VEHICLE DYNAMICS

Automotive Series

Series Editor: Thomas Kurfess

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Modeling and Control of Engines andDrivelines

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Modelling, Simulation and Control ofTwo-Wheeled Vehicles

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March 2014

Advanced Composite Materials forAutomotive Applications: StructuralIntegrity and Crashworthiness

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Guide to Load Analysis for Durabilityin Vehicle Engineering and Speckert

Johannesson November 2013

VEHICLE DYNAMICS

Martin Meywerk

Helmut-Schmidt-University (University of the Federal Armed Forces Hamburg),Germany

This edition first published 2015© 2015 John Wiley & Sons Ltd

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Library of Congress Cataloging-in-Publication Data Applied for.

ISBN: 9781118971352

A catalogue record for this book is available from the British Library.

Set in 11/13pt Times by Laserwords Private Limited, Chennai, India

1 2015

For my wife Annetteand my children Sophia, Aljoscha, Indira and Felicia

Contents

Foreword xi

Series Preface xiii

Preface xv

List of Abbreviations and Symbols xvii

1 Introduction 11.1 Introductory Remarks 31.2 Motion of the Vehicle 41.3 Questions and Exercises 8

2 The Wheel 112.1 Equations of Motion of the Wheel 112.2 Wheel Resistances 14

2.2.1 Rolling Resistance 142.2.2 Aquaplaning 182.2.3 Bearing Resistance 192.2.4 Toe-In/Toe-Out Resistance 19

2.3 Tyre Longitudinal Force Coefficient, Slip 202.4 Questions and Exercises 24

3 Driving Resistances, Power Requirement 273.1 Aerodynamic Drag 273.2 Gradient Resistance 293.3 Acceleration Resistance 293.4 Equation of Motion for the Entire Vehicle 323.5 Performance 343.6 Questions and Exercises 39

viii Contents

4 Converters 434.1 Clutch, Rotational Speed Converter 454.2 Transmission, Torque Converter 484.3 Questions and Exercises 54

5 Driving Performance Diagrams, Fuel Consumption 575.1 Maximum Speed without Gradient 605.2 Gradeability 615.3 Acceleration Capability 615.4 Fuel Consumption 635.5 Fuel Consumption Test Procedures 685.6 Questions and Exercises 70

6 Driving Limits 736.1 Equations of Motion 746.2 Braking Process 796.3 Braking Rate 846.4 Questions and Exercises 91

7 Hybrid Powertrains 937.1 Principal Functionalities 937.2 Topologies of Hybrid Powertrains 997.3 Regenerative Braking and Charging 1017.4 Questions and Exercises 106

8 Adaptive Cruise Control 1078.1 Components and Control Algorithm 1078.2 Measurement of Distances and Relative Velocities 1128.3 Approach Ability 1178.4 Questions and Exercises 118

9 Ride Dynamics 1219.1 Vibration Caused by Uneven Roads 124

9.1.1 Damped Harmonic Oscillator 1249.1.2 Assessment Criteria 1289.1.3 Stochastic Irregularities 1309.1.4 Conflict between Safety and Comfort 132

9.2 Oscillations of Powertrains 1449.2.1 Torsional Oscillators 1449.2.2 Centrifugal Pendulum Vibration Absorbers 147

9.3 Examples 1519.4 Questions and Exercises 152

Contents ix

10 Vehicle Substitute Models 15510.1 Two-mass Substitute System 15510.2 Two-axle Vehicle, Single-track Excitation 15810.3 Non-linear Characteristic Curves 16510.4 Questions and Exercises 167

11 Single-track Model, Tyre Slip Angle, Steering 16911.1 Equations of Motion of the Single-track Model 16911.2 Slip Angle 17711.3 Steering 18111.4 Linearized Equations of Motion of the Single-track Model 18511.5 Relationship between Longitudinal Forces and Lateral Forces in the

Contact Patch 18811.6 Effect of Differentials when Cornering 18911.7 Questions and Exercises 191

12 Circular Driving at a Constant Speed 19312.1 Equations 19312.2 Solution of the Equations 19512.3 Geometric Aspects 19712.4 Oversteering and Understeering 20112.5 Questions and Exercises 205

13 Dynamic Behaviour 20713.1 Stability of Steady-state Driving Conditions 20713.2 Steering Behaviour 21013.3 Crosswind Behaviour 21313.4 Questions and Exercises 216

14 Influence of Wheel Load Transfer 21714.1 Wheel Load Transfer without Considering Vehicle Roll 21714.2 Wheel Load Transfer Considering Vehicle Roll 22114.3 Questions and Exercises 228

15 Toe-in/Toe-out, Camber and Self-steering Coefficient 22915.1 Toe-in/Toe-out, Camber 22915.2 Questions and Exercises 233

16 Suspension Systems 23516.1 Questions and Exercises 245

17 Torque and Speed Converters 24717.1 Speed Converters, Clutches 247

x Contents

17.2 Transmission 25217.3 Questions and Exercises 258

18 Shock Absorbers, Springs and Brakes 25918.1 Shock Absorbers 25918.2 Ideal Active Suspension and Skyhook Damping 264

18.2.1 Ideal Active Suspension 26418.2.2 Skyhook Dampers 267

18.3 Suspension Springs 26918.4 Brake Systems 27718.5 Questions and Exercises 281

19 Active Longitudinal and Lateral Systems 28319.1 Main Components of ABS 28319.2 ABS Operations 28719.3 Build-up Delay of Yaw Moment 29019.4 Traction Control System 29319.5 Lateral Stability Systems 29419.6 Hydraulic Units for ABS and ESP 29619.7 Active Steering System 29719.8 Questions and Exercises 298

20 Multi-body Systems 30120.1 Kinematics of Rigid Bodies 30220.2 Kinetic Energy of a Rigid Body 30520.3 Components of Multi-body Systems 30920.4 Orientation of Rigid Bodies 31220.5 Derivation and Solution of the Equations 315

20.5.1 Derivation of the Equations 31520.5.2 Solution of Equations 316

20.6 Applications of MBS 31720.7 Questions and Exercises 322

Glossary 323

References 329

Index 331

Foreword

This book is an extract of lectures on vehicle dynamics and mechatronic systemsin vehicles held at the Helmut-Schmidt-University, University of the Federal ArmedForces, Hamburg, Germany. The lectures have been held since 2002 (Vehicle Dynam-ics) and 2009 (Vehicle Mechatronics). The book is an introduction to the field ofvehicle dynamics and most parts of the book should be comprehensible to under-graduate students with a knowledge of basic mathematics and engineering mechanicsat the end of their Bachelor studies in mechanical engineering. However, some partsrequire advanced methods which are taught in graduate studies (Master programmein mechanical engineering).

I wish to thank Mrs Martina Gerds for converting the pictures to Corel Draw withLaTeX labels and for typing Chapter 9. My thanks go to Mr Darrel Fernandes, B.Sc.,for the pre-translation of my German scripts. I especially wish to thank Mr ColinHawkins for checking and correcting the final version of the book with respect tothe English language. My scientific assistants, especially Dr Winfried Tomaske andDipl.-Ing. Tobias Hellberg, I thank for proofreading, especially with regard to thetechnical aspects. Special thanks for assistance in preparing a number of Solid Worksconstructions for pictures of suspensions and transmissions as well for help in prepar-ing some MATLAB diagrams go to Mr Hellberg. Last but not the least, my thanks goto my family, my wife, Dr Annette Nicolay, and my children, Sophia, Aljoscha, Indiraand Felicia, for their patience and for giving me a lot of time to prepare this book.

Series Preface

The automobile is a critical element of any society, and the dynamic performanceof the vehicle is a key aspect regarding its value proposition. Furthermore, vehicledynamics have been studied for many years, and provide a plethora of opportunitiesfor the instructor to teach her students a wide variety of concepts. Not only are thesedynamics fundamental to the transportation sector, they are quite elegant in naturelinking various aspects of kinematics, dynamics and physics, and form the basis ofsome of the most impressive machines that have ever been engineered.

Vehicle Dynamics is a comprehensive text of the dynamics, modeling and control ofnot only the entire vehicle system, but also key elements of the vehicle such as trans-missions, and hybrid systems integration. The text provides a comprehensive overviewof key classical elements of the vehicle, as well as modern twenty-first century con-cepts that have only recently been implemented on the most modern commercialvehicles. The topics covered in this text range from basic vehicle rigid body kinematicsand wheel dynamic analysis, to advanced concepts in cruise control, hybrid power-train design and analysis and multi-body systems. This text is part of the AutomotiveSeries whose primary goal is to publish practical and topical books for researchersand practitioners in industry, and post-graduate/advanced undergraduates in automo-tive engineering. The series addresses new and emerging technologies in automotiveengineering supporting the development of next generation transportation systems.The series covers a wide range of topics, including design, modelling and manufac-turing, and it provides a source of relevant information that will be of interest andbenefit to people working in the field of automotive engineering.

Vehicle Dynamics presents a number of different designs, analysis and implementa-tion considerations related to automobiles including power requirements, converters,performance, fuel consumption and vehicle dynamic models. The text is written froma very pragmatic perspective, based on the author’s extensive experience. The bookis written such that it is useful for both undergraduate and post-graduate courses, and

xiv Series Preface

is also an excellent reference text for those practicing automotive systems design andengineering, in the field. The text spans a wide spectrum of concepts that are criticalto the understanding of vehicle performance, making this book welcome addition tothe Automotive Series.

Thomas KurfessOctober 2014

Preface

This books covers the main parts of vehicle dynamics, which is divided into threetopics: longitudinal, vertical and lateral dynamics. It also explains some applications,especially those with a mechatronic background, and outlines some components.

Figure 1 provides an overview of the chapters of the book. The main parts (lon-gitudinal, vertical and lateral) as well as applications and component chapters aregrouped together. Many principal aspects of dynamics are explained by using simplemechanical models (e.g. quarter-vehicle model and single-track model). As the vir-tual development process with very complex multi-body systems (MBS) is used inthe design of modern cars, this simulation technique is described very briefly in thelast chapter. Although these MBS models are able to predict many details, the user ofsuch models should understand the principles of how vehicles behave, and the maintheory behind dynamic behaviour. It is therefore important to learn the basic dynamicbehaviour using the simple models described in this book.

Chapter 1 contains some general data for vehicles. These remarks are followed byan introduction to some of the basics of frames and axis systems. This introductionshould be read by everyone. Following this are the three groups of longitudinal, ver-tical and lateral dynamics, which are largely independent. The longitudinal and thevertical parts are completely independent of the other parts and may be read andunderstood without any knowledge of the other parts. The third group, containing thelateral dynamics part, includes a number of aspects that may be difficult to understandwithout first reading the longitudinal or the vertical part.

The application chapters can only partly be understood without reading the corre-sponding theory chapter. Readers are therefore recommended to start with the basicparts: the basics for Chapters 7 (Hybrid Powertrains), 8 (Adaptive Cruise Control) and17 (Torque and Speed Converters) can be found in the longitudinal dynamics chapters,while lateral dynamics is important for Chapter 16 (Suspension Systems), in the caseof Chapter 18 (Shock Absorbers, Springs and Brakes) a knowledge is required of verti-cal dynamics as well as some aspects of longitudinal and lateral dynamics. Chapter 19(Active Longitudinal and Lateral Systems), as the name reveals, involves longitudinaland lateral aspects. Chapter 20 is nearly independent of the theoretical considerations.

Figure 1 includes the letters B and M, which stand for Bachelor and Master, behindevery chapter. Chapters of level B should be comprehensible for undergraduate

xvi Preface

1. Introduction (B)

Longitudinal Dynamics

2. The wheel (B)

3. Driving Resistances, Power Requirement (B)

4. Converters (B)

5. Driving Performance Diagrams,Fuel Consumption (B)

6. Driving limits (B)

Vertical Dynamics,Oscillations

9. Ride Dynamics (M)

10. Vehicle substitute models (M)

Lateral Dynamics

11. Single Track Model, Tyre Sideslip Angle, Steering (B)

12. Circular Driving at a Constant Speed (B)

13. Dynamic Behaviour (M)

14. Influence of Wheel Load Transfer (B)

15. Toe-in/ Toe-out, Camber and Self-steering Coefficient (B)

20. Multibody Systems (M)

Application

7. Hybrid Powertrains (B)

8. Adaptive Cruise Control (B)

16. Suspension Systems (B)

17. Torque and Speed Converters (B)

18. Shock Absorbers, Springs and Brakes (B)

19. Active Longitudinal and Lateral Systems (B)

Vehicle Dynamics

Figure 1 Chapters of the book

Analysing Evaluating Creating

Applying

Understanding

Remembering

Figure 2 Bloom’s taxonomy of learning

students with a knowledge of engineering mechanics and mathematics at the endof their Bachelor studies in mechanical engineering. Topics covered are: algebra;trigonometric functions; differential calculus; linear algebra; vectors; coordinatesystems; force, torque, equilibrium; mass, centre of mass, moment of inertia; methodof sections, friction, Newton’s laws, Lagrange’s equation. In chapters followed bylevel M, an advanced knowledge is useful, as is usually taught to graduate students:ordinary differential equations (ODE), stability of ODEs, Laplace transformation,Fourier transformation, stochastic description of uneven roads and spectral densities.

At the end of nearly each chapter, you will find some questions and exercises. Theseare for monitoring learning progress or for applying the material learned to some smallproblems. For this reason, the questions and tasks are arranged in classes accordingto Bloom’s taxonomy of learning (cf. Figure 2).

The simplest class is Remembering, which means that you only have to rememberthe correct content (e.g. a definition or a formula). You should be able to answer thequestions of the second class Understanding if you have understood the content. Thetasks of the third class Application involve applying the content to some unknownproblem. The remaining three classes Analysing, Evaluating and Creating are moresuited to extended student works, such as Bachelor or Master theses, and are thereforerarely included in this book.

List of Abbreviations and Symbols

The tables on the following pages summarize the mathematical symbols and abbre-viations used in this book. In most cases (but not in all), the indices used with thesymbols indicate the following:

• v vehicle• b body• w wheel• t tyre• x, y, z: �ex∗, �ey∗, �ez∗

Sometimes a symbol, which is needed only in a very local part of the book, couldbe used in another meaning than it is described in the following tabular, as well assometimes the units can differ from those given in the tabular. Symbols that occuronly in a small part of the book are not listed in the tabular.

xviii List of Abbreviations and Symbols

Table 1 List of symbols

Symbol Description Units Page

a Acceleration of the vehicle a = xv m/s2 84A Aerodynamic area m2 27α Tyre slip angle rad 177

(or locally used angle)ABS Anti-lock braking system – 283Ac Clothoid parameter m 171ACC Adaptive cruise control – 107αg Angle of inclination of the road rad 11, 29αgz Progression ratio of a transmission 1 48

αgz = iz−1/iz , z = 2, . . . , Nzmaxαj Tyre slip angle (j = 1 front; j = 2 rear) rad 186αj Average tyre slip angle (j = 1 front; j = 2 rear) rad 219αji Inner tyre slip angle (j = 1 front; j = 2 rear) rad 217αjo Outer tyre slip angle (j = 1 front; j = 2 rear) rad 217ASF Active front steering – 297ASR Anti-slip regulation – 293β Vehicle sideslip angle rad 170γ Camber angle rad 229cα Cornering stiffness N/rad 180cα Mean cornering stiffness N/rad 219cd Aerodynamic drag coefficient 1 27clj Aerodynamic lift coefficient (j = 1 front; j = 2

rear)1 76

CPVA Centrifugal pendulum vibration absorber – 144cy Aerodynamic crosswind coefficient 1 214DAE Differential algebraic equations – 315δ10 Toe-in δ10 > 0, toe-out δ10 < 0 rad 229ΔFzji Wheel load change at the inner wheel j;

(j = 1 front; j = 2 rear) N 217ΔFzjo Wheel load change at the outer wheel j;

(j = 1 front; j = 2 rear) N 217(O,�eix, �eiy, �eiz) Inertial reference frame – 6ηd Efficiency of the differential 1 49ηe Efficiency of the engine 1 67ηe Mean efficiency of the engine 1 63ηt Efficiency of the

drivetrain (transmission and differential) 1 58ηt Mean efficiency of the drivetrain

(speed and torque converter, differential) 1 63ηz Efficiency of the zth gear of

the transmission 1 49ESP Electronic stability programme – 294

List of Abbreviations and Symbols xix

Table 2 List of symbols

Symbol Description Units Page

(Scp, �etx, �ety, �etz) Tyre reference frame – 7(Scm, �evx, �evy, �evz) Vehicle reference frame – 6ew Eccentricity at the wheel m 12ewj Eccentricity at the wheels of axle j

(j = 1 front; j = 2 rear) m 74(Scmw, �ewx, �ewy, �ewz) Wheel reference frame,

�ewz perpendicular to the road plane – 7Fa Aerodynamic drag in the longitudinal N 27

direction for simplified models inlongitudinal dynamics

Fax Aerodynamic drag force in the �evx-direction N 172for single track model and wheel load transfer

Fay Aerodynamic drag force in the �evy-direction N 172Faz Aerodynamic lift force in the �evz-direction N 76Fbasic Basic demand of tractive force:

Fbasic = Fr + Fa

N 35

fe Frequency of the received signal Hz 112Fg Gradient resistance N 29Fg/i Combined gradient and inertial resistance

for y = p + λxv/g N 35Fideal Ideal (demand) characteristic map of tractive

forceN 37

Fi Acceleration or inertial resistance N 33FMCW Frequency modulated continuous wave – 112fr Rolling resistance coefficient 1 15fs Frequency of the transmitted signal Hz 112fri Coefficients for fr approximation (i = 0, 1, 4) 1 16Fr Rolling resistance N 15Ftot Total tractive force demand:

Ftot = Fa + Fi + Fg + Fr

N 34

Fwsz Force supplied at the wheel from the powertrainfor gear z

N 50

Fx Section force for tyre-road N 13Fxj Section force for tyre-road N 32

(j = 1 front; j = 2 rear)Fzj Section force (wheel load) (j = 1 front; j = 2

rear)N 74

Fzj aero Wheel load aerodynamic portion (j = 1 front;j = 2 rear)

N 76

Fzj stat Wheel load static portion (j = 1 front; j = 2rear)

N 75

xx List of Abbreviations and Symbols

Table 3 List of symbols

Symbol Description Units Page

Fzj dyn Wheel load dynamic portion (j = 1 front;j = 2 rear)

N 78

Fz Wheel or axle load in the �ewz-direction N 12g Gravitational acceleration m/s2 13Gaj Weight of the axle j (j = 1 front; j = 2 rear) N 75Gb Weight of body (sprung mass) N 75h Distance: centre of mass Scm – road m 86hb Distance: centre of mass of the body – road m 75hcm Distance: centre of mass Scm – road m 77HP Pump – 297hpp Distance: Centre of pressure Spp – road m 76

for air flow in the �evx-directionHSV High-pressure selector valve – 297id Transmission ratio of the differential (final

drive)1 44

ig Transmission ratio of the gearbox; 1 30for a stepped transmission: ig = iz

it Total transmission ratio it = izid 1 58iz Transmission ratio of gear z of the

transmission, z = 1, . . . , Nz max

1 48

Jaj Moment of inertia of the axle j kg m2 30Jc Moment of inertia of gear, differential, Cardan

shaftkg m2 30

Je Moment of inertia of engine, clutch kg m2 30Jz Moment of inertia of the vehicle with respect to

the �ez-axiskg m2 172

κ Angle of rotation of the body of the vehicle rad 222κcc Instantaneous curvature (κcc = 1/ρcc) of the

vehicle path1/m 171

κw Wavenumber of an uneven road rad/m 164� Wheelbase; distance between front and rear

axlem 75

�1 Distance in the �evx direction between front axlecentre of mass and centre of mass Scm of thevehicle

m 75

�2 Distance in the �evx direction between rear axlecentre of mass and centre of mass Scm of thevehicle

m 75

λ Rotational mass factor 1 33λe Eigenvalue with respect to time 1/s 207�cm Distance: centre of gravity Scm – m 172

centre of pressure Spp in the �evx direction

List of Abbreviations and Symbols xxi

Table 4 List of symbols

Symbol Description Units Page

μ = μ(S) Tyre longitudinal force coefficient 1 21Ma Aerodynamic moment Nm 74μa Coefficient of adhesion 1 21maj Mass of the axle (j = 1 front; j = 2 rear) kg 32Maj Section moment at the axle j Nm 32

(j = 1 front; j = 2 rear)mb Mass of the body or sprung mass of the vehicle kg 31MBS Multi-body systems – 4Mcc Centre of curvature – 170Mcr Instantaneous centre of rotation – 173Me Torque supplied from the engine N 58M100 Full load moment of the engine Nm 37Mi Input moment (e.g. at input of transmission or

clutch)Nm 45

M(Pmax) Moment where the power of the engines Nm 37reaches a maximum

Ml Torque loss from the engine N 59Mmax Maximum torque of the engine Nm 55M(nmax) Full load moment of the engine at nmax Nm 37M(nmin) Full load moment of the engine at nmin Nm 37Mo Output moment (e.g. at input of transmission or

clutch)Nm 45

μs Coefficient of pure sliding 1 22mtot Total mass (sprung and unsprung mass) kg 29Mws Torque supplied at the wheel from the

powertrainN 58

nc Total caster trail nc = nkc + ntc m 188ne Engine speed (revolutions) rad/s 44ni Input speed or revolutions (e.g. at input of

transmission or clutch)rev/s 45

niz Input speed (revolutions) of transmission atgear z

rad/s 48

nkc Kinematic caster trail m 182nmax Maximum speed of the engine rpm 37nmin Minimum speed of the engine rpm 37no Output speed or revolutions (e.g. at input of

transmission or clutch)rev/s 45

n(Pmax) Engine speed where the power of the engines rpm 37reaches a maximum

nw max Maximum revolutions per minute of the wheel rpm 44noz Output speed (revolutions) of transmission at

gear zrad/s 48

xxii List of Abbreviations and Symbols

Table 5 List of symbols

Symbol Description Units Page

ntc Tyre caster trail m 177nw Wheel speed (revolutions) rad/s 44Nz max Number of gears in a transmission 1 48ODE Ordinary differential equation – 5OEM Original equipment manufacturer – 121ωi Input angular velocities (e.g. at input of

transmission or clutch)rad/s 45

ωo Output angular velocities (e.g. at input oftransmission or clutch)

rad/s 45

p Gradient (inclination) of a road p = tan αg 1 29ψ Yaw angle rad 7ϕ Roll angle rad 7P100 Full load power of the engine W = Nm/s 37Pa Power of aerodynamic drag force (S = 0:

Pa = Favv)W = Nm/s 35

ϕaj Angle of rotation of the axle j (j = 1 front;j = 2 rear)

rad 32

ϕb Pitch angle of the body rad 159Pbasic Basic demand of power: Pbasic = Pr + Pa N 35ϕc Angular velocity gear, differential, Cardan shaft rad/s 30Pe Power supplied from the engine W = Nm/s 46ϕe Angular velocity engine, clutch rad/s 30Pg/i Power of combined gradient and inertial

resistanceW = Nm/s 35

Pg Power of gradient resistance W = Nm/s 35Φh Spectral density of the stochastic road surface

irregularitym3 131

Φh(Ω0) Coefficient of roughness m3 131Pbasic Total power demand: Pbasic = Pr + Pa N 34Pideal Ideal (demand) characteristic map of power at

the wheelN 37

Pi Input power (e.g. at input of transmission) inChapter 4

W = Nm/s 45

Pi Power of inertia forces in Chapter 3 W = Nm/s 35Pmax Maximum power of the engine reaches a

maximumW = Nm/s 37

Po Output power (e.g. at input of transmission) W = Nm/s 45Pr Power of rolling resistance (S = 0: Pr = Frvv) W = Nm/s 34Ptot Total power demand:

Ptot = Pr + Pa + Pg + Pi

N 34

ϕw Rotational angle of the wheel w.r.t the �ewy-axis rad 14

List of Abbreviations and Symbols xxiii

Table 6 List of symbols

Symbol Description Units Page

Pwsi Power supplied at the wheel from thepowertrain for gear i

W = Nm/s 50

Pw Power at the wheel W = Nm/s 63ρa Mass density of air kg/m3 27ρcc Instantaneous radius of curvature of the vehicle

pathm 171

rk Scrub radius: distance between the intersection m 183of the steering axis with road and the centre of

the contact patchrσ Kingpin offset between the wheel centre and

the steering axism 184

Rw0 Dynamic rolling radius m 20Rw0j Dynamic rolling radius (j = 1 front; j = 2 rear) m 32rwst Static radius of a wheel m 14rwstj Static radius of the wheels of the axle j (j = 1

front; j = 2 rear)m 32

σ Inclination angle of the steering axis; anglefrom the �eV z direction to the projection ofthe steering axis on to the �eV z-�eV y-plane

rad 181

Sj Slip at wheels of the axle j (j = 1 front; j = 2rear)

1 32

sj Track of the axle j (j = 1 front; j = 2 rear) m 219Scm Centre of mass of the vehicle (sprung and

unsprung mass)– 6

Scmw Centre of mass of a wheel – 7SOV Switch over valve – 297Spp Centre of pressure – 76Scp Centre point of the contact patch – 7σn Normal stress distribution in the contact patch N/m2 12SSF Static stability factor – 220τ Caster angle; angle from the �evz direction to the

projection of the steering axis on to the�evz-�evx-plane

rad 181

tb Pressure build-up time s 80tfb Foot force build-up time s 80tf Duration of full braking process s 82tr Reaction time s 79tt Transmission time s 80ϑ Pitch angle rad 7v Absolute value of �vv: v = |�vv| = vv m/s 27va Velocity of wind m/s 27�va Velocity vector of wind m/s 214

xxiv List of Abbreviations and Symbols

Table 7 List of symbols

Symbol Description Units Page

vc Circumferential velocity of wheel m/s 20v2

ch Characteristic velocity squared m2/s2 197vcrit Critical velocity m/s 209�vr Resultant velocity vector (wind and vehicle) m/s 214�vv Velocity of the vehicle at Scm;

vv = |�vv|;�v = �vv

m/s 214

�v = (vvx, vvy, vvz) · (�eix, �eiy, �eiz)T

vw Velocity of wheel m/s 179w Waviness of an uneven road 1 131Ww Work at the wheel J = Nm 63X Section force between wheel and body N 13Xj Section force axle-vehicle at the axle j N 33

(j = 1 front; j = 2 rear)xv Acceleration of the vehicle m/s2 30xaj Coordinate of the axle j (j = 1 front;j = 2

rear)m 32

xv, yv, zv Scm vehicle coordinates w.r.t. (O,�eix, �eiy, �eiz) m 6xw, yw, zw Scmw wheel coordinates

w.r.t. (Scm, �evx, �evy, �evz)m 14

Y Section force between wheel and body N 178Z Section force between wheel and body N 13Z Braking ration Z = −a/g 1 84z1 Displacement of the wheel m 264z2 Displacement of the body m 264z3 Displacement of the seat m 136zb Displacement of the body m 156zw Displacement of the wheel m 156

1Introduction

Automobiles have been used for over 100 years for the transportation of people andgoods. Despite this long period, essential elements of an automobile have in principleremained the same, i.e. four wheels and an internal combustion engine with a torqueconverter drive. However, the technical details of an automobile have changed a greatdeal, and the complexity has increased substantially. This has partly gone hand in handwith general technical progress, on the one hand, and increasing customer demands,on the other. Legal requirements have also led to distinct changes in automobiles.

The importance of automobiles becomes evident when we look at the graphs inFigures 1.1–1.4. You should bear in mind that the abscissas of most graphs are parti-tioned logarithmically. The quantity, the distances travelled and the distances travelledper capita are at a very high level, or these values are increasing at a high rate. If welook at some European countries or the United States of America, we can recognizestagnation at a high level, whereas emerging economies exhibit high rates of growth.The need to develop new, economic and ecological vehicles is evident. In order todo this, engineers should be familiar with the basic properties of automobiles. As theautomobile is something which moves and which not only moves forward at a constantvelocity, but also dynamic behaviour depends on these basic properties. Consequently,the basic dynamic properties form the main topic of this book.

The ecological aspect could be a dramatic limiting factor in the development ofvehicles throughout the world. If the number of cars per 1000 inhabitants in Chinaand Hong Kong grows from 22 in the year 2007 to 816, which is the number inthe USA, then this represents a factor of 40. If we now multiply the CO2 emissionsof the USA from the year 2007 by 40, we obtain around 57 000 Mt, which is 12 timesthe world CO2 emissions from fuel combustion in road transport for the year 2007.This seems to be very high (or perhaps too high), and vehicles with lower fuel con-sumption or hybrid or electric powertrains will have to be developed and improved inthe coming decades.

Vehicle Dynamics, First Edition. Martin Meywerk.© 2015 John Wiley & Sons, Ltd. Published 2015 by John Wiley & Sons, Ltd.Companion Website: www.wiley.com/go/meywerk/vehicle

2 Vehicle Dynamics

11990 1995 2000 2001 2002 2003 2004 2005 2006 2007

10

100

1000

United States Germany France United Kingdom India Brazil China + Hong Kong

Figure 1.1 Passenger cars (and light trucks in US) per 1000 inhabitants (data from OECD2014)

5

25

20081990 1995 2000 2001 2002 2003 2004 2005 2006 2007

x10

0000

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Figure 1.2 Road passenger km (million pkm) (data from OECD 2014)

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Figure 1.3 Passenger km/capita (data from OECD 2014)

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Figure 1.4 International Energy Agency (IEA) CO2 from fuel combustion (Mt) in RoadTransport (data from OECD 2014)

Introduction 3

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Figure 1.5 Importance of purchase criteria (Braess and Seiffert 2001)

The presentation of the most important buying criteria in Figure 1.5 highlights theecological and economic aspects as well as safety, handling behaviour and comfort.The last three points, namely safety, handling behaviour and comfort, are stronglylinked with the driving dynamics and suspension, making these aspects of particu-lar importance in the automotive industry. Safety is generally subdivided into activesafety (active safety systems help to avoid accidents) and passive safety (passive safetysystems protect the occupants during an accident).

It is evident that the dynamics of the vehicle is of crucial importance because of theimpact on active safety; handling behaviour and comfort are also closely associatedwith the properties of vehicle dynamics. For this reason, particular emphasis is placedon the aspect of dynamics in this course.

The aim of this course is to define and identify the basic concepts and relationshipsthat are necessary for understanding the dynamics of a motor vehicle.

The content of this textbook is limited to the essentials, and the course closelyfollows the monograph of Mitschke and Wallentowitz 2004 (German). Further rec-ommended reading can be found in the bibliography at the end of this book, e.g.Heissing and Ersoy 2011, Dukkipati et al. 2008, Gillespie 1992, Jazar 2014, or Reim-pell et al. 2001.

1.1 Introductory Remarks

The content of this book is divided into four parts: longitudinal dynamics, verticaldynamics, lateral dynamics and structural design of vehicle components andautomotive mechatronic systems. Longitudinal dynamics is included in Chapters2–6, which discuss the process of acceleration and braking. Key importance here isgiven to the total running resistance, the demand and supply of power and the driving

4 Vehicle Dynamics

state diagrams. In Chapters 7 and 8, additional systems of longitudinal dynamics aredescribed: alternative powertrains and adaptive cruise control systems. In Chapters 9and 10, the behaviour of the vehicle when driving on an uneven surface is explainedin the context of vertical dynamics. These chapters study the basics of the theory ofoscillations and the influence of vibrations on humans. Lateral dynamics, the contentsof Chapters 11–15, describes the handling behaviour of a vehicle during cornering.Important concepts such as slip, oversteer and understeer, toe and camber angle areexplained. It deals with the influence of wheel load on the handling behaviour.

Chapters 16–19 highlight the engineering design (structural) aspects of anautomobile. In addition to speed and torque converters, they also discuss brakes andchassis elements of active safety systems, such as anti-lock braking system (ABS),anti-slip regulation (ASR) and electronic stability programme (ESP). In Chapter 20,multi-body systems (MBS) are explained. MBS are computational models whichallow more precise calculations of the dynamic behaviour of vehicles.

1.2 Motion of the Vehicle

To describe the dynamics of motor vehicles, we use, as in any other branch of engi-neering, models with a greater or lesser degree of detail. The complexity of the modelsdepends on the questions under investigation. Today the MBS models are most com-monly used in both science and research as well as in the development departmentsof the automotive industry. Multi-body systems consist of one or more rigid bodieswhich are interconnected by springs and/or shock absorbers and joints.

Figure 1.6 shows an MBS model of a vehicle. This model is taken from the commer-cial MBS programme ADAMS. Another example of a McPherson front axle is shownin Figure 1.7. These MBS models allow high accuracy in the simulation of dynamicbehaviour. A lot of details can be incorporated into these models, even flexible partscan be considered. However, the detailed simulation yields a large number of effectsin the calculated results and the engineer has to interpret and understand these results.As an example, an engineer has to distinguish between main effects and numericalphenomena. For this purpose, it is helpful to understand the basic dynamics and toknow simple models for calculating the behaviour of a vehicle in order to interpret oreven to check the MBS results. This book therefore takes vehicle dynamic behaviourand simple models as its main topics.

In a simplified view of the motor vehicle, a model could consist of five rigid bodies:the four wheels and the body structure. These are interconnected by springs, shockabsorbers and rigid body suspensions with joints. A rigid body has six degrees offreedom. This simple model would therefore have 5 × 6 = 30 degrees of freedom1.

1 We may argue that the suspension between wheel carrier and body of the vehicle locks five degrees of freedom,the wheel bearing will unlock one degree of freedom, which results all together in only two degrees of freedom forone wheel. The sum for the whole vehicle will then be 14. That is correct under the assumption that there are nocompliances in the suspension. Since modern cars have these compliances, the number of 30 is correct.