0
0
0
0
Ra1
Ra2
0
0
Lt1 mm sin t1 32
0
0
0
Ra2
Ra3
0
Lt1 mm sin t1 34
0
0
0
0
Ra3
Ra4
0
Lt2 mm sin t2 54
Ra0
0
0
0
Ra4
0
Lt2 mm sin t2 51
Appendix C 154
Kr
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Kt Kt1
Kt
0
0
0
0
0
Kt
Kt Kt2
Tk
Kbi 0( ) Ra0
0
0
0
Kbi 4( ) Ra0
Kbi 0( ) Ra1
Kbi 1( ) Ra1
0
0
0
0
Kbi 1( ) Ra2
Kbi 2( ) Ra2
0
0
0
0
Kbi 2( ) Ra3
Kbi 3( ) Ra3
0
0
0
0
Kbi 3( ) Ra4
Kbi 4( ) Ra4
0
Kbi 1( ) Lt1 mm sin t1 32
Kbi 2( ) Lt1 mm sin t1 34
0
0
0
0
0
Kbi 3( ) Lt2 mm sin t2 54
Kbi 4( ) Lt2 mm sin t2 51
Tc
Cb0
Ra0
0
0
0
Cb4
Ra0
Cb0
Ra1
Cb1
Ra1
0
0
0
0
Cb1
Ra2
Cb2
Ra2
0
0
0
0
Cb2
Ra3
Cb3
Ra3
0
0
0
0
Cb3
Ra4
Cb4
Ra4
0
Cb1
Lt1 mm sin t1 32
Cb2
Lt1 mm sin t1 34
0
0
0
0
0
Cb3
Lt2 mm sin t2 54
Cb4
Lt2 mm sin t2 51
C matrix
C Cr Rt Tc
K matrix
K Kr Rt Tk
New Equations of Motion for Dual Drive System
I K amp C matricies rearranged to place driving pulley in 1st row + 1st column and driven in 2nd row + 2nd column
IA augment I3
I0
I1
I2
I4
I5
I6
IC augment I0
I3
I1
I2
I4
I5
I6
I1kgmm2 1 106
kg m2
0 0 0 0 0 0
Ia stack I1kgmm2 IAT 0
T
IAT 1
T
IAT 2
T
IAT 4
T
IAT 5
T
IAT 6
T
Ic stack I1kgmm2 ICT 3
T
ICT 1
T
ICT 2
T
ICT 4
T
ICT 5
T
ICT 6
T
Appendix C 155
RtA augment Rt3
Rt0
Rt1
Rt2
Rt4
RtC augment Rt0
Rt3
Rt1
Rt2
Rt4
Rta stack RtAT 3
T
RtAT 0
T
RtAT 1
T
RtAT 2
T
RtAT 4
T
RtAT 5
T
RtAT 6
T
Rtc stack RtCT 0
T
RtCT 3
T
RtCT 1
T
RtCT 2
T
RtCT 4
T
RtCT 5
T
RtCT 6
T
TkA augment Tk3
Tk0
Tk1
Tk2
Tk4
Tk5
Tk6
Tka stack TkAT 3
T
TkAT 0
T
TkAT 1
T
TkAT 2
T
TkAT 4
T
TkC augment Tk0
Tk3
Tk1
Tk2
Tk4
Tk5
Tk6
Tkc stack TkCT 0
T
TkCT 3
T
TkCT 1
T
TkCT 2
T
TkCT 4
T
TcA augment Tc3
Tc0
Tc1
Tc2
Tc4
Tc5
Tc6
Tca stack TcAT 3
T
TcAT 0
T
TcAT 1
T
TcAT 2
T
TcAT 4
T
TcC augment Tc0
Tc3
Tc1
Tc2
Tc4
Tc5
Tc6
Tcc stack TcAT 0
T
TcAT 3
T
TcAT 1
T
TcAT 2
T
TcAT 4
T
Ka Kr Rta Tka Kc Kr Rtc Tkc Ca Cr Rta Tca Cc Cr Rtc Tcc
CHECK
KA augment K3
K0
K1
K2
K4
K5
K6
KC augment K0
K3
K1
K2
K4
K5
K6
CA augment C3
C0
C1
C2
C4
C5
C6
CC augment C0
C3
C1
C2
C4
C5
C6
Appendix C 156
Kacheck stack KAT 3
T
KAT 0
T
KAT 1
T
KAT 2
T
KAT 4
T
KAT 5
T
KAT 6
T
Kccheck stack KCT 0
T
KCT 3
T
KCT 1
T
KCT 2
T
KCT 4
T
KCT 5
T
KCT 6
T
Cacheck stack CAT 3
T
CAT 0
T
CAT 1
T
CAT 2
T
CAT 4
T
CAT 5
T
CAT 6
T
Cccheck stack CCT 0
T
CCT 3
T
CCT 1
T
CCT 2
T
CCT 4
T
CCT 5
T
CCT 6
T
Results for System switching from ISG as DRIVING pulley to Crankshaft as Drivi ng Pulley
Modified Submatricies for ISG Driving Phase --gt CS Driving Phase
Unit step function to provide shift from crankshaft DRIVING case (ie ISG driven case) to crankshaft DRIVEN
case (ie ISG driving case)
H n( ) 1 n 750if
0 n 750if
lt-- crankshaft DRIVING case (Phase change bw 2 cases occurs when n
reaches start speed)
I11mod n( ) Ic0 0
H n( ) 1if
Ia0 0
H n( ) 0if
I22mod n( )submatrix Ic 1 6 1 6( )
UnitsOf I( )H n( ) 1if
submatrix Ia 1 6 1 6( )
UnitsOf I( )H n( ) 0if
K11mod n( )
Kc0 0
UnitsOf K( )H n( ) 1if
Ka0 0
UnitsOf K( )H n( ) 0if
C11modn( )
Cc0 0
UnitsOf C( )H n( ) 1if
Ca0 0
UnitsOf C( )H n( ) 0if
K22mod n( )submatrix Kc 1 6 1 6( )
UnitsOf K( )H n( ) 1if
submatrix Ka 1 6 1 6( )
UnitsOf K( )H n( ) 0if
C22modn( )submatrix Cc 1 6 1 6( )
UnitsOf C( )H n( ) 1if
submatrix Ca 1 6 1 6( )
UnitsOf C( )H n( ) 0if
K21mod n( )submatrix Kc 1 6 0 0( )
UnitsOf K( )H n( ) 1if
submatrix Ka 1 6 0 0( )
UnitsOf K( )H n( ) 0if
C21modn( )submatrix Cc 1 6 0 0( )
UnitsOf C( )H n( ) 1if
submatrix Ca 1 6 0 0( )
UnitsOf C( )H n( ) 0if
K12mod n( )submatrix Kc 0 0 1 6( )
UnitsOf K( )H n( ) 1if
submatrix Ka 0 0 1 6( )
UnitsOf K( )H n( ) 0if
C12modn( )submatrix Cc 0 0 1 6( )
UnitsOf C( )H n( ) 1if
submatrix Ca 0 0 1 6( )
UnitsOf C( )H n( ) 0if
Appendix C 157
2mod n( ) I22mod n( )1
K22mod n( ) mod n( ) sort eigenvals 2mod n( ) nmod n( )mod n( )
2
EVmodn( ) augmenteigenvec 2mod n( ) mod n( )0
max eigenvec 2mod n( ) mod n( )0
eigenvec 2mod n( ) mod n( )1
max eigenvec 2mod n( ) mod n( )1
eigenvec 2mod n( ) mod n( )2
max eigenvec 2mod n( ) mod n( )2
eigenvec 2mod n( ) mod n( )3
max eigenvec 2mod n( ) mod n( )3
eigenvec 2mod n( ) mod n( )4
max eigenvec 2mod n( ) mod n( )4
eigenvec 2mod n( ) mod n( )5
max eigenvec 2mod n( ) mod n( )5
modeshapesmod n( ) stack nmod n( )T
EVmodn( )
t 0 0001 1
mode1a t( ) EVmod100( )0
sin nmod 100( )0 t mode2a t( ) EVmod100( )1
sin nmod 100( )1 t
mode1c t( ) EVmod800( )0
sin nmod 800( )0 t mode2c t( ) EVmod800( )1
sin nmod 800( )1 t
Pulley responses amp torque requirement for crankshaft amp alternator pulleys pulley1 and 4 respectively
The system equation becomes
I14q14 -double-dot + C1144 q14 -dot + K1144 q14 + C12qm-dot + K12qm = Qc
I2qm-double-dot + C22qm-dot + K22qm + C21q1-dot + K21q1 = 0
Pulley responses
Qm = - [(K22 - 2I2) + jC22 ]-1(K21 + jC21 )Q1
Torque requirement for crank shaft Pulley 1
qc = [(K11 -2I1) + jC11 ]Q1 + (K12 + jC12 )Qm
Torque requirement for alternator shaft Pulley 4
qa = [(K44 -2I4) + jC44 ]Q4 + (K12 + jC12 )Qm
Appendix C 158
Let DRIVING pulley have a unit amplitude 1 = 1 and let the system frequency be calculated based on
engine speed n
n 60 90 6000 n( )4n
60 a n( )
2n Ra0
60 Ra3
mod n( ) n( ) H n( ) 1if
a n( ) H n( ) 0if
Ymod n( ) K22mod n( ) mod n( ) 2 I22mod n( )
j mod n( ) C22modn( )
mmod n( ) Ymod n( )( )1
K21mod n( ) j mod n( ) C21modn( )
Crankshaft amp ISG required torques
Let input from DRIVING pulley be an angular displacement with constant amplitude of angular acceleration
Ac n( ) 650 1 n( )Ac n( )
n( ) 2
Let Qm = QmQ1(n) for n lt 750
and Qm = QmQ4(n) for n gt 750
Aa n( )42
I3 3
1a n( )Aa n( )
a n( ) 2
Qc0 4
qcmod n( ) K11mod n( ) mod n( ) 2
I11mod n( )
j mod n( ) C11modn( )
1 n( ) K12mod n( ) j mod n( ) C12modn( ) mmod n( ) 1 n( )
H n( ) 1if
Qc0 H n( ) 0if
qamod n( ) K11mod n( ) mod n( ) 2
I11mod n( )
j mod n( ) C11modn( )
1a n( ) K12mod n( ) jmod n( ) C12modn( ) mmod n( ) 1a n( ) Qc0
H n( ) 0if
0 H n( ) 1if
Q n( ) 48 n
Ra0
Ra3
48
18000
(ISG torque requirement alternate equation)
Appendix C 159
Dynamic tensioner arm torques
Qtt1mod n( )Kt Kt1
UnitsOf Kt( )j mod n( )
Ct Ct1
UnitsOf Cr( )
mmod n( )4 1 n( )
H n( ) 1if
Kt Kt1
UnitsOf Kt( )j mod n( )
Ct Ct1
UnitsOf Cr( )
mmod n( )4 1a n( )
H n( ) 0if
Qtt2mod n( )Kt Kt2
UnitsOf Kt( )j mod n( )
Ct Ct2
UnitsOf Cr( )
mmod n( )5 1 n( )
H n( ) 1if
Kt Kt2
UnitsOf Kt( )j mod n( )
Ct Ct2
UnitsOf Cr( )
mmod n( )5 1a n( )
H n( ) 0if
Appendix C 160
Dynamic belt span tensions
d n( ) 1 n( ) H n( ) 1if
1a n( ) H n( ) 0if
mod n( )
d n( )
mmod n( ) d n( ) 0 0
mmod n( ) d n( ) 1 0
mmod n( ) d n( ) 2 0
mmod n( ) d n( ) 3 0
mmod n( ) d n( ) 4 0
mmod n( ) d n( ) 5 0
Tm n( ) j n( )Tcc
UnitsOf Tcc( )
Tkc
UnitsOf Tkc( )
mod n( )
H n( ) 1if
j n( )Tca
UnitsOf Tca( )
Tka
UnitsOf Tka( )
mod n( )
H n( ) 0if
Tm n( ) j n( )Tcc
UnitsOf Tcc( )
Tkc
UnitsOf Tkc( )
mod n( )
H n( ) 1if
j n( )Tca
UnitsOf Tca( )
Tka
UnitsOf Tka( )
mod n( )
H n( ) 0if
(tensions for driving pulley belt spans)
Appendix C 161
C3 Static Analysis
Static Analysis using K Tk amp Q matricies amp Ts
For static case K = Q
Tension T = T0 + Tks
Thus T = K-1QTks + T0
Q1 68N m Qt1 0N m Qt2 0N m Ts 300N
Qc
Q4
Q2
Q3
Q5
Qt1
Qt2
Qc
5
2
0
0
0
0
J Qa
Q1
Q2
Q3
Q5
Qt1
Qt2
Qa
68
2
0
0
0
0
N m
cK22mod 900( )( )
1
N mQc A
K22mod 600( )1
N mQa
a
A0
A1
A2
0
A3
A4
A5
0
c1
c2
c0
c3
c4
c5
Tc Tk Ts Ta Tk a Ts
162
APPENDIX D
MATLAB Functions amp Scripts
D1 Parametric Analysis
D11 TwinMainm
The following function script performs the parametric analysis for the B-ISG system with a Twin
Tensioner It calls the function TwinTenStaticTensionm The parametric analysis perturbs a
single input parameter for the called function TwinTenStaticTensionm The main function takes
an initial input value for the Twin Tensioner‟s stiffness parameters Kto Kt1o Kt2o and
geometric parameters D3o D5o X3o Y3o X5o and Y5o An input parameter is allowed to
increment by six percent over a range from sixty percent below its initial value to sixty percent
above its initial value The coordinate parameters are incremented through a mesh of Cartesian
points with prescribed boundaries The TwinMainm function plots the parametric results
______________________________________________________________________________
clc
clear all
Static tension for single tensioner system for CS and Alt driving
Initial Conditions
Kto = 20626
Kt1o = 10314
Kt2o = 16502
D3o = 007240
D5o = 007240
X3o =0292761
Y3o =087
X5o =12057
Y5o =09193
Pertubations of initial parameters
Kt = (Kto-060Kto)006Kto(Kto+060Kto)
Kt1 = (Kt1o-060Kt1o)006Kt1o(Kt1o+060Kt1o)
Kt2 = (Kt2o-060Kt2o)006Kt2o(Kt2o+060Kt2o)
D3 = (D3o-060D3o)006D3o(D3o+060D3o)
D5 = (D5o-060D5o)006D5o(D5o+060D5o)
No of data points
s = 21
T = zeros(5s)
Ta = zeros(5s)
Parametric Plots
for i = 1s
Appendix D 163
[T(i)Ta(i)] = TwinTenStaticTension(Kt(i)Kt1oKt2oD3oD5oX3oY3oX5oY5o)
end
figure
[AXH1H2] = plotyy(Kt()T(1)Kt()Ta(4)plot) hold on
H3 = line(Kt()T(5)ParentAX(1)) hold on
H4 = line(Kt()Ta(3)Parent AX(2)) hold off
set(AXYLimModeautoYTickModeauto)
title(Static Tension vs Coupled Stiffness bw Arms 1 amp 2)
xlabel(Kt (Nmrad))
set(get(AX(1)Ylabel)StringCS Span Tension (N))
set(get(AX(2)Ylabel)StringISG Span Tension (N))
set(H1LineStyle-Colorb)
set(H2LineStyle-Colorg)
set(H3LineStyleColorget(H1Color))
set(H4LineStyleColorget(H2Color))
legend([H1H2H3H4]CS Tautest SpanISG Tautest SpanCS Slackest SpanISG
Slackest SpanLocationBest)
for i = 1s
[T(i)Ta(i)] = TwinTenStaticTension(KtoKt1(i)Kt2oD3oD5oX3oY3oX5oY5o)
end
figure
[AXH1H2] = plotyy(Kt1()T(1)Kt1()Ta(4)plot) hold on
H3 = line(Kt1()T(5)ParentAX(1)) hold on
H4 = line(Kt1()Ta(3)Parent AX(2)) hold off
set(AXYLimModeautoYTickModeauto)
title(Static Tension vs Tensioner Arm 1 Stiffness)
xlabel(Kt1 (Nmrad))
set(get(AX(1)Ylabel)StringCS Span Tension (N))
set(get(AX(2)Ylabel)StringISG Span Tension (N))
set(H1LineStyle-Colorb)
set(H2LineStyle-Colorg)
set(H3LineStyleColorget(H1Color))
set(H4LineStyleColorget(H2Color))
legend([H1H2H3H4]CS Tautest SpanISG Tautest SpanCS Slackest SpanISG
Slackest SpanLocationBest)
for i = 1s
[T(i)Ta(i)] = TwinTenStaticTension(KtoKt1oKt2(i)D3oD5oX3oY3oX5oY5o)
end
figure
[AXH1H2] = plotyy(Kt2()T(1)Kt2()Ta(4)plot) hold on
H3 = line(Kt2()T(5)ParentAX(1)) hold on
H4 = line(Kt2()Ta(3)Parent AX(2)) hold off
set(AXYLimModeautoYTickModeauto)
title(Static Tension vs Tensioner Arm 2 Stiffness)
xlabel(Kt2 (Nmrad))
set(get(AX(1)Ylabel)StringCS Span Tension (N))
Appendix D 164
set(get(AX(2)Ylabel)StringISG Span Tension (N))
set(H1LineStyle-Colorb)
set(H2LineStyle-Colorg)
set(H3LineStyleColorget(H1Color))
set(H4LineStyleColorget(H2Color))
legend([H1H2H3H4]CS Tautest SpanISG Tautest SpanCS Slackest SpanISG
Slackest SpanLocationBest)
for i = 1s
[T(i)Ta(i)] = TwinTenStaticTension(KtoKt1oKt2oD3(i)D5oX3oY3oX5oY5o)
end
figure
[AXH1H2] = plotyy(D3()T(1)D3()Ta(4)plot) hold on
H3 = line(D3()T(5)ParentAX(1)) hold on
H4 = line(D3()Ta(3)Parent AX(2)) hold off
set(AXYLimModeautoYTickModeauto)
title(Static Tension vs Tensioner Pulley 1 Diameter)
xlabel(D3 (m))
set(get(AX(1)Ylabel)StringCS Span Tension (N))
set(get(AX(2)Ylabel)StringISG Span Tension (N))
set(H1LineStyle-Colorb)
set(H2LineStyle-Colorg)
set(H3LineStyleColorget(H1Color))
set(H4LineStyleColorget(H2Color))
legend([H1H2H3H4]CS Tautest SpanISG Tautest SpanCS Slackest SpanISG
Slackest SpanLocationBest)
for i = 1s
[T(i)Ta(i)] = TwinTenStaticTension(KtoKt1oKt2oD3oD5(i)X3oY3oX5oY5o)
end
figure
[AXH1H2] = plotyy(D5()T(1)D5()Ta(4)plot) hold on
H3 = line(D5()T(5)ParentAX(1)) hold on
H4 = line(D5()Ta(3)Parent AX(2)) hold off
set(AXYLimModeautoYTickModeauto)
title(Static Tension vs Tensioner Pulley 2 Diameter)
xlabel(D5 (m))
set(get(AX(1)Ylabel)StringCS Span Tension (N))
set(get(AX(2)Ylabel)StringISG Span Tension (N))
set(H1LineStyle-Colorb)
set(H2LineStyle-Colorg)
set(H3LineStyleColorget(H1Color))
set(H4LineStyleColorget(H2Color))
legend([H1H2H3H4]CS Tautest SpanISG Tautest SpanCS Slackest SpanISG
Slackest SpanLocationBest)
Mesh points
m = 101
n = 101
Appendix D 165
T = zeros(5nm)
Ta = zeros(5nm)
[ixxiyy] = meshgrid(1m1n)
minX3 = 0260200
maxX3 = 0317677
minY3 = -0056640
maxY3 = 0228456
midX3 = 0311641
X3 = minX3 + (ixx-1)(maxX3-minX3)(m-1)
Y3 = minY3 + (iyy-1)(maxY3-minY3)(n-1)
for i = 1n
for j = 1m
if ((X3(ij)lt midX3)ampamp(Y3(ij)gt=(sqrt((0087945^2)-((X3(ij)-0224)^2)-
006395)))ampamp(Y3(ij)lt=(-1sqrt(((00703^2)-((X3(ij)-
024759)^2)))+016664)))||((X3(ij)gt=midX3)ampamp(Y3(ij)gt=(35548X3(ij)-
11134868))ampamp(Y3(ij)lt=(-1(sqrt(((00703^2)-((X3(ij)-024759)^2))))+016664))) mx+b
lt= y lt= circle4
[T(ij)Ta(ij)] =
TwinTenStaticTension(KtoKt1oKt2oD3oD5oX3(ij)Y3(ij)X5oY5o)
else
T(ij) = zeros(511)
Ta(ij) = zeros(511)
end
end
end
figure
Z1 = squeeze(T(1))
surf(X3Y3real(Z1))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(CS DRIVING Tautest Span Tension vs Tensioner Pulley 1 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(CS Span Tension (N))
figure
Z5 = squeeze(T(5))
surf(X3Y3real(Z5))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(CS DRIVING Slackest Span Tension vs Tensioner Pulley 1 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
Appendix D 166
zlabel(CS Span Tension (N))
figure
Za4 = squeeze(Ta(4))
surf(X3Y3real(Za4))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(ISG DRIVING Tautest Span Tension vs Tensioner Pulley 1 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(ISG Span Tension (N))
figure
Za3 = squeeze(Ta(3))
surf(X3Y3real(Za3))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(ISG DRIVING Slackest Span Tension vs Tensioner Pulley 1 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(ISG Span Tension (N))
minX5 = -0037093
maxX5 = 0212509
minY5 = 00362
maxY5 = 0228456
midX5a = 0131965
midX5b = 017729
X5 = minX5 + (ixx-1)(maxX5-minX5)(m-1)
Y5 = minY5 + (iyy-1)(maxY5-minY5)(n-1)
for i = 1n
for j = 1m
if
(X5(ij)ltmidX5a)ampamp(Y5(ij)lt=(0386X5(ij)+0146468))ampamp(Y5(ij)gt=(sqrt((0136525^2)-
(X5(ij)^2))))
[T(ij)Ta(ij)] =
TwinTenStaticTension(KtoKt1oKt2oD3oD5oX3oY3oX5(ij)Y5(ij))
elseif
((X5(ij)gt=midX5a)ampamp(X5(ij)ltmidX5b))ampamp(Y5(ij)gt=00362)ampamp(Y5(ij)lt=(0386X5(ij)+0
146468))
[T(ij)Ta(ij)] =
TwinTenStaticTension(KtoKt1oKt2oD3oD5oX3oY3oX5(ij)Y5(ij))
elseif (X5(ij)gt=midX5b)ampamp(Y5(ij)gt=(sqrt((00703^2)-(((X5(ij)-
024759)^2)))+016664))ampamp(Y5(ij)lt=(0386X5(ij)+0146468))
Appendix D 167
[T(ij)Ta(ij)] =
TwinTenStaticTension(KtoKt1oKt2oD3oD5oX3oY3oX5(ij)Y5(ij))
else
T(ij) = zeros(511)
Ta(ij) = zeros(511)
end
end
end
figure
Z1 = squeeze(T(1))
surf(X5Y5real(Z1))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(CS DRIVING Tautest Span Tension vs Tensioner Pulley 2 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(CS Span Tension (N))
figure
Z5 = squeeze(T(5))
surf(X5Y5real(Z5))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(CS DRIVING Slackest Span Tension vs Tensioner Pulley 2 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(CS Span Tension (N))
figure
Za4 = squeeze(Ta(4))
surf(X5Y5real(Za4))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(ISG DRIVING Tautest Span Tension vs Tensioner Pulley 2 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(ISG Span Tension (N))
figure
Za3 = squeeze(Ta(3))
surf(X5Y5real(Za3))
ZLim([50 500])
axis tight
Appendix D 168
colormap jet
colorbar
title(ISG DRIVING Slackest Span Tension vs Tensioner Pulley 2 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(ISG Span Tension (N))
D12 TwinTenStaticTensionm
The function TwinTenStaticTensionm simulates the static model of the B-ISG system with a
Twin Tensioner This function returns 3 vectors the static tension of each belt span in the
crankshaft- and ISG-driving phases of operation and the angle of displacement of each rigid
body in the ISG- driving phase It takes the input parameters kt kt1 kt2 for the tensioner arm
stiffness D3 and D5 for the tensioner pulley diameters and X3Y3 X5 and Y5 for the tensioner
arm pulley coordinates The function is called in the parametric analysis solution script
TwinMainm and in the optimization solution script OptimizationTwinm
D2 Optimization
D21 OptimizationTwinm
The following script is for the main function OptimizationTwinm It performs an optimization
search for the B-ISG system with a Twin Tensioner It takes an input for a parameter vector
containing values for the design variables The program calls the objective function
objfunTwinm and the constraint function confunTwinm The program can perform a genetic
algorithm (GA) optimization search or a hybrid GA optimization that includes a localized search
The optimal solution vector corresponding to the design variables and the optimal objective
function value is returned The program inputs the optimized values for the design variables into
TwinTenStaticTensionm This called function returns the optimized static state of tensions for
the crankshaft- and ISG- driving phases and for the angle of displacement of the rigid bodies in
the ISG driving phase
______________________________________________________________________________
clc
clear all
Initial values for variables
Kto = 20626
Kt1o = 10314
Kt2o = 16502
X3o = 0292761
Y3o = 0087
X5o = 012057
Appendix D 169
Y5o = 009193
w0 =[Kto Kt1o Kt2o X3o Y3o X5o Y5o] Start Point (row vector)
Variable ranges
minKt = Kto - 1Kto
maxKt = Kto + 1Kto
minKt1 = Kt1o - 1Kt1o
maxKt1 = Kt1o + 1Kt1o
minKt2 = Kt2o - 1Kt2o
maxKt2 = Kt2o + 1Kt2o
minX3 = 0260200
maxX3 = 0317677
minY3 = -0056640
maxY3 = 0228456
minX5 = -0037093
maxX5 = 0212509
minY5 = 00362
maxY5 = 0228456
ObjectiveFunction = objfunTwin
nvars = 7 Number of variables
ConstraintFunction = confunTwin
Uncomment next two lines (and comment the two functions after them) to use GA algorithm
for optimization
options = gaoptimset(InitialPopulationw0PopInitRange[minKt minKt1 minKt2 minX3
minY3 minX5 minY5 maxKt maxKt1 maxKt2 maxX3 maxY3 maxX5
maxY5]PopulationSize100Displayfinal)
[wfvalexitflagoutput] =
ga(ObjectiveFunctionnvars[][][][][][]ConstraintFunctionoptions)
fminconOptions = optimset(DisplayiterLargeScaleoff) Largescale off since gradient not
provided
options = gaoptimset(InitialPopulationw0PopInitRange[minKt minKt1 minKt2 minX3
minY3 minX5 minY5 maxKt maxKt1 maxKt2 maxX3 maxY3 maxX5
maxY5]PopulationSize100HybridFcnfmincon fminconOptions)
[wfvalexitflagoutput] = ga(ObjectiveFunctionnvars[][][][][][]ConstraintFunctionoptions)
[TTaThetaDegA] = TwinTenStaticTension(w(1)w(2)w(3)w(4)w(5)w(6)w(7))
D22 confunTwinm
The constraint function confunTwinm is used by the main optimization program to ensure
input values are constrained within the prescribed regions The function makes use of inequality
constraints for seven constrained variables corresponding to the design variables It takes an
input vector corresponding to the design variables and returns a data set of the vector values that
satisfy the prescribed constraints
Appendix D 170
D23 objfunTwinm
This function is the objective function for the main optimization program It outputs a value for
a weighted objective function or a non-weighted objective function relating the optimization of
the static tension The program takes an input vector containing a set of values for the design
variables that are within prescribed constraints The description of the function is similar to
TwinTenStaticTensionm but differs in the fact that it only returns a scalar value which is the
value of the objective function
171
VITA
ADEBUKOLA OLATUNDE
Email adebukolaolatundegmailcom
Adebukola Olatunde is a graduate research student at the University of Toronto in Toronto
Ontario Canada She obtained a Bachelor‟s Degree in Mechanical Engineering from McMaster
University in Hamilton Ontario Canada in 2002 Upon graduation she pursued a graduate
degree in mechanical engineering at the University of Toronto with a specialization in
mechanical systems dynamics and vibrations and environmental engineering In September
2008 she completed the requirements for the Master of Applied Science degree in Mechanical
Engineering She has held the position of teaching assistant for undergraduate courses in
dynamics and vibrations Adebukola has completed course work in professional education She
is a registered member of professional engineering organizations including the Professional
Engineer‟s of Ontario Engineer-in-Training program the Canadian Society of Mechanical
Engineers and the National Society of Black Engineers She intends to practice as a professional
engineering consultant in mechanical design
viii
D22 confunTwinm 169
D23 objfunTwinm 170
VITA 171
ix
LIST OF TABLES
21 Passive Dual Tensioner Designs from Patent Literature
31 Selected Contact Point Types on the ith Pulley and Types for the ith Belt Span
32 Coordinate Points for Pulley Centres and Twin Tensioner Pivot
33 Geometric Results of B-ISG System with Twin Tensioner
34 Data for Input Parameters used in Dynamic and Static Computations
35 Static Solution for Belt Span Tensions in Crankshaft and ISG Driving Cases for a B-ISG
Serpentine Belt Drive with a Single Tensioner
36 Static Solution for Belt Span Tensions in Crankshaft and ISG Driving Cases for a B-ISG
Serpentine Belt Drive with a Twin Tensioner
41 Initial Values Increments and Ranges for Parameters of Twin Tensioner
51 Summary of Parametric Analysis Data for Twin Tensioner Properties
52a GA Optimization Results for Twin Tensioner Parameters and Objective Function
52b Computations for Tensions and Angles from GA Optimization Results
53a Hybrid Optimization Results for Twin Tensioner Parameters and Objective Function
53b Computations for Tensions and Angles from Hybrid Optimization Results
54a Non-Weighted Optimization Results for Twin Tensioner Parameters and Objective
Function
54b Computations for Tensions and Angles from Non-Weighted Optimizations
x
55 Weighted Optimization Results for Static Tensions for Optimal B-ISG System with a
Twin Tensioner
56 Non-Weighted Optimization Results for Static Tensions for Optimal B-ISG System with a
Twin Tensioner
xi
LIST OF FIGURES
21 Hybrid Functions
31 Schematic of the Twin Tensioner
32 B-ISG Serpentine Belt Drive with Twin Tensioner
33 Angles Coordinates and Possible Belt Contact Points for the ith and i+1th Pulleys
34 Twin Tensioner Free Body Diagram of a Four-Degree of Freedom System
35 Free Body Diagram for Non-Tensioner Pulleys
36a ISG Driving Case Natural Frequency of System and Mode Shapes for Responsive Rigid
Bodies
36b ISG Driving Case First Mode Responses
36c ISG Driving Case Second Mode Responses
37a Crankshaft Driving Case Natural Frequency of System and Mode Shapes for Responsive
Rigid Bodies
37b Crankshaft Driving Case First Mode Responses
37c Crankshaft Driving Case Second Mode Responses
38 Crankshaft Pulley Dynamic Response (for crankshaft driven case)
39 ISG Pulley Dynamic Response (for ISG driven case)
310 Air Conditioner Pulley Dynamic Response
311 Tensioner Pulley 1 Dynamic Response
xii
312 Tensioner Pulley 2 Dynamic Response
313 Tensioner Arm 1 Dynamic Response
314 Tensioner Arm 2 Dynamic Response
315 Required Driving Torque for the ISG Pulley
316 Required Driving Torque for the Crankshaft Pulley
317 Dynamic Torque for Tensioner Arm 1
318 Dynamic Torque for Tensioner Arm 2
319 Span 1 (between crankshaft and air conditioner) Dynamic Belt Span Tension
320 Span 2 (between air conditioner and tensioner 1) Dynamic Belt Span Tension
321 Span 3 (between tensioner 1 and ISG) Dynamic Belt Span Tension
322 Span 4 (between ISG and tensioner 2) Dynamic Belt Span Tension
323 Span 5 (between tensioner 2 and crankshaft) Dynamic Belt Span Tension
324 B-ISG Serpentine Belt Drive with Single Tensioner
41a Regions 1 and 2 and their Associated Guidelines for Coordinates of Tensioner Pulleys 1
amp 2
41b Regions 1 and 2 in Cartesian Space
42 Parametric Analysis for Coupled Stiffness Arm Constant kt (N∙mrad)
43 Parametric Analysis for Stiffness of Arm 1 kt1 (N∙mrad)
44 Parametric Analysis for Stiffness of Arm 2 kt2 (N∙mrad)
xiii
45 Parametric Analysis for Pulley 1 Diameter D3 (m)
46 Parametric Analysis for Pulley 2 Diameter D5 (m)
47 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Tautest Span
Tension in Crankshaft Driving Case
48 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Slackest Span
Tension in Crankshaft Driving Case
49 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Tautest Span
Tension in ISG Driving Case
410 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Slackest Span
Tension in ISG Driving Case
411 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Tautest Span
Tension in Crankshaft Driving Case
412 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Slackest Span
Tension in Crankshaft Driving Case
413 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Tautest Span
Tension in ISG Driving Case
414 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Slackest Span
Tension in ISG Driving Case
51 Static Stability of the B-ISG Twin Tensioner Based on the Angular Displacement of
Tensioner Arms 1 and 2
A1 Proposed design by Bayerische Motoren Werke AG corresponding to patent nos
EP1420192-A2 and DE10253450-A1
A2a First of four proposed designs (and its various configurations) by Bosch GMBH
corresponding to patent no WO0026532-A1
A2b Second of four proposed designs by Bosch GMBH corresponding to patent no
WO0026532-A1
A2c Third of four proposed designs (and its various configurations) by Bosch GMBH
corresponding to patent no WO0026532-A1
A2d Fourth of four proposed designs (and its various configurations) by Bosch GMBH
corresponding to patent no WO0026532-A1
A3 Proposed design by Daimler Chrysler AG corresponding to patent no DE10324268-A1
A4 Proposed design by Dayco Products LLC corresponding to patent no US6942589-B2
xiv
A5 Proposed design by Gates Corp corresponding to patent no WO2003038309-A
A6 Proposed design by General Motors Corp corresponding to patent no US20060287146-
A1
A7 Proposed design by INA Schaeffler KG corresponding to patent no DE10044645-A1
A8a First of three proposed designs by INA Schaeffler KG corresponding to patent no
DE10159073-A1
A8b Second of three proposed designs by INA Schaeffler KG corresponding to patent no
DE10159073-A1
A8c Third of three proposed designs by INA Schaeffler KG corresponding to patent no
DE10159073-A1
A9 Proposed design by INA Schaeffler KG corresponding to patent no DE10359641-A1
A10 Proposed design by INA Schaeffler KG corresponding to patent no EP1723350-A1
A11 Proposed design by INA Schaeffler KG corresponding to patent no EP1738093-A1
A12 Proposed design by INA Schaeffler KG corresponding to patent no DE102004012395-
A1
A13 Proposed design by INA Schaeffler KG corresponding to patent nos DE102005017038-
A1and WO2006108461-A1
A14 Proposed design by Litens Automotive GMBH et al corresponding to patent no
US20010007839-A1
A15 Proposed design by Mitsubishi Jidosha Eng KK and Mitsubishi Motor Corp
corresponding to patent no JP2005083514-A
A16 Proposed design by Nissan Motor Co Ltd corresponding to patent no JP3565040-B2
A17 Proposed design by NTN Corp corresponding to patent no JP2006189073-A
A18 Proposed design by Valeo Equipment Electriques Moteur corresponding to patent nos
EP1658432 and WO2005015007
B1 Single Tensioner B-ISG System
B2 Free-body Diagram of ith Pulley
xv
B3 Free-body Diagram of Single Tensioner
C1 Schematic of B-ISG System with Twin Tensioner
C2 Possible Contact Points
xvi
LIST OF SYMBOLS
Latin Letters
A Belt cord cross-sectional area
C Damping matrix of the system
cb Belt damping
119888119894119887 Belt damping constant of the ith belt span
119914119946119946 Damping matrix element in the ith row and ith column
ct Damping acting between tensioner arms 1 and 2
cti Damping of the ith tensioner arm
DCS Diameter of crankshaft pulley
DISG Diameter of ISG pulley
ft Belt transition frequency
H(n) Phase change function
I Inertial matrix of the system
119920119938 Inertial matrix under ISG driving phase
119920119940 Inertial matrix under crankshaft driving phase
Ii Inertia of the ith pulley
Iti Inertia of the ith tensioner arm
119920120784120784 Submatrix of inertial matrix I
j Imaginary coordinate (ie (-1)12
)
K Stiffness matrix of the system
xvii
119896119887 Belt factor
119870119887 Belt cord stiffness
119896119894119887 Belt stiffness constant of the ith belt span
kt Spring stiffness acting between tensioner arms 1 and 2
kti Coil spring of the ith tensioner arm
119922120784120784 Submatrix of stiffness matrix K
Lfi Lbi Lengths of possible belt span connections from the ith pulley
Lti Length of the ith tensioner arm
Modeia Mode shape of the ith rigid body in the ISG driving phase
Modeic Mode shape of the ith rigid body in the crankshaft driving phase
n Engine speed
N Motor speed
nCS rpm of crankshaft pulley
NF Motor speed without load
nISG rpm of ISG pulley
Q Required torque matrix
qc Amplitude of the required crankshaft torque
QcsISG Required torque of the driving pulley (crankshaft or ISG)
Qm Required torque matrix of driven rigid bodies
Qti Dynamic torque of the ith tensioner arm
Ri Radius of the ith pulley
T Matrix of belt span static tensions
xviii
Trsquo Dynamic belt tension matrix
119931119940 Damping matrix due to the belt
119931119948 Stiffness matrix due to the belt
Ti Tension of the ith belt span
To Initial belt tension for the system
Ts Stall torque
Tti Tension for the neighbouring belt spans of the ith tensioner pulley
(XiYi) Coordinates of the ith pulley centre
XYfi XYbi XYfbi
XYbfi Possible connection points on the ith pulley leading to the ith belt span
XYf2i XYb2i
XYfb2i XYbf2i Possible connection points on the ith pulley leading to the (i-1)th belt span
Greek Letters
αi Angle between the datum and the line connecting the ith and (i+1)th pulley
centres
βji Angle of orientation for the ith belt span
120597θti(t) 120579 ti(t)
120579 ti(t)
Angular displacement velocity and acceleration (rotational coordinate) of the
ith tensioner arm
120637119938 General coordinate matrix under ISG driving phase
120637119940 General coordinate matrix under crankshaft driving phase
θfi θbi Angles between the datum and the belt connection spans with lengths Lfi and
Lbi respectively
Θi Amplitude of displacement of the ith pulley
xix
θi(t) 120579 i(t) 120579 i(t) Angular position velocity and acceleration (rotational coordinate) of the ith
pulley
θti Angle of the ith tensioner arm
θtoi Initial pivot angle of the ith tensioner arm
θm Angular displacement matrix of driven rigid bodies
Θm Amplitude of displacement of driven rigid bodies
ρ Belt cord density
120601119894 Belt wrap angle on the ith pulley
φmax Belt maximum phase angle
φ0deg Belt phase angle at zero frequency
ω Frequency of the system
ωcs Angular frequency of crankshaft pulley
ωISG Angular frequency of the ISG pulley
120654119951 Natural frequency of system
1
CHAPTER 1 INTRODUCTION
11 Background
Belt drive systems are the means of power transmission in conventional automobiles The
emergence of hybrid technologies specifically the Belt-driven Integrated Starter-generator (B-
ISG) has placed higher demands on belt drives than ever before The presence of an integrated
starter-generator (ISG) in a belt transmission places excessive strain on the belt leading to
premature belt failure This phenomenon has motivated automotive makers to design a tensioner
that is suitable for the B-ISG system
The belt drive is also known interchangeably as the front-end accessory drive-belt (FEAD) the
belt accessory-drive system (BAS) or the belt transmission system In a traditional setting the
role of this system is to transmit torque generated by an internal combustion engine (ICE) in
order to reliably drive multiple peripheral devices mounted on the engine block The high speed
torque is transmitted through a crankshaft pulley to a serpentine belt The serpentine belt is a
single continuous member that winds around the driving and driven accessory pulleys of the
drive system Serpentine belts used in automotive applications consist of several layers The
load-bearing layer is a flexible member consisting of high stiffness fibers [1] It is covered by a
protective layer to guard against mechanical damage and is bound below by a visco-elastic layer
that provides the required shock absorption and grip against the rigid pulleys [1] The accessory
devices may include an alternator power steering pump water pump and air conditioner
compressor among others
Introduction 2
The B-ISG system is a transmission system characteristic to micro-hybrid automobiles It is akin
to traditional belt drives differing in the fact that an electric motor called an integrated starter-
generator (ISG) replaces the original alternator re-starts the engine from idle speed and provides
braking regeneration [2] The re-start function of the micro-hybrid transmission is known as
stop-start In the B-ISG setting the ISG is mounted on the belt drive The ISG produces a speed
of approximately 2000 to 2500rpm in order to spin the engine at approximately 750rpm and
upwards to produce an instantaneous start in the start-stop process [3] The high rotations per
minute (rpm) produced by the ISG consistently places much higher tension requirements on the
belt than when the crankshaft is driving the belt It is preferable not to exceed a range of 600N to
800N of tension on the belt since this exceeds the safe operating conditions of belts used in most
traditional drive systems [4] The traditional belt drive system‟s tensioner a single-arm
tensioner does not suitably reduce the high belt tension nor provide enough tension in the slack
belts spans occurring in the ISG phase of operation for the B-ISG system
In order for the belt to transfer torque in a drive system its initial tension must be set to a value
that is sufficient to keep all spans rigid This value must not be too low as to allow any one span
to be slack during the drive‟s phases of operation Furthermore the belt must not be ldquoinstalled
with too high a tensionrdquo since this can lead to ldquopremature failure of the bearings supporting the
drive and driven pulleys and of the belt itselfrdquo [5] The presence of a tensioning mechanism in
an automotive belt drive allows for an enhanced belt life and performance since pre-tensioning
of the belt is normally not sufficient for all phases of belt drive operation A tensioner allows for
the system to cope with moderate to severe changes in belt span tensions
Introduction 3
Traditional automotive tensioners for belt drives of an ICE consist of a single spring-loaded
arm This type of tensioner is normally designed to provide a passive response to changes in belt
span tension The introduction of the ISG electric motor into the traditional belt drive with a
single-arm tensioner results in the presence of excessively slack spans and excessively tight
spans in the belt The tension requirements in the ISG-driving phase which differ from the
crankshaft-driving phase are poorly met by a traditional single-arm passive tensioner
Tensioners can be divided into two general classes passive and active In both classes the
single-arm tensioner design approach is the norm The passive class of tensioners employ purely
mechanical power to achieve tensioning of the belt while the active class also known as
automatic tensioners typically use some sort of electronic actuation Automatic tensioners have
been employed by various automotive manufacturers however ldquosuch devices add mass
complication and cost to each enginerdquo [5]
12 Motivation
The motivation for the research undertaken arises from the undesirable presence of high belt
tension in automotive belt drives Manufacturers of automotive belt drives have presented
numerous approaches for tension mechanism designs As mentioned in the preceding section
the automation of the traditional single-arm tensioner has disadvantages for manufacturers A
survey of the literature reveals that few quantitative investigations in comparison to the
qualitative investigations provided through patent literature have been conducted in the area of
passive and dual tensioner configurations As such the author of the research project has selected
to investigate the performance of a passive twin-arm tensioner design The theoretical tensioner
Introduction 4
configuration is motivated by research and developments of industry partner Litens
Automotivendash a manufacturer of automotive belt drive systems and components Litens‟
specialty in automotive tensioners has provided a basis for the research work conducted
13 Thesis Objectives and Scope of Research
The objective of this project is to model and investigate a system containing a passive twin-arm
tensioner in a B-ISG serpentine belt drive where the driving pulley alternates between a
crankshaft pulley and an ISG pulley The modeling of a serpentine belt drive system is in
continuation of the work done by post-doctoral fellow Zhen Mu in development of the priority
software known as FEAD at the University of Toronto Firstly for the B-ISG system with a
twin-arm tensioner the geometric state and its equations of motion (EOM) describing the
dynamic and static states are derived The modeling approach was verified by deriving the
geometric properties and the EOM of the system with a single tensioner arm and comparing its
crankshaft-phase‟s simulation results with FEAD software simulations This also provides
comparison of the new twin-arm tensioner belt drive model with the former single-arm tensioner
equipped belt drive model Secondly the model for the static system is investigated through
analysis of the tensioner parameters Thirdly the design variables selected from the parametric
analysis are used for optimization of the new system with respect to its criteria for desired
performance
Introduction 5
14 Organization and Content of the Thesis
This thesis presents the investigation of a passive twin-arm tensioner design in a B-ISG
serpentine belt drive system which is distinguished by having its driving pulley alternate
between a crankshaft pulley and an ISG pulley
Chapter 2 presents the literature reviewed relevant to the area of the thesis topic The context of
the research discusses the function and location of the ISG in hybrid technologies in order to
provide a background for the B-ISG system The attributes of the B-ISG are then discussed
Subsequently a description is given of the developments made in modeling belt drive systems
At the close of the chapter the prior art in tensioner designs and investigations are discussed
The third chapter describes the system models and theory for the B-ISG system with a twin-arm
tensioner Models for the geometric properties and the static and dynamic cases are derived The
simulation results of the system model are presented
Then the fourth chapter contains the parametric analysis The methodologies employed results
and a discussion are provided The design variables of the system to be considered in the
optimization are also discussed
The optimization of a B-ISG system with a passive twin-arm tensioner is presented in Chapter 5
The evaluation of optimization methods results of optimization and discussion of the results are
included Chapter 6 concludes the thesis work in summarizing the response to the thesis
Introduction 6
objectives and concluding the results of the investigation of the objectives Recommendations for
future work in the design and analysis of a B-ISG tensioner design are also described
7
CHAPTER 2 LITERATURE REVIEW
21 Introduction
This literature review justifies the study of the thesis research the significance of the topic and
provides the overall framework for the project The design of a tensioner for a Belt-driven
Integrated Starter-generator (B-ISG) system is a link in the chain of power transmission
developments in hybrid automobiles This chapter will begin with the context of the B-ISG
followed by a review of the hybrid classifications and the critical role of the ISG for each type
The function location and structure of the B-ISG system are then discussed Then a discussion
of the modeling of automotive belt transmissions is presented A systematic review of the prior
art and current state of tensioning mechanisms for B-ISG systems amalgamates the literature and
research evidence relevant to the thesis topic which is the design of a B-ISG tensioner
The Belt-driven Integrated Starter-generator (B-ISG) system is a part of a hybrid class that is
distinguished from other hybrid classes by the structure functions and location of its ISG The
B-ISG unit is a hybrid technology applied to traditional automotive belt drives The use of a B-
ISG system to achieve a start-stop function in the car engine is estimated to cut fuel consumption
in conventional automobiles by up to ten percent and thus reduce CO2 emissions [6]
Environmental and legislative standards for reducing CO2 emissions in vehicles have called for
carmakers to produce less polluting and more efficient vehicle powertrain systems [7] The
transition to bdquocleaner‟ cars makes room for the introduction of the ISG machine into conventional
automotive belt drives [8] The reduction of CO2 emissions and the similarity of the B-ISG
Literature Review 8
transmission to that of conventional cars provide the motivation for the thesis research
Consequently the micro-hybrid class of cars is especially discussed in the literature review since
it contains the B-ISG type of transmission system The micro-hybrid class is one of several
hybrid classes
A look at the performance of a belt-drive under the influence of an ISG is rooted in the
developments of hybrid technology The distinction of the ISG function and its location in each
hybrid class is discussed in the following section
22 B-ISG System
221 ISG in Hybrids
This section of the review discusses the standard classes of hybrid cars which are full power
mild and micro- hybrids Special attention is given to hybrid vehicle architectures involving
internal combustion engines (ICEs) as the main power source This is done for the sake of
comparison between hybrid classes since the ICE is the standard power source for B-ISG micro-
hybrids which is the focus of the research The term conventional car vehicle or automobile
henceforth refers to a vehicle powered solely by a gas or diesel ICE
A hybrid vehicle has a drive system that uses a combination of energy devices This may include
an ICE a battery and an electric motor typically an ISG Two systems exist in the classification
of hybrid vehicles The older system of classification separates hybrids into two classes series
hybrids and parallel hybrids In the older system many modern hybrid vehicles have modes of
operation matching both categories classifying them under either of the two classes [9] The
Literature Review 9
new system of classification has four classes full power mild and micro Under these classes
vehicles are more often under a sole category [9] In both systems an ICE may act as the primary
source of power otherwise it may be a fuel cell The fuel used by the ICE may be gas (petrol)
diesel or an alternative fuel such as ethanol bio-diesel or natural gas
2211 Full Hybrids
In a full hybrid car the ICE is used to power the integrated starter-generator (ISG) which stores
electrical energy in the batteries to be used to power an electric traction motor [8] The electric
traction motor is akin to a second ISG as it generates power and provides torque output It also
supplies an extra boost to the wheels during acceleration and drives up steep inclines A full
hybrid vehicle is able to move by electrical power only It can be driven by the ISG powering
the electric traction motor without the engine running This silent acceleration known as electric
launch is normally employed when accelerating from standstill [9] Full hybrids can generate
and consume energy at the same time Full hybrid vehicles also use regenerative braking [8]
The ISG allows this by converting from an electric traction motor to a generator when braking or
decelerating The kinetic energy from the car‟s motion is then turned into electricity and stored
in the batteries For full hybrids to achieve this they often use break-by-wire a form of
electronically controlled braking technology
A high-voltage (ie 36- or 42-volt) ISG is employed in full hybrids to start the ICE It spins the
engine more than 900 rpm whereas conventional 12-volt starter motors spin the engine at
approximately 250 rpm [9] Thus the full hybrid vehicle is able to have an instantaneous start In
full hybrids the ISG is placed in the position of the flywheel and can have its motion decoupled
Literature Review 10
from the engine [9] The ISG device also allows full hybrids to have engine start-stop also called
an idle-stop ability The idle-stop function refers to when the engine shuts down as soon as a
vehicle stops from its ICE driving mode which saves on the fuel it normally burns while idling
[8] The vehicle returns to the engine driving mode of operation by way of the ISG‟s start-up of
the crankshaft which restarts the engine in less than 300 milliseconds [9] In summary at
standstill the tachometer of the engine drops to 0 rpm since the engine has ceased the engine is
started only when needed which is often several seconds after acceleration has begun The
engine start-stop feature is achieved by way of an electronic control system that shuts off the ICE
when it is not needed to assist in driving the wheels or to produce electricity for recharging the
batteries The start-stop feature by itself is estimated to produce a ten percent fuel gain in hybrids
over conventional vehicles particularly in urban driving conditions [9] Since the ICE is
required to provide only the average horsepower used by the vehicle the engine is downsized in
comparison to a conventional automobile that obtains all its power from an ICE Frequently in
full hybrids the ICE uses an alternative operating strategy such as the Atkinson Cycle which has
a higher efficiency while having a lower power output Examples of full hybrids include the
Ford Escape and the Toyota Prius [9]
2212 Power Hybrids
Akin to the full hybrid the ISG of the power hybrid enables the same features electric launch
regenerative braking and engine idle-stop The distinguishing characteristic from full hybrids is
the ICE is not downsized to meet only the average power demand [9] Thus the engine of a
power hybrid is large and produces a high amount of horsepower compared to the former
Overall a power hybrid has the assist of a full size ICE and therefore has more torque and a
Literature Review 11
greater acceleration performance than a full hybrid or a conventional vehicle with the same size
ICE [9] The Lexus RX400h unit is an example of a power hybrid [9]
2213 Mild Hybrids
In the hybrid types discussed thus far the ISG is positioned between the engine and transmission
to provide traction for the wheels and for regenerative braking Often times the armature or rotor
of the electric motor-generator which is the ISG replaces the engine flywheel in full and power
hybrids [9] In the case of the mild hybrid the ISG is not decoupled from the ICE and hence it is
not able to drive the wheels apart from the engine It remains that the ISG shares the same shaft
with the ICE In this environment the electric launch feature does not exist since the ISG does
not turn the wheels independently of the engine and energy cannot be generated and consumed
at the same time However the ISG of the mild hybrid allows for the remaining features of the
full hybrid regenerative braking and engine idle-stop including the fact that the engine is
downsized to meet only the average demand for horsepower Mild hybrid vehicles include the
GMC Sierra pickup and 2003 to 2005 Honda Civic models [9]
2214 Micro Hybrids
Micro hybrid is the category of hybrids that can contain a B-ISG transmission and is also closest
to modern conventional vehicles This class normally features a gas or diesel ICE [9] The
conventional automobile is modified by installing an ISG unit on the mechanical drive in place
of or in addition to the starter motor The starter motor typically 12-volts is removed only in
the case that the ISG device passes cold start testing which is also dependent on the engine size
[10] Various mechanical drives that may be employed include chain gear or belt drives or a
Literature Review 12
clutchgear arrangement The majority of literature pertaining to mechanical driven ISG
applications does not pursue clutchgear arrangements since it is associated with greater costs
and increased speed issues Findings by Henry et al [11] show that the belt drive in
comparison to chain and gear drives has a decreased cost (especially if the ISG is mounted
directly to the accessory drive) has no need for lubrication has less restriction in the packaging
environment and produces very low noise Also mounting the ISG unit on a separate belt from
that linking the accessory pulleys is undesirable since applying the ISG directly to the accessory
belt drive requires less engine transmission or vehicle modifications
As with full power and mild hybrids the presence of the ISG allows for the start-stop feature
The automobile‟s electronic control unit (ECU) is calibrated or engine control circuitry (a
separate ECU) is added to the conventional car in order to shut down the engine when the
vehicle is stopped [12] The control system also controls the charge cycle of the ISG [9] This
entails that it dictates the field current by way of a microprocessor to allow the system to defer
battery charge cycles until the vehicle is decelerating [13] This produces electricity to recharge
the battery primarily during deceleration and braking The B-ISG transmission of a micro hybrid
and its various components are discussed in the subsequent section Examples of micro hybrid
vehicles are the PSA Group‟s Citroen C2 and C3 [14] Ford‟s Fiesta [14] and BMW‟s Mini
Cooper D and various others of BMW‟s European models [15]
Literature Review 13
Figure 21 Hybrid Functions
Source Dr Daniel Kok FFA July 2004 modified [16]
Figure 21 shows that the higher the voltage available to the ISG unit the more hybrid functions
it is capable of performing It is noted that B-ISG transmissions of the micro-hybrid class may
also exceed the typical functions of micro-hybrids For instance Ford‟s HyTrans van (developed
in partnership with Ricardo UK Ltd Valeo SA Gates Corporation and the UK Department for
Transport) uses a B-ISG system and a 42-volt battery The van is diesel-powered and has
characteristics of a mild hybrid such as cold cranks and engine assists [17]
222 B-ISG Structure Location and Function
2221 Structure and Location
The ISG is composed of an electrical machine normally of the inductive type which includes a
stator (stationary part of the ISG) and a rotor (non-stationary part of the ISG) and a converter
comprising of a regulator a modulator switches and filters There are various configurations to
integrate the ISG unit into an automobile power train One configuration situates the ISG
directly on the crankshaft in the place of the present flywheel [11] This set-up is more compact
however it results in a longer power train which becomes a potential concern for transverse-
Literature Review 14
mounted engines [18] An alternative set-up is to have a side-mounted ISG This term is used to
describe the configuration of mounting the electrical device on the side of the mechanical drive
[18] As mentioned in Section 2214 a belt drive is used as the mechanical drive for the thesis
research hence the ISG is belt-mounted and the transmission becomes a belt-driven ISG system
In this arrangement the ISG replaces the alternator [13] and in some cases the starter motor may
be removed This design allows for the functions of the ISG system mentioned in the description
of the ISG role in micro-hybrids [9] The side-mounted ISG specifically the belt-mounted ISG
is more evolutionary to the conventional car since it ldquoallows for a more traditional under-hood
layoutrdquo [11]
2222 Functionalities
The primary duty of the ISG in a micro hybrid specifically in a B-ISG setting is to bring the
engine from rest to normal operating speeds within a time span ranging from 250 to 400 ms [3]
and in some high voltage settings to provide cold starting
The cold starting operation of the ISG refers to starting the engine from its off mode rather than
idle mode andor when the engine is at a low temperature for example -29 to -50 degrees
Celsius [2] If the ISG is used for cold starting the peak torque is determined by the torque
requirement for the cold starting operation of the target vehicle since it is greater than the
nominal torque For this function the ldquomachine has to provide a breakaway torque about 15 [to]
18 times the nominal cranking torque to overcome static torque and rotate the engine from 0 to
[between] 10 [and] 20rpmrdquo [2] This remains to be a challenge for the ISG as the 12-volt
architecture most commonly found in vehicles does not supply sufficient voltage [2] The
introduction of the ISG machine and other electrical units in vehicles encourages a transition
Literature Review 15
from a 12-volt or 14-volt to a 42-volt electrical architecture [19] The transition to 42-volt
architecture brings ldquopotential higher-voltage functionalities that come with an ISG systemrdquo [20]
At present ldquowhen the [ISG] machine cannot provide enough torque for initial cold engine
cranking the conventional starter will [remain] in the system and perform only for the initial
cranking while the stop-start function is taken over by the [ISG] machinerdquo [2] The ISG‟s launch
assist torque the torque required to bring the engine from idle speed to the speed at which it can
develop a higher torque output is 2000 to 2500 rpm for most gas engines [3]
Delphi‟s Energen 5 High Output 12-volt Belt-alternator-starter (or B-ISG) was implemented by
researchers on a 53 L V-8 engine with an automatic transmission in a Chevrolet Silverado truck
[21] The ISG was applied in a belt-mounted configuration and was used only for warm engine
re-starts The results of Wezenbeek et al [21] showed that the starting torque for a re-start by the
12-Volt ISG was 42 Nm ISG‟s have also been used in 14V 36V and 42V architectures [13]
23 Belt Drive Modeling
The modeling of a serpentine belt drive and tensioning mechanism has typically involved the
application of Newtonian equilibrium equations to rigid bodies in order to derive the equations of
motion for the system There are two modes of motion in a serpentine belt drive transverse
motion and rotational motion The former can be viewed as the motion of the belt directed
normal to the direction of the beltpulley contact plane similar to the vibratory motion of a taut
string that is fixed at either end However the study of the rotational motion in a belt drive is the
focus of the thesis research
Literature Review 16
Much work on the mechanics of the belt drive was carried out by Firbank [22] Firbank‟s
models helped to understand belt performance and the influence of driving and driven pulleys on
the tension member The first description of a serpentine belt drive for automotive use was in
1979 by Cassidy et al [23] and since this time there has been an increasing body of knowledge
on the mathematical modeling of serpentine belt drives Ulsoy et al [24] presented a design
methodology to improve the dynamic performance of instability mechanisms for belt tensioner
systems The mathematical model developed by Ulsoy et al [24] coupled the equations of
motion that were obtained through a dynamic equilibrium of moments about a pivot point the
equations of motion for the transverse vibration of the belt and the equations of motion for the
belt tension variations appearing in the transverse vibrations This along with the boundary and
initial conditions were used to describe the vibration and stability of the coupled belt-tensioner
system Their system also considered the geometry of the belt drive and tensioner motion
Hereafter Beikmann et al [25] predicted the belt drive vibration for a system composed of a
driving pulley driven pulley and a dynamic tensioner The authors coupled the linear equations
of transverse motion for the respective belt spans with the equations of motion for pulleys and a
tensioner This was used to form the free response of the system and evaluate its response
through a closed-form solution of the system‟s natural frequencies and mode shapes
A complex modal analysis of a serpentine belt drive system was carried out by Kraver et al [26]
to determine the effect of damping on rotational vibration mode solutions The equations of
motion developed for a multi-pulley flat belt system with viscous damping and elastic
Literature Review 17
properties including the presence of a rotary tensioner were manipulated to carry out the modal
analysis
Beikmann et al [27] also derived a nonlinear model to predict the operating state of a belt-
tensioner system by way of nonlinear numerical methods and an approximated linear closed-
form method The authors used this strategy to develop a single design parameter referred to as
a tensioner constant to measure the effectiveness of the tensioning mechanism in relation to its
operating state from a reference state The authors considered the steady state tensions in belt
spans as a result of accessory loads belt drive geometry and tensioner properties
Zhang and Zu [28] conducted a modal analysis for the response of a linear serpentine belt drive
system A non-iterative approach was used to explicitly form the equations for the system‟s
natural frequencies An exact closed-form expression for the dynamic response of the system
using eigenfunction expansion was derived with the system under steady-state conditions and
subject to harmonic excitation
The work conducted by Balaji and Mockensturm [29] considered a front-end accessory drive
(FEAD) with a decoupler or isolator attached to a pulley The rotational response for the FEAD
was found analytically by considering the system to be piecewise linear about the equilibrium
angular deflections The effect of their nonlinear terms was considered through numerical
integration of the derived equations of motion by way of the iterative methodndash fourth order
Runge-Kutta The authors in this case considered the longitudinal (ie rotational) vibration of
the belt spans only
Literature Review 18
The first to carry out the analysis of a serpentine belt drive system containing a two-pulley
tensioner was Nouri in 2005 [30] Nouri found the closed-form analytical solution of a
serpentine belt drive with a two-pulley tensioner for the case of sinusoidal excitation He
employed Runge Kutta method as well to solve the equations of motion to find the response of
the system under a general input from the crankshaft The author‟s work also included the
optimization of the tensioner design in order to minimize belt span vibrations due to crankshaft
excitation Furthermore the author applied active control techniques to the tensioner in a belt
drive system
The works discussed have made significant contributions to the research and development into
tensioner systems for serpentine belt drives These lead into the requirements for the structure
function and location of tensioner systems particularly for B-ISG transmissions
24 Tensioners for B-ISG System
241 Tensioners Structure Function and Location
Literature shows that the improvement of a serpentine belt life in a B-ISG system centers on the
tensioning mechanism redesign This mechanism as shown by researchers including
Wezenbeek et al [21] and Henry et al [11] is crucial in establishing the least tension in the belt
(above a zero value) in order to guard against failure by way of slip due to slack spans in the belt
and oscillations during engine re-start It is noted by Firbank [22] that the mechanics of a belt-
drive ldquois based on the idea that belt behaviour is governed by the elastic extension or contraction
of the belt arising from tension variationsrdquo [22] these variations may be compensated for by an
adjustable tensioner
Literature Review 19
The two types of tensioners are passive and active tensioners The former permits an applied
initial tension and then acts as an idler and normally employs mechanical power and can include
passive hydraulic actuation This type is cheaper than the latter and easier to package The latter
type is capable of continually adjusting the belt tension since it permits a lower static tension
Active tensioners typically employ electric or magnetic-electric actuation andor a combination
of active and passive actuators such as electrical actuation of a hydraulic force
Conventional belt tensioners comprise of a single tensioner arm that is fitted with a sole idler
pulley to engage a serpentine belt [31] A radial bearing is used to rotatably connect the idler
pulley to the tensioner arm [31] The tensioner arm is mounted on a pivot pin that is wrapped by
a bushing and is free to rotate [31] The pin covered by the bushing is fixed to the engine
housing [31] A rotary spring is wrapped about the bearing pin and bushing to provide a pre-
tension force to the belt via the tensioner arm and idler pulley thus taking up the slack due to the
changes in belt length [31] When the belt undergoes stretch under a load the spring drives the
tensioner arm and idler pulley further into the belt [31] Belt tension changes under the modes of
operation which can include when the crankshaft (or driving pulley) abruptly decelerates from a
steady-state condition and auxiliary components continue to rotate still in their own inherent
inertia and thus become the primary drivers [31] These fluctuations in belt tension lead to belt
flutter or skip and slip that may damage other components present in the belt drive [31]
Locating the tensioner on the slack side of the belt is intended to lower the initial static tension
[11] In conventional vehicles the engine always drives the alternator so the tensioner is located
in the belt span that links the crankshaft and alternator pulleys In a B-ISG setting the slack span
Literature Review 20
of the belt alternates between the driving mode of the ISG and the driving mode of the crankshaft
[32] Research by Henry et al [11] and also the summary of prior art for tensioners in Table
21 show that placing the idlertensioner pulley in the slack span in the case that the ISG is
driving instead of in the slack span when the crankshaft is driving allows for easier packaging
and for the least static tension Designs shown in Table 21 place the tensioneridler pulley in the
same span as Henry et al [11] or in both the slack and taut spans if using a double
tensioneridler configuration
242 Systematic Review of Tensioner Designs for a B-ISG System
The proposals for belt tensioner devices to manage the issue of high peaks in belt tension for B-
ISG settings are largely in patent records as the re-design of a tensioner has been primarily a
concern of automotive makers thus far A systematic review of the patent literature has been
conducted in order to identify evaluate and collate relevant tensioning mechanism designs
applicable to a B-ISG setting Its research objective is to influence the selection of a tensioner
configuration for the thesis study
The predefined search strategy used by the researcher has been to consider patents dating only
post-2000 as many patents dating earlier are referred to in later patents as they are developed on
in most cases by the original inventor (eg an INA Schaeffler KG patent published in 2000 may
refer to its own earlier patent presented in 1999) Patents dating pre-2000 that do not have any
successor were also considered The inclusion and exclusion criteria and rationales that were
used to assess potential patents are as follows
Inclusion of
Literature Review 21
tensioner designs with two arms andor two pivots andor two pulleys
mechanical tensioners (ie exclusion of magnetic or electrical actuators or any
combination of active actuators) in order to minimize cost
tension devices that are an independent structure apart from the ISG structure in order to
reduce the required modification to the accessory belt drive of a conventional automobile
and
advanced designs that have not been further developed upon in a subsequent patent by the
inventor or an outside party
Table 21 provides a collation of the results for the systematic review based on the selection
criteria Illustrations of the collated patent designs may be seen in Appendix A It is noted that
the patent literature pertaining to these designs in most cases provides minimal numerical data
for belt tensions achieved by the tensioning mechanism In most cases only claims concerning
the outcome in belt performance achievable by the said tension device is stated in the patent
Table 21 Passive Dual Tensioner Designs from Patent Literature
Bayerische
Motoren Werke
AG
Patents EP1420192-A2 DE10253450-A1 [33]
Design Approach
2 tensioner pulleys (idlers) and 2 tension arms are mounted outside the periphery of the belt drive these form tiltable clamping arms around a common axis of rotation
A torsion spring is used at bearing bushings to mount tension arms at ISG shaft
Each tension arm cooperates with torsion spring mechanism to rotate through a damping
device in order to apply appropriate pressure to taut and slack spans of the belt in
different modes of operation
Bosch GMBH Patent WO0026532 et al [34]
Design Approach
2 tension pulleys each one is mounted on the return and load spans of the driven and
driving pulley respectively
Idlers (tension pulleys) each connect to a spring which is attached on one end to a fixed point
Literature Review 22
Idlers‟ motions are independent of each other and correspond to the tautness or
slackness in their respective spans
Or alternatively a spring connects the idler pulleys and one of the two idlers is fixed at
its axis of rotation
Daimler Chrysler
AG
Patents DE10324268-A1 [35]
Design Approach
2 idlers are given a working force by a self-aligning bearing
Bearing supports auxiliary unit (ISG) and is arranged concentrically with the axle
auxiliary unit pulley
Dayco Products
LLC
Patents US6942589-B2 et al [36]
Design Approach
2 tension arms are each rotatably coupled to an idler pulley
One idler pulley is on the tight belt span while the other idler pulley is on the slack belt
span
Tension arms maintain constant angle between one another
One arm forms a positive differential angle with the belt and the remaining arm forms a negative differential angle with the belt
Idler pulleys are on opposite sides of the ISG pulley
Gates Corporation Patents US20060249118-A1 WO2003038309-A [37]
Design Approach
A tensioner pulley contacts the belt at the slack span during start-up (ISG-driving mode)
A tensioner is asymmetrically biased in direction tending to cause power transmission
belt to be under tension
McVicar et al
(Firm General
Motors Corp)
Patent US20060287146-A1 [38]
Design Approach
2 tension pulleys and carrier arms with a central pivot are mounted to the engine
One tension arm and pulley moderately biases one side of belt run to take up slack
during engine start-up while other tension arm and pulley holds appropriate bias against
taut span of belt
A hydraulic strut is connected to one arm to provide moderate bias to belt during normal
engine operation and velocity sensitive resistance to increasing belt forces during engine
start-up
INA Schaeffler
KG et al
Patents DE10044645-A1 [39] DE10159073-A1 [40] EP1723350-A1 et al [41]
DE10359641-A1 et al [42] EP1738093-A1 et al [43] DE102004012395-A1 [44]
WO2006108461-A1 et al [45]
Design Approach
2 tension arms and 2 pulleys approach ndash o Mutually independent tensioning arms are supported for rotation in the same
plane of the housing part
o Idler pulley corresponding to each tensioning arm engages with different
sections of belt
o When high tension span alternates with slack span of belt drive one tension
arm will increase pressure on current slack span of belt and the other will
decrease pressure accordingly on taut span
o Or when the span under highest tension changes one tensioner arm moves out
of the belt drive periphery to a dead center due to a resulting force from the taut
span of the ISG starting mode
o Deflection of the taut span acts on associated pulley to apply a counter-moment to the other idler pulley on the slack span
Literature Review 23
o The 2 lever arms are of different lengths and each have an idler pulley of
different diameters and different wrap angles of belt (see DE10045143-A1 et
al)
1 tensioner arm and 2 pulleys approach ndash
o 2 idler pulleys are pinned to a beam arranged on a clamping arm that is tiltably
linked to the beam o The ISG machine is supported by a shock absorber
o During ISG start-up one idler pulley is induced to a dead center position while
it pulls the remaining idler pulley into a clamping position until force
equilibrium takes place
o A shock absorber is laid out such that its supporting spring action provides
necessary preloading at the idler pulley in the direction of the taut span during
ISG start-up mode
Litens Automotive
Group Ltd
Patents US6506137-B2 et al [46]
Design Approach
2 tension pulleys on opposite sides of the ISG pulley engage the belt
They are positioned such that their applied forces result in opposing directed moments with respect to the tension device‟s axis of pivot
The pivot axis varies relative to the force applied to each tension pulley
Diameters of the tensioner pulleys are approximately equal and belt wrap angles of the
tensioner pulleys are approximately equal
A limited swivel angle for the tensioner arms work cycle is permitted
Mitsubishi Jidosha
Eng KK
Mitsubishi Motor
Corp
Patents JP2005083514-A [47]
Design Approach
2 tensioners are used
1 tensioner is held on the slack span of the driving pulley in a locked condition and a
second tensioner is held on the slack side of the starting (driven) pulley in a free condition
Nissan Patents JP3565040-B2 et al [48]
Design Approach
A single tensioner is on the slack span once ISG pulley is in start-up mode
The tension device is comprised of a oil pressure tensioner and a half ratchet mechanism
(a plunger which performs retreat actuation according to the energizing force of the oil
pressure spring and load received from the ISG)
The tensioner is equipped with a relief valve to keep a predetermined load lower than the
maximum load added by the ISG device
NTN Corp Patent JP2006189073-A [49]
Design Approach
An automatic tensioner is equipped with a hydraulic damper mechanism comprised of a
screw bolt using saw-screwed teeth and a cylinder nut a return spring and a spring seat
in a pressure chamber (within the screw bolt) a rod seat (that is fitted to the lower end of
the cylinder nut) a spring support (arranged on varying diameter stepped recessed
sections of the rod seat) and a check valve with an openingclosing passage
The cylinder and screw bolt act as the rigidity buffer under excessive loads during ISG
start-up mode of operation
Valeo Equipment
Electriques
Moteur
Patents EP1658432 WO2005015007 [50]
Design Approach
ldquoThe invention relates to a system or a starter (10) in which a pulley (80) is rotationally mounted on a section (22) of a shaft which axially extends inside a pulley (80) and
Literature Review 24
forwards at least partially outside a support element (200) and is characterized in that
the free front end (23) of said shaft section (22) is carried by an arm (206) connected to
the support element (200)rdquo
The author notes that published patents and patent applications may retain patent numbers for multiple patent
offices (ie European Patent Office German Patent Office etc) In such cases the published patent number or in
the absence of such a number the published patent application number has been specified However published
patent documents in the above cases also served as the document (ie identical) to the published patent if available
Quoted from patent abstract as machine translation is poor
25 Summary
The research on tensioner designs from the patent literature demonstrates a lack of quantifiable
data for the performance of a twin tensioner particularly suited to a B-ISG system The review of
the literature for the modeling theory of serpentine belt drives and design of tensioners shows
few belt drive models that are specific to a B-ISG setting Hence the literature review supports
the thesis objective of modeling a B-ISG tensioner specifically one that has a passive twin
tensioner configuration and as well measuring the tensioner‟s performance The survey of
hybrid classes reveals that the micro-hybrid class is the only class employing a closely
conventional belt transmission and hence its B-ISG transmission is applicable for tensioner
investigation The patent designs for tensioners contribute to the development of the tensioner
design to be studied in the following chapter
25
CHAPTER 3 MODELING OF B-ISG SYSTEM
31 Overview
The derivation of a theoretical model for a B-ISG system uses real life data to explore the
conceptual system under realistic conditions The literature and prior art of tensioner designs
leads the researcher to make the following modeling contributions a proposed design for a
passive two-pulley tensioner computation of geometric attributes for a B-ISG system with the
proposed tensioner and derivation of the system‟s equations of motion (EOM) under dynamic
and static states as well as deriving the EOM for the B-ISG system with only a passive single-
pulley tensioner for comparison The principles of dynamic equilibrium are applied to the
conceptual system to derive the EOM
32 B-ISG Tensioner Design
The proposed design for a passive two pulley tensioner configures two tensioners about a single
fixed pivot point in the interior space of a serpentine belt drive One end of each tensioner arm
coincides with the centre point of a tensioner pulley and this point marks the axis of rotation of
the pulley The other end of each arm is pivoted about a point so that the arms share the same
axis of rotation This conceptual design henceforth is called a Twin Tensioner Figure 31 shows
a schematic for the proposed design
Modeling of B-ISG 26
Figure 31 Schematic of the Twin Tensioner
The tensioner pulley coordinates are described by (XiYi) their radii by Ri their arm lengths Lti
and their angles θti The rotation of the arms is resisted by stiffness kt of a coil spring acting
between the two arms and spring stiffness kti acting between each arm and the pivot point The
motion of each arm is dampened by dampers and akin to the springs a damper acts between the
two arms ct and a damper cti acts between each arm and the pivot point The result is a
tensioning mechanism with four degrees of freedom (DOF) that includes independent rotations
of the two pulleys and two arms
The following section relates the geometry of the rigid bodies in a B-ISG system equipped with a
Twin Tensioner to their respective motions
Modeling of B-ISG 27
33 Geometric Model of a B-ISG System with a Twin Tensioner
The B-ISG system with the Twin Tensioner is shown in Figure 32 The geometry of the drive
provides the lengths of the belt spans and angles of wrap for the belt and pulley contact surfaces
These variables are crucial to resolve the components of forces and moment arms acting on each
rigid body in the system and are used in the derivation of the EOM in section 34 Zhen Mu‟s
geometric modeling approach [51] used in the development of the software FEAD was applied
to the Twin Tensioner system to compute the system‟s unique geometric attributes
Figure 32 B-ISG Serpentine Belt Drive with Twin Tensioner
It is noted that in Figure 31 and Figure 32 showing the schematic of the Twin Tensioner and
the overall system respectively that for the purpose of the geometric computations the forward
direction follows the convention of the numbering order counterclockwise The numbering
order is in reverse to the actual direction of the belt motion which is in the clockwise direction in
this study The fourth pulley is identified as an ISG unit pulley However the properties used
for the ISG pulley‟s geometry inertia stiffness and damping is modeled as a conventional
Modeling of B-ISG 28
alternator pulley This pulley is conceptualized as an ISG when it is modeled as the driving
pulley at which point the requirements of the ISG are solved for and its non-inertia attributes
are not needed to be ascribed
Figure 33 shows the geometric attributes needed to resolve the wrap angle of the belt on each
pulley Variables (XiYi) and XYfi XYbi XYfbi and XYbfi are the ith pulley centre coordinates and
its possible belt connection points respectively Length Lfi is the length of the span connecting
the points XYfi and XYf(i+1) or XYbi and XYb(i+1) on the ith and (i+1)th pulleys respectively
Similarly Lbi is the length of the span between XYfbi and XYfb(i+1) or XYbfi and XYbf (i+1) on the
ith and (i+1)th pulleys respectively Angles αi θfi and θbi represent the angle between a line
connecting the ith and (i+1)th pulley centres and the angles of the belt connection spans with
lengths Lfi and Lbi respectively Ri is the radius of the ith pulley
Figure 33 Angles Coordinates and Possible Belt Contact Points for the ith and i+1th Pulleys
[modified] [51]
Modeling of B-ISG 29
The angle between the horizontal and the line connecting the ith and (i+1)th pulley centres αi is
calculated using Zhen‟s method [51] This method uses the pulley‟s coordinates and a cosine
trigonometric relation
i acos
Xi 1
Xi
Xi 1
Xi
2
Yi 1
Yi
2
Yi 1
Yi
if
(31a)
i 2 acos
Xi 1
Xi
Xi 1
Xi
2
Yi 1
Yi
2
Yi 1
Yi
if
(31b)
The lengths for connecting the possible belt spans are described by the variables Lfi and Lbi
The centre point coordinates and the radii of the pulleys are related through the solution of
triangles which they form to define values of the possible belt span lengths
Lfi
Xi 1
Xi
2
Yi 1
Yi
2
Ri 1
Ri
2
(32a)
Lbi
Xi 1
Xi
2
Yi 1
Yi
2
Ri 1
Ri
2
(32b)
The set of possible belt span lengths leads to the calculation of θfi and θbi the angles between the
line connecting the ith and (i+1)th pulley centres and the possible contact point on the pulley
perimeter
Modeling of B-ISG 30
(33a)
(33b)
The array of possible belt connection points comes about from the use of the pulley centre
coordinates and their radii and the sine of the sum or differences of αi and θfi or θbi The angle
αi is calculated in equations (31a) and (31b) and angles θfi and θbi are calculated in equations
(33a) and (33b) The formula to compute the array of points is shown in equations (34) and
(35) for the ith and (i+1)th pulleys Equation (34) describes the forward belt connection point
on the ith pulley which is in the span leading forward to the next (i+1)th pulley
(34a)
(34b)
(34c)
(34d)
bi atan
Lbi
Ri
Ri 1
Modeling of B-ISG 31
Equation (35) describes the backward belt connection point on the ith pulley This point sits on
the ith pulley in the contacting belt span which leads backward to connect with the (i-1)th
pulley
(35a)
(35b)
(35c)
(35d)
The selection of the coordinates from the array of possible connection points requires a graphic
user interface allowing for the points to be chosen based on observation This was achieved
using the MathCAD software package as demonstrated in the MathCAD scripts found in
Appendix C The belt connection points can be chosen so as to have a pulley on the interior or
exterior space of the serpentine belt drive The method used in the thesis research was to plot the
array of points in the MathCAD environment with distinct symbols used for each pair of points
and to select the belt connection points accordingly By observation of the selected point types
the type of belt span connection is also chosen Selected point and belt span types are shown in
Table 31
Modeling of B-ISG 32
Table 31 Selected Contact Point Types on the ith Pulley and Types for the ith Belt Span
Pulley Forward Contact
Point
Backwards Contact
Point
Belt Span
Connection
1 Crankshaft XYf1 XYbf21 Lf1
2 Air Conditioning XYfb2 XYf22 Lb2
3 Tensioner 1 XYbf3 XYfb23 Lb3
4 AlternatorISG XYfb4 XYbf24 Lb4
5 Tensioner 2 XYbf5 XYfb25 Lb5
The inscribed angles βji between the datum and the forward connection point on the ith pulley
and βji between the datum and its backward connection point are found through solving the
angle of the arc along the pulley circumference between the datum and specified point The
wrap angle ϕi is found as the difference between the two inscribed angles for each connection
point on the pulley The angle between each belt span and the horizontal as well as the initial
angle of the tensioner arms are found using arctangent relations Furthermore the total length of
the belt is determined by the sum of the lengths of the belt spans
34 Equations of Motion for a B-ISG System with a Twin Tensioner
341 Dynamic Model of the B-ISG System
3411 Derivation of Equations of Motion
This section derives the inertia damping stiffness and torque matrices for the entire system
Moment equilibrium equations are applied to each rigid body in the system and net force
equations are applied to each belt span From these two sets of equations the inertia damping
Modeling of B-ISG 33
and stiffness terms are grouped as factors against acceleration velocity and displacement
coordinates respectively and the torque matrix is resolved concurrently
A system whose motion can be described by n independent coordinates is called an n-DOF
system Consider the free body diagram of the Twin Tensioner in Figure 34 in which each
pulley of inertia Ii is supported on an arm of inertia Iti It is assumed that the pulleys are
constrained to rotate about their respective central axes and the arms are free to rotate about their
respective pivot points then at any time the position of each pulley can be described by a
rotational coordinate θi(t) and a coordinate θti(t) can denote the rotation of each arm Thus the
tensioner system comprises of four rigid bodies where each is described by one coordinate and
hence is a four-DOF system It is important to note that each rigid body is treated as a point
mass In addition inertial rotation in the positive direction is consistent with the direction of belt
motion The belt span tensions Ti and coupled radii Ri apply moments to the pulleys
Figure 34 Twin Tensioner Free Body Diagram of a Four-Degree of Freedom System
Modeling of B-ISG 34
For the serpentine belt system considered in the thesis research there are seven rigid bodies each
having a one-DOF of motion The EOM for a seven-DOF system form second-order coupled
differential equations meaning that each equation includes all of the general coordinates and
includes up to the second-order time derivatives of these coordinates The EOM can be
obtained by applying D‟Alembert‟s principle that the sum of the moments taken about any point
including the couples equals to zero Therefore the inertial couple the product of the inertia and
acceleration is equated to the moment sum as shown in equation (35)
I ∙ θ = ΣM (35)
The moment equilibrium equations for the Twin Tensioner in Figure 34 where the positive
direction is in the clockwise direction are shown in equations (36) through to (310) The
numbering convention used for each rigid body corresponds to the labeled serpentine belt drive
system shown in Figure 32 Qi represents the required torque of the ith rigid body ci is the
damping constant of the ith rigid body βji is the angle of orientation for the ith belt span and
120597120579119905119894 120579 119905119894 and 120579 119905119894 are the angular displacement angular velocity and angular acceleration of the ith
tensioner arm The initial angle of the ith tensioner arm is described by θtoi
minusI3 ∙ θ 3 = T3 ∙ R3 minus T2 ∙ R3 minus Q3 + c3 ∙ θ 3 (36)
minusI5 ∙ θ 5 = minusT4 ∙ R5 + T5 ∙ R5 minus Q5 + c5 ∙ θ 5 (37)
Modeling of B-ISG 35
It1 ∙ θ t1 = minusTt1 ∙ Lt1 ∙ sin θto 1 minus βj4 + sin θto 1 minus βj5 minus kt ∙ partθt1 minus partθt2 minus kt1 ∙
partθt1 minus ct ∙ partθ t1 minus partθ t2 minus ct1 ∙ partθ t1 (38)
It2 ∙ θ t2 = minusTt2 ∙ Lt2 ∙ sin θto 2 minus βj2 + sin θto 1 minus βj3 minus kt ∙ partθt2 minus partθt1 minus kt2 ∙ partθt2 minus
ct ∙ partθ t2 minus partθ t1 minus ct2 ∙ partθ t2 (39)
partθt1 = θt1 minus θto 1 (310a)
partθt2 = θt2 minus θto 2 (310b)
The free body diagrams for the remaining rigid bodies crankshaft pulley air conditioner pulley
and ISG pulley are in the general form of Figure 35 The sum of the moments about the axes of
rotation are taken for these structures in equations (311) through to (313)
Figure 35 Free Body Diagram for Non-Tensioner Pulleys
Modeling of B-ISG 36
I1 ∙ θ 1 = T5 ∙ R1 minus T1 ∙ R1 + Q1 minus c1 ∙ θ 1 (311)
I2 ∙ θ 2 = T1 ∙ R2 minus T2 ∙ R2 + Q2 minus c2 ∙ θ 2 (312)
I4 ∙ θ 4 = T3 ∙ R4 minus T4 ∙ R4 + Q4 minus c4 ∙ θ 4 (313)
The relationship between belt tensions and rigid body displacements is in the general form of
equation (314) where 119827119836 and 119827119844 are damping and stiffness matrices due to the belt respectively
with each factorized by a radial arm length This relationship is described for each span in
equations (315) through to (320) The belt damping constant for the ith belt span is cib
119827prime = 119827119836 ∙ 120521 + 119827119844 ∙ 120521 (314)
T1 = To + k1b ∙ R1 ∙ θ1 minus R2 ∙ θ2 + c1
b ∙ [R1 ∙ θ 1 minus R2 ∙ θ 2] (315)
T2 = To + k2b ∙ R2 ∙ θ2 minus R3 ∙ θ3 + Lt1 ∙ [sin θto 1 minus βj2 ] ∙ (θt1 minus θto 1) + c2
b ∙ [R2 ∙ θ 2 minus R3 ∙
θ 3 + Lt1 ∙ [sin θto 1 minus βj2 ] ∙ (θ t1)] (316)
T3 = To + k3b ∙ R3 ∙ θ3 minus R4 ∙ θ4 + Lt1 ∙ [sin θto 1 minus βj3 ] ∙ (θt1 minus θto 1) + c3
b ∙ [R3 ∙ θ 3 minus R4 ∙
θ 4 + Lt1 ∙ [sin θto 1 minus βj3 ] ∙ (θ t2)] (317)
Modeling of B-ISG 37
T4 = To + k4b ∙ R4 ∙ θ4 minus R5 ∙ θ5 + Lt2 ∙ [sin θto 2 minus βj4 ] ∙ (θt2 minus θto 2) + c4
b ∙ [R4 ∙ θ 4 minus R5 ∙
θ 5 + Lt2 ∙ [sin θto 2 minus βj4 ] ∙ (θ t1)] (318)
T5 = To + k5b ∙ R5 ∙ θ5 minus R1 ∙ θ1 + Lt2 ∙ [sin θto 2 minus βj5 ] ∙ (θt2 minus θto 2) + c5
b ∙ [R5 ∙ θ 5 minus R1 ∙
θ 1 + Lt2 ∙ [sin θto 2 minus βj5 ] ∙ (θ t2)] (319)
Tprime = Ti minus To (320)
Since the applied torques on the tensioner pulleys Q3 and Q4 are zero the static equilibrium
equation of the pulleys show that the adjacent spans of each tensioner pulley are equal to each
other Hence equations (321) and (322) are denoted as follows
Tt1 = T2 = T3 (321)
Tt2 = T4 = T5 (322)
Equations (310a) (310b) and (314) through to (322) are substituted into the EOMs described
in equations (36) to (39) and (311) to (313) The newly formed equations can be arranged
and written in matrix form as shown in equations (323) through to (328) The general
coordinate matrix 120521 and its first and second derivatives are shown in the EOM below
119816 ∙ 120521 + 119810 ∙ 120521 + 119818 ∙ 120521 = 119824 (323)
Modeling of B-ISG 38
The inertia matrix I includes the inertia of each rigid body in its diagonal elements The
damping matrix C includes variables 119888119894119887 the damping of the ith belt span 119877119894 its radius 120573119895119894 its
angle 119871119905119894 the ith tensioner arm‟s length 120579119905119900119894 its initial pivot angle and 119888119905 and 119888119905119894 the ith
tensioner arm viscous damping constants Stiffness matrix K contains 119896119894119887 the ith belt span
stiffness and 119896119905 and 119896119905119894 the ith tensioner arm stiffness constants and akin to the damping
matrix the variables 119877119894 119871119905119894 120579119905119900119894 and 120573119895119894 The belt span stiffness is computed in equation
(326b) where 119870119887 represents the belt cord stiffness 119896119887 is the belt factor obtained from
experimental data 120573119895119894 is the angle of orientation for the span between the jth and ith pulleys and
ϕi is the belt wrap angle on the ith pulley
Modeling of B-ISG 39
119816 =
I1 0 0 0 0 0 00 I2 0 0 0 0 00 0 I3 0 0 0 00 0 0 I4 0 0 00 0 0 0 I5 0 00 0 0 0 0 It1 00 0 0 0 0 0 It2
(324)
119810 =
c1
b ∙ R12 + c5
b ∙ R12 + c1 minusc1
b ∙ R1 ∙ R2 0 0 minusc5b ∙ R1 ∙ R5 0 c5
b ∙ R1 ∙ Lt2 ∙ sin θto 2 minus βj5
minusc1b ∙ R1 ∙ R2 c2
b ∙ R22 + c1
b ∙ R22 + c2 minusc2
b ∙ R2 ∙ R3 0 0 c2b ∙ R2 ∙ Lt1 ∙ sin θto 1 minus βj2 0
0 minusc2b ∙ R2 ∙ R3 c3
b ∙ R32 + c2
b ∙ R32 + c3 minusc3
b ∙ R3 ∙ R4 0 C36 0
0 0 minusc3b ∙ R3 ∙ R4 c4
b ∙ R42 + c3
b ∙ R42 + c4 minusc4
b ∙ R4 ∙ R5 minusc3b ∙ R4 ∙ Lt1 ∙ sin θto 1 minus βj3 c4
b ∙ R4 ∙ Lt2 ∙ sin θto 2 minus βj4
minusc5b ∙ R1 ∙ R5 0 0 minusc4
b ∙ R4 ∙ R5 c5b ∙ R5
2 + c4b ∙ R5
2 + c5 0 C57
0 0 0 0 0 ct +ct1 minusct
0 0 0 0 0 minusct ct +ct1
(325a)
C36 = 1198773 ∙ 1198711199051 ∙ [1198883119887 ∙ 119904119894119899 1205791199051199001 minus 1205731198953 minus 1198882
119887 ∙ 119904119894119899 1205791199051199001 minus 1205731198952 ] (325b)
C57 = 1198775 ∙ 1198711199052 ∙ [1198885119887 ∙ 119904119894119899 1205791199051199002 minus 1205731198955 minus 1198884
119887 ∙ 119904119894119899 1205791199051199002 minus 1205731198954 ] (325c)
Modeling of B-ISG 40
119818 =
k1
b ∙ R12 + k5
b ∙ R12 minusk1
b ∙ R1 ∙ R2 0 0 minusk5b ∙ R1 ∙ R5 0 k5
b ∙ R1 ∙ Lt2 ∙ sin θto 2 minus βj5
minusk1b ∙ R1 ∙ R2 k2
b ∙ R22 + k1
b ∙ R22 minusk2
b ∙ R2 ∙ R3 0 0 k2b ∙ R2 ∙ Lt1 ∙ sin θto 2 minus βj2 0
0 minusk2b ∙ R2 ∙ R3 k3
b ∙ R32 + k2
b ∙ R32 minusk3
b ∙ R3 ∙ R4 0 R3 ∙ Lt1 ∙ [k3b ∙ sin θto 1 minus βj3 minus k2
b ∙ sin θto 1 minus βj2 ] 0
0 0 minusk3b ∙ R3 ∙ R4 k4
b ∙ R42 + k3
b ∙ R42 minusk4
b ∙ R4 ∙ R5 minusk3b ∙ R4 ∙ Lt1 ∙ sin θto 1 minus βj3 k4
b ∙ R4 ∙ Lt2 ∙ sin θto 2 minus βj4
minusk5b ∙ R1 ∙ R5 0 0 minusk4
b ∙ R4 ∙ R5 k5b ∙ R5
2 + k4b ∙ R5
2 0 R5 ∙ Lt2 ∙ [k5b ∙ sin θto 2 minus βj5 minus k4
b ∙ sin θto 2 minus βj4 ]
0 0 0 0 0 kt +kt1 minuskt
0 0 0 0 0 minuskt kt +kt1
(326a)
k119894b =
Kb
Li + kb ∙ Ri ∙ϕi+1
2 + Ri ∙ϕi
2
(326b)
120521 =
θ1
θ2
θ3
θ4
θ5
partθt1
partθt2
(327)
119824 =
Q1
Q2
Q3
Q4
Q5
Qt1
Qt2
(328)
Modeling of B-ISG 41
3412 Modeling of Phase Change
The phase change from the crankshaft pulley being the driving pulley to the ISG pulley being the
driving pulley is described through a conditional equality based on a set of Boolean conditions
When the crankshaft is driving the rows and the columns of the EOM are swapped such that the
new order for rows and columns is 1 (crankshaft pulley) 4 (ISG pulley) 2 (air conditioner
pulley) 3 (tensioner 1 pulley) 5 (tensioner 2 pulley) 6 (tensioner arm 1) and 7 (tensioner arm 2)
When the ISG is driving the order is the same except that the second row and second column
terms relating to the ISG pulley become the first row and first column while the crankshaft
pulley terms (previously in the first row and first column) become the second row and second
column Hence the order for all rows and columns of the matrices making up the EOM in
equation (322) switches between 1423567 (when the crankshaft pulley is driving) and
4123567 (when the ISG pulley is driving) For example in the crankshaft driving and ISG
driving phases the general coordinate matrix and the inertia matrix become the following
120521119940 =
1205791
1205794
1205792
1205793
1205795
1205971205791199051
1205971205791199052
and 120521119938 =
1205794
1205791
1205792
1205793
1205795
1205971205791199051
1205971205791199052
(329a amp b)
119816119940 =
I1 0 0 0 0 0 00 I4 0 0 0 0 00 0 I2 0 0 0 00 0 0 I3 0 0 00 0 0 0 I5 0 00 0 0 0 0 It1 00 0 0 0 0 0 It2
and 119816119938 =
I4 0 0 0 0 0 00 I1 0 0 0 0 00 0 I2 0 0 0 00 0 0 I3 0 0 00 0 0 0 I5 0 00 0 0 0 0 It1 00 0 0 0 0 0 It2
(329c amp d)
Modeling of B-ISG 42
where subscripts c and a denote the crankshaft pulley driving phase and the ISG pulley driving
phase respectively
The condition for phase change is based on the engine speed n in units of rpm Equation (330)
demonstrates the phase change
H(n) = 1 119899 ge 750 (Crankshaft driving phase)0 119899 lt 750 (ISG driving phase)
(330)
When the crankshaft pulley is the driving pulley the ISG pulley becomes the driven pulley and
following suit when the ISG pulley is the driving pulley the crankshaft pulley becomes the
driven pulley These modes of operation mean that the system will predict two different sets of
natural frequencies and mode shapes Using a Boolean condition to allow for a swap between
the first and second rows as well as between the first and second columns of the EOM matrices
I C and K allows for a continuous plot of the dynamic response to be plotted for the ISG pulley
throughout its driving and driven phases as well as for that of the crankshaft pulley
3413 Natural Frequencies Mode Shapes and Dynamic Responses
Assuming the system undergoes simple harmonic motion its matrix of natural frequencies 120596119899
and modeshapes are found by solving the eigenvalue problem shown in equation (331a)
ωn ∙ 119816120784120784 minus 11981822 ∙ 120495m = 120782 (331a)
The displacement amplitude Θm is denoted implicitly in equation (331d)
Modeling of B-ISG 43
120521119846 = θ2 θ3 θ5 θ6 partθt1 partθt2 T for H n = 1 (331b)
120521119846 = θ1 θ3 θ5 θ6 partθt1 partθt2 T for H n = 0 (331c)
θm = 120495119846 ∙ sin(ω ∙ t) (331d)
I2 and K22 are submatrices of I and K respectively meaning the first row and column of each of
the original matrices are removed The eigenvalue problem is reached by considering the
undamped and unforced motion of the system Furthermore the dynamic responses are found by
knowing that the torque requirements in the matrixndash Qm for the driven pulleys and the tensioner
arms are zero in the dynamic case which signifies a response of the system to an input solely
from the driving pulley
I1 120782120782 119816120784120784
θ 1120521 119846
+ C11 119810120783120784119810120784120783 119810120784120784
θ 1120521 119846
+ K11 119818120783120784
119818120784120783 119818120784120784 θ1
120521119846 =
QCS ISG
119824119846 (332)
1
In the case of equation (331) θm is the submatrix identified in equations (331b) through to
(331d) Therein θ1 denotes the general coordinate for the driving pulley so that in the case the
phase change function H(n) is equal to zero θ1 becomes θ4 and the order of the rows and
columns for the remaining matrices correspond to the value of H(n) as mentioned earlier in
section 3412 For simple harmonic motion the motion of the driven pulleys are described as
1 The driving torque 119876119862119878119868119878119866 denotes the crankshaft torque 119876119862119878 when the crankshaft pulley is driving or the ISG
torque 119876119868119878119866 when the ISG pulley is in its driving function
Modeling of B-ISG 44
θm = 120495119846 ∙ sin(ω ∙ t) (333)
The dynamic response of the system to an input from the driving pulley under the assumption of
sinusoidal motion is expressed in equation (334)
120495119846 = [(119818120784120784 minusω2 ∙ 119816120784120784) + 119895ω ∙ 119810120784120784]minus1 ∙ (119818120784120783 + 119895ω ∙ 119810120784120783) ∙ Θ1 (334)
3414 Crankshaft Pulley Driving Torque Acceleration and Displacement
Subsequently the crankshaft pulley driving torque acceleration and displacement are firstly
discussed It is assumed in the thesis research for the purpose of modeling that the engine
serving the crankshaft is of the four cylinder type The input torque provided by a four-cylinder
engine is assumed to be dominated by two torque pulses per revolution of the crankshaft which
is represented by the factor of 2 on the steady component of the angular velocity in equation
(335) The torque requirement of the crankshaft pulley when it is the driving pulley is
Qc = qc ∙ sin(2 ∙ ωcs ∙ t) (335)
The amplitude of the required crankshaft torque qc is expressed in equation (336) and is
derived from equation (332)
qc = K11 minus ω2 ∙ I1 + 119895 ∙ ω ∙ C11 ∙ Θ1 + (119818120783120784 + 119895 ∙ ω ∙ 119810120783120784) ∙ 120495119846 (336)
Modeling of B-ISG 45
The angular frequency for the system in radians per second (rads) ω when the crankshaft
pulley is driving can be found as a function of the engine speed in rotations per minute (rpm) n
and by taking into account the double pulse per crankshaft revolution
ω = 2 ∙ ωcs = 4 ∙ π ∙ n
60
(337)
The system is considered when the amplitude of the crankshaft‟s angular acceleration is assumed
to be constant and equal to 650 rads2 during the crankshaft pulley driving phase The amplitude
of the excitation angular input from the engine is shown in equation (339b) and is found as a
result of (338)
θ 1CS = 650 ∙ sin(ω ∙ t) (338)
θ1CS = minus650
ω2sin(ω ∙ t)
(339a) where
Θ1CS = minus650
ω2
(339b)
Modeling of B-ISG 46
3415 ISG Pulley Driving Torque Acceleration and Displacement
Secondly the torque acceleration and the displacement of the ISG pulley in its driving phase is
discussed The torque for the ISG when it is in its driving function is assumed constant Ratings
for the ISG are taken from experiments performed by researchers Wezenbeek et al [21] on an
Energen 5 High Output Belt-alternator-starter (BAS) unit from Delphi The 12-Volt BAS which
can also be called a B-ISG was reported to have a maximum allowable speed of 18000 rpm [21]
As well it was noted that the ISG pulley was sized appropriately and the engine speed was
limited to ensure that an over-speed condition of the ISG pulley would not occur [21] The stall
torque rating for the Energen ISG was reported to be 48 Nm at the electric machine shaft [21]
The formula for the torque of a permanent magnet DC motor for any given speed (equation
(340)) is used to approximate the torque of the ISG in its driving mode[52]
QISG = Ts minus (N ∙ Ts divide NF) (340)2
Knowing the stall torque (the torque at 0 rpm) Ts and the maximum rpm of the motor when it is
not under load NF allows for the torque produced 119876119868119878119866 to be found for a given motor speed N
Experimental data from Litens Automotive Group [53] shows that for engine fire-up upon ISG
re-start the crankshaft must go from 0 rpm to an idle speed of approximately 750 rpm The
pulley installed on the ISG shaft in the case of the thesis research has a diameter of 6820 mm
(DISG) while that of the crankshaft has a diameter of 20065 mm (DCS) which makes the
2 The equation for the required driving torque for the ISG pulley may also be computed from the formula shown in
(336) Figure 315 for the driving torque of the ISG pulley shows that (336) and (340) produce similar results for
the required driving torque See Figure 315 for comparison of these results
Modeling of B-ISG 47
crankshaft to ISG pulley ratio approximately 2941 This ratio is used to determine the ISG
speed in equation (341)
nISG = nCS ∙DCS
DISG
(341)
For a crankshaft speed of 750 rpm the required ISG speed nISG is found from equation (341) to
be approximately 220656 rpm Thus the ISG torque during start-up is found from equation
(340) where N is equated to the value of nISG NF is assumed to be 18000 rpm and the stall
torque is allotted the value of 48 Nm The result is a required torque of approximately 42 Nm
for the ISG The acceleration of the ISG pulley is found by taking into account the torque
developed by the rotor and the polar moment of inertia of the pulley [54]
A1ISG = θ 1ISG = QISG IISG (342)
In torsional motion the function for angular displacement of input excitation is sinusoidal since
the electric motor is assumed to be resonating As a result of constant angular acceleration the
angular displacement of the ISG pulley in its driving mode is found in equation 343
θ1ISG = Θ1ISG ∙ sin(ωISG ∙ t) (343)
Knowing that acceleration is the second derivative of the displacement the amplitude of
displacement is solved subsequently [55]
Modeling of B-ISG 48
θ 1ISG = minusωISG2 ∙ Θ
1ISG ∙ sin(ωISG ∙ t) (344)
θ 1ISG = minusωISG2 ∙ Θ
1ISG
(345a)
Θ1ISG =minusQISG IISG
ωISG2
(345b)
In this case the angular frequency for the system 120596 is equivalent to 120596119868119878119866 that is the angular
frequency of the ISG pulley which can be expressed as a function of its speed in rpm
ω = ωISG =2 ∙ π ∙ nISG
60
(346a)
or in terms of the crankshaft rpm by substituting equation (341) into (346a)
ω =2 ∙ π
60∙ nCS ∙
DCS
DISG
(346b)
3416 Tensioner Arms Dynamic Torques
The dynamic torque for the tensioner arms are shown in equations (347) and (348)
Qt1 = kt + kt1 + 119895 ∙ ω ∙ (ct + ct1) ∙ (Θt1 ∙ Θ1) (347)
Modeling of B-ISG 49
Qt2 = kt + kt2 + 119895 ∙ ω ∙ (ct + ct2) ∙ (Θt2 ∙ Θ1) (348)
3417 Dynamic Belt Span Tensions
Furthermore the dynamic belt span tensions are derived from equation (314) and described in
matrix form in equations (349) and (350)
119827prime = 119895 ∙ ω ∙ 119827119836 + 119827119844 ∙ 120495119847 (349)
where
120495119847 = Θ1
120495119846 (350)
342 Static Model of the B-ISG System
It is fitting to pursue the derivation of the static model from the system using the dynamic EOM
For the system under static conditions equations (314) and (323) simplify to equations (351)
and (352) respectively
119827prime = 119827119844 ∙ 120521 (351)
119824 = 119818 ∙ 120521 (352)
Modeling of B-ISG 50
As noted in other chapters the focus of the B-ISG tensioner investigation especially for the
parametric and optimization studies in the subsequent chapters is to determine its effect on the
static belt span tensions Therein equations (351) and (352) are used to derive the expressions
for static tension in each belt span 119931prime is the tension solely due to deflection of the belt span
Equation (320) demonstrates the relationship between the tension due to belt response and the
initial tension also known as pre-tension The static tension 119931 is found by summing the initial
tension 1198790 with the expression for the dynamic tension shown in equations (315) through to
(319) and by substituting the expressions for the rigid bodies‟ displacements from equation
(352) and the relationship shown in equation (320) into equation (351)
119827 = 119827119844 ∙ (119818minus120783 ∙ 119824) + T0 (353)3
35 Simulations
The methods used to develop the geometric dynamic and static models of the Twin Tensioner B-
ISG system in the previous sections of this chapter were verified using the software FEAD The
input data for a single tensioner B-ISG system was entered into FEAD [51] to simulate the
crankshaft driving phase alone since the ISG phase is inapplicable in the FEAD [51] software
FEAD‟s [51] results agreed with those found in the simulation of the single tensioner system‟s
geometric model and EOMs in MathCAD software Furthermore the geometric simulation
3 For the purposes of the static tension the original order for the rows and columns of the stiffness matrix K and the
torque matrix Q are maintained as depicted in (326) and (328) In performing the inverse of K and its
multiplication with Q the first row and first column (in the case of the K matrix) are removed in the crankshaft
driving case whereas the fourth row and fourth column are removed in the ISG driving case Then the product for
the displacement120637 resulting from (119922minus120783 ∙ 119928) has a zero added to serve as the first element of the column matrix in
the crankshaft driving case or as the fourth element in the ISG driving case This is shown in detail in Appendix
C3 of MathCAD scripts
Modeling of B-ISG 51
results for both of the twin and single tensioner B-ISG systems were found to be in agreement as
well
351 Geometric Analysis
The initial coordinate inputs for the centre points of the five pulleys and the Twin Tensioner
pivot point are described as Cartesian coordinates and shown in Table 32 which also includes
the diameters for the pulleys
Table 32 Coordinate Points for Pulley Centres and Twin Tensioner Pivot [56]
Rigid Body Diameter [mm] Cartesian Coordinate [Xi Yi] [mm]
1Crankshaft Pulley 20065 [00]
2 Air Conditioner Pulley 10349 [224 -6395]
3 Tensioner Pulley 1 7240 [292761 87]
4 ISG Pulley 6820 [24759 16664]
5 Tensioner Pulley 2 7240 [12057 9193]
6 Tensioner Arm Pivot --- [201384 62516]
The geometric results for the B-ISG system are shown in Table 33
Table 33 Geometric Results of B-ISG System with Twin Tensioner
Pulley Forward
Connection Point
Backward
Connection Point
Wrap
Angle
ϕi (deg)
Angle of
Belt Span
βji (deg)
Length of
Belt Span
Li (mm)
1 Crankshaft [-6818-100093] [453889475] 202996 356103 227828
2 Air
Conditioning [275299-5717] [220484 -115575] 101425 277528 14064
3 Tensioner 1 [25887599735] [256873 82257] 28126 69403 58658
4 ISG [218374184225] [27951154644] 169554 58956 129513
5 Tensioner 2 [10419659645] [15158673262] 8585 333107 65949
Total Length of Belt (mm) 1243
Modeling of B-ISG 52
352 Dynamic Analysis
The dynamic results for the system include the natural frequencies mode shapes driven pulley
and tensioner arm responses the required torque for each driving pulley the dynamic torque for
each tensioner arm and the dynamic tension for each belt span These results for the model were
computed in equations (331a) through to (331d) for natural frequencies and mode shapes in
equation (334) for the driven pulley and tensioner arm responses in equation (336) for the
crankshaft pulley driving torque in equation (340) for the ISG pulley driving torque in
equations (347) and (348) for the tensioner arm torques and lastly in equation (349) for the
dynamic tension of each belt span Figures 36 through to 323 respectively display these
results The EOM simulations can also be contrasted with those of a similar system being a B-
ISG serpentine belt drive that is equipped with a single tensioner arm and single tensioner pulley
which interacts only in the span bridging the ISG and crankshaft pulleys The EOM for a B-ISG
with a single tensioner is presented in Appendix B
It is assumed for the sake of the dynamic and static computations that the system
does not have an isolator present on any pulley
has negligible rotational damping of the pulley shafts
has negligible belt span damping and that this damping does not differ amongst
spans (ie c1b = ∙∙∙ = ci
b = 0)
has quasi-static belt stretch where its belt experiences purely elastic deformation
has fixed axes for the pulley centres and tensioner pivot
has only one accessory pulley being modeled as an air conditioner pulley and
Modeling of B-ISG 53
has a rotational belt response that is decoupled from the transverse response of the
belt
The input parameter values of the dynamic (and static) computations as influenced by the above
assumptions for the present system equipped with a Twin Tensioner are shown in Table 34
Table 34 Data for Input Parameters used in Dynamic and Static Computations [56]
Rigid Body Data
Pulley Inertia
[kg∙mm2]
Damping
[N∙m∙srad]
Stiffness
[N∙mrad]
Required
Torque
[Nm]
Crankshaft 10 000 0 0 4
Air Conditioner 2 230 0 0 2
Tensioner 1 300 1x10-4
0 0
ISG 3000 0 0 5
Tensioner 2 300 1x10-4
0 0
Tensioner Arm 1 1500 1000 10314 0
Tensioner Arm 2 1500 1000 16502 0
Tensioner Arm
couple 1000 20626
Belt Data
Initial belt tension [N] To 300
Belt cord stiffness [Nmmmm] Kb 120 00000
Belt phase angle at zero frequency [deg] φ0deg 000
Belt transition frequency [Hz] ft 000
Belt maximum phase angle [deg] φmax 000
Belt factor [magnitude] kb 0500
Belt cord density [kgm3] ρ 1000
Belt cord cross-sectional area [mm2] A 693
Modeling of B-ISG 54
These values are for the driven cases for the ISG and crankshaft pulleys respectively In the
driving case for either pulley the inertia of the rigid body is defined as 1 kg∙mm2 and the driving
torque is determined in equations (335) and (340) for the crankshaft and ISG pulleys
respectively
It is noted that because of the belt data for the phase angle at zero frequency the transition
frequency and the maximum phase angle are all zero and hence the belt damping is assumed to
be constant between frequencies These three values are typically used to generate a phase angle
versus frequency curve for the belt where the phase angle is dependent on the frequency The
curve defined by equation (354) is normally symmetric with the lowest phase angle achieved at
0 Hz and the highest phase angle achieved at the prescribed transition frequency f The belt
damping would then be found by solving for cb in the following equation
tanφ = cb ∙ 2 ∙ π ∙ f (354)
Nevertheless the assumption for constant damping between frequencies is also in harmony with
the remaining assumptions which assume damping of the belt spans to be negligible and
constant between belt spans
3521 Natural Frequency and Mode Shape
The set of natural frequencies and mode shapes for the system are shown in Figures 36 and 37
under the cases of the ISG pulley driving and the crankshaft pulley driving The forcing
frequency for the system differs for each case due to the change in driving pulley Modeic and
Modeia denote the ith rigid body according to the numbering convention used in Figure 32 in
the crankshaft and ISG driving cases respectively
Modeling of B-ISG 55
Natural Frequency ωn [Hz]
Crankshaft Pulley ΔΘ4
Air Conditioner Pulley ΔΘ
Tensioner Pulley 1 ΔΘ
Tensioner Pulley 2 ΔΘ
Tensioner Arm 1 ΔΘ
Tensioner Arm 2 ΔΘ
Figure 36a ISG Driving Case Natural Frequency of System and Mode Shapes for Responsive
Rigid Bodies
Figure 36b ISG Driving Case First Mode Responses
4 ΔΘ signifies the term amplitude of angular displacement for the indicated rigid body
Modeling of B-ISG 56
Figure 36c ISG Driving Case Second Mode Responses
Natural Frequency ωn [Hz]
ISG Pulley ΔΘ5
Air Conditioner Pulley ΔΘ
Tensioner Pulley 1 ΔΘ
Tensioner Pulley 2 ΔΘ
Tensioner Arm 1 ΔΘ
Tensioner Arm 2 ΔΘ
Figure 37a Crankshaft Driving Case Natural Frequency of System and Mode Shapes for
Responsive Rigid Bodies
5 ΔΘ signifies the term amplitude of angular displacement for the indicated rigid body
Modeling of B-ISG 57
Figure 37b Crankshaft Driving Case First Mode Responses
Figure 37c Crankshaft Driving Case Second Mode Responses
Modeling of B-ISG 58
3522 Dynamic Response
The dynamic response specifically the magnitude of angular displacement for each rigid body is
plotted in Figures 38 through to 314 as a function of the crankshaft pulley speed n This is
fitting to the analysis since the crankshaft pulley‟s rpm decides the mode of operation for the
system in particular it determines whether the crankshaft pulley or ISG pulley is the driving
pulley
Figure 38 Crankshaft Pulley Dynamic Response (for crankshaft driven case)
Figure 39 ISG Pulley Dynamic Response (for ISG driven case)
Modeling of B-ISG 59
Figure 310 Air Conditioner Pulley Dynamic Response
Figure 311 Tensioner Pulley 1 Dynamic Response
Crankshaft Driving Phase ISG
Driving
Phase
Crankshaft Driving Phase ISG
Driving
Phase
Modeling of B-ISG 60
Figure 312 Tensioner Pulley 2 Dynamic Response
Figure 313 Tensioner Arm 1 Dynamic Response
Crankshaft Driving Phase ISG
Driving
Phase
Crankshaft Driving Phase ISG
Driving Phase
Modeling of B-ISG 61
Figure 314 Tensioner Arm 2 Dynamic Response
3523 ISG Pulley and Crankshaft Pulley Torque Requirement
Figures 315 and 316 respectively showcase the required torques for the ISG pulley in its driving
mode and the crankshaft pulley in its driving mode
Figure 315 Required Driving Torque for the ISG Pulley
Crankshaft Driving Phase ISG
Driving Phase
Modeling of B-ISG 62
Figure 315 shows two plots for the required driving torque of the ISG pulley The dashed line
labeled as Q(n) simulates the application of equation (340) which models the ISG torque as a
permanent magnet DC motor The additional solid line labeled as qamod uses the formula in
equation (336) which determines the load torque of the driving pulley based on the pulley
responses Figure 315 provides a comparison of the results
Figure 316 Required Driving Torque for the Crankshaft Pulley
3524 Tensioner Arms Torque Requirements
The torque for the tensioner arms are shown in Figures 317 and 318
Modeling of B-ISG 63
Figure 317 Dynamic Torque for Tensioner Arm 1
Figure 318 Dynamic Torque for Tensioner Arm 2
3525 Dynamic Belt Span Tension
The dynamic tensions for the belt spans are shown in Figures 319 through to 323 The values
plotted represent the magnitude of the dynamic tension
Crankshaft Driving Phase ISG
Driving Phase
Crankshaft Driving Phase ISG
Driving
Phase
Modeling of B-ISG 64
Figure 319 Span 1 (between crankshaft and air conditioner) Dynamic Belt Span Tension
Figure 320 Span 2 (between air conditioner and tensioner 1) Dynamic Belt Span Tension
Crankshaft Driving Phase ISG
Driving Phase
Crankshaft Driving Phase ISG
Driving
Phase
Modeling of B-ISG 65
Figure 321 Span 3 (between tensioner 1 and ISG) Dynamic Belt Span Tension
Figure 322 Span 4 (between ISG and tensioner 2) Dynamic Belt Span Tension
Crankshaft Driving Phase ISG
Driving
Phase
Crankshaft Driving Phase ISG
Driving Phase
Modeling of B-ISG 66
Figure 323 Span 5 (between tensioner 2 and crankshaft) Dynamic Belt Span Tension
The dynamic results for the system serve to show the conditions of the system for a set of input
parameters The following chapter targets the focus of the thesis research by analyzing the affect
of changing the input parameters on the static conditions of the system It is the static results that
are the focus of the thesis and is thus analyzed in Chapters 4 and 5 in the parametric and
optimization studies respectively The dynamic analysis has been used to complete the picture of
the system‟s state under set values for input parameters
353 Static Analysis
Before looking at the static results for the system under study in brevity the static results for a
B-ISG serpentine belt drive with a single tensioner are presented In this theoretical system the
tensioner arm and tensioner pulley that interacts with the span between the air conditioner and
ISG pulleys of the original system are removed as shown in Figure 324
Crankshaft Driving Phase ISG
Driving
Phase
Modeling of B-ISG 67
Figure 324 B-ISG Serpentine Belt Drive with Single Tensioner
The complete static model as well as the dynamic model for the system in Figure 324 is found
in Appendix B The results of the static tension for each belt span of the single tensioner system
when the crankshaft is driving and the ISG is driving are shown in Table 35
Table 35 Static Solution for Belt Span Tensions in Crankshaft and ISG Driving Cases for a B-
ISG Serpentine Belt Drive with a Single Tensioner
Belt Span between Pulleys Crankshaft Driving Case [N] ISG Driving Case [N]
Crankshaft ndash Air Conditioner 481239 -361076
Air Conditioner ndash ISG 442588 -399727
ISG ndash Tensioner 29596 316721
Tensioner ndash Crankshaft 29596 316721
The tensions in Table 35 are computed with an initial tension of 300N This value for pre-
tension allows the spans in the case that the crankshaft pulley is driving to be suitably tensioned
Modeling of B-ISG 68
Whereas in the case of the ISG pulley driving the first and second spans are excessively slack
Therein an additional pretension of approximately 400N would be required which would raise
the highest tension span to over 700N This leads to the motivation of the thesis researchndash to
reduce the static belt tensions when the ISG is driving As mentioned in Chapter 1 these
tensions should be minimized to prolong belt life preferably within the range of 600 to 800N
As well it is desirable to minimize the amount of pretension exerted on the belt The current
design uses a pre-tension of 300N The above results would lead to a required pre-tension of
more than 700N to keep all spans of the belt suitably in tension (well above 0N) in order to allow
the belt to exhibit high performance in power transmission and come near to the safe threshold
This is the rationale for investigating a Twin Tensioner configuration shown in Figure 32 for
the B-ISG serpentine belt drive under study For the theoretical system with a Twin Tensioner
the following static results in Table 36 are achieved
Table 36 Static Solution for Belt Span Tensions in Crankshaft and ISG Driving Cases for a B-
ISG Serpentine Belt Drive with a Twin Tensioner
Belt Span between Pulleys Crankshaft Driving Case [N] ISG Driving Case [N]
Crankshaft ndash Air Conditioner 465848 -284152
Air Conditioner ndash Tensioner 1 427197 -322803
Tensioner 1 ndash ISG 427197 -322803
ISG ndash Tensioner 2 28057 393645
Tensioner 2 ndash Crankshaft 28057 393645
The results in Table 36 show that the span following the ISG in the case between the Tensioner
1 and ISG pulleys is less slack than in the former single tensioner set-up However there
remains an excessive amount of pre-tension required to keep all spans suitably tensioned
Modeling of B-ISG 69
36 Summary
The simulation of the model for the B-ISG system with the Twin Tensioner shows that the mode
shapes of the rigid bodies within the system (Figures 36a to 37c) are greater in magnitude when
the ISG pulley is driving than when the crankshaft pulley is driving The dynamic responses of
the system as shown in Figures 38 and 310 to 314 is small for the crankshaft pulley and are
negligible for the remaining driven bodies when the ISG is driving For the crankshaft driving
phase there is greater dynamic response for the driven rigid bodies of the system including for
that of the ISG pulley
As the engine speed increases the torque requirement for the ISG was found to vary between
approximately 41Nm and 54Nm (before dropping steeply to approximately 3Nm at an engine
speed of about 720rpm) when modeled after equation (336) or between approximately 48Nm
and 34Nm when modeled after equation (340) In contrast the torque for the crankshaft peaks
at approximately 92Nm and 52Nm at an approximate engine speed of 1450rpm and 5000rpm
respectively The dynamic torque of the first tensioner arm was shown to peak at approximately
15Nm at the transition engine speed 750rpm and again at approximately 15Nm at an
approximate engine speed of about 1450rpm A small peak of about 3Nm was also seen at an
engine speed of 5000rpm Similarly for the second tensioner arm a torque peak of
approximately 20Nm was seen at 750rpm and 1450rpm and a smaller peak of about 8Nm was
seen at an engine speed of 5000rpm
The trend for the dynamic tensions is that the peaks are highest in the ISG driving portion of the
B-ISG operation in most cases and in a few cases they are seen to be close in magnitude to that
Modeling of B-ISG 70
of the highest peaks in the crankshaft driving portion The dynamic tension for the first belt span
peaked at approximately 780Nm 830Nm and 500Nm at engine speeds of 750rpm 1450rpm
5000rpm respectively For the dynamic tension of the second belt span peaks of approximately
1250Nm 675Nm and 760Nm were seen at the same respective engine speeds for the 3 peaks of
the former span At these same engine speeds the third belt span exhibited tension peaks at
approximately 1400Nm 650Nm and 890Nm The tension peaks of the fourth span were
approximately 165Nm 150Nm and 100Nm at engine speeds 750rpm 1450rpm and 5000rpm
The fifth span experienced peaks of approximately 165Nm 170Nm and 120Nm at the same
respective engine speeds of the fourth span
The simulation results for the static tension of the B-ISG system with the Twin Tensioner reveal
that taut spans of the crankshaft driving case are lower in the ISG driving case The largest
change is an approximate decrease of 750N in spans 1 through 3 while spans 4 and 5 increase
by approximately 113N It can be seen that the spans in highest tension (1 2 and 3) in the
crankshaft driving phase become excessively slack in the ISG driving phase There is a smaller
change between the tension values for the spans in the least tension in the crankshaft driving
phase and their corresponding span in the ISG driving phase
The summary of the simulation results are used as a benchmark for the optimized system shown
in Chapter 5 The static tension simulation results are investigated through a parametric study of
the Twin Tensioner system in Chapter 4 The optimization of the system is then based on the
selected design variables from the outcome of Chapter 4
71
CHAPTER 4 PARAMETRIC ANALYSIS OF A B-ISG
TWIN TENSIONER
41 Introduction
The parameters for the proposed Twin Tensioner for a Belt-driven Integrated Starter-generator
(B-ISG) system are investigated through a parametric analysis This analysis seeks to understand
how changing one parameter influences the static belt span tensions for the system Since the
thesis research focuses on the design of a tensioning mechanism to support static tension only
the parameters specific to the actual Twin Tensioner and applicable to the static case were
considered The parameters pertaining to accessory pulley properties such as radii or various
belt properties such as belt span stiffness are not considered In the analyses a single parameter
is varied over a prescribed range while all other parameters are held constant The pivot point
described by Cartesian Coordinates [X6Y6] for the tensioner arms is held constant in all cases
42 Methodology
The parametric study method applies to the general case of a function evaluated over changes in
one of its dependent variables The methodology is illustrated for the B-ISG system‟s function
for static tension which is evaluated for each change in one of its Twin Tensioner‟s parameters
The original data used for the system is based on sample vehicle data provided by Litens [56]
Table 41 provides the initial data for the parameters as well as the incremental change and
maxima and minima limits The increment Δi for the ith parameter is chosen arbitrarily Limits
for each parameter have been chosen to be plus or minus sixty percent of its initial value
Parametric Analysis 72
Table 41 Initial Values Increments and Ranges for Parameters of Twin Tensioner
Parameter Name Initial Value Increment (+- Δi) Minimum
value Maximum value
Coupled Spring
Stiffness kt
20626
N∙mrad 1238 N∙mrad 8250 N∙mrad 33002 N∙mrad
Tensioner Arm 1
Stiffness kt1
10314
N∙mrad 0619 N∙mrad 4126 N∙mrad 16502 N∙mrad
Tensioner Arm 2
Stiffness kt2
16502
N∙mrad 0990 N∙mrad 6601 N∙mrad 26403 N∙mrad
Tensioner Pulley 1
Diameter D3 007240 m 4344 ∙ 10
-3 m 00290 m 0116 m
Tensioner Pulley 2
Diameter D5 007240 m 4344 ∙ 10
-3 m 00290 m 0116 m
Tensioner Pulley 1
Initial Coordinates
[0292761
0087] m See Figure 41 for region of possible tensioner pulley
coordinates Tensioner Pulley 2
Initial Coordinates
[012057
009193] m
The mesh of possible points for the centre coordinates of tensioner pulley 1 and tensioner pulley
2 are designated as Region 1 and Region 2 respectively in Figures 41a and 41b
Figure 41a Regions 1 and 2 and their Associated Guidelines for Coordinates of Tensioner
Pulleys 1 amp 2
CS
AC
ISG
Ten 1
Ten 11
Region II
Region I
Parametric Analysis 73
Figure 41b Regions 1 and 2 in Cartesian Space
The selection for the minimum and maximum tensioner pulley centre coordinates and their
increments are not selected arbitrarily or without derivation as the other tensioner parameters
The coordinates for the pulley centres are identified using Intergraph‟s SmartSketch software a
graphing suite in MathCAD to model the regions of space The following descriptions are used
to describe the possible positions for the tensioner pulleys
Tensioner pulleys are situated such that they are exterior to the interior space created by
the serpentine belt thus they sit bdquooutside‟ the belt loop
The highest point on the tensioner pulley does not exceed the tangent line connecting the
upper hemispheres of the pulleys on either side of it
The tensioner pulleys may not overlap any other pulley
Parametric Analysis 74
Boundaries for regions described as Region 1 in span 2 and 3 and Region 2 in span 4
and 5 is selected based on the above criteria and their lower boundaries are selected
arbitrarily
These criteria were used to define the equation for each boundary line and leads to a set of
Boolean conditions that relate the x-coordinate and y-coordinate for each Cartesian pair The
density for the mesh of points in each region is arbitrarily selected as 101 x-points and 101 y-
points in each space for the purposes of the parametric analysis The outline of this method is
described in the MATLAB scripts contained in Appendix D
The results of the parametric analysis are shown for the slackest and tautest spans in each driving
case As was demonstrated in the literature review the tautest span immediately precedes the
driving pulley and the slackest span immediately follows the driving pulley in the direction of
the belt motion Thus in the case for the crankshaft driving the tautest span is in the first span
and the slackest span is in the fifth span Whereas in the ISG driving case the tautest span is in
the fourth span and the slackest span is in the third span Hence the parametric figures in this
chapter display only the tautest and slackest span values for both driving cases so as to describe
the maximum and minimum values for tension present in the given belt
43 Results amp Discussion
431 Influence of Tensioner Arm Stiffness on Static Tension
The parametric analysis begins with changing the stiffness value for the coil spring coupled
between tensioner arms 1 and 2 This stiffness value kt is changed over a range from sixty
percent less than its initial value kt0 to sixty percent more than its original value as shown in
Parametric Analysis 75
Table 41 The results of the static tension are shown in Figure 42 for the tautest and slackest
spans for both the crankshaft and ISG driving cases
Figure 42 Parametric Analysis for Coupled Stiffness Arm Constant kt (N∙mrad)
As kt increases in the crankshaft driving phase for the B-ISG system the highest tension
decreases from 4691N to 4646N while the lowest tension decreases from 2838N to 2793N
In the ISG driving phase the highest tension increases from 378N to 3998N and the lowest
tension increases from -3384N to -3167N Thus a change of approximately -45N is found in
the crankshaft driving case and approximately +22N is found in the ISG driving case for both the
tautest and slackest spans
Parametric Analysis 76
The second parameter analyzed is the stiffness value for tensioner arm 1 The results of this are
shown in Figure 43
Figure 43 Parametric Analysis for Stiffness of Arm 1 kt1 (N∙mrad)
In Figure 43 as kt1 increases an increase from 4628N to 4681N is observed for the tension of
the tautest span when the crankshaft is driving which is a change of +53N The same value for
net change is found in the slackest span for the same driving condition whose tension increases
from 2775N to 2828N For the case when the B-ISG system is in the ISG driving phase the
change is larger a value of -261N for the tautest span that changes from 4088N to 3827N and
for the slackest span that changes from -3077N to -3338N
Parametric Analysis 77
The change in static tension for the spans as the stiffness of arm 2 varies is demonstrated in
Figure 44
Figure 44 Parametric Analysis for Stiffness of Arm 2 kt2 (N∙mrad)
In this case it is observed that as kt2 increases the tautest span for the B-ISG system in the
crankshaft driving case decreases from 4675N to 4643N as well as the slackest span which
decreases from 2822N to 279N which is an overall change of -32N for both spans Whereas in
the ISG driving case a more noticeable change is once again found a difference of +144N
This is a result of the tautest span increasing from 3863N to 4007N and the slackest span
increasing from -3301N to -3157N
Parametric Analysis 78
432 Influence of Tensioner Pulley Diameter on Static Tension
The change in the diameter of tensioner pulley 1 D3 and its effect on static tension is shown in
Figure 45
Figure 45 Parametric Analysis for Pulley 1 Diameter D3 (m)
The change in the tautest and slackest spans for the B-ISG system‟s crankshaft driving case is
from 3248N to 425N and from 1395N to 240N respectively Peaks are seen at 4799N and
2946N for the respective spans This is a change of approximately +100N and a maximum
change of 1551N for both spans For the ISG driving case the tautest and slackest spans
decrease from 1083N to 6158N and 367N to -1006N Global minimums of 3246N and -391N
for the respective spans are seen This nets a change of approximately -467N and a maximum
change of approximately -759N
Parametric Analysis 79
The effect of changing the diameter of tensioner pulley 2 on the static tension is examined in
Figure 46
Figure 46 Parametric Analysis for Pulley 2 Diameter D5 (m)
The tautest and slackest spans in the crankshaft driving mode of the belt undergo a change from
4583N to 4721N and from 273N to 2869N respectively Therein as D5 increases the trend is
that for both spans there is an increase in tension of approximately 14N Contrastingly the spans
experience a decrease in the ISG driving case as D5 increases The tension of the tautest span
goes from 4296N to 3635N and that of the slackest span goes from -2866N to -3529N This
equals a decrease of approximately 66N for both spans
Parametric Analysis 80
433 Influence of Tensioner Pulley 1 Coordinates on Static Tension
The influence of the coordinates of tensioner pulley 1 on the value of tension in the tautest span
for the B-ISG system‟s crankshaft driving case is demonstrated in Figure 47
Figure 47 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Tautest Span
Tension in Crankshaft Driving Case
The region shown in Figure 47 corresponds to region 1 which is the realm of the positions for
tensioner pulley 1 The possible pulley coordinates in this case are represented by the non-blue
area reaching to the perimeter of the plot It is evident in the darkest red region of the plot
where the y-coordinate is between approximately 0m and 0075m and the x-coordinate is
(N)
Parametric Analysis 81
between approximately 026m and 031m that the highest value of tension is experienced in the
tautest span for the crankshaft driving case The range of tension for Region 1 in the tautest span
when the crankshaft is driving is between a maximum of approximately 500N and a minimum of
approximately 300N This equals an overall difference of 200N in tension for the tautest span by
moving the position of pulley 1 The lowest values for tension are obtained when the pulley
coordinates are approximately -0025m to 015m for the y-coordinate and approximately 031m
to 032m for the x-coordinate which corresponds to the yellow region An area of low tension is
also seen in the area where the y-coordinate is approximately 0m and the x-coordinate is
approximately between 026m and 027m
The changes in tension for the slackest span under the condition of the crankshaft pulley being
the driving pulley are shown in Figure 48
Parametric Analysis 82
Figure 48 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Slackest Span
Tension in Crankshaft Driving Case
Once again the possible coordinate points for tensioner pulley 1 in the B-ISG system are
represented by the non-blue region For the slackest span in the crankshaft driving case it is seen
that the lowest tension is approximately 125N while the highest tension is approximately 325N
This is an overall change of 200N that is achieved in the region The highest values are achieved
in the space where the y-coordinates are approximately 0m to 0075m and the x-coordinate
ranges from 026m to 031m which corresponds to the deep red region The lowest tension
values are achieved in the space where the y-coordinate ranges from approximately -0025m to
015m and the x-coordinate ranges from 031m to 032m which corresponds to the light blue-
green region of the plot The area containing a y-coordinate of approximately 0m and x-
(N)
Parametric Analysis 83
coordinates that are approximately between 026m and 027m also show minimum tension for
the slack span The regions of the x-y coordinates for the maximum and minimum tensions are
alike to the tautest span in Region 1 for the crankshaft driving case as well as was seen in Figure
47
The tension for the tautest span in the case that the ISG is driving in the B-ISG system is found
in Figure 49
Figure 49 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Tautest Span
Tension in ISG Driving Case
(N)
Parametric Analysis 84
Region 1 is represented by the coordinate values shown in the non-dark blue space of the plot in
Figure 49 The tautest span in the case of the ISG driving experiences a range of tension values
in Region 1 from 200N up to 1100N equaling a difference of 900N The minimum tension
values are achieved in the medium to light blue region This includes y-coordinates of
approximately 0m to 0075m and x-coordinates of approximately 026m to 03m The
maximum tension values are in the darkest red area inclusive of y-coordinates -0025m to 015m
and x-coordinates 031m to 032m in addition to y-coordinate of approximately 0m and x-
coordinates of approximately 026m to 027m It can be observed that aforementioned regions
for minimum and maximum tensions in Figure 49 are reverse to those seen in Figures 47 and
48 for the crankshaft driving case
The change in tension for the slackest span of the B-ISG system when it is driven by the ISG is
shown in Figure 410
Parametric Analysis 85
Figure 410 Parametric Analysis for Tensioner Pulley 1 Coordinates [X3Y3] and Slackest Span
Tension in ISG Driving Case
Figure 410 exhibits the realm of possible points for tensioner pulley 1 for the case of the ISG
driving in the non-yellow-green area The minimum tension values are achieved in the darkest
blue area where the minimum tension is approximately -500N This area corresponds to y-
coordinates from approximately 0m to 005m and x-coordinates from approximately 026m to
03m The area of a maximum tension is approximately 400N and corresponds to the darkest red
area inclusive of y-coordinates -0025m to 015m and x-coordinates 031m to 032m as well as
the coordinates for y equaling approximately 0m and for x equaling approximately 026m to
027m The difference between maximum and minimum tensions in this case is approximately
900N It is noticed once again that the space of x- and y-coordinates containing the maximum
(N)
Parametric Analysis 86
tension is in the similar location to that of the described space for minimum tension in the
crankshaft driving case in Figure 47 and 48
434 Influence of Tensioner Pulley 2 Coordinates on Static Tension
The influence of pulley 2 coordinates on the tension value for the tautest span when the
crankshaft is driving the B-ISG system is shown in Figure 411 and is represented by the values
corresponding to the non-blue area
Figure 411 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Tautest Span
Tension in Crankshaft Driving Case
In Figure 411 the possible coordinates are contained within Region 2 The maximum tension
value is approximately 500N and is found in the darkest red space including approximately y-
(N)
Parametric Analysis 87
coordinates 004m to 014m and x-coordinates 0025m to 0175m and also y-coordinates 013m
to 02m corresponding to the x-coordinate at 0175m A minimum tension value of
approximately 350N is found in the yellow space and includes approximately y-coordinates
008m to 018m and x-coordinates 016m to 02m The difference in tension values is 150N
The analysis of the change in coordinates for tension pulley 2 on the value for tension in the
slackest span is shown in Figure 412 in the non-blue region
Figure 412 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Slackest Span
Tension in Crankshaft Driving Case
The value of 325N is the highest tension for the slack span in the crankshaft driving case of the
B-ISG system and is found in the deep-red region where the y-coordinates are between
(N)
Parametric Analysis 88
approximately 004m and 013m and the x-coordinates are approximately between 0025m and
016m as well as where y is between 013m and 02m and x is approximately 0175m The
lowest tension value for the slack span is approximately 150N and is found in the green-blue
space where y-coordinates are between approximately 01m and 022m and the x-coordinates
are between approximately 016m and 021m The overall difference in minimum and maximum
tension values is 175N The spaces for the maximum and minimum tension values are similar in
location to that found in Figure 411 for the tautest span in the crankshaft driving case
Figure 413 provides the theoretical data for the tension values of the tautest span as the position
of the B-ISG system‟s tensioner pulley 2 changes in the ISG driving case Possible points are in
the space of values which correspond to the non-dark-blue region in Figure 413
Parametric Analysis 89
Figure 413 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Tautest Span
Tension in ISG Driving Case
In Figure 413 the region for high tension reaches a value of approximately 950N and the region
for low tension reaches approximately 250N This equals a difference of 700N between
maximum and minimum tension values for the tautest span in the B-ISG system‟s ISG driving
case The coordinate points within the space that maximum tension is reached is in the dark red
region and includes y-coordinates from approximately 008m to 022m and x-coordinates from
approximately 016m to 021m The coordinate points within the space that minimum tension is
reached is in the blue-green region and includes y-coordinates from approximately 004m to
013m and the corresponding x-coordinates from approximately 0025m to 015m An additional
small region of minimum tension is seen in the area where the x-coordinate is approximately
(N)
Parametric Analysis 90
0175m and the y-coordinates are approximately between 013m and 02m The location for the
area of pulley centre points that achieve maximum and minimum tension values is approximately
located in the reverse positions on the plot when compared to that of the case for the crankshaft
driving in Figures 411 and 412 Therein the trend seen for pulley coordinates for the second
tensioner pulley follows suit with that of the first tensioner pulley which is that the area of points
for maximum tension in the crankshaft driving case becomes the approximate area of points for
minimum tension in the ISG driving case and vice versa
In Figure 414 the results of the parametric analysis on the coordinates of the second tensioner
pulley and its effect on the slackest span‟s tension in the ISG driving case is shown Similar to
earlier figures the non-dark yellow region represents Region 2 that contains the possible points
for the pulley‟s Cartesian coordinates
Parametric Analysis 91
Figure 414 Parametric Analysis for Tensioner Pulley 2 Coordinates [X5Y5] and Slackest
Span Tension in ISG Driving Case
Figure 414 demonstrates a difference of approximately 725N between the highest and lowest
tension values for the slackest span of the B-ISG system in the ISG driving case The highest
tension values are approximately 225N The area of points that allow the second tension pulley
to achieve maximum tension in the belt span includes y-coordinates from approximately 01m to
022m and the corresponding x-coordinates from approximately 016m to 021m This
corresponds to the darkest red region in Figure 414 The coordinate values where the lowest
tension being approximately -500N is achieved include y-coordinate values from
approximately 004m to 013m and the corresponding x-coordinates from approximately 0025m
to 015m corresponding to the darkest blue region A dark blue region of lowest tension is also
(N)
Parametric Analysis 92
seen in the area where y is approximately between 013m and 02m and the x-coordinate is
approximately 0175m The regions for maximum and minimum tension values are observed to
be similar to those found in Figure 413 and alike to Figure 413 to be in reverse to those found
in Figure 411 and 412 for the tautest and slackest spans in the crankshaft driving case So as for
the changes in tensioner pulley 2 coordinates the areas for minimum tension in Region 2 of the
ISG driving case are similar to the areas for maximum tension in Region 2 of the crankshaft
driving case and vice versa for the maximum tension of the ISG driving case and the minimum
tension for the crankshaft driving case in Region 2
44 Conclusion
Overall the trend in the plots of Figures 47 48 411 and 412 indicate in the crankshaft driving
portion that the B-ISG system‟s belt span tensions experience the following effect
Minimum tension for the tautest span is achieved when tensioner pulley 1 centre
coordinates are located closer to the right side boundary and bottom left boundary of
Region 1 or when tensioner pulley 2 centre coordinates are within the upper right space
(near to the ISG pulley) and the space closer to the top boundary of Region 2
Maximum tension for the slackest span is achieved when the first tensioner pulley‟s
coordinates are located in the mid space and near to the bottom boundary of Region 1
and when the second tensioner pulley‟s coordinates are located near to the bottom left
boundary of Region 2 which is the boundary nearest to the crankshaft pulley
Parametric Analysis 93
The trend for minimizing the tautest span signifies that the tension for the slackest span is also
minimized at the same time As well maximizing the slackest span signifies that the tension for
the tautest span is also maximized at the same time too
The trend for the B-ISG system‟s ISG driving case as can be seen in Figures 49 410 413 and
414 is approximately in reverse to that of the crankshaft driving case for the system Wherein
points corresponding to minimum tension in Regions 1 and 2 in the ISG case are approximately
the same as points corresponding to maximum tension in the Regions for the crankshaft case and
vice versa for the ISG cases‟ areas of maximum tension
Minimum tension for the tautest span is present when the first tensioner pulley‟s
coordinates are near to mid to lower boundary of Region 1 and when the second
tensioner pulley‟s coordinates are close to the bottom left boundary of Region 2 which
is the furthest boundary from the ISG pulley and closest to the crankshaft pulley
Maximum tension for the slackest span is achieved when the first tensioner pulley is
located close to the right boundary of Region 1 and when the second tensioner pulley is
located near the right boundary and towards the top right boundary of Region 2
It is observed in Figures 47 to 414 and alike to Figures 42 to 46 the tautest and slackest
spans decrease or increase together Thus it can be assumed that the tension values in these
spans and likely the remaining spans outside of the tautest and slackest spans follow suit
Therein when parameters are changed to minimize one belt span‟s tension the remaining spans
will also have their tension values reduced Figures 42 through to 413 showed this clearly
where the overall change in the tension of the tautest and slackest spans changed by
Parametric Analysis 94
approximately the same values for each separate case of the crankshaft driving and the ISG
driving in the B-ISG system
Design variables are selected in the following chapter from the parameters that have been
analyzed in the present chapter The influence of changing parameters on the static tension
values for the various spans is further explored through an optimization study of the static belt
tension for the B-ISG system equipped with a Twin Tensioner in the following chapter Chapter
5
95
CHAPTER 5 OPTIMIZATION OF A B-ISG TENSIONER
The objective of the optimization analysis is to minimize the absolute magnitude of the static
tension in the ISG-operating mode of the serpentine belt drive The optimization seeks to
optimize the performance of the proposed Twin Tensioner design by using its properties as the
design variables for the objective function The optimization task begins with the selection of
these design variables for the objective function and then the selection of an optimization
method The results of the optimization will be compared with the results of the analytical
model for the static system and with the parametric analysis‟ data
51 Optimization Problem
511 Selection of Design Variables
The optimal system corresponds to the properties of the Twin Tensioner that result in minimized
magnitudes of static tension for the various belt spans Therein the design variables for the
optimization procedure are selected from amongst the Twin Tensioner‟s properties In the
parametric analysis of Chapter 4 the tensioner properties presented included
coupled stiffness kt
tensioner arm 1 stiffness kt1
tensioner arm 2 stiffness kt2
tensioner pulley 1 diameter D3
tensioner pulley 2 diameter D5
tensioner pulley 1 initial coordinates [X3Y3] and
Optimization 96
tensioner pulley 2 initial coordinates [X5Y5]
It was observed in the former chapter that perturbations of the stiffness and geometric parameters
caused a change between the lowest and highest values for the static tension especially in the
case of perturbations in the geometric parameters diameter and coordinates Table 51
summarizes the observed changes in the belt span tensions corresponding to the Twin Tensioner
parameters‟ maximum and minimum values
Table 51 Summary of Parametric Analysis Data for Twin Tensioner Properties
Parameter Symbol
Original Tensions in TautSlack Span (Crankshaft
Mode) [N]
Tension at
Min | Max Parameter6 for
Crankshaft Mode [N]
Percent Change from Original for
Min | Max Tensions []
Original Tension in TautSlack Span (ISG Mode)
[N]
Tension at
Min | Max Parameter Value in ISG Mode [N]
Percent Change from Original Tension for
Min | Max Tensions []
kt
465848 (taut) 4691 4646 07 -03 393645 (taut) 378 3998 -40 16
28057 (slack) 2838 2793 12 -05 -322803 (slack) -3384 -3167 -48 19
kt1
465848 (taut) 4628 4681 -07 05 393645 (taut) 4088 3827 38 -28
28057 (slack) 2775 2828 -11 08 -322803 (slack) -3077 -3338 47 -34
kt2
465848 (taut) 4675 4643 04 -03 393645 (taut) 3863 4007 -19 18
28057 (slack) 2822 279 06 -06 -322803 (slack) -3301 -3157 -23 22
D3 465848 (taut) 3248 425 -303 -88 393645 (taut) 1083 6158 1751 564
28057 (slack) 1395 240 -503 -145 -322803 (slack) 367 -1006 2137 688
D5 465848 (taut) 4583 4721 -16 13 393645 (taut) 4296 3635 91 -77
28057 (slack) 273 2869 -27 23 -322803 (slack) -2866 -3529 112 -93
[X3Y3] 465848 (taut) 300 500 -356 73 393645 (taut) 200 1100 -492 1794
28057 (slack) 125 325 -554 158 -322803 (slack) -500 400 -549 2239
6 The values for the tension for each of the taut and slack spans provided correspond to the minimum and maximum
values of the parameter listed in each case such that the columns of identical colour correspond to each other For
the coordinate parameters the minimum and maximum parameter value is inadmissible The tension values in these
cases are simply the minimum and maximum tension values achieved by the coordinate parameter listed
Optimization 97
[X5Y5] 465848 (taut) 350 500 -249 73 393645 (taut) 250 950 -365 1413
28057 (slack) 150 325 -465 158 -322803 (slack) -500 225 -549 1697
The results of the parametric analyses for the Twin Tensioner parameters show that there is a
noticeable percent change between the initial tensions and the tensions corresponding to each of
the minima and maxima parameter values or in the case of the coordinates between the
minimum and maximum tensions for the spans Thus the parametric data does not encourage
exclusion of any of the tensioner parameters from being selected as a design variable As a
theoretical experiment the optimization procedure seeks to find feasible physical solutions
Hence economic criteria are considered in the selection of the design variables from among the
Twin Tensioner‟s parameters Of the tensioner properties it is found that the diameter of the
tensioner pulleys has the largest impact on cost Adding mass to a tensioner pulley as a result of
increasing the diameter and consequently its inertia increases the cost of material Material cost
is most significant in the manufacture process of pulleys as their manufacturing is largely
automated [4] Furthermore varying the structure of a pulley requires retooling which also
increases the cost to manufacture As such the tensioner pulley diameters D3 and D5 are
excluded from being selected as design variables The remaining tensioner properties the
stiffness parameters and the initial coordinates of the pulley centres are selected as the design
variables for the objective function of the optimization process
512 Objective Function amp Constraints
In order to deal with two objective functions for a taut span and a slack span a weighted
approach was employed This emerges from the results of Chapter 3 for the static model and
Chapter 4 for the parametric study for the static system which show that a high tension span and
Optimization 98
a highly slack span exist in the ISG-driving phase of the B-ISG system Therein the first
objective function of equation (51a) is described as equaling fifty percent of the absolute tension
value of the tautest span and fifty percent of the absolute tension value of the slackest span for
the case of the ISG driving only The second objective function uses a non-weighted approach
and is described as the absolute tension of the slackest span when the ISG is driving A non-
weighted approach is motivated by the phenomenon of a fixed difference that is seen between
the slackest and tautest spans of the optimal designs found in the weighted optimization
simulations Equations (51a) through to (51c) display the objective functions
The limits for the design variables are expanded from those used in the parametric analysis for
the non-coordinate parameters kt kt1 and kt2 so that they are permitted to vary from
approximately 0 to approximately 200 of the initial value for each parameter kt0 kt10 and kt20
respectively In the case of the coordinate parameters [X3Y3] and [X5Y5] the x- and y-
coordinates are permitted to vary within the spaces Region 1 and Region 2 respectively which
were prescribed in Chapter 4 Figure 41a and 41b
Aside from the design variables design constraints on the system include the requirement for
static stability of the Twin Tensioner An optimal solution for the B-ISG system must achieve
the goal of the objective function which is to minimize the absolute tensions in the system
However for an optimal solution to be feasible the movement of the tensioner arm must remain
within an appropriate threshold In practice an automotive tensioner arm for the belt
transmission may be considered stable if its movement remains within a 10 degree range of
Optimization 99
motion [4] As such the angle of displacement for tensioner arms 1 and 2 are designated by θ t1
and θt2 respectively in the listed constraints
The optimization task is described in equations 51a to 52 Variables a through to g represent
scalar limits for the x-coordinate for corresponding ranges of the y-coordinate
Minimize 119879119908119890119894119892 119893119905119890119889 = 05 ∙ 119879119905119886119906119905 + 05 ∙ 119879119904119897119886119888119896
or119879119899119900119899 minus119908119890119894119892 119893119905119890119889 = 119879119904119897119886119888119896
(51a)
where
119879119905119886119906119905 = 119891119905119886119906119905 119896119905 1198961199051 1198961199052 1198833 1198843 1198835 1198845 (51b)
119879119904119897119886119888119896 = 119891119904119897119886119888119896 (119896119905 1198961199051 1198961199052 1198833 119884311988351198845) (51c)
Subject to
(1198961199050 minus 1 ∙ 1198961199050) le 119896119905 le (1198961199050 + 11198961199050)(11989611990510 minus 1 ∙ 11989611990510) le 1198961199051 le (11989611990510 + 111989611990510)(11989611990520 minus 1 ∙ 11989611990520) le 1198961199052 le (11989611990520 + 111989611990520)
119886 le 1198833 le 119888
1198931 1198833 le 1198843 le 1198933 1198833 119891119900119903 119886 le 1198833 lt 119887
1198932 1198833 le 1198843 le 1198933 1198833 119891119900119903 119887 le 1198833 le 119888119889 le 1198835 le 119892
1198934 1198835 le 1198845 le 1198937 1198835 for 119889 le 1198835 lt 1198901198935(1198835) le 1198845 le 1198937(1198835) for 119890 le 1198835 lt 119891
1198936 1198835 le 1198845 le 1198937 1198833 for 119891 le 1198833 le 119892 1205791199051 le 10deg 1205791199052 le 10deg
(52)
The functions for the taut and slack spans represent the fourth and third span respectively in the
case of the ISG driving The equations for the tensions of the aforementioned spans are shown
in equation 51a to 51c and are derived from equation 353 The constraints for the
optimization are described in equation 52
Optimization 100
52 Optimization Method
A twofold approach was used in the optimization method A global search alone and then a
hybrid search comprising of a global search and a local search The Genetic Algorithm is used
as the global search method and a Quadratic Sequential Programming algorithm is used for the
local search method
521 Genetic Algorithm
Genetic Algorithm (GA) is a part of the growing genre of evolutionary algorithms [57] The
optimization approach differs from classical search approaches by its ease of use and global
perspective [57] GA mimics biological evolution theory by using a ldquocross-over of heritable
information random mutation and selection on the basis of fitness between generationsrdquo [58] to
form a robust search algorithm that requires minimal problem information [57] The parameter
sets are represented as sample points modeled as bdquochromosomes‟ or data strings that are
measured against how well they allow the model to achieve the optimization task [58] GA is
stochastic which means that its algorithm uses random choices to generate subsequent sampling
points rather than using a set rule to generate the following sample This avoids the pitfall of
gradient-based techniques that may focus on local maxima or minima and end-up neglecting
regions containing higher peaks or lower valleys [57] Furthermore due to the randomness of
the GA‟s search strategy it is able to search a population (a region of possible parameter sets)
faster than other optimization techniques The GA approach is viewed as a universal
optimization approach while many classical methods viewed to be efficient for one optimization
problem may be seen as inefficient for others However because GA is a probabilistic algorithm
its solution for the objective function may only be near to a global optimum As such the current
Optimization 101
state of stochastic or global optimization methods has been to refine results of the GA with a
local search and optimization procedure
522 Hybrid Optimization Algorithm
In order to enhance the result of the objective function found by the GA a Hybrid optimization
function is implemented in MATLAB software The Hybrid optimization function combines a
global search GA with a local search Sequential Quadratic Programming (SQP) The hybrid
process refines the value of the objective function found through GA by using the final set of
points found by the algorithm as the initial point of the SQP algorithm The GA function
determines the region containing a global optimum and then the SQP algorithm uses a gradient
based technique to find a solution closer to the global optimum The MATLAB algorithm a
constrained minimization function known as fmincon uses an SQP method that approximates the
Hessian for the Lagrangian function (ie the second derivatives of the Lagrangian) by way of a
quasi-Newton approach to generate a quadratic program (QP) sub-problem [59] The solution
for the QP provides the search direction of the line search procedure used when each iteration is
performed [59]
53 Results and Discussion
531 Parameter Settings amp Stopping Criteria for Simulations
The parameter settings for the optimization procedure included setting the stall time limit to
200s This is the interval of time the GA is given to find an improvement in the value of the
objective function This is an increase from MATLAB‟s default of 20s Increasing the stall time
limit allows for the optimization search to consistently converge without being limited by time
Optimization 102
The population size used in finding the optimal solution is set at 100 This value was chosen
after varying the population size between 50 and 2000 showed no change in the value of the
objective function The max number of generations is set at 100 The time limit for the
algorithm is set at infinite The limiting factor serving as the stopping condition for the
optimization search was the function tolerance which is set at 1x10-6
This allows the program
to run until the ratio of the change in the objective function over the stall generations is less than
the value for function tolerance The stall generation setting is defined as the number of
generations since the last improvement of the objective function value and is limited to 50
532 Optimization Simulations
The results of the genetic algorithm optimization simulations performed in MATLAB are shown
in the following tables Table 52a and Table 52b
Table 52a GA Optimization Results for Twin Tensioner Parameters and Objective Function
Trial
No
Genetic Algorithm Optimization Method
Objective
Function
Value [N]
kt [N∙mrad]
kt1 [N∙mrad]
kt2 [N∙mrad]
[X3Y3]
[m] [X5Y5]
[m]
1 3582241 314069 204844 165020 [02928 00703] [01618 01036]
2 3582241 103646 205284 198901 [03009 00607] [01283 00809]
3 3582241 126431 204740 43549 [03010 00631] [01311 01147]
4 3582241 180285 206230 254870 [03095 00865] [01080 01675]
5 3582241 74757 204559 189077 [03084 00617] [01265 00718]
Original Design 20626 10314 16502 [02928 0087] [01206 00919]
Optimization 103
Table 52b Computations for Tensions and Angles from GA Optimization Results
Trial No
Genetic Algorithm Optimization Method
Slackest Tension [N] Tautest Tension [N] Tensioner Arm 1
Displacement [deg]
Tensioner Arm 2
Displacement [deg]
T3 T4 Θt1 Θt2
1 -1572307 5592176 -00025 -49748
2 -4054309 3110174 -00002 -20213
3 -3930858 3233624 -00004 -38370
4 -1309751 5854731 -00010 -49525
5 -4092446 3072036 -00000 -17703
Original Design -322803 393645 16410 -4571
For each trial above the GA function required 4 generations each consisting of 20 900 function
evaluations before finding no change in the optimal objective function value according to
stopping conditions
The results of the Hybrid function optimization are provided in Tables 53a and 53b below
Table 53a Hybrid Optimization Results for Twin Tensioner Parameters and Objective Function
Trial
No
Hybrid Optimization Method
Objective
Function
Value [N]
of
Function
Evals ( of
Iterations)
kt [N∙mrad]
kt1 [N∙mrad]
kt2 [N∙mrad]
[X3Y3]
[m] [X5Y5]
[m]
1 3582241 16 (1) 16065 205846 229494 [02780 00581] [01679 01288]
2 3582241 20 (1) 249227 205635 25218 [02901 00634] [01559 00870]
3 3582241 16 (1) 297295 204878 320479 [02962 00702] [01336 01447]
4 3582241 53 (1) 241433 204262 229683 [02912 00647] [00047 01465]
Optimization 104
5 3582241 21 (1) 379096 205548 188888 [02973 00703] [01206 01376]
Original Design 20626 10314 16502 [02928 0087] [01206 00919]
Table 53b Computations for Tensions and Angles from Hybrid Optimization Results
Trial No
Hybrid Algorithm Optimization Method
Slackest Tension [N] Tautest Tension [N] Tensioner Arm 1
Displacement [deg]
Tensioner Arm 2
Displacement [deg]
T3 T4 Θt1 Θt2
1 -2584641 4579841 -02430 67549
2 -3708747 3455736 -00023 -41068
3 -1707181 5457302 -00099 -43944
4 -269178 6895304 00006 -25366
5 -2982335 4182148 -00003 -41134
Original Design -322803 393645 16410 -4571
In Table 53a it can be seen that iterations of 16 20 21 or 53 were required for the local search
algorithm following the GA to find an optimal solution Once again the GA function
computed 4 generations which consisted of approximately 20 900 function evaluations before
securing an optimum solution
The simulation results of the non-weighted hybrid optimization approach are shown in tables
54a and 54b below
Optimization 105
Table 54a Non-Weighted Optimization Results for Twin Tensioner Parameters and Objective
Function
Trial
No
Objective
Function
Value [N]
of
Function
Evals ( of
Iterations)
kt [N∙mrad]
kt1 [N∙mrad]
kt2 [N∙mrad]
[X3Y3]
[m] [X5Y5]
[m]
Genetic Algorithm Optimization Method
1 33509e
-004 20900 (4) 321799 75530 212653 [02860 00602] [01082 01858]
Hybrid Optimization Method
1 73214e
-011 381 (13) 234881 14730 323358 [02952 00688] [00048 01466]
Original Design 20626 10314 16502 [02928 0087] [01206 00919]
Table 54b Computations for Tensions and Angles from Non-Weighted Optimizations
Trial No Slackest Tension [N] Tautest Tension [N]
Tensioner Arm 1
Displacement [deg]
Tensioner Arm 2
Displacement [deg]
T3 T4 Θt1 Θt2
Genetic Algorithm Optimization Method
1 -00003 7164479 -00588 -06213
Hybrid Optimization Method
1 -00000 7164482 15543 -16254
Original Design -322803 393645 16410 -4571
The weighted optimization data of Table 54a shows that the GA simulation again used 4
generations containing 20 900 function evaluations to conduct a global search for the optimal
system While the weighted Hybrid optimization used 13 iterations (consisting of 381 function
evaluations) after its GA run which used the same number of generations and function
evaluations as the GA run in the non-weighted simulations Tables 54a and 54b show the data
Optimization 106
for only one trial for each of the non-weighted GA and hybrid methods since only a single
optimal point exists in this case
533 Discussion
The optimal design from each search method can be selected based on the least amount of
additional pre-tension (corresponding to the largest magnitude of negative tension) that would
need to be added to the system This is in harmony with the goal of the optimization of the B-
ISG system as stated earlier to minimize the static tension for the tautest span and at the same
time minimize the absolute static tension of the slackest span for the ISG driving case As well
the angular displacements corresponding to each trial‟s results show that the Twin Tensioner is
under static stability Therein the optimal solution may be selected as the design parameters
corresponding to Trial 4 of the GA simulations to Trial 4 of the Hybrid simulations or to either
of the trials for the non-weighted optimization simulations
Given the ability of the Hybrid optimization to refine the results obtained in the GA
optimization the results of Trial 4 of the Hybrid simulations are selected as the most optimal
design from the weighted objective function approaches It is interesting to note that the Hybrid
case for the least slackness in belt span tension corresponds to a significantly larger number of
function evaluations than that of the remaining Hybrid cases This anomaly however does not
invalidate the other Hybrid cases since each still satisfy the design constraints Using the data
for the optimized system in Trial 4 (of the Hybrid optimization) the static tensions for the belt
spans in both of the B-ISG‟s phases of operation are as follows
Optimization 107
Table 55 Weighted Optimization Results for Static Tensions for Optimal B-ISG System with a
Twin Tensioner
Belt Span between Pulleys Crankshaft Driving Case [N] ISG Driving Case [N]
Optimized Original Optimized Original
Crankshaft ndash Air Conditioner 3926599 465848 117333 -284152
Air Conditioner ndash Tensioner 1 3540088 427197 -269178 -322803
Tensioner 1 ndash ISG 3540088 427197 -269178 -322803
ISG ndash Tensioner 2 2073813 28057 6895304 393645
Tensioner 2 ndash Crankshaft 2073813 28057 6895304 393645
Additional Pretension
Required (approximate) + 27000 +322803 + 27000 +322803
In Table 54b it is evident that the non-weighted class of optimization simulations achieves the
least amount of required pre-tension to be added to the system The computed tension results
corresponding to both of the non-weighted GA and Hybrid approaches are approximately
equivalent Therein either of their solution parameters may also be called the most optimal
design The Hybrid solution parameters are selected as the optimal design once again due to the
refinement of the GA output contained in the Hybrid approach and its corresponding belt
tensions are listed in Table 56 below
Optimization 108
Table 56 Non-Weighted Optimization Results for Static Tensions for Optimal B-ISG System
with a Twin Tensioner
Belt Span between Pulleys Crankshaft Driving Case [N] ISG Driving Case [N]
Optimized Original Optimized Original
Crankshaft ndash Air Conditioner 3891862 465848 386511 -284152
Air Conditioner ndash Tensioner 1 3505351 427197 -00000 -322803
Tensioner 1 ndash ISG 3505351 427197 -00000 -322803
ISG ndash Tensioner 2 2039076 28057 7164482 393645
Tensioner 2 ndash Crankshaft 2039076 28057 7164482 393645
Additional Pretension
Required (approximate) + 0000 +322803 + 00000 +322803
The results of the simulation experiments are limited by the following considerations
System equations are coupled so that a fixed difference remains between tautest and
slackest spans
A limited number of simulation trials have been performed
There are multiple optimal design points for the weighted optimization search methods
Remaining tensioner parameters tensioner pulley diameters and their stiffness have not
been included in the design variables for the experiments
The belt factor kb used in the modeling of the system‟s belt has been obtained
experimentally and may be open to further sources of error
Therein the conclusions obtained and interpretations of the simulation data can be limited by the
above noted comments on the optimization experiments
Optimization 109
54 Conclusion
The outcomes the trends in the experimental data and the optimal designs can be concluded
from the optimization simulations The simulation outcomes demonstrate that in all cases the
weighted optimization functions reached an identical value for the objective function whereas
the values reached for the parameters varied widely
The lowest tension values for the tautest and slackest span were achieved in Trial 5 of the GA
optimization approach In reiteration in the presence of slack spans the tension value of the
slackest span must be added to the initial static tension for the belt Therein for the former case
an amount of at least 409N would need to be added to the 300N of pre-tension already applied to
the system (see Table 34) The highest tension values for the spans were achieved in Trial 4 of
the weighted Hybrid optimization approach and in both trials of the non-weighted optimization
approaches In the former the weighted Hybrid trial the tension value achieved in the slackest
span was approximately -27N signifying that only at least 27N would need to be added to the
present pre-tension value for the system The tension of the slackest span in the non-weighted
approach was approximately 0N signifying that the minimum required additional tension is 0N
for the system
The optimized solutions for the tension values in each span show that there is consistently a fixed
difference of 716448N between the tautest and slackest span tension values as seen in Tables
52b 53b and 54b This difference is identical to the difference between the tautest and slackest
spans of the B-ISG system for the original values of the design parameters while in its ISG
mode As well the optimal stiffness parameters for the weighted Hybrid optimization case are
Optimization 110
greater than their original values except for that of the stiffness factor of tensioner arm 1
Likewise for the non-weighted Hybrid optimization case the stiffness parameters are above their
original values without exceptions The coordinates of the optimal solutions are within close
approximation to each other and also both match the regions for moderately low tension in
Regions 1 and 2 of the ISG driving case as is shown in Figures 49 410 413 and 414
The results of the non-weighted Hybrid optimization trial and Trial 4 of the weighted Hybrid
optimization simulations are selected as the most optimal designs for the B-ISG Twin Tensioner
In these designs the Twin Tensioner is shown in Table 53b and 54b to have static stability and
to maintain suitable tensions in the ISG driving phase The tensioner parameters for the optimal
designs allow for one of the lowest amounts of additional pre-tension to be added to the system
out of all the findings from the simulations which were conducted
111
CHAPTER 6 CONCLUSION
61 Summary
The primary aim of the thesis is to reduce the magnitude of static tension in the belt spans of a
Belt-driven Integrated Starter-generator (B-ISG) system by the design and investigation of a
Twin Tensioner It is established that the operating phases of the B-ISG system produced two
cases for static tension outcomes an ISG driving case and a crankshaft driving case The
approach taken in this thesis study includes the derivation of a system model for the geometric
properties as well as for the dynamic and static states of the B-ISG system The static state of a
B-ISG system with a single tensioner mechanism is highlighted for comparison with the static
state of the Twin Tensioner-equipped B-ISG system
It is observed that there is an overall reduction in the magnitudes of the static belt tensions with
the presence of a Twin Tensioner over that of a single tensioner The influences of the geometric
and stiffness properties of the Twin Tensioner affecting the static tensions in the system are
analyzed in a parametric study It is found that there is a notable change in the static tensions
produced as result of perturbations in each respective tensioner property This demonstrates
there are no reasons to not further consider a tensioner property based solely on its influence on
the B-ISG system‟s static tensions The phenomenon of higher magnitudes for static tensions in
the ISG mode of operation over that of the crankshaft mode of operation particularly in
excessively slack spans provides the motivation for optimizing the ISG case alone for static
tension The optimization method uses weighted and non-weighted approaches with genetic
algorithm (GA) and hybrid GA searches The most optimal design has been found to be one in
Conclusion 112
which the magnitude of tension in the excessively slack spans in the ISG driving case are
significantly lower than in that of the original B-ISG Twin Tensioner design
62 Conclusion
The conclusions that can be drawn from the study of a Twin Tensioner for a B-ISG system
include the following
1 The simulations of the dynamic model demonstrate that the mode shapes for the system
are greater in the ISG-phase of operation
2 It was observed in the output of the dynamic responses that the system‟s rigid bodies
experienced larger displacements when the crankshaft was driving over that of the ISG-
driving phase It was also noted that the transition speed marking the phase change from
the ISG driving to the crankshaft driving occurred before the system reached either of its
first natural frequencies
3 The magnitudes for static belt tensions as well as dynamic tensions for a B-ISG system
are consistently greater in its ISG operating phase than in its crankshaft operating phase
4 A Twin Tensioner is able to reduce the magnitudes of the static tension for the belt spans
of a B-ISG system in comparison to when only a single tensioner mechanism is present
5 The parametric study of the B-ISG system demonstrates that the slackest and tautest belt
spans decrease or increase together for either phase of operation
6 Perturbations in the Twin Tensioner‟s geometric and stiffness properties have a
significant influence on the magnitudes of the static tension of the slackest and tautest
belt spans The coordinate position of each pulley in the Twin Tensioner configuration
Conclusion 113
has the greatest influence on the belt span static tensions out of all the tensioner
properties considered
7 Optimization of the B-ISG system shows a fixed difference trend between the static
tension of the slackest and tautest belt spans for the B-ISG system
8 The values of the design variables for the most optimal system are found using a hybrid
algorithm approach The slackest span for the optimal system is significantly less slack
than that of the original design Therein less additional pretension is required to be added
to the system to compensate for slack spans in the ISG-driving phase of operation
63 Recommendation for Future Work
The investigation of the B-ISG Twin Tensioner encourages the following future work
1 The optimization of the B-ISG system with the inclusion of diametric Twin Tensioner
properties would provide a complete picture as to the highest possible performance
outcome that the Twin Tensioner is able to have with respect to the static tensions
achieved in the belt spans
2 A larger number of optimization trials using the genetic algorithm (GA) and hybrid GA
under weighted and other approaches would investigate the scope of optimal designs
available in the Twin Tensioner for the B-ISG system
3 A model of the system without the simplification of constant damping may produce
results that are more representative of realistic operating conditions of the serpentine belt
drive A finite element analysis of the Twin Tensioner B-ISG system may provide more
applicable findings
Conclusion 114
4 Investigation of the transverse motion coupled with the rotational belt motion in an
optimized B-ISG system equipped with a Twin Tensioner may also provide a closer look
at the system under realistic conditions In addition the affect of the Twin Tensioner on
transverse motion can determine whether significant improvements in the magnitudes of
static belt span tensions are still being achieved
5 The recommendation to conduct modal decoupling of the B-ISG system‟s static model is
motivated by the fixed difference trend between the slackest and tautest belt span
tensions shown in Chapter 5 The modal decoupling of the system would allow for its
matrices comprising the equations of motion to be diagonalized and therein to decouple
the system equations Modal analysis would transform the system from physical
coordinates into natural coordinates or modal coordinates which would lead to the
decoupling of system responses
6 An investigation and optimization of the dynamic belt span tensions for a B-ISG system
with a Twin Tensioner would increase understanding of the full impact of a Twin
Tensioner mechanism on the state of the B-ISG system It would be informative to
analyze the mode shapes of the first and second modes as well as the required torques of
the driving pulleys and the resulting torque of each of the tensioner arms The
observation of the dynamic belt span tensions would also give direction as to how
damping of the system may or may not be changed
7 Further comparison with the Twin Tensioner B-ISG system‟s dynamic and static states
including the Twin Tensioner‟s stability in each versus a B-ISG system with a single
tensioner would further demonstrate the improvements or dis-improvements in the Twin
Tensioner‟s performance on a B-ISG system
Conclusion 115
8 The influence of the belt properties on the dynamic and static tensions for a B-ISG
system with a Twin Tensioner can also be investigated This again will show the
evidence of improvements or dis-improvement in the Twin Tensioner‟s performance
within a B-ISG setting
9 Lastly an experimental apparatus of the B-ISG system with a Twin Tensioner can be
designed and constructed Suitable instrumentation can be set-up to measure belt span
tensions (both static and dynamic) belt motion and numerous other system qualities
This would provide extensive guidance as to finding the most appropriate theoretical
model for the system Experimental data would provide a bench mark for evaluating the
theoretical simulation results of the Twin Tensioner-equipped B-ISG system
116
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[3] J E Walters R J Krefta G Gallegos-Lopez and G T Fattic Technology Considerations
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[7] G Tamai Saturn Engine Stop-start System with an Automatic Transmission SAE
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[8] D Naunin Hybrid Electric Vehicles ndash the Technology of the Near Future Business
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[9] A Ebron and R Cregar Introducing Hybrid Technology NAFTC Earth Toys Inc
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[12] FEV Optimization of Hybrid Concepts through Simulation Engineering Services Hybrid
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[13] K Itagaki T Teratani K Kuramochi S Nakamura T Tachibana H Nakai and Y Kamijo
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[14] National Alternative Fuels Training Consortium (NAFTC) (2005 Oct 3) Tech stuff
NAFTC Instructors Digest 1(1) pp Nov 10 2006 Available
httpwwwnaftcwvuedutechnicalNAFTCdigestvol1vol1htm
[15] Green Car Congress BMW to Apply Start-Stop and Brake Regen to MINIs Up to 60
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[16] H Jeneacute E Scheid and H Kemper Hybrid electric vehicle (HEV) concepts - fuel savings
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[17] L Gaedt D Kok (Ford Forschungszentrum Aachen GmbH) C Goodfellow D Tonkin
(Ricardo UK Ltd) C Picod (Valeo Electrical Systems) and M Neu (Gates Corportation)
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[18] S Chen B Lequesne R R Henry Y Xue and J J Ronning Design and Testing of a
Belt-driven Induction Starter-generator IEEE Transactions on Industry Applications vol
38 pp 1525-1533 NovDec 2002
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wwwsaeorgautomagelectronics01-2001
[20] J A MacBain (Delphi Automotive Systems Energenix Center) Simulation Influence in
the Design Process of Mild Hybrid Vehicles SAE Technical Papers 2002 Document no
2002-01-1196
[21] PJ Wezenbeek (Zytec Systems Ltd) D G Evans (General Motors Powertrain) D P
Sczomak (General Motors Powertrain) J P Absmeier (Delphi Corp) and G T Fattic
(Delphi Corp) Combustion Assisted Belt-Cranking of a V-8 Engine at 12-Volts SAE
Technical Papers vol 113 pp 396-407 2004 Document no 2004-01-0569
[22] T C Firbank Mechanics of the Belt Drive International Journal of Mechanical
Sciences vol 12 pp 1053-1063 1970
[23] R L Cassidy S K Fan R S MacDonald and W F Samson Serpentine Extended Life
Accessory Drive SAE Papers 1979 Document no 760699
[24] A G Ulsoy J E Whitesell and M D Hooven Design of Belt-Tensioner Systems for
Dynamic Stability Journal of Vibration Acoustics Stress and Reliability in Design
Transactions of the ASME vol 107 pp 282-290 July 1985
[25] R S Beikmann N C Perkins and A G Ulsoy Free Vibration of Serpentine Belt Drive
Systems Journal of Vibrations and Acoustics Transactions of the ASME vol 118 pp
406-413 1996
[26] T C Kraver G W Fan and J J Shah Complex Modal Analysis of a Flat Belt Pulley
System with Belt Damping and Coulomb-Damped Tensioner Journal of Mechanical
Design Transactions of the ASME vol 118 pp 306-311 Jun 1996
[27] R S Beikmann N C Perkins and A G Ulsoy Design and Analysis of Automotive
Serpentine Belt Drive Systems for Steady State Performance Journal of Mechanical
Design Transactions of the ASME vol 119 pp 162-168 Jun 1997
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[28] L Zhang and J W Zu Modal Analysis of Serpentine Belt Drive Systems Journal of
Sound and Vibration vol 222 pp 259-279 1999
[29] R Balaji and E M Mockensturm Dynamic analysis of a front-end accessory drive with a
decouplerisolator International Journal of Vehicle Design vol 39 pp 208-231 2005
[30] M Nouri Design Optimization and Active Control of Serpentine Belt Drive Systems with
Two-pulley Tensioners University of Toronto 2005
[31] G J Spicer (Litens Automotive Inc) Tensioner for use in eg belt drive system has
electronic actuator associated with clutch spring for engaging International
WO2005119089-A1 Jun 6 2005 2005
[32] Bando Chemical Industries Ltd and Litens Automotive GmbH About belt-type starter
system Feb 27 2002
[33] H Lemberger and R Jungjohann (Bayerische Motoren Werke AG) Tension device for an
envelope drive of a device especially a belt drive of a starter generator of an internal
combustion engine comprises a support part Europe EP1420192-A2 May 19 2004 2003
[34] P Ahner and M Ackermann (Bosch GMBH) Belt drive especially for internal
combustion engines to drive accessories in an automobile Germany DE19849886-A1
May 11 2000 1998
[35] N Freisinger K Hagemann J Sievert P Struebel and M Treusch (Daimler Chrysler AG)
Belt tensioning device for belt drive between engine and starter generator of motor
vehicle has self-aligning bearing that supports auxiliary unit and provides working force to
tensioners for tensioning belt Germany DE10324268 Dec 16 2004 2003
[36] C R Rogers (Dayco Products LLC) Offset starter generator drive system for a vehicle
engine has a dual arm pivoted tensioner United States US6942589-B2 Feb 8 2005 2002
[37] A Serkh and I Ali (Gates Corp) Internal combustion engine has belt drive system with
tensioner asymmetrically biased in direction tending to cause power transmission belt to be
under tension International WO2003038309-A1 May 8 2003 2002
References 120
[38] P J Mcvicar and C A Thurston (General Motors Corp) Belt alternator starter accessory
drive with dual tensioner United States US20060287146-A1 Dec 21 2006 2005
[39] W Petri and M Bogner (INA Schaeffler KG) Traction drive especially for driving
internal combustion engine units has arrangement for demand regulated setting of tension
consisting of unit with housing with limited rotation and pulley German DE10044645-
A1 Mar 21 2002 2000
[40] M Bogner (INA Schaeffler KG) Belt drive tensioner for a starter-generator of an IC
engine has locking system for locking tensioning element in an engine operating mode
locking system is directly connected to pivot arm follows arm control movements
German DE10159073-A1 Jun 12 2003 2001
[41] R Painta M Bogner and H Graf (INA Schaeffler KG) Traction mechanism drive esp
belt drive has belt tensioning pulley mounted on generator shaft and uncoupled from it via
freewheel to dampen load peaks Europe EP1723350-A1 Nov 22 2006 2005
[42] W Petri (INA Schaeffler KG) Drive unit for a combustion engine having a starter
generator and a belt drive has tensioner with spring and counter hydraulic force Germany
DE10359641-A1 Jul 28 2005 2003
[43] H Stief M Bogner B Hartmann T Kraft and M Schmid (INA Schaeffler KG) Traction
drive especially belt drive for short-duration driving of starter generator has tensioning
device with lever arm deflectable against restoring force and with end stop limiting
deflection travel Europe EP1738093-A1 Jan 3 2007 2005
[44] M Ulm (INA Schaeffler KG DE) Tension unit eg for drive in machine such as
combustion engine has belt or chain drive with wheels turning and connected with starter
generator and unit has two idlers arranged at clamping arm with machine stored by shock
absorber Germany DE102004012395-A1 Sep 29 2005 2004
[45] M Bogner (INA Schaeffler KG) Belt drive for starter motor-generator auxiliary assembly
has limited movement at the starter belt section tensioner roller bringing it into a dead point
position on starting the motor International WO2006108461-A1 Oct 19 2006 2006
References 121
[46] W Guhr (Litens Automotive GMBH) Automotive motor and drive assembly includes
tension device positioned within belt drive system having combination starter United
States US2001007839-A1 Jul 12 2001 2001
[47] K Kuniaki K Masahiko H Kazuyuki I Shuichi and T Masaki (Mitsubishi Jidosha Eng
KK and Mitsubishi Motor Corp) Tension adjustment method of belt for starter generator
in vehicle involves shifting auto-tensioners between lock state and free state to adjust
tension of belt during driving of crank pulley Japan JP2005083514-A Mar 31 2005
2003
[48] Nissan Motor Co Ltd Winding gear for starting engine of hybrid motor vehicle has
tensioner tightening chain while cranking engine and slackens chain after start of engine
provided to span side of chain Japan JP3565040-B2 Sep 15 2004 1998
[49] S Sato and H Hayakawa (NTN Corp) Auto tensioner for ancillary drive belts has
cylinder nut and screw bolt in hydraulic damper mechanism provided in middle of cylinder
acting as start-up rigidity buffer component Japan JP2006189073-A Jul 20 2006 2005
[50] G Vadin-Michaud (Valeo Equip Electrique Moteur) Pulley and belt starting system for a
thermal engine for a motor vehicle Europe EP1658432 May 24 2006 2005
[51] M Zhen University of Toronto and Litens Automotive Group Ltd FEAD vol 50
2005
[52] W E Johns Notes on Motors [Electronic] 2003 [2008 June] Available at
httpwwwgizmologynetmotorshtm
[53] Litens Automotive Group Ltd DC BAS System - Conventional Start Input Profile Nov
23 2007
[54] Arnold Magnetic Technologies Corp General Motor Terminology [Electronic] pp 7
[2008 June] Available at httpwwwgrouparnoldcommtcpdfweb_motor_glossarypdf
[55] Douglas W Jones Stepping Motors University of Iowa - Department of Computer
Science [Electronic] Feb 14 2008 [2008 June] Available at
httpwwwcsuiowaedu~jonesstepphysicshtml
References 122
[56] Litens Automotive Group Ltd (2004 Jan 31) FEAD software input data for test project
[57] K Deb Multi-Objective Optimization using Evolutionary Algorithms Toronto John Wiley
amp Sons Ltd 2001 pp 81-85
[58] P E McSharry (2004 May 11) Department of Engineering Science University of Oxford
[httpwwwengoxacuksamppubsgawbreppdf]
[59] The MathWorks Inc MATLAB vol 750342 (R2007b) Aug 15 2007
123
APPENDIX A
Passive Dual Tensioner Designs from Patent Literature
Figure A1 Proposed design by Bayerische Motoren Werke AG corresponding to patent nos EP1420192-A2 and DE10253450-A1
Source European Patent Office espcenet (publication nos EP1420192-A2 and DE10253450-A1 accessed May 2007) epespacenetcom [33]
Figure A1 label identification 1 ndash tightner 2 ndash belt drive
3 ndash starter generator
4 ndash internal-combustion engine
4‟ ndash crankshaft-lateral drive disk
5 ndash generator housing
6 ndash common axis of rotation
7 ndash featherspring of tiltable clamping arms
8 ndash clamping arm
9 ndash clamping arm
10 11 ndash idlers
12 12‟ ndash Zugtrum 13 13‟ ndash Leertrum
14 ndash carry-hurries 15 ndash generator wave
16 ndash bush
17 ndash absorption mechanism
18 18‟ ndash support arms
19 19‟ ndash auxiliary straight lines
20 ndash pipe
21 ndash torsion bar
22 ndash breaking through
23 ndash featherspring
24 ndash friction disk
25 ndash screw connection 26 ndash Wellscheibe
(European Patent Office May 2007) [33]
Appendix A 124
Figure A2a First of four proposed designs (and its various configurations) by Bosch GMBH
corresponding to patent no WO0026532-A1
Source European Patent Office espcenet (publication no WO0026532-A1 accessed Jun 2007)
epespacenetcom [34]
Figure A2b Second of four proposed designs by Bosch GMBH corresponding to patent no
WO0026532-A1
Source European Patent Office espcenet (publication no WO0026532-A1 accessed Jun 2007) epespacenetcom [34]
Figure A2c Third of four proposed designs (and its various configurations) by Bosch GMBH
corresponding to patent no WO0026532-A1
Source European Patent Office espcenet (publication no WO0026532-A1 accessed Jun 2007)
epespacenetcom [34]
Appendix A 125
Figure A2d Fourth of four proposed designs (and its various configurations) by Bosch GMBH
corresponding to patent no WO0026532-A1
Source European Patent Office espcenet (publication no WO0026532-A1 accessed Jun 2007)
epespacenetcom [34]
Figure A2a through to A2d label identification 10 ndash engine wheel
11 ndash [generator] 13 ndash spring
14 ndash belt
16 17 ndash tensioning pulleys
18 19 ndash springs
20 21 ndash fixed points
25ab ndash carriers of idlers
25c ndash gang bolt
(European Patent Office June 2007) [34]
Figure A3 Proposed design by Daimler Chrysler AG corresponding to patent no DE10324268-A1
Source European Patent Office espcenet (publication no DE10324268-A1 accessed May 2007)
epespacenetcom [35]
Figure A3 label identification
Appendix A 126
10 12 ndash belt pulleys
14 ndash auxiliary unit
16 ndash belt
22-1 22-2 ndash belt tensioners
(European Patent Office May 2007) [35]
Figure A4 Proposed design by Dayco Products LLC corresponding to patent no US6942589-B2
Source European Patent Office espcenet (publication no US6942589-B2 accessed Jun 2007)
epespacenetcom [36]
Figure A4 label identification 12 ndash belt
14 ndash tensioner
16 ndash generator pulley
18 ndash crankshaft pulley
22 ndash slack span 24 ndash tight span
32 34 ndash arms
33 35 ndash pulley
(European Patent Office June 2007) [36]
Appendix A 127
Figure A5 Proposed design by Gates Corp corresponding to patent no WO2003038309-A
Source European Patent Office espcenet (publication no WO2003038309-A accessed Jun 2007)
epespacenetcom [37]
Figure A5 label identification 12 ndash motorgenerator
14 ndash motorgenerator pulley 26 ndash belt tensioner
28 ndash belt tensioner pulley
30 ndash transmission belt
(European Patent Office June 2007) [37]
Figure A6 Proposed design by General Motors Corp corresponding to patent no US20060287146-A1
Source European Patent Office espcenet (publication no US20060287146-A1 accessed Jun 2007)
epespacenetcom [38]
Appendix A 128
Figure A6 label identification 28 ndash tensioner
32 ndash carrier arm
34 ndash secondary carrier arm
46 ndash tensioner pulley
58 ndash secondary tensioner pulley
(European Patent Office June 2007) [38]
Figure A7 Proposed design by INA Schaeffler KG corresponding to patent no DE10044645-A1
Source European Patent Office espcenet (publication no DE10044645-A1 accessed Jun 2007)
epespacenetcom [39]
Figure A7 label identification 2 ndash internal combustion engine
3 ndash traction element
11 ndash housing with limited rotation 12 13 ndash direction changing pulleys
(European Patent Office June 2007) [39]
Appendix A 129
Figure A8a First of three proposed designs by INA Schaeffler KG corresponding to patent no
DE10159073-A1
Source European Patent Office espcenet (publication no DE10159073-A1 accessed May 2007)
epespacenetcom [40]
Figure A8b Second of three proposed designs by INA Schaeffler KG corresponding to patent no
DE10159073-A1
Source European Patent Office espcenet (publication no DE10159073-A1 accessed May 2007)
epespacenetcom [40]
Appendix A 130
Figure A8c Third of three proposed designs by INA Schaeffler KG corresponding to patent no
DE10159073-A1
Source European Patent Office espcenet (publication no DE10159073-A1 accessed May 2007)
epespacenetcom [40]
Figure A8a A8b and A8c label identification 1 ndash tightener [tensioner]
2 ndash idler
3 ndash drawing means
4 ndash swivel arm
5 ndash axis of rotation
6 ndash drawing means impulse [belt]
7 ndash crankshaft
8 ndash starter generator
9 ndash bolting volume 10a ndash bolting device system
10b ndash bolting device system
10c ndash bolting device system
11 ndash friction body
12 ndash lateral surface
13 ndash bolting tape end
14 ndash bolting tape end
15 ndash control member
16 ndash torsion bar
17 ndash base
18 ndash pylon
19 ndash hub
20 ndash annular gap
21 ndash Gleitlagerbuchse
23 ndash [nil]
23 ndash friction disk
24 ndash turning camps 25 ndash teeth
26 ndash elbow levers
27 ndash clamping wedge
28 ndash internal contour
29 ndash longitudinal guidance
30 ndash system
31 ndash sensor
32 ndash clamping gap
(European Patent Office May 2007) [40]
Appendix A 131
Figure A9 Proposed design by INA Schaeffler KG corresponding to patent no DE10359641-A1
Source European Patent Office espcenet (publication no DE10359641-A1 accessed Jun 2007)
epespacenetcom [42]
Figure A9 label identification 8 ndash cylinder
10 ndash rod
12 ndash spring plate
13 ndash spring
14 ndash pressure lead
(European Patent Office June 2007) [42]
Appendix A 132
Figure A10 Proposed design by INA Schaeffler KG corresponding to patent no EP1723350-A1
Source European Patent Office espcenet (publication no EP1723350-A1 accessed Jun 2007) epespacenetcom [41]
Figure A10 label identification 4 ndash pulley
5 ndash hydraulic element 11 ndash freewheel
12 ndash shaft
(European Patent Office June 2007) [41]
Figure A11 Proposed design by INA Schaeffler KG corresponding to patent no EP1738093-A1
Source European Patent Office espcenet (publication no EP1738093-A1 accessed Jun 2007)
epespacenetcom [43]
Figure A11 label identification 1 ndash traction drive
2 ndash traction belt
3 ndash starter generator
Appendix A 133
7 ndash tension device
9 ndash lever arm
10 ndash guide roller
16 ndash end stop
(European Patent Office June 2007) [43]
Figure A12 Proposed design by INA Schaeffler KG corresponding to patent no DE102004012395-A1
Source European Patent Office espcenet (publication no DE102004012395-A1 accessed May 2007) epespacenetcom [44]
Figure A12 label identification 1 ndash belt drive
2 ndash belts
3 ndash wheel of the internal-combustion engine
4 ndash wheel of a Nebenaggregats
5 ndash wheel of the starter generator
6 ndash clamping unit
7 ndash idler
8 ndash idler
9 ndash scale beams
10 ndash drive place
11 ndash clamping arm
12 ndash camps
13 ndash coupling point
14 ndash shock absorber element
15 ndash arrow
(European Patent Office May 2007) [44]
Appendix A 134
Figure A13 Proposed design by INA Schaeffler KG corresponding to patent nos DE102005017038-A1and WO2006108461-A1
Source European Patent Office espcenet (publication nos DE102005017038-A1and WO2006108461-A1 accessed May 2007) epespacenetcom [45]
Figure A13 label identification 1 ndash belt
2 ndash wheel of the crankshaft KW
3 ndash wheel of a climatic compressor AC
4 ndash wheel of a starter generator SG
5 ndash wheel of a water pump WP
6 ndash first clamping system
7 ndash first tension adjuster lever arm
8 ndash first tension adjuster role
9 ndash second clamping system
10 ndash second tension adjuster lever arm
11 ndash second tension adjuster role 12 ndash guide roller
13 ndash drive-conditioned Zugtrum
(generatorischer enterprise (GE))
13 ndash starter-conditioned Leertrum
(starter enterprise (SE))
14 ndash drive-conditioned Leertrum (GE)
14 ndash starter-conditioned Zugtrum (SE)
14a ndash drive-conditioned Leertrumast (GE)
14a ndash starter-conditioned Zugtrumast (SE)
14b ndash drive-conditioned Leertrumast (GE)
14b ndash starter-conditioned Zugtrumast (SE)
(European Patent Office May 2007) [45]
Figure A14 Proposed design by Litens Automotive GMBH et al corresponding to patent no
US20010007839-A1
Appendix A 135
Source European Patent Office espcenet (publication no US20010007839-A1 accessed Jun 2007)
epespacenetcom [46]
Figure A14 label identification E - belt
K - crankshaft
R1 ndash first tension pulley
R2 ndash second tension pulley
S ndash tension device
T ndash drive system
1 ndash belt pulley
4 ndash belt pulley
(European Patent Office June 2007) [46]
Figure A15 Proposed design by Mitsubishi Jidosha Eng KK and Mitsubishi Motor Corp corresponding
to patent no JP2005083514-A
Source Industrial Property Digital Library and Japanese Patent Office Patent amp Utility Model Gazette DB (document no A 2005-083514 accessed May 2007) wwwipdlinpitgojp [47]
Figure A15 label identification 1 ndash Pulley for Starting
2 ndash Crank Pulley
3 ndash AC Pulley
4a ndash 1st roller
4b ndash 2nd roller
5 ndash Idler Pulley
6 ndash Belt
7c ndash Starter generator control section
7d ndash Idle stop control means
8 ndash WP Pulley
9 ndash IG Switch
10 ndash Engine
11 ndash Starter Generator
12 ndash Driving Shaft
Appendix A 136
7 ndash ECU
7a ndash 1st auto tensioner control section (the 1st auto
tensioner control means)
7b ndash 2nd auto tensioner control section (the 2nd auto
tensioner control means)
13 ndash Air-conditioner Compressor
14a ndash 1st auto tensioner
14b ndash 2nd auto tensioner
18 ndash Water Pump
(Industrial Property Digital Library May 2007) [47]
Figure A16 Proposed design by Nissan Motor Co Ltd corresponding to patent no JP3565040-B2
Source European Patent Office espcenet (publication no JP3565040-B2 accessed Jun 2007) epespacenetcom [48]
Figure A16 label identification 3 ndash chain [or belt]
5 ndash tensioner
4 ndash belt pulley
(European Patent Office June 2007) [48]
Figure A17 Proposed design by NTN Corp corresponding to patent no JP2006189073-A
Appendix A 137
Source European Patent Office espcenet (publication no JP2006189073-A accessed Jun 2007)
epespacenetcom [49]
Figure A17 label identification 5d - flange
6 ndash tensile strength spring
10 ndash actuator
10c ndash cylinder
12 ndash rod
20 ndash hydraulic damper mechanism
21 ndash cylinder nut
22 ndash screw bolt
(European Patent Office June 2007) [49]
Figure A18 Proposed design by Valeo Equipment Electriques Moteur corresponding to patent nos
EP1658432 and WO2005015007
Source European Patent Office espcenet (publication nos EP1658432 and WO2005015007
accessed May 2007) epespacenetcom [50]
Figure A18 abbreviated list of label identifications
10 ndash starter
22 ndash shaft section
23 ndash free front end
80 ndash pulley
200 ndash support element
206 - arm
(European Patent Office May 2007) [50]
The author notes that the list of labels corresponding to Figures A1a through to A7 are generated
from machine translations translated from the documentrsquos original language (ie German)
Consequently words may be translated inaccurately or not at all
138
APPENDIX B
B-ISG Serpentine Belt Drive with Single Tensioner
Equation of Motion
The equations of motion (EOM) for a B-ISG serpentine belt drive with a single tensioner are
shown The EOM has been derived similarly to that of the same system with a twin tensioner
that was provided in Chapter 3 The assumptions for the twin tensioner B-ISG system are
applicable for the single tensioner B-ISG system as well
Figure B1 shows the B-ISG system with a single tensioner pulley and arm The pulleys are
numbered 1 through 4 and their associated belt spans are numbered accordingly
Figure B1 Single Tensioner B-ISG System
Appendix B 139
The free-body diagram for the ith non-tensioner pulley in the system shown above is found in
Figure B2 The moment of inertia for the ith pulley is designated as Ii while the angular
displacement velocity and acceleration is designated as 120579119905119894 120579 119905119894 and 120579 119905119894 respectively The
required torque is Qi the angular damping is Ci and the tension of the ith span is Ti
Figure B2 Free-body Diagram of ith Pulley
The positive motion designated is assumed to be in the clockwise direction The radius for the
ith pulley is represented by Ri The equilibrium equations for the ith pulley are as follows
I1 ∙ θ 1 = T4 ∙ R1 minus T1 ∙ R1 + Q1 minus c1 ∙ θ 1 (B1)
I2 ∙ θ 2 = T1 ∙ R2 minus T2 ∙ R2 + Q2 minus c2 ∙ θ 2 (B2)
I3 ∙ θ 3 = T2 ∙ R3 minus T3 ∙ R3 + Q3 minus c3 ∙ θ 3 (B3)
Appendix B 140
A free-body diagram for the single tensioner pulley is shown in Figure B3 The rotational
stiffness and damping for the tensioner arm is designated as kt and ct respectively The angular
rotation and velocity for the arm is 120579119905119894 and 120579 119905119894 respectively
Figure B3 Free-body Diagram of Single Tensioner
From figure B2 the equations of equilibrium are resolved for the tensioner pulley The angle of
orientation for the ith belt span is designated by 120573119895119894
minusI4 ∙ θ 4 = minusT3 ∙ R4 + T4 ∙ R4 + Q4 + c4 ∙ θ 4 (B4)
It ∙ θ t = minusTt ∙ Lt ∙ sin θto minus βj4 + sin θto 1 minus βj5 minus kt ∙ partθt minus ct ∙ partθ t
(B5)
Appendix B 141
partθt = θt minus θto (B6)
The dynamic tension matrix Trsquo is proportional to the damping (Tc) and stiffness (Tk) matrices
that are due to belt damping (119888119894119887 ) and belt stiffness (119896119894
119887 ) respectively
119827prime = 119827119836 ∙ 120521 + 119827119844 ∙ 120521 (B7)
The initial tension is represented by To and the initial angle of the tensioner arm is represented
by 120579119905119900 The equation for the tension of the ith span is shown in the following equations
T1 = To + k1b ∙ R1 ∙ θ1 minus R2 ∙ θ2 + c1
b ∙ [R1 ∙ θ 1 minus R2 ∙ θ 2] (B8)
T2 = To + k2b ∙ R2 ∙ θ2 minus R3 ∙ θ3 + c2
b ∙ [R2 ∙ θ 2 minus R3 ∙ θ 3)] (B9)
T3 = To + k3b ∙ R3 ∙ θ3 minus R4 ∙ θ4 + Lt ∙ [sin θto minus βj3 ] ∙ (θt minus θto ) + c3
b ∙ [R3 ∙ θ 3 minus R4 ∙
θ 4 + Lt ∙ [sin θto minus βj3 ] ∙ (θ t)] (B10)
T4 = To + k4b ∙ R4 ∙ θ4 minus R1 ∙ θ1 + Lt ∙ [sin θto minus βj4 ] ∙ (θt minus θto ) + c4
b ∙ [R4 ∙ θ 4 minus R1 ∙
θ 1 + Lt ∙ [sin θto minus βj4 ] ∙ (θ t)] (B11)
Tprime = Ti minus To (B12)
Tt = T3 = T4 (B13)
Appendix B 142
The EOM for the single tensioner B-ISG system is found by substitution of equations B8 to
B13 into B1 to B5 The matrices in the EOM include the inertial matrix I damping matrix C
stiffness matrix K and the required torque matrix Q as well as the angular displacement
velocity and acceleration matrices 120521 120521 and 120521 respectively
119816 ∙ 120521 + 119810 ∙ 120521 + 119818 ∙ 120521 = 119824 (B14)
119816 =
I1 0 0 0 00 I2 0 0 00 0 I3 0 00 0 0 I4 00 0 0 0 It1
(B15)
The stiffness matrix includes kb the belt factor Kb the belt cord stiffness 120601119894 the wrap angle of
the belt on the ith pulley and Kbi the stiffness factor of the ith belt span Cb represents the belt
damping for each span and βji is the angle of orientation for the span between the jth and ith
pulleys It is noted in the terms of the stiffness and damping matrices below that the numerical
subscripts refer to the (i+1)th pulley The term Qt may be found within the required torque
matrix and represents the required torque for the tensioner arm As well the term It1 represents
the moment of inertia for the tensioner arm
Appendix B 143
K =
(B16)
Kbi =Kb
Li + kb ∙ Ri ∙ϕi+1
2 + Ri ∙ϕi
2
(B17)
C =
(B18)
Appendix B 144
Appendix B 144
120521 =
θ1
θ2
θ3
θ4
partθt
(B19)
119824 =
Q1
Q2
Q3
Q4
Qt
(B20)
Simulations of the EOM for the B-ISG system with a single tensioner were performed in FEAD
[51] software for dynamic and static cases This allowed for the methodology for deriving the
EOM to be verified and then applied to the B-ISG system with a twin tensioner The natural
frequencies modes shapes dynamic responses tensioner arm torques as well as the crankshaft
required torque only and the dynamic tensions were solved from the EOM as described in (331)
to (339) of Chapter 3 and as well as for the static tension from (351) to (353) of Chapter 3
This permitted verification of the complete derivation methodology and allowed for comparison
of the static tension of the B-ISG system belt spans in the case that a single tensioner is present
and in the case that a Twin Tensioner is present [51]
145
APPENDIX C
MathCAD Scripts
C1 Geometric Analysis
1 - CS
2 - AC
4 - Alt
3 - Ten1
5 - Ten 2
6 - Ten Pivot
1
2
3
4
5
Figure C1 Schematic of B-ISG
System with Twin Tensioner
Coordinate Input DataXY1 0 0( ) XY4 24759 16664( )
XY2 224 6395( ) XY5 12057 9193( )
XY3 292761 87( ) XY6 201384 62516( )
Computations
Lt1 XY30 0
XY60 0
2
XY30 1
XY60 1
2
Lt2 XY50 0
XY60 0
2
XY50 1
XY60 1
2
t1 atan2 XY30 0
XY60 0
XY30 1
XY60 1
t2 atan2 XY50 0
XY60 0
XY50 1
XY60 1
XY
XY10 0
XY20 0
XY30 0
XY40 0
XY50 0
XY60 0
XY10 1
XY20 1
XY30 1
XY40 1
XY50 1
XY60 1
x XY
0 y XY
1
Appendix C 146
i - angle bw horizontal and l ine from ith pulley center to (i+1)th pulley center
Adjust last number in range variable p to correspond to number of pulleys
p 0 1 4
k p( ) p 1( ) p 4if
0 otherwise
condition1 p( ) acos
XYk p( ) 0
XYp 0
XYk p( ) 0
XYp 0
2
XYk p( ) 1
XYp 1
2
condition2 p( ) 2 acos
XYk p( ) 0
XYp 0
XYk p( ) 0
XYp 0
2
XYk p( ) 1
XYp 1
2
p( ) if XYk p( ) 1
XYp 1
condition1 p( ) condition2 p( )
Lfi Lbi - connection belt span lengths
D1 20065mm D2 10349mm D3 7240mm D4 6820mm D5 7240mm
D
D1
D2
D3
D4
D5
Lf p( ) XYk p( ) 0
XYp 0
2
XYk p( ) 1
XYp 1
2
1
mm
Dk p( )
2
Dp
2
2
Lb p( ) XYk p( ) 0
XYp 0
2
XYk p( ) 1
XYp 1
2
1
mm
Dk p( )
2
Dp
2
2
fi bi - angle bw ith pulley center connection l ine and contact points Pbfi (or Pfbi) and Pbi
(or Pfi) respecti vely l
f p( ) atanLf p( ) mm
Dp
2
Dk p( )
2
Dp
Dk p( )
if
atanLf p( ) mm
Dk p( )
2
Dp
2
Dp
Dk p( )
if
2D
pD
k p( )if
b p( ) atan
mmLb p( )
Dp
2
Dk p( )
2
Appendix C 147
XYfi XYbi XYfbi XYbfi - 4 possible contact points for i th pulley
XYf p( ) XYp 0
Dp
2 mmcos p( ) f p( )
XYp 1
Dp
2 mmsin p( ) f p( )
XYb p( ) XYp 0
Dp
2 mmcos p( ) f p( )
XYp 1
Dp
2 mmsin p( ) f p( )
XYfb p( ) XYp 0
Dp
2 mmcos p( ) b p( )
XYp 1
Dp
2 mmsin p( ) b p( )
XYbf p( ) XYp 0
Dp
2 mmcos p( ) b p( )
XYp 1
Dp
2 mmsin p( ) b p( )
XYfi+1 XYbi+1 XYfbi+1 XYbfi+1 - 4 possible contact points for i+1th pulley
XYf2 p( ) XYk p( ) 0
Dk p( )
2 mmcos p( ) f p( )
XYk p( ) 1
Dk p( )
2 mmsin p( ) f p( )
XYb2 p( ) XYk p( ) 0
Dk p( )
2 mmcos p( ) f p( )
XYk p( ) 1
Dk p( )
2 mmsin p( ) f p( )
XYfb2 p( ) XYk p( ) 0
Dk p( )
2 mmcos p( ) b p( ) XY
k p( ) 1
Dk p( )
2 mmsin p( ) b p( )
XYbf2 p( ) XYk p( ) 0
Dk p( )
2 mmcos p( ) b p( ) XY
k p( ) 1
Dk p( )
2 mmsin p( ) b p( )
Row 1 --gt Pulley 1 Row i --gt Pulley i
XYfi
XYf 0( )0 0
XYf 1( )0 0
XYf 2( )0 0
XYf 3( )0 0
XYf 4( )0 0
XYf 0( )0 1
XYf 1( )0 1
XYf 2( )0 1
XYf 3( )0 1
XYf 4( )0 1
XYfi
6818
269222
335325
251552
108978
100093
89099
60875
200509
207158
x1 XYfi0
y1 XYfi1
Appendix C 148
XYbi
XYb 0( )0 0
XYb 1( )0 0
XYb 2( )0 0
XYb 3( )0 0
XYb 4( )0 0
XYb 0( )0 1
XYb 1( )0 1
XYb 2( )0 1
XYb 3( )0 1
XYb 4( )0 1
XYbi
47054
18575
269403
244841
164847
88606
291
30965
132651
166182
x2 XYbi0
y2 XYbi1
XYfbi
XYfb 0( )0 0
XYfb 1( )0 0
XYfb 2( )0 0
XYfb 3( )0 0
XYfb 4( )0 0
XYfb 0( )0 1
XYfb 1( )0 1
XYfb 2( )0 1
XYfb 3( )0 1
XYfb 4( )0 1
XYfbi
42113
275543
322697
229969
9452
91058
59383
75509
195834
177002
x3 XYfbi0
y3 XYfbi1
XYbfi
XYbf 0( )0 0
XYbf 1( )0 0
XYbf 2( )0 0
XYbf 3( )0 0
XYbf 4( )0 0
XYbf 0( )0 1
XYbf 1( )0 1
XYbf 2( )0 1
XYbf 3( )0 1
XYbf 4( )0 1
XYbfi
8384
211903
266707
224592
140427
551
13639
50105
141463
143331
x4 XYbfi0
y4 XYbfi1
Row 1 --gt Pulley 2 Row i --gt Pulley i+1
XYf2i
XYf2 0( )0 0
XYf2 1( )0 0
XYf2 2( )0 0
XYf2 3( )0 0
XYf2 4( )0 0
XYf2 0( )0 1
XYf2 1( )0 1
XYf2 2( )0 1
XYf2 3( )0 1
XYf2 4( )0 1
XYf2x XYf2i0
XYf2y XYf2i1
XYb2i
XYb2 0( )0 0
XYb2 1( )0 0
XYb2 2( )0 0
XYb2 3( )0 0
XYb2 4( )0 0
XYb2 0( )0 1
XYb2 1( )0 1
XYb2 2( )0 1
XYb2 3( )0 1
XYb2 4( )0 1
XYb2x XYb2i0
XYb2y XYb2i1
Appendix C 149
XYfb2i
XYfb2 0( )0 0
XYfb2 1( )0 0
XYfb2 2( )0 0
XYfb2 3( )0 0
XYfb2 4( )0 0
XYfb2 0( )0 1
XYfb2 1( )0 1
XYfb2 2( )0 1
XYfb2 3( )0 1
XYfb2 4( )0 1
XYfb2x XYfb2i
0
XYfb2y XYfb2i1
XYbf2i
XYbf2 0( )0 0
XYbf2 1( )0 0
XYbf2 2( )0 0
XYbf2 3( )0 0
XYbf2 4( )0 0
XYbf2 0( )0 1
XYbf2 1( )0 1
XYbf2 2( )0 1
XYbf2 3( )0 1
XYbf2 4( )0 1
XYbf2x XYbf2i0
XYbf2y XYbf2i1
100 40 20 80 140 200 260 320 380 440 500150
110
70
30
10
50
90
130
170
210
250Figure C2 Possible Contact Points
250
150
y1
y2
y3
y4
y
XYf2y
XYb2y
XYfb2y
XYbf2y
500100 x1 x2 x3 x4 x XYf2x XYb2x XYfb2x XYbf2x
Appendix C 150
Xij Yij - selected contact point on ith pulley for span from ith pulley to jth pulley
XY15 XYbf2iT 4
XY12 XYfiT 0
Pulley 1 contact pts
XY21 XYf2iT 0
XY23 XYfbiT 1
Pulley 2 contact pts
XY32 XYfb2iT 1
XY34 XYbfiT 2
Pulley 3 contact pts
XY43 XYbf2iT 2
XY45 XYfbiT 3
Pulley 4 contact pts
XY54 XYfb2iT 3
XY51 XYbfiT 4
Pulley 5 contact pts
By observation the lengths of span i is the following
L1 Lf 0( ) L2 Lb 1( ) L3 Lb 2( ) L4 Lb 3( ) L5 Lb 4( ) Li
L1
L2
L3
L4
L5
mm
i Angle between horizontal and span of ith pulley
i
atan
XY121
XY211
XY12
0XY21
0
atan
XY231
XY321
XY23
0XY32
0
atan
XY341
XY431
XY34
0XY43
0
atan
XY451
XY541
XY45
0XY54
0
atan
XY511
XY151
XY51
0XY15
0
Appendix C 151
Pulley 1 Pulley 2 Pulley 3 Pulley 4 Pulley 5
12 i0 2 21 i0 32 i1 2 43 i2 54 i3
15 i4 2 23 i1 34 i2 45 i3 51 i4
15
21
32
43
54
12
23
34
45
51
Wrap angle i for the ith pulley
1 2 atan2 XY150
XY151
atan2 XY120
XY121
2 atan2 XY210
XY1 0
XY211
XY1 1
atan2 XY230
XY1 0
XY231
XY1 1
3 2 atan2 XY320
XY2 0
XY321
XY2 1
atan2 XY340
XY2 0
XY341
XY2 1
4 atan2 XY430
XY3 0
XY431
XY3 1
atan2 XY450
XY3 0
XY451
XY3 1
5 atan2 XY540
XY4 0
XY541
XY4 1
atan2 XY510
XY4 0
XY511
XY4 1
1
2
3
4
5
Lb length of belt
Lbelt Li1
2
0
4
p
Dpp
Input Data for B-ISG System
Kt 20626Nm
rad (spring stiffness between tensioner arms 1
and 2)
Kt1 10314Nm
rad (stiffness for spring attached at arm 1 only)
Kt2 16502Nm
rad (stiffness for spring attached at arm 2 only)
Appendix C 152
C2 Dynamic Analysis
I C K moment of inertia damping and stiffness matrices respectively
u 0 1 4 v 0 1 4 (new counter variables where final value = no of pulleys + no of ten arms)
RaD
2
Appendix C 153
RaD
2
Ii =gt moment of inertia for ith pulley where i-1 and i represent ten arms
Ii0
0
1
2
3
4
5
6
10000
2230
300
3000
300
1500
1500
I diag Ii( ) kg mm2
Ci =gt Rotational damping and belt damping for the ith pulley where i-1 and i represent tensioner arms
1000kg
m3
CrossArea 693mm2
0 M CrossArea Lbelt M 0086kg
cb 2 KbM
Lbelt
Cb
cb
cb
cb
cb
cb
Cri
0
0
010
0
010
N mmsec
rad
Ct 1000N mmsec
rad Ct1 1000 N mm
sec
rad Ct2 1000N mm
sec
rad
Cr
Cri0
0
0
0
0
0
0
0
Cri1
0
0
0
0
0
0
0
Cri2
0
0
0
0
0
0
0
Cri3
0
0
0
0
0
0
0
Cri4
0
0
0
0
0
0
0
Ct Ct1
Ct
0
0
0
0
0
Ct
Ct Ct2
Rt
Ra0
Ra1
0
0
0
0
0
0
Ra1
Ra2
0
0
Lt1 mm sin t1 32
0
0
0
Ra2
Ra3
0
Lt1 mm sin t1 34
0
0
0
0
Ra3
Ra4
0
Lt2 mm sin t2 54
Ra0
0
0
0
Ra4
0
Lt2 mm sin t2 51
Appendix C 154
Kr
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Kt Kt1
Kt
0
0
0
0
0
Kt
Kt Kt2
Tk
Kbi 0( ) Ra0
0
0
0
Kbi 4( ) Ra0
Kbi 0( ) Ra1
Kbi 1( ) Ra1
0
0
0
0
Kbi 1( ) Ra2
Kbi 2( ) Ra2
0
0
0
0
Kbi 2( ) Ra3
Kbi 3( ) Ra3
0
0
0
0
Kbi 3( ) Ra4
Kbi 4( ) Ra4
0
Kbi 1( ) Lt1 mm sin t1 32
Kbi 2( ) Lt1 mm sin t1 34
0
0
0
0
0
Kbi 3( ) Lt2 mm sin t2 54
Kbi 4( ) Lt2 mm sin t2 51
Tc
Cb0
Ra0
0
0
0
Cb4
Ra0
Cb0
Ra1
Cb1
Ra1
0
0
0
0
Cb1
Ra2
Cb2
Ra2
0
0
0
0
Cb2
Ra3
Cb3
Ra3
0
0
0
0
Cb3
Ra4
Cb4
Ra4
0
Cb1
Lt1 mm sin t1 32
Cb2
Lt1 mm sin t1 34
0
0
0
0
0
Cb3
Lt2 mm sin t2 54
Cb4
Lt2 mm sin t2 51
C matrix
C Cr Rt Tc
K matrix
K Kr Rt Tk
New Equations of Motion for Dual Drive System
I K amp C matricies rearranged to place driving pulley in 1st row + 1st column and driven in 2nd row + 2nd column
IA augment I3
I0
I1
I2
I4
I5
I6
IC augment I0
I3
I1
I2
I4
I5
I6
I1kgmm2 1 106
kg m2
0 0 0 0 0 0
Ia stack I1kgmm2 IAT 0
T
IAT 1
T
IAT 2
T
IAT 4
T
IAT 5
T
IAT 6
T
Ic stack I1kgmm2 ICT 3
T
ICT 1
T
ICT 2
T
ICT 4
T
ICT 5
T
ICT 6
T
Appendix C 155
RtA augment Rt3
Rt0
Rt1
Rt2
Rt4
RtC augment Rt0
Rt3
Rt1
Rt2
Rt4
Rta stack RtAT 3
T
RtAT 0
T
RtAT 1
T
RtAT 2
T
RtAT 4
T
RtAT 5
T
RtAT 6
T
Rtc stack RtCT 0
T
RtCT 3
T
RtCT 1
T
RtCT 2
T
RtCT 4
T
RtCT 5
T
RtCT 6
T
TkA augment Tk3
Tk0
Tk1
Tk2
Tk4
Tk5
Tk6
Tka stack TkAT 3
T
TkAT 0
T
TkAT 1
T
TkAT 2
T
TkAT 4
T
TkC augment Tk0
Tk3
Tk1
Tk2
Tk4
Tk5
Tk6
Tkc stack TkCT 0
T
TkCT 3
T
TkCT 1
T
TkCT 2
T
TkCT 4
T
TcA augment Tc3
Tc0
Tc1
Tc2
Tc4
Tc5
Tc6
Tca stack TcAT 3
T
TcAT 0
T
TcAT 1
T
TcAT 2
T
TcAT 4
T
TcC augment Tc0
Tc3
Tc1
Tc2
Tc4
Tc5
Tc6
Tcc stack TcAT 0
T
TcAT 3
T
TcAT 1
T
TcAT 2
T
TcAT 4
T
Ka Kr Rta Tka Kc Kr Rtc Tkc Ca Cr Rta Tca Cc Cr Rtc Tcc
CHECK
KA augment K3
K0
K1
K2
K4
K5
K6
KC augment K0
K3
K1
K2
K4
K5
K6
CA augment C3
C0
C1
C2
C4
C5
C6
CC augment C0
C3
C1
C2
C4
C5
C6
Appendix C 156
Kacheck stack KAT 3
T
KAT 0
T
KAT 1
T
KAT 2
T
KAT 4
T
KAT 5
T
KAT 6
T
Kccheck stack KCT 0
T
KCT 3
T
KCT 1
T
KCT 2
T
KCT 4
T
KCT 5
T
KCT 6
T
Cacheck stack CAT 3
T
CAT 0
T
CAT 1
T
CAT 2
T
CAT 4
T
CAT 5
T
CAT 6
T
Cccheck stack CCT 0
T
CCT 3
T
CCT 1
T
CCT 2
T
CCT 4
T
CCT 5
T
CCT 6
T
Results for System switching from ISG as DRIVING pulley to Crankshaft as Drivi ng Pulley
Modified Submatricies for ISG Driving Phase --gt CS Driving Phase
Unit step function to provide shift from crankshaft DRIVING case (ie ISG driven case) to crankshaft DRIVEN
case (ie ISG driving case)
H n( ) 1 n 750if
0 n 750if
lt-- crankshaft DRIVING case (Phase change bw 2 cases occurs when n
reaches start speed)
I11mod n( ) Ic0 0
H n( ) 1if
Ia0 0
H n( ) 0if
I22mod n( )submatrix Ic 1 6 1 6( )
UnitsOf I( )H n( ) 1if
submatrix Ia 1 6 1 6( )
UnitsOf I( )H n( ) 0if
K11mod n( )
Kc0 0
UnitsOf K( )H n( ) 1if
Ka0 0
UnitsOf K( )H n( ) 0if
C11modn( )
Cc0 0
UnitsOf C( )H n( ) 1if
Ca0 0
UnitsOf C( )H n( ) 0if
K22mod n( )submatrix Kc 1 6 1 6( )
UnitsOf K( )H n( ) 1if
submatrix Ka 1 6 1 6( )
UnitsOf K( )H n( ) 0if
C22modn( )submatrix Cc 1 6 1 6( )
UnitsOf C( )H n( ) 1if
submatrix Ca 1 6 1 6( )
UnitsOf C( )H n( ) 0if
K21mod n( )submatrix Kc 1 6 0 0( )
UnitsOf K( )H n( ) 1if
submatrix Ka 1 6 0 0( )
UnitsOf K( )H n( ) 0if
C21modn( )submatrix Cc 1 6 0 0( )
UnitsOf C( )H n( ) 1if
submatrix Ca 1 6 0 0( )
UnitsOf C( )H n( ) 0if
K12mod n( )submatrix Kc 0 0 1 6( )
UnitsOf K( )H n( ) 1if
submatrix Ka 0 0 1 6( )
UnitsOf K( )H n( ) 0if
C12modn( )submatrix Cc 0 0 1 6( )
UnitsOf C( )H n( ) 1if
submatrix Ca 0 0 1 6( )
UnitsOf C( )H n( ) 0if
Appendix C 157
2mod n( ) I22mod n( )1
K22mod n( ) mod n( ) sort eigenvals 2mod n( ) nmod n( )mod n( )
2
EVmodn( ) augmenteigenvec 2mod n( ) mod n( )0
max eigenvec 2mod n( ) mod n( )0
eigenvec 2mod n( ) mod n( )1
max eigenvec 2mod n( ) mod n( )1
eigenvec 2mod n( ) mod n( )2
max eigenvec 2mod n( ) mod n( )2
eigenvec 2mod n( ) mod n( )3
max eigenvec 2mod n( ) mod n( )3
eigenvec 2mod n( ) mod n( )4
max eigenvec 2mod n( ) mod n( )4
eigenvec 2mod n( ) mod n( )5
max eigenvec 2mod n( ) mod n( )5
modeshapesmod n( ) stack nmod n( )T
EVmodn( )
t 0 0001 1
mode1a t( ) EVmod100( )0
sin nmod 100( )0 t mode2a t( ) EVmod100( )1
sin nmod 100( )1 t
mode1c t( ) EVmod800( )0
sin nmod 800( )0 t mode2c t( ) EVmod800( )1
sin nmod 800( )1 t
Pulley responses amp torque requirement for crankshaft amp alternator pulleys pulley1 and 4 respectively
The system equation becomes
I14q14 -double-dot + C1144 q14 -dot + K1144 q14 + C12qm-dot + K12qm = Qc
I2qm-double-dot + C22qm-dot + K22qm + C21q1-dot + K21q1 = 0
Pulley responses
Qm = - [(K22 - 2I2) + jC22 ]-1(K21 + jC21 )Q1
Torque requirement for crank shaft Pulley 1
qc = [(K11 -2I1) + jC11 ]Q1 + (K12 + jC12 )Qm
Torque requirement for alternator shaft Pulley 4
qa = [(K44 -2I4) + jC44 ]Q4 + (K12 + jC12 )Qm
Appendix C 158
Let DRIVING pulley have a unit amplitude 1 = 1 and let the system frequency be calculated based on
engine speed n
n 60 90 6000 n( )4n
60 a n( )
2n Ra0
60 Ra3
mod n( ) n( ) H n( ) 1if
a n( ) H n( ) 0if
Ymod n( ) K22mod n( ) mod n( ) 2 I22mod n( )
j mod n( ) C22modn( )
mmod n( ) Ymod n( )( )1
K21mod n( ) j mod n( ) C21modn( )
Crankshaft amp ISG required torques
Let input from DRIVING pulley be an angular displacement with constant amplitude of angular acceleration
Ac n( ) 650 1 n( )Ac n( )
n( ) 2
Let Qm = QmQ1(n) for n lt 750
and Qm = QmQ4(n) for n gt 750
Aa n( )42
I3 3
1a n( )Aa n( )
a n( ) 2
Qc0 4
qcmod n( ) K11mod n( ) mod n( ) 2
I11mod n( )
j mod n( ) C11modn( )
1 n( ) K12mod n( ) j mod n( ) C12modn( ) mmod n( ) 1 n( )
H n( ) 1if
Qc0 H n( ) 0if
qamod n( ) K11mod n( ) mod n( ) 2
I11mod n( )
j mod n( ) C11modn( )
1a n( ) K12mod n( ) jmod n( ) C12modn( ) mmod n( ) 1a n( ) Qc0
H n( ) 0if
0 H n( ) 1if
Q n( ) 48 n
Ra0
Ra3
48
18000
(ISG torque requirement alternate equation)
Appendix C 159
Dynamic tensioner arm torques
Qtt1mod n( )Kt Kt1
UnitsOf Kt( )j mod n( )
Ct Ct1
UnitsOf Cr( )
mmod n( )4 1 n( )
H n( ) 1if
Kt Kt1
UnitsOf Kt( )j mod n( )
Ct Ct1
UnitsOf Cr( )
mmod n( )4 1a n( )
H n( ) 0if
Qtt2mod n( )Kt Kt2
UnitsOf Kt( )j mod n( )
Ct Ct2
UnitsOf Cr( )
mmod n( )5 1 n( )
H n( ) 1if
Kt Kt2
UnitsOf Kt( )j mod n( )
Ct Ct2
UnitsOf Cr( )
mmod n( )5 1a n( )
H n( ) 0if
Appendix C 160
Dynamic belt span tensions
d n( ) 1 n( ) H n( ) 1if
1a n( ) H n( ) 0if
mod n( )
d n( )
mmod n( ) d n( ) 0 0
mmod n( ) d n( ) 1 0
mmod n( ) d n( ) 2 0
mmod n( ) d n( ) 3 0
mmod n( ) d n( ) 4 0
mmod n( ) d n( ) 5 0
Tm n( ) j n( )Tcc
UnitsOf Tcc( )
Tkc
UnitsOf Tkc( )
mod n( )
H n( ) 1if
j n( )Tca
UnitsOf Tca( )
Tka
UnitsOf Tka( )
mod n( )
H n( ) 0if
Tm n( ) j n( )Tcc
UnitsOf Tcc( )
Tkc
UnitsOf Tkc( )
mod n( )
H n( ) 1if
j n( )Tca
UnitsOf Tca( )
Tka
UnitsOf Tka( )
mod n( )
H n( ) 0if
(tensions for driving pulley belt spans)
Appendix C 161
C3 Static Analysis
Static Analysis using K Tk amp Q matricies amp Ts
For static case K = Q
Tension T = T0 + Tks
Thus T = K-1QTks + T0
Q1 68N m Qt1 0N m Qt2 0N m Ts 300N
Qc
Q4
Q2
Q3
Q5
Qt1
Qt2
Qc
5
2
0
0
0
0
J Qa
Q1
Q2
Q3
Q5
Qt1
Qt2
Qa
68
2
0
0
0
0
N m
cK22mod 900( )( )
1
N mQc A
K22mod 600( )1
N mQa
a
A0
A1
A2
0
A3
A4
A5
0
c1
c2
c0
c3
c4
c5
Tc Tk Ts Ta Tk a Ts
162
APPENDIX D
MATLAB Functions amp Scripts
D1 Parametric Analysis
D11 TwinMainm
The following function script performs the parametric analysis for the B-ISG system with a Twin
Tensioner It calls the function TwinTenStaticTensionm The parametric analysis perturbs a
single input parameter for the called function TwinTenStaticTensionm The main function takes
an initial input value for the Twin Tensioner‟s stiffness parameters Kto Kt1o Kt2o and
geometric parameters D3o D5o X3o Y3o X5o and Y5o An input parameter is allowed to
increment by six percent over a range from sixty percent below its initial value to sixty percent
above its initial value The coordinate parameters are incremented through a mesh of Cartesian
points with prescribed boundaries The TwinMainm function plots the parametric results
______________________________________________________________________________
clc
clear all
Static tension for single tensioner system for CS and Alt driving
Initial Conditions
Kto = 20626
Kt1o = 10314
Kt2o = 16502
D3o = 007240
D5o = 007240
X3o =0292761
Y3o =087
X5o =12057
Y5o =09193
Pertubations of initial parameters
Kt = (Kto-060Kto)006Kto(Kto+060Kto)
Kt1 = (Kt1o-060Kt1o)006Kt1o(Kt1o+060Kt1o)
Kt2 = (Kt2o-060Kt2o)006Kt2o(Kt2o+060Kt2o)
D3 = (D3o-060D3o)006D3o(D3o+060D3o)
D5 = (D5o-060D5o)006D5o(D5o+060D5o)
No of data points
s = 21
T = zeros(5s)
Ta = zeros(5s)
Parametric Plots
for i = 1s
Appendix D 163
[T(i)Ta(i)] = TwinTenStaticTension(Kt(i)Kt1oKt2oD3oD5oX3oY3oX5oY5o)
end
figure
[AXH1H2] = plotyy(Kt()T(1)Kt()Ta(4)plot) hold on
H3 = line(Kt()T(5)ParentAX(1)) hold on
H4 = line(Kt()Ta(3)Parent AX(2)) hold off
set(AXYLimModeautoYTickModeauto)
title(Static Tension vs Coupled Stiffness bw Arms 1 amp 2)
xlabel(Kt (Nmrad))
set(get(AX(1)Ylabel)StringCS Span Tension (N))
set(get(AX(2)Ylabel)StringISG Span Tension (N))
set(H1LineStyle-Colorb)
set(H2LineStyle-Colorg)
set(H3LineStyleColorget(H1Color))
set(H4LineStyleColorget(H2Color))
legend([H1H2H3H4]CS Tautest SpanISG Tautest SpanCS Slackest SpanISG
Slackest SpanLocationBest)
for i = 1s
[T(i)Ta(i)] = TwinTenStaticTension(KtoKt1(i)Kt2oD3oD5oX3oY3oX5oY5o)
end
figure
[AXH1H2] = plotyy(Kt1()T(1)Kt1()Ta(4)plot) hold on
H3 = line(Kt1()T(5)ParentAX(1)) hold on
H4 = line(Kt1()Ta(3)Parent AX(2)) hold off
set(AXYLimModeautoYTickModeauto)
title(Static Tension vs Tensioner Arm 1 Stiffness)
xlabel(Kt1 (Nmrad))
set(get(AX(1)Ylabel)StringCS Span Tension (N))
set(get(AX(2)Ylabel)StringISG Span Tension (N))
set(H1LineStyle-Colorb)
set(H2LineStyle-Colorg)
set(H3LineStyleColorget(H1Color))
set(H4LineStyleColorget(H2Color))
legend([H1H2H3H4]CS Tautest SpanISG Tautest SpanCS Slackest SpanISG
Slackest SpanLocationBest)
for i = 1s
[T(i)Ta(i)] = TwinTenStaticTension(KtoKt1oKt2(i)D3oD5oX3oY3oX5oY5o)
end
figure
[AXH1H2] = plotyy(Kt2()T(1)Kt2()Ta(4)plot) hold on
H3 = line(Kt2()T(5)ParentAX(1)) hold on
H4 = line(Kt2()Ta(3)Parent AX(2)) hold off
set(AXYLimModeautoYTickModeauto)
title(Static Tension vs Tensioner Arm 2 Stiffness)
xlabel(Kt2 (Nmrad))
set(get(AX(1)Ylabel)StringCS Span Tension (N))
Appendix D 164
set(get(AX(2)Ylabel)StringISG Span Tension (N))
set(H1LineStyle-Colorb)
set(H2LineStyle-Colorg)
set(H3LineStyleColorget(H1Color))
set(H4LineStyleColorget(H2Color))
legend([H1H2H3H4]CS Tautest SpanISG Tautest SpanCS Slackest SpanISG
Slackest SpanLocationBest)
for i = 1s
[T(i)Ta(i)] = TwinTenStaticTension(KtoKt1oKt2oD3(i)D5oX3oY3oX5oY5o)
end
figure
[AXH1H2] = plotyy(D3()T(1)D3()Ta(4)plot) hold on
H3 = line(D3()T(5)ParentAX(1)) hold on
H4 = line(D3()Ta(3)Parent AX(2)) hold off
set(AXYLimModeautoYTickModeauto)
title(Static Tension vs Tensioner Pulley 1 Diameter)
xlabel(D3 (m))
set(get(AX(1)Ylabel)StringCS Span Tension (N))
set(get(AX(2)Ylabel)StringISG Span Tension (N))
set(H1LineStyle-Colorb)
set(H2LineStyle-Colorg)
set(H3LineStyleColorget(H1Color))
set(H4LineStyleColorget(H2Color))
legend([H1H2H3H4]CS Tautest SpanISG Tautest SpanCS Slackest SpanISG
Slackest SpanLocationBest)
for i = 1s
[T(i)Ta(i)] = TwinTenStaticTension(KtoKt1oKt2oD3oD5(i)X3oY3oX5oY5o)
end
figure
[AXH1H2] = plotyy(D5()T(1)D5()Ta(4)plot) hold on
H3 = line(D5()T(5)ParentAX(1)) hold on
H4 = line(D5()Ta(3)Parent AX(2)) hold off
set(AXYLimModeautoYTickModeauto)
title(Static Tension vs Tensioner Pulley 2 Diameter)
xlabel(D5 (m))
set(get(AX(1)Ylabel)StringCS Span Tension (N))
set(get(AX(2)Ylabel)StringISG Span Tension (N))
set(H1LineStyle-Colorb)
set(H2LineStyle-Colorg)
set(H3LineStyleColorget(H1Color))
set(H4LineStyleColorget(H2Color))
legend([H1H2H3H4]CS Tautest SpanISG Tautest SpanCS Slackest SpanISG
Slackest SpanLocationBest)
Mesh points
m = 101
n = 101
Appendix D 165
T = zeros(5nm)
Ta = zeros(5nm)
[ixxiyy] = meshgrid(1m1n)
minX3 = 0260200
maxX3 = 0317677
minY3 = -0056640
maxY3 = 0228456
midX3 = 0311641
X3 = minX3 + (ixx-1)(maxX3-minX3)(m-1)
Y3 = minY3 + (iyy-1)(maxY3-minY3)(n-1)
for i = 1n
for j = 1m
if ((X3(ij)lt midX3)ampamp(Y3(ij)gt=(sqrt((0087945^2)-((X3(ij)-0224)^2)-
006395)))ampamp(Y3(ij)lt=(-1sqrt(((00703^2)-((X3(ij)-
024759)^2)))+016664)))||((X3(ij)gt=midX3)ampamp(Y3(ij)gt=(35548X3(ij)-
11134868))ampamp(Y3(ij)lt=(-1(sqrt(((00703^2)-((X3(ij)-024759)^2))))+016664))) mx+b
lt= y lt= circle4
[T(ij)Ta(ij)] =
TwinTenStaticTension(KtoKt1oKt2oD3oD5oX3(ij)Y3(ij)X5oY5o)
else
T(ij) = zeros(511)
Ta(ij) = zeros(511)
end
end
end
figure
Z1 = squeeze(T(1))
surf(X3Y3real(Z1))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(CS DRIVING Tautest Span Tension vs Tensioner Pulley 1 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(CS Span Tension (N))
figure
Z5 = squeeze(T(5))
surf(X3Y3real(Z5))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(CS DRIVING Slackest Span Tension vs Tensioner Pulley 1 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
Appendix D 166
zlabel(CS Span Tension (N))
figure
Za4 = squeeze(Ta(4))
surf(X3Y3real(Za4))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(ISG DRIVING Tautest Span Tension vs Tensioner Pulley 1 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(ISG Span Tension (N))
figure
Za3 = squeeze(Ta(3))
surf(X3Y3real(Za3))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(ISG DRIVING Slackest Span Tension vs Tensioner Pulley 1 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(ISG Span Tension (N))
minX5 = -0037093
maxX5 = 0212509
minY5 = 00362
maxY5 = 0228456
midX5a = 0131965
midX5b = 017729
X5 = minX5 + (ixx-1)(maxX5-minX5)(m-1)
Y5 = minY5 + (iyy-1)(maxY5-minY5)(n-1)
for i = 1n
for j = 1m
if
(X5(ij)ltmidX5a)ampamp(Y5(ij)lt=(0386X5(ij)+0146468))ampamp(Y5(ij)gt=(sqrt((0136525^2)-
(X5(ij)^2))))
[T(ij)Ta(ij)] =
TwinTenStaticTension(KtoKt1oKt2oD3oD5oX3oY3oX5(ij)Y5(ij))
elseif
((X5(ij)gt=midX5a)ampamp(X5(ij)ltmidX5b))ampamp(Y5(ij)gt=00362)ampamp(Y5(ij)lt=(0386X5(ij)+0
146468))
[T(ij)Ta(ij)] =
TwinTenStaticTension(KtoKt1oKt2oD3oD5oX3oY3oX5(ij)Y5(ij))
elseif (X5(ij)gt=midX5b)ampamp(Y5(ij)gt=(sqrt((00703^2)-(((X5(ij)-
024759)^2)))+016664))ampamp(Y5(ij)lt=(0386X5(ij)+0146468))
Appendix D 167
[T(ij)Ta(ij)] =
TwinTenStaticTension(KtoKt1oKt2oD3oD5oX3oY3oX5(ij)Y5(ij))
else
T(ij) = zeros(511)
Ta(ij) = zeros(511)
end
end
end
figure
Z1 = squeeze(T(1))
surf(X5Y5real(Z1))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(CS DRIVING Tautest Span Tension vs Tensioner Pulley 2 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(CS Span Tension (N))
figure
Z5 = squeeze(T(5))
surf(X5Y5real(Z5))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(CS DRIVING Slackest Span Tension vs Tensioner Pulley 2 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(CS Span Tension (N))
figure
Za4 = squeeze(Ta(4))
surf(X5Y5real(Za4))
ZLim([50 500])
axis tight
colormap jet
colorbar
title(ISG DRIVING Tautest Span Tension vs Tensioner Pulley 2 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(ISG Span Tension (N))
figure
Za3 = squeeze(Ta(3))
surf(X5Y5real(Za3))
ZLim([50 500])
axis tight
Appendix D 168
colormap jet
colorbar
title(ISG DRIVING Slackest Span Tension vs Tensioner Pulley 2 Coordinates)
xlabel(X-coordinate (m))
ylabel(Y-coordinate (m))
zlabel(ISG Span Tension (N))
D12 TwinTenStaticTensionm
The function TwinTenStaticTensionm simulates the static model of the B-ISG system with a
Twin Tensioner This function returns 3 vectors the static tension of each belt span in the
crankshaft- and ISG-driving phases of operation and the angle of displacement of each rigid
body in the ISG- driving phase It takes the input parameters kt kt1 kt2 for the tensioner arm
stiffness D3 and D5 for the tensioner pulley diameters and X3Y3 X5 and Y5 for the tensioner
arm pulley coordinates The function is called in the parametric analysis solution script
TwinMainm and in the optimization solution script OptimizationTwinm
D2 Optimization
D21 OptimizationTwinm
The following script is for the main function OptimizationTwinm It performs an optimization
search for the B-ISG system with a Twin Tensioner It takes an input for a parameter vector
containing values for the design variables The program calls the objective function
objfunTwinm and the constraint function confunTwinm The program can perform a genetic
algorithm (GA) optimization search or a hybrid GA optimization that includes a localized search
The optimal solution vector corresponding to the design variables and the optimal objective
function value is returned The program inputs the optimized values for the design variables into
TwinTenStaticTensionm This called function returns the optimized static state of tensions for
the crankshaft- and ISG- driving phases and for the angle of displacement of the rigid bodies in
the ISG driving phase
______________________________________________________________________________
clc
clear all
Initial values for variables
Kto = 20626
Kt1o = 10314
Kt2o = 16502
X3o = 0292761
Y3o = 0087
X5o = 012057
Appendix D 169
Y5o = 009193
w0 =[Kto Kt1o Kt2o X3o Y3o X5o Y5o] Start Point (row vector)
Variable ranges
minKt = Kto - 1Kto
maxKt = Kto + 1Kto
minKt1 = Kt1o - 1Kt1o
maxKt1 = Kt1o + 1Kt1o
minKt2 = Kt2o - 1Kt2o
maxKt2 = Kt2o + 1Kt2o
minX3 = 0260200
maxX3 = 0317677
minY3 = -0056640
maxY3 = 0228456
minX5 = -0037093
maxX5 = 0212509
minY5 = 00362
maxY5 = 0228456
ObjectiveFunction = objfunTwin
nvars = 7 Number of variables
ConstraintFunction = confunTwin
Uncomment next two lines (and comment the two functions after them) to use GA algorithm
for optimization
options = gaoptimset(InitialPopulationw0PopInitRange[minKt minKt1 minKt2 minX3
minY3 minX5 minY5 maxKt maxKt1 maxKt2 maxX3 maxY3 maxX5
maxY5]PopulationSize100Displayfinal)
[wfvalexitflagoutput] =
ga(ObjectiveFunctionnvars[][][][][][]ConstraintFunctionoptions)
fminconOptions = optimset(DisplayiterLargeScaleoff) Largescale off since gradient not
provided
options = gaoptimset(InitialPopulationw0PopInitRange[minKt minKt1 minKt2 minX3
minY3 minX5 minY5 maxKt maxKt1 maxKt2 maxX3 maxY3 maxX5
maxY5]PopulationSize100HybridFcnfmincon fminconOptions)
[wfvalexitflagoutput] = ga(ObjectiveFunctionnvars[][][][][][]ConstraintFunctionoptions)
[TTaThetaDegA] = TwinTenStaticTension(w(1)w(2)w(3)w(4)w(5)w(6)w(7))
D22 confunTwinm
The constraint function confunTwinm is used by the main optimization program to ensure
input values are constrained within the prescribed regions The function makes use of inequality
constraints for seven constrained variables corresponding to the design variables It takes an
input vector corresponding to the design variables and returns a data set of the vector values that
satisfy the prescribed constraints
Appendix D 170
D23 objfunTwinm
This function is the objective function for the main optimization program It outputs a value for
a weighted objective function or a non-weighted objective function relating the optimization of
the static tension The program takes an input vector containing a set of values for the design
variables that are within prescribed constraints The description of the function is similar to
TwinTenStaticTensionm but differs in the fact that it only returns a scalar value which is the
value of the objective function
171
VITA
ADEBUKOLA OLATUNDE
Email adebukolaolatundegmailcom
Adebukola Olatunde is a graduate research student at the University of Toronto in Toronto
Ontario Canada She obtained a Bachelor‟s Degree in Mechanical Engineering from McMaster
University in Hamilton Ontario Canada in 2002 Upon graduation she pursued a graduate
degree in mechanical engineering at the University of Toronto with a specialization in
mechanical systems dynamics and vibrations and environmental engineering In September
2008 she completed the requirements for the Master of Applied Science degree in Mechanical
Engineering She has held the position of teaching assistant for undergraduate courses in
dynamics and vibrations Adebukola has completed course work in professional education She
is a registered member of professional engineering organizations including the Professional
Engineer‟s of Ontario Engineer-in-Training program the Canadian Society of Mechanical
Engineers and the National Society of Black Engineers She intends to practice as a professional
engineering consultant in mechanical design
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