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Application of suspension derivative formulation to ground vehicle modeling and simulation
Maalej, Aref Younes, Ph.D.
The Ohio State University, 1988
Copyright ©1988 by Maalej, Aref Younes. All rights reserved.
UMI 300 N. Zeeb Rd. Ann Aibor, MI 48106
APPLICATION OF SUSPENSION DERIVATIVE FORMULATION
TO GROUND VEHICLE MODELING AND SIMULATION
DISSERTATION
Presented in Partial Fulfillment of the Requirements for
the Degree Doctor of Philosophy in the Graduate
School of the Ohio State University
By
Aref Younes Maalej, B.S.M.E, M.S.M.E, M.S.Math
*****
The Ohio State University
1988
Dessertation Committee; Approved by Q
Dr. Dennis A. Guenther Dr. Jack Collins AdvisoçX Dr. Chia-Hsiang Menq Department of mechanical Dr. Michael Foster Engineering Copyright by
Aref Younes Maalej
1988 ACKNOWLEGEMENTS
The author is very Greatfull to all who helped directly or indirictly in the completion of this work. The list includes many which makes it impossible to list all individuals in this acknowledgement statement. However special individuals merit special commendation.
The author wishes to express his gratitude to professors Dennis Guenther and John Ellis for their invaluable support, assistance and encouragement throughout the entire period of the work. I extend my thanks also to the staff of Systems Engineering Associates (SEA Inc.), particularly to Betty, for their personnel support and their help in the typing of this document.
Finally, I extend my appreciation to all my family members, particularly to Dad and Mom, and to my brothers in the OMAR IBN ALKHATTAB Mosque for their personnel support which was necsssry for the completion of this work.
IX VITA
February 3, 1963 Born - Sfax, Tunisia
December, 1984 . B.S.M.E., The Ohio State University,. Columbus, Ohio
1984-1988 Research Associate Department of Mechanical Engineering The Ohio State University Columbus, Ohio
March, 1986 ...... M.S.M.E., The Ohio State University, Columbus, Ohio
1986 ...... Assistant Project Engineer Engineering Department Shell Tunirex Tunis, Tunisia
1987 ...... Assistant Project Engineer Systems Engineering Assoc. Worthington, Ohio
June, 1988 ...... M.S.Math, The Ohio State University, Columbus, Ohio
1988 ...... Teaching Accociate Department of Mechanical Engineering The Ohio State University Columbus, Ohio
PUBLICATIONS
Maalej, A., " Modeling of Mechanical Friction at the Piston Cylinder INterface”, M.S. Thesis
Maalej, A., Guenther, D., Singh, R., " Effect of Friction on Dynamic Response of an Actuator", NFPC Journal Vol.l, No.l, June 1988, and 42nd National Fluid Power Conference - ASME, March 1987
111 Maalej, A., " A Unified Method of Large scale System Model Reduction", Les Annales de L'ENIT, Vol.2, No.2, Juillet 1988
Maalej, A., Guenther, D., Ellis, J., " Numerical Stability of Vehicle Dynamics Simulations", LASTED International Symposium in Applied Simulation and Modeling, Texas, May 1988
Maalej, A., Guenther, D., Ellis, J., "Modeling of Tire Friction Forces and Moments" Accepted by the International Journal of Vehicle System Dynamics. FIELDS OF STUDY Major Fields: Mechanical Engineering Mathematics
Areas of Specialization: System Modeling Simulation and Analysis . Mechanics, Advanced Dynamics, and Vibration Control Systems, Process Control, Digital Control, Large Scale Systems, Stochastic Processes, Signal Processing Machine Design, Advanced Mechanical Design Automation, Fluid Power Control Mathematical Methods in Engineering, Abstract and Linear Algebra, Theory of Real and Complex Functions, Integral Equations.
ÏV CONTENTS
______Page
ACKNOWLEDGEMENTS ...... Ü
VITA ...... iii
LIST OF TABLES...... viii
LIST OF FIGURES ...... ix
CHAPTER I. INTRODUCTION ...... 1
1.1 General ...... 1 1.2 Scope and Objective ...... 2 1.3 Research Overview ...... 3
CHAPTER II. LITERATURE REVIEW ...... 4
2.1 Introduction ...... 4 2.2 Multi-body Methods in Vehicle Dynamics ...... 4 2.3 Classical Lumped Methods in Vehicle Dynamics .. 12 2.4 Review of Previous Work for the Current Research ...... 14 2.4.1 Vehicle Body Subsystem and Inertial Properties ...... 14 2.4.1.1 Weight Distribution ...... 17 2.4.1.2 Center of Gravity Location ...... 17 2.4.1.3 Inertia Parameters ...... 18 2.4.2 Suspension Subsystem of the Vehicle ...... 20 2.4.3 Steering System of the Vehicle ...... 26 2.4.4 Measurement of Vehicle Parameters ...... 28 2.4.4.1 Base Structure of the SPMD ...... 32 2.4.4.2 Suspension Support Frame ...... 32 2.4. 4. 3 Wheel Pads ...... 32 2.4.4.4 Hydraulics ...... 34 2.4.4.5 Instrumentation ...... 34
CHAPTER III. EVALUATION OF TIRE MODELS AND FORMULATION OF SUSPENSION DERIVATIVES ...... 41
3.1 Introduction ...... 41 3.2 Evaluation of Tire Models ...... 41 3.2.1 Analytical Models ...... 42 3.2.2 Empirical Models ...... 45 3.2.2.1 Lateral Force ...... 45
V 3.2.2.2 Longitudinal Force ...... 49 3.2.2.3 Alignment Moment ...... 50 3.2.3 Combined Braking and Steering ...... 58 3.2.3.1 Friction Ellipse Concept ...... 58 3.2.3.2 Shaping Function ...... 58 3.2.3.2 Friction Cake Concept ..... 59 3.2.4 Computation Method and Model Evaluation .. 64 3.3 Suspension Derivatives for Kinematic Suspension 71 3.3.1 Suspension Derivatives of a^2-D Suspension 73 3.3.2 Suspension Derivatives of a 3-D Suspension 83
CHAPTER IV. DEVELOPMENT OF VEHICLE MATHEMATICAL MODELS ...... 95
4.1 Introduction ...... 95 4.2 Lumped Parameter Vehicle Model ...... 96 4.2.1 Equations of Motion ...... 96 4.2.2 Determination of Forces and Moments ..... 100 4.2.3 Camber Effect on Forces and Moments ...... 102 4.2.4 Longitudinal and Lateral Slip ...... 104 4.2.5 Wheel Dynamics ...... 105 4.3 Vehicle Model with Kinematic Suspension ...... 105 4.3.1 Formulation of Vehicle Model with Suspension Derivatives ...... 108 4.3.1.1 Body Motion Equations ...... 108 4.3.1.2 Suspension Kinematic Equations ... 108 4.3.1.3 Generalized Forces ...... 109
CHAPTER VI. MODEL SIMULATION '...... Ill
5.1 Introduction ...... Ill 5.2 ACSL Simulation Software ...... Ill 5.3 Numerical Stability of Vehicle Dynamics Simulations ...... 113 5.3.1 General Formulation of Ground Vehicle Dynamics ...... 113 5.3.2 Integration Methods in Vehicle Dynamics Simulations ...... 114 5.3.3 Open Loop Versus Closed Loop Integration . 118 5.3.4 Numerical Illustration ...... 119 5.3.5 Analytic Analysis of Simulation Stability. 121 5.3.6 Numerical Stability and Vehicle Simulation Formulation ...... 129 5.4 Simulation Model Organization ...... 129 5.5 Vehicle Handling Maneuvers ...... 136
VI CHAPTER VI. SIMULATION RESULTS AND DISCUSSION .... 140
6.1 Introduction ...... 140 6.2 Vehicle Suspension Derivatives ...... 140 6.3 simulation Maneuver Response ...... 161 6.4 Simulation Times and Costs ...... 182 6.5 Introduction of Developed Simulation to the SPMD Software Package ...... 190
CHAPTER VII. CONCLUSIONS AND RECOMMENDATIONS ...... 200
7 .1 Conclusions ...... 200 7.2 Recommendations ...... 203 APPENDIX:
A. Data used in Chapter III ...... 204
LIST OF REFERENCES ...... 207
VII LIST OF TABLES
TABLE PAGE
2.1 MBS Simulation Software ...... 11
2.2 Design Specification for the S P M D ...... 33
2.3 List of Basic Language Programs for the SPMD 36
2.4 Suspension.Parameters Measured by the SPMD . 37
3.1 Tire Lateral Tire Force Models ...... 46 i 3.2 Empirical functions of Sine Friction Model . 47 I 3.3 Empirical Constants for Tire Friction Models 66 ! 3.4 Classification of Tire Friction Models .... 72
5.1 Time Step for Stability ...... 128
6.1 Suspension Derivatives for 2-D Suspension ... 157
6.2 Suspension Derivatives for 3-D Suspension ... 158
6.3 Simulation CPU Times ...... 189 I 6.4 Suspension Derivatives Measured by the SPMD . 199 /
V Ï 1 1 LIST OF FIGURES
FIGURES PAGE
2.1 Multibody System Structure ...... 5
2.2 Automobile and a Possible Multibody Model. 7
2.3 Menu with Options of the Progràm MEDYNA .. 9
2.4 ADAMS Representation of a Front Suspension 10
2.5 Analytical Representation of Four Wheeled Vehicle ...... 15
2.6 Hypothetical Vehicle Control System ...... 16
2.7 Dynamic Moment of Inertia of a Moving Vehicle ...... 19
2.8 Short/Long Arm Suspension ...... 21
2.9 Independent MacPherson Strut Suspension 21
2.10 Suspension Load-Deflection Characteristics 23
2.11 Shock Absorber Force-Velocity Characteristics ...... 24
2.12 Photograph of the SPMD ...... 29
2.13 Sketch of the SPMD's Top View ...... 30
2.14 Sketch of SPMD's Right Side View ...... 31
2.15 Hysteresis Loop in the Suspension Test Results ...... 38
2.16 Suspension Force Versus Spring Deflection . 39
3.1 SAE Tire Axes and Terminology ...... 43 3.2 Deformation and Friction Force for a Tire Model ...... 44
3.3 Longitudinal Coefficient of Friction .... 51
IX 3.4 Longitudinal Tire Force for Various Models 52
3.5 Alignment Torque Versus Slip Angle ...... 54
3.6 Resultant Friction Force for Combined Braking and Cornering ...... 60
3.7 Tire Forces for Combined Braking and C o r n e r i n g ...... 61
3.8 Lateral Force Versus Longitudinal Slip ... 63
3.9 Lateral Force for Various Models ...... 67
3.10 Axes System for the Vehicle with Kinematic S u s p e n s i o n ...... 74
3.11 Short/Long Arm Suspension ...... 75
3.12 Instantaneous Centers of Rotations ...... 77
3.13 SLA 3-D Suspension ...... 85 4.1 Vehicle Model with Lumped Mechanical Characteristics ...... 97
4.2 Force and Moment Components in a Vehicle Model ...... 103
4.3 Vehicle Model with SLA S u s p e n s i o n ...... 106
4.4 Schematic View of a Vehicle Model with kinematic Suspension ...... 107
5.1 Explicit Structure of ACSL Program ...... 112
5.2 Mechanical Second Order System ...... 115
5.3 Second Order Vehicle Model ...... 120
5.4 Open Loop and Closed Loop Block Diagrams .. 122
5.5 Critical Time Step for Convergence...... 123
5.6 Vehicle Dynamics Block Diagram for Time Domain Simulation ...... 130 5.7 Applied Torque in Braking Maneuver ...... 137
5.8 Steering Input in TM and CLM Maneuvers .... 139
6.1 Suspension Derivative of Fore and Aft with Respect to Roll ...... 141
6.2 Suspension Derivative of Fore.and Aft with Respect to Pitch ...... 142
6.3 Suspension Derivative of Fore and Aft with Respect to Bounce ...... 143
6.4 Suspension Derivative of Fore and Aft with Respect to Yaw ...... 144
6.5 Suspension Derivative of Steer with Respect to Pitch ...... 145
6.6 Suspension Derivative of Steer with Respect to Roll ...... 146
6.7 Suspension Derivative of Steer with Respect to Bounce ...... 14?
6.8 Suspension Derivative of Steer with Respect to Y a w ...... 148
6.9 Suspension Derivative of Camber with Respect to Yaw ...... 149
6.10 Suspension Derivative of Camber with Respect to Pitch ...... 150
6.11 Suspension Derivative of Camber with Respect to roll ...... 151
6.12 Suspension Derivative of Camber with Respect to Bounce ...... 152
6.13 Suspension Derivative of Scrub with Respect to Bounce ...... 153
6.14 Suspension Derivative of Scrub with Respect to Yaw ...... 154
6.15 Suspension Derivative of Scrub with Respect to Pitch ...... 155
XI 6.16 Suspension Derivative of Scrub with Respect to Roll ...... 156
6.17 Vehicle Parameter and Data ...... 159
6.18 Camber and Steering Data ...... 160
6.19 Forward Velocity in a Braking Maneuver ... 162
6.20 Longitudinal Slip in a Braking‘“Maneuver ... 163
6.21 Longitudinal Force in a Braking Maneuver .. 164
6.22 Vehicle Tragectory in a Braking Maneuver . 165
6.23 Lateral Slip in a Braking Maneuver ...... 166
6.24 Lateral Velocity in a Braking Maneuver .... 167
6.25 Pitch Velocity Response in a Braking Maneuver ...... 168
6.26 Suspension Deflection in a Braking Maneuver 169
6.27 Simulated Lateral Slip in a CLM Maneuver .. 170
6.28 Lateral Velocity in a CLM Maneuver .... 171
6.29 Yaw Response in a CLM Maneuver ...... 172
6.30 Pitch Velocity in CLM Maneuver ...... 173
6.31 Roll Response in CLM Maneuver ...... 174
6.32 Suspension Deflection in CLM Maneuver .... 175
6.33 Lateral Velocity in CLM and Braking ...... 176
6.34 Forward Velocity in CLM and Braking ...... 177
6.35 Lateral Velocity in CLM and Braking ...... 178
6.36 Longitudinal Slip in CLM and Braking ...... 179
6.37 Pitch Response in CLM and Braking ...... 180
6.38 Vehicle Trajectory in CLM and Braking .... 181
Xll 6.39 Lateral Slip in a Turning Maneuver ...... 183
6.40 Lateral Velocity in a Turning Maneuver ... 184
6.41 Roll Response in a Turning Maneuver ...... 185
6.42 Pitch Velocity Response in a Turning M a n e u v e r ...... 186
6.43 Yaw Response in a Turning Maneuver ...... 187
6.44 Vehicle Trajectory in a Turning Maneuver . 188
6.45 Wheel Camber Angle as a Function of Roll Angle ...... 191
6.46 Variation of Steer Angle with the Roll Angle ...... 192
6.47 Variation of Tire Scrub with Roll Angle .. 193
6.48 The Normal Tire Force as a Function of Roll 194
6.49 Variation of Camber Angle with Bounce .... 195
6.50 Variation of Steer Angle with Bounce ...... 196
6.51 Tire Scrub as a Function of Bounce ...... 197
6.52 Variation of Tire Normal Force with Bounce 198
XÏX1 CHAPTER I
INTRODUCTION
1.1 General
The development of mathematical models for vehicle dynamics studies _ started in the mid-sixties as the flourishing automobile industry entered a new era of design and manufacturing. The vehicle dynamic analysis is aimed to understand and improve the handling and ride performance of vehicles and reduce their vibratory behavior so that maximum comfort may be achieved by the driver. Although several vehicle models were developed for this purpose, they all emphasized some particular characteristics as related to specific applications, such as the vehicle response to accidental crush. This rendered the models more complex and more difficult to understand. Indeed the vehicle system was modeled as a huge mechanical machine with many degrees of freedom. The high order mechanical coupling used in relating these various degrees of freedom not only caused an increase in the complexity of the models,_ and thus poorer and lower understanding of the simulation mechanism, but also became an annoying factor in controlling the numerical stability of the simulations.
The available simulations of vehicle dynamics were developed on the basis of body centered Euler equations of motion which are basically in terms of velocities (momentum considerations). The acceleration subsequently derived from these velocities might cause instability to the numerical computation due to the integration process, and the vehicle may numerically loose contact with ground if no feedback is present to retain the stability of the computation. This problem of numerical instability as related to the vehicle dynamics simulations is investigated in this study based on the newly introduced concept of open loop and closed loop integration.
Recently a Suspension Parameter Measuring Device (SPMD) was developed in the Vehicle Research and Test Center (VRTC) of the National Highway Traffic Safety Administration at East Liberty, Ohio. This device allows the measurement of various parameters related to the suspension and steering systems of the vehicle based on quasi-static testing. However, due to the variety of data and measurements that can be collected by the SPMD, existing vehicle dynamics simulations do not take full advantage of these data. For example, the SPMD measures kinematically induced motions of the wheels in the longitudinal, lateral and vertical directions, camber and steer directions of the wheels, vertical suspension forces, and caster rotation of Kingpin steering axis. Though many suspensions observe significant wheel motion in the x and y directions, the current vehicle dynamics simulations do not utilize data on these motions.
A vehicle model that takes full advantage of the SPMD's measurement capabilities and makes use ,of*, the substantial amount of suspension information that is not utilized by currently available vehicle dynamics simulations, and which prevents the possibility of any numerical instability, is the subject of this work.
1.2 Scope and Objective
The objective of this research is to focus on the understanding and the development of suspension derivatives and to develop a mathematical model for a ground vehicle with kinematic suspension. A self supported software package for the analysis of vehicle dynamics of vehicle models with kinematic suspension is also to be developed. The other objectives are the evaluation of the tire friction models based on specific criteria and the investigation of the numerical instability problem encountered in many vehicle dynamics simulation routines. Specifically, the main objectives of this study are:
1. To investigate the vehicle system components and the available models for every functional block of the vehicle. Special emphasis is given to the tire model since the controlling forces and moment are generated at the tire/ground interface. Tire models are evaluated based on model accuracy, practicality of model parameter determination and the CPU time of computation
2. To study the deficiency of large scale vehicle models from the numerical stability point of view, necessary useful conclusions are drawn to limit this inconvenience.
3. To develop two vehicle mathematical models, one using the classical large scale approach and the other introducing a new concept of suspension derivatives reducing the system to its fundamental degrees of freedom. 4. Create computer simulations for the two developed models in light of the conclusions obtained from the preceding objectives.
5. To investigate the introduction of the SPMD's measurable suspension data to the developed simulation for the model with kinematic suspension.
6. To compare results obtained from the simulation developed for the proposed models.
1.3 Research Overview
This investigation is organized in seven chapters. The introductory chapter includes a general discussion of the problem and specification of the objectives.In the second chapter the available literature in the domain of vehicle modeling and simulation is reviewed. Both the multi-body and lumped parameter approaches are discussed and compared. Previous work completed in the domain of this research is also reviewed in this chapter. In the third chapter special emphasis is put on the tire friction model, since it represents the cornerstone for the directional and stability performance of the vehicle model. Tire friction models are evaluated using collected tire data. The suspension derivatives of kinematic suspension are reviewed and developed with respect to the bounce, pitch, roll and yaw modes of the vehicle. In the fourth chapter two approaches in vehicle modeling were adapted in the development of mathematical models. The first model is developed based on the classical, lumped parameter method where the suspensions are modeled with their independent degrees of freedom (unsprung masses), whereas in the second model the tire and suspension systems are taken to be kinematically linked to the vehicle body. A thorough analysis of the suspension kinematics is included, and the corresponding suspension derivatives are developed. In the fifth chapter, a discussion of the critical factors influencing the stability of vehicle simulations is included. The simulations corresponding to the two vehicle models of chapter four are also developed. The sixth chapter includes the simulation results for several test maneuvers using the two developed models. The response is then analyzed and compared. The incorporation of these simulations with the SPMD available software is then discussed. The last chapter includes some concluding remarks, a summary of the achieved objectives, and recommendations for future work. CHAPTER II
LITERATURE REVIEW AND PREVIOUS WORK
2.1 Introduction
Two different approaches have been used to model and study the dynamics of ground vehicles. The lumped model approach consists of breaking the major parts of'the vehicle into mechanical subsystems composed of three fundamental mechanical elements: inertia, damping, and stiffness. Appropriate coupling is then introduced to connect these parts and form multi-degrees of freedom vehicle mechanical model. In this method the suspension and steering systems are treated as "black boxes" where no flexible members are included in the model and the mechanical properties are taken to be lumped properties.
The second approach is the elastic or flexible multi-body system approach (MBS) playing the same role for vehicle systems as finite element methods play for elastic structures. In the multi-body system, the individual bodies may be interconnected by joints and linkages constraining their relative motion, as well as by coupling or compliant elements resulting in coupling or interaction forces between neighboring bodies. The major advantage of the multi-body system approach is the elimination of the tedious and susceptible to error stage of the equation derivation since the system equations are generated by the computer starting from basic data for the elements. For instance, the bodies are characterized by their geometries and inertia properties, while their interconnections are described by their kinematic constraints .
2.2 Multi-body Method in Vehicle Dynamics
The multi-body approach is commonly used for complex systems which contain rigid and flexible members. The complete structure is modeled by mixing the equations obtained for each subsystem and the constraint equations. A general multi-body structure is given in Figure 2.1. The formulation of the problem is achieved in three stages:
1. The kinematic stage: where the position of the free rigid body can be described by the translation vector of its center of mass. Frequently the vector components are written with respect to a unit vector basis of the inertial reference system. The attitude, or the Bearing
Spring
Rigid Body i
Damper
Actuator
Spring
Rigid Body (i+ 1)
Figure 2.1 Multibody System Structure rotational orientation, of the body fixed axis X ., y., 2. with respect to the inertial axes X_, Y^, Zj
are generally described by a 3 x 3 rotation tensor using Euler angles.
2. The kinetic, or dynamic equations can be obtained using Lagrange's equations, once the energy expressions are determined. This stage results in the differential equations’governing the motion of the bodies.
3. The suspension dynamics stage: this is particularly important in the vehicle dynamics formulation, where the dynamic equations have to be described by additional constraint equations.
In general, the system equations are written in state space form. Therefore, a minimal order representation is expressed by first order differential equations resulting in
Xi = Xg
X 2 = M X 2 +kx^+GU+PW+ F(X^, X2 , U, W) where M is the interstate coupling matrix, k is the stiffness matrix, G is the input coupling matrix, and P is the noise transmission matrix. The input vector U may include the steering angle, the braking or driving torque and the aerodynamic forces. The noise or external disturbance vectors may include the road irregularities and the non deterministic aerodynamic effect. The non linear .function F represents the coupled non linear effect of the various contributing variables. A multi-body model for a ground vehicle is shown in Figure 2.2.
There are two important software in the domain of multi-body analysis : the Automatic Dynamic Analysis of Mechanical Systems (ADAMS), developed in the USA [2], and the Multi-body Dynamics program, MEDYNA, developed in West Germany [3]. The strategy in the first is to establish a maximal set of equations of motion for the free body diagram, and then incorporate the algebraic constraint equations. The strategy associated with the MEDYNA Program deals with a minimal set of differential equations by reducing the order through incorporating the constraint equations. This latter approach is expected to be more efficient in reducing the numerical problems associated with the integrations, as will be seen later in this rt«* f t ’tuSSMf'. »***
J, u c mwf##* ti»»* «yiB»*!**** 5. »ytst» ^ ^ ît.#* 4. »■«' ««ISffnatBA t . ttJS
FIGURE 2.2 Automobile and a Possible Multibody Model research. Furthermore, only the MEDYNA software can treat mechanical systems with closed loops, though the constraint equations have to be linearly independent. ADAMS handles large displacements while MEDYNA allows only small displacements relative to moving reference frames. In MEDYNA the elastic behavior of some of the vehicle parts may also be included. The elastic behavior of ground vehicles are particularly important since the elastic vibration mode of the car body may cause significant contribution to mass accelerations. The simulation languages used with the MBS software, provide a number of integration routines, including variable step size routines and routines for stiff differential equations. Animated graphics output is also provided with ADAMS. MEDYNA, standing for "MEhrkorper DYNAmik", the German expression for Multi-body Dynamics, is an MBS general purpose software. It can generate system equations, system solutions, and evaluate a design vehicle dynamic behavior. Bodies can be kinematically connected in tree or closed loop configuration. The coupling elements can be linear or non linear, algebraic, or non algebraic (dynamic). Passive or active force relations can be generated.
The MEDYNA software provides a library of elements where the user has only to supply location and the parameters (spring, dampers, etc.). It also has available specific vehicle models describing the primary lift and guidance suspension mechanics for road vehicles with pneumatic tires. MEDYNA establishes the dynamic equations, together with the holonomie constraints in the form;
MX + Dx + Kx = g (x, X , t, ...)
Cx(t) = 2 (t) where M, D,and K are the mass, damping and stiffness matrices, and g comprises all internal and external forces. C is the constraint matrix and z(t) is the kinematic excitation vector.
MEDYNA allows the linear system, vibratory and stochastic analysis. It is an integrated interactive program where the offered menu of options is shown in Figure 2.3.
The Automated Dynamic Analysis of Mechanical Systems, ADAMS, was initially developed to analyze the static, transient, and vibrational behavior of non linear mechanical systems. The linkages are described by their mass, inertial moments, and initial generalized coordinates; the joints, by their types and linkage adjacencies, and the springs and dampers by their force M I Ë I D Ï Ï M â MULTIBODY DYNAMICS
I. SYSTEM II. GENERATION OF III. ANALYSIS OF"'^ , IV. ANALYSIS OF DEFINITION EQUATIONS STATICS DYNAMICS
> MODEL > SYSTEM > NOMINAL > EIGENVALUES, IDENTIFIER MATRICES INTERACTION EIGENVECTORS FORCES > MULTIBODY > OUTPUT > FRE Q U EN C Y R E - . CONFICURÀTION MATRICES > S T A T IC SPONSE (GRAPHICS) EQUILIBRIUM > GYROSTATS > TIME-DEPENDENT > SP E C TR A L D E N S IT Y AND NONLINEAR (GRAPHICS) > ELA STIC TERMS STRUCTURES > COVARIANCE > MODELS FOR EXTER ANALYSIS > INTERCONNECTIONS NAL SIMULATION LANGUAGES (ACSL) > NUMERICAL > MOTION OF INTEGRATION REFERENCE FRAME > EVALUATION OF TIME HISTORIES > EXCITATIONS (GRAPHICS) > S Y S T E M OUTPUTS
• FIGURE 2.3 Menu With Options of the Program MEDYNA 10
*^12' *13' *14' ^15' *16^_
\ y --- / xl_/ /(X, •20' *21' *22' *23) % / s-Xm A. . - y f * ':
DIRECTION OF ^ Q l \ REACTION FORCES ^ d ir e c t io n OF r e a c t io n to r q u e s ■I ZF
Figure 2.4 ADAMS Representation of a Front Suspension 11
TABLE 2.1 MBS Simulation Softwares
Software Name Characteristics Applications (language)
ADAMS MBS with large displacements Machine (Fortran open loop (tree formulation) Dynamics Cartesian coordinates Interactive data input Vehicle Impact effect included Dynamics
MEDYNA MBS with small displacement Machine (Fortran) Moving reference frame Dynamics Closed loop allowed Lagrangian Coordinates Vehicle Interactive data input Dynamics Non linear force characteristics,Flexible bodies
DADS MBS with large displacement Machine (Fortran) Cartesian Coordinates Dynamics Interactive Data input Flexible bodies (in Vehicle preparation) Dynamics 12 coefficients and their attachment points relative to the links. ADAMS has the advantage of retaining all angular and displacement variables as solution variables which aids the modeling process since new inter-body constraints can be accommodated just by adding algebraic equations relating corresponding variables. In addition, the joint reactions forces are explicit solution variables, and the friction effects are routinely related to reaction forces. These advantages are at the expense of increased time of computation and the numerical difficulties which may arise from the coupled dynamic and algebraic equations.
Further development of the ADAMS method has resulted in a new program named DADS (Dynamic Analysis and Design Sensitivity). In this program new contributions, such as the Euler parameters, and the Generalized Partitioning Method are introduced. An example of ADAMS representation of a front suspensions is shown in Figure 2.4.
These multi-body simulation software are quite recent, and many other simulations are in the process of being developed. Table 2.1 summarizes the main available software discussed in this section.
2.3 Classical Lumped Methods in Vehicle Dynamics
The Lumped Vehicle System (LVS) approach has been the common approach used in the modeling and analysis of vehicle systems since the early stages of vehicle design.
It was in 1966 when the Calspan Corporation (formally Cornell Aeronautical Laboratory Inc.) introduced a model which included the general three dimensional motion resulting from vehicle control input, traversal of irregular terrain, or from collisions with simple road side barriers [6]. The model was subsequently named the Highway Vehicle Object Simulation Model (HVOSM), which has since then observed several developmental modifications in order to include the various data and parameters that have become accessible as highly developed devices become more and more introduced in the domain of vehicle instrumentation, testing and measurement. The HVOSM simulation now possesses several versions which account for various conditions and applications. For instance, version V-4 is extensively used for study of roadway and roadside geometries. Version V-7 of HVOSM is more suitable for the study of complex dynamics resulting from accident avoidance evasive maneuvers.
The vehicle is commonly modeled with an independent front suspension and a solid axle or independent rear suspension. 13
The vehicle is assumed to consist of a sprung mass, two front unsprung masses, and one or two rear unsprung masses. The sprung mass is assumed to have the standard six degrees of freedom (three translational and three rotational) that are normally associated with rigid body motion and other degrees of freedom associated with the motion of the unsprung masses. The number of degrees of freedom may vary from one simulation to another. The dynamics of the steering system is also sometimes included in the model, as well as the movements of the wheels. No longitudinal motion of the wheels is allowed by the model. The suspension system is treated as a "black box" where the inputs are the tire forces and moments acting on each vehicle wheel and the vertical motion of the unsprung masses. The outputs are the vertical forces of suspension exerted on the sprung masses, and the position and angular orientation of each of the vehicle wheels.
Besides the Highway-Vehicle-Object Simulation Model (HVOSM), there are few other models of the same category. These are the Hybrid Computer Vehicle Handling Program (HVHP) and the Improved Digital Simulation Fully Comprehensive (DSFC), both of which have similar suspension models as the HVOSM and differ just slightly from it. For instance, the HVHP simulation was developed on the basis of the HVOSM by the Bendix Research Laboratory, where the impact and irregular terrain routines were removed and simplifications were made to equations of motions of sprung and unsprung mass. Additional degrees of freedom associated with the wheel rotation and steering system were introduced to allow the investigation of complex vehicle dynamics resulting from accident avoidance maneuvers.
The development of the Hybrid Computer Vehicle Handling Program allowed the improvement of the calculation methods and the data presentation for the Vehicle Handling Test Procedure (VHTP) maneuvers developed by the University of Michigan Highway Safety Research Institute. The Calspan Corporation has also intensively used the HVHP simulation in their study of the influence of tire properties on passenger vehicle handling.
The mathematical model of the HVHP Simulation consists of seventeen non linear degrees of freedom. Four degrees of freedom are associated with the rotational motions of the four wheels. The steering system is represented by the lumped parameter model with three degrees of freedom corresponding to rotational motion of each of front wheel about its steering pivot and translational motion of the connecting steering rod and associated mass elements. The other ten degrees of freedom include the six rotational and 14 translational degrees of freedom associated with three dimensional rigid body motions of the sprung mass, two degrees of freedom for the vertical motion of each of front wheel, and two for the vertical and rotational motion of the rear axle. The program contains also the option of independent rear suspension. The vehicle mechanical model is illustrated in Figure 2.5.
An investigation conducted to evaluate several programs for vehicle dynamics ended with the following conclusions [ 4 ] : •.. '<1.
1. Many programs lack complete documentation.
2. They were developed for specific tasks and thus are not suitable for general application.
3. They do not use efficient algorithms and standard codes.
In addition most programs are usually not well structured, not fully tested, and their use is restricted to the party who has developed the code.
2.4 Review of Previous Work for the Current Research
The first stage in the development of vehicle models consists of understanding the constituting functional blocks of the system. The vehicle system has four main functional blocks. These are the vehicle body, the suspension system, the tire system, and the steering system. These four major subsystems of a ground vehicle determine the performance of ride and handling. This can be seen from the block diagram of a hypothetical driven-vehicle system shown in Figure 2.6.
Several models of these vehicle functional blocks have developed and introduced to vehicle mathematical models. An emphasis is put on the suspension force model whose parameters are measurable by the Suspension Parameter Measuring Device.
2.4.1 Vehicle Body Subsystem and Inertial Properties
Any vehicle ride or handling study requires a knowledge of the mass of the vehicle and its mass distribution properties. The mass distribution may be described by the inertia parameter values relative to a centroidal axis system and the location of the center of gravity of the vehicle. 15
9
SPACE-riXED AXIS SYSTEM 0
FIGURE 2.5 Analytical Representation of Four-Wheeled Vehicle ' 16
Road Road conditions vibration Aerodynamic ahead of and noise inputs vehicle
Position Steering Heading angle system Force Speed Road position Driver Vehicle
Brakes Accelerator
Steering feedback
Vibration, noise to driver
FIG 1 Hypothetical vehicle control loop including driver
Figure . 2.6 Hypothetical Vehicle Control System [7] 17
2.4.1.1 Weight Distributions
For the classical approach, the vehicle is seen as a mechanical system with lumped masses corresponding to the sprung weight and the front and rear unsprung weights. For an independent suspension the unsprung weight is assumed to be equally divided and centered at the wheel centers. Although there is no standard mass distribution formula, estimation of this distribution has been studied and several expressions were proposed to compute the sprung and unsprung masses. ■.
In a report based on General Motor vehicle dynamics research where hundreds of domestic passenger cars were tested [8] it was suggested that the unsprung weights can be reasonably estimated by the linear functions:
Wuf =0.04 W^ + 60
Wur = 0.067 W^ + 90 where is the total vehicle curb weight (lb) (i.e. full gas tanK and no passengers or other loading). In the statistical analysis of collected data performed by the National Aeronautical Establishment in Canada [9], linear expressions for the sprung and unsprung masses, were derived using the least square fit. Typical relations are given as follows:
Wg = 0.856 W^
^Uf = 0.385 Wut Where W is the sprung mass and W . , W ^ and W - are the total and front unsprung masses. 2.4.1.2 Center of Gravitv Location
The vehicle center of gravity location depends on each particular vehicle in question, and a complete set of data on that vehicle must be collected for this purpose. The center of gravity coordinates in the horizontal plane depends on the fore-aft weight distribution. The center of gravity height is also a property of each given car and it generally ranges from 19 to 24 inches above ground. For the lateral location of the center of gravity it is always 18 assumed that a vertical plane of symmetry exists and contains the vehicle longitudinal center line.
2.4.1.3 Inertia Parameters
The inertia parameters are determined with respect to a selected axis system. In the dynamic analysis of the total vehicle system the moment and product of inertia values are given relative to the centroidal axis systems. The centroidal inertia values can also be estimated based on vehicle weights, since they only depend "bn the vehicle mass and its distribution. These inertia values remain practically constant during ride or handling maneuver since the effect of the rotation angles of the vehicle body remain essentially negligible. To observe this effect the inertia values were expressed as a function of the rotation angles in a steering with braking maneuver, using the math model developed in Chapter 4. The variation of the total moments of inertia is shown in Figure 2.7. It is reasonable to assume these values constant throughout the simulation to reduce the computation time.
In [ 7 ] , it was proposed that the radii of gyration of the vehicle are “ 0.925 for the yaw rotation, (K^)roll " 0.0330 for the roll rotation and pitch “ 216 for the pitch rotation. Therefore, the corresponding total moments of intertia are given by I = (W^/g)(K^).
In the statistical data analysis performed by General Motors, the following linear correlations were also suggested:
ly^ = 1.13 - 2020
I^t = 1.26 - 1750
Ixs = 0.16 - 256 where the moments of inertia are in slug-ft^ and the total vehicle curb weight is in lb. Since the sprung mass moment of inertia are also required in dynamic handling analysis, empirical expression are developed and given by: 19
o CMo
§
(n
O O
g s 0.00 1.00 2.00 3.00 5.4.00 T (SEC)
FIGURE 2.7 Dynamic Moment of Inertia of a Moving Vehicle 20
^ys ^yt “ ^yu Igg = 1.05 - 1470
where is the unsprung mass pitch moment of inertia
to the total vehicle centroidal axes (slug-ft^). It is given by:
^yu = (^uf/9) ^ + (^ur/9) The roll-yaw product of inertia of the vehicle is a measure of its mass distribution in the xz plane. It appears in the directional control equations of motion and produces inertial coupling between roll and yaw motions. The correlation formula for this inertial property was found not to be achievable. However, it was proposed in [9] that I is a function of the roll and yaw moments of inertia ana an angle ( \ ) measuring the inclination of the principal system of axes relative to the x-axis. The centroidal system of axes used in Figure 2.4 is not, by definition, a principal system. The angle would define the principal system of axes.
The roll-yaw product moment of inertia is given by:
Ixz = V 2 (Ig - 1^) tan (2X ).
2.4.2 Suspension Subsvstem of the Vehicle
In American passenger cars the front suspension is an .independent system with transverse links using coil springs or torsion bars. The rear suspension is often a solid axle with a multi-control arm system using coil springs or semi elliptic leaf springs. The two types of independent suspension, which account for the majority of cars are the Short/Long Arm suspension and the MacPherson strut and link. The first is one of the earliest independent suspension designs; it has the upper control arm much shorter than the lower control arm. An example of this type of suspension is shown in Figure 2.8. In the strut and link suspension the anti-roll rod operating arm provides the fore and aft tie for the link. An example of the MacPherson suspension is shown in Figure 2.9.
One of the functions of the suspension is to absorb and damp the transmitted movement from the wheels to the vehicle body. This controls the vibration of the vehicle 21
M
Figure 2.8 Short/Long Arm Suspension [10]
Figure 2.9 Independent MacPherson Strut Suspension 22 and improves the ride performance. For this purpose the suspension is considered to have stiffness and damping effects. The suspension inputs are the tire forces and moments acting at each of the vehicle wheels, the vertical motions of the unsprung masses for independent suspension, and the vertical motion of the axle center and the axle roll angle for a solid axle. The suspension output is the vertical force the suspension exerts on the sprung mass and the position and orientation of the vehicle wheels.
A model which was proved to be satisfactory was adopted for the vehicle math model developed in the next chapter. In this suspension model the vertical force transmitted to the vehicle body is composed of several components [6]:
1. The static sprung mass weight (W .) acting at each wheel i. SI
For independent front and rear suspensions, it is given by;
Wsi = M^g b(a+b)/2 for i = 1,2 (2 .1)
= M^g a(a+b)/2 for i = 3,4
2. The Coulomb friction force (F^^) is computed by:
Fii = C'i sign(d^) (2.2)
where d . is the vertical suspension velocity and C • is the Coulomb friction coefficient.
3. The spring force acting at wheel i, is produced by the deflection of the spring and the suspension bump stop. These suspension bumpers are used to prevent metal to metal contact with excessive wheel vertical travel and are designed to minimize and control the effects of impact on the sprung mass. They are assumed to possess constant load deflection rates. The total spring force is given by:
^2i = + Fas! (2.3) The bump stop force component of the spring force is calculated by:
^Bsi ~ ^si^ “ 1) (dg^ - fiip) for dg^ > (2.4)
= kgi ( ^c - l)(dsi - Oc) ‘^si < ^c = 0 for Rg < dgj^ < 23
Suspension Force
Rebound
Suspension Deflection
Jounce
Figure 2 .1 0 Suspension Load—Deflection Characteristics 24
Damping Force
Suspension Velocity
Figure 2.11 Shock Absorber Force-Velocity Characteristics 25
where d . is the suspension deflection from the position of static equilibrium at no-load condition.
d^i = d. + diiH
This assumed form of the suspension load deflection characteristics is shown in Figure 2.10.
4. The Viscous damping is provided by the shock absorber. The restoring force of the shock absorber is approximated by four straight line segments as illustrated in Figure 2.11. The viscous damping force is expressed as:
^3i = Cl di + (Cg - C^) dg for < dg (2.5) • » • = C_ ds for d_ < dj < o. Z 1 C — i = C3 d^ for 0 < d^ < dg
= C4 di - (C4 - C3) dg for dj, > de
where denotes the slope of the segments and d and d are the break point velocities for compression and extension.
5. The anti-sway bar force at wheels i is the suspension force due to the auxiliary roll stiffness. It is given by:
^4i = (-1)^ Rp (dg - di)/Tp2 for i = 1,2 (2.6) = (-1)1+1 - d3)/T^^ for i = 3,4
where and are the front and rear roll stiffness. They comprise two parts; one corresponding to wheel rates and an auxiliary quantity which is due to the linkage elements of the suspension system and, if employed, a roll or an anti-sway bar, a stabilizer bar, included in many conventional designs of independent front suspension. 26
6. The anti-pitch force effective at wheel i is defined by;
'’API = where P.'s are empirical constants and F i_is the component of the tire force in the %-direction. The brake load is thought to be transmitted to the body through the suspension spring as well as through the link instant center. If the load force is fully transferred through the equivalent link instant center, the suspension is said to be 100% anti-pitch. Since the suspension instant center is a function of vehicle trim height, the anti-pitch coefficient is dependent on suspension deflection. 7. The anti-roll effect results from the fact that the lateral load transfer force can be resisted by the suspension linkages. Similarly to the anti-pitch force, the anti-roll force is given by: Pari = (^o + *1*1 + «2 ^i^) Pyui (z-s) where F . is the component of the tire force on wheel i along^tne y-axis. The total force exerted by the suspension on the sprung mass is given by: = "si + Pli + P2i + P3i + P4I + PAPi + PARi (2-9) 2.4.3 Steering Svstem The function of the suspension system was described as to allow the vehicle to accept vertical irregularities in the road. The steering system of a car allows the driver to control the direction of the vehicle with sufficient accuracy. It is proved to be reasonable to model the steering system as a lumped mechanical system with three degrees of freedom. This comprise the rotational motion of the right and left front wheels about kingpin axis and the translational motion of the connecting steering rod and associated mass elements. This translational degree of freedom is included to introduce the influence of coulomb friction and the free play in the steering system. The 27 equation of motion for the steered wheels of a vehicle yields: ^wi ( r + Sj_) + - Mggi i = 1,2 (2.10) where C . is the viscous damping, M„. the moment acting about the kingpin axis due to tire road contact forces, and Mgg^ is the torque applied by the connecting steering rod. "ssi = KsLi [(Si - w ® l i > - (^pl/2) Signes.)] for I / aiil > ^ pi/2 = 0 otherwise. where is the flexibility in the steering linkage; e_j corresponds to the free play at the front wheels due to ^ ball point, tie-rod end, and wheel bearing looseness. Y is the translational motion of the connecting road. The motion equation for this rod is given by: ^cr ^cr ^cr ^cr ~ ^ssl^^ll *^ss2/^12 where T is the steering gear box output torque and a is the length of the pitman arm. T can be expressed as^ K NgS where is the steering Box ratio and K is the flexibility in the steering rod and the steering gear box. The tire road reaction moment M„. is obtained from moment summation about the kingpin axis. It can be written in the form: ^Ti ” ^xui ^sai ” ^yui where Y . is the distance between the kingpin axis and wheel center line, measured along the wheel spin axis at front wheel i, and PT is the front wheel caster offset. 28 2.4.4 Measurement of Vehicle Parameters Since the search for vehicle and tire parameters represents an important step in the quantitative understanding of the dynamic motions of road vehicles, intensive research has been directed to achieve this task. Many machines and measuring devices were built for this purpose. Some of these are restricted to measure specific parameters while others are of general purpose. Among the well known devices is the machine built by the Calspan Corporation to collect data from the tire behavior. This machine has been intensively used for data collection and analysis by the Cornell Aeronautical Department. The University of Michigan Transportation Research Institute has also developed an apparatus to measure the characteristics of truck suspensions. However, the only existing device for measuring the suspension characteristics of light vehicles that can handle both kinematic and compliance inputs is at General Motors. Recently an apparatus called the Suspension Parameter Measurement Device (SPMD) was developed [11]. Since this current investigation is aimed to develop a vehicle math model and a simulation compatible with the SPMD objectives and capabilities a detailed description of this device is introduced in this section. The SPMD measures the movements and forces which occur at the road wheels of a vehicle as the body moves, or as longitudinal and lateral forces are applied at the tire road interface. These measurements allow the determination of many vehicle parameters which are required for the dynamic analysis. The SPMD is capable of measuring the movements of the wheel stub axles, the forces normal to the road surface, and movements, forces and moments occurring at the tire/road interface during the operation of a vehicle. These movements, forces and moments are assumed to be due either to movements of the vehicle body in bounce, pitch, and/or roll or to the forces and moments acting in the plane of the road and generated by steering, driving, and/or braking one or more wheels. The movements of the body allows the kinematic characteristics of the suspension to be measured, while the loads applied at the road surfaces allows the compliance characteristics to be measured. An overview photograph of the SPMD is shown in Figure 2.12, while the top view and right side view are shown in Figures 2.13 and 2.14. The variables measurable by the SPMD are: 1. Bounce displacement 2. Roll angle FIGURE 2.12 Photograph of the SPMD [11] to \D 30 Suspension Support Frame Left Wheel Pad Foil Axis — W///A [E l Pitch Axis Right Wheel Pad Detail of B. FIGURE 2.13 Sketch of the SPMD's Top View [111 Showing Its Coordinate Systems 31 Suspension Support Frame Boll Axis -X-X' Base Base . Structure Structure Wheel Pad Detail of A. Q FIGURE 2.14 Sketch of the SPMD's Right Side View [11] Showing Its Coordinate Systems 32 3. Lateral displacement of wheel center 4. Vertical displacement of wheel center 5. Wheel camber angle 6. Wheel caster angle 7. Wheel steer angle 8. Vertical suspension deflection 9. Lateral force acting on wheel 10. Normal force acting on wheel 11. Aligning moment acting on wheel 12. Overturning moment acting on wheel Table 2.2 shows a complete listing of the movements and forces that can be measured by the SPMD. The SPMD consists of five major subsystems: the base structure, the support frame, the wheel pads, the hydraulics, and the instrumentation. These subsystem components are described in the following paragraphs. 2.4.4.1 Base Structure This is a triangulated structure which carries the guides for the moving framework and supports the wheel pads. Its purpose is to react to the longitudinal, lateral and normal forces generated during suspension testing. 2.4.4.2 Suspension Support Frame This is a rectangular welded structure upon which the tested suspension is mounted. Three sets of bearings provide the three body degrees of freedom. There are two bearings which the mounting structure pivots about located along the axis of the main member that allow the structure to roll. Pitch motion is provided by the pitch bearing located at the rear of the frame assembly.Both the linear bearing at the rear of the frame and the plane bearing at the front allow the bounce degree of freedom. 2.4.4.3 Wheel Pads Two identical wheel pads support the right and left wheels of a suspension. These pads act as a ground plane and allow the tire to either move frictionlessly in the horizontal plane during kinematic testing or to apply forces in the horizontal plane during compliance testing. A wheel pad consists of three moving plates stacked above a base plate. The plates are free to move in either the x or y direction or to rotate about the z axis. This allows the measurement of the lateral, longitudinal, displacement of the tire as well as the steering rotation. A bearing 33 TABLE 2 . 2 Design Specifications for the SFHD L ocation Movement/Force Amplitude Body Bounce 500 mm R o ll ± 12 deg P itc h ± 10 deg T ire/R oad Longitudinal Displacement ± 100 mm Interface Lateral Displacement ± 100 mm Steering Angle ± 30 deg Longitudinal Force ± 20 kN Lateral Force ± 20 kN Normal Force + 4 0 /-2 0 kl Moment About x ' A xis ± 500 Nm Moment About y ' A xis ± 500 Nm Moment About z ' A xis ±1000 Nm Wheel D isk Longitudinal Displacement ± 100 mm To Ground Lateral Displacement ± 100 mm Normal Displacement ± 100 mm Steering Angle ± 30 deg Camber Angle ± 30 deg Caster Angle ± 10 deg Steering Angle ± 540 deg Hand Wheel Torque ± 100 Nm Suspension Displacement ± 250 mm Spring/Damper 34 assembly, between the force and steering plates, transfers forces from the tire to the force platform while allowing the steer plate to rotate. The x and y plate can either be driven by their hydraulic cylinders or by frictionless motion of the pads in the horizontal plane. 2.4.4.4 Hydraulics The SPMD components are driven by seven hydraulic cylinders. Three of them move the suspension support frame while each pad has two cylinders allowing perpendicular horizontal motions. The steering torques can be generated at the wheel pads by varying the location at which these cylinders act. 2.4.4.5 Instrumentation The SPMD's instrumentation consists of the transducers used to make measurements during tests, and the position reference system which is used for suspension set up. and transducer calibration. There are forty-one transducers on the SPMD. Sixteen of the transducers are contained in the two Mistier multi-component force platforms of the wheel pads. Each platform contains four, three-component piezo-electric load cells. The twelve other transducers are potentiometers used to measure the position of the left and right wheel centers and the spring/ damper deflection. Six transducers are rotary potentiometers. Four of these are used to measure the x and y motions of the wheel pads. The other two rotary potentiometers measure the rotation of the steering plates on the left and right wheel pads. Two transducers are inclinometers used to measure the road wheel caster angle. One inclinometer is connected to the rigid wheel structure on each side of the suspension. Five transducers are used to measure the length of each of the three hydraulic cylinders and the rotation of the steering column and the steering wheel torque. The Suspension Parameter Measurement Device is supported by a substantial amount of software developed for the SPMD's AIM-65/40 micro computer to perform the test control and data acquisition functions. This software, written in Basic and Assembly language, allows the SPMD to perform multiple tests of a suspension with differing tests types and initial conditions. It consists of 13 independent programs divided in three groups. The first is the manual control group of programs used to manually control the SPMD and to perform transducer calibration. The Data file manipulation group is used to create and edit 35 identification data files for the suspension and the calibration data file. The last group is the testing group which performs the various tests. A list of these various files is shown in Table 2.3. SPMD Measurement of the Suspension Parameters With the exception of the viscous damping force parameters, all the other suspension parameters are measurable by the SPMD. This is because the viscous damping is an inherently dynamic effect while the SPMD can only measure quasirstatic quantities. However this can be derived from force velocity measurement supplied from the shock absorber manufacturers. An installation factor is sometimes introduced [9]. All the other suspension parameters required for a simulation are listed in Table 2.3. The strategy for measuring these parameters on the SPMD is to select the type of suspension test cycle that excites the fewest of the component forces in Equation (2.9). Additional tests are then chosen so that each new test excites just one more of the Equation (2.9) forces. The SPMD test cycle that excites the fewest of the components is the bounce test, while leaving the wheels free to move laterally and longitudinally to eliminate the anti pitch and anti roll effect since F . and F . are zero under this condition. The viscous force is^aiso not present since the test is performed quasi-statically. Therefore, Equation (2.9) reduces to; ®i = ^si + ^li + ^2i The coulomb friction force depends only on the value of C j ' which can readily be determined from SPMD test data collected from the bounce test. In fact, the height of the hysteresis loop generated by plotting the total vertical force versus the vertical spring deflection d. is twice C .'. A typical plot of this hysteresis loop measured by the SPMD is shown in Figure 2.15. Once the contribution of the Coulomb friction and of the sprung weight is removed from the measured suspension force, the resulting force corresponds to the spring force component of S.. Figure 2.16 shows a case in which the compression (rebound) bump stop has been impacted. The impact point, at which the slope of the force curve diverges from its central straight line, corresponds to the 36 TABLE 2 . 3 L ist of BASIC Language Programs for the SPMD Program Name Program Function Manual Control Group UTIL. Used to manually control the SPMD. CALI. Used to perform transducer calibrations. Data F ile Manipulation Group IDFILF. Creates/edits an Identification Data file for a front suspension. IDFILR. Creates/edits an Identification Data file for a rear suspension. CAFILA. Creates/edits all data In a Calibration Data file. CAFILB. Edits only the slope and offset data In a Calibration D ata f i l e . Testing Group FORCE. Determines Information needed to balance forces between the left and right wheels of a suspension being tested. - DISKl. Performs the Disk 1 test program. DISKS. Performs the Disk 2 test program. DISK3. Performs the Disk 3 test program. DISK4. Performs the Disk 4 test program. DISKS. Performs the Disk 5 test program. STEER. Performs the Steering Wheel test program. 37 TABLE 2.4 Suspension Parameters Measured by the SPMD Symbol Definition Geometric Parameters T Lateral distance betwëen the effective spring centers of a solid axle. Z Static vertical distance between wheel center of gravity and sprung mass center of gravity. Vertical Force Parameters Suspension Coulomb friction at wheel i. K Linear Suspension stiffness. K Linear Coefficient of the suspension compression (jounce) bump stop stiffness per side. Kg Linear coefficient of the suspension extension (rebound) bump stop stiffness. Vertical spring deflection at which the compression bump stop is contacted. Vertical spring defection at which the extension bump stop is contacted. R Suspension auxiliary roll stiffness F,p.• Table entries relating the antipitch force acting in the suspension to the vertical spring deflection. Compliance Parameters Kg^m •' Lateral force - camber angle compliance. Kgm Overturning moment - camber angle compliance. Kg„ Aligning torque - steer angle compliance. Lateral Force - steer angle compliance. RIGHT o in 0.075 - 0.05 - 0.025 0.0 0.025 0.05 0. 075 BOUNCE PO SITIO N Figure 2.15 Hysteresis Loop in the Suspension Test Results [11] W 00 Q LEFT BIGHT ++ + Ü,n lU CJ ooc u . 0.65 0.69 0.73 0.77 0.81 0.85 SPRING DISPLACEMENT (m) w FIGURE 2.16 Vertical Suspension Force Versus 111] VO Vertical Spring Deflection 40 value of . This also allows the computation of K • and An additional test where the vertical force reaches its lower level is needed to determine and The next test considered in the test cycle is the roll test. The suspension deflections on the right and left sides of the suspension are equal but opposite. All the other conditions of the test remain unchanged. The only additional excited component is the auxiliary roll stiffness force (P..) which is obtained.by subtracting W Fli and F-. from the total force measured in this test. Curve fitting techniques can be used to determine R, the only suspension parameter needed to compute The next test is a bounce test where the wheels are not free to move laterally or longitudinally. A constant longitudinal force is also applied during the test. This test allows the isolation of the anti-pitch force, and the determination of the anti-pitch force parameters using curve fitting technique. The anti-roll force may be determined by the same procedure except a lateral force is exerted in this test instead of the longitudinal force. Curve fitting techniques are then used to find the parameters R^^, Rj^g and Rj^g for each wheel CHAPTER III EVALUATION OF TIRE MODELS AND FORMULATION OF SUSPENSION DERIVATIVES 3.1 Introduction This chapter includes an evaluation of analytical and empirical tire friction models using collected tire data. The evaluation of the models is based on accuracy, CPU time of computation and solvability of tire model parameters and the availability of curve fitting software for this purpose. This is followed by the formulation of suspension derivatives corresponding to 2-dimensional and 3-dimensional suspensions. The derivatives for the suspension spring deflection are also developed. 3.2 Evaluation of Tire Models The accuracy of response in the mathematical models for vehicle handling depends strongly on the type of formulations adopted for the tire friction forces. Unfortunately, the complexity of the tire structure and behavior prevents the development of a complete and reasonable theory which may govern the tire characteristics and performance. This complexity arises from the symmetrical (in-plane) and anti-symmetrical (out-of-plane) aspects of the tire with reference to their role of supporting the vertical axle load, transmitting longitudinal braking or driving forces and supplying the lateral cornering and camber forces which are necessary for the directional control of the vehicles. Further complexity results from the (quasi)-steady state behavior of tires (i.e. load carrying; braking/driving, rolling resistance, cornering, pneumatic trail and lateral shift on F ), and the vibratory state behavior (i.e. cushioning, dynamic coupling, phase shift and destabilization).The radial deflection, the tangential slip and the lateral slip, all have steady state and vibratory aspects. The complexity of the tire friction has forced researchers to idealize the tire characteristics and develop simplified mathematical models each for specific purpose and limited application. In these models relationships between the longitudinal, lateral, and normal forces as well as tire moments were developed. Available models can be classified 41 42 into three distinct categories: empirical and analytical models, and models based on the finite element method. More emphasis is put on the discussion of the empirical models given their advantage over the other models. This is because the empirical models are developed from dynamic data of rolling tires, while the other models are mainly developed from static modeling of the tire structure. A significant difference in the static and dynamic behavior of tires exists due to the various influencing factors listed above. Consequently, the use of the empirical models would result in a more realistic response in the simulation. Several empirical models have been suggested for the tire forces and moments, each with its specific advantages and defects. The main models, along with a new proposed model, were compared and evaluated based on suggested unified criteria. 3.2.1 Analytical Models Several analytical models have been developed using various approaches in formulating and solving the differential equations governing the deformation of the tire material [12]. The change in the tire orientation and position, which produces the deformation of the tire tread and carcass is shown in Figure 3.1. Theoretical tire models generally consist of an elastic structure (the carcass) provided with a large number of tread elements, both of which observe elastic and frictional deformation as the tire rotates and slides on the road. The relation between the deformation of tread base, tread surface and the direction of road surface are shown in Figure 3.2. Each point of the road surface moves along the path ABC, whereas points of the tread base follow the line AC. Points of the tread rubber move along the line AB, then slide to C. The lines AB and BC correspond to the slip and adhesive zones respectively. The deformation in these non-symmetric regions produces stresses whose integration over the contact area represents the lateral force, located at a distance t (pneumatic trail) from'the center of contact area. This offset in F produces the self aligning torque M . Many of these ^ analytical models have never been used in simulations due to the lack of data and/or the ambiguity of some of the model parameters. 43 4 POSITIVE 7 INCllKiATIO / ANGIE ALIGNING' TORQUE Y; IM I WHEEl^WAOINC . X' ^RCE POSITIVE SLIP ANGIE SPIN AXIS LATERAL FORCE NORMAL' FORCE V FIGURE 3.1 SAE Tire Axes and Terminology [12] Carcass Structure OC Tread Element Sliding Adhesion FIGURE 3.2 Deformation and Friction Force of a Tire Model •1^ 0^ 45 One simple analytical model, which produced satisfying results in vehicle simulations, is the model developed by Dugoff et. al. [13,14]. Closed form expressions for the lateral and longitudinal forces F and F , as functions of both a and s are obtained as given by equation (3.1) in Table 3.1, and equation (3.2) below respectively: Fx = -((Cx S)/(l-S)) f ( X) (3.2) The various variables and functions are defined in Table 3.1. 3.2.2 Empirical Models 3.2.2.1 Lateral Force Empirical models aro primarily based on the statistical analysis of data collected from tire testing. A variety of measuring devices have been designed for accurate measurement of forces and moments arising at tire/road interface due to tire deflection and distortion. The majority of these testings were done inside laboratories where the tire rolls on a large drum on a moving surface. This type of testing was found to have several limitations, such as the significant differences in tire behavior induced by the curvature of the drum [15]. The direct derivation of the tire friction models from experimental data using statistical approach and regression techniques generally results in non linear models which do not provide any insight into the physical nature of the tire. In 1972, Segel developed an empirical tire friction model which was later adopted in several vehicle dynamic simulations [6], [16]. The model, given by Equations (3.3), in Table 3.1 and (3.4) below, has never been verified for a variety of tire data. The constants A,., A,, and A_ are the coefficients of the quadratic curve approximating the small angle cornering stiffness given by: Cy = A. + - (A/Az) F/ (3.4) The determination of the side force is based on the small angle properties of the tire which are saturated at large angles. The cornering stiffness for small angles is taken to be the partial derivative of the lateral force with 46 Table 3.1 Tire Lateral Force Models Model Lateral Force Model Eqn. Associated Designation • No. Constants Dugoff Fy = -Cy tanaf(X)/(l-s) (3.1) Cy, X = y F2(l-s)/2{(CxS)2 + (Cy tan a) 2)1/2 f (X) = (2- X ) X if X < 1 =1 if X > 1 y = yo(l-eV(s2 + (tan a) 2 ) 1/2) segel F y = g(s.) F y ^ ^ ^ (3.3) g(s^) = s^ - (l/3)Sj^|s^I + (1/27) s^2 if |Si| < 3 = |Si I /Sf if Si > 3 ^i = (AlFz(Fz-A2 ) - V g ) o/Ag y y F , Paceijka Fy = Ay Sin (By arctan (3.5) (CyOy)) + S^ Ay, By, Cy, Oy = (1 - Ey)(a - Sjj) + (Ey/Cy) arctan (Cy ( « - g^j) P^î^noSal = =o * =1 “ + =2“l“l + (=.6) c„, c, + C, a 2 47 Table 3.2 Empirical Functions of Sine Friction Model ®y ^3 sin (kg arctan (kgF^)) (1 - k? Y ) / AyBy kg + k. "y 9 ^2 ^10^2 %11 Sv (^12 Eg Ax ^14^2 ■ ^15^2 Bx %16 Cx (^17 P,2 + ^18 ^2^) / (Ax (%19 ?%)) Ex ^20 + )"21^2 ^22^2 Am ^23Ez ■*■ ■ ^24^2 Em %25. Cm (^26 E;2 ^27^2 ) (^ ” ^28 1^1) / (Am®m ®^P (^29 ^2^^ = Em (^30 ■*■ ] = Eh %34 = Ev (%35Ez - "36^X2 ) 48 respect to slip angle as measured at zero slip angle for various tire normal loads. A more recent tire model was developed by H. B. Paceijka et al., [17]. The lateral and longitudinal forces and the alignment torque were all approximated by sine functions which are in effect an infinite series polynomial approximations. The lateral force, F , is given by Equation (3.5) in Table 3.1. ^ The various parameters are functions of the normal load,_ slip angle, side slip, and camber angle; they are given in Table 3.2. The emperical parameters k* (i=l,13) can be obtained by statistical analysis of experimental data. Polynomial models of different orders and forms were also proposed to describe the tire frictional response. A cubic fit for the lateral force as a function of the slip angle is found to be convenient to present tire test data. The cubic fit can compensate for the main trends of variation of the tire force. Consequently, the lateral force may be approximated by a third order polynomial as given by Equation (3.6) in Table 3.1. The constants C. (i = o, 1, 2, 3) are themselves functional parameters, defined by second order polynomials of the normal force F^. 2 (3.7) ^i “ ^io + AilFz + Ai2^Z The presence of the zeroth order term C is attributed to the deviation of the lateral force curve from the origin at zero slip angle. This may be due to tire construction distortion. The coefficients can be determined by two curve fitting procedures based on the least squares method. They are given by the following two matrix equations; m Za^ Za^2 Za . Za 2 Za 3 Za 4 i . i i i ^ 2 4 _ 5 (3.8) Za i Zai^ Za i Ztt£ t Za^3 Za^4 Za j, ^ Za^G n :Fzi :Fzi' 2 (3.9) Cfzi :Fzi ZFzi' : fylFzi 3 :Fzi ZFzi' 49 where m represents the number of data points considered in the computation at each of the n levels of normal force F^. There are many other empirical models which give the lateral force as functions of several variables, depending on the type of application. In general, the tire friction force is considered as a function of five of the most important variables: y , a, s, 6 , and F . The influence of speed on the lateral force F is considered to be rather unimportant on dry roads ^ Several other tire friction models and their influence on vehicle response can be followed in reference [18]. A detailed evaluation of these lateral force models is included in Section 3.2.4, where real tire data are used to investigate the accuracy of the above models. 3.2.2.2 Longitudinal Force The longitudinal tire force results from traction or braking and depends primarily on the longitudinal slip of the rotating tires. The longitudinal slip is the ratio of the difference between the tire translational speed and the free rolling speed to the translational speed v. s = 1 - (r w/v) (3.10) where: V = cos a + V^. Sin a r = rolling radius of the tire w = rotational speed of the tire The slip ratio influences the longitudinal force F through its effect on the circumferential coefficient of friction. In general, the force F is taken to be also proportional to the normal load, and therefore given by: ~ ^2 (3.11) where y is approximated by a functional relationship representing the friction coefficient slip behavior. In the HCVHP simulation, the coefficient of friction is approximated by two straight lines given by the following equation: y% = s for s < Sp = Mg s + Uq for s > s (3.12) 50 where s is the slip at which the peak coefficient of friction occurs.The slopes m, and m_ are evaluated as functions of the peak braking friction coefficient, mainly depending on the tire normal load, the sliding coefficient of friction, the skid number, and the slip angle. A typical variation of longitudinal friction coefficient is shown in Figure 3.3. The discontinuity of the derivative assumed in this formulation is not justifiable due to the smooth variation of the longitudinal force with the longitudinal slip. Given the similarity in the frictional phenomena at the longitudinal and lateral directions of the tire, a similar smooth polynomial function can be assumed: F% = A% + B^s + C^s |s| + D^sS (3.13) the parameters A , B , C and D can be obtained in a similar way as for tne case of lateral force. Another non linear model for the tire longitudinal force was also proposed by Paceijka. It is given by: = A^ Sin [Bjj tan(C% 0%)] (3.14) where + V = x the constants A , B , C and E can be in their turn curve fitted with respect to the normal load. The corresponding functions are listed in Table 3.2. To compare these two models, tire data [19] obtained at constant normal load (F = 800 lbs) were fitted to equations (3.13) and (3714). A listing of these data, collected, for the purpose of tire modeling is given in the Appendix. The observed data curve and the two model curves are shown in Figure 3.4, along with the curve corresponding to the friction cake model (discussed in Section 3.1.3) proposed for combined breaking and cornering. 3.2.2.3. Alignment Moment Several empirical formulations for the alignment moment were also proposed. The alignment moment is modeled in 51 Sp 1 S FIGURE 3.3 Longitudinal Coefficient of Friction 52 LONGITUDINAL FORCE MODELS -6 OEG SLIP ANGLE- 9 0 0 -j 800- L 0 N C I 700- T U D I N A 600- L F A I C 500 - T I 0 N F WOO- 0 R C E 300- L B S 2 00 - 00 - T - I— '— r longituokjml ’iLir f i g u r e 3.4 Longitudinal Tire Force for Various Models 53 [16] as a nonlinear function of lateral force, and vertical load. It is calculated as follows: «2 = <*3 '■z + *4 |Fy I ) Fy (3.15) The model proposed by Paceijka is also sinusiodai as the lateral force. It is given by: “ z = \ <®m Cretan (c^ o^) + Sy (3.16) Where °m = ( 1 - V < “ + ®h> + ( W "=>«<“ + S%)) The corresponding coefficients are listed in Table 3.2. A polynomial model can also be adopted for the alignment moment variation with the lateral slip. The moment M is given by: Mg = Ag + BjjS + CgS |s| + DgS^ (3.17) Sometimes, the alignment moment is taken as the product of the lateral force F and the pneumatic trail t: M = F t. The pneumatic trail^t is the displacement of the lateral force from the tire center. It depends on the slip angle and reaches its maximum at around zero, then tends to zero as the slip angle increases. When the tire is operating under combined lateral and longitudinal slip, the contact path is shifted both longitudinally and laterally, and the alignment moment becomes a function of both F and F . Therefore for combined braking and steering, the center of pressure will shift rearward relative to the wheel center causing an increase in the negative component due to F while the component due to F is positive. This explains the change in sign of the resultant moment that may occur. At driving condition, the relative force increases the restoring moment. The performance of these models with respect to the observed behavior is shown in Figure 3.5. The shown curves correspond to three levels of normal force (F = 500, 1000, 1500). The superiority of the sine and ^ polynomial models is remarkable, whereas the model given by Equation (3.15) and that computed using a fitted pneumatic trail resulted in noticeable deviation from the observed data. 54 ALIGNMENT TORQUE VS SLIP ANGLE MODELS 200-1 A L I 0 N M E N T T 0 R 0 U E L B S F T -100 -20 -10 0 10 20 SLIP flNCLE-DEG .:OBSERVED OATA q ;POLYNOMIAL MODEL X-.PACEIJKA MOOEL +:SE1GEL MODEL FIGURE 3.5 Alignment Torque Versus Slip Angle for Various Models 55 ALIGNMENT TORQUE VS SLIP ANGLE MODELS 200 H A L C 100 N M E N T T 0 H 0 U E XQ L B S F T -100 -20 -10 0 10 20 SLIP IIMGLE-DEG .:0BSERVE0 DATA otPOLYNOHlAL MODEL X:TOnOUE COMPUTED USING FITTED TRAIL FIGURE 3.5 (cont') 56 ALIGNMENT TORQUE VS SLIP ANGLE MODELS 200 - A L I G N M E N T T 0 R 0 U E L B S F T 'N.. -100 ■ -20 >v -10 0 10 20 SLIP flNCLE-DEG ,:OBSERVED DATA □ •.POL'tNOMlAL MODEL X : PACEIJKA MODEL +:SE1GEL MODEL FIGURE 3.5 (cont') 57 ALIGNMENT TORQUE VS SLIP ANGLE MODELS 200 4 A L G 100 N M E N T T 0 R 0 U E L 8 S F T 100 • -20 10 0 10 20 SLIP ANGLE-OEG ,;0BSERVED DATA q :POLTNOMIAL MODEL x:.PACElJKA MODEL +:SE1GEL MODEL FIGURE 3.5 (cont') 58. 3.2.3 Combined Braking and Steering Under practical conditions, the vehicle tires observe simultaneous cornering and braking effects. Under such conditions, the tire force models have to be modified to include this effect. Several methods have been proposed in this prospect; Friction Ellipse Concept The assumption here is that under constant normal loading, the vectorial sum of the lateral and longitudinal friction force vary along the contour of an ellipse whose vertices correspond to the maximum values of F and F . This gives the following equation for the lateral force^under combined braking and cornering: Fy = fyo where F is the maximum longitudinal force obtainable under acruSl normal loading, and F is the lateral force at zero longitudinal slip and actual normal load and slip angle. Shaping Function A non dimensional shaping function was proposed as a modified friction ellipse concept to account for the inter dependence between the lateral and longitudinal force [16]. This function is given in tabulated form as shown below: Side Force Shaping Function Slip 0 .05 .1 .15 .21 .30 .40 .60 .8 1.0 g(s) 1.0 .99 .97 .93 .86 .72 .56 .34 .25 0.18 Linear interpolation can be used to obtain intermediate function values. When the longitudinal slip is s, the lateral force is given by: Fy = F y g(s) F y ( a , s) = F y (a , 0) g (s) (3.19) where g(s) is the value of the shaping function at slip s. 59 Friction Cctjfe Concept The friction cake concept, introduced by R. Weber, [20], is based on the assumption that the tire forces act the same way as the corresponding slip components; Fy / a = Pys = R/k (3.20) where R is the resultant force (vectorial sum of F and F ) in the contact surface, and k is the vectorial addition or a and s. We have: R^ = Fy^ + Fjj^ ; k^ = + s“ (3.21) It is also assumed that the curve R(k) can be rotated to form a surface of revolution over the basic circle with the radius k = 1. The procedure of determining the tire forces is as follows: given the longitudinal and lateral slips (s and a ) , and the basic curve of R^versus^.the instantaneous resultant force is R(k), k = (a + s ) ' . The force R is then resolved into its components F^ and Fy: F% = R Sin (3) ; Fy = R Cos (3) (3.22) where = arctan (a /s) To compare these methods of introducing the effect of combined braking and cornering tire data are examined using the proposed methods. The resultant force R, measured at constant values of slip angle is shown in Figure 3.6. .Based _on the friction cake concept these curves should be identical. Therefore tire data indicates that this concept is not totally correct and that the assumption that the tire forces acting the same way as the corresponding slip components is an idealization of the actual behavior of tires. However, since the deviation between these curves is not very large, this method may be useful for certain applications. In fact,.this method is proved to be superior over the friction ellipse method adopted in many applications. It is also comparable to the shaping function method as shown in Figure 3.7. The effect of combined breaking and cornering on the lateral force is also shown in Figure 3.8. 60 RESULTANT FRICTION FORCE -FRICTION CAKE MOOEL- 1000 900 -K 600 WOO 300 B 200 100 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1 . 1 RESULTAMl SLIP *:R AT 0 DEC SLIP ANGLE d :R AT 8 DEC SLIP ANGLE x-.R AT 12 DEC SLIP ANGLE +:R AT 16 DEC SLIP ANGLE FIGURE 3.6 Resultant Friction Force for Combined Braking and Cornering 61 COMBINED CORNERING AND BRAKING 900 800 L 700 A T E A A 600 F A C 500 T I 0 ^ WOO F 0 A C E 300 L B S 200 100 100 200 300 GOO l’oÜ COO 900 LONGnuoINAL FRICTION FORCt-lSS , -.OBSERVED ORTA □ ;FfllC7I0N CAKE MODEL X ;FRICT10N ELLIPSE MODEL + : MODEL WITH SHAPING FUNCTION FIGURE 3.7 Tire Forces for Combined Cornering and Braking 62 COMBINED BREAKING AND CORNERING 900 H 3'»-, 800 - 700- Î 600- q SCO- t UQO- 300 - 200 - 100 ■ 0 ICO 200 300 >100 GOO 700 800 900 TIRE LONGtTOOIXAL FORCE «;OBSERVEO DATA ozSHAPING FUNCTION x:FflJCT10N ELLIPSE FIGURE 3.7 (conf ) 63 LATERAL FORCE VS LONGITUDINAL SLIP 8 DEC LRTERRL SLIP ANCLE 900 H 800- 700 600- O.WOO- 300- 2 0 0 - *0 ., 'Cp ... 100 - 0.0 O.I 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 LONCITUDIMIIL SLIP , -.OBSERVED DATA o :FR1CT10N CAKE MODEL :SHAP1NC FUNCTION MODEL FIGURE 3,8 Lateral Force Versus Longitudinal Slip 64 3.2.4 Computation Methods and Model Evaluation One of the advantages of the polynomial models, besides the fact that they preserve the shape of the friction function, is that the model constants are simple to obtain from the experimental data by many available methods such as the least squares method. The non linear sinusoidal model with an arctangent function in its argument is so complex that no numerical methods can be used to find the model constants by using the crude tire data alone. However, regression analysis can be used to successfully approximate the constants if a close approximation of the constants is initially supplied. This last requirement is vital for the convergence of the regression method, due to the high non linearity of the model and its consequent high sensitivity to the variations of the model constants. A fair initial prediction of the constant values requires a knowledge of the physical significance of the model constants. The tire friction model is given by: Fy “ Ay Sin (By arctan (CyOy)) + S^ where 0y = (1 - Ey) ( a + S^) + Ey arctan (Cy ( a + S^)) where S^ and S are the horizontal and the vertical shifts due to the presence of camber rotation. All A , B , C , E , S^, S are empirical constants that have to be^ F determined. The constant A^ represents the peak value of the side force, while the product A ^ B „ C represents the slip stiffness at zero slip. The constant E influences the curvature of the curve. To solve for these constants, a non linear regression program in the SAS (Statistical Analysis System) Library was used. Initial approximations were supplied to "NLXN” program which used Marquardt's method to successively approximate the model constants. Marquardt's method is a method which combines Newton's method with the method of Steepest (Gradient) Descent. The CPU time required for the computation of the model constants (convergence) of the program are shown below: Polynomial Model Sine Model CPU 1:60 9:73 Time The computation times of the sinusoidal model is much larger due to its high non linearity as compared to the polynomial model. The investigated data are collected in the Firestone Tire and Rubber Company (Akron, Ohio) where tire testing facilities are available. The tire is of the P195/70R14 size. The inflation pressure was 26 psi, and 65 the test consists of a slow sine sweep with respect to slip angle between + 15 degrees at nominal radial loads which are approximately 500, 1000, and 1500 pounds. Data sets are obtained for constant camber angles. Typical values for the empirical constants, obtained for tire test at zero camber angle are shown in Table 3.3. Fitting these constants as functions of the normal load F the constant coefficients can be determined. For example, constants A, and C, are found to vary with F by the following relationships: ® -2073.4 + 8.044 F, - 0.0029 F,^ z z Cy = 300 sin (1.83 arctan (0.013 F^)) A second criteria that has to be considered in the evaluation of the models,is the actual computation time required to compute the side force given the normal load and the orientation of the wheel. For few data points, the CPU time required by the 4 discussed models is shown in the table below: Model Paceijka Segel Dugoff Polynomial CPU time 00:30 00:26 00:30 00:27 This shows that Segel and the polynomial models are more efficient as far as the computation time is concerned. The third criterion in comparing the proposed models is the convergence to the actual data. The experimental data, along with the curves obtained by the empirical models and Dugoff analytical model, were superimposed as shown in Figure 3.9. This permits a direct comparison of the various models. The following remarks can be deduced. 1. The sinusoidal models allows the best fit of the side force Fy, especially at higher normal loads. 2. The peak value of the tire lateral force was not reached by the third order polynomial fit. This would underestimate the frictional response of the tire in the simulations, which in turn may overlook some of the instability problems in the simulation. The polynomial fit can be ameliorated by considering a fourth order equation of the form. Fy = Cg + c^S + Cg S IS I + C 3 + C 4 1 S I 66 Table 3.3 Empirical Constants for Tire Friction Models Ay Cy :y 462 -1009.72 .36 0.31 -1.51 0 0 995 -2946 .22 0.33 -0.5 0 0 1065 -3348 .27 0.25 -1.23 0 0 Fz Cp Cl C2 C3 Polynomialal 462 15.6 -138.12 12.01 -0.38 995 22.8 -248.67 22.88 -0.67 1455 28.8 -256.3 17.5 -0.48 67 SIDE FORCE vs SLIP ANGLE MODELS 2000 4 1000 -1000 -2000 T*T- I I I I ' I "T—T- I I I I I I I -20 -10 10 20 SLIP HNGLE-ULG m :O 0SERVED OflTfl o:POLYNOMIAL MODEL X:PACEIJKA MODEL +:SEIGEL MODEL *:DUGGOFF MODEL FIGURE 3.9 Tie Lateral Force for Various Models 68 SIDE FORCE VS SLIP ANGLE MODELS 1000 T 1 m E s I 0 E F 0 R C E L 8 S -1000 - 2000 - -20 -10 0 10 20 SLIP fiNCLE-UEO ,:OBSERVEO DATA o:POLYNOMIAL MODEL x:PRCElJKA model +:SEIGEL MODEL *;DUGG0FF MODEL FIGURE 3.9 (cont') 69 SIDE FORCE VS SLIP ANGLE MODELS 2000 1000 T I R E 3 I 0 E F 0 R C E 1000 4 -2000 ■ Ill'll " I ' ■ ' “ I—I—I— I—I—I—I— I—I—I—I—I—I—I— I—I—I—I— I—I— I— I—I—r I I -20 ■10 0 10 20 SLIP RNCLE-OEG OBSERVED DATA m:POLTNOMIAL HODEL viPRCEIJKA MODEL * iSETOEL MODEL «tOUGGOFF MODEL FIGURE 3.9 (cont') 70 SIDE FORCE VS SLIP ANGLE MODELS 2000 1000 T 1 R E S I D E F 0 R C E -1030 4 -2000 ■ "T"r I f T-1- r i- i—r-1 -p “T—r-t-t -r -r-1-r -20 -10 0 10 20 SLIP MNGLE-OCÜ ,:OBSERVED DATA o:POLTNOMIAL MODEL x:PACEIJKA MODEL * :SEICEL MODEL . :DUGGOFF MODEL f i g u r e 3.9 (conf ) 71 3. All the models fit reasonably well, with no significant difference, at small values of the slip angle. As the curves approach the peak area, they observe different deviations from the actual data. A model which does not reach the peak result in a smoother response, and therefore it results, in higher stability prediction. 4. The four discussed models, can be classified in order of convergence to actual data, as follows: (i) Sine Model Cii) Polynomial Model (iii) Segel Model ^v) Dugoff Model As the various criteria suggest inconsistent conclusions, a trade off has to be made in the selection of the appropriate model. This trade off is basically between the need of a lower CPU time or a better fitting model. . In conclusion, the choice of a friction model should be based of specific criteria. The evaluation of these models may be based on one of the following three criteria, which can be of significant importance in complex vehicle dynamics simulation: (i) Accuracy of the model (ii) Solvability of model constants and software availability (iii) Evaluation time (or CPU time) required by the model. Table 3.4 shows the classification of the friction models with respect to the above criteria. The model with rank one has the advantage over the other models on the considered criterion. Obviously, no model has the. absolute advantage over the others. However, since accuracy is generally given more consideration in the evaluation of tire friction models, Paceijka sinusoidal model is preferable. 3.3 Suspension Derivatives for Kinematic Suspension Instead of the concept of sprung and unsprung masses coupled together inertially, the movements of the tire and suspension are considered to be geared to the vehicle body kinematically [21,10]. This reduces the dynamics of the vehicle system and reduces the numerical problems in the 72 Table 3.4 Classification of Tire Friction Models Model Accuracy Parameter CPU Determination Computation Time Paceijka 1 4 3 Polynomial 2 3 2 Segel 4 2 1 Analytic: Dugoff 3 l 3 73 model simulation. It is reasonable to assume that if the wheel remains in contact with the ground, the movement of the wheels may be described in terms of displacements of vehicle body, since the suspension systems connecting the wheels to the body constitute of rigid links. Kinematic analysis of the body-wheel movements yields the functional relationships relating movements of the tire-ground contact patch to the vehicle body movements. These functional relations are called the suspension derivatives and describe the ratio of the rate of change of wheel movement to the vehicle body movement. The suspension derivatives are developed first for a Long/Short Arm suspension. Then, a vehicle model based on the concept of suspension derivatives will be developed in Chapter 5. The suspension derivatives are obtained with respect to the secondary modes of the vehicle ( bounce, pitch, roll and yaw motions). The derivatives for the bounce, roll and pitch modes were discussed in [10]. The derivatives for the suspension spring deflection are also derived for 2-dimensional and 3-dimensional suspension. The movement of wheels is controlled by the mechanical guidance system which is usually a three-dimensional linkage. However, it is also reasonable to reduce the linkage mechanism to a two dimensional system since the significant wheel center movement occurs in the YZ plane. Only independent suspensions, in which the movement of one wheel on an axle does not affect the opposing wheel, are considered in the analysis. The new axes systems are shown in Figure 3.10. Typical long/short arm suspension is shown in Figure 3.11. SLA suspensions are controlled by independent linkages allowing enough freedom to the suspension mechanism. For the determination of suspension derivatives, only two cases of suspension are analyzed: a two dimensional suspension and a three dimensional SLA suspensions. 3.3.1 Suspension Derivatives of a 2-D Suspension The performance of suspensions is defined by the main five variables: normal force of the tire road interface plane, fore and aft movement of the wheel, lateral wheel movement, steer rotation, and camber rotation. These suspension local movements are coupled to the vehicle body movements through the linking mechanisms attaching the suspension to the rigid body of the vehicle. If the links are assumed rigid, and the wheels are replaced by triangular linkages. 74 l|J Figure 3.10 Axes Systems for the Vehicle with Kinematic Suspension. 75 Ç Car 9.1 ,3.13 L 7.5 18.1» FIGURE 3.11 Short/Long Arm Suspension [20] 76 the wheel displacement resulting from the body motion can be determined analytically by using kinematic techniques. The function of relationships which relate the suspension local movements to the body movements are the suspension derivatives. They are called derivatives since they describe the quasi-static rate of change of a suspension displacement or rotation with respect to the basic body movements. This formulation allows the elimination of the excess degrees of freedom associated with the unsprung masses, and their replacement by movements proportional to the fundamental degrees of freedom associated with the vehicle body. The suspension derivatives can be determined either by the instantaneous center of rotation method, or by using a vectorial approach. The 2-D suspension-body assembly is presented by the mechanism shown in Figure 3.12. If the instantaneous center of rotation method is applied, the instantaneous centers of rotation are colinear when the three bodies move in a plane. If only bounce motion is excited and z is the velocity of 0?=, the rate of change in camber is (z3^ 9 z) = z/(0356317, and therefore: (a* i/ 9 Z) = 1 / The suspension derivative for the tire scrub with respect to bounce is: ( 3Yi/ 3z) » 033^ O35/ O35 03]^ For the case of roll rotation the center 0,- is at the origin of body axes. Based on Figure 4,6(07, the lateral and camber velocities of the suspension become: Pi =ii = P(5ir°35/°35 “31> Yi - P (0l5035)(°13°36)/(°35°3l) Hence the suspension derivatives with respect to roll became: 9(})j^/9(}) = °15°3s/°35°31 “ (°15°35)(°13°36)/(°35°3l) Alternatively, a vectorial approach may be used to develop the suspension derivatives. This analytic method is more suitable for numerical computation, since it makes use of 77 00 f ®16 23J O f .15 43 45 (a) The Instantaneous Centers of Rotation For A Verticle Velocity of the Body Origin. CO f ®16 I ©35 -I (b) The instantaneous Centers of Rotation For Roll Rotation of Body Origin. FIGURE 3.12 Instantaneous Centers of Rotation [7] 78 the actual length of the suspension links. When a vertical velocity and a roll rotation are impressed on the body, the suspension react with lateral and rotational motion. The nodes A and D are fixed to the body while B and C are fixed to the suspension unit. The tire-road contact point 0* is assumed temporarily fixed in the ground. Since the point A of link (AB) is attached to the car body, its velocity is: Va = + (z + y^p)k (3.23) The velocity of B, with respect to the body axis origin is: Vg = (ÿi - Zg' p^) j + (y'gPi) k (3.24) The coordinates with the prime superscripts are measured with respect to an axis system, similar to the body axis and fixed at O' (e.g.: z'_ = z_ - z_, , y'g = y« -, y., , Yo' = Yi) The cosine derivatives associated with the link (AB) are: Cl = (yg - and = (Zg - where: Li = [(yg - y»)^ + (zg - and a unit vector along (AB) is âb = (C^ j + s^ k). Since the link AB is assumed rigid, no relative motion exists from A to B. Therefore, the projection of the link end point velocities in the direction of the link must be equal in magnitude and opposite in signs. Hence: Vg.ab = (3.25) Substituting the velocities above in this equation the following vectorial equation is obtained: C(Yi - (Zg - Zq , ) Pi) j + (yg - yQ,) Pi k] . [Cl j + k] = [(-z^p) j + (z + y^p) k] [Ci j + Si k] (3.26) A similar equation can be derived for the link (CD): [(Yi - (Zg - ZQ,)Pi) j + (Yc “ Y q i ) Pi k] . [Cg j + Sg k] = C(-ZgP) j + (z + YgP) k] [Cg j + Sg k] (3.27) 79 where =2 - (Vd - yc)/I-2'' ®2 ’ <*D - :cl/&2 and Lj > [(ÏC - yo>^ + ('d - »c)^] by equating the corresponding components in the above equations, the following matrix equation is obtained: (Yb -Yo ')Si -(Zb - Zo')Ci C]L Pi" ^A^i - Safi Si p (Yc -YgJSg -(Zc - Zo*>^2 C. h ^0^2 - Sg i (3.28) The suspension derivatives can now be defined as: 9 Yj_/ 9 Z = A ; 3 y^/ 3 t ( Y b “ Ygil^i - (Zg - Z q i ) C ^ ] A = det .[(yc - yo<)S2 - <‘c - =2. or: (ÏB- yo') ®1 =2 - • =0') =1 =2 - (yc - yo'> ®2 (^c " Cj (Y b “ Y q i ) ( Z f l ” Z q i ) A^ = det (yc - yo<) h - <^o ' »o') =2 (yg yo') ®i “ (*B ” ^0 ') y& ®i - zafi Ag = det (yc - yo'> Sj - (2c - Zo')=2 )'d ®2 ■ "^2 80 Ag = det Sl^2 - SgCi “a =1 det “d ®2 It may be noted that the suspension derivatives, determined by this method, are also instantaneous, since the node positions change continuously as the vehicle motion progresses. However, some of the algebraic lengths are bounded by maximum limits. For instance: BO' BA ; and - 2'C' \Yb - Ya I < DC i^D - yc The suspension spring and damper are assumed to be fixed to the suspension at a point E along the link (CD) and to the body at a point F (Figure 4.6). The derivatives associated with the bounce mode are developed first. If the angular rotation of the link (CD) is a, the velocity of the node C may be written as: V q = Vp + a DC = Vp + a (-(Zg - Zjj) j + (yg - y^) k) (3.29) or 81 Vg = (Ÿi - (Zg - Zg) Pj^) j + (yg - Yg,) ?! k (3.30) Now, since: Ÿi = O y ^ / 3z) W ; p^ = (o^^/ 3z) W , and Vg = W k equating components yields : j component: à(Zg - Zg) = [By^/3 z - (Zg - Zg,) (34^/3 z) ] w Hence aa/ gz = 3a/ W =[(Zg -Zg,) (3(j)^/3 z) -OY j^/ 3z)]/(Zg - Zg) (3.31) k component: 3 a/ 3 z = C(Yq - Yqi) (S^i/Bz) - l]/(Yg - Yg) (3.32) The velocity of the spring contact point for the bounce mode is: Vg = Vg + (3a/ 3z)C-(Zj. - Zg) j + (Yg - Yg) k ] W (3.33) The velocity of the spring contact point with the vehicle rigid body is: Vp = W k Therefore, the rate of extension or compression of the spring is the projection of the velocity vector Vp and Vp on the line of action of the spring: ^ef = (^D - V • (Cgj + 83k) (3.34) where c_ and S_ are thé cosine derivatives of the link (EF) : ^ C3 = (Ye - yp)/Lef ; S3 = (Zg - Zp)/L^f ^ef ~ “ y?) simplifying equation (3.34): 3Lgf / 3z = [(Yg - Yg) S3 - (Zg - Zg) C3](3a/3z) (3.35) 82 The spring suspension derivative with respect to the roll mode can be obtained in a similar way. The velocities of node C computed in both axes systems are: V q = [9yi/3<|) - (Zg - Zq ,) (3(1) 3(j))] p j + [(Yc - yo')(3ii/34)P]k (3.36) Vg = [-ZgP - a (Zg - Zp)] j + [y^ p + a (yg - y^)] k (3.37) Hence, by equating both components the derivative ( 3a/ 3()) ) becomes: j component: 3a/3 (Zg-Zj^) (3.38) k component: 3a/3* =[(yg -yQi) (S't'i/9<1>) -yjj] / (y^ - y^) (3.39) It should be clear that these two expressions are equivalent and reduce to the same expression if the derivatives (3^,/3^) and (3yj/3<^) are replaced with their basic expressions. However, since the link CD is near the horizontal or may pass through that condition, the second equation should be used for a division by zero to be avoided. Using the same procedure as for the bounce mode, the suspension derivative associated with the spring and damper deformation becomes: 3Lgf/3(|>= [(Zp - Zjj) + (Zj, - Zg)( 3a/3(1) )] C3 + [(y^ - y^) + (Yg “ Yd) ( 3 a / 3,}, ) ] S3 (3.40) Once these suspension derivatives are determined, the position and orientation of the suspension nodes due to body motion can be computed using the following discrete equations : Ayj^ = -z^ A(j) A z ^ = A z + y^A(}> Ay^ = -Zj3 A(t Azp = A z + y^A^ 83 For the wheel-ground contact point; Û y ’q = (3y^/az)Az+ (ay^/a*!») A(|» For the outboard nodes: A y g = (3yj_/az) A z + (3yj^/a + (B^i/az) Az] A Z g = ( Y g - yg.) [ ( 3't') A(|)+ (3(|| ^/3z) A Z] A Yc = (3Yi/az) A z + (syj/at) - (z^ - Zg,) [(a(|)^/a + ( 3(f)^/3z) A z] A Z g = (Yg - Yg,) [( 3((,j^/a(|) )A<|) + (a^i/Bz) A z] For the suspension spring attachment points: AYp = -Zp A<{) ; A z p = A z + Zp A(|) A Y g = AYg - (Zg - Zg) [ (3a/3z) As + (3 a/3;}) ) Af. ] ) A Z g = AZg + (Yg - Yg) [(aa/3Z)A z + (3a/3(j) )A* ]) It is assumed in the computation that the linkage position and orientation remain fixed during each period of computation time. An appropriate time increment in the simulation has to be selected. Furthermore, the suspension derivatives have to be updated in function of the new position and orientation of the suspension nodes, at each time increment. 3.3.2 3-Dimensional Suspension Derivatives The suspension mechanism may be modeled as a three dimensional suspension where the movement of the suspension is influenced by the bounce, roll , pitch and yaw motions of the vehicle body. This allows obtaining a better and 84 more realistic coupling between the motion of the body and the suspension. An SLA 3-Dim suspension is shown in Figure 3.13. The velocity of the inboard points of the suspension fixed to the vehicle body are: = (qzj^-ry^) i + {-pz^+rx^) j + (z + py^ - g x^)k (3.41) Vg = (qzg-ryg) i + (-pZg+rXg) j + (z + py^ - q Xg)k (3.42) Vg = (qZg-ryg) i + (-pz^+rXg) j + (z + py^ - q Xg)k (3.43) The velocity of the points of the suspension fixed to the tire frame mechanism and computed with respect to the fixed point of contact between the tire and the road are: Ve = [%i - Si (yg- Yo,)] i + [ÿi + Si(Xg-XQ.) - Pi (2g - ZQ,)]j + [Pi (yg - yg,)] k (3.44) where y^, = y^ and X q , = x^ Vp = [Xi - Si (yp - yg,)] i + CYi + Si (Xp - Xg.) - Pi (Zp - Zg,)] j + [Pi (Yp - Yg,)] k (3.45) V g = [%i - Si (Yg - Yg,)] i + CYi + Si (Xg - Xg.) + Pi (Zg - z'g)] j + [Pi (yg - yg,)] k (3.46) Vp = [%i - Si (yg - yg,)] 1 + [Yi + Si (Xg - Xg,) + Pi (Zq “ Zg,)] j + [Pi ^yg — Yg,)] k (3.47) since the links are assumed rigid, the relative velocity in the direction of the links are all zeros. Therefore, the projection of the velocities of the extremum points of each link in the direction of the link must be equal. These directions are expressed as unit vectors as follows: Link (AE) AE = 1^ i + m^ j + n^ k where 1^ = (Xg - x^ )/ L^ ; m^ = (yg - y^ ) /L^ ; "l = (2e - 2^)/Li and Li = [(Xg - X^)2 + (yg . y^)2 + (z^ _ Z ^ ) 2 ] V 2 Link (CF) CF = Ig i + mg j + ng k 85 FIGURE 3.13 SLA 3-D Suspension 86 = ( Xj. - Xg) /Lg ; mg = (Yp - Yg) /Lg ; "2 “ (Zp - Zq) /Lg Lg = ((%F - %c)^ + (Yp - Yc)^ + (Zp - Link (DG) DG = I3 i + m^ j + k I3 - (Xg - Xp)/ L3 ; m 3 = (Yg - Yq) /L3 ; "3 “ (Zg - Zp)/ L3 Lj . [(Xg - xg)2 + (yg - yg): + (Zg - Zg):]l/: Link (BE) BE = 1^ i + j + k I4 “ (^E - ^B^/ ^4 '■ “ 4 “ (Yp - Yg)/ L4 "4 = (Zp - Zg)/L4 L4 = [(Xp - Xp)2 + (Yp - Yg)^ + (Zp - Zp)2] The equalitY of the projection components in the direction of link (AE) produces (Za l^) q + (-“iZ^) P + ni(z) + (n^Y^) P " (q x^) n^ (3.48) SimilarlY/ for the points fixed to the suspension frame: [ %i - Si (Yp - Y q .)] li + CYi + ®i (^F - ^e ) - Pi (Zg - Z q ,)] + n^ [Pi(Yp - Yq)) (3.49) similar equations can be obtained for the other links, and bY combining them, the following matrix equation is formed: Xi w Yi P [S] = CQ] (3.50) ®i q Pi r where the matrices [S] and [Q] are given bY: 87 where the determinants are given by; A = det ®ij 3 ? det i = 1,2,3,4 j = 1,2,3 ^X2 “ "i : ®ij+l ] det i “ 1,2,3,4 j = 1,2,3 Ax i “ ^i : ®ij+l ] i — 1,2,3,4 j = 1,2,3 A x G = det hi : *ij+l 3 i — 1,2,3,4 j = 1,2,3 Ax'P = det ki : *ij+l 3 det : n^ : i = 1,2,3,4 j = 1,2 Ayz = ^il *ij+2 3 : ti : i = 1,2,3,4 Ay<|> “ det ®il ®ij+2 3 j = 1,2 det i = 1,2,3,4 AyG = ®il : hi : ^ij+2 3 j = 1,2 d e t : k i : i = 1,2,3,4 Ay'i' “ ^ i l *ij+2 3 j = 1,2 det : Ui : Si4 3 i = 1,2,3,4 i = 1,2 ^SZ = ®ij Agij) = det i = 1,2,3,4 j = 1,2 ^ij : : ®i4 ] AgG = det : hi : i = 1,2,3,4 j = 1,2 ®ij ®i4 ] det : ki : i = 1,2,3,4 j = 1,2 às'P = ®ij H a ] det : Hi ] i — 1,2,3,4 j = 1,2,3 A^Z = ®ij A# = det ®ij : ] i = 1,2,3,4 j = 1,2,3 A(^8 = det i ~ 1,2,3,4 j = 1,2,3 ^ij : hi ] d e t i A^i|i = ^ij : h i 3 = 1,2,3,4 j = 1,2,3 The instantaneous position and orientation of the suspension nodes due to body motion are computed in function of the suspension derivatives. For the suspension nodes fixed to the vehicle rigid body, these are given by: &Xy. = 46 à A({) 4(^ - Xjç 48 = -^k Where k corresponds to the node in question (i.e. k = A,B,C,D). The instantaneous coordinates of the outboard nodes of the suspension are: AXj = ( 3 Xj^/ 3 z) A z + ( 3x^/3<|> ) A(|) + (3 x^/SG ) AG + ( 3 3t|, ) All; - ( Yj - Yo,) A (3.53) 88 lQi (yE-yo)ii +(XE-*o)mi -(ZE-Zo)mi + (yE"V"l m. ■(yp“yQ)i2 "*'(Xp'”3CQ)in2 -(Zp-Zo)iii2 + (yp-yolng [S] = -(ZQ-Zo)ni3 + (yg-yolng ‘(y£~yo)^4 '*‘tXg“XQ)m^ -(Zp-Zo)ni4 + (yg-ygln* (3.51)'“ and mi -=A*i+yA*i *A^l“^z"l -yA^i+^A^i [Q] ”2 -2c®2+yc®2 Zcl2-Xc»2 -Ycl2+%C*2 m3 -=D*3+yD*3 =Dl3'" * D * 3 ~yD^3‘’'^D®3 ”4 -2B®4+yB®4 =3^4'" * B * 4 "yB^4‘*'^B®4 (3.52) The following notation is used for simplification: [S] = [a^j] and [Q] = [ n, t, h, k. ] ; i = 1,2,3,4 ^ ^ ^ ^ j = 1,2,3,4 where the matrix elements equals to their corresponding element in the matrices above. The suspension derivatives may now be defined as follows: 3 x^/3z = ^ ' 3x^/3 3yi/3z “ Ayg/A; 3Yi/3(j. = A y 4 / A . 3 y i / 3 8 = Ay/0|A; 3y^/ 3i|f* Ayij;/ A 3s./3z = A g / A ; 3S./34> = A . 3 3 1 /39 = L^/ 3Si/9 # A 3 Page(s) missing in number only; text follows. Filmed as received. UMI 90 Vp is also given by: V, * [Xj^ - s^ (yp - yg,)] i + [y^ - (Zp - Zq .) + (Xp - Xq.) Sj^] j + C(yp - Y q ) p^] k (3.61) The velocity is given by: Vg = (qzg - y ^ r ) i + (-pZg + x^r) j + (W + py^ - qXg) k by equating components of both expressions of Vp, the following matrix equation is obtained. (Zp-Zg) -(yp-yg) -(Zp-=c) 0 (Xp-Xg) (3.62) (yp-Yc) -(Xp-Xg) 0 where: (3X/3Z) - (3 s/ 3z) (yp-yg,) (3Ÿ/3Z) - (3 s/ 3Z) (Xp-Xg,) - ( 3p/3 z)(Zp-Zg,) W ( 3 P/ 3 z) (yp - yg,) - 1 ( 3 X/ 3 p) - ( 3 s/ 3 p) (yp-yg, ) ( 3 y/ 3 P)-( 3 s/ 3 P) (Xp-Xg,)-( 3p/ 9p) (Zp-Zg,) ( 3 P/ 3 P) (Yp-Yoi) - Yc ( 3 X/ aq)-( 3 s/ 3 q) (Yp-Ygi) ( 3 y/ 3 q)-( 3 s/ 3 q) (Xp-Xg,)-( 3p/ 3q) (Zp-Zg,) ( 3 P/ 3 q)(Yp-Yot) + Xg ( 3X/ 3 r) - ( 3 s/ 3 r) (Yp-Yg, ) + Yg ( 3ÿ/ 3 r)-( 3 s/ 3 r)(Xp-Xg,)-( 3p/ 3r)(Zp-Zg,)-Xg ( 3 P/ 3 r) (Yp-Yn, ) . (3.63) 91 Since the above matrix is singular; one of the angular velocity components must be linearly dependent on the other two components. For the problem to be well posed, the following condition is required: (Xp-Xg) + Fg (yp - yg) + Fg(Zp-Zg) = 0 (3.64) By including this linear dependence, the problem is reduced to a second order problem, where the angular velocities and their corresponding derivatives can be easily determined by Cramer's rule. Now, the rate of change of the length of the suspension spring is found as follows: Vg = Vg + [à^i + àyj + âjjk] X [(Xjj-Xg)i + (y^-yg) j + (Zg-Zg) k] = Vg + [ày(zg-zg) - â2(yg-yg)]i + [-âx(Zg-Zg) + (Xg-Xg)]j + [-ây (Xg-Xg) + (yg-yg)]k (3.65) The rate of extension or compression of the spring is the projection of the velocity vector V„ and V_ on the line of action of the spring: ^ V = [ Igi + ”sj + Os*] , Vg = Wk Where 1 , m and n are the cosine derivatives of the line of action of the spring: Ig = («H - Xg) /Lg ; Bg = (yg - Yg) /Lg ; "s “ Lg = c (Xg-Xg)^ + (Xg-yg)^ + (zg-zg:= Now the suspension derivatives corresponding to the various secondary modes of the vehicle motion can be obtained by isolating the mode in question and setting all the others to zero. For instance, for the bounce mode: I'hq = {[( aay/sz) (Zg-Zg) - Oa^/az) (Yg-yg)] 1^ + C-(3ajj/3Z) (Zg-Zg) + (3ag/3z) (Xg-Xg) ] m^ + [ (3 a y 32) (yg-Yg) - (3ay/3z) (Yg-Yg) ] n^) W 92 the corresponding suspension derivative becomes Olhq/^z) = 1‘hq / W = C(yH-yc)ns - (ZH-Zc) ®s^ O V^ z ) + C(Zjj-2ç) Ig - (Xjj - Xç) ng] Oay/az) ®S "* ” y^) ^g] (3&g/3Z) (3.66) For the roll mode: Vc = -pzgj + py^k Vg = -pZqj + p y q k Ihq * (Isi + ” s^ * " s ’') - ([-(Zc-Zq) + (-®x,p(ZH-Zo) + =Z,P (='h - ’‘c > ) l“s + [ây,p(ZH-Zc) - S,p t-Zy,p(>'H-='c> + ®x,p ( % - ycl) "s ) P then ^ ^^hq/ ^ 4 ) = ["(Zc_2Q) + &z,p ” ®x,p ^Zjj-Zq)]»^ + [=y,p <=H-'c> - "z,p ( % - y c ) ) Is + [ (yc-y@) + "x.ptVH-yc) - *y,p (%-Xc) ) "s’ which can be equivalently written as; (Slhq/H) = C-(Zh "Zc )“ s + "s (yH-Yc) ^ + [(Zjj-z^) Ig - (Xjj-Xç,) n g ] (9ay/3(|,) + [ (X^-X^) m^ - (yy-yc) ^sl (Bag/ 3*) + [-(Zc-Zqimg + (yc “ yq) "g] (3.67) 93 For the Pitch mode: Vg = (qzg) i + (-qxg) k V q = (qzq) i + (-qXQ) k I'hq = (Vg-Vq) (Igi + + »sk) » ([(Zc -^q ) + ay,q (Zh -'c > - =z,q(yH-yc): * f'®x,q ( ' « r ' c ) * ] “ s - ây,q + ®x,q'yH-yc” "s > « therefore: 3 Ihq/ 3® “ ( [-(3g-Zc)*s ■*■ (yg-yc)*s] V ® ® ) + [(Zg-Zc) <3®y/39) + [-(Yg-Yclls + (3Cjj-Xg)»g] (aa^/ae) + [(Zg-Zg)lg - (Xg-Xg)) ]. ) (3.68) For the vaw mode: Vg - (Xj,r) j + t-y^r) i \ = (x@r) j + (-ygX) i Ihq = (Vg - Vg) [Igi + Xg j + n^K] = {[-(yc-yg) + &y,r (2H-=c) - »z,r (%-yc) llg + [ (Xg-Kg) - âj; j.(Zg-Zg) + &2 p (Xg-Xg) IXg + t-ây,r (=‘h -'‘c > + =x,r (yn-yc) ]"s > ^ then; (9lhq/®^) “ t"(ZH~Zc^™s ■*■ (Yg-yc^^s] > + C(Zh “Zc )1s - (Xjj-Xg)ng](aay/3,),) + [-(yH-yg)!^ + (Xjj-Xg)mg] (aa^/a,)) + [-(■yc~yQ>is + (Xg-XQ)^^] (3.69) 94 The instantaneous positions of the suspension spring nodes are given by: A + Xq A A X q = Z q a 6 - Y q A * AZg = AZ + XgA0 AXjj =A Xg + (Zy-Zg) A ay - (y^-yg) Aag A Yg =A Yg + (Xg-Xg) Aag - (Zg-Zg) Aa% A Zg =A Zg + (yg-yg) Aa% - (X g -X g ) A a y where: A = (3ay az) A z + (aaya<|,) A(|) + ( B a y 38) A6 + (3a^/3ij,)ATj, A 3y = (Bay/ 3z) A 2 + (Bay/3(j))A4 + (3ay/38) AS + (3ay/3^)Aij; ^®z “ (*&%/ ^z) ^ z + (9a /3i)A4 + (3a_/36) A6 + (3a /3if>)A^ CHAPTER IV VEHICLE MATHEMATICAL MODEL 4.1 Introduction Vehicle handling is one of the most complex phenomena in the field of vehicle dynamics. This is due to the non linear behavior of vehicle components and the complex coupling between the various functional blocks. Huber [23] was the first to recognize that the stability of vehicles depends strongly on the non linearity between lateral tire force and the wheel load. W. Kamm [24] in his thesis of directional stability provided an analysis of static and dynamic stability with particular emphasis on the effect of side winds on vehicle stability. Schilling [25] made a classical analysis of an automobile by treating the vehicle as a linear dynamic system with three degrees of freedom. The roll motion of the vehicle body was also included. Later, Segel [26] extended Schilling's ideas by adding more design parameters in the equations of motions and introduced the concept of static stability margin as a criteria of vehicle stability. Kohr [27], employed Segel's equations of motion for an extensive study of vehicle motion which was simulated on analog and digital computers. Goland and Jindra [28] introduced the effect of dynamic wheel loads on lateral tire force in their treatment of motion of an automobile. They used the empirical equations developed by Smiley and Horne [29]. The size of the vehicle model was enlarged by Fosley and Slibar [30] to include six degrees of freedom of the vehicle dynamic system. In their theoretical analysis, the authors considered inertia forces and all tire forces including traction on each individual wheel. Enke [31], in his vehicle dynamic analysis, considered various types of suspension and the effect of suspension guidance on wheel rate. He also used iteration techniques in his computations and tire curves instead of equations. The effect of flexibility of the steering system and of the body roll on vehicle stability was first discussed by Fiala [32] who determined deflections of the wheels by considering not only caster but also aligning torque and overturning moment. The primary handling motions of the vehicle are the sideslip and longitudinal slip, the movement along the y 95 96 axis (i.e. scrub), the rotations about the x and z axis (i.e., roll and yaw). The other three degrees of freedom (fore and aft, pitch, and bounce) are coupled to the handling motions by the tire reactions, ride steer, and fore and aft weight transfer. In this chapter, two mathematical models will be developed. The first is based on the classical, lumped parameter approach, while the second uses the new concept of kinematic coupling and suspension derivatives introduced in the vehicle modeling simulation. 4.2 Lumped Parameter Vehicle Model The mathematical model of vehicle is described here in terms of the fourteen degrees of freedom where the vehicle is treated as an assembly of five lumped masses consisting of the sprung mass, at the center of gravity of the vehicle body and four unsprung masses at the center of each of the four wheels. This configuration considers independent front and rear suspensions. These unsprung masses are coupled to the vehicle body by the suspension systems and to ground by the equivalent stiffness and damping of the tire structure. The vehicle model is illustrated in Figure 4,1. The model consists of seven translational degrees of freedom (i.e. three associated with the rigid body motions of the sprung mass and four degrees of freedom corresponding to the vertical motion of each unsprung mass) and seven rotational degree of freedom (i.e., three usual rotations of the rigid body motions of sprung mass and the four rotational motions of the wheels). The forces included in the analysis are the suspension forces transmitted between the sprung and unsprung masses, the gravity forces, and the tire forces and moments generated at the tire road interface. The tire reactions are considered to be the only important external forces and moments acting on the vehicle. Both the rolling resistance and the aerodynamic effects are neglected. The inertial coupling between the sprung mass and the front and rear masses is taken into account, while the gyroscopic effects of the rotating parts are neglected. The steering input to the vehicle is introduced algebraically to the wheel total steer angle. 4.2.1 Equations of Motion The Equations of Motion are developed based on Euler's equations of motion written with respect to a moving axis 97 FIGURE 4.1 Vehicle Model with Lumped Mechanical Characteristics 98 system to avoid including the derivatives of the moments and products of inertia of the vehicle sprung mass in the equations. For a point A, having coordinates x,* y. and z, in the fixed frame system (ijk), the velocity or thi point A due to the rotation of the axis is the cross product of the rotation vector with the coordinate vector: = fix R = [pi + qj + rk] x [x^i + y^j + z ^ k ] = (qz^ - r y ^ ) i + (rx^ - pz^) j + (py^ - q x ^ ) k (4.1) therefore the total velocity of point A is given by the components: x& = u + qz^ - ry^ (4.2) = V - PZ;^ + rx^ (4.3) z& = W + py^ - qx^ (4.4) The acceleration of point A is obtained by differentiating each term of the above equations: = Û +qZj^ + qz;^ - ry^^ - r ÿ ^ (4.5) = V - pz^ - pz^ + fx^ + rx^ (4.6) Za = W + py^ + py^ - qx^ - qx^ (4.7) by substituting the velocity x., y- and z. in the acceleration equations, the total accelerations becomes = U + (q + rp)z^ + (pq - r)y^ - (q^ + r^)Xj^ + qW - rV + 2qz^ - 2ry^ + x^ (4.8) 99 Ya = V + (p -rq)Zi + (pq + i)x^ - (p^ + r^)y^ - pW + rU +2rx^ - 2pz^ + (4.9) \ ** w + (p + rq)y^ + (pr - q)x^ - (p^ + q^)z^ + pV -qU + 2py^ - 2qx^ + z^ (4.10) where x , ÿ , z and x_, y_, z* are the local velocities and accelerations of tne point A. Based on the above equations we deduce the acceleration components of the vehicle center of gravity and of the unsprung masses: a% g = Û + Wq - Vr (4.11) ay g = V + Ur - Wp (4.12) a% g = W + Vp - Uq (4.13) and a^j^ = Ù - Vr + Wq + 2qd^ - x^(q^ + r^) + Yj/pg - f) + Zj^(rp + q) (4.14) ayj^ = V - pW + rU - 2pd^ - Yi(P^ + r^) + x^(pq + r) + z^(rq - p) (4.15) Sgl = w + pV - qU - (pf + q2)Zi + (p + rq)yi + (pr - q)Xi (4.16) The the Euler equations of motion of the rigid body can be written in the form (i.e. derivation of these equations is included in Chapter 5): M ( Ù + Wq - Vr ) = ZFjj (4.17) M ( V + Ur - Wp ) = ZFy (4.18) M ( W + Vp - Uq ) = ZFg (4.19) 1 0 0 + qhg - rh,y = :^x (4.20) z = ZMy (4.21) (4.22) hg + phy - qhX = :%z where (h , h , h ) are the scalar components of the angular momentum, and (F®, F , F ) and (M , M ) are the external forces and moments about the^vehlcle body axis. Since the vehicles are symmetrical with respect to the (xz) plane, the two products of inertia I and I are both zero and the Euler's equations for angular momentum become: f (4.23) Ix P - ^xz + (Ig - ly) qr - ^ x z - ^ “ x q + - I,) pr + (p: - r^) = (4.23) 4 (Ix Z M y r - ^Z ^xz p + (ly - 1%) pq + ^xz = ZMz (4.24) 4.2.2 Determination of Forces and Moments The sums of forces in the x and y directions of motion consist of the inertial forces of the unsprung masses F , and F -, the x and y components of the gravity force F-y and Fiy and the forces generated at the tire-ground contact patch in the x and y directions of the fixed frame F and F . The forces are given by: ^ Zm^ Fix = ZFlxi = ®xi (4.25) Zm^ (4.26) F i y = ZFlyi = ^yi ( M + Z m^)g sin Fg x = (8 ) (4.27) ( M + Z m^)g ^ G y - cos ( 8 ) sin ( <}> ) (4.28) Fxu " ^ Fxui (4.29) Fyu “ ^ Fyui (4.30) 1 0 1 since the road is treated as flat, horizontal plane and the pitch angle 0 and the roll angle 0 of the sprung mass are assumed to be small angles, the summation of the tire force components acting in the direction of the x and y axis of the vehicle is given by ^xui “ ®’zi sin ( e ) + cos ( ^ sin ( (4.31) Fyui = - Fgi sin ( (|) ) + Fjji sin ( cos ( il> (4.32) where F j is the normal force acting on the tire, and (F F j) are the tire force components in the longitudinal ana lateral directions, discussed in the previous chapter. The acceleration components are computed from Equations 4.14 and 4.15 where the coordinates of the unsprung masses are substituted by their corresponding values shown in the table below: Front/Right Front/Left Rear/Right Rear/Left Index 1 2 3 4 ^i A A-B -B ^ i . ; Tf/2 - Tf/2 Tj/2 - V 2 Zf + di 2f + *2 Zr + dg Zr + *4 The bounce equation of the sprung mass is inertially uncoupled with the unsprung masses which have their independent vertical motion. Therefore the external forces exciting the bounce mode are just the normal reactions generated by tire radial deflection. ^zui “ ^ the normal load F^^ is calculated from ' =wi hi + ( Rwi - hi) if hj^ < = C„i l»i i* hi > E„. (4.33) 1 0 2 where hi is the instantaneous rolling radius of the tire given by the positions of the individual wheel centers (hi = -Z^), The coordinates are calculated by: Z Z "• X 0 + y^ where Z is the location of the center of gravity measured from the ground. The condition hi > R,,i provides an indication for wheel lift-off. ^ The resultant moments about the vehicle axes are generated by the internal and external forces developed in the vehicle system. The various force components are shown in Figure 4.2. The summation of moments about their respective vehicle axes yields: = (Sg - Si)(Tf/2) +(8, - S^)(Tj/2) - Z^ + d^ + h^^) ^yu2 ( ^f * ^2 * ^2^ “ ^yu3 ( + d^ + hg) - Fyu4 ( :r + *4 + h4> + ^ “xi <^.35) 2 My = (Si + S2)A - (S3 + S^)B + F ^ i (Z^ + d^ + h^) ^XU2 (^f ^2 ■*’ ^ 2) ^XU3 (^r * ^3 * ^ 3) + Fxu4 (Zr + <4 + ^ 4) (4-36) “ z = (Fyul + - (Pyu3 + ^ 4 ^ ® + (^xu2 " (^f/Z) + (^XU4 - ^XU3) (V2) + ^ Mzi (4.37) Where M i and M i are the tire alignment torque and overturning moments respectively. The alignment torque was discussed in Chapter 3. The overturning moment acts about the longitudinal axis and is computed by the following expression [33]; “xi = °0 + <°1 + °2 tl' Fyi J'ai + ° 3 ♦i (•>•38) Where the constants 0-, O., 0_ and 0_ are empirically derived coefficients. 4.2.3 Camber Effect on Forces and Moments The camber angle is the inclination angle which the tire makes with the xz plane. It has an important influence on Z U l X U 1 XUg ZU2 Figure 4 . 2 Force end Moment Components in a Vehicle Model o w 104 the vehicle handling since a cambered tire develops a lateral force or a camber thrust particularly for an independent suspension. However the camber generated lateral force is relatively small when compared with the lateral force resulting from slip angle. J. Ellis [10] proposed a ratio of 1 to 20 in the lateral force, resulting from the same angle of camber or side slip. The camber angles and the wheel steering angles which characterize the wheel-to-vehicle orientation are required as tabular data of suspension deflection. These data are collected by e^erimental testing of the particular type of tire used with the vehicle. They are then fed to the simulation which determine the correct instantaneous angles by interpolation. 4.2.4 Longitudinal and Lateral Slip The slip angle at each tire is calculated by: a i = - tan'l ^ sign ( (4.39) where the forward velocity and the lateral velocity V„. of the contact point in the ground plane are given by: Umi = U. + 8 Wj V = V - (|) w (4'40) where Uj and Vj^ire tie forward and lateral velocities of the tiré contact points along the vehicle x and y axes. They are computed by: u^ = U - y^r + z^q (4.41) Vi = V + x^r - z^p The vertical velocities of the tire contact points along the vehicle z axis are: wj = W - Xj^r + y^p + (4.42) The.longitudinal slip is defined as the change in distance traveled per revolution due to driving or braking conditions, to the distance traveled under the free rolling condition. Therefore it is computed by: 105 Si » 1 - (üi hi)/(U^i COS (*i) + Vji sin (4^)) if -1 < Si < 1 = 1 if Si > 1 => -1 if Si < -1 (4.43) 4.2,5 Wheel Dynamics A simple model for the wheel spin which satisfies the purpose of this investigation is developed for the vehicle model. It is assumed that the wheels consist of only their rotational inertia are influenced only by the braking or driving torque and the reaction torque developed by the longitudinal force F Since all the suspension are independent, no coupling effect exists between the front or rear wheels. Then, the equations of motion of the wheels can be written as follows: Jf =» h^ Fjjj^ - T^ if -1 < S^ < 1 = 0 otherwise (4.44) where J, is the polar moment of inertia of wheel i about kingpin axis, and T, is the torque applied at wheel i. For a braking maneuver the Torque T j can be idealized as a function of time and the forward velocity. The initial rotational velocity of the wheel, corresponds to a zero slip and is given by: aJj,(t=0) = [ U^^(O) cos(w^(0)) + V^^(O) sin (o)^(O))]/ hj,(0) (4.45) 4.3 Vehicle Model with Kinematic Suspension In this section, a vehicle model based on the concept of suspension derivatives is developed. The vehicle model with the Long/Short Arm suspension is shown in Figure 4.3. Figure 4.4 shows the vehicle model with the suspension presented schematically. In this model, all the degrees of freedom associated with the vehicle body motion are retained; whearas, the suspension movements become algebraically dependent on the vehicle body motion. The model also includes the rotational degrees of freedom associated with the wheels. In reference [10], a three degree of freedom model of a ground vehicle, using suspension derivatives, was developed to study the influence of roll on handling characteristics and the steady state state response of the vehicle model on yaw, roll and side slip. 106 'A / \ — — -i/ FIGURE 4.3 Vehicle Model With SLA Suspension FIGURE 4.4 Schematic View of a Vehicle Model With Kinematic Suspension o 108 4.3.1 Formulation of Vehicle Model with Suspension The equations of motion are derived in a similar way as for the previous model. For the vehicle model shown in Figure 4.3, the equations of motion are derived for the vehicle body only. Since the unsprung masses are not included in this model. 4.3.1.1 Bodv Equations A typical point of mass gm and of coordinates x, y, z, has the following velocities: u = U - ry + qz V = V - pz + rx w = W + py - qx A plane of symmetry was assumed to exist through the center of the vehicle (xz plane). The governing equations of motion of the vehicle body are (i.e the center of mass is located at the axes origin) : M (Ù - Vr + qW) M (V - pw + Ur) M (W - pV + qU) - ZMx :xP - (ly l^)rq - Z M y l y q - C z V r p - ÉMz Iz' - C x ^y) pq The generalized forces and moments will be determined later. 4.3.1.2 Suspension Kinematic Equations The suspension units are considered kinematically linked to the vehicle body through the suspension derivatives. Therefore, the total velocity of the center of the wheel-ground contact patch is the velocity of the vehicle body center of gravity increased by the velocity components induced by the suspension derivatives. Hence: Ut i = U + (3x^/3z)W + (ax^/3(j.)p + (3Xi/38)q + (Sx^/S^) r (4.46) 109 Vfi = V + O y y a z ) W + (9yj/9*)p + (97^/90 )q + (97^/9,;, ) r (4.47) The change in the relative position and orientation of the suspension unit is also determined in function of the suspension derivative. The change in fore and aft and in tire scrub are denoted: and yj while the tire rotations in camber and steer directions are denoted (}> • and \jj j. Since the local movement of the suspension with respect to the vehicle body are small and slow, when compared to the overall dynamics of the vehicle, the corresponding motions are assumed quasi-static and no relative inertial effect is considered. The change in the local position and orientation of the vehicle are determined by Equations (3.56) through (3.59). 4.3.1.3 Generalized Forces The generalized forces for the vehicle model with suspension derivatives are identical to the forces developed for the first model. However some of the secondary variables have to be redefined to accommodate with the new formulation of the problem. The total deflection of the suspension spring is determined by: Al^ = (91j/9z)Az + (91j^/9<{>) A(f) + (91^/90) A0 + (91^/9i|;) Aip (4.48) The change in the relative velocity of the shock absorber of the suspension is given by: Al^ = (91^/9z)A W + (91i/3*)Ap + (9l^/90)A q + (9l^/9(|;)A r (4.49) The total sideslip angle of each suspension is given by: = -tan”^(Vj^/U^j^) + (1/20) (4.50) where \p j and <|>, are the total steer and camber rotation induced by the suspension derivatives. The rolling radius of the wheels is determined by analogy with the classical model as follows: hi — hiQ - 1^ (Kg / K^) (4.51) 1 1 0 where K and K, are the suspension and tire stiffness, all the other variables are determined with the same expression as for the first model. CHAPTER V MODEL SIMULATION 5.1 Introduction It was stated in the second chapter that available simulations for vehicle dynamics models are either written in special non standard codes or based on general purpose software such as ADAMS. This prevented the users from having a complementary understanding of the simulation mechanics and of the system response from the vehicle models and the simulation algorithms. Reported numerical instability [34] observed with some of the simulations could not be explained until an investigation of the system equations and the computer algorithm is possible. Some of the results related to the vehicle dynamics simulation are presented in this chapter. The simulations developed for the proposed models are based on the time-simulation software package Advanced Continuous Simulation Language (ACSL) . 5.2 ACSL Simulation Software The Advanced Continuous Simulation Language is designed for modeling and evaluating the performance of continuous systems described by time dependent, non linear differential equations and/or transfer functions. ACSL was developed for application areas such as control system design, missile and aircraft simulations and vehicle handling. An important feature of ACSL is its sorting of continuous model equations, in contrast to digital programming languages such as FORTRAN where program execution depends critically on statement order. THE ACSL language consists of a set of arithmetic operators, standard functions, and other ASCL Special functions. In the implicit structure, the simulation model in ACSL can be written with PROGRAM and END statements. The more flexible explicit structure includes INITIAL, DYNAMICS and TERMINAL sections. The flow of an ACSL Program with explicit structure is shown in Figure 5.1. In the initial section, the calculations needed before the dynamic model begins are performed. The integration routine is initialized when control transfers out of the INITIAL section. All initial condition values are transferred into the corresponding states. The program transfers control to 1 1 1 PROGRAM INITIAL I Statements performed before the run begins. State variables do not contain / the initial conditions yet. END DYNAMIC DERIVATIVE ) Statements needed to calculate derivatives of each INTEG / statement. The dynamic model. END DISCRETE I Statements executed periodically, t END f Statements executed every communication interval. END TERMINAL ) Statements executed when the termination condition TERMT becomes ) true. END END Figure 5.1 Explicit Structure of ACSL Program [35] to 113 the terminal section whenever any of the termination conditions is satisfied. 5.3. Numerical Stability of Vehicle Dynamics Simulations The accuracy and convergence of computer simulations to solve a system of coupled and nonlinear differential equations depend on the numerical method used in the simulation. For vehicle dynamics simulations, the convergence of computation depends primarily on the method used in deriving the governing equations of motion. It is proposed in this investigation, that the application of Euler equations which are generally adopted in vehicle dynamic problems, may have negative effects on the computational performance of the simulations. Infact, it is only if the forcing functions are external inputs, which are independent of the fundamental states (positions and orientation),that the numerical problems may be reduced in the computational procedure. Such numerically stable simulations can be predicted in the analysis of space vehicle dynamics. The dynamics of all aircraft systems and flying bodies (holding no fixed contact with the ground) depends solely on the acceleration or the velocity of the body. However, for the case of ground vehicles, the displacement and angular rotation come to play a significant role in the dynamics of the system, which bring a significant effect on the stability of the numerical method adopted in the simulation. 5.3.1 General Formulation of Ground Vehicle Dynamics From a dynamic point of view, ground vehicles are classified as multi-body systems characterized by their rigid body with mass and inertia (sprung and unsprung masses). These are interconnected by elements causing reaction forces (i.e. springs, dampers) or by rigid joints and bearings which constraint the motion and forces. In the most common available simulations [36], the governing equation of motion is given by: M(x) X = F (X, X, t) (5.1) where M is the equivalent mass matrix and F is the equivalent forcing function. The force vector F is frequently described by additional differential equations characterizing the suspension dynamics and depending on generalized coordinates as well as additional state components. 114 Several modes of dependence on velocity and displacement can be deduced from the generalized equations of motion. For instance, a velocity dependent mode is characteristic of the yaw rotation of the vehicle where the exciting forces are functions of the velocity dependent side slip and the slip angle. The roll rotation is an example of a mode which depends on the basic state variable characterizing the position and orientation. 5.3.2 Integration Methods in Vehicle Dynamics Simulations As indicated in the previous section, the governing equation of motion of vehicle dynamics is of the form: M(x) X = F (X, X, t) + Q (5.2) where the equivalent forcing function is broken into two state dependent and state independent functions. The vector Q is the external forcing function, and x is the vector of generalized coordinates. This equation can be written in state variable form as follows: X = Y Ÿ « M “^ (X) [F (X, Y, t) + Q] (5.3) For ground vehicles, the use of Euler equations, which treat the vehicle as a free body in space, requires the availability of constraint equations relating algebraically various degrees of freedom of the vehicle. These constraint equations can be either considered as additional equations to the problem description or may be taken care of in the development of equations of motion. For illustration, consider the case of two point masses at the ends of a rigid massless bar, and supported by two springs connected to the ground (Figure 5.2). The equations of motion of the system are: ™1 “ ^1 ” ^1 ^2 ^2 ” ^2 ” ^2 1*9 = M - a + Rg b (5.4) These equations treat the system as a free body in space where R, and R- can be any reaction forces (e.g., air resistance) opposing the motion of the body, since the system is fixed to the ground, two more complications are added to the problem for its solvability. First, the reactions R^ and Rg become state dependent disturbances. 115 Ni/ a. Two Mass System \y Xl b. Free Body Diagram Figure 5.2 Mechanical Second Order System 116 In fact, since R, and R. depend on the basic states and their first derivatives? The dynamics of the problem is increased from zeroth order (excluding the integrators) to a standard second order system. Second, the fixation of the system to the ground, results in a constraint equation relating the various states of the system: x^ - Xg - (a+b) Sin 8=0 (5.5) or: x^ = Xg + (a+b) Sin G x^ = Xg + (a+b) 0 Cos 0 ^1 “ ^*2 (a+b) [9 Cos 0 - sin 8 ] The presence of constraints is generally due to the relative motion between connected links. However, by eliminating the constraints and reducing the generalized coordinates to their minimal number, the transformation of the system equation to the state space form and the application of classical integration method would be a feasible approach. To observe the effect of integration on the computational procedure the simple Euler method (i.e. rectangular integration) is adopted. This allows the tracking of the process of numerical computation. Euler Method of Integration The general state space form of the governing differential equation of motion can be written as: X = F(X,t) + Q(t) The Euler (rectangular) method transforms the derivative of the states to the first order difference quantity. There are two types of transformation: 1. Backward difference (BD): = (X^ - %n_i)/h 2. Forward difference (FD): x^ = (X^^^ - x^^/h By introducing the forward difference of X, the state equation becomes: Xn+i = + h(F(%n,tn)) + b(Q(t„)) (5.6) As should be predicted, the use of the forward difference method (FD) resulted in a difference equation of the 117 explicit form while the use of BD method yields an implicit difference equation: ^n = %n-l + h(F (X^,t^)) + h(Q(tn)) (5.7) The implicit method generally requires a substantially greater computational effort than the explicit method [37]. The latter yields the current value X directly in terms of previously computed values of X^ (k Besides the computation time factor, these two methods of integration may have a significantly different effect on the stability of the integration procedure. A simple case is considered next, to illustrate analytically the difference between the FD method and the BD method. Consider the case where F(X,t) = - XX and Q * 0 ( X is positive since the continuous system is assumed to be stable): the difference equations of motion become: FD: = (1 - Xh) X^ (5.8) BD: = (1 + X h)-l X^ (5.9) where h is the computation time step. In the discrete domain, the stability of difference equations requires that the eigenvalues of the discrete system be within the unit circle. Hence the conditions for stability of computation for both cases become: FD -1 < 1 - h < 1 (5.10) BD 1 + h > 1 (5.11) The time steps required for stability of FD and BD methods are respectively; h < 2/ X and h > 0. This indicates the superiority of the BD method over the FD method from the stability view point, while the FD method was previously shown to be preferable from the computation time view point. 118 Error of Integration in Euler Method As can be shown from the Taylor expansion of X , the truncation error in approximating X by ((%n+l - Xn)/h) is given by; ” ®ex f^n+l> = (hf/2!) + (hf/S!) + ... (5.12) where X^(^) = D^x at t = nh The theoretical error for the implicit integration method is H n t V l > = (x^+2 - 2 + %n)/h + This obviously indicates a difference in the truncation error between the two methods, which depends on the increasing or decreasing behavior of the velocity X. 5.3.3 Open Loop vs. Closed Loop Integration in Vehicle Dynamics Simulation The theory presented in the previous section is directly applicable to the concepts of open loop and closed loop ■ integration approaches in the vehicle dynamics simulations. In the open loop approach, the vehicle is assumed to be a free body in space and the ground reactions are considered as external excitation forces computed as a function of vehicle fundamental states and their derivatives. The .closed loop integration approach treats the vehicle as a rigid body elastically fixed to the ground. Therefore; the elastic reactions at the contact points are automatically included in the equations of motion. These two approaches are analogous to the case of control systems subjected to either external disturbances or to a direct feedback both of which are functions of the states and their derivatives. To observe the influence of both integration methods in the stability and accuracy of computation and how these methods are related to the discussion in the previous section, an example is examined next. 119 5.3.4 Numerical Illustration A simple vehicle model, as shown in Figure 5.3, is used to investigate the effect of Open Loop and closed Loop Integration Methods on the stability of computation. The model consists of a second order system where only the bounce and pitch degrees of freedom are retained. If the closed loop formulation approach is used, the executable statements in the computation., algorithm would be ; Z = (1/m) [F - (Z - a sin9 ) - kg (Z + b sin 8) - (Z - ae ) - Cj (Z + be ) ] (5.14) ê = (l/J) [M + (Z - a sinè ) a - kg (Z + b sin8 ) b + (Z - aé ) a - Cg (Z + be ) b] Z = J 'z dt+ Z(tg) ; Z = J'z dt+ Z (t^) 0 = Jè' dt + 8(tg) ; 6 = J ê dt + 8 (tjj) If the open loop approach is used, the computer algorithm would contain the executable statements listed below: Z = (1/m) [F - Fg^ - Fg]^ - Fg^ - Fai] (5.15) 0 = (l/J) [M + Fgp b -Fg]^ a + F^j,b - Fa^a] Z = J"z dt + Z (tg) ; Z = J'z dt + Z (t^) 8 = J ”é dt + 0 (tg) ; 6 = J'é dt + 6 (t^) Fg2 = (Z - a sin 0) ; Fg^ = (Z + b sin 0 ) Fdi = Cl (Z - a 8) ; Faj, = Cg (Z + b 8) For this simple problem case, the closed loop integration is generally the method adopted in solving the problem. However, complex vehicle models, where coupling plays a F M m,J Figure 5.3 Second Order Vehicle Model oN i 1 2 1 significant role in the interaction of various equations, the open loop approach can not be easily avoided. These two methods of simulation can be best seen by their functional block diagrams, shown in Figure 5.4, when only the bounce mode is considered. The transfer function G is the constant (1/m). For the open loop feedback case, the generated function F can be thought of as simulated with some time delay. Although one might expect no differences’'between these two algorithms, it was found that the first method is always stable, for mechanically stable systems, whereas the open loop method is very sensitive to computational parameters, particularly the time step of the integration. This sensitivity was found to be dependent on the mechanical dynamic characteristics of the system (natural frequency and damping ratio). The collected results are shown in Figures 5.5.a and 5.5.b which illustrate this observation when the Euler (rectangular) method of integration is used. The time step A t (i.e., h), corresponds to that which allows fast convergence ofXzhe open loop and closed loop methods to the same solution, while t is the critical time step of integration beyond which any increment in the time step causes partial (some states only), or full instability in the numerical computation for the open loop method. 5.3.5 Analvtic Analysis of Simulation Stability By tracking the steps of integration performed by both methods, the closed loop approach was found to result in an implicit method of integration while the open loop approach results in an explicit method. This explains the superiority of the closed loop integration over the open loop integration from the computation stability view point. For a time step which preserved the stability of both methods, the CPU time for the closed loop simulation was 00:21 seconds, whereas it was 00:39 seconds for the open loop simulation. This is also in agreement with what was predicted earlier. The effect of dynamic characteristics of the system can also be predicted by performing an analytical analysis of the corresponding difference equations of both methods. To illustrate this effect, an analytic investigation is carried out using the simple Euler Integration Method. The system is further simplified by linearizing the model and ignoring the forcing functions F and M in the analysis. 1 2 2 Q + X X X 1 / S ------l / S - > X X a. Open Loop Feedback (2 stages) X Q b. Closed Loop Feedback Figure 5.4 Open Loop and Closed Loop Block Diagrams 123 CRITICAL TIME STEP FOR CONVERGENCE OPEN LOOP INTEGRATION METHOD -ZETfl=0.7C6- 3- T I K E S T E 2 P • « . —Q 0 - 1 NhTVF.rTi. F = 5C.':NC1 W-REC , : UNCOUPLED CASE- o : COUPLED CASE- FIGURE 5.5.a Critical Time Step for Convergence 124 CRITICAL TIME STEP FOR CONVERGENCE OPEN LOOP INTEGRATION METHOD 0.9 T I M E S T E P S b C 0.3 0.0 0.2 0.!f 0.6 0.6 1.0 1.2 1.4 1. 5 6 DAMPING RATIO UNCOUPLED CASE- q ; COUPLED CflSE- FIGURE 5.5.b Critical Time Step for Convergence 125 The difference equation, obtained with the closed loop formulation, is of the implicit form. It is given, in state space form by: [T R}\ = a, [T R] Where: = [z Z] ; af = [ 08 3 and 1 -h 0 hWg^ 1 + ZhSgWg - ahWg - 2 G h GgWg -h Q6hWg- - 2 n h Çg w g hW( 1 + 2hggWg _j (5.16) where w 2 = (%1 + kg)/m ; Wg^ = (k^ a^ + k^ bf)/j^; 2 g'gWg = (C^ + Cgi/m; 2Cg Wg = a^ + C^h^)/J ; a = (k^ a - kg b)/(k^ + kg);G = (C^ a - Cg b)/(C^ +Cg) 6 = (ki a - kg b)/(kj^a^ + kgb^) ; n = (c^ a - Cgb)/ (c^ a^ + Cg b^) when the open loop formulation is used the difference equation is of the explicit form. The resulting dynamic matrix A is given by: -hw. 1 - 2h Ç gWg a hWg 28h5^w- 6 hw 2 n h Ç g w Q -hwg l-2h ^0 Wg (5.17) 126 The stability of computation requires that the eigenvalues of the discrete dynamic matrix fall within the unit circle. To reduce the order of analytical analysis, the following assumption is made to eliminate the coupling between the bounce and pitch motions: k^a = kgb; c^a = Cgb Without loss of generality, we can assume that the bounce dynamics has a dominant effect on the stability behavior. The corresponding dynamic matrices are: 1 -h 1 h "c,b » &o,b - JiWg2 1 + 2h Î j-hWg2 The characteristic equations are obtained by taking det C X I “ A] =0 . For the open loop case, this reduces to: X^ - 2 X (1 - h G gWg) + (1 » 2h Ç gWg + h^ WgZ) = 0 (5.18) The roots of this equation are the eigenvalues of the system. They are given by: 2 _ 1)1/2 A 1,2 = (1 - h ^ ± h (5.19) For stability we require that: 1^1, 2 I <1*0 Since the eigenvalues depend on three parameters (h, Ç , Wg) the following two cases can be deduced: ^ 1st case: when C < 1, the roots 1, and 1_ are complex conjugates. Their magnitude is: ^ |Xl = ( d - h G w )2 + (1 - G 2) h^ W_2)l/2 (5.20) The condition for stability becomes: h < 2 Cg/Wg (5.21) 127 2nd case; if ç > 1, all the eigenvalues are real, they are given by: X 1,2 = 1 - hWg (Cz ± ( - 1)1/2) (5.22) The requirement for stability becomes: h < (2/Wg) ( Sz + ( - 1) . (5.23) For the closed loop case, the characteristic equation is a second order polynomial in X . The eigenvalues are given by: X 1^2 = [(1 + h Gz*z) ± i hWg (Ç g2 _ 1)1/2 ^ (5.24) The critical value of h, corresponds to what brings the magnitude of the roots to 1 or larger. If gg2 < i; the roots are complex and their magnitude is given by: |X| = 1/ [ 1 + 2h GgWg + h2wg2]l/2 (5.25) which is always less than 1 for all time step values. Therefore, no restriction is required on h, if g < 1, for the closed loop formulation. 2 .If Ç 2 > 1, the root magnitude becomes: |X| = 1/[1 + h Wg (Eg + ( Çg2 - 1) 1/2)]1/2 (5.26) It is obvious that |X( is always less than 1, since ( 5z ± (S - 1)^/^) > 0 for Ç g > 1. The conditions for stability, for this case, are summarized in Table 5.1. If the coupling between bounce and pitch modes is present, the ranges of stability of the open loop and closed loop methods, will be affected, but the latter remains always superior over the first. Moreover, when the coupling is present and/or the system is of higher order; the analytic investigation of stability in function of the complex roots becomes tedious. However, the stability of 128 Table 5.1 Time Steps for Stability Open Loop he < 2/Wg ( Çg + Method (Euler) ( - 1 )1/ 2) Closed Loop h_ > 0 he > 0 Method ° Open Loop ij. < 2hg [-1 + hr < 2h^ [-1 + Method (Runge- (1 + 2hg)l/2]-l (1 + 2h.) V 23-I Kutta) 129 the system can still be studied without solving for the complex roots of the characteristic equation. The results obtained, so far, correspond to the case where the derivatives are expressed in forward or backward differences. However, differential equations can also be expressed by simple central difference approximations, it was shown in [38] that these approximations are stable if the time step h is less then 2/w where w. is the highest natural frequency of the system. It was^. further suggested that a time step considerably smaller than 2/w has to be used to retain the desirable detail of the solution. 5.3.6 Numerical Stabilitv and Vehicle Simulation Formulation In vehicle models, the suspension forces depend partially on the fundamental states of the vehicle systems. Moreover, these forces contribute to the forcing loads in the bounce, pitch and roll equations of the system. . On the other hand, the yaw motion is influenced only by the longitudinal and lateral forces, or equivalently by steering and vector velocity. Therefore, it appears that the most satisfactory model to use in ground vehicle simulations is one with the pitch, bounce and roll as second order modes connected to the road by the suspension and sideslips, and with the yaw rotation as a first order mode with typical tire force data which is dependent on steering and vector velocity of the contact patch on the road. When it is desirable for the vehicle to leave the ground, the flight portions should be simulated using Euler equations, resuming the second order modes as the car returns to earth and including structural deformation modes. 5.4 Simulation Model Organization The vehicle dynamic mechanism can be described by a block diagram as shown in Figure 5.6. The vehicle model simulations are developed based on this block description. The simulation inputs required for any particular run are; 1. vehicle characteristics which include the inertial and kinematic properties of the vehicle (e.g. masses, moments of inertia, dimensions ... ) 2. Suspension functions for the determination of the spring and shock absorber forces. 130 GRAVITY ROAD PROFILE . Co»(f. of Fricficn, 6 DOF SUSPENSION SPRUNG DEFLECTION MASS (Wheel Hop) Suipcniion Ftarco WHEEL/ROAD Tire Normal ' INTERACTION Force, Rolling Fit -- Rodius, FORCES AND Comber, yy TIRE MODEL' MOMENTS «T' FORCES AND APPLIED TO . MOMENTS SPRUNG MASS WHEEL STEER Tire MODEL Slip Steering Wheel Anglo Angle »T WHEEL SLIP Tire Slip MODEL R o IIq Figure 5.6 Vehicle Dynamics Slock Diagram for Time Domain simulation 131 3. Front and rear steer and camber data table for the interpolation of wheel orientation in function of the tire deflection (for lumped-parameter model). 4. The suspension derivatives for the computation of suspension displacement and orientation (for the model with kinematic suspension). 5. Tire friction model constants, required for the determination of the circumferential and lateral forces, the alignment torques and the overturning moments. 6. Friction coefficient characterizing the type of the road (i.e. dry, wet, concrete, soft, ...). The other inputs which determine the type of maneuver for the simulation run are the brake torques and the steer angle. The front and rear brake torques determine the rate of change of the angular velocities of the wheel which produce the preferred longitudinal slip for the braking maneuver. In general, the longitudinal slip is fully controllable by the driver. The steer angle transmitted from the steering wheel to the vehicle wheels controls the directional destination of the vehicle. Several functions of steer angle can be input to the vehicle to perform the directional maneuver needed. Initial values of some of the variables, such as the forward velocity have to be defined in the program. Some other initial values are also required to allow successful sorting of the program by the ACSL software. The simulation outputs are the time history of the vehicle position and orientation and the translational and angular velocities of the vehicle. Supplementary possible outputs include position and orientation of the suspension, longitudinal and lateral slip, the angular velocities of the wheels, and the forces and moments generated at the tire/road contact patch. Many other secondary variables are also available. The system equations developed in the previous chapter can be formulated in the matrix form. For the case of vehicle model with lumped parameter suspension, the matrix equation is: 132 U V W P q r [ M ] = [ F ] (5.27) 031I 0)2 0)3 0)4 where [M] and [F] are the equivalent inertia and load matrices respectively. They are given by: [A] = 0 0 0 0 ^t =1 0 0 0 0 0 0 0 0 0 0 “ t -Cl 0 C2 0 0 0 0 0 0 0 0 0 0 M 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -I 0 -Cl Ixt xzt 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 :yt 0 0 0 C2 -^xzt :zt 0 0 0 0 0 0 0 0 0 0 nil -m^A 0 mi 0 0 0 0 0 0 0 0 0 ni2 -mjA 0 0 m2 0 0 0 0 0 0 0 0 m 3 *3Tr/2 0 0 0 m 3 0 0 0 0 0 0 0 -m^B 0 0 0 0 ™4 0 0 0 0 0 0 0 0 0 0 0 c 0 0 ^W1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Jw2 0 0 0 0 0 0 0 p 0 0 0 0 0 0 ^W3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 "^W4 (5.28) 133 and ^^xui - Wq - g sin(0)) ^^yui + Mt - Ur + g cos(0) sin(^)) - Z ( Ug - Vp + g cos(0) cos((j))) (Sg - Si)Tf/2 + (S4 - Sg)T^/2 -..^(Wp - Ur + geos ( 0 ) sin(0 ) ) -Z F . ii2 yui - Ï 3,4 + «1 + “ri> +Z"xi (Sj^ + SjJA - (S3 + S^)B -CjCWg - Vr + g sin(e)) + + di + «fi > + L « x u i < 2r + «1 + «ri> '£/yui )* - (f,/yui )« -Cz'Or - «P - 9 <=°=(0) [F] = sin(+) ) + (F^ 3 - F^„3>V2 + (^xu4 “ ^XU3 ) V 2 +Z ^zul + Si + mi g cos (0) COS((j>) - mi (Vp - Ug) m^ g cos COS( -Fxf2 * %f2 " ^q2 ”^xr3 * »r3 " ^q3 ”^xr4 * »r4 “ ^q4 (5.23) where; Cl = +(m^ + BgiZf + (m3 + m^)Zj. C2 = (m^ + MgjA + (m3 + m^)B I j j t = i x ^ "*■ '*■ ^2^^] 134 Igt = Iz + ("'i + "»2> + t / / 4 ) + (m^ + mg) (B^ + Tj,V4) Ixzt == ^xz + (% + molAZf + (m-, + m.)BZ all tnë other variables,forces, and moments are as defined previously. For the vehicle model with kinematic suspension, the system matrix equation is given by: U V W P [ M ]' q = C F ]' (5.30) r “1 0)2 0)3 0)4 where: [ M ] ' = M 0 0 0 0 0 0 0 0 0 0 M 0 0 0 0 0 0 0 0 0 0 M 0 0 0 0 0 0 0 0 0 0 0 Ix "Ixz 0 0 0 0 0 0 0 0 0 0 0 0 0 "y 0 0 0 “Ixz 0 Iz 0 0 0 0 0 0 0 0 0 0 Jwl 0 0 0 0 0 0 0 0 0 0 ^W2 0 0 0 0 0 0 0 0 0 0 ^W3 0 0 0 0 0 0 0 0 0 0 ^W4 (5.31) 135 and ^^xui ^ - Wq - g sin(9)) ^^yui + M (Wp - Ur + g cos (9) sin( -Z + M ( Uq - Vp + g cos (9) cos (({>)) (Sg - S^)Tf/2 + (S4 - S^)T^/2 Z(Fyui( By/ B*)i) + 2M, XI (Si + SglA - (S3 + 84)5 - [F]' = (Fxui( 3y/ »+)i) ( E/yui - % / y u i + (^XU2 - Fxul)^f/2 + (Fxu4 — ) V 2 ^XU3 + : Mzi * -?xfl »fl ■ ^qi * ”^xf2 ^f2 * “^xra ^r3 " ^q3 * ”^xr4 %r4 ■ (5.32) With the exception of the acceleration computation and the suspension derivative calculations, all other calculations are done within the derivative segment of the ACSL program. This allows sorting of the program statements and creates an efficient order for computation. The execution of the simulation can be done interactively once the model is translated and gone through the FORTRAN compilation and load operation. The user control the execution by a sequential set of commands which exercise the model. One of the special features of ACSL is that during run time the data values, entered in the CONSTANT statement, may be changed allowing the analysis of vehicle response with several characteristics with the same program and in the same run. Execution termination conditions can also be changed leaving more control of the time of execution with the user. The outputs needed to be investigated have to be specified with the PREPAR statement in order to be prepared and plotted. 136 The structure of the ACSL program is the same for both models. It is composed of the following sections: 1. The INITIAL section: it includes the declaration of variable types and the CONSTANT statements for the variables which do not change their values during the run of the program. All initial calculations needed before the dynamic model begins are performed (e.g. initial rolling radius of the tires). 2. The DYNAMIC section: it includes the needed steps to compute the derivative of the states. The derivative subsection, within the DYNAMIC block allows the outputs to be calculated before they are used. This provides an automatic ordering of the algorithm statements in the derivative subsection, so that functional role of each statement and the corresponding variable can be better observed. Both the acceleration statements and the kinematic computations for the suspension derivatives are performed outside the derivative block. All the other calculations including the tire friction forces and moments, the contact patch velocities, the suspension deflections and forces, the wheel's orientation and normal force, and the tire rolling radius are performed in the derivative section. 5.5 Vehicle Handling Maneuvers The vehicle dynamic simulations developed in this investigation are capable of executing the following maneuvers. 1. Straight Line Braking Maneuver (SLBM): this allows the ^ investigation of vehicle behavior and performance during braking. The deceleration rate can be varied by changing the peak braking torque. The rates of change of the input torques change when the forward velocity decreases to fractional predetermined ranges of magnitude. This permits a smooth response for the vehicle, and prevents any unpredictable excitation from occurring. An example of a simulated braking torque is shown in Figure 5.7. No steering angle is transmitted to the wheels during this maneuver. 2. Braking in a Turning Maneuver (BTM): In this phase the braking may be complete bringing the vehicle to a stop or just to reduce the vehicle forward velocity to a steady state value. The turning of the vehicle is obtained by a steer angle input whose magnitude determines the curvature radius of the rotation. 137 SIMULATED BRAKING TORQUE o CO CÛ cr> ?§. CM' a o o o d m 2.00 4.00 6.00 8.00 10.0 0.00 T (SEC) Figure 5.7 Applied Torque in Braking Maneuver 138 3. Turning Maneuver (TM): this differs from the previous maneuver by keeping the forward velocity of the vehicle constant. The extent of turning, measured by the radius of curvature is controllable by the peak value of the steer angle input simulated by a triangular function (Figure 5.8.a). The steer angle generates a lateral force which produces a steady state lateral velocity and the change of direction. 4. Change of Lane Maneuver (CLM) : Thi'S' maneuver is identical to the TM except the steer angle input function which has to allow the vehicle just to change the lane of forward motion. This is obtained by a triangular harmonic function during a specified period of time. Such function will cause the lateral velocity to increase to a peak value than a steady state zero magnitude allowing the vehicle to be displaced lateral to a steady state position. The input function is shown in Figure 5.8.b. Other maneuvers can be developed and introduced to the program. This may include trapezoidal or sinusoidal steer or drastic steer and brake to investigate the vehicle roll-over tendency. 139 o o STEER INPUT FOR CLM & TM o to Li- •. Li en o CO o ’ o ' 0.00 1.00 2.00 3.00 5.0C Figure 5.8 Steer Input in TM and CLM Maneuvers 140 CHAPTER VI SIMULATION RESULTS AND DISCUSSION 6.1 Introduction In this chapter some of the simulation results are presented and a qualitative critique of ..the vehicle model with kinematic suspension is included. Results from both models are compared, as well as with other results obtained from available simulations. 6.2 Vehicle Suspension Derivatives The suspension derivatives measure the coupling between the suspension motion and the fundamental states of the vehicle. They are determined by the kinematic structure linking the suspension units to the vehicle body. They are computed using the formulations developed in Chapter 4. The time history of the simulated suspension derivatives are shown in Figures 6.1 through 6.16 where the suspension derivatives (SD) are designated by four letters and one character for the suspension number. The third and fourth letters correspond the suspension movement (i.e. Y for scrub) and the vehicle reference mode (i.e. P for roll) respectively. For instance SDYPl corresponds to the suspension derivatives of the suspension scrub with respect to the roll mode for the front right suspension. It can be observed that these functional derivatives remain practically constant during the simulation time. Therefore it may be assumed that the derivatives are constant and they just depend on the kinematic structure of the suspension linkages. Hence they can be considered as vehicle characteristic parameters and introduced in the INITIAL section of the ACSL program. This allows a significant reduction in the time of computation during each run. For the kinematic Long/Short Arm suspension shown in Figures 4.7 (2-Dimensional) and 4.8 (3-Dimensional) the steady state suspension derivatives are shown in Tables 6.1 and 6.2 respectively. For the case of 2-Dimensional suspension, the influence of pitch and yaw rotations in the suspension movements is considered through their effect on the vertical displacement of the point at which the suspension is linked to the vehicle body. 141 o FORE&AFT/ROLL DERIVATIVES o 8 Q_ X o o CO o s ' 0.00 1.00 2.00 3.00 4.00 5.00 T (SEC) FIGURE 6.1 Suspension Derivative of Fore and Aft Movement of the Tire with Repsect to Vehicle Body Roll 142 o o FORE&AFT/PITCH DERIVATIVES o CD OCD o 1 7 X Q CO O . X'f eno S O' 0.00 1.00 2.00 3.00 4.00 5.00 T (SEC) FIGURE 6.2 Suspension Derivative of Fore and Aft Movement of the Tire with Respect to Vehicle Body Pitch Rotation 143 o o FORE&AFT/BOUNCE DERIVATIVE S 5^ X Q .CO s ? en o CO 0 1 o 0.00 1.00 2.00 3.00 4.00 5.00 T (SEC) FIGURE 6.3 Suspension Derivative of Fore and Aft Movement of the Tire with Respect to Vehicle Body Bounce 144 o o FORE&AFT/YAW DERIVATIVE o (\jo a en isTo Q ' e n (TJo. I O.rr- 0.00 1.00 00 3.00 4.00 5.00 (SEC) FIGURE 6.4 Suspension Derivative of Fore and Aft Movement of the Tire with Respect to Vehicle Yaw Rotation 145 STEER/PITCH DERIVATIVE to o ’ o o o CO too o ' 0.00 1.00 2.00 3.00 4.00 5.00 I (SEC) FIGURE 6.5 Suspension Derivative of Tire Steer Rotation with Respect to Vehicle Pitch Rotation l46 O o •STEER/ROLL DERIVATIVE O CD R Û_ CO Q, CD ? O CO 0 1 s 0.00 1.00 2 .00 3.00 4.00 5.00 T (SEC: FIGURE 6.6 Suspension Derivative of Tire Steer Rotation with Respect to Vehicle Body Roll 147 o o STEER/BOUNCE DERIVATIVE o CO s rvi en CSJo g 0.00 0.80 1.60 2. 40 3.20 4.00 FIGURE 6.7 Suspension Derivative of Tire Steer Rotation with Respect to Vehicle Body Bounce 148 S o O STEER/YAW DERIVATIVE O O 8 O O CD (Tien CD men œ en en 8 0.00 0.80 1.60 2.40 3.20 4,00 FIGURE 6.8 Suspension Derivative of Tire Steer Rotation with Respect to Vehicle Yaw Rotation 149 o o CAMBER/YAW DERIVATIVE o U 3 ' R o_ wr^#Ajvwvr-wru*f— v O* I O tD ? 8 " --- 0.00 1.00 00 3.00 4.00 5.00 (SEC) FIGURE 6.9 Suspension Derivative of Tire Camber with Repsect to Vehicle Yaw Rotation 150 o o CAMBER/PITCH DERIVATIVE CD C\J O CL. CD O ' 0.00 1.00 2.00 3.00 4.00 5.00 T (SEC) Suspsnsion D©irivativs of Tirs Cambsir with f i g u r e 6.10 Respect to Vehicle Pitch Rotation 151 o CAMBER/ROLL DERIVATIVE cJ 00 o U 3 Q_ Q_ o C\J s 0.00 1.00 2.00 3.00 4.00 5.00 T (SEC) FIGURE 6.11 Suspension Derivative of Tire Camber with Respect to Vehicle Body Roll 152 o o CAMBER/BOUNCE DERIVATIVE o CO s IVJ û _ r\)o S 0.00 0.80 1.60 2.40 3.20 4.00 FIGURE 6.12 Suspension Derivative of Tire Camber With Respect to Vehicle Body Bounce 153 o o SCRUB/BOUNCE DERIVATIVE o CO o CO o C\J o 0*1--- 0.00 1.00 2.00 3.00 4^0 5.00 T (SEC) FIGURE 6.13 Suspension Derivative of Tire Scrub with Respect to Vehicle Body Bounce 154 o o SCRUB/YfiW DERIVATIVE o CO g rsio Ë:. CO® (\lo ! $ 0.00 1.00 2.00 3.00 4.00 5.00 T (SEC) FIGURE 6.14 Suspension Derivative of Tire Scrub with Respect to Vehicle Yaw Rotation 155 o o SCRUB/PITCH DERIVATIVE • o s C 3 ° 8 cJ O ro S ' 0.00 0.80 1.60 2.40 3.20 4.00 FIGURE 6.15 Suspension Derivative of Tire Scrub with Repsect to Vehicle Pitch Rotation 156 o o SCRUB/ROLL DERIVATIVES o •H (\i2 C L. I > - O en a _ o en RI o o . ^ 0.00 1.00 2.00 3.00 4.00 5.00 T (SEC) FIGURE 6.16 Suspension Derivative of Tire Scrub With Respect to Vehicle Body Roll TABLE 6.1 (Units: v.z.l- inch ; & - radian) Suspension # 1 2 3 4 3 y/ 3(|, -23.9428 -23.9403 -23.9403 -23.9428 37/ 3 2 0.02013 - 0.0202 - 0.0202 0.02013 34/3 2 0.0014 - 0.0014 - 0.0014 0.0014 3 4»/ 3 4 1.0429 1.0435 1.0435 1.0429 31/ 34» -17.54 17.56 17.56 -17.5 31/3 z - 0.55073 - 0.55133 - 0.55133 - 0.55043 158 TABLE 6.2 f Units: X .V .2 - inch ; (j), 0 , i{)- radian) 1 2 3 4 3S/9 2 0.00102 - 0.00102 - 0.00102 0.00102 3s/3(|) 0.032 0.032 0.032 0.032 3s/30 - 0.162 0.162 - 0.095 0.095 ds/dljJ 0.999 0.999 0.999 0,999 3 y/e z 0.0441 - 0.0441 0.01495 - 0.01495 3y/3 3y/3 0 - 2.98485 2.9836 1.18556 - 1.18565 3y/3* 63.098 63.098 -62.922 -62.922 3X/3 2 - 0.0189 - 0.0189 - 0.0184 - 0.0184 3X/3& - 0.5994 0.5996 0.58514 - 0.5848 3X/30 7.082 7.084 4.32 4.32 3X/3^ -31.72 31.72 31.72 -31.72 3(|)/ 3 2 0.0057 - 0.0057 0.0017 - 0.0017 3 (j)/3 3^/30 - 0.35 0.35 0.094 - 0.094 3 11.6 Ib/in/sec^ 1800 mi 0.315 %wr 1800 = 0.315 "^2 kgf 160 m 3 = 0.5015 160 X r 0.5015 iti4 Cwf 0.0 10.8 in Cwr 0.0 = 10.8 58 2r Ccf A 63 = 58 Ccr B 63 14.5 = 6800 lb in/sec^ 14.5 Rwr 42551 28.8 "y ^wf 43465 *^wr 28.8 : 1790 ^xz Tf 63.5 ^r : 63.5 FIGURE 6.17 Vehicle Parameters and Data 160 Chevrolet Brookwood (Static Displacement = 3.0) Displacement Camber Steering 0 0 0 1 0.85 -0.24 2 1.68 -0.53 3 2.18 -0.73 4 2.43 -0.89 5 2.47 - 1.01 6 2.29 - 1.10 7 1.96 -1.17 FIGURE 6.18 Camber and Steering Data 161 6.3 Simulation Maneuver Response All the simulation maneuvers are generated with vehicle data characteristic of a 1971 Chevrolet Brookwood. A listing of the vehicle parameters are shown in Figure 6.17. The camber and steering data for the interpolation of wheel orientation in function of the suspension vertical displacement are shown in Figure 6.18. A large number of comparison runs were made for several vehicle maneuvers. The first set involves a straight line braking maneuver. The initial speed was taken to be 48 mpli;* For the braking maneuver, the simulation results using both models are shown in Figures 6.19 through 6.26. In Figure 6.19 the forward velocity U is plotted versus time and an excellent agreement can be seen. Figure 6.20 shows the simulated longitudinal slip; while. Figure 6.21 shows the longitudinal force F . The vehicle model with kinematic suspension resulted In a lower steady state magnitude of longitudinal slip. The vehicle trajectory (i.e., Y versus X) is given by Figure 6.22. Since the maneuver in question is a straight line braking, ideally y and v should be zero. However, the deviation in y and that of the lateral slip and the lateral velocity V, shown in Figure 6.23 and 6.24, indicates that the vehicle is also excited in the lateral direction during a braking maneuver. This excitation is due to the complex influence of the vehicle body motion on the suspension orientation. The pitch response is shown in Figure 6.25 where excellent agreement of the two models is again observed. The suspension deflection obtained by both models is shown in Figure 3.26. The oscillations observed in the model with dynamic suspension is due to the dynamic behavior of the unsprung masses. The second simulated maneuver involves a change of line, at a constant forward speed of 700 in/sec (approximately 48 mph)._ The change of line is obtained by a steer input function as shown in Figure 5.8.b. The simulation results are shown in Figures 6.27 through 6.32. The lateral slip and lateral velocity are given by Figures 6.27 and 6.28 respectively. The model response is practically identical. Similar observation can also be made for the model response in yaw, pitch and roll velocities (Figures 6.29, 6.30 and 6.31). The suspension deflection in this maneuver is shown in Figure 6.32. The deflection magnitude in a CLM maneuver is much lower than that of braking maneuver. A combined maneuver of change of line and braking characterize a parking maneuver. The input to the simulation are a braking torque and a steer angle. The vehicle response to this maneuver is shown in Figures 6.33 to 6.38. 162 o o Vehicle Model with Dynamic Suspension Vehicle Model with Kinematic Suspension CO O \ o ' o tv- o o o 0.00 2.00 4.00 6.00 8.00 T (SEC) 10.0 FIGURE 6.19 Forward Velocity in a Braking Maneuver 163 o CM CO O CM t S ü_ Vehiols Model with Dynamic Suspension CO o Vehicle Model with CD Kinematic Suspension O' O I 0 6 0.00 2.00 4.00 6.00 8.00 T (SEC) 10.0 FIGURE 6.20 Longitudinal Slip in a Braking Maneuver 164 (M en X X U.O cû O CD 0.00 0.80 2.40 3.20 4.00 FIGURE 6.21 Longitudinal Force in a Braking Maneuver 165 o o CD ta Vehicle Model with ~ Dynamic Suspension Vehicle Model with o Kinematic Suspension CD o X CDo O' s CD 2.00 - 1.20 -0.40 0.40 1.20 2.00 FIGURE 6.22 Vehicle Trajectory in a Braking Maneuver 166 o o CO O CJ Vehicle Model with Dynamic Suspension o CO o' 1.00 2.00 3.00 4.00 0.00 T (SEC) 5.00 FIGURE 6.23 Lateral Slip in a Braking Maneuver 167 _ Vehicle Model with ~ Dynamic suspension Vehicle Model with o Kinematic Suspension o csi CM O O O O COT 2.00 4.00 6.00 8.00 JO.C ' 0.00 T (SEC) FIGURE 6.24 Lateral Velocity in a Braking Maneuver 168 Vehicle Model with Dynamic Suspension _L__ Vehicle Model with oCO Kinematic Suspension o oCO o 0.00 2.00 4.00 6.00 8.00 T (SEC) lO.O FIGURE 6.25 Pitch Velocity Response in a Braking Maneuver 169 o o R Vehicle Model with ~ Dynamic Suspension Vehicle Model with o ~ Kinematic Suspension CM o CO CO o o 0.00 0.80 1.60 2.40 3.20 T (SEC) 4.00 FIGURE 6.26 Suspension Deflection in a Braking Maneuver 170 o o Vehicle Model with Dynamic Suspension Vehicle Model with Kinematic Suspension U 7 li ar œo s 1.60 2.40 3.20 4.00 0.00 0.80 T (SEC) FIGURE 6.27 Simulated Lateral Slip in a CLM Maneuver 171 o o CD Vehicle Model with Dynamic suspension Vehicle Model with Kinematic Suspension O rj s 0.00 0.80 1.60 T (SEC) 2.40 3.20 4.00 FIGURE 6.28 Lateral Velocity in a CLM Maneuver 172 o o CD Vehicle Model with Dynamic Suspension Vehicle Model with o CD Kinematic Suspension CO CO CO o ' 0.00 1.00 2.00 3.00 4.00 T (SEC) 5.00 FIGURE 6.29 Yaw Response in a CLM Maneuver 173 o o Vehicle Model with Dynamic Suspension Vehicle Model with CM Kinematic suspension o o (M 0.00 0.80 1.60 2.40 3.20 4.00 FIGURE 6.30 Pitch Velocity in CLM Maneuver 174 o o Vehicle Model with " Dynamic Suspension Vehicle Model with o Kinematic suspension o C\J o CM O f S CL, g o cdI-- ' 0.00 0.80 1.60 2. 40 4.00 FIGURE 6.31 Roll Response in CLM Maneuver 175 Vehicle Model with Dynamic Suspension Vehicle Model with o Kinematic Suspension (M o s , o Q_ O o o 1.00 2.00 3.00 T (SEC) 4.00 5.00 FIGURE 6.32 Suspension Deflection in CLM Maneuver 176 o LD vehicle Model with Dynamic Suspension Vehicle Model with Kinematic Suspension CO o o o CO s in 0.00 1.00 2.00 3.00 4.00 5.00 FIGURE 6.33 Lateral Velocity in CLM and Braking 177 oa Vehicle Model with Dynamic Suspension Vehicle Model with o Kinematic Suspension CO o s o tno o o C\J o' ao o 0.00 1.00 3.00 5.00'1.00 f T s e c ) FIGURE 6.34 Forward Velocity in a CLM and Braking Maneuver 178 o o Vehicle Model with Dynamic Suspension Vehicle Model with o CD Kinematic Suspension o (NJ Ü- 0.00 1.00 2.00 3.00 4.00 T (SEC) 5.00 FIGURE 6.35 Lateral Slip in a CLM and Braking Maneuver 179 o o o CM ro" o CM ü_ Vehicle Model with Dynamic Suspension CD Vehicle Model with o Kinematic Suspension CO o o 0.00 1.00 2.00 3.00 4.00 T (SEC) 5.00 FIGURE 6.36 Longitudinal Slip in a CLM and Braking Maneuver 180 o Vehicle Model with Dynamic Suspension ____ Vehicle Model with CO Kinematic Suspension o o a oCD o ' 0.00 2.00 4.00 6.00 8.00 10.0 FIGURE 6.37 Pitch Response in a CLM and Braking Maneuver 181 o o Vehicle Model with Dynamic Suspension _ Vehicle Model with o Kinematic Suspension m o CM o CD o CO 0.00 0.80 2.10 3.20 4.00 FIGURE 6.38 Vehicle Trajectory in a CLM and Braking Maneuver 182 The vehicle lateral and forward velocities are shown in Figures 6.33 and 6.34, while the simulated corresponding slip is shown in Figures 6.35 and 6.36. The lateral velocity in the dynamic suspension model observes a higher overshoot in the second half of the period before reaching the zero steady state magnitude. The pitch velocity response is shown in Figure 6.37. The second overshoot after the steady state values correspond to drastic change in the torque input as the forward velocity approaching the braking point. The vehicle trajectory i,s shown in Figure 6.38. The last set of comparison maneuver involved steering without braking. This turning maneuver is generated by a half cycle steer angle, as shown in Figure 5.8.a. Figures 6.39 to 6.44 show the simulation results for this turning maneuver. The forward velocity of the vehicle is kept at 48 mph, while a positive half cycle of steering input excites the vehicle. The peak value of the steer input is 0.01 radians. The vehicle response in lateral slip is shown in Figure 6.39, while the lateral velocity output is shown in Figure 6.40. The vehicle response in roll, pitch, and yaw rotations is shown in Figures 6.41, 6.42 and 6.43 respectively. The vehicle trajectory is shown in Figure 6.44. 6.4 Simulation Times and Costs One of the factors used to compare the two investigated models is the CPU time required to run the simulation. When the maximum simulation time is set at 5 seconds. The CPU time use by both simulations for two different maneuvers is shown in Table 6.3. These results show that the substitution of the dynamic suspension by a kinematic suspension results in a cost saving of about 54.57% with no significant loss in accuracy. This result is expected since the dynamic suspension increases the order of the vehicle system which causes an increase of time consumed by the integration routines. The minimum integration step size required for the stability also differs for the two simulations. It was found to be 0.009 seconds for the model with dynamic suspension and 0.02 seconds for the vehicle model with kinematic suspension. This result is expected since the wheel frequency is higher then the vehicle body frequencies. This requires a smaller time step for numerical stability. 183 o o Vehicle Model with Dynamic Suspension Vehicle Model with Kinematic suspension U3 o ’ o U_ to o o ' 0.00 1.00 2.00 3.00 T (SEC) 4.00 5.00 FIGURE 6.39 Lateral slip in a Turning Maneuver 184 o o ■S o Vehicle Model with Dynamic Suspension Vehicle Model with Kinematic Suspension s 0.00 1.00 2.00 3.00 4.00 5.00 T (SEC) FIGURE 6.4 0 Lateral Velocity in a Turning Maneuver 185 o o Vehicle Model with Dynamic Suspension Vehicle Model with Kinematic Suspension CM o o ' 0.00 2.00 4.00 6.00 8.00 T (SEC) 10. 0 ' FIGURE 6.41 Roll Response in a Turning Maneuver 186 <\j Vehicle Model with Dynamic Suspension Vehicle Model with Kinematic Suspension C\J o o ' 0.00 2.00 4.00 6.00 T (SEC) 8.00 10.0 FIGURE 6.42 Pitch Velocity Response in a Turning Maneuver 187 Q O _ Vehicle Model with Dynamic Suspension _ Vehicle Model with Kinematic Suspension CD (M (\i o o o o' o ' 0.00 1.00 2.00 3.00 4.00 T (SEC) 5.00 FIGURE 6.43 Yaw Response in a Turning Maneuver 188 o Vehicle Model with Dynamic suspension Vehicle Model with o Kinematic Suspension C\J (D CO 0.00 0.80 1.60 2.40 .20 4.00 FIGURE .6.44 Vehicle Trajectory in a Turning Maneuver 189 TABLE 6.3 Simulation CPU Times Braking Maneuver CLM maneuver Lumped Vehicle Model 00:23.38 00:23.39 Model with Kinematic 00:10.62 00:12.38 Suspension 190 6.5 Introduction of developed Simulation to the SPMD Software Package Available simulations do not take full advantage of the suspension data measured by the SPMD. For instance, the SPMD measures the kinematically induced motion of wheels in longitudinal, lateral and vertical displacement of the suspension, as well as the camber and steer rotation of the wheels. The Highway Vehicle Simulation,.Model, the Hibrid Computer Vehicle Handling Program and the-Digital Simulation Fully Comprehensive do not utilize all these generated data. The unavailability of such data encouraged the modeling of the suspension units with independent degrees of freedom and inertial coupling to the vehicle body. Since the SPMD was designed to measure the suspension characteristics, the suspension derivatives as defined in this investigation can be computed using statistical analysis of the generated data. By feeding these vehicle suspension characteristics to the developed model, the vehicle ride and handling performance can be investigated just after the measurement phase is completed. The introduction of the vehicle model with kinematic suspension to the SPMD software facilities makes a significant change and improvement of the SPMD machine by transforming it from a measuring device to a unit which combines vehicle parameter measurements with ride and handling performance prediction. The determination of suspension derivatives is based on the generated results for kinematic and compliance tests performed for the suspension. The suspension displacements and orientations obtained for a 1983 Toyota Camry front suspension are shown in Figures 6.45 through 6.52. Figures 6.45 through 6.48 show plots of the outputs of the roll test. Each of these plots shows the measured data values of one of the dependent variables plotted as a function of the independent variable, roll angle. As should be expected, the camber angle varies rapidly with the roll angle. The relationship is essentially linear, with the values of the camber angles at zero roll angle being the initial settings of the wheels. Similar observations can be made for the other suspension relationships which justifies the constant suspension derivative assumption. For bounce test, the suspension response measured by the SPMD is shown in Figures 6.49 through 6.52. The suspension derivative can be directly determined from the above plot. A summary of these measured derivatives are shown in Table 6.4. a LD + LEFT X RIGHT >—'cni LT) . (U CVJ ■D ÜJ -J o CD . z o Œ ÛC ÜJ m GO • Œ I (_) ______ i n . I ^ . 0 - 2.0 0.0 2.0 4.0 ROLL ANGLE (deg) • FIGURE 6.45 Wheel Camber Angle as a Function of Roll Angle [111 VO H* in + LEFT X RIGHT in 2 ÜJ CD z CE H44+4++++HH-t- ÛC in LU LU I— en in I -4.0 0.0 2.0 4.0 ROLL ANGLE (deg) FIGURE 6.46 Variation of Steer Angle With the Roll Angle [11] H VD N> o 4- LEFT in o CO 3 O cc .. u o en cc LD 1— 4 • 1 III 11 un II II 1 )|i III O -5.0 -2.5 0.0 2.5 5.0 ROLL ANGLE (deg) FIGURE 6.47 Variation of Tire Scrub With Roll Angle [11] V £ > W XX%/XX RIGHT LEFT -4.0 • 2.0 0.0 2.0 ROLL ANGLE (deg) FIGURE 6.48 The Tire Normal Force as a Function of Roll AMgle [111 VD 45k ______1______1______ + LEFT ) X RIGHT ». aj ) u •- LU _j CD Z ------■ Œ OC LU CD Z Œ ------,------,------1 1 / 0.075 BOUNCE POSITION (m) FIGURE 6.49 Variation of Camber Angle With the Bounce Position [11] VO in R GHT -0.5 -0.U5 -0.4 -0.35 -0.3 BOUNCE POSITION (m) FIGURE 6.50 Tire Scrub as a Function of Bounce Position [il] KO a\ o m (S) CM TJ ÜJ Œ ca LU G en--- '-0.5 -0.45 -0.4 -0.35 -0.3 BOUNCE POSITION (m) FIGURE 6.51 Variation of Steer Angle With Bounce Position [11] VO o o a. en o 4- LEFT o X RIGHT m. r- 5 g O- Ü J ^ ë g o tn X* ü_ 3* o Œ o* i g IT)_ -0.075 -0.05 -0.025 ÔTo o'. 025 Ôu 05 .= 0.075 BOUNCE POSITION . " FIGURE 6.52 Variation of the Tire Normal Force with Bounce Motion [11] H* VO 00 199 TABLE 6.4 Suspension Derivatives Measured bv the SPMD fl983 Toyota Camrv; Units; v.z - inch, è - radian ) Roll Test ( 3(j) / 3(}i ) 0.903 ( 3 s / 3(j> ) 0.0948 ( 3 y / 3({) ) -36.207 Bounce Test ( 3(j)/ 3 z) 0.0062 ( 3 s/ S z ) 0.0022 { 3 y / 3 z) 0.0192 CHAPTER VII CONCLUSIONS AND RECOMMENDATIONS 7.1 Conclusions This investigation focused on four main areas in the domain of vehicle dynamics modeling and simulation. First the major functional block components of the"Vehicle system are analyzed. Special emphasis is put on the discussion and evaluation of tire models. The numerical stability question in vehicle dynamics simulations is also studied using the concepts of open and closed loop integration. Conditions for numerical stability are derived and recommendations for the simulation formulation are proposed to prevent numerical instability during severe maneuvers. The third topic is the application of suspension derivatives in vehicle modeling by modeling the suspension system kinematically rather than dynamically. A vehicle model with kinematic suspension is developed using the suspension derivatives. The corresponding computer simulation is also developed and the simulation results are compared to those obtained from the simulation of a lumped parameter vehicle model. Finally the relevance of the results obtained from the developed model and simulation to the Suspension Parameter Measuring Device (SPMD) is discussed. The most noticeable achievements in this investigation are discussed below. Vehicle Functional Block Modeling The vehicle functional blocks have been developed and improved with time since the initiation of vehicle modeling. However, there have been inconsistencies in formulating and predicting the tire friction behavior due to the complexity of the tire friction phenomenon. 'The most current tire models are investigated using tire data. The evaluation of these models is based on the solvability of model parameters, the accuracy of the models and the CPU time used by the models. It was proposed in this investigation that a simple polynomial model may be used to describe, fairly accurately, the tire behavior. The polynomial model, whose parameters are determined from the dynamic data of rolling tires, describes the tire behavior better than an analytical model whose constants are determined from static modeling of tire structure. The advantage of this model over other empirical models is its simplicity and the solvability of its parameters using 200 201 available polynomial fitting routines. Some of the available models, which are complex, require an initial close guess of the parameters as a requirement for model convergence. In many cases convergence is unachievable. Moreover, the polynomial model requires less CPU time when compared to the other empirical models. The proposed polynomial model fits, fairly closely, the measured data of tire friction forces and moments. The model can be further improved by increasing the order of the polynomial. Stability Question in Vehicle simulation'^ It is suggested in this investigation that the instability problem commonly appearing in several vehicle dynamics simulations may not necessarily be due to the numerical method used in the simulations. It was found that the application of the Euler equations in vehicle dynamics problems can have a negative effect on the computational performance of the simulation, particularly under severe maneuver conditions as the vehicle loses contact with the ground. Under these conditions, the Euler equations preserve the order of the modes although some of these modes loose dependence on their zeroth order derivatives. This increases the risk of error accumulation and eventually causes instability. This problem of instability is discussed by introducing the concept of open/closed loop formulation of the simulation. The lift off condition transforms the formulation of the problem from a closed loop to an open loop formulation. The problem of numerical instability can be avoided by preserving the closed loop formulation of the vehicle dynamics simulation under all conditions. Therefore, The dependence of the tire forces and moments on the position and orientation of the vehicle necessitates that the pitch, bounce, and roll be used as second order modes connected to the road by the suspension and side slip and the yaw rotation as a first order mode with typical tire force data, which is dependent on steering and vector velocity of the contact patch on the road. When it is desirable for the vehicle to leave the ground, the flight portion should be simulated using Euler equations resuming the second order modes as the car returns to ground. Vehicle Models with Kinematic Versus Dynamic Suspensions In the traditional approach of modeling the vehicle system, the suspensions are assumed to have their independent degrees of freedom dynamically linked to the vehicle body motion. This not only increases the order of the vehicle 202 model and the subsequent numerical difficulties, but also undermines the effect of body motion on the suspension position and orientation in and about the x and y axes directions. It is proposed that the introduction of the suspension derivative concept in vehicle modeling not only simplifies the vehicle model but also represents a more realistic description of the vehicle suspension body coupling. The dynamic coupling, due to the unsprung masses, is replaced by a kinematic coupling, where the suspension is assumed to be kinematically linked to the vehicle body through the suspension linkages. The suspension derivatives which measure the coupling between the vehicle body motion and suspension position and orientation are then computed algebraically as functions of the linkage node positions. The simulated values of these derivatives are found to be practically constant during the simulation time. In this research two simulation models were developed. The first model is based on the lumped parameter approach where the suspension is modeled with its independent degree of freedom; whereas, in the second model the suspension is assumed kinematic. This second model allows not only a reduction in the order of the vehicle system but also a significant reduction in the CPU time used by the simulation. On the other hand, the vehicle response obtained by both models, with dynamic and kinematic suspension respectively, was found to vary slightly in the fundamental states of the system (i.e. forward and lateral velocity). However, the differences become more noticeable in the intermediate variables such as the secondary states (i.e. roll, pitch, ... ) and the suspension position and orientation. Relevance of this Investigation to the SPMD The introduction of suspension derivatives to the vehicle model and simulation necessitates the availability of these derivatives which are characteristics of the suspension system. A direct measurement of these derivatives always yield a much better estimate of their values than their determination by analytic calculation using a model and dimensions for the kinematic structure. The measurement of these derivatives is possible by using the Suspension Parameter Measuring Device. The SPMD allows the measurement of all the characteristics of the suspensions of production vehicles. This includes the suspension force parameters and the suspension derivatives. The computer library of the SPMD possesses all the software necessary for the tests required for data measurement and collection. 203 Additional software may be introduced to calculate the force parameters and suspension derivatives as outlined in sections 2.4.4 and 6.5. The developed simulation for the vehicle model with kinematic suspension can then be introduced to the SPMD computer library, making use of the computed suspension characteristics to simulate the vehicle handling and ride performance. This permits the SPMD to widen its capability from just a measuring device, to a system that produces vehicle ride and handling performance predictions. This significantly improves the capability of the SPMD and renders it a more comprehensive device for the testing and analysis of ground vehicles. 7.2 Recommendations The logical extension of this investigation would be to introduce model and simulation developed into the SPMD. Using the data set measured by the SPMD, this simulation would be used to predict vehicle performance in a variety of maneuvers. These predictions could be compared with the data collected during full scale field testing of actual vehicles. APPENDIX A TIRE DATA 204 Tire Friction Experimenul Data (aller Segel (1?) (0170A Oiat Ply Tire. Dry Aiphaîf. Pressure = 28 psi. N * 800 Ibs., V * 2 0 mph F . (Ibs.) s 0. .025 .05 .075 . 1 .135 .15 .173 J .235 Q p" 0. 312. 497. C33. 721. 778. 613. 831. 8«4. 857. 8® 0. 174. 310 4 1 3.' 505. 576. 633 670. 703. 734. 12® 0. 110. 30G. 284. 374. 4 4 2 . SOI. 541. 587. 630 16® 0. 63. 134. 305. 275. 332. 385. 4 2 9 . 4 7 5 . 5 1 4 . 7 0 9 . 8 0 9 . 603. 67 0 . 6 9 7 . 771. 700. ,776. 740 59G 7 4 3 * caira;\o(4StO to O en Fy (ibi.l Q 02s OS .075 .1 * :s .15 .175 2 225 2* 279 271 ' . 2 6 2 ." 253 • • 244. 2 3 1 ." 2 1 7 ." 2 0 4 . " 190 181 " 4" 15*. 533 " 515 •• 496 •• 478. 4 5 2 ." 426 " 400 " 374 356 " s" e i9 613. 765 758. 730. 703. 670 629. 537. 556 12* 032. SO) 688. 664. 839. 813. 783. 752. 718. 683. 16* 93S. 926 917. 603. 877. 859. 835. 809. 779. 750. 25 375 .3 .4 .5 .6 .7 .8 .9 1 0 2* 1 7 2 "" 163 154. 127. 114 " 100. 9 2 .'" 83. 7 3 . " 0 3 " 4* 3 3 8 ." 319." 301. 249. 2 1 6 . - 182. 167 "" 151, 133 " 1 2 5 / 6* 514. 4BB. 460. 374, 309. 271. 231. 197. 167." 1 4 5 / 1:' 644. 609. 572. 488. 409. 354, 296. 263. 211." 1 9 6 / 16* 725. 6 9 1 667. 502. 499, 440. . 3BS. 342. 392." 272." • IttUpOlltK) •• inu>pcu:rJ »o O a> 207 LIST OF REFERENCES [1] Kortum, W., "Introduction of System Dynamics to Ground Vehicles", 3rd Seminar on Advanced Vehicle System Dynamics, Supplement to Vehicle System Dynamics, Vol. 16,1986. [2] Wallrap, O., " MEDINA, an Interactive Analysis and Design Program for Flexible Multibody Systems ", 3rd Seminar on Advanced Vehicle System Dynamics, Volume 16, 1986. [3] Orlandea, N.V., " ADAMS, Theory and Application ", 3rd Seminar on advanced Vehicle System Dynamics on Roads and oh Tracks, Supplement of Vehicle System Dynamics, Vol. 16, 1986. [4] Kortum, W. and Schiehlen, W., " General Purpose Vehicle System Dynamics Software based on Multibody Formalisms ", Vehicle System Dynamics, 14, pp. 229-236 (1986). 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