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Dynamics of Tapered Roller Bearings:
Modelling and Analysis
Gil Nogueira dos Santos
Thesis to obtain the Master in Science Degree in
Mechanical Engineering
Supervisor: Prof. Jorge Alberto Cadete Ambrósio
Examination Committee
Chairperson: Prof. João Orlando Marques Gameiro Folgado
Supervisor: Prof. Jorge Alberto Cadete Ambrósio
Member of the Committee: Prof. Marta Isabel Pimenta Verdete da Silva Carvalho
November 2017
i
ACKNOWLEDGMENTS
To my supervisor, Prof. Jorge Ambrósio, I would like to express my profound gratitude for
all the guidance and teaching that he provided throughout this work. His knowledge and constant
teaching and were crucial to keep me motivated and persistently working to overcome many
obstacles.
To my colleagues from the research group, I thank you all the patience and help provided.
Almost as a secondary supervisor team, you followed my work and without a doubt played a
crucial role in the soft skills that helped me create better content. To Hugo Magalhães, I express
my gratitude with the help and experience in Contact Detection, Force Models and SAGA
visualization tools, to Pedro Antunes, for his support in most computational aspects, and finally
to João Costa, helping me with mathematical problems and in the making of this document.
To Diana, for the constant motivation and believe in me. Lastly, to my friend and family,
for always being there for me when I most needed.
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ABSTRACT
The understanding of the dynamic performance of roller bearings, used in railway vehicles for
instance, is fundamental to support the evaluation of the bearings performance via monitoring
systems. The vibration output of the axleboxes is the measurable outcome of the bearing dynamic
response, under operating conditions, that is characterized in this work. The main goal of this
work is to develop a dynamic analysis tool, referred to as BearDyn, in MATLAB®, able to handle
models representative of actual railway axle bearings, by using a multibody formulation to
describe the mechanical elements of the bearing and their interactions, using realistic bearing
geometric data is obtained by precise measurements of tapered bearings. Online contact
detection is studied in steady state general geometries. A dynamic analysis program of two
colliding bodies is developed to validate the contact detection methods. In BearDyn, continuous
contact force models based on the Hertz elastic contact theory and modified according to
experimental evidence describe the interactions between the elements. Tribological lubrication
models are applied to describe the tangential forces in the presence of lubricant. Finally, the
BearDyn is demonstrated in the framework of realistic train operations that include the bearing
loading due to the wheel-rail contact and the supporting mechanisms. The bearing dynamic
response is obtained in terms of forces, kinematic quantities and different interaction measures,
in the time and frequency domain. Two visualization tools are developed using MATLAB® and
SAGA, to visually identify the geometries used and contact points locations.
Keywords
Tapered Roller Bearings
Multibody Dynamics
Contact Detection
Railway Dynamics
Hertz Contact
Elastohydrodynamic lubrication
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RESUMO
A resposta dinâmica de rolamentos usados em veículos ferroviários é o principal objetivo do
projeto MAXBE, que motiva o trabalho aqui apresentado. A monitorização de rolamentos via
sistemas adequados usa a informação em termos de vibração para avaliar a condição dos seus
componentes mecânicos. O principal objetivo deste trabalho é desenvolver uma ferramenta de
analise dinâmica, referida como BearDyn, em MATLAB®, capaz de avaliar o movimento de
modelos representativos de rolamentos usados em aplicações ferroviárias, através de uma
formulação multicorpo para descrever os elementos mecânicos do rolamento e as suas
interações. Métodos de deteção de contacto são estudados para geometrias em estado
estacionário. Um programa de análise dinâmica com dois corpos é desenvolvido para validar os
métodos de deteção de contacto em corpos com movimento. No BearDyn, modelos de força de
contacto baseadas na teoria de contacto elástico de Hertz são utilizados. Modelos tribológicos
de lubrificação são aplicados para descrever as forças tangenciais na presença de lubrificante.
Por fim, BearDyn é testado para condições ferroviárias realísticas, considerando a carga
proveniente do contacto roda-carril e mecanismos de suporte. A resposta dinâmica é obtida em
termos de forças, quantidades cinemáticas e diferentes medidas de interações no domínio do
tempo e frequência. Duas ferramentas de visualização são desenvolvidas utilizando MATLAB® e
SAGA, para observar as geometrias usadas e localização dos pontos de contacto.
Palavras-Chave
Rolamentos Cónicos
Dinâmica Multicorpo
Detecção de Contacto
Dinamica ferroviária
Contacto Hertziano
Lubrificação elastohidrodinâmica
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TABLE OF CONTENTS
Acknowledgments……………………………………………………………………………..i
Abstract…………………………………………………………………………………………iii
Resumo…………..……………………………………………………………………………...v
Table of Contents……………………………………………………………………….….…vii
List of Figures………………………………………………………………………………….ix
List of Tables…………………………………………………………………………………..xii
List of Symbols……………………………………………………………………………….xiii
1. Introduction .......................................................................................................... 1
1.1. State of Art ................................................................................................... 4
1.2. Work Description ......................................................................................... 5
2. Dynamic Analysis ................................................................................................. 7
2.1. Multibody Dynamics Formulation ................................................................ 7
3. Contact Modelling in a Dynamic Analysis ....................................................... 11
3.1. Contact Points Detection ........................................................................... 11
3.2. Solution of a system of nonlinear equations .............................................. 17
3.3. Contact detection demonstration with simple geometries ......................... 20
3.4. Dynamic Response for Demonstration Cases .......................................... 26
3.5. Summary and conclusions ........................................................................ 32
4. Formulation for the Bearing Contact Detection .............................................. 33
4.1. Roller Bearing Contacts ............................................................................. 33
4.2. Surface definition ....................................................................................... 34
4.3. Raceways to roller contacts....................................................................... 34
4.4. Flange to roller contacts ............................................................................ 36
4.5. Cage to roller contacts ............................................................................... 40
5. Contact Forces ................................................................................................... 45
5.1. Normal contact forces ................................................................................ 45
5.2. Tangencial forces ...................................................................................... 51
6. Computational Implementation ........................................................................ 55
6.1. Contact Implementation ............................................................................. 56
6.2. Time integration method ............................................................................ 59
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6.3. Verification Methods .................................................................................. 60
7. Results and Discussion ..................................................................................... 63
7.1. Contact detection with visualization tool .................................................... 63
7.2. Simulation Conditions of the Dynamic Analysis ........................................ 64
7.3. Results of the Dynamic Analysis ............................................................... 65
8. Conclusions ........................................................................................................ 73
8.1. Future work ................................................................................................ 74
References………………………………………………………………………….…….……77
Appendix A - Input Data for Bearing Model………………………………………………80
Appendix B - Visualization Tools Developed…….………………………………………86
ix
LIST OF FIGURES
Figure 1: Typical elements in a roller bearing ............................................................................... 1
Figure 2: Axlebox for railway applications with (a) Spherical, (b) Cylindrical and (c)
Tapered Roller Bearings (Figure adapted from [3]) ....................................................... 2
Figure 3: Double row configuration for Tapered Roler Bearings: (a) back-to-back and (b)
face-to-face .................................................................................................................... 2
Figure 4: Extruded view of an Axlebox with Tapered Roller Bearings (adapted from [3]) ............ 3
Figure 5: Overview of BearDyn Structure .................................................................................... 6
Figure 6: Definition of the Cartesian coordinates for a rigid body ................................................. 8
Figure 7: (a) Representation of a system of uncontrained bodies; (b) Free-body diagram
of an uncontrained body ................................................................................................ 9
Figure 8: Candidates to contact points between two parametric surfaces (adapted from
[27]) .............................................................................................................................. 12
Figure 9: Contact Points Detected. (a) Correct pair of points; (b) Incorrect pair of points .......... 13
Figure 10: (a) Roller Approaching a surface, (b) Roller penetrating a surface ........................... 14
Figure 11: Roller divided in slices ................................................................................................ 15
Figure 12: Geometric relations between a point P in a circumference and a point Q in a
generic surface ............................................................................................................ 15
Figure 13: Contacting points between a circle of the roller and a line ........................................ 16
Figure 14: Geometric relations describing the positions of point P and Q in two circles ........... 21
Figure 15: Possible solutions to the system of equations that defines the contact
detection....................................................................................................................... 22
Figure 16: Representation of the geometries generated with the local axis and: (a)
Incorrect Pair of Points detected; (b) correct pair of contact points ............................. 22
Figure 17: MATLAB® representation of two circles with local axis and the correspondent
contact points ............................................................................................................... 23
Figure 18: (a) Auxiliary vector to define (b) the three vectors in point Q in a sphere .................. 24
Figure 19: MATLAB® representation of two spheres and correspondent contact point .............. 25
Figure 20 - Geometric relations between: (a) point P in a circle; (b) point Q in a cylinder.......... 26
Figure 21: MATLAB® representation of a circle, a cylinder surface and the contact point ......... 26
Figure 22: DAP Coin scheme ..................................................................................................... 27
Figure 23: Initialization of DAP Coin ........................................................................................... 27
Figure 24: Slice method applied to a cylindrical roller ................................................................. 28
Figure 25: Velocity vectors in the contact points necessary to evaluate the normal and
tangential forces ........................................................................................................... 29
Figure 26: Kelvin Voigt normal force versus the relative penetration depth and relative
normal velocity of the colliding bodies ......................................................................... 30
Figure 27: Tangencial force versus tangencial velocity using the Threfall model ....................... 31
Figure 28: Frames from the DAP Coin analysis .......................................................................... 31
x
Figure 29: 3D view of the starting point for the dynamic coin and roller with the local axis
for both bodies ............................................................................................................. 32
Figure 30: Contacts considered in BearDyn .............................................................................. 33
Figure 31: Axisymmetric surface obtained as the sweep of a line about an axis: (a)
Parametric representation of the line and its tangential and normal vectors in
point Q; (b) Surface of revolution, with the sweep angle 𝜃2 and the surface
defining vectors at point Q ........................................................................................... 34
Figure 32: Typical tapered roller bearing with a highlighted cross section ................................. 35
Figure 33: Contact Point Q and surface normal and tangent vectors in a conical surface
for: (a) contact point in the body fixed 𝜉𝜂 plane; (b) external contact, as in the
inner raceway; (c) internal contact, as in the outer raceway ....................................... 35
Figure 34: (a) Contact geometry between rollers and flanges, in which point Q refers to
the contact point for a tapered roller; (b) Detailed geometry of the flanges for
tapered roller bearings inner raceway ......................................................................... 36
Figure 35: Contact detection between left flange and left circular landmark .............................. 37
Figure 36: (a) Typical contact between spherical cap and conical surface; (b) Contact
point in the local plane for the spherical cap of tapered roller bearing end ................. 38
Figure 37: Contact detection between a spherical cap and a conical flange .............................. 38
Figure 38: Roller side view with different spherical cap radius: (a) Previous value
implemented; (b) new radius applied to BearDyn....................................................... 39
Figure 39: Roller with basic dimensions and necessary angles to project 𝑅𝑒𝑟 ........................... 39
Figure 40: Contact of circle with line, as in the contact between the roller and the side of
the pocket .................................................................................................................... 40
Figure 41: Contact detection between roller and right side of the cage pocket .......................... 41
Figure 42: Contact detection between spherical cap in roller and large top of the cage
pocket........................................................................................................................... 43
Figure 43: Contact detection between left circumference in roller and small top of the
cage pocket .................................................................................................................. 43
Figure 44: Contact patches of Hertzian contact force models: (a) Elliptical contact; (b)
Point Contact; (c) Elliptical Contact; (d) Line Contact (Adapted from [21]) ................. 46
Figure 45: (a) Geometry of contacting elastic solids; (b) Stress distribution and patch
geometry (Adapted from [21]) ...................................................................................... 47
Figure 46: Ideal Line Contact between two bodies ..................................................................... 49
Figure 47: Geometric radius in a line contact .............................................................................. 50
Figure 48: Types of contact ( Boundary lubricant layer, Lubricant): (a) Dry
contact; (b) Boundary mode; (c) Mixed mode; (d) Full fluid mode (Adapted from
[21]) .............................................................................................................................. 51
Figure 49: Scheme of BearDyn code main structure ................................................................. 55
Figure 50: F_Contact function structure ................................................................................... 57
Figure 51: CostFunction function scheme .............................................................................. 58
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Figure 52: Visualization Tool developed in MATLAB® ................................................................ 60
Figure 53: Roller bearing displayed in SAGA ............................................................................... 61
Figure 54: SAGA representation of the contact points in one roller for the cage, top flange
and inner raceway ........................................................................................................ 61
Figure 55: Representation of the geometries used in BearDyn: (a) Roller geometries; (b)
Inner raceway and cage geometries ............................................................................ 63
Figure 56: Roller with all the geometries used in contact detection represented (with
exception to the Outer Raceway) with the contact points for: (a) Inner Raceway
and Flanges; (b) Cage Sides and Cage Tops ............................................................. 64
Figure 57: Initial position of the roller bearing represented in SAGA with contact points
from Body 15 ................................................................................................................ 65
Figure 58: c .................................................................................................................................. 66
Figure 59: Contact Points with forces being applied in the Outer Raceway and Right
Flange at t=0.02s ......................................................................................................... 66
Figure 60: Contact Forces in simulation t=0.03s from Body 14: (a) all the timesteps; (b)
from t=0.023s until t=0.03s .......................................................................................... 67
Figure 61: Contact forces applied to: (a) spherical large end of the roller; (b) Outer
raceway ........................................................................................................................ 67
Figure 62: Contact Forces in simulation t=0.1s without cage from: (a) Body 14; (b) Outer
Raceway ...................................................................................................................... 68
Figure 63: Final frame in the simulation without cage ................................................................. 68
Figure 64: (a) Representation bearing at timestep t=0.03s; (b) force introduced to
maintain the alignment ................................................................................................. 70
Figure 65: Initial and final frames in the simulation for t=0.2s ..................................................... 70
Figure 66: Contact Forces in the Inner Raceway during the simulation ..................................... 71
Figure 67: Frequency response of total forces acting on the inner raceway, resulting from
the simulation of a tapered roller bearing with complete contact detection, load
of 5kN and tangential forces applied, from t=0.01s ..................................................... 71
Figure 69: Radius in the roller direction of two contacting surfaces ............................................ 83
Figure 70: Scheme of the video maker for DAP Coin and DAP Roller ....................................... 85
Figure 71: Scheme of the visualization tool ................................................................................ 85
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LIST OF TABLES
Table 1: Results from contact point detection for two random circles......................................... 22
Table 2: Results from contact point detection for two random spheres ...................................... 25
Table 3: Results from contact point detection for circle to cylinder contact ................................ 26
Table 4: Stiffness for point contact used in BearDyn ................................................................. 48
Table 5: Stifness for line contact used in BearDyn .................................................................... 50
Table 6: Flags DataBase for vectors from geometries implemented .......................................... 58
Table 7: Time elapsed for the same t=0.03s simulation using three different methods to
solve contact ................................................................................................................ 68
xiii
LIST OF SYMBOLS
Convention
a, A, α Scalar a Vector A Matrix
Overscores
a First time derivative a Second time derivative a Skew-symmetric matrix
Superscripts
0a Initial condition
Ta Matrix or vector transpose
1a Inverse a Vector expressed in the body-fixed reference frame *
a Vector expressed in the ( ) plane
Subscripts
ia
Refers to rigid body i in a system
ja
Refers to rigid body j in a system
ab Refers to the roller in a roller bearing system
,c cagea a
Refers to the cage in a roller bearing system
innera
Refers to the inner raceway in a roller bearing system
outera
Refers to the outer raceway in a roller bearing system
na
Refers to normal force
Pa
Refers to point P on body i
Qa
Refers to point Q on body j
sa
Refers to the number of the slice on a roller
ta
Refers to tangential force
gravitationala
Force caused by gravity
contacta
Force caused by contact
gyroscopica
Force caused by gyroscopic forces
loada
Force caused by external load
Latin Symbols
a,b Semi-axes of the contact ellipse, for point contact, or contact patch, for line contact
a,b,n,Y
Constants for the calculation of normal contact force, for line contact
A Generic transformation matrix
2A
Rotation matrix
b Binormal vector C Generic clearance
,d D Generic diameter
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d Distance vector
0 1 2 3, , ,e e e e
Euler parameters
E Young’s modulus E Equivalent modulus of elasticity *E Composite modulus of elasticity
f
Generic force
f Vector of generic forces g
Vector of generalized forces G Material parameter h Generic height
ch Lubricant film central thickness
isoh Isothermal central lubricant film thickness
minh
Minimum lubricant film thickness
H Film thickness parameter
I Inertia J Inertia tensor
fK
Thermal conductivity
ptK
Contact stiffness
,l L Generic length
efL
Effective contact length, for line contact
L Auxiliary matrix, function of Euler parameters m Mass of the rigid body
mi
Sum of all moments respective to body i
M Global mass matrix nb Total number of bodies in the system nc Total number of coordinates ,n m Constants for calculations with norm ASTM D341
n Normal vector
slN
Number of slices in which the roller is divided
N Diagonal matrix of masses O Origin p
Pressure p
Orientation of the rigid body built with Euler parameters p Time derivatives of Euler parameters
P Generic point on body i q
Normal compressive load q
Vector of generalized coordinates q Vector of generalized velocities
*q
Vector of generalized velocities in Euler-parameter space
q Vector of generalized accelerations Q Generic point on body j ,r R Generic radius
, , ,ax ay bx byr r r r
Characteristic radii of contacting surfaces
r Translational position vector
r Translational velocity vector s Slip ratio ,s t Constants for the calculation of the starvation factor
s Position vector
S Mean dimensionless shear stress t Time t Tangent vector û Average velocity of the contacting surfaces u Sliding velocity of the bodies in the contact point U Speed parameter
xv
v Generic linear velocity
v Generic linear velocity vector
W Generic width, or load parameter for the calculation of the lubricant film thickness
cx
Dimensionless length of the EHL contact area
, ,x y z Global coordinates , ,X Y Z Global reference frame
y, y Auxiliary vectors used in the integration process
Greek symbols
1 Pressure coefficient of viscosity
r Angle of the raceway surface
m Pitch angle of a roller in the pitch circle
Angular position of a contact point on a cage pocket top, for contact detection
Right-hand-side of acceleration equations Penetration, or interference, during contact
t Viscosity temperature index
0 Lubricant viscosity at reference conditions
Lubricant viscosity
Angular position of a contact point on a circle of the roller, for contact detection
2 Angular position of a contact point on a surface of revolution, for contact detection
, , , Constants for the calculation of the contact stiffness, for point contact Vector of Lagrange multipliers Lubricant film parameter Equivalent friction coefficient bd Boundary mode equivalent friction coefficient
fm Full-film mode equivalent friction coefficient
*
Viscosity of the lubricant for the calculation of the full-film equivalent friction coefficient
Poisson’s ratio; Kinematic viscosity
, 1 2 Roughness of contacting surfaces
avg Mean normal stress
max
Maximum stress
p Constant for the calculation of the contact patch dimensions, for line contact
iso Dimensionless shear velocity of the lubricant
0 Lubricant characteristic stress
Angular position of a contact point on a raceway of the spherical bearing, for contact
S Starvation factor
T Thermal reduction factor
Angular position of a contact point on a spherical cap
Vector of kinematic position constraints
Modified factor for side-leakage
Parameter for the calculation of the full-film equivalent friction coefficient Angular velocity
Angular velocity vector
Angular acceleration vector
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* * *
0
1 2 3
1 2 3
, , ,
, , ,
, ,
V V V
B B B
Experimental constants for the calculation of the full-film equivalent friction coefficient
, , Local, or body-fixed, reference frame
Abbreviations
MAXBE Interoperable Monitoring, Diagnosis and Maintenance Strategies for Axle Bearings
BearDyn Bearing Dynamic Analysis Program ODE Ordinary Differential Equation SAGA System Animation for Graphical Analysis EHL Elastohydrodynamic Lubrication
1
1. Introduction
Transportation has many modes which have developed over time. Since late 18th century,
Railway systems have played a significant role in the human development. They provided the first
non-animal powered transportation method and due to the technological advances, trains have
developed into a fast, comfortable and safe way to travel. One of the subsystems responsible for
the behaviour of the train is the axle bearing. If the component is not working properly it will have
a direct impact in the performance and safety of the train. For this reason, axle bearing damage
has been the object of intense attention by railway operators and transportation authorities,
leading to a variety of approaches and systems in place to deal with the problem.
In an attempt to standardize the condition monitoring and early diagnosis of axle bearings
by technology integration, a collaborative project between several European organizations, which
uses the acronym MAXBE was created. The current European documents do not present any
guidelines for the maintenance management of the rolling stock with regards to axle bearings,
which make the association of monitoring, diagnosis and maintenance an interesting approach.
The work here presented is within the project objectives and gets closer to a realistic dynamic
simulation of tapered roller bearings.
The purpose of a bearing is to support a load while permitting relative motion between
two elements of a machine. The term rolling contact bearings refers to the wide variety of bearings
that use spherical balls or some other type of roller between the stationary and the moving
elements allowing for a reduced friction between moving parts. As seen in Figure 1, the
components of a typical rolling contact bearing are the inner raceway, the outer raceway, the
rolling elements and usually a cage, which maintains the spacing between the rolling elements
and prevents contact between them [1].
Figure 1: Typical elements in a roller bearing
The most common type of bearing supports a rotating shaft, resisting purely radial loads
or a combination of radial and axial (thrust) loads. In Railway applications, the rolling stock axle
bearings are subjected to radial impact loads caused by rail joints, switches and sometimes wheel
Outer Raceway
CageRollers
Inner Raceway
2
flats, as well as to the static and dynamic radial loads of vehicle weight. They are responsible to
resist hunting axial loads generated by lateral movement as trains run on curved rails or due to
snaking motion. All these loads together form complex combinations that act on axle bearings.
Therefore, axle bearings must be designed on the basis of not only dimensional requirements of
the axle journal and bearing box geometry, but also to support the complex load conditions. In
local trains, the principal requirement is to have an as low as possible floor design and multiple
units. Therefore, the bearings used must be particularly compact design [2]. Figure 2 shows three
axleboxes with different bearing types. All of them are reliable and provide good operational
security, whether the vehicle is a tram or a high speed train [2], [3].
(a) (b) (c)
Figure 2: Axlebox for railway applications with (a) Spherical, (b) Cylindrical and (c) Tapered Roller Bearings (Figure adapted from [3])
This work focus on axleboxes with tapered roller bearings. These systems can carry
combinations of large radial and axial forces. Because of the difference between the inner and
outer raceway contact angles, there is a force component that drives the tapered rollers against
the guide flange. This leads to a relatively large sliding friction generated at this flange, meaning
that the bearing requires special attention with respect to cooling and lubrication. To achieve a
greater radial load-carrying capacity and to eliminate problems of axial adjustment due to distance
between bearings, tapered roller bearings may be combined, as shown in Figure 3, to form a two-
row bearing [4].
(a) (b)
Figure 3: Double row configuration for Tapered Roler Bearings: (a) back-to-back and (b) face-to-face
There are two types of duplex arrangements of tapered roller bearings: back-to-back,
Figure 3 (a), and face-to-face, Figure 3 (b). For rolling stock axle applications where heavy
ep
LiLi Si LiLi Si Li LiSi
3
moment loads are expected, the back-to-back arrangement, which provides a greater distance
between load centres, is preferable. When the rollers are loaded, part of their load is transferred
to the large rib of the inner ring. The rollers maintain sliding contact with the rib while are being
guided. This results in the friction force of these bearings, which is higher than that of cylindrical
bearings. Recently, improvements in surface roughness and contact geometry have virtually
eliminated the friction problems associated with tapered roller bearings for axles [2], [5]. Figure 4
shows the extruded view of an axlebox with double-row tapered roller bearings in a back-to-back
configuration.
Figure 4: Extruded view of an Axlebox with Tapered Roller Bearings (adapted from [3])
A rolling element has a finite life. The bearing will normally fail due to fatigue, caused by
the high cyclic stresses between the rolling elements and raceways, even if operated under ideal
design conditions. However, most manufacturers state that 95 percent of the rolling element
bearings fail prematurely from external sources including chemical contamination, corrosion,
improper installation or brinelling, which is the formation of indentations in the raceways caused
by static overloading, exposition to vibrations while being stationary, or even by improper
lubrification [6] [7]. Zerbst [8] did an extended overview of potential innovative safe life and
damage tolerance methods for railway axles and concluded that fatigue crack initiation and
propagation have a big impact in failure of high cycle railway axles.
As the failure of components in the axle bearing is one of the most significant factor in the
safety incidents, it’s important to have monitoring and diagnosing systems able to deliver early
warnings that help to prevent accidents and reduce maintenance costs. A number of tools and
techniques of condition monitoring can be used to check a series of properties, such as Vibration
analysis, Acoustic monitoring, Thermography and Lubrication sampling [6,7].
The practical application of railway axleboxes bearings condition monitoring is done by
using vehicle-based sensors. This on-board monitoring systems that uses the vibration or thermal
information to infer the health of the mechanical components and, consequently, to trigger the
maintenance or operation actions. Modern rolling stock is fitted with high-capacity communication
buses and multiple sensors which require advance processing units for data collection and
management [7]. There has been a wide number of experimental researches to study the
bearings dynamics, as seen in [6], [9] and [10]. Due to the difficulties that arise from using data
4
from experimental methods, to predict the life of a bearing, theoretical methods are an
advantageous alternative. The noise and vibration output of the axleboxes of the rolling stock is
in fact a measurable outcome of the bearing dynamic response, under the operating conditions,
towards which the methodology developed here is aimed. Developing a computational tool that
simulates the roller bearing dynamics can lead to an innovative way to predict flaws, reduce
maintenance costs and better understand the behavior of the roller bearings under working
conditions.
1.1. State of Art
The method presented here is the development of a dynamic analysis tool, referred to as
BearDyn which serves as the acronym for Bearing Dynamic Analysis Program, able to handle
bearing models representative of the actual tapered roller bearings used in railway operations.
In Computer Aided Engineering for roller bearings, the first approach to computer
modeling of roller bearings was with quasi-static models. This type of work is based on static
equilibrium formulation, with the centrifugal forces and gyroscopic moments encountered at high
speeds added as additional external forces and moments [11]. One of the more popular quasi-
static models are credited to the work of Crecelius and Privics at SKF, with the software
SHABERTH [12], to Poplawski with the software COBRA [13] and to Amaraki with the software
BRAIN [14].
With the continued advancement in computer industry, some dynamic bearing analysis
programs were developed. One of the first approaches of a generalized dynamics model to solve
the differential equations of motion of the cage in an angular contact ball bearing with constrained
ball motion was developed by Walters [15], implemented in a computer code, BASDAP. Gupta
developed ADORE [16], where the classical differential equations of motion are integrated as a
function of time to provide real-time dynamic performance simulation of rolling bearings, including
ball, cylindrical, and tapered roller bearings.
Advancement in rolling bearing dynamics modeling continued with Stacke and Fritzson
[17] at SKF. Using multibody techniques, with particular emphasis on contact problems between
bearing elements, Stacke and Fritzson modeled overall dynamic behavior of rolling bearings,
resulting in the computer model Bearing Simulation Tool BEAST. With particular emphasis on
prediction of rolling element slip and the resulting cage forces in planetary application of rolling
bearings, Houpert [18] has published a code, CAGEDYN, to model overall bearing dynamics.
In the recent years, Roller Bearing Company Schaeffler Technologies [19] developed an
industrial software named CABA3D for the simulation of dynamic processes in rolling bearings.
CABA3D has multi-body formulation with a hydrodynamic friction and contact model specially
designed for rolling bearings. Kiekbusch [20] developed dynamic simulation models using
commercial multibody software ADAMS and SIMPACK with advanced lubrication models, cage
modeling and elastic structures.
5
Practical implementation of very sophisticated mathematical techniques is gradually
becoming a reality as the available computing power continually unfolds, motivating the work here
presented. The goal is to find more realistic ways to represent the dynamic behaviour of the roller
bearings with multibody formulation and get the simulation results closer to the reality.
1.2. Work Description
The objective of the MAXBE project was the understanding of the performance of the axle
bearings in actual operation conditions and the development of a dynamic analysis tool able to
handle bearing models representative of the actual bearings used in railway operations. The final
goal of the work described in this document is to obtain a dynamic analysis tool BearDyn which
is able to perform a study on tapered roller bearings, whose models are described by data in
dedicated files. The dynamic response of the forces is to be post-processed to obtain the
Frequency Response Functions (FRF), which serve as the basis for the evaluation of the bearing
health.
In order to allow realistic simulations and prepare the program for the purpose intended,
the inputs and models used for the formulation and functioning of BearDyn must use the state-
of-art features and methodology, preferably by being compatible with methods and models used
by other partners of the project MAXBE. In this sense, the work now presented includes not only
the developments by the author but also the background knowledge is made accessible. The
structure of the models and theoretical knowledge, in which the body of the work is done, is
available in the work by Ambrósio [21] delivered to project MAXBE.
Following the work of Ambrósio [21], Lima [22] applied the methodologies and the first
version of BearDyn was developed. The focus was on Spherical and Tapered Roller Bearings,
using a multibody formulation to describe the elements of the bearing and their interactions.
BearDyn uses different geometries to describe the rollers, cage and raceways and evaluates
the existence of contact between them. The interactions between elements were described by
continuous contact force models based on the Hertz elastic contact theory [23] and a simplified
model to describe the tangential forces was applied.
Lima [22] explained the formulation needed for BearDyn but did not implement all the
necessary interactions for one full bearing. For this reason, the results of the dynamic analysis
were not trustworthy and since very few verification methods were used during the
implementation. A study of better fitter numerical procedures was found necessary for BearDyn
in order to better trust the models and successfully implement them in the Dynamic Analysis
Program. Since the previous results show that BearDyn was not able to properly simulate the
dynamic response of a full bearing, it is important to verify the methods used and verify if they are
working. Figure 5 shows a general overview of the work developed by Lima [22] and highlights
the parts were further attention is needed, among which the Contact Detection is deemed as the
most critical part. BearDyn uses different geometries to describe the elements and detects the
contact between them. If one contact detection is not well implemented, it can compromise the
6
complete simulation either by the wrong evaluation of contact forces or by the existence of
vibrations that lead to the stall of the integrator algorithms.
Figure 5: Overview of BearDyn Structure
Contact Detection in the first version of BearDyn was not reliable, failing in numerous
situations. For this reason, the first part of this work is the study and testing of the Contact
Detection methods for different geometries in order to better understand how they work and what
their limitations are. The multibody formulation used is described in Chapter 2, with special detail
to their implications in the implementation of the contact forces. In Chapter 3 Contact Detection
is addressed. The methods are explained for general geometries and applied to a demonstration
involving general surfaces. After analysing and correcting the formulation to roller bearing
geometries, a dynamic analysis tool with two contacting bodies is developed to test the contact
detection in moving geometries. These two initial steps are crucial to gain sensibility and detect
the flaws existing in the BearDyn formulation. After having reliable results from these auxiliary
programs, the main Bearing Dynamic Analysis Program is finally addressed.
BearDyn uses different geometries to describe the elements and evaluates the contact
between them. This requires particular and complicated formulation for each type of roller bearing.
In Chapter 4 the formulation for the geometries used for Contact Detection in BearDyn is
presented. In this work, all the formulation is described, implemented and verified by example,
leading to some significant changes to the formulation originaly selected by Lima [22]. Chapter 5
shows the contact force models used in BearDyn. Then the formulation described in this work is
applied being the implementation explained in detail in Chapter 6. Finally, BearDyn results are
presented in Chapter 7 and the conclusions and future developments are detailed in Chapter 8.
Input Datageometry, material,
lubrication
MBS solver
MBS tool
Initialization
Contact Detection
Normal Contact Force
Friction
7
2. Dynamic Analysis
The dynamic analysis of a multibody system allows the study of its motion and forces transmitted
for a given time period, as a function of the initial conditions, external applied forces and/or
prescribed motions. This chapter presents the formulation of the general equations of motion to
spatial dynamic analysis implemented in BearDyn. The multibody formulation is supported by the
classic mechanics theories [24].
2.1. Multibody Dynamics Formulation
Cartesian coordinates and Newton-Euler’s method are used to formulate the Equations of motion
of the spatial multibody systems [25]. The kinematics of a single rigid body, as that shown in
Figure 6, is described by a set of coordinates 𝐪𝑖 = [𝐫𝑖𝑇 𝐩𝑖
𝑇]𝑖𝑇, in which the position of the body
with respect to global coordinate system XYZ is defined by the coordinate vector 𝐫𝑖 = [𝑥 𝑦 𝑧]𝑖𝑇
that represents the location of the local reference frame (𝜉𝜂𝜁)𝑖 and the orientation of the body is
described by the rotational coordinates vector 𝐩𝑖 = [𝑒1 𝑒2 𝑒3 𝑒4]𝑖𝑇, which is made with the
Euler parameters for the rigid body [25]. The complete multibody system, made of nb bodies, is
described by a set of coordinates 𝐪 in the form,
1 2, , ,
TT T T
nb q q q q (2.1)
Let a point P, shown in Figure 6, be defined in the rigid body. The vector 𝐬𝑖𝑃 represents
the location of the point P with respect to the origin of the local reference frame of the body (𝜉𝜂𝜁)𝑖.
The position of point P, with respect to the global reference frame, is defined by vector 𝐫𝑖𝑃, as
expressed in,
'P P P
i i i i i i r r s r A s (2.2)
where 𝐀𝑖 is the transformation matrix for body 𝑖 which defines the orientation of the referential
(𝜉𝜂𝜁)𝑖 with respect to the global coordinate system XYZ. The transformation matrix is expressed
as,
2
0 0 1 2 32 1 I 2 ; T
i e e e e e A ee e eT (2.3)
The multibody system generally includes a set of kinematic constraints denoted as 𝚽(𝐪, 𝑡) = 𝟎,
which represent the kinematic joints or any type of relations between the coordinates. Using the
Lagrange multipliers, the constraint reaction forces are added to the equations of motion. These
are written together with the second time derivative of the constraint equations. Thus, the set of
equations that describe the motion of the multibody system is,
T
q
q
M Φ q g
Φ 0 λ γ (2.4)
8
where λ is the vector of Lagrange multipliers and 𝛄 is the vector that groups all the terms of the
acceleration constraint equations that depend on the velocities only, that is,
- - - 2q tt qtqγ Φ q q Φ Φ q (2.5)
and the Jacobian matrix is denoted by 𝚽𝑞 while the subscript 𝑡 means time derivative. The
Lagrange multipliers, associated with the kinematic constraints, are physically related to the
reaction forces and moments generated between the bodies interconnected by kinematic joints.
This system of equations is solved for �� and 𝛌. Then, in each integration time step, the
accelerations vector, ��, together with velocities vector, ��, are integrated in order to obtain the
system velocities and positions for the next time step [26].
Figure 6: Definition of the Cartesian coordinates for a rigid body
In the computational tool BearDyn developed for this work, kinematic constraints are not
introduced as its range of application includes isolated roller bearings only. Consequently, the
dynamic response of the system depends only on the forces applied to the bodies. The system is
defined as a system of unconstrained bodies, where the equations of motion are used in a
simplified form, for one body, described as
i i iM q g (2.6)
where,
i ; ;
'i
i i i
N 0 r fM q g
n0 J ω (2.7)
in which the sum of all forces acting on the body is 𝐟𝑖, the sum of all moments by 𝐧𝑖′, 𝐉i
′ the inertia
tensor for the body and 𝐍i is the diagonal matrix with the mass of the body.
In a system of unconstrained bodies, as represented in Figure 7 (a), it is assumed that
there are nb bodies acted upon various force elements. The motion of the bodies is confined by
contact being the outer raceway fixed in space. A free-body diagram representative of a generic
body from the system is shown in Figure 7 (b).
Oi
i
i i
P
(i)X
Y
Z ri
r P
i
sP
i
9
(a) (b) Figure 7: (a) Representation of a system of uncontrained bodies; (b) Free-body diagram of an uncontrained body
Of particular interest in the application of the multibody dynamics formulation to the
representation and solution of bearing rolling element dynamics is the construction of the force
vector g. This vector includes all the external and internal applied forces in the system, namely,
the springs, dampers and actuator forces, the gravitational forces, the normal, hydrodynamic,
elastohydrodynamic and friction forces between bearing rolling elements and the gyroscopic
forces of the rigid bodies. The internal forces in the bearing rolling elements, in particular the
normal forces, friction, hydrodynamic and elastohydrodynamic forces, presented and discussed
in this work, need to be applied in the contact points identified during the contact detection
process. In what follows, the vector of forces, g, present in Eq.(2.6), includes the forces applied
in all the bodies of the system as 𝐠 = [𝐠1𝑇 , 𝐠2
𝑇 , … , 𝐠𝑛𝑏𝑇 ]𝑇, being each of the individual body force
vector written as
i '
ii
i
T P
app i i i i i
fg
n A s f ω J ω' ' '
(2.8)
where it is supposed that force 𝐟𝑖 is applied on point P of body i, shown in Figure 6, 𝐧𝑎𝑝𝑝𝑖 is a
vector with the moments directly applied, 𝝎𝑖′ is the body angular velocity, expressed in body
coordinates and 𝐉𝑖 ’ is the inertia tensor, also expressed in body fixed frame. Therefore, when
contact is detected, the coordinates of the contact points, in each surface, 𝐬𝑖𝑃 need to be identified
during the contact detection process. Afterwards, by using appropriate normal contact force
models and tangential force models the vector of applied forces 𝐟𝑖 is calculated. Note that the
term −��𝑖𝐉𝑖′𝛚𝑖
′ is the gyroscopic force, and therefore, it is not strictly an applied force, but still
included in the body force vector.
The dynamic analysis is performed by solving the system composed by the equations of
motion of all bodies presented in Eq.(2.6). Then, the acceleration and velocity vectors, �� and ��,
are integrated to obtain the velocities and positions of the bodies in the next timestep. Further
detail on the general integration algorithm is given in [25].
X
Y
Z
(i)Oi
ii
i
1f
2f
3f
n
1P
3P
2P
X
Y
Z
n
1f
2f
10
11
3. Contact Modelling in a Dynamic Analysis
In a roller bearing simulation all the interactions between bodies are described as contact forces.
This means that there are no constrains in the system and all the dynamic behaviour is
consequence of the forces that develop due to contact. One incorrect contact is enough to
compromise the whole simulation. For this reason, the first approach to the problem of correct
contact detection is to solve simple contact geometries interactions to guarantee that all the
methods used in BearDyn are validated, i.e., they are correct and reliable.
The formulation of contact is structured as a two-stage problem. In the first stage the positions
and geometries of the bodies are evaluated to identify the points of proximity and the eventual
existence of contact. The second stage consists in the evaluation of the normal opposite contact
forces that develop between the surfaces when the contact occurs, as well as all the friction forces
that may exist. This chapter focus, firstly, on applying contact detection methods to different
geometries and understanding how to generate and solve the necessary system of equations.
Contact detection is tested with simple demonstration geometries. Finally, a dynamic analysis
program with two bodies colliding is developed, with simplified normal and tangential force
models, to test the methodologies developed.
3.1. Contact Points Detection
It is crucial to identify a reliable way of identifying if contact takes place, because a deficient
detection leads to errors in the calculation of the forces applied eventually compromising the
complete dynamic response of the system. For two moving bodies, there is always a pair of points,
one in each surface, where the contact is more likely to occur. Depending on the forces acting in
the system, the bodies have a translational and rotational movement, which leads to a position
change of the contact point in each body, meaning the position of two points must be updated in
each time step. Over time, depending on the geometries and positions of the bodies, the program
will find the two points that are in contact, or closer to each other, and identify the interference
between the surfaces, i.e., the amount of deformation of the contact surfaces on their points of
contact. For this reason, this procedure is considered to be an online contact detection method.
In realistic mechanical systems, it is likely that a rigid body contacts with other bodies in
more than one pair of points, at the same time. This is why in a dynamic analysis simulation all
the possible pair of contact points must be considered. When evaluating the existence of contact,
it is necessary to study every pair of contact points individually, since the value of force developed
in each point can be different depending on the geometry and kinematics of the body. Every
contact point has a persistent method, which is solved to calculate its position in each body. This
method consists in finding the two closest points between each body, in a certain area where
contact is prone to develop, and identifying the pseudo-interference, or local deformation,
between the contacting surfaces. If penetration occurs, the contact point on one body has to be
located inside the volume of the other body. The contact points are defined as those that
12
correspond to maximum indentation, i.e., the points of maximum elastic deformation, measured
along the normal to the contact patch.
Contact between two generic surfaces
Let it be assumed that two bodies are in motion and approaching each other. These bodies can
be described as generic surfaces, represented in Figure 8. Points P and Q represent the closer
proximity points, each belonging to a generic surface on body i and j, respectively, with the
position in the global frame being 𝐫𝑃 and 𝐫𝑄 as defined in Eq.(2.2).
Figure 8: Candidates to contact points between two parametric surfaces (adapted from [27])
The distance between points is described as vector d given by:
P Q d r r (3.1)
On point P the vector normal to the surface is 𝐧𝑃, while 𝐭𝑃 and 𝐛𝑃 are the tangent and bitangent
vectors to the surface, forming an orthogonal basis. The same applies to 𝐧𝑄, 𝐭𝑄 and 𝐛𝑄, vectors
evaluated on point Q. All the vector previously defined as well as the local positions of each point,
depend on the parametric description of each surface, reason why for each specific geometry
they are described individually in the forthcoming sections of this work. The relation between the
body fixed and inertial coordinates of the vectors that define the normal, tangent, binormal and
position of the contact point in the surfaces is written as
' '
' '
P P P P
P P
i i
i PiP i
b b
t
n A n A
A r A nt r (3.2)
where 𝐀𝑖 is the transformation matrix and 𝐫i is the position of centre of mass for the body i. The
same applies to point Q, located in body j.
(j)
( i )
nQ
nP
Q
d
tQ
bQ
sQ
rQ
rj
rPri
sP
bP P tP
i i
i
j
jj
Y
Z
X
13
A point and all the respective vectors in a general three-dimensional surface can be
described with two parameters. The search for the two closest points of contact, or close
proximity, consists in solving a non-linear system with four equations to find the four parameters
(two for each body) associated with the two contact points. The conditions for minimal distance
between the two surfaces are generically described by [27]
0 0
; 0 0
T TQ P
T
Q
TQP P
n t d b
n t d b (3.3)
which means that not only the normal to each surface must be collinear with the vector that
connects the two points in closer proximity but also perpendicular to the tangent and binormal
vectors on each point. The system of equations must include at least one of the normal vectors
to the surfaces, because if all the conditions include the vector d, in case of pseudo-penetration,
the trivial case of d = 0 would satisfy the conditions, without representing correctly the
identification of the contact points. In Figure 9-(a) the correct contact point is represented and in
Figure 9-(b) the two points with d = 0 are represented.
(a) (b)
Figure 9: Contact Points Detected. (a) Correct pair of points; (b) Incorrect pair of points
Effective contact occurs if, besides the fulfilment of Eq.(3.3), penetration also exists,
which is expressed by,
0T
Q d n (3.4)
otherwise, the points are in close proximity, but not in contact.
Note that the geometric boundaries of the surfaces are not described, which implies that
the contact search can extend beyond such boundaries, therefore a special care is necessary to
ensure that the pair of points is always detected inside the domain that virtually limits the surface.
In addition, other surface geometries play the role of the geometric limits to ensure the realistic
behaviour. For example, the tapered roller bearing has flanges that limit the relative travel of the
𝜂
𝜁
𝜂𝑖
𝜁𝑖
𝜂
𝜁
𝜂𝑖
𝜁𝑖
14
rollers along the raceways, thus preventing that its location with respect to the raceways extends
beyond functional limits.
To solve the system of nonlinear equations, some methods require the Jacobian matrix,
reason why the matrix needs to be formulated. For general definition, the system of nonlinear
equations is defined as 𝐅(𝐱) where 𝐱 = [𝑥1, 𝑥2, 𝑥3, 𝑥4], being (𝑥1, 𝑥2) the parameters that define
the point P in body i and(𝑥3, 𝑥4) the parameters that define the point Q in body j. The Jacobian
matrix to solve the contact detection between two general surfaces is:
1 2 3 4
1 2 3 4
1 2 3 3 4 4
1 1 2 2 3 4
'( )
T T
Q QT TP PQ Q P P
T T
Q QT TP PQ Q P P
T T T TQ QT T
Q Q Q Q
T T T T
T TP PP P P P
x x x x
x x x x
x x x x x x
x x x x x x
n nt tn n t t
t tn nt t n n
F xb bd d d d
b b b d b d
b bd d d db d b d b b
(3.5)
where the partial derivatives are specific to each geometry. When implementing the contact
detection, it is important to ensure that the vector of variables x is correctly formulated, i.e., that
the variables are inserted in the correct order, otherwise the Jacobian is different and a general
formulation for all surface to surface contacts is not possible.
Contact between a surface and a line
When applying the procedure to the case of a cylindrical body, such as a roller, approaching a
generic surface, let it be assumed that in the course of its motion the roller actually contacts the
surface. The situation is numerically perceived as contact, as illustrated in Figure 10-(b), being
the shaded volume a representation of the penetration of the roller in the surface, i.e., the
interference between the two bodies, designated also as penetration.
(a) (b)
Figure 10: (a) Roller Approaching a surface, (b) Roller penetrating a surface
The contact patch between the two bodies is described as a line contact along the
longitudinal direction, distributed over a small area. This means that the approach of two generic
i
ii
j
j
ji
ii
j
j
j
15
surface contacting in one point is insufficient, since this contact area cannot be simplified into a
single point, being a line, eventually with varying interference depth. Furthermore, the geometry
of the surfaces may not be cylindrical or conical and/or the axis of the rollers may be misaligned
with the axis of the raceways resulting in skewing between the two rolling elements. Any of the
conditions mentioned lead to a local deformation of the contact line between the roller and the
contacting surface that varies along the roller axis. Consequently, instead of defining a common
penetration depth for the complete roller, the penetration can only be defined for each particular
cross-section of the roller.
Figure 11: Roller divided in slices
Let a roller with the shape represented in Figure 11 represent any generic roller, i.e.,
cylindrical, spherical, toroidal, tapered or spherical tapered roller. Consider now that the roller is
divided in a user defined Nsl number of strips, i.e., cylindrical segments, which act as rigid bodies
without any relative motion between them. Now the contact problem of the complete roller can be
described as Nsl independent contact problems of thin cylinders, in which the contacting
penetration depth is constant throughout the slice, or strip [20]. Therefore, each one of the contact
problems, required to represent the roller to surface contact, is described by the contact of the
central cross-section of the slice.
As a result of the approach followed here, the search for contact of two bodies that the
contact path is line is reduced to the identification of the minimum distance between the central
cross-section of each slice and the surface. By identifying the proper interaction conditions it is
possible to verify if such distance corresponds to separation or to effective contact. In what follows
the central cross-section of the slice is designated by circular cross section or simply by circle.
The representation of the two contacting points between a circle of the roller and a surface can
be seen in Figure 12. For the line, only the tangential vector 𝒕𝑃 is needed, since the normal vector
𝒏𝑄 from the surface gives the direction of the contact force if the interference exists.
16
Figure 12: Geometric relations between a point P in a circumference and a point Q in a generic surface
Only one parameter describes the position of the contact point on the circular line and
two parameters continue to be required for the surface. This results in one of the equations of
Eq.(3.3) to be no longer needed, so the conditions for minimal distance between the circle and a
surface are now described by,
0
0
0
Q P
T
Q
T
Q
n
d t
d b
b
(3.6)
Effective contact occurs with the circle and surface if both Eq.(3.4) and Eq. (3.6) are fulfilled, i.e.,
if besides being P and Q the points representing the point of more proximity, they also lead to a
penetration between the surface. For the system of three nonlinear equations 𝐅(𝐱), the variable
vector 𝐱 = [𝑥1, 𝑥2, 𝑥3] has 𝑥1 the parameter to define point P in body i, and (𝑥2, 𝑥3) the parameters
to define the point Q in body j. The general Jacobian matrix is defined as,
1 2 3
1 2 2 3 3
1 2 2 3 3
'( )
T T
Q QT PQ P P
T T TQ QT T
Q Q Q
T T TQ QT T
Q Q Q
x x x
x x x x x
x x x x x
n nbn b b
t td d dF x t t d t d
b bd d db b d b d
(3.7)
Contact between two lines
Some surfaces of contact in a roller bearing have a small height to length ratio, such as the cage
side and cage top that contact the roller, meaning that the surface can be approximated by a line
in order to simplify the contact detection. The representation of the two contacting points between
two lines is depicted in Figure 13.
i
**
j
j
j
i
i
ri
sP
rj
X
YZ
sQ
Q
P
Pb nQ tQ
bQ
d
17
Figure 13: Contacting points between a circle of the roller and a line
The number of parameters that need to be identified with respect to the general surface
to surface contact decreases to two: one parameter is required to define each line. In this case,
the equations to be fulfilled to find the closest points depend on the vectors defining the
geometries. If each geometry is characterized with one tangential vector 𝐛𝑄 and 𝐛𝑃, the contact
system is given as:
0
0
T
Q
T
P
d b
d b (3.8)
If instead of one tangent vector to each surface, two tangent vectors to one surface are defined,
the system is, alternatively, written:
0
0
T
Q
T
Q
d b
d t (3.9)
The discussion on the most suitable version of the contact equations is done via suitable
application cases. Each case is demonstrated and applied for particular tapered roller bearing
contacts in Chapter 4. For both cases, the system of nonlinear equations 𝐅(𝐱) will have the
parameters vector 𝐱 = [𝑥1, 𝑥2] with 𝑥1 defining the point P and 𝑥2 the point Q. The Jacobian matrix
are, for the first system,
1 1 2'
1
1 2 2
( )
T T
T PP P
T TQT
Q Q
x x x
x x x
bd db d b
F xbd d
b b d
(3.10)
and for the second system of nonlinear equations,
i
**
X
YZ
ii
j
j
j
sQ
sP
bP
P
Q
nQ
Qb
d
rj
ri
Qt
18
2
1 2 2'
1 1 2
( )
T TQT
Q Q
T TQT
Q Q
x x x
x x x
bd db b d
F xtd d
t d t
(3.11)
3.2. Solution of a system of nonlinear equations
In all types of contacts there is a system of nonlinear equations that needs to be solved. The
system of equations to solve depends on the case in study, but the methods used are general,
meaning that can be applied to any of them. Since the contact points to be identified are
responsible for the magnitude and direction of the contact forces, it is important to have a reliable
way of detecting the correct pair of points and avoid that other solutions of the system of nonlinear
equations are selected instead.
It has been seen that the identification of the closest points between the geometrical
features of two bodies must fulfil several conditions, namely Equations (3.3), (3.6), (3.8) and (3.9)
depending on the case in study. These equations define sets of equations that involve the inner
product between vectors that depend on the geometry and kinematics of each interacting body.
The priority is to find a method that always converge to the best solution, i.e., the solution that
ensures the closest proximity of the points in the surfaces. Since the contact problem is solved
online, the method should also be fast. Note that for a single roller bearing used in a railway
application, more than 1000 contact need to be solved every time step in BearDyn, which means
that the simulation can get very time consuming.
Solving the system of nonlinear equations means finding a set of design parameters, 𝐱 =
{x1, x2, … , xnp}, that can, in some way, be defined as optimal, using optimization techniques. An
efficient and accurate solution to this problem depends not only on the size of the problem in
terms of the number of constraints and design variables but also on characteristics of the objective
function and constraints. Since the systems to be solved are composed by nonlinear equations,
their solution requires Nonlinear Programming, in which the objective function and constraints
can be nonlinear functions of the design variables [28]. Alternatively, standard methods to solve
systems of nonlinear equations, such as the Newton-Raphson method can be used.
All methods implemented use an initial estimation of the parameters, calculated in the
initialization of the dynamic analysis program before calling the integration method. In an iterative
procedure, the method establishes a direction of search until the value of these parameters,
satisfying the given equations, are identified, i.e., when the step size of the objective function
between iterations is smaller than the specified tolerance, or the maximum iteration number is
reached. For other methods tolerances must also be set with a mathematical significance proper
to the specific method.
Optimization tools
19
The Optimization toolbox in MATLAB® provides functions for finding parameters that minimize or
maximize objectives while satisfying constraints. Toolbox solvers can be used to find optimal
solutions to continuous problems and perform design optimization tasks, including parameter
estimation. Two functions presented in this toolbox are tested, namely the fsolve and fmincon.
The MATLAB® function fsolve is such that given a set of n nonlinear functions 𝐅𝑖(𝐱),
being n the number of components of the vector x, the goal of equation solving is to find a
vector x that ensures all 𝐅𝑖(𝐱) = 0. fsolve is used to solve systems of nonlinear equations by
minimizing the sum of squares of the components based on the initial values given. If the sum of
squares is null, the system of equation is solved. fsolve has three algorithms: Trust-region,
Trust-region dogleg and Levenberg-Marquardt that can be tested independently. Supposing a
starting point x in n-space, it is required to move to a point that leads to a lower function value.
The basic idea is to approximate F with a simpler function Q, which reasonably reflects the
behaviour of function F in a neighbourhood n around the point x. This neighbourhood is the trust
region. A trial step is computed by minimizing, or approximately minimizing, over n. For additional
understanding on this method, the reader is directed to references [29] [30].
Within the methods tested, fsolve is the more reliable of all the described methods, but
takes more computational time. It is a good method for a first test on solving the system of
equations, but for large dynamic analysis with many contact detections it becomes slow.
The MATLAB® function fmincon is tested as an alternative method to solve the system of
equations. fmincon finds a constrained local minimum of the objective function of several
variables near an initial estimate. Starting at x0, the method finds a minimum x to the function
described as objective function, subject to the constraints defined. This constraints can be linear
equalities, defined set of lower and upper bounds on the design variables, x, or nonlinear
inequalities. The function to be minimized and the constraints must both be continuous. When the
problem is infeasible, fmincon attempts to minimize the maximum constraint value. fmincon
has five algorithm options: interior-point, trust-region-reflective, sqp, sqp-legacy and active-set as
explained in [28] [29]. The objective function to minimize is the distance modulus, subject to the
conditions defined in (3.3), (3.6), (3.8) and (3.9), depending on the contact geometries in each
specific case. For two general surfaces, the optimal problem is defined as:
0
0min . .
0
0
P
T
Q P
T
QT
T
P
T
Q
s t
n t
n tf d d
d b
d b
(3.12)
This method proved to be as reliable as fsolve but with higher computation time,
therefore it is used only when everything else fails, or for cases in which other methods are
unreliable.
20
Newton-Raphson
The Newton-Raphson method for solving a system of equations 𝐅(𝐱) = 0 is based on the
convergence, under suitable conditions, of the sequence
1 lim 1,2, ,
'
m
m
m
m m m F x
x xF x
(3.13)
where the counter m refers to the iteration number, x is a vector containing the desired variables,
𝐅(𝐱𝑚) is the system of functions to be evaluated, defined in (3.3), (3.6), (3.8) and (3.9), and 𝐅′(𝐱𝑚)
is the Jacobian matrix, the matrix of all first-order partial derivatives, as seen in
1 1
1
1
'
np
m
m m
np
f f
x x
f f
x x
F x (3.14)
that can be evaluated from the analytical first order derivative or obtained from the function 𝐅(𝐱)
in a discrete interval ∆𝐱 defined by finite differences, i.e.,
1
1
m n m nm
np n n
f x f xf
x x x
(3.15)
Both methods available for the evaluation of the Jacobian Matrix are applied and
compared in this work. The idea is to identify not only their computational cost but also their
precision. When the Jacobian Matrix is obtained with the analytical first order derivatives, the
method is referred as Analytical Newton-Raphson. If the Jacobian matrix is obtained with first
order derivatives from the finite differences the method is referred as Computational Newton-
Raphson.
3.3. Contact detection demonstration with simple geometries
The main goal of this work is the development of the dynamic tool BearDyn. Being this a very
complex program, some intermediate steps are necessary to test and validate the supporting
numerical methods, individually. The main challenges are the contact detection solution, i.e.,
methods to solve the nonlinear system of equations for contact detection and then the contact
force calculation, i.e., evaluation of the normal and tangential forces. Testing the alternative
methods individually allows to better understand the numerical issues allowing an easier way to
interpret results, identifying problems and to favouring decisions about corrections and
modifications.
For this purpose, different general geometries are formulated and implemented to find
points of close proximity. This allows to detect numerical and formulation problems that are
21
corrected before the methods are used with more complex bearing geometries. Three programs
of geometries creation and contact point detection are developed in MATLAB®, one for each
contact type described, demonstrated in this section.
22
Contact between two circles
As a first approach, the method described is applied to two circles in a two-dimensional space,
being this the easiest geometry to implement and test. As showed in Figure 14, the local position
of each point in the two approaching surfaces can be defined as:
' '
cos cos ;
sin sin
P
j
P j Q iP
j
Q
i
Q
i
R R
s s (3.16)
being 𝑅𝑖 and 𝑅 the circle radios. The global positions in the inertial frame (XY) are already
expressed in Eq.(2.2).
Figure 14: Geometric relations describing the positions of point P and Q in two circles
For the surface containing point P, two vectors are defined: 𝐧𝑃 is the normal vector to the
surface of the circle and 𝐭𝑃 the tangent vector. The same applies for the surface containing point
Q,
cos cos
; sin sin
P P
j j
P P
j
jP P
j
j
n A t A (3.17)
where 𝐀 is the transformation matrix from body j coordinates to (XY). For finding the two
parameters (𝜑 𝑃 , 𝜑𝑖
𝑄) that define the two contact points two non-linear equations are needed. The
conditions for minimal distance are described as:
0
0
T
j i
T
i
n t
d t (3.18)
Since this system of equations has four roots, there are four potential combinations of points that
can be the solution to the system, as seen in Figure 15. In reality only (P-Q) represents the true
contact point, but the system also admits (H-P), (Q-L) and (L-H). To minimize the problem, a
Q
c
c
P
c
c
𝜃
𝜃𝑖
X
Y
𝜑
𝜑𝑖
nQ
nP
Qt
tP
'sQ
'sP
23
discrete method for finding a close guess to the solution is implemented and the solution obtained
is used as an initial guess for the solvers applied in this case. In this way the solver is guided to
the combination of points relevant to this case study.
Figure 15: Possible solutions to the system of equations that defines the contact detection
To test the different methods to solve the nonlinear system of equations, two circles of
the same radius where generated in random positions and orientations. To these geometries,
Newton-Raphson, fsolve and fmincon methods are used and the initial guess for each method
is given as the pair of points in closest proximity that truly represent the contact penetration or
that are closer to contact. The geometries and points found were represented for visual
confirmation of the results. Figure 16 shows two solutions from the MATLAB® code implemented,
with the detected pair of contact points identified.
(a) (b)
Figure 16: Representation of the geometries generated with the local axis and: (a) Incorrect Pair of Points detected; (b) correct pair of contact points
There are 100 locations and orientations of the circles, randomly selected, being these
solved with all methods. For each method and the results are presented in the Table 1:
Method Success rate (%) Time Interval (s)
Newton-Raphson 66.7 0.10 fsolve 100 0.15 fmincon 100 0.37
Table 1: Results from contact point detection for two random circles
H
L
QP
24
This analysis clearly shows that Newton-Raphson is not always capable of finding the
correct result in a reliable way, being the incorrect solution detected for the cases were the two
circles are positioned with large penetrations, as seen in Figure 16-(a). Since in a roller bearing
analysis all the indentations are very small, a new analysis is done with the two circles either in
contact or separated by a maximum distance of 1% of the radius. One example of an analysis
result is seen in the Figure 17. This limitation for the relative location of the surfaces eliminates
the errors in Newton-Raphson method and demonstrates its reliability, being a method to
implement in BearDyn.
Figure 17: MATLAB® representation of two circles with local axis and the correspondent contact points
The Newton-Raphson method shows the importance of defining the initial guess close to
the true solution to avoid the convergence to an unwanted pair of points, which despite being a
valid solution of the system of equations, does not represent the true contact points. Visual
confirmation is enough to find if the correct solution is being evaluated in the test case. However,
for complex simulations of roller bearings the method must be absolutely reliable as no visual
confirmation of all the contact detection correctness is possible.
Contact between two spheres
Spheres are the 3D geometries with more possible pair of solutions to the nonlinear system of
nonlinear equations. Finding a reliable way of detecting always the correct contact point in this
case is ensuring that for other geometries the method also works. Going from a 2D to a 3D
geometry means that a point in a surface is now characterized with two tangent vectors and one
normal to the surface, all perpendicular to each other. As represented in Figure 18 (a), the three
vectors and the local distance 𝐬𝑄′ of a point from the surface is function of two variables (𝜙, 𝜃).
To define the nonlinear equations them two auxiliary vectors 𝐧𝜙′ and 𝐭𝜙
′ are defined, seen in
Figure 18 (b):
' 'cos sin 0 ; sin cos 0T T
n t (3.19)
25
Vectors 𝐧𝑄 and 𝐭𝑄 are defined by a single rotation of an angle 𝛽 about one axis defined
with the vector 𝐮. The transformation matrix 𝐀𝜃, correspondent to this rotation, can be obtained
from the Euler parameters, computed with the vector and angle. In this case, the parameters used
for the rotation matrix are:
1 2 3
0
with cos
22
Te e e sin
e
u te u
(3.20)
Now, the transformation matrix 𝐀𝜃 is built with Eq. (2.3). The vectors and that describe the
distance in point Q and the surface normal and tangents are
' '
' ' '
Q Q Q
Q Q s QR
n A n b t n
t A t s n
(3.21)
being 𝑅𝑠 the radius of the sphere. The relation between the body fixed and inertial coordinates of
the vectors are expressed in Eq.(3.2).
(a) (b)
Figure 18: (a) Auxiliary vector to define (b) the three vectors in point Q in a sphere
Following the procedure for the point P in the other sphere, the same vectors are obtained
for a point in that body. The contact points are found solving the system of equations for two
generic surfaces, defined in Eq.(3.3). Using the same procedures applied to the circles, a
MATLAB® program for generating spheres in space and to identify the contact points using
fsolve and fmincon is implemented. The center position of the two bodies are defined
randomly independently of the spheres to be in penetration or separated by a maximum distance
of 1% of the radius. Figure 19 shows an image of two spheres generated in a 3D space with the
contact points successfully detected. The initial guess for both is identified close to the correct
pair of points that represent the contact.
i
ii
i
Q
i
i
bQ
tQ
nQ
t
n
t
26
Figure 19: MATLAB® representation of two spheres and correspondent contact point
With the increment in the number of equations to be solve, the computational time varies
more with the solution method than in the previous analysis, being more dependent of the initial
guess for the position of points. The results for two methods to solve the system of equations are
represented in Table 2.
Method Success rate (%) Time Interval (s) fsolve 100 0.2 – 0.3 fmincon 100 0.4 – 10
Table 2: Results from contact point detection for two random spheres
Contact between circle and cylinder
The contact between a cylinder and a circle represents the collision between a 3D geometry and
a 2D geometry although positioned in space. As showed in Figure 20 (a) the three vectors and
local position of a point Q in the cylinder are evaluated as function of two variables (𝜙𝑄 , 𝜉𝑄), i. e.,
s
0 cos sin 0 sin cos
1 0 0 cos R sin
T T
Q Q Q Q Q Q
TT
Q Q Q s Q QR
n b
t s
' '
' '
(3.22)
For the circle in Figure 20 (b), only one parameter defines de position and vectors of point P
cos sin 0 sin cos 0
cos sin 0
T T
P P P P P P
T
P c P c PR R
n t
s
' '
' (3.23)
Since the circle only has one parameter to characterize a candidate contact and the
cylinder has another two parameters, three equations are needed to define the contact point, i.e.,
in both surfaces the three nonlinear equations defined in Eq.(3.6). Note that the normal vector in
the equations must be the defined in the 3D space, in this case 𝐧𝑄, because is the only vector
that truly represent the normal direction of the pseudo penetration.
27
Figure 20 - Geometric relations between: (a) point P in a circle; (b) point Q in a cylinder
In Figure 21 a circle penetrating a cylinder is represented, with the pair of contact points
marked. The normal vector to the cylindrical surface 𝐧𝑄 allows calculating the penetration, when the
normal vector to the circle 𝐧𝑃 is only in the plane of the circle it cannot be used to calculate the
penetration.
Figure 21: MATLAB® representation of a circle, a cylinder surface and the contact point
For this case, a cylindrical surface was generated always in the same position, with the
circle changing his position and orientation. For all the tests made, both fsolve and fmincon were
able to successfully detect the contact points, in a relatively quick time. In Table 3 the graphical
results from one simulation with the Contact Detection program are represented.
Method Success rate (%) Time (s) fsolve 100 0.15 fmincon 100 0.41
Table 3: Results from contact point detection for circle to cylinder contact
3.4. Dynamic Response for Demonstration Cases
Since the contact detection must be evaluated every timestep, it is important to test the
robustness of the methods applied in a Dynamic Analysis Program. The contact between circle
and cylinder is tested. First a single circle in the rolling body, representing a coin is simulated and,
afterwards, multiple circles bodies, simulating a cylindrical roller defined with the slice method is
simulated. Thus, a Dynamic Analysis Program with two bodies, named DAP_Coin, was created
to simulate a coin rolling along a cylindrical surface, as showed in Figure 22. The geometry of the
coin, Body 2, is formulated as a circle and the floor surface, Body 1, is a cylinder, described in
Section 3.3.3.
i
i
i
Q
i
i
i
P
bQ
tQ
nP
tP sP
'sQ
nQQ
P
Q
28
The contact detection between a line and a surface is implemented being the contact
force modelled with the Kelvin-Voigt Contact Model for the Normal Forces and Threall Friction
Model to represent the Tangential Forces. Note that the dissipative parts of the Kelvin-Voigt model
is modified in order to avoid the development of normal forces that resist the separation of the
surfaces during rebound.
Figure 22: DAP Coin scheme
Figure 23 shows the initial position of the coin, placed close to the surface in a higher
position. It is expected that due to gravity, the coin starts rolling down, and eventually fall to the
side. The program DAP_Coin is implemented in a similar structure from BearDyn.
Figure 23: Initialization of DAP Coin
To test the movement of a roller along a cylindrical surface, DAP_Coin is adapted to
represent the movement of a roller against a cylindrical surface. Initial positions maintained as in
the same and the roller is defined with six slices, each being a circle with the same radius. This
new Dynamic Analysis Program was named DAP_Roller, being the slice method, explained in
Section 3.1.2, used to describe the roller geometry showed in Figure 24.
nf
tf
i
i
i
Z,
Y, 𝜂1
𝜁1
‘𝜂2
𝜁2𝜃𝑖
𝑟1
29
Roller contacts
The contact point on a roller, for each pair of contacting points considered in this work, always
belongs to a circumference representing the central cross-section of a slice. With reference to
Figure 24, the coordinates of point P, belonging to the circumference, and the normal and tangent
vectors to the circle, are expressed in the body i reference frame as,
' ' '
0 0
cos cos sin
sin sin cos
P
P s P P
s
R
R
s n t (3.24)
in which angle 𝜃 is measured in a plane (𝜂𝜁)∗ parallel to plane (𝜂𝜁), 𝑅𝑠 is the radius of the
circumference and 𝜉𝑃 is the circumference’s 𝜉 coordinate. Note that the radius 𝑅𝑆 can change
between slices, allowing to correctly represent the roller bearing surface for all bearing types,
including tapered rollers used in BearDyn. The vector’s position in the inertial frame (XYZ) is
expressed in Eq. (3.2).
Figure 24: Slice method applied to a cylindrical roller
Contact Forces
During the contact detection it is possible to identify the location of the points in contact with the
surfaces of the rolling element, the relative indentation and orientation between the contacting
surfaces. Once contact is detected, the forces are applied in each pair of contacting points P and
Q. Some kinematic variables are required by the models of the normal contact and friction forces.
It is necessary to calculate the relative velocity between the bodies, namely its projection on the
surface tangent to the contacting bodies on the contact points and its projection on the normal
vector to the surface, as seen in Figure 25. The velocities of the contact points P and Q are
obtained by taking the time derivative of the position vectors, 𝐫𝑃 and 𝐫𝑄, written as,
'
'
P i i i P
Q j j j Q
r r A ω s
r r A ω s (3.25)
where ��𝑖 and �� are the velocities of the origin of the body fixed referential and ��𝑖′ and ��
′ the skew-
symmetric matrices associated to the angular velocities of body i and j, respectively, expressed
nP
bP'sP
P
Pi
i
i
30
in the body fixed coordinate systems. The relative velocity between bodies i and j in the point of
contact is obtained as,
PQ P Q r r r (3.26)
The relative normal velocity for the colliding bodies is the projection of the relative velocity
vector ��𝑃𝑄 in the normal vector to the surfaces in the contact point, as,
Q PQ n r (3.27)
and the sliding relative velocity between the bodies, in the contact point, is just the projection of
the same vector in the tangent plane to the contacting surfaces, as,
PQ Q u r n (3.28)
The contact force vectors, applied in bodies i and j, are related to each normal vectors by,
Q Q
Q
n
P
f
f n
f f (3.29)
where 𝑓𝑛 is the normal contact force developed during contact, which is a value related to the
indentation developed between the two bodies, calculated in Eq.(3.4). Note that all the formulation
for contact detection used throughout this document always considers the normal 𝐧𝑄 as the true
contact force direction, since 𝐧𝑃 is not always possible to obtain and sometimes does not
represent the true direction of the indentation, as for line contact.
Figure 25: Velocity vectors in the contact points necessary to evaluate the normal and tangential forces
The simplest contact force relationship, known as Kelvin-Voigt viscous-elastic model, is
modelled by a parallel spring-damper element. The spring represents the elasticity of the
contacting bodies while the damper describes the loss of kinetic energy during the impact [31].
The normal Kelvin-Voigt contact force, 𝑓𝑁, is calculated for a given penetration depth, 𝛿, as:
2 3
0
0
1 3 2 0
0
n
n
n
e e
n
e
K
f K c c r r
c K
(3.30)
j
j
j
X
YZ P
Q
nQ
rPQ
u
31
where K is the stiffness, defined as 105 N/m in the current study case, 𝛿 is the relative penetration
depth, 𝑐𝑒 is the restitution coefficient, and �� is the relative normal velocity of the colliding bodies,
��0 is the maximum penetration velocity, defined as 0.1 m/s, the exponent n is equal to 1.5 for
circular and elliptical contacts and r is the ratio ��/��0. Figure 26 shows the resultant normal force
according to different values of 𝛿 and ��.
Figure 26: Kelvin Voigt normal force versus the relative penetration depth and relative normal velocity of the colliding bodies
The value for stiffness K and n used in this chapter is merely a reference number, being
studied in detail in Chapter 5. In this phase the value used is only for demonstration purposes,
since the realistic value depends on many factors, such as material properties and contact patch
type, which is detailed when addressing BearDyn contact forces.
Besides the normal forces that develop during contact, also tangential forces due to friction
develop between the contacting bodies. When contacting, bodies slide or tend to slide relative to
each other, there are forces generated which are tangential to the surfaces of contact, applied in
each pair of contacting points P and Q. These forces are usually referred to as friction forces. The
Coulomb friction law of sliding friction can represent the most fundamental and simplest model of
friction between dry contacting surfaces. This model states that the tangential friction force 𝑓𝑡 is
proportional to the magnitude of the normal contact force, 𝑓𝑛, by introducing a coefficient of friction
𝜇, applied at the contact point in the opposing direction of the tangential velocity 𝒗𝑡 [31]
tnt
tvf f
v (3.31)
Furthermore, the application of the original Coulomb friction law in a general purpose
computational program may lead to numerical difficulties because it is a highly non-linear
phenomenon that may involve switching between sliding and rolling conditions. Threlfall [33]
proposed another friction force model, in which the transition between −𝑓𝑡 and 𝑓𝑡 is made using
a curve as follows,
0
0
3 /
0
1 t
t
t
v vt
N t
t
t
N t
f if v vv
f e if v vv
f
v
v (3.32)
nf
32
where 𝑣0 is a small characteristic velocity as compared to the maximum relative tangential velocity
encountered during the simulation, defined as 0.1 m/s. Figure 27 shows the resultant tangential
force versus the tangential velocity.
Figure 27: Tangencial force versus tangencial velocity using the Threfall model
As in the normal force, BearDyn accounts for lubrication model when defining the friction
coefficient 𝜇 , which is addressed in Chapter 5.
Results
In order to visualize the evolution of the bodies position over time, a small representation function
is developed in MATLAB® for both DAP_Coin and DAP_Roller, using two generic MATLAB®
functions getframe and VideoWriter. The function receives as input the positions from the
bodies evaluated during the dynamic analysis and for all the timesteps the representation of the
rigid bodies was made and the image created saved as a frame by the function getframe. After
that, the saved frames for all the timesteps are put together in a animation movie with the function
VideoWriter. The method is better explained in Appendix B. Some frames from the rolling coin
analysis are seen in Figure 28. The coin rolls down and, after reaching the bottom, starts losing
velocity, stopping at the same height as it started. Then the direction of the movement changes
and the coin returns to the initial position. The same analysis is made for a roller with six slices
and the results were similar. Figure 29 shows a 3D view of both DAP_Coin and DAP_Roller
starting positions.
Figure 28: Frames from the DAP Coin analysis
tf
tv0v 0v
nf
nf
33
Figure 29: 3D view of the starting point for the dynamic coin and roller with the local axis for both bodies
The representation allowed to better understand the influence of the model parameters
for the Normal and Tangential model, as well as to validate the contact detection and the direction
of contact forces during the analysis. After the correct visualization of an expected behaviour for
a roller moving along a cylindrical surface, the Tapered Roller Bearing problem is addressed.
3.5. Summary and conclusions
The work developed in this chapter enables some selected conclusions that impact the methods
selected in BearDyn software. Finding the correct contact pair of points is crucial for the success
of the Dynamic Analysis. Since a deficient detection leads to errors in the simulation, it is important
to find a reliable method that always detect the correct pair of contact points where the contact
force is applied.
For all the geometries tested, Newton-Raphson, fsolve and fmincon methods are able
to always detect the correct pair of contact points if the initial guess is close to the final solution.
In the demonstration geometries generated, the initial guess is selected with the help of a pre-
processing discrete search for the pair of points with the smallest distance. In a Dynamic Analysis
Program, the initial estimates are the results from the previous time step, because they are close
enough to the correct solution the solver is able to converge.
fsolve takes less computation time than fmincon but it is more costly than Newton-
Raphson. However, the Newton-Raphson method requires more a intensive implementation
work. Since in a bearing analysis there are over 1000 different contact searches happening in
each timestep, the fsolve method is more attractive to implement as a first approach. After
working with fsolve, the Newton-Raphson must be implemented in order to improve the
computational time.
34
4. Formulation for the Bearing Contact Detection
In this chapter, the formulation for the geometries used to create the system of nonlinear
equations used to solve the contact detection problem is explained. Different geometries are used
to represent the different contact surfaces. Some surfaces are described with conical or spherical
geometries, such as the raceways, flanges and spherical cap. To reduce the computational effort,
some surfaces are reduced into lines, such as the cage pocket and roller slices. For each contact
type, the geometries that represent the two surfaces are evaluated and, depending on the type
of contact, the system of equations is made of two, three or four equations to solve. All the
necessary dimensions for the models that use the formulation developed here are presented in
Appendix A. The partial derivatives of the vectors formulated in this chapter, needed to build the
Analytical Jacobian Matrices from Eq.(3.5), Eq.(3.7), Eq.(3.10) and Eq.(3.11), are specific for each
geometry and are not presented in this document. Even though the derivatives are time
consuming to calculate, they are not complicated and only require patience by the author.
4.1. Roller Bearing Contacts
The most important roller contacts, in any type of roller bearing, take place between the roller side
and the raceways or flange surfaces and between rollers and cage pockets. In this work, only the
most common and important roller contacts are considered, reason why contact is studied only
between the rollers and each of the bodies directly surrounding them and not between rollers or
between cage and raceways. The contacts considered in this work, illustrated in Figure 30, are:
• Contact between roller and raceways; Contact between roller and inner raceway (Inner); Contact between roller and outer raceway (Outer);
• Contact between roller and flanges; Contact between roller top and right flange (FR); Contact between roller top and left flange (FL);
• Contact between roller and cage; Contact between roller side and cage pocket tops (C2.1, C4.1, C4.2); Contact between roller side and cage pocket sides (C3.1, C3.2).
Figure 30: Contacts considered in BearDyn
Inner
FRFL
Outer
2.1C
3.1C3.1C4.2C
4.1C
3.2C3.2C
c
cc
ii
i
35
4.2. Surface definition
All bearing surfaces are characterized by geometries of revolution, i.e., they are obtained by
sweeping a plane line about an axis of revolution, as illustrated in Figure 31. Therefore, the
coordinates of any point in the surface can be expressed in terms of the parameters that define
the planar line and the sweep angle. For instance, in Figure 31 (a) the line is described as a
function of 𝜉 , defined as 𝑅(𝜉 ) and the sweep angle is 𝜃2.
(a) (b)
Figure 31: Axisymmetric surface obtained as the sweep of a line about an axis: (a) Parametric representation of the line and its tangential and normal vectors in point Q; (b) Surface of revolution, with the sweep angle 𝜃2 and the surface
defining vectors at point Q
In the (𝜉𝜂) plane the position and normal and tangent vectors of point Q are a function
of a single parameter, which in the case illustrated in Figure 31 (a) is the coordinate 𝜉𝑄. The
surface revolution is obtained by sweeping the line around the axis of revolution with the
coordinate 𝜃2. The components of the position, normal and tangent vectors of point Q are
' ' '
2
* *
* * *
*
2 2 2
2 2 2
*
2
;
0
cos( ) cos( ) cos( ) sin( )
sin( ) sin( ) sin( )
; ;
c s( )
o
Q Q Q
Q Q Q Q Q Q
Q Q
R
R
n b
s n n b b t
n b
(4.1)
in which 𝑅 = 𝑅(𝜉𝑄), 𝐧𝑄∗ = 𝐧𝑄
∗ (𝜉𝑄) and 𝐛𝑄∗ = 𝐛𝑄
∗ (𝜉𝑄) are a function of the parameter defining the
sweep line.
4.3. Raceways to roller contacts The first contact is between the raceway and the rollers. Using the slice method, the roller is
reduced to six circles with different radius and longitudinal positions, all defined in the initialization.
The contact happens between a surface and a line. As described in Section 3.4.2, the vectors 𝐬𝑃,
𝐧𝑃 and 𝐭𝑃 are defined for a point P located in a circle representing a roller slice. The inner and
outer ring raceways are segments of cones that, if projected, would meet at a common point on
the main axis of the bearing, as seen in Figure 32. In the inner raceway, the contact takes place
with an external conical surface while for the outer raceway the contact is with an internal conical
surface.
j
j
j
*
*
bQ
nQ
tQ
sQ
Q
2
j
j Q *bQQ
*nQQ
Rr(j)
Q
*sQQ
36
Figure 32: Typical tapered roller bearing with a highlighted cross section
As in Figure 33(a), the coordinate of point Q in the sweep line is related with the apex
position L and r is the raceway angle.
tanr j rR L (4.2)
At point Q, the position of a point in the sweep line that defines the contact cross-section of the
conical surface and the normal and tangent vectors, in the body fixed coordinate system, are
written as
* * *
sin cos
tan ; cos ; sin
0 00
j r r
Q j r Q r Q rL
s n b (4.3)
The body fixed vectors are obtained by sweeping the line around the axis of revolution
with Eq.(4.1) and the inertial coordinates with Eq.(3.2). For the outer raceway, the contact is
interior to the conical surface and the negative of the normal vector must be used.
(a) (b) (c) Figure 33: Contact Point Q and surface normal and tangent vectors in a conical surface for: (a) contact point in the body
fixed 𝜉𝜂 plane; (b) external contact, as in the inner raceway; (c) internal contact, as in the outer raceway
Since one parameter defines de circle and two parameters the conical surface, the
contact points between raceways and roller slices is obtained by solving the system from Eq.(3.6)
DgnDc
c
md
r
r r r
r r
37
4.4. Flange to roller contacts In roller bearings, the dynamics of the rollers leads to occasional contacts between the roller tops
and the flanges that limit the land length of the roller. In tapered roller bearings, the centrifugal
force of the rotating elements forces the roller to a constant contact against the right flange
represented in Figure 34 (a), making this a defining contact of the dynamic response. To the roller
end diameters, in the case of the small end of a tapered roller, the contact with the flange is
achieved with the circular landmark that limits the roller. Due to its spherical shape, the large end
contact with the flange is between a spherical surface and a conical solid.
(a) (b) Figure 34: (a) Contact geometry between rollers and flanges, in which point Q refers to the contact point for a tapered
roller; (b) Detailed geometry of the flanges for tapered roller bearings inner raceway
The flanges contact surface results from sweeping a line segment about the axis of the
roller, with the flange angles and heights provided by the general bearing user data. To
characterize the segment let the radius of the raceway flanges shown in Figure 34 (b) be,
1
/ cos2
i m b kR d D (4.4)
being the remaining geometrical features defined in [22]. The flange angles, in the definition of
the sweeping surface obtained from the revolution of the line segments about the axis of the
raceway, are referred to the orientation of 𝜂 . The intermediate angles and radius necessary to
define such flanges are,
;
sin ; sin2 2
R iR i L iL i
i iiL i i iR i i
l lR R R R
(4.5)
For the right inner flange, the point of contact position and the vectors are written as,
38
* * *
sin1
2 cos cos sin
; sin cos
0 0 0
Ri j iL
R R R
Q j Q R Q R
l R
s n b ; (4.6)
being 𝛼𝑅, 𝑙𝑖 and 𝑅𝑖𝐿 parameters defined in the initialization. For the left inner flange, the equations
are,
* * *
sin1
2 cos cos sin
sin cos
0 0 0
Li j iL
L L L
Q j Q L Q L
l R
n bs ; ; (4.7)
The body fixed vectors are obtained by sweeping the line around the axis of revolution with
Eq.(4.1) and the inertial coordinates of the contact point with Eq.(3.2).
Flange Left
In the case of the small end of a tapered roller, the circular landmark that limits the roller is
obtained with the same equations used for the roller slice, defined in Eq.(3.24) , with the specific
𝜉 position and radius,
/ 2 ; / 2b c LeL R D (4.8)
being 𝐿𝑏 the length of the roller and 𝐷𝐿𝑒 the diameter of the small end. As seen in Figure 35, the
flange depends on two parameters, (𝜂𝑄, 𝜃𝑄) while the circle is defined with only 𝜃𝑃. The system
of nonlinear equations that needs to be solved is a surface to line composed with three equations,
as defined in Eq.(3.6).
Figure 35: Contact detection between left flange and left circular landmark
c
i
ii
P
c
c
Q
d
PbtQ
bQnQ
P
39
Flange Right
The tapered roller top end is, in fact, a spherical surface, with a small curvature, making this
contact different from that of the small top with the other flange. The contact between the tapered
roller large end with the inner raceway flange, is represented as a contact between a spherical
cap and a conical surface, as depicted by Figure 36 (a).
(a) (b)
Figure 36: (a) Typical contact between spherical cap and conical surface; (b) Contact point in the local plane for the spherical cap of tapered roller bearing end
The geometry of the spherical cap is obtained as the surface resulting from sweeping the
arc of circumference, depicted in Figure 36 (b), about the roller axis. In this case, both surfaces
are defined with two parameters: the spherical cap is defined with the angles 𝜑𝑄 and 𝜃𝑄 and the
conical flange with one angle 𝜃𝑃 and a dimension 𝜂𝑃, as seen in Figure 37.
Figure 37: Contact detection between a spherical cap and a conical flange
The position of any point in the arc of circle, and the corresponding normal and tangent
vectors, are written as,
0
* * *
cos cos sin
sin ; sin ; cos
0 0 0
er Q Q Q
Q er Q Q Q Q Q
R
R
s n b (4.9)
The body fixed vectors are obtained by sweeping the line around the axis of revolution
with Eq.(4.1) and the inertial coordinates of the point with Eq.(3.2). In the dynamic analysis, it is
P
bP
tP
nP
dP
P
tQ
nQ
i
ii
Q
Q
bQQ
c
c
c
40
expected that the misalignment of the rollers during the simulation affects the contact points
position. Since these two surfaces are in continuous contact during a simulation of tapered roller
bearings, the normal and tangential forces applied have a big impact in the dynamic response of
the roller.
Previous work [22] shows this contact to be prone to convergence errors due to the large
value of the curvature radius of the roller large top. This problem is due to the radius of the
spherical surface being too large. In Figure 38 (a) the roller bearing with the previous radius of
the spherical cap is represented. As explained in Section 3.1.1 the geometries are not limited to
numerically, which make it possible for the contact point to be detected in a location where there
no material exists. If this situation occurs, the bodies continue to approach and the contact force
is not applied.
Figure 38: Roller side view with different spherical cap radius: (a) Previous value implemented; (b) new radius applied to BearDyn
Figure 39: Roller with basic dimensions and necessary angles to project 𝑅𝑒𝑟
To solve this problem, the radius of curvature 𝑅𝑒𝑟 is projected in order to guarantee that
the contact point is initially in the correct position between flange and sphere, as showed in Figure
38 (b). With the dimensions depicted in Figure 39, some auxiliary angles were defined
01
02
arcsin 2 2 2
2 2 2
i
er
ifr i o
lD
R
(4.10)
P
Rer
Q
PQ
Rer
erR fr
2
1
o ik
i
R
i
41
with 𝐷𝑙 as the diameter in the large end of the roller. The radius 𝑅𝑒𝑟 must be such that the following
condition is verified,
1 2fr (4.11)
The corrected value obtained for radius 𝑅𝑒𝑟 is 𝑅𝑒𝑟 = 0.05 m.
When applying contacts with this formulation for the spherical cap, an additional problem
occurs when trying to solve the system of equations. The contact point in the geometry described
by Eq. (4.9) depends on two angles, 𝜑𝑄 and 𝜃𝑄, that are close to zero and since the curvature
radius is still considerably large. This leads to problems when formulating the Jacobian matrix,
since after a few timesteps a full line in the Jacobian is composed with zeros only, making
convergence to a solution impossible. The conclusion is that then using this geometry, only
fsolve will be able to solve the system. Since this optimization function has multiple methods to
reach a solution, when one fails, the function passes to the next method until it is able to solve
the problem. Both Computational and Analytical Newton-Raphson methods are incapable to
reach to a solution because they depend only on the Jacobian matrix.
4.5. Cage to roller contacts Contacts between roller tops and lines, or very narrow rectangular patches, describe well the
contact between the tops of tapered rollers and the roller pocket of the cage. In what follows it is
assumed that the thickness of the cage is small enough so that the pocket shapes at the cage
mid-thickness are lines that represent the potential contact surfaces. As showed in Figure 40 (a),
the large top of the roller is a spherical surface, reason why its contact is described as between a
surface and a line. The small top of the roller is assumed to be flat, reason why its contact can be
detected as if between a circle and a line with the circumference located at each end of the roller.
This end of the roller can contact the top of the cage pocket either in one point or in a line along
the end as shown in Figure 40 (b). The circle to line contact evaluation is required when checking
for the collisions between the roller, represented by slices, and the side of the cage pocket. In this
case, the circle, representing the slice, contacts with the pocket long side, which is represented
by the line of its mid-thickness. The situation is illustrated in Figure 40-(c).
Figure 40: Contact of circle with line, as in the contact between the roller and the side of the pocket
42
Side contacts
The contact situation between the roller and the side of the pocket is depicted in Figure 41, where
the circle corresponding to the central section of the roller slice approaches the line representing
the mid-thickness of the side pocket. Note that in Figure 41 only the pair of contact of the right
side of the cage pocket with the roller is fully represented. For the left cage pocket side the
situation is identical, and, therefore, not repeated here.
Figure 41: Contact detection between roller and right side of the cage pocket
To describe the line for the side cage the quantities that need to be defined are despicted
in Figure 41, being
1
tan2
/ 22sin
1 / 2
ou inc l c
wp
c
c m p
R RR W
P
R
i
(4.12)
where 𝛼𝑚, 𝛼𝑐, 𝑃𝑤 and 𝑊𝑙 are dimensions given as features of the roller bearing. The position of
the contact point along the line that represents the left side of the cage pocket and its tangent
vector are given as a function of the parameter 𝜉𝑐, which is also the coordinate of the point in the
body fixed coordinate frame. The general equation for position and tangent vector of a line along
an axis are,
'' ''0 0 ; 1 0 0TT
P c P
s b (4.13)
By transposing and rotating the line, the position and tangent vector in the cage local axis for the
left line are obtained as,
i
ii
Q
d
c
c
cR
c
0c
sQ
p
bP
bQ
PsP
c
c
Q
43
' ''
' ''
left
left
P l lef
lef
t
t
P
left PP
s s A s
bAb (4.14)
The same procedure is applied for the right line, using 𝐒𝑙𝑟𝑖𝑔ℎ𝑡
and 𝐀𝑟𝑖𝑔ℎ𝑡. The position vectors 𝐬𝑙𝑙𝑒𝑓𝑡
and 𝐬𝑙𝑟𝑖𝑔ℎ𝑡
are,
00
cos ; cos
sin sin
left right
l ou c l ou c p
ou cou c p
R R
R R
s s (4.15)
and the direction of the rotation is given by the transformation matrixes 𝐀𝑙𝑒𝑓𝑡 and 𝐀𝑟𝑖𝑔ℎ𝑡, created
with the Euler parameters as in Eq. (2.3). The vectors and angles to compute the parameters are,
coscos
sin cos sin cos
sin sin sin sin
right left
cc
c c c c p
c cc c p
right c left c
A Au u (4.16)
The definition for the side cage lines allows to simulate bearings with different numbers
of rollers. The contact to solve is the between two lines, with the vectors 𝐬𝑄′ and 𝐛𝑄
′ as defined in
Eq. (3.24) and the system is defined by Eq.(3.8), with the parameters (𝛽𝑄, 𝜃𝑃).
Pocket top contacts
The potential contact configuration between the tapered roller large end and the top of the cage
pocket is depicted in Figure 42, where the spherical cap corresponding to the end section of the
roller approaches the line representing the mid-thickness of the top of the pocket. The spherical
geometry is the same used in the right flange contact, described in Section 4.4.2., where vectors
𝐬𝑄′ , 𝐧𝑄
′ , 𝐭𝑄′ and 𝐛𝑄
′ are obtained. In order to develop the cage pocket configuration, it is assumed
that the center of mass of the cage, to which the body fixed coordinate frame (𝜉𝜂𝜁)𝑐 is attached,
is located in its geometrical center. With reference to Figure 42, 𝜉0𝑐 is half of the pocket long side
dimension, 𝑅𝑐 is the mid-thickness radius at the level of the pocket top and 𝛽𝑃 is the angular
positions of the contact point P on the pocket top, with respect to the body fixed direction 𝜂𝑐.
In order to define the contact conditions, the positions of the potential contact points in
the line that defines the pocket top and the tangent vectors at such points are written as,
0
' '
0
cos ; cos
sin sin
P ou P P P
ou P P
c
R
R
s t (4.17)
44
being 𝑅𝑜𝑢 the cage outer diameter and 𝜉0𝑐 half the cage width. The system of equations that need
solving is a surface to line contact composed with three equations, as defined in Eq.(3.6), with
the spherical cap depending on two parameters (𝜑𝑄 , 𝜃𝑄) and the circular line defining the cage
with one parameter, 𝛽𝑃.
Figure 42: Contact detection between spherical cap in roller and large top of the cage pocket
The small top of the roller is assumed as flat, reason why its contact can be identified as
a contact between a circle and a line, with the circumference located at each end of the roller.
The potential contact configuration between the tapered roller small end and the top of the cage
pocket is depicted in Figure 43, where the circle corresponding to the end section of the roller
approaches the line representing the mid-thickness of the top of the pocket. Two potential points
of contact, 𝑄1 and 𝑄2, may develop.
Figure 43: Contact detection between left circumference in roller and small top of the cage pocket
For the circumference definition, only 𝐬𝑃′ is needed, which is formulated in Eq.(3.24) for a
general slice in a roller. The cage small top pocket coordinates and vectors are formulated
tQ
nQ
i
ii
Q
Q
bQ
Q
c
cc
P
ouR
P
bP
0c
d
c
c
i
i
i
2Q
1Q
2d
2Q
tQ22P
1P 1P
Q2s
2bQ
bQ1
2P
1d
1Q
tQ1
nQ
45
similarly to those of contact with the large end of the roller. Two additional vectors are added: 𝐭𝑄′
is necessary to formulate the system of equations and 𝐧𝑄′ to give the direction of the normal
contact force, if interference exists. These quantities are given by,
0
' ' ' '
0 0
cos ; ; sin ; cos
sin 0 cos s
1
0
in
in Q Q
c
Q Q Q Q Q
in Q Q Q
R
R
s n b t (4.18)
To solve the contact between two lines, the equations must be formulated according to
Eq.(3.9), with the parameters (𝜃𝑃 , 𝛽𝑄). The same system is used to find both pair of points,
meaning that the equations are solved two times, expecting different results in each. To make
sure that the solvers don not converge to the same solution, twice, leaving one possible pair of
contact points undetected, the initial guess for the parameters must lead to different solutions
each time that the system is solved. By giving in the initialization an initial guess of the parameters
close to the wanted solution, the system converges correctly and this issue is not problematic
during the simulation.
46
5. Contact Forces
In the roller bearing contact problem, the dimension of the contact area is small when compared
with the typical dimensions of the contacting bodies. Hence, the normal loads that develop in the
contact patch can be replaced by normal forces. According to the Hertz theory [23], used here to
study the contact problem, the dimensions of the contact area are only dependent of the normal
force, the material properties and the surfaces curvature of the contacting bodies, being
independent of the tangential forces that develop in the contact interface. Therefore, the normal and
the tangential contact problems are decoupled and their solutions are treated sequentially. This is
a common approach for solving multibody systems with contact, as seen in the works [19], [20] and
[34], [35].
Besides the normal forces that develop during contact, also tangential forces due to
friction or to the lubrication fluid develop between the contacting bodies. The tangential forces are
also applied in each pair of contacting points P and Q and their value is proportional to the normal
force developed during contact. For the evaluation of normal and tangential forces it is necessary
to calculate the relative velocity between the bodies, namely its projection on the surface tangent
to the contacting bodies on the points of contact and its projection on the normal vector to the
surface, as defined for DAP_Roller in Section 3.4.3.
5.1. Normal contact forces
Before studying the contact between two bodies, some definitions have to be introduced. A
contact is said to be conforming if the surfaces of the two bodies fit exactly or even closely together
with deformation. Bodies that have dissimilar surfaces, such as rolling-element bearings, are said
to be non-conforming. When two non-conforming solids are brought into contact, they touch
initially at a single point or along a line [34]. Under the action of a normal load, the non-conforming
elastic bodies deform in the vicinity of the first contact point so that they touch over an area. Figure
44 shows typical contact patches of wheel-rail contact and roller bearings.
Generally, this contact area is small when compared with the typical dimensions of the
contacting bodies, such as the radius of curvature of the surfaces near the contact. The contact
stresses are highly concentrated in the region close to the contact and decrease rapidly in
intensity with the distance from the contact point. The stress distribution over the contact area, or
contact patch, is described by Hertz elastic contact theory if some conditions are met [23].
In order for the Hertz elastic contact theory to be valid it is required that:
• All deformations must be within the linear elastic limits, i.e., the strains are small and the
body stresses can be described by a linear elastic constitutive relation;
• All shear stresses are neglected, i.e., the loading is assumed normal to the contacting
surfaces and, therefore, the contact is frictionless;
47
• The dimensions of the contact patch are small when compared to the dimensions of the
surfaces in contact, i.e., the contact area is much smaller than the characteristic radii that define
the surfaces curvatures, implying in turn that the surfaces are continuous and non-conforming.
According to Hertz, the contact force 𝑓𝑛 follows the relation,
n
nf K (5.1)
where 𝛿 is the amount of penetration, or indentation, between the surfaces. In such theory of
contact is required to predict the shape of the contact area, its growth with increasing loads, the
magnitude and distribution of the surface normal stresses and, eventually, the tangential tractions
that are transmitted across the interface. Different contact patches generate different contact
forces. Since it is very likely that multiple contacts occur at same time, it is important to guarantee
that the system remains stable and the variation of the contact forces applied are not too steep
and generate high frequencies that the ODE solver is not able to integrate.
(a) (b)
(c) (d) Figure 44: Contact patches of Hertzian contact force models: (a) Elliptical contact; (b) Point Contact; (c) Elliptical
Contact; (d) Line Contact (Adapted from [21])
Consider now a situation for which the contact between the two spheres is caused by a
direct central collision. In general, the two spheres do not rebound with the same initial velocities,
because part of the initial kinetic energy is dissipated in the form of permanent deformation, heat
develops, etc. It is evident that the contact force model of Eq.(5.1) cannot be used during both
phases of contact, i.e. compression and restitution, since this would suggest that no energy is
dissipated in the process of impact. To model such effect, the Kelvin Voigt Contact Model is
considered, as described in Eq.(3.30). Several values for restitution coefficient, 𝑐𝑒, are used being
concluded that only a small portion of the kinetic energy is dissipated during contact, meaning that
the coefficient is higher than 0.95. More complex models for contact force such as Lankarani et
al.[35] contact model, considering not only the restitution coefficient but also the relative approach
velocity between the bodies just before the impact can be considered in general.
48
Point Contact
In BearDyn, the point contact between two surfaces appears when the right spherical cap from
the roller interferes with the right flange and cage large top, and the circular landmark that limits
the roller with the left flange. The undeformed geometry of nonconformal contacting solids can be
represented in general terms by two ellipsoids, as shown in Figure 45.
(a) (b) Figure 45: (a) Geometry of contacting elastic solids; (b) Stress distribution and patch geometry (Adapted from [21])
Let the interference, or compression, between two surfaces be described by 𝛿, which is
found using Eq.(3.4). The normal contact force, in the case of point contact is
3/2
n ptf K (5.2)
where the contact stiffness 𝐾𝑝𝑡 is given by [36],
1/2
34.5ptK E
(5.3)
being the characteristic radii of the contacting surfaces described in Figure 45 (b), 𝜅 = 𝑎/𝑏, with
a and b being the semi-axis of the contact ellipse and 𝔼, 𝔽 are complete elliptical integrals of first
and second kind [4]. Using a least-square relation, Brew and Hamrock [37] find an approximate
expression for the elliptical integrals and for 𝜅 as,
0.0630
1.0339
1.5277 0.6023ln
1.0003 0.5968ln
y
x
y
x
x
y
R
R
R
R
R
R
(5.4)
49
where,
11
1 1 1 1 ; x y
ax bx ay by
R Rr r r r
(5.5)
and,
x
y
R
R (5.6)
The equivalent modulus 𝐸′ is given as,
12 2
1 2
1 2
1 12E
E E
(5.7)
where 𝐸1, 𝜈1, 𝐸2 and 𝜈2 are the Young’s modulus and Poisson’s ratio of the material of each body
in contact, respectively.
For simplification, contact patches are assumed as circular, 𝑎 = 𝑏, for all the point contacts.
Two different materials exist in the Roller Bearing, with different properties: one type of steel for the
raceways and rollers, referred to as material 1, and the other for the cage, with a different Young
modulus and Poisson’s ratio, material 2. As a result, two stiffness for the point contact are computed,
for contacts between the same material that happen between flanges and rollers and for contact
between the roller and cage. The resulting stiffness are presented in Table 4.
Type 𝑲𝒆𝒑
material 1 – material 1 1 2.2523 x 1011 material 1 – material 2 1 2.8565 x 109
Table 4: Stiffness for point contact used in BearDyn
The evaluation of the maximum stress developed during contact is also of interest since
it is used in the evaluation of tangential forces caused by lubricated rolling contact. For a point
contact the maximum stress in the contact patch, shown in Figure 43, is written as [38]
3
2
nmax
f
ab
(5.8)
with the dimensions of the semi-axis of the elliptical contact patch found as [36]
1/32
1/3
6
'
6
'
aE
bE
(5.9)
where a is referred to as the semimajor and b the semiminor axes of the contact ellipse developed.
50
Line Contact
The ideal line contact, illustrated in Figure 46, can be understood simply as a case where 𝑏 ≫ 𝑎.
However, the relation between indentation and normal force becomes, in this case, a nonlinear
relation that requires an iterative procedure to obtain the normal force when the indentation is
known. Several elastic contact models for cylinders with parallel axis based on the Hertz elastic
contact theory have been proposed in [21] and [22].
Figure 46: Ideal Line Contact between two bodies
In this work, the method used to calculate the normal force in lines is the more compatible
with the slice method, used for contact detection, that does not require large computational effort.
Based on laboratory testing, Palmgren [39] developed a relation, which is the basis of current
contact models in roller bearing line contact, written as
99
2 1010
8
10
2 13.81 n
ef
f
EL
(5.10)
where 𝐿𝑒𝑓 is the length of the line. Rearranged and having the normal force written as a function
of the indentation, or relative elastic approach, is
* 8/9 10/90.71069n eff E L (5.11)
In order to apply the normal contact force model for line contact to the roller contact with
the inner and outer raceways and with the sides of the cage pocket, the roller is discretized in 𝑁𝑠𝑙
strips and the contact force is evaluated based on the strip indentation. The normal contact force
of a strip of the roller is given by Palmgren’s simplified equation,
1/9 8/9 10/9 1,...0. ,356ns ef lsl sf sL NE N (5.12)
where the counter s refers to the number of the slice in each of the rollers. Note that this equation
can be rearranged in the form 𝑓𝑛𝑠 = 𝐾𝑒𝑙𝛿𝑛 where
1/9 8/90.356el sl efK E N L (5.13)
a
b
51
As described in the case of the point contact, two different materials exist in the Roller
Bearing with different properties, leading to different values for the stiffnesses. In the contact
between raceway and roller the material is the same, while in the contact between side cage and
roller the two bodies have different materials. The used stiffnesses in BearDyn are presented in
Table 5.
Type 𝑲𝒆𝒍
material 1 – material 1 4.1205 x 109 material 1 – material 2 5.225 x 107
Table 5: Stifness for line contact used in BearDyn
In the case of line contact, the evaluation of the maximum stress developed during contact
is given by the equation,
2
nmax
ef
f
L b
(5.14)
With the dimensions of the dimension b of the contact patch found as,
1
2
0.00335 n
ef p
fb
L
(5.15)
where,
1 1 1 1
p
ax ay bx byr r r r (5.16)
In a line contact, between cylindrical surfaces, 𝑟𝑎𝑦 and 𝑟𝑏𝑦 have a high value and are considered
as infinite. The values used in BearDyn for the radius of each surface implemented are presented
in Appendix A, meaning that the contacts considered have different values for maximum stress.
As for the case of the point contact, the evaluation of the contact stress does not play a role in
the evaluation of the normal contact force but it is required for the evaluation of the tangential
contact forces when in the presence elastohydrodynamics lubrication.
Figure 47: Geometric radius in a line contact
axr
bxr
52
5.2. Tangencial forces
Surfaces in rolling contact, subjected to normal contact loads are also subject to tangential forces
due to friction, in the case of dry contact such as for rail-wheel interaction, or due to the lubricant,
in the case of lubricated contact as in normal roller bearing applications. Regardless of the type
of contact, the relation between the tangential forces and the normal contact forces is given by,
t nf f (5.17)
where 𝜇 is the equivalent friction coefficient. Note that the tangential force is applied in the
opposite direction of the relative velocity between the contacting surfaces, obtained by Eq. (3.28)
The simple form of Eq. (5.17) hides the complexity of the calculation of 𝜇 for many
important tangential forces, as in the case of lubricated contact. In the case of lubrication between
the contacting surfaces, depending on the lubricant film thickness and on the roughness of the
contacting surfaces the type of contact is different and the equivalent friction coefficient has to be
evaluated differently. Figure 45 shows the different contact modes, from dry contact through full
fluid film lubricated mode.
(a) (b)
(c) (d)
Figure 48: Types of contact ( Boundary lubricant layer, Lubricant): (a) Dry contact; (b) Boundary mode; (c) Mixed mode; (d) Full fluid mode (Adapted from [21])
For lubricated contact, the relation between the fluid film thickness and the roughness of
the contacting surfaces defines the type of contact mode that is taking place. Let the lubricant film
parameter be defined as,
2 2
1 2
Λ ch
(5.18)
where ℎ𝑐 is the lubricant film central thickness and 𝜎1 and 𝜎2 the roughness of the contacting
surfaces. Assuming for each lubrication mode a different equivalent friction coefficient, the relation
between in Eq. (5.30), such lubrication mode equivalent friction coefficients is written as [40,41]
53
bd6
fm bd fm6
bd fmfm
Λ Λ
Λ Λ Λ Λ ΛΛ Λ
Λ Λ
bd
bd fm
fm
fm
(5.19)
The typical values for the lubricant film parameter used to define the transitions between
the different lubrication modes are Λbd = 0.01 and Λfm = 1.5 [40]. The evaluation of the equivalent
friction coefficient required the prior evaluation of the lubricant film parameter, which in turn imply
the calculation of the lubricant film thickness, and the equivalent friction coefficients for each mode
of lubrication, 𝜇𝑏𝑑 and 𝜇𝑓𝑚. A sequence of the steps to be taken to theoretically calculate the
equivalent friction coefficient can be resumed in the following sections.
Calculation of the lubricant film thickness, 𝒉𝒄
Being the contacting surface roughness known, either from direct measurement of the rolling
elements or by published data, the calculation of the lubricant fluid filme thickness plays the
central role in the decision on the lubrication mode experienced by the rolling elements.The
lubricant film thickness varies along the contact region, being generally of importance its
calculation in the center, designated by ℎ𝑐
c iso t sh h (5.20)
where ℎ𝑖𝑠𝑜 is the isothermal central lubricant fluid thickness for fully flooded lubrication and 𝜙𝑡 and
𝜙𝑠 are the thermal reduction factor and starvation factor, respectively. In this work both factors
are not considered, since the information on the temperature and flooding conditions are
inexistent. Further information on both factors can be found in [21] and must be used in future
works.
Calculation of the adimensional parameters
The evolution of the lubrification forces requires the knowledge of the velocity, normal force,
material and geometry of the contacting surfaces. To this, the following non-dimensional
quantities are defined:
Speed parameter 0
0
uU
E R
(5.21)
Load parameter
0
qW
E R (5.22)
Material parameter 1 0G E (5.23)
54
where R is the effective radius in the rolling direction, 𝐸0 the equivalent modulus of elasticity and
û the average velocity of the contacting surfaces, all defined as
2 2
1 20
1 2
1 2
1 12
1
2
ax bx
ax bx
r rR
r r
EE E
û u u
(5.24)
with 𝑢1 and 𝑢2 the velocities evaluated at points P and Q by Eq. (3.25) and the curvature radius
𝑟𝑎𝑥 and 𝑟𝑏𝑥 defined in Appendix A.
Calculation of isothermal central lubricant film thickness
The isothermal central lubricant film thickness is given by,
iso ch H R (5.25)
according to the recommendations by Harada and Sakaguchi [40] the central lubricant film
thickness is calculated by the maximum of
max ,c em rmH H H (5.26)
Under the assumption of the elastohydrodynamic lubrication theory, Chittenden et al. [42] propose
that the central lubricant film thickness parameter is evaluated as
2/30.68 0.49 0.073 1.234.31 1 k
emH U G W e (5.27)
in which 𝑘 = 𝑎/𝑏. For convenience, the ratio between the major and minor semi-axis of the
contacting ellipse may be written as [43]
0.6361.0339 rk (5.28)
with 𝛼𝑟 = 𝑅𝑦/𝑅𝑥. For hydrodynamic lubrication, in which the contacting surfaces are assumed
rigid, Brewe et al. suggest that the contact film thickness parameter is written as [44]
2
1128 0.131tan 1.6832
rrm r
UH
W
(5.29)
where the modified factor for side-leakage, 𝜑 is given by
1
21
3 r
(5.30)
55
Calculation of the equivalent friction coefficients
The evaluation of the tangential forces at contact, using Eq. (5.17) requires the evaluation of the
equivalent friction coefficient using Eq. (5.19). In order to evaluate the equivalent friction
coefficient for the specific type of contact it is necessary to evaluate the equivalent friction
coefficient for boundary lubrication mode, 𝜇𝑏𝑑, if the lubricant film parameter is Λ < Λ𝑏𝑑 and the
equivalent friction coefficient for full film mode, 𝜇𝑓𝑚, if the lubricant film parameter is Λ > Λ𝑓𝑚.
In the lubrication boundary mode the equivalent friction coefficient, 𝜇𝑏𝑑, proposed by Kragelskii
[45] as,
181.460.1 0.1 22.28 s
bd s e (5.31)
where the slip ratio s is given as,
1 2u u
sû
(5.32)
Note that for a slip ratio 𝑠 = 0 the equivalent friction coefficient is 𝜇𝑏𝑑 = 0 while for very large slip
ratios it tends to 𝜇𝑏𝑑 → 0.1.
The full-film mode equivalent friction coefficient, µ𝑓𝑚, is given as
* *
2*
0 1 2 2* 2 *
83 2sinh Φ Φ
max max
f
fm
max max
Ke e
h
(5.33)
where the dimensionless parameter 𝜓 is given by,
* *
8
o
f
sK
(5.34)
the maximum stress, 𝜎𝑚𝑎𝑥 , given by Eq. (5.8) for point contacts and Eq. (5.14) for line contacts,
𝛼∗, 𝐾𝑓 , 𝛽 and 𝜇0 are parameter from the roller bearing information and the function Φ(𝜓) is defined
as,
21
0
sinhΦ '
'
y
d
(5.35)
In the computer applications the integral in Eq. (5.35) can be tabulated being such table
interpolated during the dynamic analysis.
56
6. Computational Implementation
After validating the results from DAP_Coin and DAP_Roller, the more complex geometries in
BearDyn are approached and all the formulation for contact detection in a single row tapered
roller bearing implemented. The dynamic analysis created in MATLAB® includes functions
developed by M. Lima [22], functions form the MATLAB® database such as ODE45 and fsolve,
and functions developed by the author.
The BearDyn program is complex due to the multiple contact detection needed at each
time step, but, in general, follows a simple structure. The core of the program is the integration
process, which evaluates a number of functions to obtain the vector �� to be integrated at each
time step, as described in Chapter 2. A scheme of the program main structure can be seen in
Figure 49, where the functions highlighted are the ones developed in this work, Force Vector,
Write Output, or the ones corrected, i.e., Initialization. The program is developed with the
possibility to add more bearing types, such as cylindrical bearings, with single or double rows.
Figure 49: Scheme of BearDyn code main structure
The BearDyn program uses text files as input, giving the information about the geometry
of the bearing, integration parameters, material properties, lubricant parameters, as well as the
forces resulting from the wheel-rail contact to be applied on the inner raceway. With this
information, all the initial conditions are settled, such as initial positions, orientations and velocities
of the bodies. These initializations are used as starting conditions for the integration, which calls
the function Funceval, where the accelerations and velocities of the bodies are evaluated and
then integrated by ODE45, leading to the positions and velocities. All this implementation work is
Entry
Files
Read Files
Initialization
Integration FunEval
Auxiliary vector q
Mass matrix M
Obtain
�� =1
2 𝛚
′
Force Vector g
Solve System
Form auxilary
vector ��
Write Output
Output
Files
Mq g
57
developed by Lima [22], and some functions from the initialization are updated here to introduce
new initial estimations of contact points and some necessary information for the contact force
evaluation.
At each timestep, all the forces applied to the bodies are evaluated and placed in the
force vector. The forces involved in the analysis are Gravitational Forces, Gyroscopic Forces,
Load Forces and Contact Forces, being the latter more complex requiring most of the
computational effort.
6.1. Contact Implementation
Each roller has 29 possible contact points that need to be detected individually in each timestep.
When working with a full single row roller bearing, the number rises to 319 possible contact points
to evaluate. In addition, as showed in Chapter 4, different geometries involved in the contact need
different nonlinear system of equations. For all these reasons, it is important to implement all the
particularities in the most simple and efficient way, with a special attention to reduce the
computational effort involved. All the previous work with contact detection and dynamic analysis
is crucial to reach this phase with a better understanding of the contact detection models and how
to implement them in the best way possible.
The function F_Contact is where all the contact detection and forces application occurs,
being its structure presented in Figure 50. Since each contact point pair is calculated individually,
first the program defines a set of variables necessary to formulate de system of equations and
selects the solver to be used. The program, depending on the variable method_flag entered by
the user, has three different methods to solve the system: fsolve, Numerical Newton-Raphson
and Analytical Newton-Raphson. The difference between the two Newton-Raphson methods is
the calculation of the Jacobian Matrix. In the numerical method, the matrix is obtained with Eq.
(3.15) by calculating the derivatives in a selected timespan, while in the Analytical Newton-
Raphson uses the analytical derivatives from the system, programmed specifically for each
contact type into the code.
Each timestep, F_Contact starts by solving each contact detection individually and if
the contact exists (𝛿 < 0), proceeds to calculate the normal and tangential contact forces, based
on the indentation and lubrication conditions, and applying the forces and resultant torque to the
force vector. The function then proceeds to the next contact points until all of them are evaluated
for that timestep.
Even though there are three methods implemented to solve the contact detection
equations, the system formulation is the same and some common general functions are used in
different methods. Both fsolve and Computational Newton-Raphson use the function
CostFunction, were the system of equations is formulated. As seen in Figure 51, the function
starts by calling the function Geometry two times, one for each geometry involved in the contact.
58
Figure 50: F_Contact function structure
The function Geometry is created to store the vectors required for the formulations for
the different geometries, required to create the cost function, and, in the case of the Analytical
Newton-Raphson method the vectors derivatives that are included in the Jacobian matrix. As
input, the function receives three flags (Type, Rows and Position) that identify the geometry
required for that particular contact and one flag method_flag identifying if the vectors derivatives
are needed. As output, the function delivers the necessary information to formulate de system of
equations and if needed, the Jacobian matrix. Table 6 shows the geometries implemented and
the correspondent flags.
All rollers
evalutated
Enter F_Contact
Define bodies involved i1,i2,
positions p1,p2, method_flag and
system_flag
fsolveNumerical Newton
RaphsonAnalitical Newton
Raphson
𝛿 < 0
yes
No
method_flag
Evaluate and Apply
Contact Forces for the
Points
All contacts
evaluated
Return
yes
No
No
yes
59
Figure 51: CostFunction function scheme
Type Flag Rows Flag Position Flag
Tapered 1 1 Row 0 Circle Slice 0 Inner Raceway 1 Outer Raceway 2 Spherical Cap 3 Right Flange 4 Left Flange 5 Left Circle to Flange 6 Cage Side Left 7 Cage Side Right 8 Cage Large Top 9 Cage Small Top 10 Left Circle to Cage Small Top 11
Table 6: Flags DataBase for vectors from geometries implemented
Depending on the system_flag decided in F_Contact, the function will formulate the
correct system of equations for this specific contact detection. As seen in Chapter 4, between all
contacts formulated there are only four systems of nonlinear equations to be solved for the
different geometries, depending on the number of variables and the contact nature.
In the Analytical Newton-Raphson, the function Jacobian is used to output the cost
function and Jacobian to solve. This function has a structure similar to CostFunction, presented
in Figure 51, with an additional process: when calling Geometry, the vectors derivatives are also
evaluated, and with that, depending on the system_flag, the function evaluates not only the cost
function but also de Jacobian Matrix for the system under analysis.
Enter CostFunction
Calculate vectors in Geometry for
position 1, p1
Calculate vectors in Geometry for
position 2, p2
system_flag
Solve system:
𝐧 ∗ 𝐭𝑃
𝑇 ∗ 𝐭
𝑇 ∗ 𝐛𝑄
=0
Solve system:
𝐧 ∗ 𝐭
𝐧 ∗ 𝐭
∗ 𝐛
𝑇 ∗ 𝐛𝑃
=0
Solve system: 𝑇 ∗ 𝐛𝑄
𝑇 ∗ 𝐛𝑃
= 0
Solve system: 𝑇 ∗ 𝐛𝑄
𝑇 ∗ 𝐭𝑄= 0
Return
60
6.2. Time integration method
The time integration method used to solve the equations of motions over time of the multibody
system in BearDyn is selected among the ordinary differential equation solvers available in
MATLAB®. The time integration method used from MATLAB® is an ordinary differential equation
solver known as ODE45, which is recognized for self-adjusting the integration time-steps over time
and realizing several integration searches to obtain successful results. Even though the user can
define the time-steps from which to select results, it does not mean that the integrator exclusively
calculates results in those points.
In ODE45, it is known that the time step adjusts itself to the frequency contents of the
system time response, using the predictor-corrector method. The initial, "prediction" step, starts
from a function fitted to the function-values and derivative-values at a preceding set of points to
extrapolate ("anticipate") this function's value at a subsequent, new point. The next, "corrector"
step refines the initial approximation by using the predicted value of the function and another
method to interpolate that unknown function value at the same subsequent point. A detailed
overview of the different integration methods and their use is presented in [26]. The consequence
is having time steps which are calculated being called trials, opposed to those which are selected
from the integrator and called successful time steps.
In order to report forces and contact points happening during the simulation, it is important
to identify the successful timesteps and guarantee that the values obtain in the so-called
“prediction” steps are not considered. This is achieved by setting a solver output property with
odeset, given as OutputFcn. This property controls the output that the solver generates, as a
function handle which is called after every successful integration steps only. In this form, a handle
function is created to retrieve the struct CP_Results from Funceval when the integration step
is successful and ignores its update when the contrary occurs. The struct CP_Results reports
the contact points happening in BearDyn for all bearings, as well as the correspondent Normal
and Tangent forces in the vector form.
This problem is characterized by a highly non-linear behavior, since forces are
proportional to the penetration between two bodies and multiple forces can happen rapidly in
multiple contact points at once. If the solver tries to make a large time step it can happen that the
penetration at a time is extremely high and the force applied unrealistic, which ultimately
destabilizes the system. This situation can be avoided by defining a maximum permitted time
step, so contact is detected at its early phase, allowing only for small penetrations. This value is
also set in ODE45 options, with the value of 10−5 s. Similar works use timesteps between 10−4
and 10−5 seconds [20]. With this value, the number of predictive steps also reduces and less time
is spent doing such calculations.
61
6.3. Verification Methods Since there are no other available similar programs or studies which can be used to compare
results and experimental procedures, the verification is made in two parts: first with MATLAB® a
visualization tool, developed to represent the geometries in the roller bearing and contact points
detected in the initial timestep, then with a dynamic visualization tool named SAGA. The
verification methods follow a sequential order, were the next problem is only addressed after the
previous one is successfully implemented and verified.
Contact Detection is verified using MATLAB® representation for the geometries and the
points of close proximity. A representation function is developed to represent a complete single
row tapered roller bearing with four rollers in the initial position, with small spheres representing
the pair of points of close proximity, evaluated in BearDyn for the first timestep. The input
structures and contact detection functions used are the same as in BearDyn, meaning that in
reality this function serves as an incubator for the development of the contact detection for the
dynamic analysis program. Step by step, all the implemented contact formulations in the function
geometry are tested with this tool and once the functions work correctly, they are copied to the
main program. In this way it is possible to validate the geometries used and detect flaws and
correct them. The code structure for this program is presented in Appendix B. Figure 52 shows
the representation of the complete roller bearing without contact points represented. The rollers
are represented with blue lines, including the circular lines used in the slice methods and the
spherical cap for used the large top, at grey the inner raceway with the two flanges and the cage
is represented with the black lines, including the two circles for the tops and the lines for the side
pockets.
Figure 52: Visualization Tool developed in MATLAB®
After validating the initial position for the bodies and initial contact point positions,
the dynamic analysis can be carried. The most helpful and adequate method encountered
to help verify the veracity of the simulation results for more than one timestep is the
dynamic visualization tool SAGA, which stands for System Animation for Graphical
62
Analysis. SAGA is a tool which uses files with information on position and orientation of
each body of a system over several steps of time for the display of geometric models
created in SolidWorks®, or in an equivalent program. For this purpose, a single row
tapered roller bearing model is modelled in SolidWorks® using the real geometric
properties, presented in Appendix A. The model created for SAGA is shown in Figure 53.
The flanges in the inner raceway are omitted for a better visualization of the rollers and,
posteriorly, the pair of contact points.
Figure 53: Roller bearing displayed in SAGA
In order to add more detail to the visualization, the pair of possible contact points are also
represented for one roller, an approach first developed in [32] for Railway Contacts and now
adapted for Roller Bearings . A display example of the tapered bearing model obtained with SAGA
at the initial timestep with the potential contact points represented is showed in Figure 54, with
the cage geometry not being displayed, but the contact points for the side and top represented. If
that pair of points is in indentation, the spheres are represented in red. If the points are only the
points of close proximity and not being in contact, the spheres are represented in green.
Figure 54: SAGA representation of the contact points in one roller for the cage, top flange and inner raceway
Figure 54 is the frame for the initial timestep, reason why all contact points represented
are green. Since the distances between points are very small, the two spheres from one contact
63
detection in some cases appear to be only one sphere. Adding the colour difference allows to
identify interference and where the contact forces are being evaluated. The necessary files and
method used for the point detection representation are explained in Appendix B.
SAGA is a helpful tool to verify the results qualitatively, since it is possible to see if the
bodies shift position and orientation according to what is expected and with the points
visualization, if contacts occur at the expected areas where these are detected and the forces are
well calculated. With this method, a solid understanding of the bearing behaviour and of the
conditions applied to the system is obtained, as well as a good critical capacity to interpret results,
identify problems and to make decisions about corrections and modifications.
64
7. Results and Discussion
The BearDyn program results from the implementation of the different methodologies proposed
and developed in this work. Different preliminary results and verification methods for BearDyn
allow for a better understanding on how the implemented models work together and how to
achieve the computational effectiveness of BearDyn on limited computational resources. Even
though the program still has stability problems, it is still possible to obtain results for different
cases and with the results to illustrate problems which ultimately help understanding the
difficulties. For this purpose, the contact detection is verified with the representation of the
obtained contact points and the dynamic analysis of a tapered bearing is carried here with
BearDyn, being results discussed.
7.1. Contact detection with visualization tool
With the visualization tool it is possible to represent all the geometries and all the pairs of contact
points for one roller. Figure 55 shows the geometries implemented in BearDyn, with exception to
the Outer Raceway. With the blue lines, all the geometries describing surfaces in the roller are
presented, discretised Figure 55 (a). In grey lines, the surfaces of the inner raceway and in black
lines the cage, both explained in Figure 55 (b).
(a) (b)
Figure 55: Representation of the geometries used in BearDyn: (a) Roller geometries; (b) Inner raceway and cage
geometries
After the evaluation of contact at the first timestep, the results are saved and represented
in the visualization tool. All the contact points implemented, with exception to the contact between
roller slices and outer raceway, are represented in Figure 56. The possible interferences from
roller slice to inner raceway, left flange to roller left circular landmark and roller spherical cap right
to flange right are displayed with a red dot in Figure 56 (a). Note that the right flange point is being
evaluated in the correct position, meaning that the spherical cap radius is well calculated, as
explained in Section 4.4.2. Figure 56 (b) was the representation of the points evaluated in the
cage large top, right side, left side and the two contact points in the small top, all possible contacts
with the roller geometries.
Flange Left
Cage Large Top
Cage Small Top
Flange RightCage Side
Left Circle
Roller Slice
Spherical Cap
65
(a) (b) Figure 56: Roller with all the geometries used in contact detection represented (with exception to the Outer Raceway)
with the contact points for: (a) Inner Raceway and Flanges; (b) Cage Sides and Cage Tops
7.2. Simulation Conditions of the Dynamic Analysis
After showing the feasibility of the contact detection, the dynamic analysis is addressed. Firstly,
defining the conditions of the simulation is necessary. The load resulting from forces external to
the roller bearing considered here is caused by the weight of the train car, while travelling at the
operational velocity of 50 km/h. This loading, which represents the average vertical force on each
axle bearing, must in future studies, be replaced by a more realistic force resulting from the wheel-
rail interaction and transferred from the wheel to the axle. Knowing that each railway vehicle is
supported by two boggie, which distribute the vehicle weight over its components, the resulting
load is considered as being applied equally and directly over the axles.
The inner raceways of the roller bearings are rigidly fixed to the shaft of the wheelsets, being
assumed that the proper fraction of load resulting from the weight of the train is applied directly
to the center of mass of the roller bearing inner raceway. For a common passenger train car,
weighing approximately 40 tons, with 8 axle bearings, a force of 50 kN is applied downwards in
the vertical direction, Z, in the inner raceway of the bearing. Usual axleboxes are fitted with
double-row rollers, but this simulation only considers a single row bearing. For this reason, the
total load under a single row bearing can be approached to half of the total load, 25 kN. Since the
inner raceway of the roller bearing is fixed to the wheel shaft, while the outer raceway to the
axlebox, a constant angular velocity is applied to the inner raceway and the outer raceway is fixed
to the inertia frame. Considering the operational velocity of 50 km/h of the train, as the wheel shaft
is part of the wheelset and considering this as a rigid body where the wheel has a mean radius of
0.45 meters, the inner raceway angular velocity is initialized with a value of 30 rad/s and
maintained constant throughout the complete simulation.
All geometric data required to create realistic models of the tapered bearing is acquired
by measurements of actual bearings used in project MAXBE. The acquisition of the necessary
information to fulfil the input data tables relative to tapered bearings is achieved by the tribology
group at the Faculty of Engineering of University of Porto, also partner in the MAXBE project. The
66
data obtained, for the tapered bearing with the reference BT2-7088 (SKF), with the dimensions
130×230×160, is presented in the Appendix A. The rheological characteristic values of the
lubricant are given also in the Appendix A, for the lubricant oil Total Carter EP220.
7.3. Results of the Dynamic Analysis
BearDyn is first tested for a tapered single row roller bearing with 4 rollers. This preliminary
simulations allows fast results to understand the dynamic response that is extremely useful during
the development of the program. However, the results have no significant value and are not
presented in this work. The results from simulations for a complete roller bearing are presented
and discussed.
Simulation without load applied
In this work, the results of the dynamic response of a single row tapered roller bearing in a railway
operation scenario are illustrated. For the tapered bearing, its kinematics is illustrated in Figure
57 by using the animation program SAGA [46]. The visualization of the roller bearing kinematics
allows appraising for the correct detection and expected kinematics. In order to show some results
from each simulation, a roller is selected to include the representation of the contact points. This
roller has the index Body 15, is located in the lower part of the bearing.
Figure 57: Initial position of the roller bearing represented in SAGA with contact points from Body 15
As a first approach, a simulation of the non-loaded tapered roller bearing with tangential
forces is addressed. This simulation allows to test the contact detection inside a dynamic analysis,
as well as the models used for the calculation of the contact forces. The analysis is performed for
a total integration time of 0.03 seconds, where the roller progressed about 1/4 of the complete
pitch circle, requiring 8h30 hours of computational time, using the Newton-Raphson method for
solving the systems used in contact detection. Figure 58 shows three frames from this simulation.
Note that Body 15 is represented with the colour blue, which is merely a representation for better
perception of the movement. During the simulation, the bodies start to change position and the
surfaces between them start to interact. Due to the initial velocity and gravity force, the rollers
start to contact with the outer raceway and are pushed to one side, until the spherical surface
67
from the large top of the roller start to contact the right flange. Figure 59 shows the timestep
t=0.02s, where the red dots for the contact points indicate that there is contact between roller and
outer raceway and roller and right flange. Contact with the cage side starts at t=0.021s, followed
by the cage tops. After t=0.03s the simulation starts to require computational effort that is too
large and the time to obtain another timestep grows exponentially.
Figure 58: c
Figure 59: Contact Points with forces being applied in the Outer Raceway and Right Flange at t=0.02s
The total force acting in Body 15 are presented in Figure 60 (a). It is clear that the first 0.025s of
the simulation correspond to the transient period in which the dynamics of the system has to be
disregarded, while the system is achieving equilibrium. For this reason, the plot in Figure 60 (b)
discards the first 0.023 seconds of simulation. After that, it is possible to see that the normal force
is stabilizing into a value in the interval [10, 15] N, which is roughly the force due to gravity of a
roller with the mass 0.14 kg and the kinetic energy from the initial velocity. The tangential force is
always 10 percent of the normal force, which means that the contact is dry and the lubricant is
unable to decrease the friction. As explained in Section 5.2, the friction coefficient calculated
depends on the lubricant film parameter Λ, with Equation (5.19). For this magnitude of load
applied it is necessary to increase the lubricant film thickness in the contact points. The first
approach to solve this problem is increase the angular velocity of the inner raceway to 60 rad/s.
By increasing the relative velocity between contact points, the lubricant film parameter increased
but the increment was insufficient to achieve the mixed lubrication mode. The next solution is
changing the lubricant used. The oil Total Carter EP220 is not suitable for high pressure roller
bearings from axleboxes. Instead, a lubricant grease must be selected, depending on the
68
temperature of operation, speed factor and load factor [47]. It is expected that the new lubricant
properties allow the contact to be at least in the mixed mode (since the full film mode is hard to
obtain in this type of loads) and reduce the friction between surfaces.
(a) (b) Figure 60: Contact Forces in simulation t=0.03s from Body 14: (a) all the timesteps; (b) from t=0.023s until t=0.03s
The force acting on the roller large end is displayed in Figure 61(a) ensures that the roller is
continuously being pushed against the right flange. In Figure 61(b) it is possible to see the total force
in the outer raceway, caused by the rollers, also converging after the initial transient period.
(a) (b) Figure 61: Contact forces applied to: (a) spherical large end of the roller; (b) Outer raceway
Simulation without cage
By taking the cage contacts out of the simulation not only the computational time reduces
to half, which is expected since the number of systems of equations to solve in the contact
detection reduces, but also the integrator is capable of evaluating after t=0.03s in a reasonable
time. In Figure 62 the results for a simulation without cage for t=0.1s are presented. Figure 62 (a)
shows the forces acting on body 14. The force stabilizes after t=0.035s and starts loosing intensity
with the movement. A full revolution simulation is possible, but since the results do not have
physical value, this simulation is not carried. Figure 63 shows the final frame from the simulation.
Note that the interference between rollers is expected since the cage that prevents the contact
0.023 0.024 0.025 0.026 0.027 0.028 0.029 0.030
1
2
3
4
5
6
7
T (s)
Forc
e (
x10
2 N
)
0 0.005 0.01 0.015 0.02 0.025 0.030
1
2
3
4
5
6
T (s)
Forc
e (
x10
3 N
)
Normal Force
TangentialForceNormal Force
TangentialForce
0 0.005 0.01 0.015 0.02 0.025 0.030
0.5
1
1.5
2
2.5
T (s)
Forc
e (
N)
0.01 0.015 0.02 0.025 0.030
5
10
15
20
25
T (s)
Forc
e (
x 1
03 N
)
69
between rollers is not considered. The rollers stay inside the bearing inner raceway, which gives
confidence in the flange contact implementations.
(a) (b) Figure 62: Contact Forces in simulation t=0.1s without cage from: (a) Body 14; (b) Outer Raceway
Figure 63: Final frame in the simulation without cage
Computational Efficiency
The most computational effort in BearDyn is by far the contact detection. Around 80% of the time
spent in the simulation in the evaluation of the contact points. For that reason, three methods to
solve the contact systems of equations were implemented, as explained in Section 6.1. To
compare them, BearDyn simulated, under the same conditions as above, the movement for
t=0.03s using the different methods. The elapsed time is showed in Table 7.
Method Time fsolve 24h
Computational Newton-Raphson 8h Analytical Newton-Raphson 9h30
Table 7: Time elapsed for the same t=0.03s simulation using three different methods to solve contact
It is clear that both Newton-Raphson methods are quicker than the optimization function.
Since both functions are implemented specifically for this problem, they have a quicker response,
but not having other algorithms to solve when Newton-Raphson fails turns out to be problematic
when dealing with the spherical top formulation. It is expected that the Newton-Raphson with the
0 0.02 0.04 0.06 0.08 0.10
5
10
15
20
25
30
35
40
45
50
T (s)
Forc
e (
N)
0 0.02 0.04 0.06 0.08 0.10
2
4
6
8
10
12
14
16
18
20
T (s)
Forc
e (
x10
2 N
)
Normal Force
Tangential Force
70
analytical Jacobian matrix installed in the program, specific for each contact, is the fastest of the
three methods, since the derivatives are more precise and lead to the solution in fewer steps.
This is not verified and is explained by the extra computational effort involved in creating the
analytical Jacobian matrix. For an easier implementation, to create the Jacobian matrix BearDyn
needs to always call two functions: Jacobian and Geometry. These extra steps for solving
around 400 systems per timestep increased the computational effort.
After t=0.03s, in the complete roller simulation, it was perceived that the integration process
required extremely high computational resources. During the project of BearDyn, whenever a
simulation in a specific timestep started to decrease the timestep exponentially with the
computational time increasing, usually meant that some implemented model started to act in that
timestep and caused the system to destabilize. This is seen as a consequence of the existence of
high frequency contents in the interaction of the contacting bodies, which causes the integration
process to reduce the time step and, thus, requiring a much higher number of time steps to obtain
successful results. The source of that high frequency noticed after t=0.03s is the instability in the
cage. This body is subject to multiple forces coming from all the rollers constantly contacting and
creating small perturbations in opposite directions. When added all together, these forces make the
movement hard to predict and difficult to compute. This claim is sustained by the results obtained
from a simulation without the cage contacts involved, where this instability is no longer encountered.
In order to solve this problem, some alternatives are studied. In an attempt to reduce the
high frequency in the cage, Kelvin Voigt Contact Model is implemented and the coefficient of
restitution changed to try to maximize the possible simulation time. This method increases the
computational effort for values of coefficient of restitution below 0.95 and for higher values does
not improve the response. For this reason, the model is discarded and the coefficient of restitution
defined as 1. Different ODEs were tested, such as ODE15s and ODE23t. The first one considers
the problem as stiff, meaning that the integrator takes smaller steps due to the high variation of
the nearby solutions. In theory this has similarities to the roller bearing problem, but the solver is
unable to deal with the contact detection forces and stops when the first contact occurs. ODE23t
is used to solve moderately stiff problems with low order accuracy, resulting in more
computational difficulties than the ODE45.
Simulation with load
When adding the load, the frequency of the system increases and the integrator proceeds at a
slower pace. For this reason, the obtained results are limited by the time of the simulation. As a
first approach for trying to understand the behaviour of the roller bearing under load, the force
magnitude is reduced to 10%, with 5kN. After t=0.03s, it’s possible to observe the misalignment
of the bearing as seen in Figure 64 (a). This behaviour is explained by the forces applied to the
inner raceway. Since this body is not constrained in any way, the load applied generates an axial
force reacting on the raceway, leading to the misalignment. Tapered roller bearings usually work
in double-row back-to-back configurations to balance the lateral component of the force, resulting
71
in a more stable performance. Since this double row bearing is not possible to implement in the
given time, an axial force is computed by penalizing this axial displacement, as seen in Figure 64
(b). In addition, a constraint to the cage movement in the surface (YZ) is added, ensuring that the
cage also stays with the right alignment during the simulation.
(a) (b)
Figure 64: (a) Representation bearing at timestep t=0.03s; (b) force introduced to maintain the alignment
This new force and constraint allow the simulation to run any given total time. A simulation
was completed for t=0.2s, giving a more significant report on kinematic positions evolution and
forces applied. The performance time of the simulation is not ideal, since it is not enough for the
rollers to complete one revolution around the bearing center. However, this simulation allows the
broadest results possible to be obtained due to the computational time required for a full
revolution.
Figure 65 shows the initial and final position of the simulation. Even though lateral forces
are being applied to the cage tops, the rollers do not leave the cage pockets, meaning that the
contact forces are being applied correctly. With load, the rollers tend to slide to the left flange,
while when no load is applied, the rollers slide to the right flange.
Figure 65: Initial and final frames in the simulation for t=0.2s
𝛿
𝑓𝑛 = 𝐾𝛿𝑛
72
The forces in the inner raceway are displayed in Figure 66. The system shows a transient
behaviour in the beginning and converges after t=0.01s to the value 5kN, which is the reaction to
the load force being applied with the same magnitude.
Figure 66: Contact Forces in the Inner Raceway during the simulation
To obtain the frequency response of the system, the total forces actuating on the inner
raceway over time are converted into a function of amplitude and frequency, using a Fast-Fourier
Transform algorithm, where the results relative to the first 0.01 seconds of the simulation are
ignored. The resulting graphic is presented in Figure 67.
Figure 67: Frequency response of total forces acting on the inner raceway, resulting from the simulation of a tapered roller bearing with complete contact detection, load of 5kN and tangential forces applied, from t=0.01s
The expected frequencies for the tapered roller bearing studied are calculated according
to the procedure reported in references [48] and [49]. The expected working frequencies of the
inner raceway, BPFI, are calculated as,
cos
DPFI 12
b k
S
m
DNRF
d
(7.1)
where NR is the number of rollers in the system and Fs the inner raceway rotational frequency.
For the conditions considered in the simulation shown in Figure 67, the working frequencies of
the inner raceway are expected to have the value 400 Hz. According to Li [49], a bearing with no
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20
5
10
15
20
25
30
T (s)
Forc
e (
x 1
03 N
)
0 10 20 30 40 50 60 35 40 45 500
2
4
6
8
10
12
Frequency (kHz)
Am
plit
ude (
x10)
73
defects can have a response with this frequency value and its harmonics, with a small and even
frequency, where no frequency stands out. This does not correspond to the case presented, since
it should be noted that only with the dynamic response obtained for several complete revolutions
of the roller bearing are the FFT reliable for the frequency analysis. Due to the computational cost
of the current implementation of the methodology, the data obtained is not sufficient for any
reliable study.
74
8. Conclusions
The work presented here addresses the development of methods and implementation of an
efficient bearing dynamic analysis tool, called BearDyn, which allows the analysis of tapered
roller bearings, following the work developed by Lima [22]. The inherited program has a structure
defined for the data setup of spherical and tapered roller bearings; the initial position, orientation
and velocities of each roller bearing components, i.e., rollers, cages, inner and outer raceways,
defined ensuring kinematic consistency and a dynamic analysis procedure devised and
implemented. This work focused on the remaining parts fundamental to the dynamic tool: first,
being the contact detection, based on the kinematics and geometry of each body, while the
second consists in the evaluation of the normal and tangential forces developed. To successfully
implement this two steps in BearDyn, models were first tested in simpler problems. The general
equations of motion for the dynamic analysis and necessary kinematics are firstly defined. The
contact problem for a wide number of geometries is addressed and contact detection methods
are tested in steady state geometries and a simpler dynamic analysis with a roller and a surface.
The deficient contact detection that lead to errors in the previous version of BearDyn are
corrected with the introduction of the normal vector of the surface to the system of equations and
tested in these preliminary tests. This correction is a crucial step since one incorrect pair of points
detected in enough to compromise the complete simulation.
The formulation to describe all the geometries needed in BearDyn is explained and
validated with a developed MATLAB® visualization tool, with necessary modifications. Contact
detection is implemented in BearDyn and the first timestep possible contact points are
represented and validated. The system assembly for contact detection was implemented with a
simple and organized method involving only two functions and one flag to create any system of
an implemented contact, allowing for future contact detections in Spherical Roller Bearings to be
introduced in the same functions. Three different methods to solve the systems are implemented
in order to find the fastest and most efficient of them all, reducing the simulation time to more than
half. Normal and tangential forces, due to normal contact and to lubrication, are modelled. The
contact forces are detected and applied between the roller and the raceways, flanges and cage.
The necessary models to calculate the tangential forces caused by lubrication are explained and
tested with the oil Total Carter EP220.
Several dynamic analyses under different conditions are simulated. First, BearDyn is
tested with no load applied, a contact velocity of the inner raceway and a constrained outer
raceway. Contact detection is validated in the analysis with a SAGA visualization of the movement
between bodies and contact points representation. The lubricant used resulted as improper to
roller bearings subject to high loads, leading only to a dry contact. A new grease lubricant with
different properties must be considered in order to obtain at least mixed mode lubrication.
75
When performing dynamic simulations of the models with BearDyn for a longer timespan,
it is identified an extremely high computational effort. This difficulty clearly identifies the need for
the development of robust and efficient computational algorithms, eventually based on the use of
parallel computation strategies. The integrator used has a variable timestep, resulting in a step
choice depending on the behaviour of the system. The unpredictable nature of the problem
requires for a maximum step size defined to 10-5 s.
When introducing the load, a simulation is carried with 10% of the expected value and
the results confirmed. The frequency response is a fundamental piece to support the development
of any monitoring systems in particular. A preliminary dynamic analysis of a single-row tapered
roller bearing model allows extracting the kinematic quantities of importance to its
characterization. In any case, the length of the time responses obtained are not yet enough to
draw conclusions about the general dynamic performance of the roller bearing models, but are
enough to verify the proper functioning of the dynamic analysis.
8.1. Future work
While working for project MAXBE, the final goal of creating a computational tool that allows
obtaining the full dynamic performance of a tapered roller bearing in working railway conditions
was partially achieved. In order to achieve the final state of the dynamic analysis computational
tool, extra time and effort should be spent in its creation. To do so, the BearDyn program should
be submitted to some more testing and development. Efforts must be done to gain computational
power to run the simulations, eventually using parallel computational strategies. New models
considering damping, such as Lankarani Model [35] used in wheel rail contact, can reduce the
frequency of the problem.
To allow more realistic simulation, the lubricant must be replaced. The correct grease
lubricant must be selected from manufacturers and the lubricant properties changed. The
obtained 𝜇 for the contact points must be reported as an output in the struct CP_Results and
studied to guarantee the lubricant working conditions. In the lubricant model, temperature and
starvation factors must also be considered to improve the realism of the simulation, including
realistic data from operating conditions.
BearDyn should be prepared to run simulations for enough time to obtain realistic results,
after leaving the transient period and the when stabilization of the system occurs. The system
needs about 2 or 3 complete revolutions of the rollers around the center of the bearing to allow
for its dynamic performance evaluation.
After having a full single-row tapered bearing simulation for at least two revolutions, the
code must be adapted to simulate a double-row tapered bearing. Since this is the roller bearing
used in the axlebox, only in double-row bearings the simulation can be compared with
experimental results.
76
To complete the code, also forces resulting from wheel-rail contact should be correctly
introduced into the calculation of force vector, either by resorting to a known timely response and
the external forces entry table, or by full detection and calculation implemented in the dynamic
analysis tool code.
When BearDyn is functioning properly with these modifications, the code should be
updated to also allow simulations of spherical roller bearings. Some of the necessary
implementations are already present in BearDyn code. Contact detection between rollers and
raceways, flanges and cage need to be implemented. All the specific geometrical information from
the contacts must be introduced into geometry and the contact system formulated with the
function cost_function.
BearDyn should be prepared to run simulations for enough time to obtain realistic results,
after leaving the transient period and the when stabilization of the system occurs. The system
needs about 2 or 3 complete revolutions of the rollers around the center of the bearing to allow
for its dynamic performance evaluation.
To allow the monitoring of the bearings performance in railway operating conditions, as
the final goal desired for project MAXBE, the dynamic response of the bearing should be
converted to a vibration response using post-processing tools in order to be used as a basis for
comparison with the dynamic response of bearings with defects. A final approach requires typical
defects on bearings to be modelled in the code of the dynamic analysis tool. When this is
achieved, the main goal of this project is reached, where the collection of vibration response data
obtained with BearDyn for different bearings with or without defects can be used to infer the
health of axle bearings, by comparison with the responses of bearings obtained via wayside or
on-board monitoring systems.
77
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80
Appendix A – Input Data for Bearing Model
The present appendix presents the geometric data to be delivered to BearDyn by the user, in the
format of a text file with a defined number of records. The structure to be followed is described
for the necessary data for each type of roller bearing, defined by the properties detailed in the
figures.
A.1. General bearing data
Tapered Units
Bearing.Type BType 4 Bearing.NumberRollers BNumbRoller 21 Bearing.OuterDiameter do BOuterDiameter 0.2300 [m] Bearing.InnerDiameter di BInnerDiameter 0.1300 [m] Bearing.PitchDiameter dm BPitchDiameter 0.1800 [m] Bearing.Width L BWidth 0.1600 [m]
Figure 1: Tapered Bearing general dimensions
Record 1 of the input deck:
Type; NumberRollers; OuterDiameter; InnerDiameter; PitchDiameter; Width
A.3. Tapered bearing geometry, surface and mass data
Units
Tapered.RollerDiameterLargeEnd Dl 0.0238 [m] Tapered.RollerDiameterSmallEnd Ds 0.0222 [m] Tapered.RollerLenght Lb 0.0456 [m] Tapered.RollerLandLenght Lc 0.0432 [m] Tapered.RollerCrownRadius Rcr 0.0024 [m] Tapered.RollerEndRadiusLargeEnd RerR 0.0500 [m] Tapered.RollerEndRadiusSmallEnd RerL Inf [m] Tapered.OuterRaceSemiConeAngle o 0.1541 [rad]
Tapered.InnerRaceSemiConeAngle i 0.1192 [rad]
Tapered.OuterRaceLandLenght lo 0.0489 [m] Tapered.InnerRaceLandLenght li 0.0460 [m] Tapered.OuterRaceWidth Lo 0.0670 [m] Tapered.InnerRaceWidth Li 0.0794 [m] Tapered.InnerRaceLandStart ei 0.0112 [m] Tapered.endplay ep 0 If ep <0, preload is
reported here [m]
do
Lo
di
Li
dm
j
j
81
Tapered.NumberRows Nrow 2 If NRow<0, face-to-face mounting
Tapered.InterRaceSpacing Si 0.0179 If the bearing has two rows, Si is used
[m]
Tapered.RollerMass mb 0.1460 [kg] Tapered.RollerInertia Ib, Ib, Ib [1.2630E-6; 4.0326E-6;
4.0326E-6] [kg m2]
Tapered.RollerSurfaceRoughness sb 0.0600 [µm] Tapered.InnerRaceMass mi 9.7680 [kg] Tapered.InnerRaceInertia Ii, Ii, Ii [6.3070E-3; 6.3070E-3;
5.7590E-3] [kg m2]
Tapered.InnerRaceSurfaceRoughness si 0.2100 [µm] Tapered.OuterRaceMass mo 10.499 [kg] Tapered.OuterRaceInertia Io, Io, Io [1.0617E-2; 1.0617E-2;
1.6764E-2] [kg m2]
Tapered.OuterRaceSurfaceRoughness so 0.2200 [µm]
(a) (b) (c) Figure 2: Tapered roller bearing: (a) Perspective view; (b) Bearing dimensions; (c) Roller dimensions.
Record 2b of the input deck:
RollerDiameterLargeEnd; RollerDiameterSmallEnd; RollerLenght; RollerLandLenght; RollerCrownRadius; RollerEndRadiusLargeEnd; RollerEndRadiusSmallEnd
Record 3b of the input deck:
OuterRaceSemiConeAngle; InnerRaceSemiConeAngle; OuterRaceLandLenght; InnerRaceLandLenght; OuterRaceWidth; InnerRaceWidth; InnerRaceLandStart; endplay; NumberRows; InterRaceSpacing
Record 4b of the input deck:
RollerMass; RollerInertia; RollerSurfaceRoughness; InnerRaceMass; InnerRaceInertia; InnerRaceSurfaceRoughness; OuterRaceMass; OuterRaceInertia; OuterRaceSurfaceRoughness
j
j
j
Ds
Rcr
Lc
Lr
Dl
RerR
RerL
Lb
½ Lr
dm
lo
ep
eido
Lo
o
di
i
Li
li
j
j
82
(a) (b) (c) Figure 3: Tapered roller bearing mounting: (a) Back to back with endplay, ep>0 and Nrow=+2; (b) Back to back, ep≤0 and
Nrow=+2; (c) Face to face, ep≤0 and Nrow=-2.
A.4. Race flange geometry
Tapered Units
Flange.OuterLeftAngle oL Does not apply [rad]
Flange.OuterRightAngle oR Does not apply [rad]
Flange.InnerLeftAngle iL 0.4355 [rad]
Flange.InnerRightAngle iR 0.1192 [rad]
Flange.OuterLeftHeight hoL Does not apply [m]
Flange.OuterRightHeight hoR Does not apply [m]
Flange.InnerLeftHeight hiL 0.0018 [m]
Flange.InnerRightHeight hiR 0.0074 [m]
(a) (b) (c) Figure 4: Flanges in roller bearings: (a) Angle and height definitions; (b) Tapered roller bearing; (c) Detail of the flanges
in the tapered bearing.
Record 5 of the input deck:
OuterLeftAngle;OuterRightAngle; InnerLeftAngle;InnerRightAngle; OuterLeftHeight;OuterRightHeight; InnerLeftHeight; InnerRightHeight
A.5. Cage geometry, surface and mass data
It is important to notice that some of the data relative to the tapered cage geometry is only
available when the bearing is fully mounted. This results in some of the data entries of the
following table to not be available, since these were not measured. Also, the cage of the spherical
bearing used has a geometry different to the one expected, which is not able to be described with
the data predicted in the following table. This way, in future studies where the cage of the spherical
bearing is considered, this table should suffer the appropriate modifications.
ep
LiLi Si LiLi Si Li LiSi
iR
hiRiL
hiL
iL
iR
hoL=hoR=0
hiR
hiL
iL iR
oRoL
hoRhoL
hiRhiL
83
Tapered Units
Cage.Guidance CGuide 0 Cage.OuterDiameter Rou 0.0950 [m] Cage.InnerDiameter Rin 0.0860 [m] Cage.Width Wl 0.0465 [m] Cage.OuterRaceClearance Cou No info. available [m] Cage.InnerRaceClearance Cin No info. available [m] Cage.SemiConeAngle c 0.15 [rad]
Cage.GuideLandRadiusRight RgR 0.0991 [m] Cage.GuideLandWidthRight WgR 0.0086 [m] Cage.GuideLandPositionRight LgR No info. available [m] Cage.GuideLandClearanceRight CgR No info. available [m] Cage.GuideLandRadiusLeft RgL 0.0860 [m] Cage.GuideLandWidthLeft WgL 0.0074 [m] Cage.GuideLandPositionLeft LgL No info. available [m] Cage.GuideLandClearanceLeft CgL No info. available [m] Cage.PocketShape PType 2 Cage.PocketLenght Pl 0.0465 [m] Cage.PocketWidth Pw 0.0230 [m] Cage.PocketDimension1 P1 No info. available [m] Cage.PocketDimension2 P2 No info. available [m] Cage.PocketDimension3 P3 No info. available [m] Cage.PocketDimension3 P4 No info. available [m] Cage.Mass mc 0.2000 [kg] Cage.Inertia Ic, Ic, Ic [5.2172E-5;
9.4550E-5; 5.2172E-5]
[kg m2]
Cage.PocketSurfaceRoughness sc 2.5500 [µm]
Cage guidance:
0 No guidance 1 Outer race guidance 2 Inner race guidance
Cage pocket shape:
1 Cylindrical 2 Rectangular 3 Guided Surface
Figure 5: Cage for Tapered roller bearings. Note that a cage will not have both outer and inner guide lands.
Rou Rin
Wl
RgRRgL
CgRCgL
WgRWgL
LgRLgL
c
84
(a) (b) (c) Figure 6: Cage pockets types: (a) Cylindrical; (b) Rectangular; (c) Guided surfaces.
Record 6 of the input deck:
Guidance;OuterDiameter;InnerDiameter;Width;OuterRaceClearance; InnerRaceClearance; SemiConeAngle
Record 7 of the input deck:
GuideLandRadiusRight; GuideLandWidthRight; GuideLandPositionRight; GuideLandClearanceRight; GuideLandRadiusLeft; GuideLandWidthLeft; GuideLandPositionLeft; GuideLandClearanceLeft
Record 8 of the input deck:
PocketLenght; PocketWidth; PocketDimension1; PocketDimension2; PocketDimension3; PocketDimension3
Record 9 of the input deck:
Mass; Inertia; PocketSurfaceRoughness
A.6. Geometries radius in the roller direction for tangential forces
Units
r_x.Roller 0.0115 [m] r_x.InnerRaceway 0.0784 [m] r_x.OuterRaceway 0.1016 [m] r_x.RollerEndRadiusLargeEnd 0.05 [m] r_x.FlangeLeft 0.0886 [m] r_x.FlangeRight Inf [m] r_x.CageSide Inf [m] r_x.CageSmallTop Inf [m] r_x.CageLargeTop Inf [m]
Figure 69: Radius in the roller direction of two contacting surfaces
Pl
Pw
P3
P2P1 P3
P2
P4
P1
85
A.7. Rheological properties of the lubricant oil Total Carter EP220
Name Symbol Value Description/Application Units
Lubricant.alpha α* 9.4275E-9
Regression coefficients for Gupta Type I traction model [41]
[1/Pa] Lubricant.V1 V1
4.0063E-1 [Pa s] Lubricant.V2 V2 5.0459E-2 [1/K] Lubricant.V3 V3 1.5590E-2 [s/m] Lubricant.B1 B1 3.4104E-1 [1/K] Lubricant.B2 B2 1.0930 Lubricant.B3 B3 2.5647E-1 Lubricant.T T 3.7315E2 Temperature in operating conditions [K] Lubricant.T0 T0 3.5315E2 Reference temperature [K] Lubricant.Kf1 Kf1 1.4870 Thermal conductivity of material 1 [W/(m K)] Lubricant.Kf2 Kf2 14870 Thermal conductivity of material 2 [W/(m K)] Lubricant.P P 1.0000E9 Pressure in operating conditions [Pa] Lubricant.P0 P0 2.0000E8 Reference pressure [Pa] Lubricant.Visco 44.2300 Viscosity in operating conditions [cSt]
Lubricant.s s 0.9904 Experimental constants for Gold et. Al [50] expression
Lubricant.t t 0.1390 Lubricant.Visco1 1 319.2200
2 points of known temperature and kinematic viscosity for the use of norm ASTM D341 [51].
[cSt]
Lubricant.T1 T1 313.0000 [K] Lubricant.Visco2 2 65.2800 [cSt]
Lubricant.T2 T2 343.0000 [K]
Structure of Lubricant.txt file:
Record 1 of the input deck:
Lubricant.alpha; Lubricant.V1; Lubricant.V2; Lubricant.V3; Lubricant.B1; Lubricant.B2; Lubricant.B3
Record 2 of the input deck:
Lubricant.T; Lubricant.T0; Lubricant.Kf1; Lubricant.Kf2
Record 3 of the input deck:
Lubricant.P; Lubricant.P0; Lubricant.Visco; Lubricant.s; Lubricant.t
Record 4 of the input deck:
Lubricant.Visco1; Lubricant.T1; Lubricant.Visco2; Lubricant.T2
86
Appendix B – Visualization Tools Developed
B.1. Video Representation for DAP Coin and DAP Roller
The positions of the center of mass from the bodies, obtained over time from the integration in
the dynamic analysis, are used to represent the geometries and create a video simulating
the obtained movement. For all the timesteps, a figure with the bodies represented in the
position is generated and then saved as a matrix with the function getframe. Finally, all
the saved images are compiled into a video with the function VideoWriter. It’s
important to maintain a constant viewpoint when generating the figure frames, in order to
obtain a smooth transition in the video. This view point can be controlled with the option
view, in order to get different perspectives of the movement. Figure 69 shows the general
scheme of the program
Figure 70
Figure 70: Scheme of the video maker for DAP Coin and DAP Roller
B.2. MATLAB Visualization Tool for BearDyn
The visualization tool uses functions from BearDyn to detect the contact points and proceeds to
represent the geometries and contact points specified by the user. It is necessary to run BearDyn
for at least one timestep in order to generate the initialization data structures necessary in the
visualization tool. These results are saved in out.mat and must be copied to the Visualization Tool
folder. The program proceeds to call the function F_Contact and subsequent functions
necessary to the contact point evaluation for all rollers and stores the positions of the points. After
that, all the geometries specified are represented with the information obtained directly from the
function geometry from BearDyn. Figure 70 shows the general scheme.
Figure 71: Scheme of the visualization tool
DynamicAnalysis
y, t Represent
cylinder and
body 2
frames
matrixWrite Video
Call F_Contact.m
to obtain all
contact points
Load ‘out.mat’Represent the i
rollers wanted
Represent the
Raceway and
Cage geometries
Represent the
contact points for
the i rollers
87
B.3. SAGA Visualization Data
After the dynamic analysis of a mechanical system, the SAGA Program is used for the visualization
and animation of the system. This program receives an input Simulation file, .str, which contains
the history of the position and orientation of the rigid bodies of the vehicle model. This information
is defined using homogenous transformation matrix (HTM),
0 1
T
t
i
A rH i i
(B.1)
where 𝐀𝑖 is the transformation matrix and 𝐫𝑖 is the position from body i, in a specific timestep t.
The homogenous transformation matrix for a timestep, 𝐇𝑡, is described by the sequence of
matrices 𝐇𝑖t for i = 1,…,NBodies. The motion of the full system is described by the time sequence
of matrices 𝐇𝑡 until the final timestep is reached. For instance, considering a simulation with 1
second and the time step of report is 0.1 seconds, there are 11 frames.
The number of bodies used in SAGA is not the same as the number of bodies from the
dynamic analysis. Additional bodies are used to represent the white background and the
candidates to be points of contact. For each pair of points, 4 new bodies are needed: two green
spheres and two red spheres, also referred to as dots. If in a given timestep the pair of points
represents contact and indentation exists, the red spheres will be represented in the contact
points location and the green spheres will be represented outside the user’s camera view. If the
points are not in contact and only represent the points of more proximity between surfaces, the
green dots will be placed in the location of the points and the red dots outside the frame.
A code to generate an input file which contains appropriate bodies information to visualize
is developed and implemented in this work. The .str file is generated with the information stored
in the struct CP_Results generated in BearDyn to store the information form the successful
timesteps.
Bodies used in SAGA:
Nr of Bodies
Background 1 Inner Raceway 1
Rollers 21 Cage 1
Outer Raceway 1
Contact Points Pairs of Points Nr of Bodies
Inner Raceway 6 24 Outer Raceway 6 24
Flange Left 1 4 Flange Right 1 4
Cage Left Side 6 24 Cage Right Side 6 24 Cage Small Top 2 8 Cage Large Top 1 4
Total 141
88
Homogeneous Transformation Matrices for the bodies used in SAGA:
BearDyn bodies 1,...,0 1
T
t i
body i NBodies
A rH i
(B.2)
Background 0 1
0backgrou
t
nd
IH
(B.3)
Sphere in Contact Point 0 1
T
CP
CP
t
rH
I
(B.4)
Sphere outside Contact Point inf
inf0 1
T
t
I rH
(B.5)
100 100 100T
inf r
Besides the .str file, other files are necessary to run SAGA. They are:
• Geometry file, .geo, containing the name of the files where the geometry of all
objects of the multibody system is defined
• Geometry files for each object, .g, for visual representation of each body,
generated from 3D Solidworks models
• Transformation file, .tra, Storing the point of view for the visualization
• Colourmap file, .shd, used to define the colour map for each object geometric
file for polygons fill and frame