crankshaft modeling for cylinder pressure estimation
TRANSCRIPT
Crankshaft Modeling & Identi�cation
for Cylinder Pressure Estimation
- con�dential -
Aart-Jan van der Hoeven
CST 2012.111
Internship Report
Advisors: dr.ir. M.C.F. Donkers (TNO)
dr.ir. F.P.T. Willems (TU/e)
Eindhoven University of TechnologyDepartment of Mechanical EngineeringControl Systems Technology
Eindhoven, December, 2012
Summary
The control problem of (heavy-duty diesel) engines consists of meeting the driver's torque request,while minimizing fuel consumption and staying within emission legislation constraints. With theintroduction of cylinder pressure sensors, it becomes possible to control the combustion process usingclosed-loop control, which enables advanced combustion concepts, improves transient performanceand is more robust to uncertainties. The cylinder pressure sensors necessary for this approach arerelatively expensive and not yet in mass-production for heavy-duty diesel applications. At TNO, avirtual cylinder pressure sensor concept has been developed in which only one cylinder pressure sensor isused and the other cylinder pressures are estimated by using the crankshaft position signal. A dynamicmodel of the crankshaft and piston system, that provides the relation between cylinder pressures andangular velocity, plays a crucial role in this algorithm and is developed within this internship project.Several dynamic crankshaft models can be found in literature. They are either used for structuraldesign [6], combustion phasing estimation [9] or indication of cylinder health [4]. The work presentedin this report is aimed for application in an on-line estimation of heavy-duty diesel cylinder pressure.
The presented model does use six cylinder pressure signals as input and provides the angular velocityat the location of the position encoder as output. The crankshaft model consists of nine bodies; six ofthem represent the cylinders, including a crank-slider mechanism and static friction model. The rearof the crankshaft is supplemented with a �ywheel body, which also contains an amount of lumpedmass due to components connected to it. The front of the crankshaft is supplemented with a torsionaldamper and front pulley body. All bodies are interconnected by springs and dampers, which representthe sti�ness and damping of the material.
A parameter identi�cation is performed using the least squares error �tting algorithm lsqnonlin
of the MATLAB Optimization Toolbox. High-accuracy measurement data of cylinder pressures andangular velocity were measured at TNO and used for identi�cation. After identifying appropriatevalues for the parameters of the model, the model is analyzed in terms of accuracy, complexity andsensitivity. For nine operating points throughout the operating region of the engine, the average RMSvelocity output error is about 0.5 rad/s (0.3%) and the model performs well in predicting the velocitywaveform. After evaluating multiple model extensions and their performance, the presented model isconsidered as a good compromise between complexity, performance and robustness.
Two candidates for model simpli�cation are proposed. The �rst proposal is using a constant massmatrix in the equations of motions, and is shown to be feasible in terms of model output. The secondproposal is a simpli�ed friction modeling in which the instantaneous friction torque is replaced bya constant average friction torque. This reduces the amount of parameters to be identi�ed at eachoperating point. The proposal might be feasible, but should be evaluated using a dedicated parameteridenti�cation. Reducing the crankshaft model to a single rigid-body, which is commonly the approachfor light-duty engines, is shown to be an invalid approach for heavy-duty engines.
A model describing the dynamic behavior of a heavy-duty diesel engine has become available as aresult of this work. Since this control oriented model was required for the next step in the developmentand validation of the virtual cylinder pressure sensor concept, this algorithm might now actually proveitself on real engine data. The virtual sensor will make implementation of closed-loop combustioncontrol much more attractive and will contribute to cleaner and more fuel e�cient vehicles.
2
Contents
1. Introduction 4
2. Engine Description 62.1. Engine Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2. Crankshaft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3. Torsional Damper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4. Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3. Crankshaft Modeling 93.1. Crankshaft Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2. Crank-Slider Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.3. Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.4. Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.5. Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4. Parameter Identi�cation 164.1. Identi�cation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.2. Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.3. Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.4. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5. Analysis 215.1. Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.2. Model Simpli�cations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.3. Parameter Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
6. Conclusions, Recommendations and Implications 396.1. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396.2. Recommendations for Future Research . . . . . . . . . . . . . . . . . . . . . . . . . 406.3. Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Bibliography 40
A. DAF MX Engine Data 42
B. Connecting Rod Inertial Properties 45
C. Testbench Dynamics 47
D. MATLAB Implementation 49
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1. Introduction
The control problem of (heavy-duty diesel) engines consists of meeting the driver's torque requestwhile minimizing fuel consumption and staying within emission legislation constraints. In traditionaldiesel engine control systems this problem is solved by a separate air path and fuel path controller.The air path is regulated in closed-loop with indirect measurement of performance quantities, butthe fuel path is mainly relying on feed-forward with compensations for transients. A more robustapproach became available with the introduction of cylinder pressure sensors. Performance quantitieslike indicated mean e�ective pressure (IMEP), 50% heat release crank-angle (CA50), cylinder pressurerise (dp/dα) and peak cylinder pressure (pmax) can be determined and can be used in closed-loopcombustion control [11]. The bene�ts of this type of control are enhanced combustion stability,which is needed to enable advanced combustion concepts, improved dynamic operating performance interms of emissions, fuel consumption and torque response, robustness for variations in the combustionprocess due to production tolerances and aging [12] and for monitoring and diagnostics.
The cylinder pressure sensors needed for closed-loop combustion control are relatively expensive andnot yet in mass-production for heavy-duty diesel applications. Research can be found in which thenumber of necessary sensors is reduced or in which virtual sensing concepts are applied [9, 13, 4]. Byvirtual sensing, we mean that the cylinder pressure is not directly measured, but estimated from otheroutputs using an algorithm that includes a mathematical model. At TNO, a virtual cylinder pressuresensor concept has been developed whereby only one cylinder pressure sensor is used and the othercylinder pressures are estimated by using the crankshaft position signal. The problem of estimatingmultiple unknown cylinder pressures from one measured pressure and crankshaft position is, roughlyspeaking, solved in three steps, as schematically shown in Figure 1.1. First, the measured cylinderpressure is used to derive initial estimates of the induced torques on the crankshaft. Then, the initialestimates are improved, using crankshaft position information, by solving a LQ tracking problem.Finally, the cylinder pressures are reconstructed from the estimated induced torques. A dynamicmodel of the crankshaft and piston system, that provides the relation between cylinder pressures andangular velocity, plays a crucial role in this procedure and is developed within this internship project.
Figure 1.1.: Overview of the virtual cylinder pressure sensor algorithm.
Using numerical simulations, the idea of the virtual sensor concept is shown to have potential. Yet, ithas to be validated using measurements and a dynamic crankshaft model of a real heavy-duty diesel
4
engine. For this validation, six high-frequency cylinder pressures and a crankshaft position signalare recorded on a DAF MX 6-cylinder engine. The crankshaft model, that is developed within thisinternship project, enables validation of the virtual sensor concept on the DAF engine. Also, thedeveloped crankshaft model structure can be easily applied to other (heavy-duty diesel) engines.
Several dynamic crankshaft models can be found in literature. They are either used for structuraldesign [6], combustion phasing estimation [9] or indication of cylinder health [4]. The work presentedin this report is aimed for application in an on-line estimation of heavy-duty diesel cylinder pressureand should predict the angular velocity with a small error margin (5%).
This report is organized as follows. First a description of the engine being used for the measurementsis given in Chapter 2. A derivation of the crankshaft model structure and its implementation is givenin Chapter 3. In Chapter 4, the identi�cation of the model parameters is outlined and the simulationresults are given. In Chapter 5, the results and parameter sensitivity are analyzed and three simpli�edmodels are proposed. Finally, Chapter 6 provides the conclusions and recommendations for this work.
5
2. Engine Description
In this chapter, the heavy-duty diesel engine being used for this work is described. This engine hasbeen used to measure the cylinder pressures and crankshaft position with high accuracy. The structuralinformation of the crankshaft is provided by the manufacturer. This enables the development andvalidation of a crankshaft model, which will be done in next chapter.
2.1. Engine Overview
A DAF MX 6-cylinder diesel engine is used to represent a heavy-duty diesel engine with exhaustgas recirculation (EGR), variable turbine geometry (VTG) and common rail fuel injection. At TNO,a demonstrator engine of this type has been used to do experiments on new combustion controlconcepts. These experiments required this engine to be equipped with six cylinder pressure sensorsand a high-resolution crankshaft position encoder. A schematic overview of the engine and someimportant parameters are shown in Figure 2.1. The crankshaft drives the oil-pump and camshaft bymeans of a gear transmission at the rear (�ywheel-side). The camshaft, in turn, actuates both thecylinder valves and fuel unit-pumps. The front of the crankshaft is equipped with a torsional damperof which the housing also serves as a pulley for the accessory belt. Some of the accessories are thewater pump, air-conditioner compressor and alternator.
Number of cylinders 6
Cylinder volume 12.9 L
Compression ratio 17.45
Bore 130 mm
Stroke 162 mm
Connecting rod length 262 mm
Piston o�-set 0.8 mm
Firing order 1-5-3-6-2-4
Max. torque 2500 Nm
Max. power 375 kW
Figure 2.1.: Overview of a DAF MX 6-cylinder diesel engine and some characteristic parameters.
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2.2. Crankshaft
A schematic of the crankshaft is shown in Figure 2.2. The front of the crankshaft (left) is supplementedwith a torsional damper. The rear of the crankshaft (right) is supplemented with a gear-wheel and�ywheel (not shown). The cylinder numbering is 1 to 6 from front to rear.
Figure 2.2.: Overview of the DAF MX 6-cylinder crankshaft.
The crankshaft is partitioned into six cylinder segments that all have the same structure, as shown inFigure 2.3. Each cylinder segments consists of half of the crankshaft mains, two crankshaft webs anda crankshaft pin to which the connecting rod is mounted. The length, inertia and torsional sti�nessof each segment is provided by DAF, resulting from a FEM analysis of a CAD model, and can befound in Appendix A.
Figure 2.3.: A schematic of the crankshaft segments per cylinder (left) and a schematic of the torsionaldamper (right).
2.3. Torsional Damper
Crankshafts are often equipped with a torsional damper, in order to remove energy from torsionalvibration of the system. The engine being used for the measurements is equipped with a damper ofthe viscous �uid type and it is attached to the front side of the crankshaft. A hollow housing is rigidlyconnected to the crankshaft and an inertia ring is close-�tted inside it, while being suspended in asmall layer of viscous silicon �uid. The housing rotates along with the crankshaft and due to shearforces generated by a relative velocity between ring and housing, energy is dissipated in the form ofheat [7]. This allows the damper to be modeled as a damping between two bodies. A schematicoverview of the damper is shown in the right-hand �gure of Figure 2.3. The outer circumference ofthe housing also serves as a pulley for the accessories belt.
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2.4. Measurements
High-frequency measurements are conducted on a testbench at TNO. The measurements are sampledwith a resolution of 0.1 crankshaft angle degree (CAD). The measured signals include the six individualcylinder pressures, which are measured using Kistler piezoelectric pressure transducers, and crankshaftposition, which is measured using an optical position encoder. The position encoder was mounted tothe torsional damper housing at the front of the crankshaft, while the load was applied to the �ywheelat the rear of the crankshaft. The load was generated with an electric drive connected to the �ywheelby means of a shaft and rubber coupling. A schematic overview of the measurement set-up can beseen in Figure 2.4. All cylinder pressure signals are sampled with reference to the encoder positionand pegged, using the polytropic coe�cient method, to obtain absolute pressure levels.
Figure 2.4.: Measurement set-up with engine, shaft, coupling and electric drive.
Measurements are conducted at several combinations of engine speed and load. At each operatingpoint, 200 cycles are measured. This allows to analyze, identify and validate the crankshaft modelat multiple operating points. The model should be robust enough to match the measurement datawith a comparable accuracy at all operating points. One set of intermediate points will be only usedfor model validation, to prevent an over-�t to the measurement data and to prove robustness of theidenti�ed model. The operating points being used are shown in the engine operating map in Figure2.5. The {A, B, C} operating points are at {1213, 1525, 1838} rpm.
1000 1200 1400 1600 1800 20000
500
1000
1500
2000
2500
3000
3500
Engine speed [rpm]
Eng
ine
torq
ue [N
m]
A20
A60
A100
B10
B50
B100
C0
C40
C100
max. valid. identify
Figure 2.5.: Engine operating points for identi�cation and validation.
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3. Crankshaft Modeling
In this chapter, we will develop the dynamic crankshaft model of the engine discussed in previouschapter. First, an overview of the developed crankshaft model is given. Subsequently, the crank-sliderkinematics are studied, to obtain expressions for the piston acceleration and combustion torque asa function of crankshaft angle and angular velocity. Then, an instantaneous representation of thepiston friction is de�ned. Finally, the dynamic behavior of the multi-body model of a crankshaft ispresented, its equations of motion are derived and a few notes on the numerical implementation aregiven.
3.1. Crankshaft Model
For modeling the dynamic behavior of the crankshaft, it is split into nine bodies as can be seen inFigure 3.1. Body J1 at the front represents the torsional damper ring. Body J2 represents the frontof the crankshaft including the torsional damper housing and lumped accessories. For each cylinderi ∈ {1, ..., 6}, a body Ji+2 is added, which have a position dependent kinetic energy due to theoscillating mass. At the rear of the crankshaft, body J9 represents the �ywheel and lumped massesof gear-wheel, camshaft, testbench coupling, etc. All bodies are interconnected with springs anddampers, which represent the material sti�ness and damping. The external load Tload provided by thetestbench is applied at the �ywheel. At the front of the crankshaft, a load Tf represents the load ofthe accessories. Furthermore, each cylinder body is actuated by a torque resulting from combustionand friction. The states of the model are the angular position and velocity of each body, i ∈ {1, ..., 9}.The dynamics of the testbench are not included in the model, for reasons discussed in Appendix C.
Figure 3.1.: Schematic representation of the crankshaft model.
9
3.2. Crank-Slider Kinematics
Based on the geometry of the crank-slider mechanism, the equations for the piston velocity andcombustion torque are derived in this section. Both equations can be expressed using a so-callede�ective radius of the crankshaft, which is a function of crankshaft angular position. The pistonvelocity and cylinder pressure torque are used in Section 3.4 for describing the dynamics of thecrankshaft-piston system.
3.2.1. Piston Velocity
The piston velocity is derived from the geometry of the crank-slider mechanism, which is shown inFigure 3.2. First, the relation between the connecting rod angle and crankshaft angle is derived.Second, the e�ective height of the connecting rod is approximated, which is used in an expressionfor the angular velocity of the connecting rod. The position of the piston is de�ned as a functionof crankshaft angle. Next, using the previously derived expressions, the piston velocity can also bede�ned as a function of crankshaft position.
Figure 3.2.: Schematic overview of a crank-slider mechanism.
The angle of the connecting rod ψ is a function of the crankshaft angle θ, and satis�es
L sinψ = R sin θ (3.1)
in which L is the connecting rod length and R is the crankshaft radius. The e�ective height of theconnecting rod, as seen from the cylinder axis, is given by
L cosψ =√L2 −R2 sin2 θ ≈ L (3.2)
in which the Pythagoras Theorem is used to derive the unknown quantity. For the small connecting rodangles ψ, that typically occur in internal combustion engines, the e�ective height can be approximatedby the length of the connecting rod itself. The angular velocity of the connecting rod, obtained bytaking the time derivative of (3.1) and rearranging terms and substituting (3.2), is a function ofcrankshaft position and crankshaft velocity and is given by
ψ = θR cos θ
L cosψ≈ θR
Lcos θ (3.3)
10
The position of the piston Xp is derived from the geometry of the crank-slider mechanism as
Xp = R cos θ + L cosψ (3.4)
The velocity of the piston is obtained by taking the time derivative of the piston position, as given by
Xp = −Rθ sin θ − Lψ sinψ = −(R sin θ +
R2
2Lsin 2θ
)θ = −re�θ (3.5)
in which (3.3) is substituted and the identity sin θ cos θ = 12 sin 2θ is used. Hence, the angular and
linear velocity are related through the e�ective radius re� of the crank-slider mechanism.
3.2.2. Cylinder Pressure Torque
The induced torque due to pressure in the cylinder Tcomb is also derived from the geometry of thecrank-slider mechanism, which is shown in Figure 3.3 with the relevant forces for this derivation.
Figure 3.3.: Schematic overview of a crank-slider mechanism with relevant forces.
The cylinder pressure pcyl multiplied with the piston area A results in a combustion force Fcomb inline with the cylinder axis towards the rotation axis of the crankshaft. The combustion force can bedecomposed into a normal force and force in the direction of the connecting rod and is given by
Fcr =Fcombcosψ
=pcylA
cosψ(3.6)
The length R′ perpendicular to Fcr towards the rotation axis of the crankshaft, is given by
R′ = R sin (θ + ψ) = R (sin θ cosψ + cos θ sinψ) (3.7)
The cylinder pressure torque can thus be expressed by the product of the force in the direction of theconnecting rod Fcomb (3.6) and the length R′ (3.7), i.e.,
Tcomb (θ) = R′Fcr =
(R sin θ +
R2
2Lcos 2θ
)Fcomb = re�Fcomb (3.8)
in which again the e�ective radius is used, as in (3.5).
11
3.3. Friction
The engine is subjected to several sources of friction at its components. In general, detailed modelsare necessary to calculate the friction amplitudes. The main source of friction is the piston and linerinteraction per cylinder [1, 3, 8]. As an approximation of the friction characteristics that have beenexperimentally observed and reported in literature, the total friction torque can be modeled by threeterms that are applied at each cylinder, being a constant, an absolute piston velocity dependent termand an absolute combustion torque dependent term [3]. The magnitude of these three terms arecharacterized by the friction coe�cients c1, c2 and c3, resulting in the friction torque equation usedin this report, which is
Tfric,i = c1 + c2
∣∣∣Xp
∣∣∣+ c3 · |Tcomb| (3.9)
0 120 240 360 480 600 7200
20
40
60
80
100
120
Crankshaft position [deg]
Fric
tion
torq
ue [N
m]
instantaneousaverage
Figure 3.4.: Typical friction torque waveform per cylinder.
A typical waveform of the total friction torque per cylinder as shown in Figure 3.4, is calculated using(3.9) and some typical cylinder pressure and velocity pro�les. The two large peaks are mainly relatedto the load dependent term which is dominant around top-dead-center at 360 deg, while the othersare mainly speed dependent.
The friction coe�cients may vary among the operating points of the engine. Using cylinder pressureand velocity measurements of an engine, it might be possible to identify them as a function of enginespeed and load and derive the describing relation of these maps. More details about this approachare given in Section 5.2.2.
3.4. Equations of motion
In this report, we will derive the equations of motion of the system by using the Lagrangian mechanicsmethod [2, 6]. The equations of motions can be obtained by solving
d
dt
(∂Ekin
∂θi
)− ∂Ekin
∂θi+∂Epot∂θi
= Q (3.10)
12
where Ekin is the kinetic energy, Epot is the potential energy and Q the vector of all nonconservativetorques. The total kinetic energy is given by
Ekin =
9∑i=1
12Jcs,iθ
2i +
8∑i=3
12Jcr,rotθ
2i +
8∑i=3
12moscX
2p,i (ϕi) (3.11)
which is a summation of the kinetic energy of the rotating crankshaft segments and the connectingrod bodies, and the oscillating mass. The inertia of each cylinder segment consists of two times halfthe crankshaft mains that surround the cylinder, two crankshaft webs and one crankshaft pin, as wasexplained in Section 2.2, and is denoted by Jcs,i. The rotating part of the connecting rod is representedby Jcr,rot. The oscillating mass mosc consists of all piston related components, but also includes theoscillating part of the connecting rod. A correction for obtaining the correct equivalent connectingrod inertia is neglected here, as explained in Appendix B. The kinetic energy of the oscillating massis expressed in terms of its linear velocity (3.5), which is a function of its corresponding absolutecrankshaft position as de�ned by
Xp,i (ϕi) = −re� (ϕi) θi (3.12)
in which the absolute position, because of the �ring order 1-5-3-6-2-4, is given by
ϕi = θi −[
0 4 2 5 1 3] 2π
3(3.13)
The total potential energy Epot is given by
Epot =8∑i=1
12ki (θi+1 − θi)2 (3.14)
in which ki is the sti�ness between the bodies. Qi contains all nonconservative torques
Q =[
0 Tf T1 T2 T3 T4 T5 T6 Tload
]T− ∂D
∂θi(3.15)
in which D contains the material damping of the crankshaft model
D =8∑i=1
12di
(θi+1 − θi
)2(3.16)
and the torques Ti applied at the cylinder bodies Ji, i ∈ {3, ..., 8} consist of the cylinder pressuretorque (3.8) minus the friction torque (3.9)
Ti = Tcomb,i − Tfric,i (3.17)
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With the provided information, (3.10) can be solved to obtain nine equations of motion, which canbe written in the form
M(θ)θ +Kθ = F (θ, θ) (3.18)
in which the nonconstant mass matrix M(θ) = {Mi,j} consists of elements
Mi,j =
Jcs,i for i = j ∈ {1, 2, 9}Jcs,i + Jcr,rot +moscr
2e�
(ϕi) for i = j ∈ {3, ..., 8}0 elsewhere
(3.19)
with ϕi as in (3.13). The sti�ness matrix K = {Ki,j} has elements
Ki,j =
−ki − ki+1 for i = j, i ∈ {1, ..., 9}ki−1 for i = j − 1, i ∈ {2, ..., 9}ki for i = j + 1, i ∈ {1, ..., 8}0 elsewhere
(3.20)
with k0 = k9 := 0, and the input torque vector F (θ, θ) = {Fi} consists of elements
Fi =
{Qi for i ∈ {1, 2, 9}Qi −moscre� (ϕi) re� (ϕi) θi for i ∈ {3, ..., 8}
(3.21)
with ϕi as in (3.13). The last term of the input torque for the cylinder bodies comes from the variationin kinetic energy as a function crankshaft position and results in a static nonlinear input function.The numerical values of the entries of the nonconstant mass matrix M are given in Appendix A, butcan also be seen in Figure 3.5 in which the terms of matrix entry M3,3 of cylinder segment #1 aregiven as a function of crankshaft position. The variation is dependent on one of the system statesand therefore results in nonlinear system dynamics. The analysis of the approximation of the massmatrix by a constant matrix will be given in Section 5.2.1.
0 120 240 360 480 600 7200
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Crankshaft position [deg]
M3,
3 [kgm
²]
crankshaft connecting rod oscillating mass
Figure 3.5.: Matrix entry M3,3 terms of cylinder segment #1.
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3.5. Implementation
In general, the crankshaft model can be represented by the block diagram shown in Figure 3.6. Theinput to the model consists of the cylinder pressures per cylinder i ∈ {1, ..., 6} and the loads appliedby the testbench (Tload) and accessories (Tf ). Since the cylinder pressures were measured with theposition encoder at the front pulley as a reference, the cylinder pressures have to be applied based onthe same reference. The output of the model consists of the velocity of the crankshaft, as determinedat the front pulley, which is at the same location as the position encoder during the measurements.
Figure 3.6.: Block diagram of the crankshaft model.
The model can be represented by a state-space description with nonconstant A matrix and a staticnonlinear input function, as given by
x = A(x)x+B f(x, pcyl,i)
y = C x(3.22)
in which the system states consist of the body angular velocities and positions de�ned by
x =[θ1 θ1 . . . θ9 θ9
]T(3.23)
and the static nonlinear input function consists of the input torque vector (3.21), as de�ned by
f(x, pcyl,i) = F (3.24)
The di�erential equations are implemented in MATLAB and numerically solved by an ordinary dif-ferential equation (ODE) solver. Using the default ode45 solver results in relatively long simulationtimes, since small time steps are necessary to obtain a stable solution using the explicit method,thereby indicating that the problem is sti�. Therefore the implicit ode15s solver was chosen to per-form the simulations. For the variable time steps, a relative tolerance of 2 · 105 was chosen as acompromise between accuracy and simulation time. The front pulley velocity θ2 is selected as outputof the model. After simulation, the output can be used to calculate the error with the measuredoutput for identi�cation purposes or other analysis. The MATLAB code of the implementation isgiven in Appendix D.
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4. Parameter Identi�cation
The model structure presented in the previous chapter contains a set of parameters, which still needto be determined. For most of the parameters, an initial estimate is available from data that hasbeen provided by the engine manufacturer. However, this data was generated by simulation softwareand might contain some uncertainty. Furthermore, there are also unknown parameters that initiallycan only be estimated by rules of thumb. This makes it necessary to do parameter identi�cation inorder to �nd appropriate values for the parameters, such that the model reproduces the output of thereal engine as closely as possible. This chapter will give a description of the identi�cation procedurethat has been applied, an overview of the parameters to be identi�ed and some observations on thein�uence of the initial conditions of the simulations. Finally, the identi�cation results are discussedand simulation results are given for the crankshaft model with identi�ed parameters. The simulationresults will be analyzed in next chapter.
4.1. Identi�cation Method
As already mentioned, the inputs u of the model consist of the measured cylinder pressure signalsand the constant load applied by the testbench, which is obtained by averaging a low frequencytorque measurement over the 200 measured cycles. The load due to the accessories is neglected, inorder to reduce the amount of unknowns. By choosing a certain set of parameters in the model,its front pulley velocity output y can be compared to the measured velocity output y, as shown inFigure 4.1. The error e between those two signals is used in a least-squares error �tting algorithm tosearch for the set of parameters that minimizes the output error. The lsqnonlin function providedby the MATLAB Optimization Toolbox is used to do this. This function uses a trust-region-re�ectivemethod for determining the direction at which the parameters should be modi�ed in order to obtainthe minimal model error. The measured output velocity is zero-phase �ltered, using the MATLABfiltfilt function with a low-pass Butterworth �lter, in order to contain only frequencies up to the36th harmonic. The �ltered signal is then used in the parameter identi�cation process.
Figure 4.1.: Overview of the parameter identi�cation process.
4.2. Parameters
The parameters of the model can be grouped into inertia, mass, sti�ness, damping, friction, geometry,velocity and load related parameters. A full list of parameters and how they are identi�ed is shown
16
in Table 4.1. The data provided by the engine manufacturer, given in Appendix A, was used here asinitial estimates of the parameters.
Model parameter Unit Initial est. Ident. parameter Ident. model parameter
Inertia
J1 kgm² -
J2 kgm² scale factor p1
J3,rot kgm² -
J4,rot kgm² -
J5,rot kgm² -
J6,rot kgm² -
J7,rot kgm² -
J8,rot kgm² -
J9 kgm² scale factor p2
Mass mosc kg -
Sti�ness
k1 MNm/rad -
k2 MNm/rad scale factor p3
k3,...,k7 MNm/rad scale factor p4
k8 MNm/rad scale factor p5
Dampingd1 Ns/m scale factor p6
cd s²/m² cd = p7/100
Friction
c1 Nm c1 = p8+3(k−1) see Table 5.3
c2 Ns c2 = p9+3(k−1)/100 see Table 5.3
c3 - c3 = p10+3(k−1)/100 see Table 5.3
Geometry
R m -
L m -
A m² -
Velocity θ0,k rad/s N0/30 · π θ0,k = p8+3n+(k−1) see Appendix D
Load Tload Nm - - From measurements
Table 4.1.: Overview of the parameters in the crankshaft model.
The uncertainty for the inertia model parameters is mainly present at the inertia of the front J2and rear J9 body of the crankshaft, since the amount of lumped mass is not directly known. Theidenti�cation parameters p1 and p2 are used to multiply the initial estimates during identi�cation.The initial estimates of the torsional damper ring inertia J1, the constant terms of the cylinder bodyinertias Ji,rot with i ∈ {3, ..., 8} , and the oscillating mass mosc are considered to be accurate enoughto be used directly in simulation. The uncertainty in their values, for instance due to production
17
tolerances, is assumed to be too small to add additional identi�cation parameters. The oscillatingmass is assumed to be equal for each cylinder.
The sti�ness coe�cient k1 is neglected, since the viscous silicon �uid of the torsional damper is notexpected to have any torsional sti�ness. If the model is used to simulate the behavior of an enginewith tuned-mass torsional damper, consisting of a spring-supported �oating mass, then k1 can begiven a nonzero value. All cylinder bodies do have a similar structural layout. The sti�ness betweenthose bodies is expected to have the same amount of uncertainty and are therefore scaled with thesame identi�cation parameter p4. The sti�ness coe�cients that connect the front and rear body tothe cylinders are scaled with identi�cation parameters p3 and p5. They are expected to have eachtheir own uncertainty, since the e�ective length of these bodies may in practice deviate from thelengths that were used to obtain the initial estimates in FEM simulation.
For the internal material damping that is modeled by the relative damping coe�cients di, withi ∈ {2, ..., 8}, the values are determined by the relation provided in [6] and is given by di = cd ki/ωin which the parameter cd is used to �t the model to the data. The factor ki is of the correspondingsti�ness coe�cient and ω is the crankshaft velocity. As can be seen, the material damping decreaseswhen the velocity increases and increases when the sti�ness increases. Using the given relation, onlyone parameter is needed to �t the internal damping at multiple operating speeds. The damping ofthe torsional damper is scaled by identi�cation parameter p6.
The friction coe�cients c1, c2 and c3 are identi�ed for each individual operating point. The number ofidenti�cation parameters therefore varies with the number n of provided operating points. The sameholds for the initial velocities θ0,k of the bodies. The operating points are indexed by k ∈ {1, ..., n}.
Additional pre-scaling was applied to some identi�cation parameters in order to obtain similar mag-nitudes of range in which the parameters are allowed to be varied by the parameter identi�cationprocess. The implementation of the actual parameter identi�cation process in MATLAB, using thelsqnonlin function, is given in Appendix D.
4.3. Initial Conditions
At all the measured operating points, the system is at the same angular velocity after each cycle of tworevolutions. Still, the simulations have to be started with non-zero initial conditions. The velocitiesθi, i ∈ {1, ..., 9} may be approximated by the average crankshaft velocity, but the angular positionsθi, i ∈ {1, ..., 9} are more di�cult. However, it is observed that the angular positions converge to astable set of positions, when evaluating them in subsequent cycles at the same crankshaft position. Bystarting a simulation with the approximated velocities and zero-valued positions as initial conditions,the states at the end of a cycle can be used as initial conditions for the next simulation. As long asthe system is kept at the steady-state operating point, the �nal condition of one cycle can be usedas initial condition for the next cycle. This approach can be used in simulation both for parameteridenti�cation and analysis of the results as done in the next chapter.
The two-step identi�cation procedure is shown in Figure 4.2. At the �rst step, the simulation is runfor one cycle of 720 degrees. The state determined at the end of this cycle is used to start thesimulation of the second step, which is a series of three subsequent cycles. In this step, the errorbetween measured and simulated output of the second cycle and half of the �rst and third cycle isused to identify the parameters. The identi�cation data therefore consists of two cycles that cover thecombustion events in the cylinder order 1-5-3-6-2-4 and the range of data is long enough to ensure aparameter �t that keeps the system at its operating point.
18
Figure 4.2.: Two-step identi�cation procedure.
The identi�cation data consists of multiple operating points. The outlined two-step procedure isperformed for every individual operating point. The error output of each operating point is combinedinto one set of error data, which is used within the nonlinear least-squares error �tting algorithm.By sampling the error output at a crankshaft position interval of one degree, all operating points areequally weighted by having the same number of error values.
4.4. Results
The parameter values resulting from the identi�cation process are given in Table 4.1. The identi�edamount of inertia for the front body J2 is approximately equal to the initial estimate. The accessories,which are connected to the front body by means of a belt, therefore seem to be dynamically decoupledfrom the crankshaft. The identi�ed amount of inertia for the rear body J9 is substantially larger thanthe initial estimate, since the the camshaft is connected to the crankshaft by a gear transmissionand inertia is added by the shaft and coupling of the testbench. The sti�ness parameters have beenunderestimated in FEM analysis, with increasing underestimation towards the front of the crankshaft.
Using the identi�ed set of model parameters, it is possible to perform simulations in MATLAB. Thesimulation results given here are the angular velocity θ2 at the front body, as a function of the angularposition θ2 of the front body. Results are shown for nine operating points, discussed in Section 2.4:
� Operating points for identi�cation (A100, B100, C100, A20, B10, C0)Simulations are performed using the identi�ed set of model parameters.
� Operating points for validation (A60, B50, C40)The same model parameters are used. The friction coe�cients are interpolated between theones that have been identi�ed. Using the cylinder pressure signals of the validation operatingpoints and correct estimates of the initial velocity, it is possible to perform simulations.
All graphs do have a 120 degree grid interval, to indicate the 6 cylinder events. Measurements wereperformed with the combustion event of cylinder 1 located at 360 CAD. The same event location isused in simulations, such that the combustion event order is 6-2-4-1-5-3 for 0,120,240,360,480,600degrees crankshaft position. The determination of these event locations is described in Section 5.2.3.
With the same set of model parameters, the model is able to capture most dynamics at the selectedoperating points. As expected, the amount of variation in speed is mostly related to the load condi-tions. The waveform is mostly related to the engine speed. While the variation in speed is about 60rpm at A100, the variation increases up to 160 rpm at C100. The results will be analyzed in the nextchapter.
19
0 120 240 360 480 600 7201170
1180
1190
1200
1210
1220
1230
1240
1250
Crankshaft position [deg]
Spe
ed [r
pm]
A20
measurementfilteredsimulation
0 120 240 360 480 600 7201170
1180
1190
1200
1210
1220
1230
1240
1250
Crankshaft position [deg]
Spe
ed [r
pm]
A60
0 120 240 360 480 600 7201170
1180
1190
1200
1210
1220
1230
1240
1250
Crankshaft position [deg]
Spe
ed [r
pm]
A100
0 120 240 360 480 600 7201480
1490
1500
1510
1520
1530
1540
1550
1560
1570
1580
Crankshaft position [deg]
Spe
ed [r
pm]
B10
0 120 240 360 480 600 7201480
1490
1500
1510
1520
1530
1540
1550
1560
1570
1580
Crankshaft position [deg]
Spe
ed [r
pm]
B50
0 120 240 360 480 600 7201480
1490
1500
1510
1520
1530
1540
1550
1560
1570
1580
Crankshaft position [deg]
Spe
ed [r
pm]
B100
0 120 240 360 480 600 720
1760
1780
1800
1820
1840
1860
1880
1900
1920
Crankshaft position [deg]
Spe
ed [r
pm]
C0
0 120 240 360 480 600 720
1760
1780
1800
1820
1840
1860
1880
1900
1920
Crankshaft position [deg]
Spe
ed [r
pm]
C40
0 120 240 360 480 600 720
1760
1780
1800
1820
1840
1860
1880
1900
1920
Crankshaft position [deg]
Spe
ed [r
pm]
C100
Figure 4.3.: Front body angular velocity θ2 for the studied operating points.
20
5. Analysis
After identifying appropriate values for the parameters of the model, the model can be analyzed interms of accuracy, complexity and sensitivity. In the �rst section of this chapter, the accuracy of themodel is determined in both time and frequency domain. In the second section, the complexity ofthe model is reduced by proposing three simpli�cations and analyzing the consequences for modelingaccuracy. These simpli�cations include using a constant mass matrix, a simpli�ed friction modeland a reduction to a single rigid-body crankshaft model. In the third section, the sensitivity of thecrankshaft model to variation of parameters is studied in order to obtain insight in the importance ofthe individual parameters for identi�cation and robustness.
5.1. Accuracy
The accuracy of the identi�ed crankshaft model is determined by analyzing the error between themeasured and simulated angular velocity in both time and frequency domain, for all studied operatingpoints. In the frequency domain, the error is expressed as a function of engine harmonics. To do so,�rst a de�nition of the engine harmonics is given and the frequency content of the input and outputdata is determined for two operating points. Second, the resonance frequencies of the crankshaftsystem are determined and related to the observed output behavior. Based on the knowledge gainedin the �rst two sections, the model error is �nally analyzed to determine the accuracy of the identi�edcrankshaft model.
5.1.1. Engine Harmonics
The cyclic nature of internal combustion engines can be described in terms of engine harmonics. Fora four-stroke engine with two revolutions per cycle, the harmonic frequencies occur at multiples ofthe cycle duration, and have frequencies
fi =N
60 · 2kh , kh ∈ N (5.1)
Since the studied engine has six combustion events per cycle, every 6th harmonic is of special interest.The frequency components of the induced torque due to combustion are shown in Figure 5.1 foran A100 and C100 operating point. These torques were determined from the un�ltered cylinderpressure measurements. For each harmonic frequency, the frequency content is determined usingthe MATLAB function goertzel. This function implements a discrete Fourier transform using theGoertzel algorithm and returns the Fourier transform for a selected set of frequencies. The frequencyaxis is normalized to engine harmonics. It is clearly visible that every 6th, 12th, 18th and 24th harmonicis dominant in the system input. Frequency components above the 36th harmonic are not signi�cantanymore, when compared to the most dominant harmonics.
21
0 6 12 18 24 30 36 42 48
100
102
104 A100
Harmonic: fi/ f
1 [−]
Com
bust
ion
torq
ue |T
(f)|
[Nm
]
0 6 12 18 24 30 36 42 48
100
102
104 C100
Harmonic: fi/ f
1 [−]
Com
bust
ion
torq
ue |T
(f)|
[Nm
]
Figure 5.1.: Frequency components of the combustion torque.
The frequency components of the output angular velocity, measured at the pulley at the front ofthe engine, are shown in Figure 5.2. Although the frequency components of the combustion torquelook quite the same for the two given operating points, the output speed of the C100 operating pointshows excessive response at the 12th harmonic. This suggests a resonance frequency of the crankshaftsystem in the neighborhood of 184 Hz.
0 6 12 18 24 30 36 42 4810
−3
10−2
10−1
100
101 A100
Harmonic: fi/ f
1 [−]
Out
put s
peed
|dθ/
dt(f
)| [r
ad/s
]
0 6 12 18 24 30 36 42 4810
−3
10−2
10−1
100
101 C100
Harmonic: fi/ f
1 [−]
Out
put s
peed
|dθ/
dt(f
)| [r
ad/s
]
Figure 5.2.: Frequency components of the output speed.
5.1.2. Resonance Frequencies
With the identi�ed model parameters, it is possible to determine the system resonance frequencies.Due to the nonlinear nature of the system, the model does not have resonance frequencies at isolatedfrequencies. To still have an idea of the resonances of the system, it is linearized at di�erent crankshaftpositions, namely at θ2 = {0, 45, 90, 135, 180, 225, 270, 315} degrees. For each crankshaft position,the state-space representation of the system has constant matrices and is therefore regarded as locallylinearized. The Bode diagram of the transfer functions between torque inputs at the cylinder bodiesand front body position output is shown in Figure 5.3.
The location of the resonance varies between 196.5 and 203.6 Hz and is the same for every inputlocation. The location of the resonance agrees with the earlier rough estimation, in previous section,
22
based upon the measurement data. The anti-resonance however is also dependent on the inputlocation and for cylinder 1 to 6, it is located around 61, 67, 76, 89, 113 and 168 Hz respectively. Ascan be seen, the nonlinear e�ects do not cause a very large range of (linearized) dynamic behavior. Alinearization of the system is not expected to have dramatic e�ects on the performance of the model.A partial linearization, by averaging the mass matrix, will be therefore studied in Section 5.2.1.
100
102
−240
−220
−200
−180
−160
−140
−120
−100
−80
−60
−40From: T1
To: θ
2
100
102
From: T2
100
102
From: T3
100
102
From: T4
100
102
From: T5
100
102
From: T6
Frequency (Hz)
Magnitude (dB)
Figure 5.3.: Bode diagram of the locally linearized system, from torque inputs at the cylinder bodiesto angular position at the front body.
The excitation frequencies, of the combustion torque, can be shown as a function of engine speed.By also showing the system resonance frequency band, a so-called Campbell diagram is obtained, seeFigure 5.4. The A, B and C operating speeds, with their corresponding speed variations, are indicatedby the red areas. The resonance frequency band is highlighted in blue. The harmonics are shownby the dashed lines, whereby every 6th harmonic is solid. As can be seen, for operating speed Athe 20th harmonic does coincide with the resonance and one of the main harmonics (18th) is nearby,while at operating speed C the 13th harmonic does coincide with the resonance and one of its mainharmonics (12th) is nearby. This explains why at operating speed C a large �uctuation in output speedis observed.
0 500 1000 1500 2000 25000
50
100
150
200
250
Engine speed [rpm]
Fre
quen
cy [H
z]
Figure 5.4.: Campbell diagram with excitation and resonance frequencies.
23
5.1.3. Model Error
The accuracy of the model is determined by evaluating a number of properties in time and frequencydomain, at the operating points for validation. In the time-domain, the properties to evaluate modelaccuracy are the minimum, maximum and root mean square (RMS) error between measured andsimulated crankshaft velocity over one cycle. The relative RMS error is obtained by relating the RMS
error to the average crankshaft velocity θ2 at the front body. The time-domain properties are shownin Table 5.1.
θ2 [rad/s]Model error
Min [rad/s] Max [rad/s] RMS [rad/s] RMS [%]
A100 127.0 -0.9477 1.4441 0.4331 0.34
B100 159.7 -1.2659 1.9229 0.4993 0.31
C100 192.5 -2.0587 2.7798 0.7279 0.38
A60 127.0 -1.2627 1.3517 0.4129 0.33
B50 159.7 -1.5294 1.7083 0.4873 0.31
C40 192.5 -2.1657 2.1888 0.5619 0.29
A20 127.0 -0.9469 1.4293 0.3617 0.28
B10 159.7 -1.4739 1.8382 0.4725 0.30
C0 192.5 -2.0860 1.8624 0.6053 0.31
average -1.5263 1.8362 0.5069 0.32
Table 5.1.: Model accuracy at various operating points.
In general, the RMS error increases with increasing engine speed and slightly with increasing load. Thecrankshaft speed variation amplitude itself shows the same behavior for the given operating points,making the observed error characteristic feasible. Relating the RMS error to the average crankshaftvelocity does result in an average model error of 0.32%. Even when considering that peak values ofthe error might be three times larger than the RMS value, the relative error is very acceptable.
In order to obtain more information on the source of the model error, its frequency components aredetermined in terms of the engine harmonics. These frequency components are shown in Figure 5.5.At all operating points, the model error shows harmonic frequency components around 0.05 rad/s. Atthe A (1213 rpm) operating speed, the most dominant model error harmonics are around the 18th and36th harmonic. At the B (1525 rpm) and C (1838 rpm) operating speeds, the most dominant modelerror harmonics are mostly around the 12th harmonic. These are quite the same frequencies as thosethat are dominant in the measured output because of the resonance frequency that is excited, thusmaking it hard to determine a possible error source. The dynamics around the 6th main harmonic aregenerally captured well. As shown by the relative RMS error, the performance of the identi�ed modelis good. In order to improve the result, more attention should be given to identifying the systembehavior around the resonance frequency. This can be accomplished by increasing the importance ofthe C operating speeds during identi�cation and should result in more equally distributed frequencycontent of the velocity error.
24
0 6 12 18 24 30 36 42 480
0.1
0.2
0.3
0.4
0.5
Harmonic: fi / f
1 [−]
Vel
ocity
err
or: |
e(f)
| [ra
d/s]
A20
0 6 12 18 24 30 36 42 480
0.1
0.2
0.3
0.4
0.5
Harmonic: fi / f
1 [−]
Vel
ocity
err
or: |
e(f)
| [ra
d/s]
A60
0 6 12 18 24 30 36 42 480
0.1
0.2
0.3
0.4
0.5
Harmonic: fi / f
1 [−]
Vel
ocity
err
or: |
e(f)
| [ra
d/s]
A100
0 6 12 18 24 30 36 42 480
0.1
0.2
0.3
0.4
0.5
Harmonic: fi / f
1 [−]
Vel
ocity
err
or: |
e(f)
| [ra
d/s]
B10
0 6 12 18 24 30 36 42 480
0.1
0.2
0.3
0.4
0.5
Harmonic: fi / f
1 [−]
Vel
ocity
err
or: |
e(f)
| [ra
d/s]
B50
0 6 12 18 24 30 36 42 480
0.1
0.2
0.3
0.4
0.5
Harmonic: fi / f
1 [−]
Vel
ocity
err
or: |
e(f)
| [ra
d/s]
B100
0 6 12 18 24 30 36 42 480
0.1
0.2
0.3
0.4
0.5
Harmonic: fi / f
1 [−]
Vel
ocity
err
or: |
e(f)
| [ra
d/s]
C0
0 6 12 18 24 30 36 42 480
0.1
0.2
0.3
0.4
0.5
Harmonic: fi / f
1 [−]
Vel
ocity
err
or: |
e(f)
| [ra
d/s]
C40
0 6 12 18 24 30 36 42 480
0.1
0.2
0.3
0.4
0.5
Harmonic: fi / f
1 [−]
Vel
ocity
err
or: |
e(f)
| [ra
d/s]
C100
Figure 5.5.: Frequency components of the model error.
25
5.2. Model Simpli�cations
In previous section, the accuracy of the proposed model with its identi�ed parameters has beenshown. The proposed model structure is nonlinear and contains a signi�cant number of parameters.To study whether this amount of model complexity is really necessary, some simpli�cations of themodel are investigated in this section. Three simpli�cations are proposed, which are based on usinga constant mass matrix, a simpli�ed friction model and by representing the crankshaft as a singlerigid-body. Reducing the model complexity is bene�cial for reducing the computational e�ort duringidenti�cation and in the implementation of the virtual sensor algorithm on the ECU.
5.2.1. Constant mass matrix
The crankshaft model contains a crank-slider mechanism for every cylinder body. The kinetic energyof these bodies therefore varies as a function of its angular position and is one of the two reasons whythe system is nonlinear. Namely, it causes the mass matrix (3.19) to become a function of angularposition. The second reason why the system is nonlinear is because the input torque (3.21) is astatic nonlinear function of angular position. In this section, the consequences of averaging the massmatrix is evaluated. The reason for investigating this simpli�cation is that it would allow the systemto be decomposed into an interconnection of a linear time-invariant system and a static nonlinearfunction. The current version of the virtual cylinder pressure sensor algorithm is developed for thistype of system structure, where the constant mass matrix was originally obtained by neglecting thevariation due to the oscillating mass.
In Figure 3.5, the variation of the mass matrix element M3,3 is shown. The nonconstant contributionof the oscillating mass can be averaged by taking the average value of moscr
2e�. As a result, the
averaged mass matrix is given by
Mi,j =
Jcs,i for i = j ∈ {1, 2, 9}Jcs,i + Jcr,rot +mosc
(R2
2 + R4
8L2
)for i = j ∈ {3, ..., 8}
0 elsewhere
(5.2)
which is independent of the crankshaft position. The input torque remains the same static nonlinearfunction. As can be seen in Figure 5.6, the proposed constant mass matrix shows very little di�erencein output at the A100 and C100 operating points.
26
0 120 240 360 480 600 7201140
1150
1160
1170
1180
1190
1200
1210
1220
1230
1240
1250
Crankshaft position [deg]
Spe
ed [r
pm]
A100
measurementfilterednonconstantconstant
0 120 240 360 480 600 720
1650
1700
1750
1800
1850
1900
1950
Crankshaft position [deg]
Spe
ed [r
pm]
C100
measurementfilterednonconstantconstant
Figure 5.6.: Simulation output for the system with nonconstant and constant mass matrix.
The di�erence in output is hardly visible. The model error of both the simulation output withnonconstant and constant mass matrix is therefore shown in Figure 5.7.
0 120 240 360 480 600 720−8
−6
−4
−2
0
2
4
6
8
10
Crankshaft position [deg]
Spe
ed e
rror
[rpm
]
A100
nonconstant constant
0 120 240 360 480 600 720
−20
−15
−10
−5
0
5
10
15
20
Crankshaft position [deg]
Spe
ed e
rror
[rpm
]
C100
nonconstant constant
Figure 5.7.: Model error for the system with nonconstant and constant mass matrix.
The RMS model error for the operating points used for identi�cation is given in Table 5.2 for boththe model with nonconstant and constant mass matrix. In general, the di�erence in performance issmall and both positive and negative. At high loads the error increases slightly, but not excessively.The crankshaft model with the proposed constant mass matrix is therefore a good candidate forimplementation in the virtual cylinder pressure sensor algorithm. This removes the necessity to increasethe complexity and computational e�ort of the algorithm to deal with a non-LTI crankshaft model.
27
However, the actual performance should be evaluated in the intended application and a parameteridenti�cation dedicated to the simpli�ed model may improve the results.
Model error (RMS) [rad/s]
Nonconstant Constant Di�erence
A100 0.4331 0.4443 0.0112 (2.6%)
B100 0.4993 0.5165 0.0172 (3.4%)
C100 0.7279 0.7375 0.0096 (1.3%)
A20 0.3617 0.3598 -0.0019 (-0.5%)
B10 0.4725 0.4775 0.0050 (1.1%)
C0 0.6053 0.6149 0.0096 (1.6%)
Table 5.2.: Comparison in model error for the system with nonconstant and constant mass matrix
5.2.2. Simpli�ed Friction Modeling
The friction coe�cients, as de�ned in Section 3.3, are identi�ed for every single operating point. Forthe given set of operating points for identi�cation, this results in a set of six values for every coe�cient.This amount of values is not enough to determine a detailed trend or engine characteristic throughoutthe map of operating points. The identi�ed values are listed in Table 5.3.
Constant Velocity Load
c1 c2 · 100 c3 · 100
A100
B100
C100
A20
B10
C0
Table 5.3.: Identi�ed friction coe�cients.
A visualization of the friction coe�cients within the engine map is shown in Figure 5.8. The identi�edcoe�cients are shown in blue, while the interpolated coe�cients for the validation operating pointsare shown in red. The friction coe�cient c1 for the constant term is in general a correction for theapplied load by the testbench, and does not show a clear trend. Moreover, the constant term doesnot directly contribute to the dynamic behavior of the crankshaft. The friction coe�cient c2 for thedependency on absolute piston velocity and c3 for the dependency on absolute combustion torqueboth show a little bit more consistent trend.
28
10001500
2000
0
2000
4000−2
0
2
4
N [rpm]T [Nm]
c 1
10001500
2000
0
2000
40000
0.01
0.02
0.03
N [rpm]T [Nm]
c 2
10001500
2000
0
2000
4000−0.05
0
0.05
N [rpm]T [Nm]
c 3
Figure 5.8.: Identi�ed friction coe�cients in the engine map.
For the friction coe�cients at the validation operating points, the approach is to use the interpolatedvalues for c2 and c3, while c1 is tuned for correct steady-state operation. As can be seen in Section 4.4,this approach performs well.
Due to the lack of detail in the friction coe�cient characterization, caused by not having a largeset of operating points for identi�cation, it could be questioned whether it is realistic to include theproposed amount of detail in friction modeling. The instantaneous nature of the friction torque addsto the dynamics of the system, but will a constant friction deviate much in performance? To furtherinvestigate this question, additional simulations have been performed, whereby the average value ofthe instantaneous friction torque is used to replace the friction torque at the cylinder bodies. For theA100 operating point, where piston velocities are relatively low and combustion torques are large, acomparison of simulation results is shown in Figure 5.9. A slightly di�erent response can be seen, butin general the improvements and degradations cancel each other and a similar change in response isalso possible by parameter tuning, as will be shown in Section 5.3.
0 120 240 360 480 600 7201170
1180
1190
1200
1210
1220
1230
1240
1250
Crankshaft position [deg]
Spe
ed [r
pm]
measured filtered instantaneous constant
Figure 5.9.: Simulation comparison for instantaneous and constant friction torque at A100.
For the other operating points, the comparison is made by evaluating the di�erence in model error,as shown in Table 5.4. Overall, the di�erence in performance is small and both positive and negative.The C100 operating point is the only one that shows a large di�erence in model error, but the relative
29
RMS error is still only (0.794/192.5=) 0.41%, which is acceptable. It should be noted however thatthe model parameters were identi�ed using the instantaneous friction model and it would be better tocompare the two models after performing individual identi�cations. Furthermore, actual performanceshould also be evaluated in the intended application, namely the virtual cylinder pressure sensor.
Model error (RMS) [rad/s]
Instantaneous Constant Di�erence
A100 0.4331 0.4536 0.0205 (4.7%)
B100 0.4993 0.4855 -0.0138 (-2.8%)
C100 0.7279 0.7940 0.0661 (9.1%)
A20 0.3617 0.3541 -0.0076 (-2.1%)
B10 0.4725 0.4619 -0.0106 (-2.2%)
C0 0.6053 0.6298 0.0245 (4.0%)
Table 5.4.: Comparison in model error for instantaneous and constant friction torque.
5.2.3. Rigid-Body Approximation
The proposed model structure consists of nine separate bodies. The question arises whether thisamount of bodies is necessary for describing the dynamic torsional behavior of the crankshaft. Acommon approach for light-duty gasoline engines is to use a rigid-body model in which no de�ectionbetween the cylinder bodies is present [13]. This approach is used here to determine whether thereduction to a lower order model can also be used for this particular heavy-duty diesel application.All bodies are lumped into one body with a nonconstant mass, due to a summation of the individualcontributions of (3.19). The applied torque is a summation of all torques (3.21) that were originallyapplied per body. The equations of motion are given by M(θ)θ = F (θ, θ) in which the nonconstantmass M(θ) is given by M =
∑9i=1 Jcs,i + 6Jcr,rot +
∑8i=3moscr
2e�
(ϕi) and the input torque F (θ, θ)
is given by F =∑9
i=1Qi −∑8
i=3moscre� (ϕi) re� (ϕi) θi.
The MATLAB implementation of the rigid-body model is given in Appendix D. Rigid-body simulationshave been performed for the A20, A100, C0 and C100 operating point, of which the results are shownin Figure 5.10. As can be seen, the simulation output is dominated by the 6th harmonic frequency,where the measured output especially at the C operating speed does mainly contains the 12th harmonicfrequency. For the A operating points, the amplitude and waveform of the simulation output roughly�ts the measurement. At the C operating points, there is a large di�erence in amplitude and lesscomparable waveform. It is clear that the rigid-body crankshaft model does not contain enoughdynamics to describe the measured velocity output.
30
0 120 240 360 480 600 7201170
1180
1190
1200
1210
1220
1230
1240
1250
Crankshaft position [deg]
Spe
ed [r
pm]
A20
measurementfilteredsimulation
0 120 240 360 480 600 7201170
1180
1190
1200
1210
1220
1230
1240
1250
Crankshaft position [deg]
Spe
ed [r
pm]
A100
0 120 240 360 480 600 720
1760
1780
1800
1820
1840
1860
1880
1900
1920
Crankshaft position [deg]
Spe
ed [r
pm]
C0
0 120 240 360 480 600 720
1760
1780
1800
1820
1840
1860
1880
1900
1920
Crankshaft position [deg]
Spe
ed [r
pm]
C100
Figure 5.10.: Rigid-body simulation results for the A20, A100, C0 and C100 operating points.
Another way of looking at the reduction to a rigid-body model is by observing the de�ection of eachbody with relation to the front body, de�ned as θi−θ2 . Fairly large de�ections are present at full load,as shown in Figure 5.11 on next page. It should also be noted that the crankshaft passes through analmost non-deformed state every 120 degrees, which is mainly the result of the zero e�ective radiusat cylinder top-dead-center. For cylinder 6, which is at largest distance from the front pulley at whichthe position is measured, the de�ection reaches up to 0.8 degree. Neglecting these de�ections byreducing the system to a rigid-body is therefore not realistic, as was already shown by the rigid-bodysimulations.
31
0 120 240 360 480 600 720−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
Crankshaft position [deg]
Def
lect
ion
w.r
.t. fr
ont [
deg]
A20
0 120 240 360 480 600 720−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
Crankshaft position [deg]
Def
lect
ion
w.r
.t. fr
ont [
deg]
A60
0 120 240 360 480 600 720−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
Crankshaft position [deg]
Def
lect
ion
w.r
.t. fr
ont [
deg]
A100
0 120 240 360 480 600 720−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
Crankshaft position [deg]
Def
lect
ion
w.r
.t. fr
ont [
deg]
B10
0 120 240 360 480 600 720−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
Crankshaft position [deg]
Def
lect
ion
w.r
.t. fr
ont [
deg]
B50
0 120 240 360 480 600 720−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
Crankshaft position [deg]
Def
lect
ion
w.r
.t. fr
ont [
deg]
B100
0 120 240 360 480 600 720−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
Crankshaft position [deg]
Def
lect
ion
w.r
.t. fr
ont [
deg]
C0
1
2
3
4
5
6
rear
0 120 240 360 480 600 720−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
Crankshaft position [deg]
Def
lect
ion
w.r
.t. fr
ont [
deg]
C40
0 120 240 360 480 600 720−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
Crankshaft position [deg]
Def
lect
ion
w.r
.t. fr
ont [
deg]
C100
Figure 5.11.: De�ection angles with respect to the front body θi − θ2 for all operating points.
32
5.3. Parameter Sensitivity
In order to determine the relative importance of the model parameters, a parameter sensitivity studyhas been performed. This study gives an insight in which model parameters are important duringidenti�cation and for robustness. It also gives an indication of which parameters are important forimproving the performance at a particular operating point. The sensitivity is determined by varyingeach individual parameter at all operating points for identi�cation and observing the change in RMSspeed error. The parameter variation is done by scaling its identi�ed value in a range of 90% to 110%.The change in RMS speed error is then expressed as the relative change to the RMS speed error ofthe initial parameter value.
The parameters being studied are the parameters that were identi�ed, see Table 4.1. Also J1, Ji,rotand mosc are varied to observe their sensitivity. The parameter sensitivity results are given in thegraphs below.
Front body inertia
The parameter sensitivity of the front body inertia J2 is shown in Figure 5.12. All operating pointsshow a decrease in speed error RMS when the inertia of the front body is increased, except for theC100 operating point which shows a very sensitive response in the opposite direction. The C100operating point, with one of its main excitation harmonics close to the system resonance frequency,is sensitive for the exact location of this resonance frequency. This location is dependent on the frontbody inertia. The optimal value of the inertia is not directly visible as a minimum for all operatingpoints, but is a compromise between the performance at the various operating points.
0.9 0.95 1 1.05 1.1
0.4
0.5
0.6
0.7
0.8
0.9
1
J2 / J
2,identified [−]
Spe
ed e
rror
RM
S [r
ad/s
]
0.9 0.95 1 1.05 1.10.95
1
1.05
1.1
1.15
1.2
J2 / J
2,identified [−]
Spe
ed e
rror
RM
S in
crea
se [−
]
A100
B100
C100
A20
B10
C0
Figure 5.12.: Model error for variation of the front body inertia.
Rear body inertia
The parameter sensitivity of the rear body inertia J9 is shown in Figure 5.13. The variation in rearbody inertia results in optimal values for each operating point, where the value resulting from theidenti�cation process is the overall optimum. The A100 operating point is most sensitive to variation
33
of the rear body inertia, as would be expected since this operating point has the largest low frequencyexcitation. The large rear body inertia is relatively important for the low frequency system behavior.
0.9 0.95 1 1.05 1.1
0.4
0.5
0.6
0.7
0.8
0.9
1
J9 / J
9,identified [−]
Spe
ed e
rror
RM
S [r
ad/s
]
0.9 0.95 1 1.05 1.10.95
1
1.05
1.1
1.15
1.2
1.25
J9 / J
9,identified [−]
Spe
ed e
rror
RM
S in
crea
se [−
]
A100
B100
C100
A20
B10
C0
Figure 5.13.: Model error for variation of the rear body inertia.
Front sti�ness
The parameter sensitivity of the front sti�ness k2 is shown in Figure 5.14. All operating points tendto be quite insensitive for variation in front sti�ness and there is no clear overall trend. The frontsti�ness is quite large when compared to the sti�ness between the cylinders, thus making the systembehavior relatively insensitive to variation of this parameter. From the identi�cation process, theidenti�ed value is very close to the initial estimate from the available data.
0.9 0.95 1 1.05 1.10.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
k2 / k
2,identified [−]
Spe
ed e
rror
RM
S [r
ad/s
]
0.9 0.95 1 1.05 1.1
0.98
1
1.02
1.04
1.06
1.08
1.1
k2 / k
2,identified [−]
Spe
ed e
rror
RM
S in
crea
se [−
]
A100
B100
C100
A20
B10
C0
Figure 5.14.: Model error for variation of the front sti�ness.
Sti�ness between cylinder bodies
The parameter sensitivity of the sti�ness ki,i ∈ {3, ..., 7} between the cylinder bodies is shown inFigure 5.15. The C100 operating point is strongly sensitive to variation of the sti�ness, due to
34
variation in the location of the system resonance frequency. All other operating points show a lesssensitive response and do have an optimum just below the identi�ed value. The identi�ed value isagain a compromise between all operating points.
0.9 0.95 1 1.05 1.10
0.5
1
1.5
2
ki / k
i,identified [−]
Spe
ed e
rror
RM
S [r
ad/s
]
0.9 0.95 1 1.05 1.10.5
1
1.5
2
2.5
3
ki / k
i,identified [−]
Spe
ed e
rror
RM
S in
crea
se [−
]
A100B100C100A20B10C0
Figure 5.15.: Model error for variation of the sti�ness between the cylinder bodies.
Rear sti�ness
The parameter sensitivity of the rear sti�ness k8 is shown in Figure 5.16. The variation of the rearsti�ness shows the same sensitivity behavior as the front body inertia, but in opposite direction, aswould be expected from the resonance frequency ωn ≈
√k/m. That is, lowering the rear sti�ness
improves the performance at all operating points, except the C100 operating point which showsexcessive sensitivity in the other direction.
0.9 0.95 1 1.05 1.1
0.4
0.5
0.6
0.7
0.8
0.9
1
k8 / k
8,identified [−]
Spe
ed e
rror
RM
S [r
ad/s
]
0.9 0.95 1 1.05 1.10.95
1
1.05
1.1
1.15
1.2
1.25
k8 / k
8,identified [−]
Spe
ed e
rror
RM
S in
crea
se [−
]
A100B100C100A20B10C0
Figure 5.16.: Model error for variation of the rear sti�ness.
Torsional damper, damping coe�cient
The parameter sensitivity of the damping coe�cient d1 of the torsional damper is shown in Figure5.17. Within the applied range of variation, the performance of the model is not very sensitive for
35
the damping coe�cient of the torsional damper. This observation holds for all operating points.
0.9 0.95 1 1.05 1.10.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
d1 / d
1,identified [−]
Spe
ed e
rror
RM
S [r
ad/s
]
0.9 0.95 1 1.05 1.1
0.98
1
1.02
1.04
1.06
1.08
1.1
1.12
d1 / d
1,identified [−]
Spe
ed e
rror
RM
S in
crea
se [−
]
A100B100C100A20B10C0
Figure 5.17.: Model error for variation of the damping coe�cient of the torsional damper.
Torsional damper, inertia
The parameter sensitivity of the inertia J1 of the torsional damper is shown in Figure 5.18. In general,an increase of the inertia of the torsional damper causes better performance of the model. The C100operating point shows again an opposite and more sensitive behavior, comparable with the front bodyinertia. The optimal value for the inertia is a compromise between the performance of all operatingpoints.
0.9 0.95 1 1.05 1.1
0.4
0.5
0.6
0.7
0.8
0.9
1
J1 / J
1,identified [−]
Spe
ed e
rror
RM
S [r
ad/s
]
0.9 0.95 1 1.05 1.10.96
0.98
1
1.02
1.04
1.06
1.08
1.1
1.12
J1 / J
1,identified [−]
Spe
ed e
rror
RM
S in
crea
se [−
]
A100B100C100A20B10C0
Figure 5.18.: Model error for variation of the inertia of the torsional damper.
Damping coe�cient
The parameter sensitivity of the damping coe�cient cd is shown in Figure 5.19. For most operatingpoints, a somewhat increased damping coe�cient increases the performance of the model. For the
36
C100 operating point a decrease in damping coe�cient is bene�cial, again in contrast to the otheroperating points.
0.9 0.95 1 1.05 1.1
0.4
0.5
0.6
0.7
0.8
0.9
1
cd / c
d,identified [−]
Spe
ed e
rror
RM
S [r
ad/s
]
0.9 0.95 1 1.05 1.10.95
1
1.05
1.1
1.15
1.2
cd / c
d,identified [−]
Spe
ed e
rror
RM
S in
crea
se [−
]
A100B100C100A20B10C0
Figure 5.19.: Model error for variation of the damping coe�cient.
Cylinder body inertia
The parameter sensitivity of the cylinder body inertia Jcs,i is shown in Figure 5.20. The C100 operatingpoint shows an optimum at 94% of the identi�ed amount of cylinder body inertia. All other operatingpoints show an improvement with increased cylinder body inertia.
0.9 0.95 1 1.05 1.1
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
Ji / J
i,identified [−]
Spe
ed e
rror
RM
S [r
ad/s
]
0.9 0.95 1 1.05 1.10.9
1
1.1
1.2
1.3
1.4
Ji / J
i,identified [−]
Spe
ed e
rror
RM
S in
crea
se [−
]
A100B100C100A20B10C0
Figure 5.20.: Model error for variation of the cylinder body inertia.
Oscillating mass
The parameter sensitivity of the oscillating mass mosc is shown in Figure 5.21. In general, the modelperformance is increased with a decrease in oscillating mass. The C0 operating point shows a di�erentbehavior with an optimum at the original amount of oscillating mass.
37
0.9 0.95 1 1.05 1.1
0.4
0.5
0.6
0.7
0.8
mosc
/ mosc,identified
[−]
Spe
ed e
rror
RM
S [r
ad/s
]
0.9 0.95 1 1.05 1.1
1
1.05
1.1
1.15
mosc
/ mosc,identified
[−]
Spe
ed e
rror
RM
S in
crea
se [−
]
A100B100C100A20B10C0
Figure 5.21.: Model error for variation of the oscillating mass.
Observations on the sensitivity study
For all studied parameters, although not shown here, the summation of the speed error RMS valuesof each operating point does show a minimum at or nearby the identi�ed value. This indicatesthat the identi�cation process has successfully determined an optimal set of parameters. However,combinations of parameter variation are also possible and the identi�ed set of parameters mighttherefore also be a local minimum and dependent on the initial estimates of the parameter values.The initial estimates are based on data that was determined already quite accurately and these initialestimates are considered to be a good starting point. For most parameters, the C100 operatingpoint shows quite di�erent and more sensitive behavior under in�uence of parameter variation, dueto its excitation at the system resonance frequency. This often leads to a compromise between themodel performance of individual operating points. Since the C0 and C100 operating points are nearbythe system resonance frequency with one of their main excitation harmonics and do show oppositebehavior in their sensitivity to parameter variation, it is worth trying to identify the model parametersat only these two operating points. This would decrease the computational e�ort of the identi�cationprocess, because the other operating points are only necessary for identifying their friction coe�cients.
The parameters k2 for the front sti�ness and d1 for the damping coe�cient of the torsional damperare relatively insensitive to variation. They may need a larger range of parameter values duringthe identi�cation process in order to determine their optimal values. Within the range of variationof 90% to 110%, the parameters of the rear body inertia J9 and sti�ness between the cylinderski,i ∈ {3, ..., 7} do show the largest sensitivity and may be considered as most important parametersduring identi�cation and for robustness. Both parameters do show a large di�erence between theirinitial estimate and identi�ed value. Since they are important for the model performance, it isrecommended to provide a better initial estimate for the parameter identi�cation.
38
6. Conclusions, Recommendations and
Implications
In this report, a dynamic crankshaft model is presented, which is a crucial part in the virtual cylinderpressure sensor concept of TNO. In this chapter, conclusions are drawn from the work discussed inthis report, some observations for future research are made and the implications of this work arediscussed.
6.1. Conclusions
A new �exible crankshaft model has been successfully developed and experimentally validated. Thecontrol oriented model does use six cylinder pressure signals as input and provides the angular velocityat the location of the position encoder as output. In contrast to crankshaft models found in literature,which are mainly used for structural design and diagnosis, the new model is developed for on-lineestimation of individual cylinder pressure traces to be used in closed-loop combustion control. Thepresented work is dedicated to a DAF MX 6-cylinder diesel engine, which represents a typical heavy-duty diesel engine application. Compared to previous work at TNO, an improved approximation ofthe system is proposed, to obtain a LTI system with static nonlinear function which �ts the currentvirtual sensor algorithm.
A parameter identi�cation is performed using the least squares error �tting algorithm lsqnonlin of theMATLAB Optimization Toolbox. The initial model parameter values are based on FEM data providedby the engine manufacturer and are proven to be close to the values identi�ed from measurementdata. A parameter sensitivity analysis does show that the cylinder body inertia Jcs,i,i ∈ {3, ..., 8}, rearbody inertia J9 and cylinder sti�ness ki,i ∈ {3, ..., 7} are the most sensitive parameters and thereforemost important during identi�cation and for robustness.
For nine operating points throughout the operating region of the engine, the average RMS errorbetween simulated and measured velocity is about 0.5 rad/s. Expressed as a relative error this isabout 0.3% of the crankshaft velocity, which is a good result. At the C operating points (angularvelocity equal to 1838 rpm), the �rst natural frequency of the system is strongly excited. This resultsin a fairly simple velocity waveform, since it is dominated mainly by a single frequency. At lowerengine speeds, the velocity waveform is dominated by more frequencies. Especially at the B operatingpoints and at low loads, in which case the resonance frequencies are hardly excited, there is a largerdi�erence in simulated and measured velocity waveform, although not noticeable in the RMS error.Overall, the crankshaft model and single set of identi�ed parameters performs well at all operatingpoints in predicting the angular velocity with a small error, at the position encoder location.
Two candidates for model simpli�cation are proposed. The �rst proposal is an averaging of the massmatrix of the equations of motion. Using a constant mass matrix does not signi�cantly increase theoutput error and is therefore a good candidate for implementation in the virtual cylinder pressure sensoralgorithm. The second proposal is to use a simple friction model in which the instantaneous friction
39
torque is replaced by a constant average friction torque. This reduces the amount of parameters tobe identi�ed at each operating point. For the considered set of operating points, the output error isboth increased and decreased, but only with small amounts.
A reduction to a single rigid-body, which is commonly the approach for light-duty engines, is not acandidate for model simpli�cation. It was shown that this is not a valid approach for the heavy-dutyengine being considered here, since the amount of angular de�ection between the bodies cannot beneglected and will result in a large output error.
6.2. Recommendations for Future Research
The crankshaft model presented predicts the angular velocity with small error. As such, it is rec-ommended to implement the crankshaft model in the virtual cylinder pressure sensor algorithm todetermine the actual performance of the model and identi�ed set of parameters. In the virtual sensoralgorithm, the required accuracy of the crankshaft model may be de�ned as crankshaft position de-pendent. The required accuracy around top-dead-center of each piston is larger, as a consequence ofthe method used by the virtual sensor algorithm to determine the six cylinder pressure traces from oneangular velocity signal. This may be exploited by using a crankshaft position dependent weightingfunction during identi�cation, in order to improve the performance of the virtual cylinder pressuresensor algorithm.
The intended application is in a heavy-duty vehicle drivetrain. It is expected that the vehicle drivetrainsystem may be regarded as dynamically decoupled from the crankshaft system, but additional researchhas to be done to prove this statement. A similar analysis as for the testbench dynamics can be used.If the drivetrain is not decoupled from the crankshaft system, this will introduce a transmission-geardependent rear body inertia. Another problem to solve is knowledge about the applied load at therear of the crankshaft system. The load can be estimated by the ECU of the engine, or has to bemeasured at the driveshaft. The load applied by the accessories also has to be estimated by theECU or has to be measured at the front pulley system. For the real-time implementation of thealgorithm, it is necessary to evaluate whether the calculations can be performed within the availabletime-span. Since production engines are equipped with low-resolution position sensors, it has to bedetermined whether interpolation of this signal is accurate enough for the virtual sensor algorithm.Future research may also focus on online estimation of the model parameters and on model accuracyduring transients where the applied loads are not constant.
6.3. Implications
A model describing the dynamic behavior of a heavy-duty diesel engine has become available as a resultof this work. Since this control oriented model was required for the next step in the development andvalidation of the virtual cylinder pressure sensor concept, the algorithm might actually prove itself onreal engine data. The virtual sensor will make implementation of closed-loop combustion control muchmore attractive, because the amount of required cylinder pressure sensors is reduced to a single one.Closed-loop combustion control will, in turn, contribute to cleaner and more fuel e�cient vehicles.
40
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[2] B. de Kraker. Mechanical Vibrations, chapter 2: Lagrangian Mechanics, pages 79�82. ShakerPublishing, 2009.
[3] C.D. Rakopoulos E.G. Giakoumis A.M. Dimaratos. Evaluation of various dynamic issues duringtransient operation of turbocharged diesel engine with special reference to friction development.SAE International, page 20, 2007.
[4] M. Geveci A.W. Osburn M.A. Franchek. An investigation of crankshaft oscillations for cylinderhealth diagnostics. Mechanical Systems and Signal Processing, 19, 2005.
[5] G. Genta. Vibration Dynamics and Control, chapter 29: Torsional Vibration of Crankshafts.Springer, 2009.
[6] H. Ying Y. Shouping Z. Fujun Z. Changlu L. Qiang W. Haiyan. Non-linear torsional vibra-tion characteristics of an internal combustion engine crankshaft assembly. Chinese Journal of
Mechanical Engineering, 25:12, 2012.
[7] M. Harrison. Controlling Noise and Vibrations, chapter 6: Sources of vibration and their control,page 326. SAE International, 2004.
[8] J.B. Heywood. Internal Combustion Engine Fundamentals, chapter 13: Engine Friction andLubrication, pages 712�747. McGraw-Hill, 1988.
[9] I. Andersson T. McKelvey. A system inversion approach on a crankshaft of an internal combustionengine. IEEE Conference on Decision and Control, 2004.
[10] A.J. Martyr M.A. Plint. Engine Testing, Theory and Practice, chapter 9: Coupling the engineto the dynamometer, pages 170�196. Elsevier, 2007.
[11] F.P.T. Willems E. Doosje F. Engels X. Seykens. Cylinder pressure-based control in heavy-dutyegr diesel engines using a virtual heat release and emission sensor. SAE International, 2010.
[12] F.P.T. Willems. Cylinder pressure-based control for heavy-duty diesel engines: potential andcontrol system speci�cation. Technical report, TNO-060-HM-2011-00247, 2011.
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Control Conference, 2011.
41
B. Connecting Rod Inertial Properties
The inertial properties of the connecting rod are given by its mass mcr, inertia Jcr and location ofthe center of gravity (COG), as depicted in Figure B.1. The length of the connecting rod is de�nedby L. The COG is positioned by length a and b.
Figure B.1.: Representation of the connecting rod.
It is common practice to divide the connecting rod into a rotating mass mrot at the crankshaft pinand an oscillating mass mosc at the piston pin. This division is performed based on the length ratioof the COG position as shown by
mrot =b
Lmcr ≈ 3.35 [kg] (B.1)
mosc =a
Lmcr ≈ 1.44 [kg] (B.2)
However, for a correct equivalent of the connecting rod in terms of its kinetic energy, a correctionJcr,ψ has to be made to the resulting inertia of the two masses, as shown in [5] and here repeated by
Jcr,ψ = Jcr −(mrota
2 +moscb2)≈ −0.0047 [kgm²] (B.3)
The correction Jcr,ψ is de�ned around the COG of the connecting rod. For calculations, it is moreconvenient to express the correction around the rotation axis of the crankshaft as Jcr,θ, such that itcan be added to the inertia of the cylinder segment body. The conversion is based upon conservationof kinetic energy between the two considered points, as shown in
12Jcr,θθ
2 = 12Jcr,ψψ
2 → Jcr,θ = Jcr,ψ
(ψ
θ
)2
(B.4)
By substituting (3.3) into (B.4), the correction is expressed as
45
Jcr,θ = Jcr,ψR2
L2cos2 θ (B.5)
However, by evaluating the maximum absolute value of the correction at θ = 0
max |Jcr,θ| = −Jcr,ψR2
L2≈ 4.51 · 10−4 [kgm2] (B.6)
and comparing it to the constant part of the cylinder body inertia
max |Jcr,θ|Jcyl
≈ 0.5% (B.7)
the conclusion is drawn that it is not necessary to include the correction.
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C. Testbench Dynamics
The measurements on the DAF MX engine are performed at a testbench, where the engine wasconnected to an electric motor using a shaft and rubber coupling. It is common practice to designthe sti�ness of this coupling for obtaining a resonance frequency outside the normal operating rangeof engine speeds, as outlined in [10]. The testbench inertia can therefore be assumed to be decoupledfrom the crankshaft system and may be neglected in the dynamic analysis of the crankshaft. Sincethe sti�ness of the coupling is known for this particular set of measurements, a veri�cation can bemade whether this assumption is valid.
A comparison is made between a rigid-body crankshaft (system A) and a rigid-body crankshaft witha testbench inertia connected to it by means of a sti�ness (system B). A schematic diagram of bothsystems is shown in Figure C.1.
Figure C.1.: Testbench dynamic model comparison.
For system A, the equation of motion and its Laplace transform is given by
Jcsθcs = Tcs + Tload,1 →(Jcss
2)
Θcs = Tcs + Tload,1 (C.1)
The angular position output can be written as
Θout = Θcs =1
Jcss2Tcs +
1
Jcss2Tload,1 (C.2)
For system B, the equations of motion and their Laplace transform are given by
Jcsθcs = Tcs − k (θcs − θtb) →(Jcss
2 + k)
Θcs = Tcs + kΘtb (C.3)
andJtbθtb = Tload,2 + k (θcs − θtb) →
(Jtbs
2 + k)
Θtb = Tload,2 + kΘcs (C.4)
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By substituting (C.3) in (C.4), the angular position output can be written as
Θout = Θtb =
[Jcss
2 + k
(Jcss2 + k) (Jtbs2 + k)− k2
]Tload,2 +
[k
(Jcss2 + k) (Jtbs2 + k)− k2
]Tcs (C.5)
The ampli�cation between input torque Tcs and output position Θout is used to determine the similarityof both models in the frequency domain. By multiplying the ampli�cation derived from (C.2) withthe ampli�cation and its inverse derived from (C.5), the model comparison is expressed as
k
(Jcss2 + k) (Jtbs2 + k)− k2=
1
Jcss2Hcompare (C.6)
in which Hcompare is given by
Hcompare =Jcss
2k
(Jcss2 + k) (Jtbs2 + k)− k2(C.7)
The amplitude of Hcompare is a measure for frequencies where the crankshaft and testbench maybe regarded as decoupled. Around the resonance frequency of this transfer function, the behavior isdominated by the multi-body dynamics of system B. At other frequencies, the e�ect of the testbenchinertia may be neglected when analyzing the crankshaft dynamics. The location of the resonance canbe determined by solving for the poles of Hcompare, which are given by
fn =1
2π
√Jcs + JtbJcsJtb
k (C.8)
When the testbench inertia increases, the resonance frequency decreases. There is an asymptoticminimum of the resonance frequency that can be achieved, which is de�ned by
fn,min =1
2π
√k
Jcs(C.9)
For the engine being used in this report, the asymptotic boundary is at about 11Hz. Since only theamount of inertia of the crankshaft system and coupling components are known, and informationabout the electric motor is not available, it is not possible to de�ne an exact resonance frequency.Using the minimal amount of known inertia, an upper boundary at 32Hz can be determined using(C.8). The actual resonance frequency will be somewhere between these two limits. Whether thismakes it necessary to include the testbench dynamics depends on the harmonic excitation frequenciesof the engine. At a low engine speed of 1000rpm, the �rst harmonic is located at 81/3Hz. For a6-cylinder engine, mainly the 6th harmonic (50Hz) and its multiples are of importance, as was shownin Section 5.1.1. It is therefore not necessary to include the testbench dynamics into the model ofthe crankshaft.
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