UNIVERSITY OF CINCINNATI
DATE: December 6, 2002
I, César A. Grau , hereby submit this as part of the requirements for the degree of: Doctor of Philosophy in: Mechanical Engineering It is entitled: A Parametric Study of the Lateral Dynamics of a Nonlinear Four-wheel Road-Vehicle model
Approved by: Ronald L. Huston Christopher McCord David Richardson David Thompson
A PARAMETRIC STUDY OF THE LATERAL DYNAMICS OF A NONLINEAR FOUR-WHEEL ROAD-VEHICLE MODEL
A dissertation submitted to
Division of research and Advanced Studies of the University of Cincinnati
in partial fulfillment of the requirements for the degree of
DOCTORATE OF PHILOSOPHY (Ph.D)
In the Department of Mechanical, Industrial, and Nuclear Engineering of the College of Engineering
2002
By
César A. Grau
M.S. University of Cincinnati 1998 B.S. Universidad Simón Bolívar 1995
Committee Chair: Ronald L. Huston, Ph.D., P.E.
ABSTRACT
This study presents the influence of basic vehicle parameters on the lateral dynamics of a four-wheel road-vehicle model. To encompass the majority of the vehicles currently on public roads in the US and other countries, the road-vehicle model in this study uses three different nominal parameter sets corresponding to small, medium and large (S.U.V.) vehicles. Parameter values from each set are changed individually from their nominal values through a finite range, and the effect of these changes on the dynamics of the model are observed. A step-steer and total vehicle velocity of constant magnitude are used as inputs for the model. An analysis where several vehicle parameters are changed simultaneously is used to study parameter interaction. Response characteristics, both transient and steady-state, are chosen as indicators of the lateral dynamics.
The governing differential equations of the road-vehicle model are solved using a fourth- order Runge-Kutta integration programmed in MATLAB - a mathematical programming software. The road-vehicle model in this study is a rigid body approximation that considers the effects of lateral weight transfer on the normal load of the tires. The rigid- body approximation allows this study to have broad scope and applications because the vehicle parameters considered in this study are basic and general to any road-vehicle. The model used in this study does not include characteristics that have to be particular to a specific vehicle, such as suspension system architecture, axle kinematics, compliance, and suspension component stiffness. A piece-wise tire approximation represents the nonlinear lateral force produced by the tires. This tire model is compared with the Magic
I Formula nonlinear tire model. Simple and multiple regression analysis techniques are used in this study to quantify the observations derived from the results.
The results of this study show that the longitudinal mass center position is the most influential vehicle parameter and that its influence on the response characteristics chosen for this study is almost always exponential in nature. Further results suggest that the other vehicle parameters considered in this study influence the response characteristics steady state lateral velocity, steady state angular velocity, and angular velocity overshoot in a way that is approximately linear. The other response characteristics appear to be influenced in a nonlinear way by the simultaneous variation of the vehicle parameters.
II
III ACKNOWLEDGMENTS
I would like to thank the members of my committee, Dr. McCord, Dr. Richardson, Dr.
Thompson, and my advisor, Dr. Huston, for their valuable input during this dissertation. I
especially want to thank Dr. Huston for his kind support and friendship during all these
years. I feel fortunate to have him as an advisor and friend.
I would also like to thank the Society of Automotive Engineers for allowing me to attend
their Professional Development Seminars as a student member. This generous benefit
allowed me to attend all the seminars related to vehicle dynamics and learn directly from
some of the most renowned expert in the field.
I would also like to thank my friends and family for their encouragement and
understanding, especially when times were not easy.
Finally, I want to dedicate this dissertation to my parents, whose loving support has been my strongest inspiration.
IV TABLE OF CONTENTS
Page
ABSTRACT ...... I
ACKNOWLEDGMENTS...... IV
TABLE OF CONTENTS...... 1
LIST OF FIGURES AND TABLES ...... 3
LIST OF SYMBOLS...... 5
1. INTRODUCTION...... 6
1.1 Background and motivation...... 6 1.2 Scope and Objective...... 9 1.3 Solution methodology ...... 11 1.4 Overview of the Dissertation...... 11
2. HISTORICAL REVIEW ...... 15
2.1 Vehicle dynamics. The beginnings...... 16 2.2 The computer age. Vehicle simulation history...... 17
3. CONCEPTS IN VEHICLE DYNAMICS ...... 19
3.1 The Bicycle Model...... 19 3.2 Limitations of the Bicycle Model...... 23 3.3 The nonlinear tire ...... 24
4. METHODOLOGY ...... 31
4.1 The Four-wheel Road-vehicle Model...... 31 4.2 Simplifications of the Four-wheel Road-vehicle Model...... 32 4.3 Solution Procedure ...... 33
5. RESULTS ...... 39
5.1 Variation plots ...... 39 5.2 Simple Linear Regression Analysis...... 59 5.3 Multiple Regression Analysis...... 61
6. DISCUSSION OF RESULTS AND CONCLUSIONS...... 67
6.1 Summary ...... 67 6.2 Discussion of Results ...... 68 6.3 Contributions and advances from the research...... 75
1 6.4 Future directions...... 77
BIBLIOGRAPHY ...... 79
APPENDIX A: THE TIRE MODEL ...... 91
A.1 The Piece-wise Tire Model...... 91 A.2 The Magic Formula for tire lateral force...... 101 A.3 The coefficient of friction µ...... 105
APPENDIX B: BASIC MATLAB CODE FOR VARIATION ANALYSIS...... 106
2 LIST OF FIGURES AND TABLES
Page
FIGURE 3.1. BICYCLE MODEL ...... 19 FIGURE 3.2. TIRE LATERAL FORCE...... 26 FIGURE 3.3. LATERAL FORCE VS. SLIP ANGLE FOR SEVERAL NORMAL LOADS ...... 29 FIGURE 3.4. THE TIRE LOAD SENSITIVITY EFFECT...... 30 FIGURE 4.1. FOUR-WHEEL ROAD VEHICLE MODEL ...... 31 FIGURE 4.2. RESPONSE DATA...... 37 FIGURE 5.1. VARIATION OF STEADY STATE LATERAL VELOCITY VS. VARIATION OF B...... 39 FIGURE 5.2. VARIATION OF STEADY STATE LATERAL VELOCITY VS. VARIATION OF M ..... 40 FIGURE 5.3. VARIATION OF STEADY STATE LATERAL VELOCITY VS. VARIATION OF IZ...... 40 FIGURE 5.4. VARIATION OF STEADY STATE LATERAL VELOCITY VS. VARIATION OF S.F... 41 FIGURE 5.5. VARIATION OF STEADY STATE LATERAL VELOCITY VS. VARIATION OF KRS. 41 FIGURE 5.6. VARIATION OF STEADY STATE ANGULAR VELOCITY VS. VARIATION OF B..... 42 FIGURE 5.7. VARIATION OF STEADY STATE ANGULAR VELOCITY VS. VARIATION OF M.... 42 FIGURE 5.8. VARIATION OF STEADY STATE ANGULAR VELOCITY VS. VARIATION OF IZ. ... 43 FIGURE 5.9. VARIATION OF STEADY STATE ANGULAR VELOCITY VS. VARIATION OF S.F. 43 FIGURE 5.10. VARIATION OF STEADY STATE ANGULAR VELOCITY VS. VARIATION OF KRS...... 44 FIGURE 5.11. VARIATION OF LATERAL VELOCITY RISE TIME VS. VARIATION OF B...... 44 FIGURE 5.12. VARIATION OF LATERAL VELOCITY RISE TIME VS. VARIATION OF M...... 45 FIGURE 5.13. VARIATION OF LATERAL VELOCITY RISE TIME VS. VARIATION OF IZ...... 45 FIGURE 5.14. VARIATION OF LATERAL VELOCITY RISE TIME VS. VARIATION OF S.F...... 46 FIGURE 5.15. VARIATION OF LATERAL VELOCITY RISE TIME VS. VARIATION OF KRS...... 46 FIGURE 5.16. VARIATION OF ANGULAR VELOCITY RISE TIME VS. VARIATION OF B...... 47 FIGURE 5.17. VARIATION OF ANGULAR VELOCITY RISE TIME VS. VARIATION OF M...... 47 FIGURE 5.18. VARIATION OF ANGULAR VELOCITY RISE TIME VS. VARIATION OF IZ...... 48 FIGURE 5.19. VARIATION OF ANGULAR VELOCITY RISE TIME VS. VARIATION OF S.F...... 48 FIGURE 5.20. VARIATION OF ANGULAR VELOCITY RISE TIME VS. VARIATION OF KRS...... 49 FIGURE 5.21. VARIATION OF LATERAL VELOCITY PEAK RESPONSE TIME VS. VARIATION OF B...... 49 FIGURE 5.22. VARIATION OF LATERAL VELOCITY PEAK RESPONSE TIME VS. VARIATION OF M...... 50 FIGURE 5.23. VARIATION OF LATERAL VELOCITY PEAK RESPONSE TIME VS. VARIATION OF IZ...... 50 FIGURE 5.24. VARIATION OF LATERAL VELOCITY PEAK RESPONSE TIME VS. VARIATION OF S.F...... 51 FIGURE 5.25. VARIATION OF LATERAL VELOCITY PEAK RESPONSE TIME VS. VARIATION OF KRS...... 51 FIGURE 5.26. VARIATION OF ANGULAR VELOCITY PEAK RESPONSE TIME VS. VARIATION OF B...... 52 FIGURE 5.27. VARIATION OF ANGULAR VELOCITY PEAK RESPONSE TIME VS. VARIATION OF M...... 52 FIGURE 5.28. VARIATION OF ANGULAR VELOCITY PEAK RESPONSE TIME VS. VARIATION OF IZ...... 53 FIGURE 5.29. VARIATION OF ANGULAR VELOCITY PEAK RESPONSE TIME VS. VARIATION OF S.F...... 53 FIGURE 5.30. VARIATION OF ANGULAR VELOCITY PEAK RESPONSE TIME VS. VARIATION OF KRS...... 54 FIGURE 5.31. VARIATION OF LATERAL VELOCITY OVERSHOOT VS. VARIATION OF B...... 54
3 FIGURE 5.32. VARIATION OF LATERAL VELOCITY OVERSHOOT VS. VARIATION OF M...... 55 FIGURE 5.33. VARIATION OF LATERAL VELOCITY OVERSHOOT VS. VARIATION OF IZ...... 55 FIGURE 5.34. VARIATION OF LATERAL VELOCITY OVERSHOOT VS. VARIATION OF S.F...... 56 FIGURE 5.35. VARIATION OF LATERAL VELOCITY OVERSHOOT VS. VARIATION OF KRS. ... 56 FIGURE 5.36. VARIATION OF ANGULAR VELOCITY OVERSHOOT VS. VARIATION OF B...... 57 FIGURE 5.37. VARIATION OF ANGULAR VELOCITY OVERSHOOT VS. VARIATION OF M...... 57 FIGURE 5.38. VARIATION OF ANGULAR VELOCITY OVERSHOOT VS. VARIATION OF IZ...... 58 FIGURE 5.39. VARIATION OF ANGULAR VELOCITY OVERSHOOT VS. VARIATION OF S.F. . 58 FIGURE 5.40. VARIATION OF ANGULAR VELOCITY OVERSHOOT VS. VARIATION OF KRS. 59 TABLE 5.1. LINEAR REGRESSION RESULTS...... 63 TABLE 5.2. MULTIPLE REGRESSION RESULTS...... 64 TABLE 5.3. MULTIPLE REGRESSION RESULTS. REVISED...... 65 TABLE 5.4 SHOWS MULTIPLE REGRESSION MODELS THAT INCLUDE CROSS PRODUCTS OF THE FOUR VEHICLE PARAMETERS...... 65 TABLE 5.4. MULTIPLE REGRESSION RESULTS. CROSS PRODUCTS OF PARAMETERS...... 66 FIGURE A.1. PIECE-WISE TIRE MODEL...... 93 FIGURE A.2. THE PIECE-WISE TIRE MODEL...... 98 FIGURE A.3. LATERAL FORCE VS. SLIP ANGLE FOR SEVERAL NORMAL LOADS. PIECE-WISE TIRE MODEL...... 99 FIGURE A.4. THE TIRE LOAD SENSITIVITY EFFECT. PIECE-WISE TIRE MODEL...... 100 FIGURE A.5. COMPARISON BETWEEN THE MAGIC FORMULA AND THE PIECE-WISE TIRE MODEL...... 102 FIGURE A.6. LATERAL VELOCITY. COMPARISON BETWEEN PIECE-WISE AND MAGIC FORMULA TIRE MODELS ...... 103 FIGURE A.7. ANGULAR VELOCITY. COMPARISON BETWEEN PIECE-WISE AND MAGIC FORMULA TIRE MODELS ...... 104 FIGURE A.8. COEFFICIENT OF FRICTION...... 105
4 LIST OF SYMBOLS
vy Lateral velocity of vehicle with respect to vehicle reference frame. Ω Angular velocity of Vehicle with respect to vehicle reference frame.
Cα Tire cornering stiffness.
Fy Lateral force. α Tire slip angle. δ Steering angle. a Distance from mass center to front axle. b Distance from mass center to rear axle.
Cαf Front tire cornering stiffness.
Cαr Rear tire cornering stiffness. S.F. Stability factor. c Vehicle track width. h Mass center height. m Vehicle mass.
Vx Vehicle forward velocity.
Fz Force in the z-direction. Normal force on the tires. g Acceleration due to gravity. krs Coefficient of rear roll stiffness. µ Tire-road coefficient of friction. f Normalized tire lateral force. W Static normal load on tire. V Total vehicle velocity.
Iz Vehicle yaw moment of inertia. l Vehicle wheelbase length.
5 1. INTRODUCTION
1.1 Background and motivation
Road-vehicle safety can be divided into two parts: passive safety and active safety. The first deals with the challenge of reducing passenger injuries after the collision has occurred. Topics such as vehicle crashworthiness and collision energy management, passenger restraint systems, pedestrian impact protection, and post-impact fire prevention are all addressed by passive safety. Active safety, instead, considers aspects that help avoid the collision or crash in the first place. Active safety covers a broad range of vehicle design and research topics, such as braking performance, comfort and ergonomics, vehicle lighting and visibility, and of course vehicle handling, steering response and lateral stability, that is, the field of road-vehicle lateral dynamics.
A recent compilation of motor vehicle crash data (NHTSA, 1997) provides the following facts:
• More than 6.8 million road-vehicle crashes were reported in the U.S. in 1996.
• Fifty-six percent of all fatal crashes in 1996 involved only one vehicle.
• More than half of fatal crashes occurred on roads with posted limits of 55 mph or
more.
6 These facts suggest that at least a significantly large proportion of road-vehicle crashes are the result of some form of loss of control of the vehicle. This is especially true at higher speeds, such as in highway travel, where vehicle stability and control are of higher importance.
It can be inferred from the above statements that the nature of the response of a road- vehicle after a steering input is important in terms of active safety. A vehicle that exhibits an unstable or abrupt response will likely lead the driver to lose control and possibly leave the road or hit another object. On the other extreme, a road-vehicle with a steering response that is too slow or insufficient will not change its trajectory quickly or thoroughly enough to prevent a collision or to allow the driver to remain on the road. A road-vehicle loses its capacity to accept driver input when the limits of its dynamical capabilities are reached and exceeded. It is the goal of the designer to position the dynamic limits of the vehicle as far from normal operating conditions as possible, within the design considerations of the vehicle. Not less importantly, the designer must also strive to produce a controllable vehicle. The modern road-vehicle has become immensely successful for many reasons and one of the most important is that it has evolved into a device whose response is progressive, stable and compatible with the reaction times of normal drivers. These characteristics allow safe, continuous operation that requires only a reasonable amount of skill and concentration, therefore helping put the modern automobile within reach of a vast segment of the world’s population.
7 The subject of driving enjoyment is another important aspect of the road-vehicle which has helped promote a better understanding of lateral dynamics. Ever since the first automobile traveled under its own power, there have been auto enthusiasts and people who have derived pleasure from the dynamics of driving a road-vehicle. The precision of the controls and the progressiveness of the responses are fundamental qualities in this subject. A vehicle that exhibits poor response characteristics is not only potentially unsafe but also less satisfying to drive because the driver’s input is not transformed into a proportional or desired response by the road-vehicle. Manufacturers today fully understand the importance of this subject and develop their road-vehicles to offer acceptable levels of driving satisfaction. In terms of lateral dynamics, a vehicle should exhibit reasonably short transients with small overshoots and a steady state behavior that is neither excessively understeer or oversteer in nature and that at the limit, the vehicle attitude remains as close to neutral as possible, allowing the most effective use of the cornering capability of each axle.
To achieve the desired lateral dynamic characteristics, it is necessary to have a full understanding of the vehicle parameters that affect those characteristics. Great advances in computer simulation have resulted in complex multibody road-vehicle models. Due to their complexities, these models are used mainly to simulate specific road-vehicles and not as tools for general or fundamental level research. Multibody road-vehicle models are not adequate for the objectives of this dissertation because the results that can be obtained are vehicle-specific and therefore not general in nature, that is, the results have a narrow scope and are applicable only to the vehicle studied. In the other extreme, highly
8 simplified mathematical models are simply not complete enough to provide useful insight on the dynamic effect of vehicle parameters.
The value of studying the dynamic influence of basic vehicle parameters is that their relative influence can be observed and understood, further advancing the knowledge in the field of vehicle dynamics. Understanding the effect of basic vehicle parameter in design and research allows decisions to be made either early in the design process or after vehicle modifications that result from marketing decisions. For vehicle designers, it allows a more solid basis for selecting suspension or tire parameters to compensate or address the changes to the vehicle dynamics that certain basic vehicle parameters may produce. For vehicle dynamics researchers it is valuable and broadly applicable knowledge that builds on the theory developed in this field for over half a century.
1.2 Scope and Objective
The scope of this research is to study and understand the influence each of the selected basic vehicle parameters has on the lateral dynamics of a four-wheel road-vehicle model.
The objective of this research is to accomplish this study with a road-vehicle model that includes the effects of lateral weight transfer and the changes it produces on the normal loads on each wheel, and that also takes into account the nonlinear nature of the lateral force developed by the tires. The results of this study should be of a general nature so that they have the broadest application possible. The parameters that are subjected to study in this research are 1) longitudinal position of the mass-center, 2) total vehicle mass, 3) yaw
9 moment of inertia, 4) stability factor, and 5) rear roll-stiffness coefficient. Three different sets of parameters are selected as nominal and represent the three major categories of road-vehicles in the world. The first of the three parameter sets represents a small, light car, such as commonly used in Europe, Asia and developing countries. The second parameter set represents a typical mid-size sedan common all over the world. The third
parameter set represents a large SUV (Sport Utility Vehicle), also sold throughout the
world and increasingly popular in the US.
This research intends to provide insight and advance the knowledge in the field of lateral
vehicle dynamics by using a nonlinear road-vehicle model that captures the basic features
of real road-vehicles while keeping the results general. The use of a nonlinear model is an
advance over previous efforts at generalized results that have used the Bicycle Model, a basic mathematical model that is only valid in the linear range of lateral vehicle dynamics.
The objective of this research is to understand the contribution of each parameter individually and in combination with each other. The study of the individual contribution of each parameter is beneficial because most of the parameters selected for this study are totally independent of one another and therefore can be changed, in practice, individually without affecting the other parameters. For example, the Stability Factor has no general
connection with the position of the mass-center. On the other hand, observing the effects
of the yaw moment of inertia independently from the total vehicle mass can be difficult in
practice. This study provides observations of these two basic vehicle parameters
10 individually. As said before, in this study a multiple regression analysis is included to understand possible interactions between parameters.
1.3 Solution methodology
The differential equations of motion are solved in MATLAB using a fourth-order Runge-
Kutta integration algorithm. For this study, several MATLAB programs have been written and are used to develop the piece-wise tire model, to compare the tire model with the Magic Formula and to develop the integration process. Some of these MATLAB programs helped develop the parameter sets, assisted in finding the dynamic limits of the vehicle model, and were used to develop the parametric study. MATLAB was also used for the regression analysis and the generation of all the plots.
1.4 Overview of the Dissertation
The need for a systematic estimation of the influence of basic road-vehicle parameters was identified. A level of complexity of the four-wheel road-vehicle model was chosen that would result in conclusions that had broad applications and significant value. This included a mathematical model to approximate the nonlinear behavior of pneumatic tires as they generate lateral forces. Three basic vehicle parameter sets were selected so that they cover the majority of current road-vehicles. Finally, a set of initial conditions was selected that would keep the tire model working in the nonlinear region of the lateral force curve.
11
The input data for this research includes:
1. Road vehicle parameters: These include mass, wheelbase, yaw moment of inertia,
position of the mass center, wheel track width, and rear roll-stiffness coefficient.
2. Initial conditions: These include constant total vehicle velocity, initial lateral
velocity vy, initial angular velocity Ω, and initial steering angle.
3. Tire model parameters: The cornering stiffness Cα is selected. Thirteen constants
are selected to produce the piece-wise tire model.
4. Parameter variation range: For parameter variation studies, a range is selected for
the variation of each parameter from its nominal value. For the longitudinal
position of the mass center, the range is limited to +/- 15% of nominal due to
dynamical limitations. The rest of the parameters have a variation range of +/-
50%.
5. Parameter variation step number: This quantifies the number of steps taken by
each parameter as it is varied individually from its nominal value.
The output consists basically of the time-history of the lateral and angular velocities,
but other outputs are produced from the basic time-history, as follows:
1. Lateral velocity response data: This consists of rise time, peak-response time,
overshoot as a percentage over the steady-state value, and the steady-state value
itself.
12 2. Angular velocity response data: This consists of rise time, peak-response time,
overshoot as a percentage over the steady-state value, and the steady-state value
itself.
3. Variation of the response data: This consists of the values of the response data
described above that result from the individual variation of the parameters.
Other output values have been used to aid the development of this research, but are
not part of the results of this dissertation:
4. Time history of the lateral acceleration: The rate of change of the lateral velocity
of the reference frame attached to the model.
5. Time history of the centripetal acceleration: The acceleration, perpendicular to the
longitudinal velocity of the road-vehicle model, due to the curved trajectory.
6. Time history of the normal load on the tires.
7. Time history of the tire slip angles.
8. Time history of the normalized tire lateral forces.
9. Time history of the steering angle.
To perform the parameter variation study, three MATLAB programs were developed for each or the parameter sets. These programs compute the values of the response data that result from varying each parameter individually from its nominal value. The result is a finite number of response data values that illustrate how the response changes as a result of an individual variation of the basic model parameters.
13 All the result plots in this research study are made in MATLAB.
A further refinement of the parametric variation study consists of a simple linear
regression analysis of the curves of response change vs. parameter variation. The simple
regression analysis is performed in MATLAB. The slope and y-intercept point of the
regression line are obtained directly by using the MATLAB “Basic Curve Fitting”
toolbox. The values of r2 are calculated from the known formulas. A multiple regression
analysis is performed in MS Excel and involves four of the vehicle parameters
simultaneously.
Chapter 2 of this dissertation provides a historical review of the field of Lateral Vehicle
Dynamics. Chapter 3 presents a discussion of the concepts used in this research. Chapter
4 consists of the solution technique and validation of the tire model. Chapter 5 presents
and describes the results. Chapter 6 provides a discussion of the results, conclusions and future directions. Appendix A describes the Piece-wise Tire Model, Appendix B presents the results of the double-variation analysis, and Appendix C presents the basic MATLAB code for the variation analysis.
14 2. HISTORICAL REVIEW
The current body of knowledge in the field of vehicle dynamics is the result of a fascinating journey along the development of the automobile. Ever since the first automobile huffed and puffed noisily under its own power, there has been an interest on controlling it, on making sure it follows the driver’s command faithfully and predictably.
During the early years of the automobile, development of vehicle dynamics sometimes lagged behind other essential aspects such as reliability and cost. Legend has it that after complaints from customers about lackluster brakes, Ettore Bugatti, maker of the famous
Bugatti race cars of the 1920s replied ‘I make my cars to go, not to stop’. By the time of
Mr. Bugatti’s famous quote, however, there were already groups of engineers, mechanics and general tinkerers that had already visualized the need for understanding how automobiles change directions, how they can be more stable and safe. The roads available in the dawn of the automobile were normally stone or gravel; sometimes no road at all. It is not surprising that the first systemized efforts in vehicle dynamics were focused on ride. These beginnings, however, fueled the interest on lateral vehicle dynamics until it quickly became an essential subject in vehicle design. In the following pages, a brief account of the development of vehicle dynamics is presented. It is divided into two parts.
The first part is dedicated to the beginnings of this field and the second is a historical account of modern advancements in this field.
15 2.1 Vehicle dynamics. The beginnings.
In 1908 Frederick William Lanchester (1868-1946) published a paper where he observed that an automobile with a tiller steering “oversteers” if the centrifugal force on the driver’s hands pushes toward greater steer angle. This quaint statement reflects the beginning stages of vehicle dynamics awareness in the dawn of the automobile age. More substantial findings were, however, just a few years away.
Vehicle dynamics is closely related to the development of the pneumatic tire. The tire is the interface between the car and the road and transmits the dynamic forces that make the vehicle change directions. The first account of the concept of tire slip angle was presented in 1925 by George Broulhiet of France, who published a paper entitled “The
Suspension and the Automobile Steering Mechanism”. Shortly after, Becker, Fromm, and
Maruhn, of Germany published an investigation of the vibration of automobile steering mechanisms, but more importantly, they showed the relationship between tire lateral forces, normal load, and tire slip angle by performing measurements of tires rolling on a rotating steel drum. In 1934, Maurice Olley, from England, while working for Cadillac in the US, produced a small internal report that is credited with being the earliest discussion of the concepts of oversteer and understeer as are generally known. By then, the seed for the beginning of systemized vehicle dynamics research had been planted. Mr. Olley, in the 1930s went on to develop several concepts that are today fundamental, such as tire aligning torque, stabilizer bars and their effects, tire load and load transfer sensitivities and others. Bob Schilling, who was an associate of Olley’s, described the response of an
16 automobile to a step steer input in a 1938 paper entitled “Handling Factors”. In the 1940s, a linear model for the lateral tire forces was presented.
By the 1950s, the field of vehicle dynamics was advancing rapidly. In 1956, David W.
Whitcomb presented a two-degree of freedom linear road-vehicle model, which was later known as the Bicycle model and used it to develop a mathematical analysis of road- vehicle stability and control (Milliken, Milliken, 1995). In the same year, Leonard Segel, who obtained an Aeronautical Engineering degree from the University of Cincinnati in
1947, had developed a linear four-wheel mathematical model of a road-vehicle (Segel,
1956) where he presented an eigenvalue analysis of a road-vehicle directional stability and an angular velocity frequency response analysis, among other advancements. In the following year, Albert G. Fonda (Fonda, 1956), while working at the Cornell Aeronautic
Laboratory introduced the concept of friction circle to describe combined longitudinal and lateral tire forces.
2.2 The computer age. Vehicle simulation history.
The beginning of the 1960s saw the first efforts in computer simulation of road-vehicle dynamics. In 1960 Charles E. Carrig et al. of Ford Motor Company performed one of the earliest uses of computer-aided design in the industry by developing a suspension- simulation program in FORTRAN to aid the design of the mid-engined Ford Mustang I show car (Classic Car Quarterly, 1998). That same year, at General Motors, Bob Kohr used an electronic analog computer to solve transient directional control equations. But
17 the computer age really started when the first multibody dynamics codes were introduced
in the 1980s. In that decade, researchers and auto manufacturers started using the
numerical codes ADAMS, DADS, and MEDYNA, among others (Kortüm, 1993). In
1989 Pacejka et al, presented the Magic Formula tire model, an empirical approximation
to tire behavior (Pacejka, 1993). The Magic Formula has gained acceptance worldwide as
a tire model for computer simulation and since its creation it has been continually updated.
18 3. CONCEPTS IN VEHICLE DYNAMICS
3.1 The Bicycle Model.
The basic analytical study of vehicle dynamics begins with the formulation of the Bicycle
Model, so called because it models a four-wheeled vehicle by a planar two-wheeled
device as represented in Figure 1, where δ is the steering angle, Fy1 and Fy2 are front and
rear tire lateral force, respectively, and C.G. is the vehicle center of gravity or mass center. In this model, the lateral load transfer due to lateral acceleration is neglected;
therefore the vehicle remains upright and does not pitch or roll. It has only two degrees of
freedom.
δ
Fy1 a
C.G.
b
Fy2
Figure 3.1. Bicycle Model
19
The track width of this model is zero and the normal forces on the wheels remain
constant throughout the motion of the model. The tires on the model are assumed to
generate lateral forces directly proportional to the slip angle α relative to the direction of
travel of the wheel. Therefore the lateral force developed by the tire is given by
Fy = Cαα (3.1)
where Cα is the cornering stiffness.
The slip angle α of a tire (Milliken, Milliken, 1995) is defined as the angle between the
direction of travel of the tire and the plane of the wheel, perpendicular to its axis. Slip
angle in a tire is produced by the elastic deformation of the tire tread in contact with the
road, or contact patch, when the tire is subjected to a lateral load while in a rolling
motion. When the lateral load on the tire (or produced by it) increases, the elastic
deformation of the tire tread increases until it starts sliding. The onset of sliding is progressive and when it is at its fullest, the lateral force on the tire is no longer a function
of slip angle but of coefficient of friction µ.
The term slip angle is misleading because there is no sliding involved except near the
trailing edge of the contact patch, when the tire tread is leaving the road. The term slip
angle is borrowed from aircraft dynamics, where it is related to when an aircraft slips at
an angle to its trajectory.
20
Equation (3.1) provides a valid and reasonably accurate representation of lateral tire
forces for small slip angles and lateral accelerations up to 0.4g. Above these conditions,
the behavior of the tire becomes increasingly nonlinear and the lateral force developed by
the tire is no longer proportional to its slip angle. For these conditions, Equation (3.1)
becomes inaccurate.
The two degrees of freedom of the bicycle model are represented by the lateral velocity
component vy and the angular velocity Ω. The forward speed vx is assumed to be constant
and the steering angle δ is a given input variable.
With these assumptions, the governing dynamical equations, for small steering angle δ,
are found to be:
m(v& y + vxΩ) − Fy1 − Fy2 = 0 (3.2) and
I zzΩ& − aFy1 + bFy2 = 0 (3.3)
where m is the mass of the vehicle, I is the central moment of inertia about the vertical
axis, or yaw inertia, and a and b are geometrical parameters locating the mass center, as
shown in Figure 3.1.
21 In the bicycle model with equal tires front and rear, when the mass-center is forward of the geometrical center of the wheelbase and the vehicle is negotiating a steady state turn, it will understeer. When the mass-center is located aft of the geometrical center of the wheelbase, the vehicle will oversteer. Finally, when the mass-center coincides with the geometrical center of the wheelbase, the vehicle will neutral steer. The term understeer means that if the speed is increased on a constant radius turn, the steering angle must increase to continue on the same trajectory. The term oversteer means that under the same conditions, the steering angle must decrease or even change sign, in order to continue on the same trajectory. Neutral steer means that the steering angle does not need to be modified in a steady state turn if the speed is increased. An alternate definition for understeer and oversteer is that in the former the rate of change of the slip angle of the front tires is greater than that of the rear tires; in the latter, the opposite holds, that is, the rate of change of the slip angle of the rear tires is greater than that of the front tires. In a neutral steer condition, the rate of change of the slip angles is the same, front and rear.
Oversteer vehicles are said to be unstable because above a certain forward speed in steady state motion, any input will result in a divergent response. Stability analyses using the bicycle model are well established and are part of the basic theory of automobile vehicle dynamics. By using the Routh-Hurwitz criteria analysts have developed the well known critical speed equation
2 2Cαf Cαr (a + b) vcritical = (3.5) m(Cαf a − Cαrb)
22
where a and b are the distances from the mass-center to the front and rear axles, Cαf and
Cαr are the cornering stiffness for the front and rear axles and m is the total vehicle mass.
The critical speed vcritical is the speed at which an oversteer vehicle response becomes divergent regardless of the input, unless a correction input, or phase-reversal is introduced. The critical speed is defined only for oversteer vehicles. In equation (3.4), if
Cαfa = Cαrb (neutral steer vehicle), then the critical speed is infinitely large and the vehicle is therefore stable. If Cαfa < Cαrb (understeer vehicle), then the critical speed is not defined.
Normal mass-production vehicles usually have equal tires on all four wheels, therefore
Cαf = Cαr and Equation (4) simplifies,
2C (a + b)2 v = α (3.6) critical m(a − b)
3.2 Limitations of the Bicycle Model
As mentioned in the previous section, the Bicycle Model is a simplified model of the four-wheel road vehicle. This model is meant to represent the dynamics of the vehicle at low lateral acceleration values, typically no more that 0.3 - 0.4g. Under this restriction,
23 the lateral load transfer is small enough so that it can be neglected and the lateral forces developed by the tires can be assumed to be directly proportional to the tire slip angle α, as shown in equation 1. The the applications of the Bicycle Model are therefore restricted to small inputs and low lateral acceleration. Equations (3.1) through (3.6) are valid only within these restrictions.
If the dynamics of a vehicle at high lateral accelerations (such as in the case of emergency situations) are to be studied, then a nonlinear tire model should be used, along with a road-vehicle model that takes into account the effects of dynamic load transfer.
The relationship between the vehicle track width and the height of the mass-center becomes important. This relationship is defined as the Stability Factor (S.F.),
c StabilityFactor = (3.7) 2h
where c is the wheel track width and h is the mass-center height.
3.3 The nonlinear tire
As it was mentioned above, a tire develops lateral force in a nonlinear way, as can be seen in Figure 3.2. Near the origin, the lateral force is basically proportional to the slip angle. This is called the linear region and it is where the linear dynamic analysis described above is valid. Further from the origin the slope decreases as the lateral force and slip angle increase. At the peak value of lateral force the slope is zero and then it becomes negative until it reaches an asymptotic value that is slightly lower than the peak
24 value. This last region is called the sliding mode and here the lateral force generated by the tire is dominated by the friction coefficient between the tire and the road.
25
Tire Lateral Force 5000
4000
3000
2000
1000
0 Lateral force, N Lateral -1000
-2000
-3000
-4000
-5000 -30 -20 -10 0 10 20 30 Slip angle, deg
Figure 3.2. Tire Lateral Force
26
Another important aspect of the nonlinear behavior of tires is the load sensitivity. The lateral force developed by a tire increases with normal load, but in a nonlinear way.
Figure 3.3 shows how as the normal load increases, the lateral force increases but not proportionally. The friction coefficient between the tire and the road decreases for higher normal loads, therefore the increase of lateral load is partially offset by the reduction in friction coefficient. Additionally, the peak lateral force moves toward the right (larger slip angles) as the normal load increases.
Figure 3.4 shows the effects of tire load sensitivity when comparing lateral force with normal load for a constant slip angle value.
The tire load sensitivity is an important behavior of tires and it determines the effects of dynamic weight transfer. Consider an axle that has a static normal load of 5000 N. When subjected to the lateral acceleration produced by a turn, the normal load on the outside tire increases and the normal load on the inside tire decreases by the same amount, therefore the total normal load supported by the axle is still 5000 N. Since the relationship between lateral force and normal load is not linear but rather convex down, the sum of the lateral force available for both the inside and outside tires is less that that available in the absence of lateral weight transfer. It is said that the axle has reduced its cornering capability. The lateral weight transfer in an axle can be altered by reducing or increasing the roll stiffness, that is, the resistance to the roll couple produced by the lateral inertia force multiplied by the height of the mass-center. If for example, the roll stiffness of the rear axle is increased relative to that of the front axle, the cornering
27 capability of the rear axle is decreased and the vehicle becomes more oversteer or less understeer. Altering the roll stiffness can be accomplished by suspension system tuning or by an anti-roll bar. It is important to point out that lateral weight transfer is not the product of roll, but rather roll is produced by lateral weight transfer. This allows the study of lateral weight transfer without adding a roll degree of freedom to the road-vehicle model.
28
Lateral force vs. Slip Angle for Several Normal Loads 5000
4000
3000
2000
1000 Increasing Normal Load
0 Lateral force, N Lateral -1000
-2000
-3000
-4000
-5000 -50 -40 -30 -20 -10 0 10 20 30 40 50 Slip angle, deg
Figure 3.3. Lateral Force vs. Slip Angle for Several Normal Loads
29
Lateral force vs. Normal force for several slip angles 4500
4000
8 deg 3500 Increasing Slip Angle
6 deg 3000
2500 4 deg
Lateral force, N Lateral 2000
1500 2 deg
1000
500
0 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Normal force, N
Figure 3.4. The Tire Load Sensitivity Effect
30 4. METHODOLOGY
4.1 The Four-wheel Road-vehicle Model
x
Vx y
Fy1 Fy2
a m V&y mVxΩ m V&y h b Fy4 Fy3 Fy1+ Fy4 Fy2+ Fy3
mg Fz1+ Fz4 Fz2+ Fz3
c
Figure 4.1. Four-Wheel road Vehicle Model
Figure 4.1 shows the free body diagrams for the proposed four-wheel road vehicle model.
The coordinate reference frame is attached to the vehicle. The equations of motion that result from this model are
m(V&y +VxΩ) − Fy1 − Fy2 − Fy3 − Fy4 = 0 (4.1)
I z Ω& − a(Fy1 + Fy2 ) + b(Fy3 + Fy4 ) = 0 (4.2)
31 Where Fy1, Fy2, Fy3, and Fy4 are the lateral forces developed by the tires, Vy and Vx are the vehicle lateral and longitudinal velocities, and Ω is the vehicle angular or yaw velocity; all referred to the vehicle reference frame. The parameters a and b define the vehicle mass center longitudinal position as shown in Figure 4.1 The normal forces, Fzi, on the tires are
mb g h(V& +V Ω) F = + y x (4.3) z1 a + b 2 c
mb g h(V& +V Ω) F = − y x (4.4) z2 a + b 2 c
ma g hkrs(V& +V Ω) F = − y x (4.5) z3 a + b 2 c
ma g hkrs(V& +V Ω) F = + y x (4.6) z4 a + b 2 c
where the term krs, in equations (4.5) and (4.6) is defined as the rear roll stiffness coefficient. This coefficient is described in Section 4.6.
4.2 Simplifications of the Four-wheel Road-vehicle Model.
The mathematical modeling of physical systems is always a compromise between accuracy and complexity. In the case of this research an additional compromise needs to be made: the need to conserve generality. The objective of this study is to obtain a general insight on the individual effects of basic vehicle parameters on lateral vehicle
32 dynamics. The results and conclusions should have wide application. For this reason, a more complex vehicle model was deemed inappropriate. The details of suspension geometry, component stiffness and other aspects that are specific to any vehicle would have influenced the results in such a way that generalized conclusions would not have been possible. The simplifications of this model are:
1. The model has no suspension. This means, there is no camber change, compliance
steer or roll steer effects. Also, there is no rolling motion (or rolling degree of
freedom) to consider.
2. The steering angle is equal for each front wheel.
3. There are no compliances and inertias in the steering system.
4. The effects of tire aligning torque are not considered.
5. The model is symmetric along the longitudinal axis.
6. A simplified tire model represents the nonlinear characteristics of the tire.
7. Both tires at each axle share the same slip angle.
8. Total vehicle velocity is constant.
9. The track width of the front and rear axles are equal.
4.3 Solution Procedure
A step steer is chosen as the input for each simulation. Additionally, the initial values for the variables lateral velocity Vy and angular velocity Ω are both zero. The following
33 parameters are the initial conditions and vehicle parameters that remain constant throughout the study:
1. Total vehicle velocity V (m/s): 20
2. Step steer angle at the front wheels (deg): 2
3. Initial lateral velocity Vy0 (m/s): 0
4. Initial angular velocity Ω0 (rad/s): 0
5. Tire cornering stiffness Ca (N/rad): 50,000
6. Road/tire coefficient of friction at zero normal load µ0: 0.95
The vehicle parameters that are modified in this study are:
1. Longitudinal position of the mass-center: Distance b from mass-center to rear
axle.
2. Total vehicle mass, m: Total vehicle mass.
3. Yaw moment of inertia, Iz: Moment of inertia about the vertical axis of the
vehicle.
c 4. Stability factor, SF: The quotient , where c is the vehicle track width and h is 2h
the mass-center height.
5. Rear roll stiffness coefficient, krs: A coefficient defined for this study that
modifies the amount of lateral weight transfer in the rear axle. A krs value of 1
means that the lateral weight transfer in the rear axle is a result only of the lateral
34 acceleration. Values of krs greater than 1 enhance the lateral weight transfer in the
rear axle, therefore representing the effect of an anti-roll bar.
Three different sets of parameters are selected as nominal and represent the three major categories of road-vehicles in the world. The first of the three parameter sets represents a small, light car, such as commonly used in Europe, Asia and developing countries. The second parameter set represents a typical mid-size sedan common all over the world. The third parameter set represents a large SUV (Sport Utility Vehicle), also sold throughout the world and increasingly popular in the US.
The nominal values for the three parameter sets considered in this study are:
1. Parameter set 1: Light car:
1.1 Wheelbase, l (m): 2.6
1.2 Distance from mass-center to rear axle, b (m): 1.6
1.3 Mass, m (kg): 1000
2 1.4 Yaw moment of inertia, Iz, (kg-m ): 1400
1.5 Stability Factor, SF: 1.500
1.6 Rear roll stiffness coefficient, krs: 1
2. Parameter set 2: Medium car:
2.1 Wheelbase, l (m): 2.75
2.2 Distance from mass-center to rear axle, b (m): 1.65
35 2.3 Mass, m (kg): 1500
2 2.4 Yaw moment of inertia, Iz, (kg-m ): 2200
2.5 Stability Factor, SF: 1.410
2.6 Rear roll stiffness coefficient, krs: 1
3. Parameter set 3: Large SUV:
3.1 Wheelbase, l (m): 3.00
3.2 Distance from mass-center to rear axle, b (m): 1.80
3.3 Mass, m (kg): 2000
2 3.4 Yaw moment of inertia, Iz, (kg-m ): 2800
3.5 Stability Factor, SF: 1.125
3.6 Rear roll stiffness coefficient, krs: 1
The response data to be studied in this research consists of the following concepts, which are visualized in Figure 4.8:
1. Steady-state final value: This is the value of the response when the system has
reached steady-state conditions.
2. Rise time: It is defined as the time the system takes to first reach 90% of the
steady-state response value (Milliken, Milliken 1995).
3. Peak response time: It is the time the system takes to reach the maximum value of
the response.
36 4. Overshoot: For this study, it is defined as the difference between the peak value
and the steady state value, as a percentage of the steady-state value.
Angular velocity Omega 0.35
Peak response time Steady-state value
0.3
Overshoot
0.25
0.2
0.15
Angular velocity Omega. rad/s Omega. velocity Angular Rise time
0.1
90% of steady state response
0.05
0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Time. s
Figure 4.2. Response Data
Three MATLAB programs are developed for the parametric study. Each program calculates the response for one of the three parameter sets described above. The differential equation integration method used is the fourth-order Runge-Kutta method with a fixed step size of 1.25x 10-3 s. In the vehicle model, a step-steer magnitude of 2 deg is chosen because for the total vehicle velocity V of 20ms or 44.7 MPH, a step steer
37 of 2 deg corresponds to a medium severity maneuver. For a typical steering gear ratio of
20:1, the chosen step steer value results in a steering wheel angle of 40 deg. In each
MATLAB program, each basic vehicle parameter is modified individually in eleven steps and the vehicle response is calculated in every step. The parameter b (distance from mass-center to rear axle) is modified through a range of -15% to +15% of it nominal value for each parameter set. The range is limited to +/-15% because values of b that are more than 15% smaller than nominal tend to result in an unstable oversteer response, unless the total vehicle velocity V is reduced substantially. The rest of the basic vehicle parameters are modified individually +/- 50% of their nominal value in eleven steps.
The results are visualized by plotting, for each response characteristic, the normalized response, against the parameter variation. The normalized response is defined as the response value that results from the parameter variation, divided by the response that results from the nominal parameter set, as shown in Equation (4.7).
response normalized response = (4.7) nom. response
A normalized response value of 1 means that changing the parameter does not result in a variation of the response from its nominal value. Normalized response values of less than
1 and greater than 1 mean the variation of that particular parameter results in a decrease or increase in the value of the response, respectively, relative to the response from the nominal parameter set.
38 5. RESULTS
This chapter presents the results of this study. It is divided into three sections; the first presents the variation plots, the second introduces and presents the linear regression analysis and the third section presents a multiple regression analysis using four vehicle parameters.
5.1 Variation plots
The following variation plots show the relationship between the normalized response characteristic as defined in Equation 4.7 of Section 4.3, and the variation of the vehicle characteristic as a percentage of its nominal value.
Variation of Steady State Lateral Velocity 4.5 Small car SUV 4 Medium car
3.5
3
2.5
2
1.5 Normalized SteadyState LateralVelocity 1
0.5
0 -15 -10 -5 0 5 10 15 Variation of b, %
Figure 5.1. Variation of Steady State Lateral Velocity vs. Variation of b
39 Variation of Steady State Lateral Velocity 1.6 Small car SUV Medium car 1.4
1.2
1
0.8
0.6 Normalized SteadyState LateralVelocity
0.4
0.2 -50 -40 -30 -20 -10 0 10 20 30 40 50 Variation of m, %
Figure 5.2. Variation of Steady State Lateral Velocity vs. Variation of m
Variation of Steady State Lateral Velocity 1.05 Small car SUV 1.04 Medium car
1.03
1.02
1.01
1
0.99
0.98 Normalized SteadyState LateralVelocity 0.97
0.96
0.95 -50 -40 -30 -20 -10 0 10 20 30 40 50 Variation of Iz, %
Figure 5.3. Variation of Steady State Lateral Velocity vs. Variation of Iz
40 Variation of Steady State Lateral Velocity 1.08 Small car SUV Medium car 1.06
1.04
1.02
1 Normalized SteadyState LateralVelocity
0.98
0.96 -50 -40 -30 -20 -10 0 10 20 30 40 50 Variation of S.F., %
Figure 5.4. Variation of Steady State Lateral Velocity vs. Variation of S.F.
Variation of Steady State Lateral Velocity 1.1 Small car SUV 1.08 Medium car
1.06
1.04
1.02
1
Normalized SteadyState LateralVelocity 0.98
0.96
0.94 -50 -40 -30 -20 -10 0 10 20 30 40 50 Variation of krs, %
Figure 5.5. Variation of Steady State Lateral Velocity vs. Variation of krs.
41 Variation of Steady State Angular Velocity 2 Small car SUV Medium car 1.8
1.6
1.4
1.2
1 Normalized SteadyState AngularVelocity
0.8
0.6 -15 -10 -5 0 5 10 15 Variation of b, %
Figure 5.6. Variation of Steady State Angular Velocity vs. Variation of b.
Variation of Steady State Angular Velocity 1.5 Small car SUV Medium car 1.4
1.3
1.2
1.1
1 Normalized Steady State Angular Velocity Angular State Steady Normalized
0.9
0.8 -50 -40 -30 -20 -10 0 10 20 30 40 50 Variation of m, %
Figure 5.7. Variation of Steady State Angular Velocity vs. Variation of m.
42 Variation of Steady State Angular Velocity 1.05 Small car SUV 1.04 Medium car
1.03
1.02
1.01
1
0.99
0.98 Normalized SteadyState AngularVelocity 0.97
0.96
0.95 -50 -40 -30 -20 -10 0 10 20 30 40 50 Variation of Iz, %
Figure 5.8. Variation of Steady State Angular Velocity vs. Variation of Iz.
Variation of Steady State Angular Velocity 1.05 Small car SUV 1.04 Medium car
1.03
1.02
1.01
1
0.99
0.98 Normalized Steady State Angular Velocity Angular State Steady Normalized 0.97
0.96
0.95 -50 -40 -30 -20 -10 0 10 20 30 40 50 Variation of SF, %
Figure 5.9. Variation of Steady State Angular Velocity vs. Variation of S.F.
43 Variation of Steady State Angular Velocity 1.05 Small car SUV 1.04 Medium car
1.03
1.02
1.01
1
0.99
0.98 Normalized SteadyState AngularVelocity 0.97
0.96
0.95 -50 -40 -30 -20 -10 0 10 20 30 40 50 Variation of krs, %
Figure 5.10. Variation of Steady State Angular Velocity vs. Variation of krs.
Variation of Lateral Velocity Rise Time 3.5 Small car SUV Medium car 3
2.5
2
1.5 Normalized Lateral Velocity Rise Time
1
0.5 -15 -10 -5 0 5 10 15 Variation of b, %
Figure 5.11. Variation of Lateral Velocity Rise Time vs. Variation of b.
44 Variation of Lateral Velocity Rise Time 1.4 Small car SUV 1.35 Medium car
1.3
1.25
1.2
1.15
1.1
1.05 Normalized Lateral Velocity Rise Time
1
0.95
0.9 -50 -40 -30 -20 -10 0 10 20 30 40 50 Variation of m, %
Figure 5.12. Variation of Lateral Velocity Rise Time vs. Variation of m.
Variation of Lateral Velocity Rise Time 1.25
1.2
1.15
1.1
1.05
1
0.95
0.9 Normalized Lateral Velocity Rise Time
0.85 Small car SUV 0.8 Medium car
0.75 -50 -40 -30 -20 -10 0 10 20 30 40 50 Variation of Iz, %
Figure 5.13. Variation of Lateral Velocity Rise Time vs. Variation of Iz.
45 Variation of Lateral Velocity Rise Time 1.06 Small car SUV 1.05 Medium car
1.04
1.03
1.02
1.01
1 Normalized Lateral Velocity Rise Time 0.99
0.98
0.97 -50 -40 -30 -20 -10 0 10 20 30 40 50 Variation of SF, %
Figure 5.14. Variation of Lateral Velocity Rise Time vs. Variation of S.F.
Variation of Lateral Velocity Rise Time 1.12
1.1
1.08
1.06
1.04
1.02
1 Normalized Lateral Velocity Rise Time 0.98
Small car SUV 0.96 Medium car
0.94 -50 -40 -30 -20 -10 0 10 20 30 40 50 Variation of krs, %
Figure 5.15. Variation of Lateral Velocity Rise Time vs. Variation of krs.
46 Variation of Angular Velocity Rise Time 2.2 Small car SUV Medium car 2
1.8
1.6
1.4
1.2 Normalized Angular Velocity Rise Time
1
0.8 -15 -10 -5 0 5 10 15 Variation of b, %
Figure 5.16. Variation of Angular Velocity Rise Time vs. Variation of b.
Variation of Angular Velocity Rise Time 1.7 Small car SUV 1.6 Medium car
1.5
1.4
1.3
1.2
1.1
1 Normalized Lateral Velocity Rise Time
0.9
0.8
0.7 -50 -40 -30 -20 -10 0 10 20 30 40 50 Variation of m, %
Figure 5.17. Variation of Angular Velocity Rise Time vs. Variation of m.
47 Variation of Angular Velocity Rise Time 1.5
1.4
1.3
1.2
1.1
1
0.9
0.8 Normalized Lateral Velocity Rise Time
0.7 Small car SUV 0.6 Medium car
0.5 -50 -40 -30 -20 -10 0 10 20 30 40 50 Variation of Iz, %
Figure 5.18. Variation of Angular Velocity Rise Time vs. Variation of Iz.
Variation of Angular Velocity Rise Time 1.05 Small car SUV 1.04 Medium car
1.03
1.02
1.01
1
0.99
0.98 Normalized Lateral Velocity Rise Time
0.97
0.96
0.95 -50 -40 -30 -20 -10 0 10 20 30 40 50 Variation of SF, %
Figure 5.19. Variation of Angular Velocity Rise Time vs. Variation of S.F.
48 Variation of Angular Velocity Rise Time 1.05
1.04
1.03
1.02
1.01
1
0.99
0.98 Normalized Lateral Velocity Rise Time
0.97 Small car SUV 0.96 Medium car
0.95 -50 -40 -30 -20 -10 0 10 20 30 40 50 Variation of krs, %
Figure 5.20. Variation of Angular Velocity Rise Time vs. Variation of krs.
Variation of Laterar Velocity Peak Response Time 5 Small car SUV 4.5 Medium car
4
3.5
3
2.5
2
1.5 Normalized Lateral Velocity Peak ResponseTime
1
0.5 -15 -10 -5 0 5 10 15 Variation of b, %
Figure 5.21. Variation of Lateral Velocity Peak Response Time vs. Variation of b.
49 Variation of Lateral Velocity Peak Response Time 1.3 Small car SUV Medium car 1.25
1.2
1.15
1.1
1.05
1 Normalized Lateral Velocity Peak ResponseTime
0.95
0.9 -50 -40 -30 -20 -10 0 10 20 30 40 50 Variation of m, %
Figure 5.22. Variation of Lateral Velocity Peak Response Time vs. Variation of m.
Variation of Lateral Velocity Peak Response Time 1.25
1.2
1.15
1.1
1.05
1
0.95
0.9
Normalized Lateral Velocity Peak ResponseTime 0.85 Small car SUV 0.8 Medium car
0.75 -50 -40 -30 -20 -10 0 10 20 30 40 50 Variation of Iz, %
Figure 5.23. Variation of Lateral Velocity Peak Response Time vs. Variation of Iz.
50 Variation of Lateral Velocity Peak Response Time
Small car SUV 1.08 Medium car
1.06
1.04
1.02 Normalized Lateral Velocity Peak ResponseTime
1
-50 -40 -30 -20 -10 0 10 20 30 40 50 Variation of SF, %
Figure 5.24. Variation of Lateral Velocity Peak Response Time vs. Variation of S.F.
Variation of Lateral Velocity Peak Response Time 1.25 Small car SUV Medium car 1.2
1.15
1.1
1.05 Normalized Lateral Velocity Peak ResponseTime 1
0.95 -50 -40 -30 -20 -10 0 10 20 30 40 50 Variation of krs, %
Figure 5.25. Variation of Lateral Velocity Peak Response Time vs. Variation of krs.
51 Variation of Angular Velocity Peak Response Time 2.6 Small car SUV 2.4 Medium car
2.2
2
1.8
1.6
1.4
1.2 Normalized Angular Velocity Peak Response Time
1
0.8 -15 -10 -5 0 5 10 15 Variation of b, %
Figure 5.26. Variation of Angular Velocity Peak Response Time vs. Variation of b.
Variation of Angular Velocity Peak Response Time 1.6 Small car SUV Medium car 1.5
1.4
1.3
1.2
1.1
1 Normalized Angular Velocity Peak Response Time Response Peak Velocity Angular Normalized
0.9
0.8 -50 -40 -30 -20 -10 0 10 20 30 40 50 Variation of m, %
Figure 5.27. Variation of Angular Velocity Peak Response Time vs. Variation of m.
52 Variation of Angular Velocity Peak Response Time 1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6 Normalized Angular Velocity Peak Response Time Small car SUV 0.5 Medium car
0.4 -50 -40 -30 -20 -10 0 10 20 30 40 50 Variation of Iz, %
Figure 5.28. Variation of Angular Velocity Peak Response Time vs. Variation of Iz.
Variation of Angular Velocity Peak Response Time 2 Small car SUV 1.8 Medium car
1.6
1.4
1.2
1
0.8
0.6
0.4 Normalized Angular Velocity Peak Response Time Response Peak Velocity Angular Normalized
0.2
0 -50 -40 -30 -20 -10 0 10 20 30 40 50 Variation of SF, %
Figure 5.29. Variation of Angular Velocity Peak Response Time vs. Variation of S.F.
53 Variation of Angular Velocity Peak Response Time 1.05
1.04
1.03
1.02
1.01
1
0.99
0.98
0.97
Normalized Angular Velocity Peak Response Time Small car SUV 0.96 Medium car
0.95 -50 -40 -30 -20 -10 0 10 20 30 40 50 Variation of krs, %
Figure 5.30. Variation of Angular Velocity Peak Response Time vs. Variation of krs.
Variation of Lateral Velocity Overshoot
1.1
1.05
1
0.95
0.9 Normalized Lateral Velocity Overshoot Velocity Lateral Normalized
Small car 0.85 SUV Medium car
0.8 -15 -10 -5 0 5 10 15 Variation of b, %
Figure 5.31. Variation of Lateral Velocity Overshoot vs. Variation of b.
54 Variation of Lateral Velocity Overshoot 1.1 Small car SUV 1.08 Medium car
1.06
1.04
1.02
1
0.98
0.96 Normalized Lateral Velocity Overshoot
0.94
0.92
0.9 -50 -40 -30 -20 -10 0 10 20 30 40 50 Variation of m, %
Figure 5.32. Variation of Lateral Velocity Overshoot vs. Variation of m.
Variation of Lateral Velocity Overshoot 1.05 Small car SUV 1.04 Medium car
1.03
1.02
1.01
1
0.99
0.98 Normalized Lateral Velocity Overshoot Velocity Lateral Normalized
0.97
0.96
0.95 -50 -40 -30 -20 -10 0 10 20 30 40 50 Variation of Iz, %
Figure 5.33. Variation of Lateral Velocity Overshoot vs. Variation of Iz.
55 Variation of Lateral Velocity Overshoot 1.05
1.04
1.03
1.02
1.01
1
0.99
0.98 Normalized Lateral Velocity Overshoot
0.97 Small car SUV Medium car 0.96
0.95 -50 -40 -30 -20 -10 0 10 20 30 40 50 Variation of SF, %
Figure 5.34. Variation of Lateral Velocity Overshoot vs. Variation of S.F.
Variation of Lateral Velocity Overshoot 1.05 Small car SUV 1.04 Medium car
1.03
1.02
1.01
1
0.99
0.98 Normalized Lateral Velocity Overshoot Velocity Lateral Normalized
0.97
0.96
0.95 -50 -40 -30 -20 -10 0 10 20 30 40 50 Variation of krs, %
Figure 5.35. Variation of Lateral Velocity Overshoot vs. Variation of krs.
56 Variation of Angular Velocity Overshoot 1.05
1
0.95
Normalized Angular Velocity Overshoot Velocity Angular Normalized 0.9
Small car SUV Medium car
0.85 -15 -10 -5 0 5 10 15 Variation of b, %
Figure 5.36. Variation of Angular Velocity Overshoot vs. Variation of b.
Variation of Angular Velocity Overshoot 1.25
1.2
1.15
1.1
1.05
1
Normalized Angular Velocity Overshoot Velocity Angular Normalized 0.95
Small car 0.9 SUV Medium car
0.85 -50 -40 -30 -20 -10 0 10 20 30 40 50 Variation of m, %
Figure 5.37. Variation of Angular Velocity Overshoot vs. Variation of m.
57 Variation of Angular Velocity Overshoot 1.1 Small car SUV Medium car 1.08
1.06
1.04
1.02
1
Normalized Angular Velocity Overshoot Velocity Angular Normalized 0.98
0.96
0.94 -50 -40 -30 -20 -10 0 10 20 30 40 50 Variation of Iz, %
Figure 5.38. Variation of Angular Velocity Overshoot vs. Variation of Iz.
Variation of Angular Velocity Overshoot 1.05 Small car SUV 1.04 Medium car
1.03
1.02
1.01
1
0.99
0.98 Normalized Angular Velocity Overshoot Velocity Angular Normalized
0.97
0.96
0.95 -50 -40 -30 -20 -10 0 10 20 30 40 50 Variation of SF, %
Figure 5.39. Variation of Angular Velocity Overshoot vs. Variation of S.F.
58 Variation of Angular Velocity Overshoot 1.05 Small car SUV 1.04 Medium car
1.03
1.02
1.01
1
0.99
0.98 Normalized Angular Velocity Overshoot Velocity Angular Normalized
0.97
0.96
0.95 -50 -40 -30 -20 -10 0 10 20 30 40 50 Variation of krs, %
Figure 5.40. Variation of Angular Velocity Overshoot vs. Variation of krs.
5.2 Simple Linear Regression Analysis.
The variation analysis illustrated in the previous section suggests that changes in all response characteristics due to the variation of the parameter b (the distance from the rear axle to the mass center) exhibit an exponential nature. Individual variation of the other vehicle parameters results in changes to the response characteristics that appear close to linear in nature. The following simple linear regression analysis allows these observations to be quantified.
For the parameter b, the data has to be plotted in a semi-log fashion before a linear regression analysis can be performed. The regression analysis of the semi-log data also
59 allows the computation of r2. After that, the equations for the regression lines are transformed back into the exponential form and are shown in equations (5.1) to (5.8).
−0.07559b y1 = 105.3e (5.1)
−0..02583b y2 = 103.2e (5.2)
−0.03566b y3 = 115.3e (5.3)
−0.02053b y4 =107.6e (5.4)
−0.04688b y5 = 120.3e (5.5)
−0.02844b y6 = 110.8e (5.6)
0.00161b y7 = 100.8e (5.7)
0.00319b y8 = 98.5e (5.8)
Where,
y1 = Steady State Lateral Velocity
y2 = Steady State Angular Velocity
y3 = Lateral Velocity Rise Time
y4 = Angular Velocity Rise Time
y5 = Lateral Velocity Peak-Response Time
y6 = Angular Velocity Peak-Response Time
y7 = Lateral Velocity Overshoot
y8 = Angular Velocity Overshoot
60
The simple linear regression analysis for the rest of the parameters is performed in a straight-forward fashion.
Table 5.1 shows the results of the simple linear regression analysis. The slope and y- intercept of the regression lines are obtained with the curve-fit toolbox available on
MATLAB. The values of r2 were calculated from the known formulas.
5.3 Multiple Regression Analysis.
The models used in the multiple regression analysis have the following general form:
yi = β0 + β1x1 + β2x2 + β3x3 + β4x4 + ε (5.9)
where yi is a response characteristic such as steady-state lateral velocity or angular velocity overshoot. The value of the coefficient βi determines the contribution of the independent variable xi, and β0 is the y-intercept. The random error coefficient ε makes the model probabilistic rather than deterministic. In this analysis the independent variables are the vehicle mass m, the yaw moment of inertia Iz, the stability factor S.F., and the rear roll-stiffness coefficient krs. The distance from the rear axle to the mass- center b is not included because the objective of this analysis is to determine the relationship between the vehicle parameters that appear to behave linearly, as seen in the previous sections of this chapter.
61
As in section 5.2, a MATLAB routine generates the data for the analysis. In the multiple regression case, the amount of data can be too large, so care has to be taken in the sampling process. For this reason, a concession was made in the data generation process.
The number of steps taken to vary each vehicle parameter from -50% of nominal value to
+50% was reduced from eleven steps used in the previous sections to seven steps. The result was a reduction of the data generation computing time from a theoretical 33 hours to a more reasonable 5 hours. This decision reduced the number of events in the analysis from 14,641 to 2,401, based on Equation 5.10,
NstepsNpar = Nevents (5.10)
Where Nsteps is the number of steps taken to vary each vehicle parameter, Npar is the number of vehicle parameters involved in the analysis, and Nevents is the number of events in the regression analysis.
Table 5.2 presents a summary of the results of the multiple regression analysis, showing the regression coefficients, R2 values, and F-values.
The data used in this section comes from the medium size vehicle parameter set (see section 4.3). This parameter set was selected for this multiple regression analysis because it is between the two extremes of small car and large SUV.
62 STEADY STATE LATERAL VELOCITY 2 Slope (β1) r DISTANCE FROM MASS CENTER TO REAR AXLE (b) -7.96 0.9762 MASS (m) 0.9154 0.9570 YAW MOMENT OF INERTIA (Iz) 0 0* STABILITY FACTOR -0.04123 0.6533 REAR ROLL-STIFFNESS COEFFICIENT 0.08028 0.7929 STEADY STATE ANGULAR VELOCITY 2 Slope (β1) r DISTANCE FROM MASS CENTER TO REAR AXLE (b) -2.67 0.9560 MASS (m) -0.4371 0.9706 YAW MOMENT OF INERTIA (Iz) 0 0* STABILITY FACTOR 0.01732 0.6601 REAR ROLL-STIFFNESS COEFFICIENT 0.02188 0.8979 LATERAL VELOCITY RISE TIME 2 Slope (β1) r DISTANCE FROM MASS CENTER TO REAR AXLE (b) -4.11 0.7937 MASS (m) -0.2747 0.7158 YAW MOMENT OF INERTIA (Iz) 0.4723 0.9974 STABILITY FACTOR -0.03537 0.6869 REAR ROLL-STIFFNESS COEFFICIENT 0.09552 0.8250 ANGULAR VELOCITY RISE TIME 2 Slope (β1) r DISTANCE FROM MASS CENTER TO REAR AXLE (b) -2.21 0.7958 MASS (m) -0.8072 0.9482 YAW MOMENT OF INERTIA (Iz) 0.9700 0.9990 STABILITY FACTOR 0.0255 0.6164 REAR ROLL-STIFFNESS COEFFICIENT 0.0385 0.8222 LATERAL VELOCITY PEAK-RESPONSE TIME 2 Slope (β1) r DISTANCE FROM MASS CENTER TO REAR AXLE (b) -5.64 0.7977 MASS (m) -0.2224 0.7402 YAW MOMENT OF INERTIA (Iz) 0.4369 0.9955 STABILITY FACTOR -0.0580 0.7202 REAR ROLL-STIFFNESS COEFFICIENT 0.1305 0.8254 ANGULAR VELOCITY PEAK-RESPONSE TIME 2 Slope (β1) r DISTANCE FROM MASS CENTER TO REAR AXLE (b) -3.15 0.8068 MASS (m) -0.5498 0.8616 YAW MOMENT OF INERTIA (Iz) 0.799 0.9987 STABILITY FACTOR 0 0* REAR ROLL-STIFFNESS COEFFICIENT 0.02332 0.7466 LATERAL VELOCITY OVERSHOOT 2 Slope (β1) r DISTANCE FROM MASS CENTER TO REAR AXLE (b) 0.1625 0.4200 MASS (m) -0.007569 0.0300 YAW MOMENT OF INERTIA (Iz) 0.04105 0.8530 STABILITY FACTOR -0.003924 0.0643 REAR ROLL-STIFFNESS COEFFICIENT -0.02621 0.7867 ANGULAR VELOCITY OVERSHOOT 2 Slope (β1) r DISTANCE FROM MASS CENTER TO REAR AXLE (b) 0.3138 0.6013 MASS (m) 0.2582 0.9567 YAW MOMENT OF INERTIA (Iz) -0.1067 0.9477 STABILITY FACTOR -0.01662 0.6524 REAR ROLL-STIFFNESS COEFFICIENT -0.01967 0.9075 *All the data points are along a horizontal line.
Table 5.1. Linear Regression Results
63
STEADY STATE LATERAL VELOCITY (y1) MULTIPLE REGRESSION MODEL F R2 -2 -4 -9 -2 -2 y1 = -7.59x10 - 2.51x10 m+1.45x10 Iz+4.82x10 S.F. - 9.95x10 krs + ε 1754.9 0.7455
STEADY STATE ANGULAR VELOCITY (y2) MULTIPLE REGRESSION MODEL F R2 -5 -18 -4 -3 y2 = 0.1793- 3.75x10 m -1.1x10 Iz - 2.75x10 S.F. + 6.84x10 krs + ε 10510.7 0.9461
LATERAL VELOCITY RISE TIME (y3) MULTIPLE REGRESSION MODEL F R2 -4 -5 y3 = 0.3296 - 1.03x10 m+8.35x10 Iz - 0.109S.F. + 0.201krs + ε 117.36 0.1638
ANGULAR VELOCITY RISE TIME (y4) MULTIPLE REGRESSION MODEL F R2 -2 -5 -5 -3 -2 y4 = 8.07x10 – 5.94x10 m+5.32x10 Iz - 9.23x10 S.F. – 2.47x10 krs + ε 490.68 0.4503
LATERAL VELOCITY PEAK-RESPONSE TIME (y5) MULTIPLE REGRESSION MODEL F R2 -4 -4 y5 = 0.7198 – 2.16x10 m+1.03x10 Iz – 0.3675 S.F. + 0.6597 krs + ε 75.57 0.1081
ANGULAR VELOCITY PEAK-RESPONSE TIME (y6) MULTIPLE REGRESSION MODEL F R2 -4 -4 y6 = 0.2724 – 1.45x10 m+1.11x10 Iz – 0.1386 S.F. + 0.2359 krs + ε 46.71 0.07234
LATERAL VELOCITY OVERSHOOT (y7) MULTIPLE REGRESSION MODEL F R2 -6 -5 -3 -2 y7 = 1.0602 + 5.34x10 m+1.54x10 Iz – 2.31x10 S.F. – 4.04x10 krs + ε 557.82 0.4822
ANGULAR VELOCITY OVERSHOOT (y8) MULTIPLE REGRESSION MODEL F R2 -4 -5 -3 -2 y8 = 1.0587 + 2.07x10 m – 5.9x10 Iz – 3.63x10 S.F. – 4.09x10 krs + ε 11351.1 0.9499
Table 5.2. Multiple Regression Results
64
Table 5.3 shows the results of a multiple regression analysis involving only the parameters vehicle mass m and yaw moment of inertia Iz.
LATERAL VELOCITY RISE TIME (y3) MULTIPLE REGRESSION MODEL F R2 -5 -5 y3 = 0.2649 – 6.10x10 m+8.07x10 Iz + ε 204.20 0.8988
LATERAL VELOCITY PEAK-RESPONSE TIME (y5) MULTIPLE REGRESSION MODEL F R2 -5 -4 y5 = 0.4176 – 8.20x10 m+1.23x10 Iz + ε 208.18 0.9005
ANGULAR VELOCITY PEAK-RESPONSE TIME (y6) MULTIPLE REGRESSION MODEL F R2 -5 -4 y6 = 0.18.42 – 9.40x10 m+1.02x10 Iz + ε 250.0 0.9159
Table 5.3. Multiple Regression Results. Revised.
Table 5.4 shows multiple regression models that include cross products of the four vehicle parameters.
65
STEADY STATE LATERAL VELOCITY (y1) MULTIPLE REGRESSION MODEL F R2 -4 -9 -2 -20 y1 = 0.26- 2.67x10 m+1.45x10 Iz-7.32x10 S.F. – 0.56krs -1.94x10 mIz- -5 -5 -9 -9 1061.3 0.8162 -5.32x10 mS.F.+9.84x10 mkrs-4.2x10 IzS.F.+6.52x10 Izkrs+0.20S.F.krs + ε
STEADY STATE ANGULAR VELOCITY (y2) MULTIPLE REGRESSION MODEL F R2 -5 -17 -21 y2 = 0.15- 3.04x10 m-9.91x10 Iz+0.011S.F. +0.037krs -3.95x10 mIz+ -7 -6 -17 -17 6031.7 0.9619 +2.09x10 mS.F.-7.48x10 mkrs+7.93x10 IzS.F.-1.77x10 Izkrs-0.012S.F.krs + ε
LATERAL VELOCITY RISE TIME (y3) MULTIPLE REGRESSION MODEL F R2 -5 -4 -8 y3 = -0.60+ 4.51x10 m+1.43x10 Iz+0.22S.F. + 1.20krs -4.56x10 mIz+ -4 -4 -6 -5 86.62 0.2660 +1.04x10 mS.F.-2.10x10 mkrs-7.0x10 IzS.F.+1.94x10 Izkrs-0.47S.F.krs + ε
ANGULAR VELOCITY RISE TIME (y4) MULTIPLE REGRESSION MODEL F R2 + -5 -5 -2 -8 y4 = -0.10 1.38x10 m+8.66x10 Iz+5.86x10 S.F. + 0.11krs -2.61x10 mIz+ -6 -5 -6 -5 -2 234.57 0.4953 +6.89x10 mS.F.-2.63x10 mkrs-9.9x10 IzS.F.+2.12x10 Izkrs-5.6x10 S.F.krs + ε
LATERAL VELOCITY PEAK-RESPONSE TIME (y5) MULTIPLE REGRESSION MODEL F R2 -5 -4 -8 y5 = 2.01+ 6.12x10 m+1.54x10 Iz+0.60S.F. + 4.10krs -5.44x10 mIz+ -4 -4 -5 -5 59.98 0.2006 +2.89x10 mS.F.-6.05x10 mkrs+6.38x10 IzS.F.-6.89x10 Izkrs-1.54S.F.krs + ε
ANGULAR VELOCITY PEAK-RESPONSE TIME (y6) MULTIPLE REGRESSION MODEL F R2 -5 -4 -8 y6 = -0.97+ 6.56x10 m+2.0x10 Iz+0.35S.F. + 1.45krs -7.08x10 mIz+ -4 -4 -5 -5 31.97 0.1180 +1.12x10 mS.F.-2.3x10 mkrs-1.61x10 IzS.F.+4.18x10 Izkrs-0.62S.F.krs + ε
LATERAL VELOCITY OVERSHOOT (y7) MULTIPLE REGRESSION MODEL F R2 -5 -5 -2 -8 y7 = 1.3- 9.32x10 m-1.88x10 Iz-7.47x10 S.F. – 0.18krs +2.4x10 mIz+ -5 -5 -6 -8 -2 1009.1 0.8085 +1.0x10 mS.F.+2.95x10 mkrs-1.3x10 IzS.F.+9.08x10 Izkrs+6.0x10 S.F.krs + ε
ANGULAR VELOCITY OVERSHOOT (y8) MULTIPLE REGRESSION MODEL F R2
-4 -6 -2 -8 y8 = 1.07+ 2.76x10 m-3.43x10 Iz-7.81x10 S.F. – 0.18krs -4.12x10 mIz+ -6 -5 -7 -6 10490.7 0.9777 +5.55x10 mS.F.+1.31x10 mkrs-8.49x10 IzS.F.+7.44x10 Izkrs+0.07S.F.krs + ε
Table 5.4. Multiple Regression Results. Cross products of parameters.
66 6. DISCUSSION OF RESULTS AND CONCLUSIONS
6.1 Summary
This dissertation presents a parametric study of the lateral dynamics using a newly- developed four-wheel road-vehicle model. Included in this model is a tire model which accounts for the nonlinear behavior of pneumatic tires. This model is intended to have a broad range of applicability so that it may be applied with the majority of personal vehicle types currently in production throughout the world.
The governing equations of motion produced by the model are programmed in MATLAB and solved using a fourth-order Runge-Kutta integrator.
The parametric study is believed to be the most comprehensive study undertaken to date with the results providing new insight into the complex phenomena of lateral vehicle dynamics. The study quantifies the effect of a large class of geometric and physical vehicle parameters.
For this study, several response characteristics are chosen as indicators of the effects of individual vehicle parameter changes on the lateral dynamics of on-highway passenger vehicles. Basic vehicle parameters are modified individually and the variation plots in
Chapter 5 illustrate the effects of these changes. The case where several vehicle parameters are modified simultaneously is also studied to determine if unexpected
67 interactive effects appear. The longitudinal position of the mass center turns out to be the most influential parameter for the response characteristics. Indeed when the longitudinal position of the mass center is varied most of the response characteristics seem to change exponentially. The other vehicle parameters have either important effects, small effects, or no appreciable effects on response characteristics. These effects are nearly linear in some cases. A regression analysis helps quantify these observations.
6.2 Discussion of Results
Chapter 5 presents the results of this research. A variation analysis was performed, which consisted of computing the effects of individual parameter variation on the dynamic response characteristics of a four-wheel road-vehicle model. The results of the variation analysis show that the longitudinal position of the mass center, represented by the distance b from the mass center to the rear axle, is by far the most influential parameter.
This observation is consistent with earlier theories developed with the Bicycle Model.
The variation of the response characteristics due to the parameter b is discovered to be exponential – a valuable finding of the study. The variation analysis on the lateral and angular velocity overshoot show that for these response characteristics, the effect of the longitudinal position of the mass center is of the same order of magnitude as the effects of the rest of the vehicle parameters. However, the data for the parameter b shows excessive scatter in the lateral velocity overshoot analysis, as seen in Figure 5.31, which prevents the formulation of reliable conclusions.
68 It can be seen in the variation plots of Section 5.1 that the variation of response characteristics due to vehicle parameters other than b, appear to follow a linear behavior.
A simple linear regression analysis provides quantitative insight on the observations mentioned above. A strong relationship between cause and effect in the data is reflected by a value of r2 close to one. A high r2 value in a linear regression analysis means that the data can be explained by a linear representation. In the case of the parameter b, the regression analysis is performed after the data has been transformed by applying the logarithm function. The exponential formulas in Chapter 5 are obtained by applying the exponential function to the regression line formulas.
The simple linear regression analysis performed for the parameter b shows that for six of the eight response characteristics, the r2 values range from 0.9762 to 0.7937. These results show that for most response characteristics, the parameter b does indeed follow an exponential behavior. The two remaining response characteristics have somewhat lower r2 values, preventing any strong conclusions. However, these r2 values are high enough to suggest an exponential tendency for these two response characteristics.
The slope β1 in Table 5.1 is the slope of the regression line and it allows us to compare the magnitude of the effects of each parameter on the response characteristics. The slope
β1 is nondimensional, therefore it allows these comparisons to be made. Since the parameter b has an exponential behavior, the slope of each exponential regression line is not constant. For comparison purposes, the slope β1 for the parameter b is taken at the
69 nominal value of b. As can be observed in Table 5.1, the slope corresponding to the parameter b is much higher than that of the other parameters, except in the case of the lateral and angular overshoot. In these cases, the slope β1 corresponding to the parameter b is of the same order of magnitude as the rest of the parameters, but still larger in both cases. The response characteristic most affected by the parameter b is the steady state lateral velocity, with a slope of -7.96. This slope is many times greater in absolute value than the next largest of 0.9154, which corresponds to the mass of the vehicle. This result shows the overwhelming effect of the mass center longitudinal position on the steady state lateral velocity. It is interesting to note that the effect of b on this response characteristic is opposite to the effect of vehicle mass. Similar observations can be derived from Table 5.1. It is important to note that the angular velocity overshoot is the only response characteristic for which the effects of b near its nominal values, are of the same order of magnitude as vehicle mass, which is the next most influential parameter.
For all parameters other than b, most of the linear regression analysis produces values of r2 greater than 0.65 and in many cases greater than 0.80. There are a few cases in which the excessive scatter of data and constant (slope equal to 0) behavior result in low values of r2, preventing strong conclusions to be made about the trends. This scatter of data appears to be a consequence of resolution limit in the numerical computations.
The following is a discussion of the individual effects on the response characteristics of the vehicle parameters that behave linearly (those different from b), based on the results of the linear regression analysis:
70
• Steady State Lateral Velocity: For this response characteristic, the most influential
vehicle parameter is the vehicle mass m, with a regression line slope of 0.9154 which
is over twenty times greater than the slope of the stability factor S.F. and over ten
times greater than that of the rear roll stiffness coefficient krs. The yaw moment of
inertia Iz has no appreciable effect on this response characteristic.
• Steady State Angular Velocity: The most influential parameter for this response
characteristic is the vehicle mass m. The stability factor S.F. and the rear roll stiffness
coefficient krs are similar in magnitude and about twenty times less influential than
the vehicle mass regarding this response characteristic. The steady state angular
velocity appears not to be affected by the yaw moment of inertia Iz.
• Lateral Velocity Rise Time: The yaw moment of inertia Iz has the most effect on this
response characteristic as noted by the regression line slope β1, which is the largest in
this group. It also exhibits the least amount of scatter in the data and the strongest
relationship between cause and effect, as indicated by the high value of r2 of 0.9974.
The second most influential vehicle parameter is the vehicle mass m, with a
regression line slope 42% smaller than that of the yaw moment of inertia Iz. The
stability factor SF and the rear roll stiffness coefficient krs have effects that are about
an order of magnitude lower than the most influential parameter in this group.
• Angular Velocity Rise Time: for this response characteristic the vehicle mass m and
the yaw moment of inertia Iz have similar regression line slope magnitude but
different signs, which suggest similar levels of influence on angular velocity response
time, but in opposite directions. The values of r2 for these parameters are also close to
71 one. The effect of the other two vehicle parameters is inconclusive because of the low
r2 values obtained in the regression analysis.
• Lateral Velocity Peak-Response Time: The yaw moment of inertia Iz is again the most
influential parameter in this group and the one with r2 closest to one. The vehicle
mass m presents an influence level about 19% lower than the yaw moment of inertia
Iz. The rear roll stiffness coefficient krs presents an influence about half the
magnitude of the vehicle mass m, but about double that of the stability factor S.F, the
least influential vehicle parameter in this group.
• Angular Velocity Peak-Response Time: As above, the yaw moment of inertia Iz is the
most influential vehicle parameter concerning this response characteristic, and also
the one with the highest r2 value. The effect of the vehicle mass m is about 32%
smaller in magnitude and also of opposite sign. The stability factor S.F. shows no
effect on this response characteristic. The rear roll stiffness coefficient krs shows an
effect about 20 times smaller than that of vehicle mass.
• Lateral Velocity Overshoot: The data points for the vehicle mass m present
substantial scatter in the region to the left of Figure 5.32 (below the nominal value),
therefore a cause and effect relationship cannot be inferred, at least in that region. The
rest of the vehicle parameters show a little influence on this response characteristic,
although the yaw moment of inertia Iz seems to be the strongest of the group and the
slope of its regression line is about 36% larger than that of the rear roll stiffness
coefficient krs.
• Angular Velocity Overshoot: The most influential vehicle parameter for this response
characteristic is vehicle mass m, with a regression line slope more than double in
72 magnitude from that of the yaw moment of inertia Iz. The other two vehicle
parameters are similar in their influence on this response characteristic. It is important
to note that for this response characteristic, the effect corresponding to parameter b is
of the same order of magnitude as the rest of the parameters.
As seen above, certain vehicle parameters are more influential than others in affecting a particular response characteristic when these parameters change individually. For example, if the lateral velocity rise time is of importance, the yaw moment of inertia Iz will be the most effective parameter. Alternatively, if the angular velocity overshoot is chosen for study, the vehicle mass m will be about 2.5 times more influential than the yaw moment of inertia mentioned above. Finally, for the steady state lateral and angular velocity the yaw moment of inertia Iz has no effect, but the stability factor S.F. and the rear roll-stiffness coefficient krs have equivalent levels of influence on these response characteristics.
The linear regression analysis revealed some cases in which r2 was too low. This means that there was a very low cause and effect relationship, explained in some cases by the fact that some parameters were of little or no influence to some response characteristics, or alternatively, that in some of these cases, there was excessive data point scatter.
The multiple regression analysis described in Chapter 5 show that the regression models for steady-state lateral and angular velocity, and angular velocity overshoot fit the data very well. The R2 value (equivalent to the r2 value in the simple linear regression) for
73 these three models is close to one. For the other regression models, the R2 value is not high which suggests that the data does not follow a linear trend. For all regression models, however, the F value is always higher than the F0.05 value, which is obtained from tables in the literature. For the number of degrees of freedom of the data used in this analysis, the value of F0.05 is equal to 2.37. A value of F higher than the F0.05 value indicates the global usefulness of the model with a confidence of 100(1-0.05)%. This is called the analysis of variance F-test [65]. Based on this test, all models generated by the multiple regression analysis appear to be useful for predicting their respective response characteristics. Additionally, the results of the F-test provide strong evidence that, in all cases, at least one of the model coefficients is nonzero.
Tables 5.3 and 5.4 show the results of an additional multiple regression analysis. The objective was to obtain more insight on the nature of the response characteristics for which the regression models have low R2 values, as seen in Table 5.2. The results of the multiple regression analysis involving only the vehicle parameters mass and yaw moment of inertia (m and Iz, respectively) and shown in Table 5.3 reveal that when ignoring all but these two parameters, the R2 values increase to satisfactory levels. This however does not necessarily mean that regression models in Table 5.3 are valid, because if regression analysis values change dramatically when ignoring certain parameters, this means that those ignored parameters may have an appreciable effect on the data and should not be left out. Furthermore, these results suggest the existence of parameter interaction in the cases where there are low R2 and F-test values. Table 5.4 shows the results of a regression analysis involving cross products of the four vehicle parameters included in
74 this analysis. The cross product of parameters is an attempt to address parameter interaction. The new R2 and F-test values in general, did not increase any more than is expected when increasing the number of parameters in the regression model (the cross product of parameters are essentially new parameters). In view of these results, attempting other forms of nonlinearities and parameter interaction in the regression analysis, in the search for better values of R2 and F-test, is both impractical and not part of the objectives of this study. Therefore, the conclusion that can be made about the regression models with low R2 and F-test values is that the response characteristics
Lateral Velocity Rise Time, Angular Velocity Rise Time, Lateral Velocity Peak-
Response Time, Angular Velocity Peak-Response Time, and Lateral Velocity Overshoot are influenced in a nonlinear way when simultaneously varying the vehicle parameters.
This reveals a nonlinear parameter interaction for these cases.
6.3 Contributions and advances from the research.
• This dissertation presents results of a comprehensive parametric study which
includes the nonlinear characteristics of pneumatic tires on the lateral dynamics of
a road-vehicle. These results provide new insight into the effects of lateral weight
transfer on lateral vehicle dynamics. Such effects cannot be studied with simpler,
previously used models such as the bicycle model.
• The parametric study in this research revealed that in the nonlinear region of tire
lateral load, the longitudinal position of the mass center is the most influential
parameter in the lateral dynamics of a road-vehicle model. Additionally, the
75 regression analysis shows that this parameter affects vehicle response
exponentially.
• The vehicle parameters other than the longitudinal position of the mass center
influence the response characteristics in a way that is nearly linear, even in the
nonlinear region of tire lateral load. A multiple regression analysis shows that for
some response characteristics, the effects of the different vehicle parameters can
be determined and studied independently.
• This study provides valuable insight on the influence of a broad range of basic
vehicle parameters on the lateral dynamics of road-vehicles. The research results
provide both a qualitative and a quantitative understanding of the relationship
between basic vehicle parameters and their effects on the lateral dynamics of
road-vehicles. These results allow estimations to be made about both magnitude
and direction of the effects of different vehicle parameters on response
characteristics, even in the nonlinear region of tire lateral load.
• Due to the broad nature of the road-vehicle model developed in this research, the
results obtained have a wide range of application with insight not previously
available.
• The parametric insight developed in this study further advances the knowledge in
the field of Vehicle Dynamics thus providing a major advance over the commonly
used bicycle model.
76 6.4 Future directions.
The mathematical models and procedures developed and used in this study have proven to be adequate for the objectives and scope formulated in the beginning of this dissertation. However, as is usually the case when developing any research topic, many new directions and possibilities are found. Aspects that were once unknown are now motivation for increasing the scope of the research efforts and improving the results. This has been the case in this study. Several possibilities have been identified and are described below.
• The insight obtained by this parametric study can be enhanced substantially by
extending into the frequency domain. The effect of vehicle parameters on the
frequency response can be assessed and quantified. Studies in the frequency
domain would include modifying input characteristics such as frequency and
amplitude, in addition to vehicle parameters.
• A roll-stiffness coefficient for the front axle can be included in addition to the
current rear roll-stiffness coefficient krs. This enhancement would allow the study
of the effect of roll stiffness on both axles.
• Including steering system parameters such as variable steering ratio or steering
compliance would increase the number of vehicle parameters without losing
generality in the results.
• The road-vehicle model used in this study simulates a parallel steering system,
that is, one where the front wheels share the same steering angle. While this is a
valid approximation due to the relatively small steering angles involved, adding a
77 vehicle parameter that quantifies the amount of Ackermann steering geometry in
the front wheels would enhance the results of this study.
• An analytical approach to this topic, as opposed to a simulation-based approach,
would enhance the insight obtained in this study and build a more fundamental
body of knowledge.
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90 APPENDIX A: THE TIRE MODEL
A.1 The Piece-wise Tire Model
The lateral forces developed by the tires, Fy, can be represented in a nondimensional form,
Fy = fµFz (A.8)
where µ is the coefficient of friction between the tire and the road, Fz is the normal force on the tire and f is the nondimensional or normalized tire lateral force. The normalized slip angle is defined as,
C tanα α = α (A.9) µFz
where tanα is the tangent of the tire slip angle. For the front wheels:
−Vy − aΩ tanα1,2 = (A.10) Vx
For the rear wheels:
91 −Vy + bΩ tanα1,2 = (A.11) Vx
To approximate the normalized tire lateral force f, the author developed a piece-wise tire model that consists of a finite number of linear segments that approximates the nonlinear nature of f, as described by the Magic Formula (Pacejka, 1993). The nonlinear nature of f can be seen in Figure A.1. The Magic Formula is described in the next section of this appendix.
The piece-wise tire model used in this research consists of 26 linear segments. The constants that define these segments are:
p01=0.1 p05=0.545 p09=1.38 q1=0.1 q5=0.5 q9=0.9
p02=0.20 p06=0.681 p1=2.0 q2=0.2 q6=0.6 qp2=0.9
p03=0.31 p07=0.85 p c=3.0 q3=0.3 q7=0.7
p04=0.42 p08=1.06 p2=8.7 q4=0.4 q8=0.8
92
Piece-wise Tire Model
1
0.8
0.6
0.4
0.2
0
Normalized lateral force lateral Normalized -0.2
-0.4
-0.6
-0.8
-1
-10 -8 -6 -4 -2 0 2 4 6 8 10 Normalized slip angle
Figure A.1. Piece-wise Tire Model
93
To generate the positive portion of the piece-wise tire model, as seen in Figure A.1, the following equations were used,
for α ≤ p01 ,
q1 f = α (A.12) p01 for α ≤ p02
q2 − q1 f = ()α − p01 + q1 (A.13) p02 − p01
for α ≤ p03
q3 − q2 f = ()α − p02 + q2 (A.14) p03 − p02
for α ≤ p04
q4 − q3 f = ()α − p03 + q3 (A.15) p04 − p03
94 for α ≤ p05
q5 − q4 f = ()α − p04 + q4 (A.16) p05 − p04
for α ≤ p06
q6 − q5 f = ()α − p05 + q5 (A.17) p06 − p05
for α ≤ p07
q7 − q6 f = ()α − p06 + q6 (A.18) p07 − p06
for α ≤ p08
q8 − q7 f = ()α − p07 + q7 (A.19) p08 − p07
for α ≤ p09
q9 − q8 f = ()α − p08 + q8 (A.20) p09 − p08
95
for α ≤ p1
1− q9 f = ()α − p09 + q9 (A.21) p1− p09
for α ≤ pc
f = 1 (A.22)
for α ≤ p2
qp2 −1 f = ()α − pc +1 (A.23) p2 − pc
for α > p2
f = qp2 (A.24)
The negative portion of the model is obtained by sign changes in the above equations.
The piece-wise model approximates the basic behavior of the lateral force curve of a real tire. The slope of f decreases as the normalized slip angle increases, up to a maximum
96 value of 1, corresponding to a slip angle value of p1=2 in the model. After the maximum value is reached, the slope of f becomes negative and increases in negative value as the slip angle increases, until it reaches an inflection point at p2=8.7 in the model used for
Figure A.1.
Figure A.2 shows the piece-wise tire model without the normalization of slip angle and lateral force.
Another important aspect of nonlinear tire behavior that the piece-wise model approximates is the tire load sensitivity effect. As mentioned in Chapter 3, the peak lateral friction coefficient, Fy/Fz of the tire decreases as the normal load Fz increases. This effect makes the lateral force generated by an axle decrease as lateral weight transfer changes the normal load on the inside and outside tires, when the vehicle is on a curved path. The tire load sensitivity for the piece-wise tire model is shown in Figures A.3 and
A.4.
The discontinuous nature of the piece-wise tire model does not affect numerical computation when using the Fourth-order Runge-Kutta integration method with a fixed step size. This is because there is no differentiation or integration of the tire model in this method. The solution process programmed into the MATLAB script simply takes one value of the piece-wise tire model for every step in the integration, including the mid- steps. Therefore, even at the endpoints of the linear segments in the tire model, there are no discontinuities and hence no chance for singularities or values that tend to infinity.
97
Piece-wise linear tire model 6000
4000
2000
0 lateral force, N force, lateral
-2000
-4000
-6000 -80 -60 -40 -20 0 20 40 60 80 slip angle, deg
Figure A.2. The Piece-wise Tire Model
98
Lateral force vs. Slip Angle for Several Normal Loads. Piece-wise Tire Model 5000
4000
3000
2000
1000 Increasing Normal Load
0 Lateral force, N Lateral -1000
-2000
-3000
-4000
-5000 -50 -40 -30 -20 -10 0 10 20 30 40 50 Slip angle, deg
Figure A.3. Lateral Force vs. Slip Angle for Several Normal Loads. Piece-wise Tire
Model
99
Lateral force vs. Normal force for several slip angles. Piece-wise Tire Model 4500
4000
8 deg 3500
Increasing Slip Angle 6 deg 3000
2500 4 deg
Lateral force, N Lateral 2000
1500 2 deg
1000
500
0 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Normal force, N
Figure A.4. The Tire Load Sensitivity Effect. Piece-wise Tire Model
100 A.2 The Magic Formula for tire lateral force.
The Magic Formula is an empirical curve fitting technique developed by Pacejka et al
(Pacejka, 1993). and it is widely used for vehicle dynamics studies. The basic mathematical expressions for the Magic Formula are,
f = D′sinθ (A.25)
θ = C′arctan(B′φ) (A.26)
E′ φ = ()1− E′ α + arctan()B′α (A.27) B′
The parameters B’, C’, D’ and E’ have values of 0.714, 1.40, 1.00 and -0.20 respectively for a particular tire of size P195/70R-14. Figure A.5 shows a comparison between the curves for f generated by the Magic Formula and the piece-wise linear tire model. It can be seen in Figure A.5 that the piece-wise linear model closely approximates the curve generated by the Magic Formula. The advantage of the piece-wise tire model is that each segment can be positioned arbitrarily close to the experimental curve or data by individually modifying the constant values described in section A.1, until a satisfactory fit is obtained throughout the entire range of interest. For the purpose of this research, the
Magic Formula was used to fit the segments of the piece-wise model because experimental tire data is difficult to obtain and often proprietary.
101
Comparison Between Magic Formula and Piece-wise Tire models
1
0.8
0.6 Normalized lateral force lateral Normalized 0.4
Magic Formula 0.2 Piece-wise Model
0 0 1 2 3 4 5 6 7 8 9 10 Normalized slip angle
Figure A.5. Comparison between the Magic Formula and the Piece-wise Tire Model
To establish the validity of the piece-wise tire model for lateral dynamics simulation, a comparison is made with the same simulation using the Magic formula tire model. For this validation, a five second simulation is run with a 2 deg step steer and 20 m/s forward velocity. The data set chosen is that of the mid-size road-vehicle. Figures A.6 and A.7 show the time history of the lateral velocity and angular velocity, respectively. It is evident that the results using both tire models are almost identical.
102 Lateral Velocity. Piece-wise Model and Magic Formula 0.1 Piece-wise Model Magic Formula
0
-0.1
-0.2
Lateral velocity, velocity, m/s Lateral -0.3
-0.4
-0.5
-0.6 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time, s
Figure A.6. Lateral Velocity. Comparison Between Piece-wise and Magic Formula
Tire Models
103 Angular Velocity. Piece-wise Model and Magic Formula 9 Piece-wise Model Magic Formula 8
7
6
5
4 Angular velocity, deg/s Angular velocity,
3
2
1
0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time, s
Figure A.7. Angular Velocity. Comparison Between Piece-wise and Magic Formula
Tire Models
104 A.3 The coefficient of friction µ.
The coefficient of friction µ, between the tire and the road is not a constant value, but rather decreases monotonically as the normal load is increased in the case of a dry tire/road interface. The expression for the coefficient of friction µ used throughout this research is the linear approximation
F µ = 0.95 − 0.2 z (A.28) W
where Fz is the instantaneous normal load on the tire and W is the static normal load on the tire. This is shown in Figure A.8.
Tire/Road Coefficient of Friction 1
0.95
0.9
0.85
0.8 Coefficient of Friction of Coefficient
0.75
0.7
0.65 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Normal Load on Tire, N
Figure A.8. Coefficient of Friction
105 APPENDIX B: Basic MATLAB Code for Variation Analysis
%Fourth order Runge-Kutta
%clear all
tic
%Parameter set 1:
g=9.81;
m=1000;
V=20;
Delta=2;
t=10;
N=8000;
h=t/N;
y=zeros(N+1,2);
%Initial conditions
y(1,1)=0;
y(1,2)=0;
b=1.6;
a=2.6-b;
hh=0.5;
c=1.5;
krs=1;
106 mu=0.95; su=0.2;
Iz=1400;
%Nominal Calpha
Calpha=50000;
%Piece-wise linear tire model parameters
p01=0.1; p02=0.20; p03=0.31; p04=0.42; p05=0.545; p06=0.681; p07=0.85; p08=1.06; p09=1.38; p1=2.0; pc=3.0; p2=8.7; q1=0.1;
107 q2=0.2; q3=0.3; q4=0.4; q5=0.5; q6=0.6; q7=0.7; q8=0.8; q9=0.9; qp2=0.9;
dVy=zeros(N+1,1);
IntOmega=zeros(1,N+1);
var=11; for kk=0:var-1
per=15;
b=1.6*(1-per/100)+(kk*(1.6*(2*per/100)))/(var-1);
a=2.6-b;
kk
for ii=1:N
108 Vx=sqrt((V^2)-(y(ii,1)^2));
Vxx(ii)=Vx;
%Step steer input
B=Delta*pi/180;
BBB(ii)=B;
if ii==1
dVy(ii)=0; else
dVy(ii)=((y(ii,1)-y(ii-1,1))/h); end
Fz1=(m*b*g/((a+b)*2)) + (m*b*hh*(dVy(ii)+Vx*y(ii,2))/((a+b)*c)); if Fz1<0
Fz1=0; end
Fz2=(m*b*g/((a+b)*2)) - (m*b*hh*(dVy(ii)+Vx*y(ii,2))/((a+b)*c)); if Fz2<0
Fz2=0; end
109 Fz3=(m*a*g/((a+b)*2)) - (m*a*hh*krs*(dVy(ii)+Vx*y(ii,2))/((a+b)*c)); if Fz3<0
Fz3=0; end
FFz3(ii)=Fz3;
Fz4=(m*a*g/((a+b)*2)) + (m*a*hh*krs*(dVy(ii)+Vx*y(ii,2))/((a+b)*c)); if Fz4<0
Fz4=0; end
FFz4(ii)=Fz4;
if Fz1==0 & Fz4==0
('Two-wheel lift (Left side: Fz1 and Fz4). Discard result')
pause else end
if Fz2==0 & Fz3==0
('Two-wheel lift (Right side: Fz2 and Fz3). Discard result')
pause
110 else end
if Fz1==0
Fz2=m*b*g/((a+b)); else end
if Fz2==0
Fz1=m*b*g/((a+b)); else end
if Fz3==0
Fz4=m*a*g/((a+b)); else end
if Fz4==0
111 Fz3=m*a*g/((a+b)); else end
%Friction coefficient formula
u1=mu - (su/(m*g))*Fz1; u2=mu - (su/(m*g))*Fz2; u3=mu - (su/(m*g))*Fz3; u4=mu - (su/(m*g))*Fz4;
if Fz1==0
a_1=0; else
a_1=(Calpha/(u1*Fz1))*(((-y(ii,1)-a*y(ii,2))/Vx)+B); end a_11(ii)=a_1;
if Fz2==0
a_2=0; else
a_2=(Calpha/(u2*Fz2))*(((-y(ii,1)-a*y(ii,2))/Vx)+B);
112 end a_22(ii)=a_2;
if Fz3==0
a_3=0; else
a_3=(Calpha/(u3*Fz3))*((-y(ii,1)+b*y(ii,2))/Vx); end a_33(ii)=a_3;
if Fz4==0
a_4=0; else
a_4=(Calpha/(u4*Fz4))*((-y(ii,1)+b*y(ii,2))/Vx); end a_44(ii)=a_4;
if a_1<0
113 if abs(a_1) <= p01
f1=-((q1/p01)*abs(a_1));
elseif abs(a_1) <= p02
f1=-(((q2-q1)/(p02-p01))*(abs(a_1)-p01)+q1);
elseif abs(a_1) <= p03
f1=-(((q3-q2)/(p03-p02))*(abs(a_1)-p02)+q2);
elseif abs(a_1) <= p04
f1=-(((q4-q3)/(p04-p03))*(abs(a_1)-p03)+q3);
elseif abs(a_1) <= p05
f1=-(((q5-q4)/(p05-p04))*(abs(a_1)-p04)+q4);
elseif abs(a_1) <= p06
f1=-(((q6-q5)/(p06-p05))*(abs(a_1)-p05)+q5);
elseif abs(a_1) <= p07
f1=-(((q7-q6)/(p07-p06))*(abs(a_1)-p06)+q6);
114
elseif abs(a_1) <= p08
f1=-(((q8-q7)/(p08-p07))*(abs(a_1)-p07)+q7);
elseif abs(a_1) <= p09
f1=-(((q9-q8)/(p09-p08))*(abs(a_1)-p08)+q8);
elseif abs(a_1) <= p1
f1=-(((1-q9)/(p1-p09))*(abs(a_1)-p09)+q9);
elseif abs(a_1) <= pc
f1=-1;
elseif abs(a_1) <= p2
f1=-(((qp2-1)/(p2-pc))*(abs(a_1)-pc)+1);
elseif abs(a_1) >p2
115 f1=-qp2;
end
elseif a_1>=0
if a_1 <= p01
f1=(q1/p01)*a_1;
elseif a_1 <= p02
f1=((q2-q1)/(p02-p01))*(a_1-p01)+q1;
elseif a_1 <= p03
f1=((q3-q2)/(p03-p02))*(a_1-p02)+q2;
elseif a_1 <= p04
116 f1=((q4-q3)/(p04-p03))*(a_1-p03)+q3;
elseif a_1 <= p05
f1=((q5-q4)/(p05-p04))*(a_1-p04)+q4;
elseif a_1 <= p06
f1=((q6-q5)/(p06-p05))*(a_1-p05)+q5;
elseif a_1 <= p07
f1=((q7-q6)/(p07-p06))*(a_1-p06)+q6;
elseif a_1 <= p08
f1=((q8-q7)/(p08-p07))*(a_1-p07)+q7;
elseif a_1 <= p09
f1=((q9-q8)/(p09-p08))*(a_1-p08)+q8;
117 elseif a_1 <= p1
f1=((1-q9)/(p1-p09))*(a_1-p09)+q9;
elseif a_1 <= pc
f1=1;
elseif a_1 <= p2
f1=((qp2-1)/(p2-pc))*(a_1-pc)+1;
elseif a_1 >p2
f1=qp2;
end end
ff1(ii)=f1;
if a_2<0
118
if abs(a_2) <= p01
f2=-((q1/p01)*abs(a_2));
elseif abs(a_2) <= p02
f2=-(((q2-q1)/(p02-p01))*(abs(a_2)-p01)+q1);
elseif abs(a_2) <= p03
f2=-(((q3-q2)/(p03-p02))*(abs(a_2)-p02)+q2);
elseif abs(a_2) <= p04
f2=-(((q4-q3)/(p04-p03))*(abs(a_2)-p03)+q3);
elseif abs(a_2) <= p05
f2=-(((q5-q4)/(p05-p04))*(abs(a_2)-p04)+q4);
elseif abs(a_2) <= p06
119 f2=-(((q6-q5)/(p06-p05))*(abs(a_2)-p05)+q5);
elseif abs(a_2) <= p07
f2=-(((q7-q6)/(p07-p06))*(abs(a_2)-p06)+q6);
elseif abs(a_2) <= p08
f2=-(((q8-q7)/(p08-p07))*(abs(a_2)-p07)+q7);
elseif abs(a_2) <= p09
f2=-(((q9-q8)/(p09-p08))*(abs(a_2)-p08)+q8);
elseif abs(a_2) <= p1
f2=-(((1-q9)/(p1-p09))*(abs(a_2)-p09)+q9);
elseif abs(a_2) <= pc
f2=-1;
120 elseif abs(a_2) <= p2
f2=-(((qp2-1)/(p2-pc))*(abs(a_2)-pc)+1);
elseif abs(a_2) >p2
f2=-qp2;
end
elseif a_2>=0
if a_2 <= p01
f2=(q1/p01)*a_2;
elseif a_2 <= p02
f2=((q2-q1)/(p02-p01))*(a_2-p01)+q1;
121 elseif a_2 <= p03
f2=((q3-q2)/(p03-p02))*(a_2-p02)+q2;
elseif a_2 <= p04
f2=((q4-q3)/(p04-p03))*(a_2-p03)+q3;
elseif a_2 <= p05
f2=((q5-q4)/(p05-p04))*(a_2-p04)+q4;
elseif a_2 <= p06
f2=((q6-q5)/(p06-p05))*(a_2-p05)+q5;
elseif a_2 <= p07
f2=((q7-q6)/(p07-p06))*(a_2-p06)+q6;
elseif a_2 <= p08
f2=((q8-q7)/(p08-p07))*(a_2-p07)+q7;
122
elseif a_2 <= p09
f2=((q9-q8)/(p09-p08))*(a_2-p08)+q8;
elseif a_2 <= p1
f2=((1-q9)/(p1-p09))*(a_2-p09)+q9;
elseif a_2 <= pc
f2=1;
elseif a_2 <= p2
f2=((qp2-1)/(p2-pc))*(a_2-pc)+1;
elseif a_2 >p2
f2=qp2;
end end
123
ff2(ii)=f2;
if a_3<0
if abs(a_3) <= p01
f3=-((q1/p01)*abs(a_3));
elseif abs(a_3) <= p02
f3=-(((q2-q1)/(p02-p01))*(abs(a_3)-p01)+q1);
elseif abs(a_3) <= p03
f3=-(((q3-q2)/(p03-p02))*(abs(a_3)-p02)+q2);
elseif abs(a_3) <= p04
f3=-(((q4-q3)/(p04-p03))*(abs(a_3)-p03)+q3);
124 elseif abs(a_3) <= p05
f3=-(((q5-q4)/(p05-p04))*(abs(a_3)-p04)+q4);
elseif abs(a_3) <= p06
f3=-(((q6-q5)/(p06-p05))*(abs(a_3)-p05)+q5);
elseif abs(a_3) <= p07
f3=-(((q7-q6)/(p07-p06))*(abs(a_3)-p06)+q6);
elseif abs(a_3) <= p08
f3=-(((q8-q7)/(p08-p07))*(abs(a_3)-p07)+q7);
elseif abs(a_3) <= p09
f3=-(((q9-q8)/(p09-p08))*(abs(a_3)-p08)+q8);
elseif abs(a_3) <= p1
f3=-(((1-q9)/(p1-p09))*(abs(a_3)-p09)+q9);
125
elseif abs(a_3) <= pc
f3=-1;
elseif abs(a_3) <= p2
f3=-(((qp2-1)/(p2-pc))*(abs(a_3)-pc)+1);
elseif abs(a_3) >p2
f3=-qp2;
end
elseif a_3>=0
if a_3 <= p01
f3=(q1/p01)*a_3;
126
elseif a_3 <= p02
f3=((q2-q1)/(p02-p01))*(a_3-p01)+q1;
elseif a_3 <= p03
f3=((q3-q2)/(p03-p02))*(a_3-p02)+q2;
elseif a_3 <= p04
f3=((q4-q3)/(p04-p03))*(a_3-p03)+q3;
elseif a_3 <= p05
f3=((q5-q4)/(p05-p04))*(a_3-p04)+q4;
elseif a_3 <= p06
f3=((q6-q5)/(p06-p05))*(a_3-p05)+q5;
elseif a_3 <= p07
f3=((q7-q6)/(p07-p06))*(a_3-p06)+q6;
127
elseif a_3 <= p08
f3=((q8-q7)/(p08-p07))*(a_3-p07)+q7;
elseif a_3 <= p09
f3=((q9-q8)/(p09-p08))*(a_3-p08)+q8;
elseif a_3 <= p1
f3=((1-q9)/(p1-p09))*(a_3-p09)+q9;
elseif a_3 <= pc
f3=1;
elseif a_3 <= p2
f3=((qp2-1)/(p2-pc))*(a_3-pc)+1;
elseif a_3 >p2
128 f3=qp2;
end end
ff3(ii)=f3;
if a_4<0
if abs(a_4) <= p01
f4=-((q1/p01)*abs(a_4));
elseif abs(a_4) <= p02
f4=-(((q2-q1)/(p02-p01))*(abs(a_4)-p01)+q1);
elseif abs(a_4) <= p03
f4=-(((q3-q2)/(p03-p02))*(abs(a_4)-p02)+q2);
129
elseif abs(a_4) <= p04
f4=-(((q4-q3)/(p04-p03))*(abs(a_4)-p03)+q3);
elseif abs(a_4) <= p05
f4=-(((q5-q4)/(p05-p04))*(abs(a_4)-p04)+q4);
elseif abs(a_4) <= p06
f4=-(((q6-q5)/(p06-p05))*(abs(a_4)-p05)+q5);
elseif abs(a_4) <= p07
f4=-(((q7-q6)/(p07-p06))*(abs(a_4)-p06)+q6);
elseif abs(a_4) <= p08
f4=-(((q8-q7)/(p08-p07))*(abs(a_4)-p07)+q7);
elseif abs(a_4) <= p09
f4=-(((q9-q8)/(p09-p08))*(abs(a_4)-p08)+q8);
130
elseif abs(a_4) <= p1
f4=-(((1-q9)/(p1-p09))*(abs(a_4)-p09)+q9);
elseif abs(a_4) <= pc
f4=-1;
elseif abs(a_4) <= p2
f4=-(((qp2-1)/(p2-pc))*(abs(a_4)-pc)+1);
elseif abs(a_4) >p2
f4=-qp2;
end
elseif a_4>=0
131
if a_4 <= p01
f4=(q1/p01)*a_4;
elseif a_4 <= p02
f4=((q2-q1)/(p02-p01))*(a_4-p01)+q1;
elseif a_4 <= p03
f4=((q3-q2)/(p03-p02))*(a_4-p02)+q2;
elseif a_4 <= p04
f4=((q4-q3)/(p04-p03))*(a_4-p03)+q3;
elseif a_4 <= p05
f4=((q5-q4)/(p05-p04))*(a_4-p04)+q4;
elseif a_4 <= p06
132 f4=((q6-q5)/(p06-p05))*(a_4-p05)+q5;
elseif a_4 <= p07
f4=((q7-q6)/(p07-p06))*(a_4-p06)+q6;
elseif a_4 <= p08
f4=((q8-q7)/(p08-p07))*(a_4-p07)+q7;
elseif a_4 <= p09
f4=((q9-q8)/(p09-p08))*(a_4-p08)+q8;
elseif a_4 <= p1
f4=((1-q9)/(p1-p09))*(a_4-p09)+q9;
elseif a_4 <= pc
f4=1;
133 elseif a_4 <= p2
f4=((qp2-1)/(p2-pc))*(a_4-pc)+1;
elseif a_4 >p2
f4=qp2;
end
end
ff4(ii)=f4;
A=
(su*(b^2)*(hh^2)/((c^2)*g*((a+b)^2)))*(f1+f2)+(su*(a^2)*(hh^2)*(krs^2)/((c^2)*g*((a
+b)^2)))*(f3+f4);
B= 1-(((mu*b*hh/(c*(a+b)))-(su*(b^2)*hh/(c*((a+b)^2))))*(f1-f2)-
((mu*a*hh*krs/(c*(a+b)))-(su*(a^2)*hh*krs/(c*((a+b)^2))))*(f3-f4));
C_= (((su/4)*g*(b^2)/((a+b)^2))-
((mu/2)*g*b/(a+b)))*(f1+f2)+(((su/4)*g*(a^2)/((a+b)^2))-((mu/2)*g*a/(a+b)))*(f3+f4);
134 D= (m*a*b/((a+b)*Iz))*(((mu/2)*g-((su/4)*g*b/(a+b))-
(su*b*(hh^2)*(dVy(ii)^2)/((c^2)*g*(a+b))))*(f1+f2)+(hh*dVy(ii)/c)*(- mu+(su*b/(a+b)))*(f2-f1));%-Betadd(ii);
E= (m*a*b/((a+b)*Iz))*(((mu/2)*g-((su/4)*g*a/(a+b))-
(su*a*(hh^2)*(krs^2)*(dVy(ii)^2)/((c^2)*g*(a+b))))*(f3+f4)+(hh*krs*dVy(ii)/c)*(mu-
(su*a/(a+b)))*(f4-f3));
BN=2*A*Vx*y(ii,2)+B;
CN=C_+A*(Vx^2)*(y(ii,2)^2)+Vx*y(ii,2)*(B+1);
DN=(m*a*b/((a+b)*Iz))*(((mu/2)*g-((su/4)*g*b/(a+b))-
(su*b*(hh^2)*((dVy(ii)+Vx*y(ii,2))^2)/((c^2)*g*(a+b))))*(f1+f2)+(hh*(dVy(ii)+Vx*y( ii,2))/c)*(-mu+(su*b/(a+b)))*(f2-f1));%-Betadd(ii);
EN=(m*a*b/((a+b)*Iz))*(((mu/2)*g-((su/4)*g*a/(a+b))-
(su*a*(hh^2)*(krs^2)*((dVy(ii)+Vx*y(ii,2))^2)/((c^2)*g*(a+b))))*(f3+f4)+(hh*krs*(dV y(ii)+Vx*y(ii,2))/c)*(mu-(su*a/(a+b)))*(f4-f3));
if abs(A)<0.00001
dVk1(ii)=(-CN)/BN; %-((Vx^2)/R)
k1=y(ii,:)+(h/2)*[dVk1(ii);DN-EN]';
BNk1=2*A*Vx*k1(1,2)+B;
CNk1=C_+A*(Vx^2)*(k1(1,2)^2)+Vx*k1(1,2)*(B+1);
135 DNk1=(m*a*b/((a+b)*Iz))*(((mu/2)*g-((su/4)*g*b/(a+b))-
(su*b*(hh^2)*((dVy(ii)+Vx*k1(1,2))^2)/((c^2)*g*(a+b))))*(f1+f2)+(hh*(dVy(ii)+Vx*k
1(1,2))/c)*(-mu+(su*b/(a+b)))*(f2-f1));%-Betadd(ii);
ENk1=(m*a*b/((a+b)*Iz))*(((mu/2)*g-((su/4)*g*a/(a+b))-
(su*a*(hh^2)*(krs^2)*((dVy(ii)+Vx*k1(1,2))^2)/((c^2)*g*(a+b))))*(f3+f4)+(hh*krs*(d
Vy(ii)+Vx*k1(1,2))/c)*(mu-(su*a/(a+b)))*(f4-f3));
k2=y(ii,:)+(h/2)*[-CNk1/BNk1;DNk1-ENk1]';
BNk2=2*A*Vx*k2(1,2)+B;
CNk2=C_+A*(Vx^2)*(k2(1,2)^2)+Vx*k2(1,2)*(B+1);
DNk2=(m*a*b/((a+b)*Iz))*(((mu/2)*g-((su/4)*g*b/(a+b))-
(su*b*(hh^2)*((dVy(ii)+Vx*k2(1,2))^2)/((c^2)*g*(a+b))))*(f1+f2)+(hh*(dVy(ii)+Vx*k
2(1,2))/c)*(-mu+(su*b/(a+b)))*(f2-f1));%-Betadd(ii);
ENk2=(m*a*b/((a+b)*Iz))*(((mu/2)*g-((su/4)*g*a/(a+b))-
(su*a*(hh^2)*(krs^2)*((dVy(ii)+Vx*k2(1,2))^2)/((c^2)*g*(a+b))))*(f3+f4)+(hh*krs*(d
Vy(ii)+Vx*k2(1,2))/c)*(mu-(su*a/(a+b)))*(f4-f3));
k3=y(ii,:)+h*[-CNk2/BNk2;DNk2-ENk2]';
BNk3=2*A*Vx*k3(1,2)+B;
CNk3=C_+A*(Vx^2)*(k3(1,2)^2)+Vx*k3(1,2)*(B+1);
DNk3=(m*a*b/((a+b)*Iz))*(((mu/2)*g-((su/4)*g*b/(a+b))-
(su*b*(hh^2)*((dVy(ii)+Vx*k3(1,2))^2)/((c^2)*g*(a+b))))*(f1+f2)+(hh*(dVy(ii)+Vx*k
3(1,2))/c)*(-mu+(su*b/(a+b)))*(f2-f1));%-Betadd(ii);
136 ENk3=(m*a*b/((a+b)*Iz))*(((mu/2)*g-((su/4)*g*a/(a+b))-
(su*a*(hh^2)*(krs^2)*((dVy(ii)+Vx*k3(1,2))^2)/((c^2)*g*(a+b))))*(f3+f4)+(hh*krs*(d
Vy(ii)+Vx*k3(1,2))/c)*(mu-(su*a/(a+b)))*(f4-f3));
k4=y(ii,:)+h*[-CNk3/BNk3;DNk3-ENk3]';
y(ii+1,:)=y(ii,:) + (1/3)*k1 - (1/3)*y(ii,:) + (2/3)*k2 - (2/3)*y(ii,:) + (1/3)*k3 -
(1/3)*y(ii,:) + (1/6)*k4 - (1/6)*y(ii,:);
else
dVk1(ii)=((-BN+sqrt((BN^2)-4*A*CN))/(2*A));
k1=y(ii,:)+(h/2)*[dVk1(ii);DN-EN]';
BNk1=2*A*Vx*k1(1,2)+B;
CNk1=C_+A*(Vx^2)*(k1(1,2)^2)+Vx*k1(1,2)*(B+1);
DNk1=(m*a*b/((a+b)*Iz))*(((mu/2)*g-((su/4)*g*b/(a+b))-
(su*b*(hh^2)*((dVy(ii)+Vx*k1(1,2))^2)/((c^2)*g*(a+b))))*(f1+f2)+(hh*(dVy(ii)+Vx*k
1(1,2))/c)*(-mu+(su*b/(a+b)))*(f2-f1));%-Betadd(ii);
ENk1=(m*a*b/((a+b)*Iz))*(((mu/2)*g-((su/4)*g*a/(a+b))-
(su*a*(hh^2)*(krs^2)*((dVy(ii)+Vx*k1(1,2))^2)/((c^2)*g*(a+b))))*(f3+f4)+(hh*krs*(d
Vy(ii)+Vx*k1(1,2))/c)*(mu-(su*a/(a+b)))*(f4-f3));
k2=y(ii,:)+(h/2)*[((-BNk1+sqrt((BNk1^2)-4*A*CNk1))/(2*A));DNk1-ENk1]';
BNk2=2*A*Vx*k2(1,2)+B;
137 CNk2=C_+A*(Vx^2)*(k2(1,2)^2)+Vx*k2(1,2)*(B+1);
DNk2=(m*a*b/((a+b)*Iz))*(((mu/2)*g-((su/4)*g*b/(a+b))-
(su*b*(hh^2)*((dVy(ii)+Vx*k2(1,2))^2)/((c^2)*g*(a+b))))*(f1+f2)+(hh*(dVy(ii)+Vx*k
2(1,2))/c)*(-mu+(su*b/(a+b)))*(f2-f1));%-Betadd(ii);
ENk2=(m*a*b/((a+b)*Iz))*(((mu/2)*g-((su/4)*g*a/(a+b))-
(su*a*(hh^2)*(krs^2)*((dVy(ii)+Vx*k2(1,2))^2)/((c^2)*g*(a+b))))*(f3+f4)+(hh*krs*(d
Vy(ii)+Vx*k2(1,2))/c)*(mu-(su*a/(a+b)))*(f4-f3));
k3=y(ii,:)+h*[((-BNk2+sqrt((BNk2^2)-4*A*CNk2))/(2*A));DNk2-ENk2]';
BNk3=2*A*Vx*k3(1,2)+B;
CNk3=C_+A*(Vx^2)*(k3(1,2)^2)+Vx*k3(1,2)*(B+1);
DNk3=(m*a*b/((a+b)*Iz))*(((mu/2)*g-((su/4)*g*b/(a+b))-
(su*b*(hh^2)*((dVy(ii)+Vx*k3(1,2))^2)/((c^2)*g*(a+b))))*(f1+f2)+(hh*(dVy(ii)+Vx*k
3(1,2))/c)*(-mu+(su*b/(a+b)))*(f2-f1));%-Betadd(ii);
ENk3=(m*a*b/((a+b)*Iz))*(((mu/2)*g-((su/4)*g*a/(a+b))-
(su*a*(hh^2)*(krs^2)*((dVy(ii)+Vx*k3(1,2))^2)/((c^2)*g*(a+b))))*(f3+f4)+(hh*krs*(d
Vy(ii)+Vx*k3(1,2))/c)*(mu-(su*a/(a+b)))*(f4-f3));
k4=y(ii,:)+h*[((-BNk3+sqrt((BNk3^2)-4*A*CNk3))/(2*A));DNk3-ENk3]';
y(ii+1,:)=y(ii,:) + (1/3)*k1 - (1/3)*y(ii,:) + (2/3)*k2 - (2/3)*y(ii,:) + (1/3)*k3 -
(1/3)*y(ii,:) + (1/6)*k4 - (1/6)*y(ii,:);
138 end end yb1(:,kk+1)=y(:,1); yb2(:,kk+1)=y(:,2);
%Rise time (only for +Delta)
RTb1(kk+1)=0; for ii=1:2001
if yb1(ii,kk+1)<0.9*yb1(N,kk+1)
T(ii)=ii*h;
else
T(ii)=100;
end end
RTb1(kk+1)=min(T);
%Overshoot. Only for US case if abs(min(yb1(:,kk+1)))>abs(yb1(N,kk+1))
OSHb1(kk+1)=0;
for ii=1:2001
if abs(yb1(ii,kk+1))==abs(min(yb1(:,kk+1)))
S(ii)=ii*h;
else
139 S(ii)=100;
end
end
OSHb1(kk+1)=min(S); else
('No overshoot') end
%Rise time
RTb2(kk+1)=0; for ii=1:2001
if abs(yb2(ii,kk+1))>0.9*abs(yb2(N,kk+1))
T(ii)=ii*h;
else
T(ii)=100;
end end
RTb2(kk+1)=min(T);
%Overshoot. Only for US case if abs(max(yb2(:,kk+1)))>abs(yb2(N,kk+1))
OSHb2(kk+1)=0;
for ii=1:2001
140 if abs(yb2(ii,kk+1))==abs(max(yb2(:,kk+1)))
S(ii)=ii*h;
else
S(ii)=100;
end
end
OSHb2(kk+1)=min(S);
else
('No overshoot')
end
%Percent overshoot1
PERb1(kk+1)=(abs(min(yb1(:,kk+1)))*180/pi)/(abs(yb1(N,kk+1))*180/pi);
%Percent overshoot2
PERb2(kk+1)=(max(abs(yb2(:,kk+1)))*180/pi)/(abs(yb2(N,kk+1))*180/pi); end toc
141