the science of vehicle dynamics - gbv

Massimo Guiggiani

The Science of Vehicle

Dynamics

Handling, Braking, and Ride of Road

and Race Cars

Second Edition

Springer

Contents

1 Introduction 1

1.1 Vehicle Definition 2

1.2 Vehicle Basic Scheme 3

References 6

2 Mechanics of the Wheel with Tire 7

2.1 The Tire as a Vehicle Component 9

2.2 Carcass Features 9

2.3 Contact Patch 10

2.4 Rim Position and Motion 12

2.4.1 Reference System 13

2.4.2 Rim Kinematics 13

2.5 Footprint Force 16

2.5.1 Perfectly Flat Road Surface 18

2.6 Global Mechanical Behavior 20

2.6.1 Tire Transient Behavior 20

2.6.2 Tire Steady-State Behavior 20

2.6.3 Simplifications Based on Tire Tests 21

2.7 Rolling Resistance Moment 23

2.8 Definition of Pure Rolling for Tires 25

2.8.1 Zero Longitudinal Force 26

2.8.2 Zero Lateral Force 28

2.8.3 Zero Vertical Moment 28

2.8.4 Zero Lateral Force and Zero Vertical Moment 28

2.8.5 Pure Rolling Summary 29

2.8.6 Rolling Velocity and Rolling Yaw Rate 31

2.9 Definition of Tire Slips 33

2.9.1 Theoretical Slips 34

2.9.2 The Simple Case (No Camber) 35

2.9.3 From Slips to Velocities 35

ix

t Contents

2.9.4 (Not So) Practical Slips 36

2.9.5 Tire Slips Are Rim Slips Indeed 36

2.9.6 Slip Angle 37

2.10 Grip Forces and Tire Slips 38

2.11 Tire Tests 39

2.11.1 Tests with Pure Longitudinal Slip 41

2.11.2 Tests with Pure Lateral Slip 42

2.12 Magic Formula 45

2.12.1 Magic Formula Properties 46

2.12.2 Fitting of Experimental Data 47

2.12.3 Vertical Load Dependence 47

2.12.4 Horizontal and Vertical Shifts 50

2.12.5 Camber Dependence 50

2.13 Mechanics of the Wheel with Tire 50

2.13.1 Braking/Driving 51

2.13.2 Cornering 51

2.13.3 Combined 53

2.13.4 Camber 55

2.13.5 Grip 56

2.13.6 Vertical Moment 57

2.14 Exercises 58

2.14.1 Pure Rolling 58

2.14.2 Theoretical and Practical Slips 58

2.14.3 Tire Translational Slips and Slip Angle 58

2.14.4 Tire Spin Slip and Camber Angle 59

2.14.5 Motorcycle Tire 59

2.14.6 Finding the Magic Formula Coefficients 60

2.15 Summary 63

2.16 List of Some Relevant Concepts 63

2.17 Key Symbols 63

References 64

3 Vehicle Model for Handling and Performance 67

3.1 Mathematical Framework 68

3.1.1 Vehicle Axis System 68

3.2 Vehicle Congruence (Kinematic) Equations 69

3.2.1 Velocity of G, and Yaw Rate of the Vehicle 69

3.2.2 Yaw Angle of the Vehicle, and Trajectory of G . . . .70

3.2.3 Velocity Center C 72

3.2.4 Fundamental Ratios /? and p 73

3.2.5 Acceleration of G and Angular Acceleration

of the Vehicle 73

3.2.6 Radius of Curvature of the Trajectory of G 76

Contents x*

3.2.7 Radius of Curvature of the Trajectoryof a Generic Point 78

3.2.8 Telemetry Data and Mathematical Channels 78

3.2.9 Acceleration Center K 79

3.2.10 Inflection Circle 80

3.3 Tire Kinematics (Tire Slips) 81

3.3.1 Translational Slips 84

3.3.2 Spin Slips 85

3.4 Steering Geometry (Ackermann) 85

3.4.1 Ackermann Steering Kinematics 87

3.4.2 Best Steering Geometry 89

3.4.3 Position of Velocity Center and Relative

Slip Angles 89

3.5 Vehicle Constitutive (Tire) Equations 90

3.6 Vehicle Equilibrium Equations 91

3.6.1 Inertial Terms 92

3.6.2 External Force and Moment 92

3.7 Forces Acting on the Vehicle 93

3.7.1 Weight 93

3.7.2 Aerodynamic Force 93

3.7.3 Road-Tire Friction Forces 95

3.7.4 Road-Tire Vertical Forces 99

3.8 Vehicle Equilibrium Equations (More Explicit Form) .......100

3.9 Vertical Loads and Load Transfers 102

3.9.1 Longitudinal Load Transfer 102

3.9.2 Lateral Load Transfers 103

3.9.3 Vertical Load on Each Tire 103

3.10 Suspension First-Order Analysis 104

3.10.1 Suspension Reference Configuration 105

3.10.2 Suspension Internal Coordinates 106

3.10.3 Kinematic Camber Variation 107

3.10.4 Kinematic Track Width Variation 108

3.10.5 Vehicle Internal Coordinates 109

3.10.6 Definition of Roll and Vertical Stiffnesses 109

3.10.7 Suspension Internal Equilibrium 113

3.10.8 Effects of a Lateral Force 113

3.10.9 No-Roll Centers and No-Roll Axis 115

3.10.10 Suspension Jacking 118

3.10.11 Roll Moment. 118

3.10.12 Roll Angles and Lateral Load Transfers 120

3.10.13 Explicit Expressions of the Lateral Load

Transfers 122

3.10.14 Lateral Load Transfers with Rigid Tires 124

xii Contents

3.11 Sprang and Unsprung Masses 124

3.12 Dependent Suspensions (Solid Axle) 125

3.13 Linked Suspensions 128

3.14 Differential Mechanisms 128

3.14.1 Relative Angular Speeds 130

3.14.2 Torque Balance 130

3.14.3 Internal Efficiency and TBR 131

3.14.4 Locking Coefficient 135

3.14.5 Rule of Thumb 136

3.14.6 A Simple Mathematical Model 138

3.14.7 Alternative Governing Equations 138

3.14.8 Open Differential 139

3.14.9 Limited-Slip Differentials 139

3.14.10 Geared Differentials 140

3.14.11 Clutch-Pack Differentials 141

3.14.12 Spindle Axle 144

3.14.13 Differential-Tire Interaction 144

3.14.14 Informal Summary About the Differential

Behavior 150

3.15 Vehicle Model for Handling and Performance 150

3.15.1 Equilibrium Equations 150

3.15.2 Camber Variations 152

3.15.3 Roll Angles 153

3.15.4 Steer Angles 153

3.15.5 Tire Slips 154

3.15.6 Tire Constitutive Equations 155

3.15.7 Differential Mechanism Equations 156

3.15.8 Summary 156

3.16 The Structure of This Vehicle Model 157

3.17 Three-Axle Vehicles 157

3.18 Exercises 160

3.18.1 Center of Curvature QG of the Trajectory of G 160

3.18.2 Track Variation 160

3.18.3 Camber Variation 160

3.18.4 Power Loss in a Self-locking Differential 161

3.18.5 Differential-Tires Interaction 161

3.19 Summary 164


3.21 Key Symbols 165

References 167

Contents xiii

4 Braking Performance 169

4.1 Pure Braking 170

4.2 Vehicle Model for Braking Performance 170

4.3 Equilibrium Equations 171

4.4 Longitudinal Load Transfer 172

4.5 Maximum Deceleration 172

4.6 Brake Balance 173

4.7 All Possible Braking Combinations 173

4.8 Changing the Grip 175

4.9 Changing the Weight Distribution 176

4.10 A Numerical Example 176

4.11 Braking Performance of Formula Cars 177


4.11.2 Vertical Loads 178

4.11.3 Maximum Deceleration 179

4.11.4 Brake Balance 180

4.11.5 Speed Independent Brake Balance 181

4.11.6 Typical Formula 1 Braking Performance 181

4.12 Braking, Stopping, and Safe Distances 183

4.13 Exercises 183

4.13.1 Minimum Braking Distance 183

4.13.2 Braking with Aerodynamic Downforces 185

4.13.3 GP2 Brake Balance 185

4.13.4 Speed Independent Brake Balance 186

4.14 Summary 187



References 188

5 The Kinematics of Cornering 189

5.1 Planar Kinematics of a Rigid Body 189

5.1.1 Velocity Field and Velocity Center 190

5.1.2 Acceleration Field and Acceleration Center 192

5.1.3 Inflection Circle and Radii of Curvature 193

5.2 The Kinematics of a Turning Vehicle 196

5.2.1 Moving and Fixed Centrodes of a TurningVehicle 197

5.2.2 Inflection Circle of a Turning Vehicle 201

5.2.3 Tracking the Curvatures of Front and Rear

Midpoints 205

5.2.4 Evolutes 210

xiv Contents

5.3 Exercises 210

5.3.1 Front and Rear Radii of Curvature 210

5.3.2 Drawing Centrodes 211

5.4 Key Symbols 211

References 212

6 Handling of Road Cars 213

6.1 Additional Simplifying Assumptions for Road

Car Modeling 214

6.1.1 Negligible Vertical Aerodynamic Loads 214

6.1.2 Almost Constant Forward Speed 214

6.1.3 Open Differential 215

6.2 Mathematical Model for Road Car Handling 215

6.2.1 Global Equilibrium 216

6.2.2 Approximate Lateral Forces 217

6.2.3 Lateral Load Transfers and Vertical Loads 218


6.2.5 Camber Angle Variations 220



6.2.8 Simplified Tire Slips 224

6.2.9 Tire Lateral Forces 226

6.3 Double Track Model 227

6.3.1 Governing Equations of the Double

Track Model 227

6.3.2 Dynamical Equations of the Double

Track Model 228

6.3.3 Alternative State Variables (fi and p) 228

6.4 Vehicle in Steady-State Conditions 229

6.5 Single Track Model 231

6.5.1 From Double to Single 231

6.5.2 "Forcing" the Lateral Forces 234

6.5.3 Axle Characteristics 235

6.5.4 Governing Equations of the SingleTrack Model 244

6.5.5 Dynamical Equations of the SingleTrack Model 246

6.5.6 Alternative State Variables (p and p) 247

6.5.7 Inverse Congruence Equations. . .

248

6.5.8 & and jg2 as State Variables 248

6.5.9 Driving Force 250

Contents xv

6.5.10 The Role of the Steady-State Lateral

Acceleration 251

6.5.11 Slopes of the Axle Characteristics 252

6.6 Double Track, or Single Track? 252

6.7 Steady-State Maps 253

6.7.1 Steady-State Gradients 255

6.7.2 Alternative Steady-State Gradients 256

6.7.3 Understeer and Oversteer 256

6.7.4 Handling Diagram 259

6.8 Map of Achievable Performance (MAP) 261

6.8.1 MAP Fundamentals 262

6.8.2 MAP Curvature p Versus Steer Angle 5 268

6.8.3 Other Possible MAPs 273

6.9 Weak Concepts in Classical Vehicle Dynamics 274

6.9.1 The Understeer Gradient 275

6.9.2 Popular Definitions of Understeer/Oversteer 276

6.10 Double Track Model in Transient Conditions 276

6.10.1 Equilibrium Points 277

6.10.2 Free Oscillations (No Driver Action) 277

6.10.3 MAP for Transient Behavior 281

6.10.4 Stability of the Equilibrium 282

6.10.5 Forced Oscillations (Driver Action) 282

6.11 Relationship Between Steady-State Data and Transient

Behavior 284

6.11.1 Stability Derivatives from Steady-StateGradients 285

6.11.2 Equations of Motion 287

6.11.3 Estimation of the Control Derivatives 288

6.11.4 Objective Evaluation of Car Handling 288

6.12 Stability (Again) 290

6.13 New Understeer Gradient 291

6.14 The Nonlinear Single Track Model Revisited 292

6.14.1 Different Vehicles with Identical Handling 295

6.15 Linear Single Track Model 298

6.15.1 Governing Equations 299

6.15.2 Solution for Constant Forward Speed 301

6.15.3 Critical Speed 303

6.15.4 Transient Vehicle Behavior 303

6.15.5 Steady-State Behavior: Steering Pad 306

6.15.6 Lateral Wind Gust 307

6.15.7 Banked Road 311

xvi Contents

6.16 Compliant Steering System 312


6.16.2 Effects of Steer Compliance 314

6.17 Road Vehicles with Locked or Limited Slip Differential 315

6.18 Exercises 315

6.18.1 Camber Variations 315

6.18.2 Ackermann Coefficient 315

6.18.3 Toe-In 316

6.18.4 Steering Angles 316

6.18.5 Axle Characteristics 316

6.18.6 Playing with Linear Differential Equations 317

6.18.7 Static Margin 317

6.18.8 Banked Road 317

6.18.9 Rear Steer 318

6.18.10 Wind Gust 318

6.19 Summary 319



References 322

7 Handling of Race Cars 323

7.1 Assumptions for Race Car Handling 323

7.1.1 Aerodynamic Downloads 324

7.1.2 Limited-Slip Differential 324

7.2 Vehicle Model for Race Car Handling 325


7.2.2 Lateral Forces for Dynamic Equilibrium 328

7.2.3 Tire Forces 328


7.2.5 Camber Angles 330


7.2.7 Vertical Loads on Each Wheel 332

7.2.8 Lateral Load Transfers 333


7.2.10 Behavior of the Limited-Slip Differential 334

7.2.11 Reducing the Number of Equations 3357.3 Double Track Race Car Model 337

7.3.1 Single Track? 3377.4 Basics for Steady-State Handling Analysis 338

7.5 The Handling Diagram Becomes the Handling Surface 339

7.5.1 Handling with Locked Differential

(and No Wings) 339

Contents xvii

7.6 Handling of Formula Cars 352

7.6.1 Handling Surface 353

7.6.2 Map of Achievable Performance (MAP) 354

7.7 Exercises 363

7.7.1 Vehicle Kinematic Equations 363

7.7.2 Spin Slip Contributions 367

7.7.3 Acceleration Center K and Acceleration

of the Velocity Center C 368

7.7.4 Aerodynamic Downforces 368

7.7.5 Roll Stiffnesses in Formula Cars 369

7.7.6 Lateral Load Transfers in Formula Cars 370

7.7.7 Centrifugal Force not Applied at the Center

of Mass 371

7.7.8 Global Aerodynamic Force 371

7.8 Summary 372



References 375

8 Map of Achievable Performance (MAP) 377

8.1 MAP Fundamental Idea 377

8.2 Achievable Regions 378

8.2.1 Input Achievable Region 378

8.2.2 Output Achievable Regions 382

8.2.3 Mixed I/O Achievable Regions 384

8.3 Achievable Performances on Input Regions 384

8.4 Achievable Performances on Output Regions 386

8.5 Achievable Performances on Mixed I/O Regions 387

8.6 MAP from Slowly Increasing Steer Tests 388

8.7 MAP from Constant Steer Tests 390

8.8 Concluding Remarks 392

8.9 Key Symbols 392

9 Handling with Roll Motion 393

9.1 Vehicle Position and Orientation 393

9.2 Yaw, Pitch and Roll 394

9.3 Angular Velocity 397

9.4 Angular Acceleration 399

9.5 Vehicle Lateral Velocity 399

9.5.1 Track Invariant Points 399

9.5.2 Vehicle Invariant Point (VIP) 403

9.5.3 Lateral Velocity and Acceleration 404

xviii Contents

9.6 Three-Dimensional Vehicle Dynamics 405

9.6.1 Velocity and Acceleration of G 405

9.6.2 Rate of Change of the Angular Momentum 407

9.6.3 Completing the Torque Equation 408


9.6.5 Including the Unsprung Mass 409

9.7 Handling with Roll Motion 410


9.7.2 Load Transfers 410

9.7.3 Constitutive (Tire) Equations 411

9.7.4 Congruence (Kinematic) Equations 411

9.8 Steady-State and Transient Analysis 412

9.9 Exercise 412

9.9.1 Roll Motion and Camber Variation 412

9.10 Summary . .

413



References 415

10 Ride Comfort and Road Holding 417

10.1 Vehicle Models for Ride and Road Holding 418

10.2 Quarter Car Model 422

10.2.1 The Inerter as a Spring Softener 426

10.2.2 Quarter Car Natural Frequencies and Modes 426

10.3 Damper Tuning 430

10.3.1 Optimal Damper for Comfort 430

10.3.2 Optimal Damper for Road Holding 432

10.3.3 The Inerter as a Tool for Road Holding

Tuning 433

10.4 More General Suspension Layouts 435

10.5 Road Profiles 436

10.6 Free Vibrations of Road Cars 437


10.6.2 Proportional Viscous Damping 440

10.6.3 Vehicle with Proportional Viscous Damping 441

10.6.4 Principal Coordinates 443

10.6.5 Selection of Front and Rear Suspension

Vertical Stiffnesses 445

10.7 Tuning of Suspension Stiffnesses 448

10.7.1 Optimality of Proportional Damping 449

10.7.2 A Numerical Example. 450

10.8 Non-proportional Damping 452

10.9 Interconnected Suspensions 453

Contents X1X

10.10 Exercises 456

10.10.1 Playing with rj 456

10.10.2 Playing with p 456

10.11 Summary 457



References 459

11 Tire Models 461

11.1 Brush Model Definition 461

11.1.1 Roadway and Rim 462

11.1.2 Shape of the Contact Patch 463

11.1.3 Pressure Distribution and Vertical Load 464

11.1.4 Force-Couple Resultant 466

11.1.5 Elastic Compliance of the Tire Carcass 467

11.1.6 Friction 468

11.1.7 Constitutive Relationship 469

11.1.8 Kinematics 470

11.1.9 Brush Model Slips 472

11.1.10 Sliding Velocity of the Bristle Tips 473

11.1.11 Summary of Relevant Velocities 474

11.2 General Governing Equations of the Brush Model 475

11.2.1 Data for Numerical Examples 478

11.3 Brush Model Steady-State Behavior 478

11.3.1 Steady-State Governing Equations 479

11.3.2 Adhesion and Sliding Zones 479

11.3.3 Force-Couple Resultant 483

11.3.4 Examples of Tangential Stress Distributions 484

11.4 Adhesion Everywhere (Linear Behavior) 488

11.5 Translational Slip Only (<r + 0, cp = 0) 491

11.5.1 Rectangular Contact Patch 498

11.5.2 Elliptical Contact Patch 507

11.6 Wheel with Pure Spin Slip (a = 0, q> / 0) 510

11.7 Wheel with Both Translational and Spin Slips 513

11.7.1 Rectangular Contact Patch 513

11.7.2 Elliptical Contact Patch.

515

11.8 Brush Model Transient Behavior 519

11.8.1 Transient Models with Carcass ComplianceOnly 521

11.8.2 Transient Model with Carcass and Tread

Compliance 525

xx Contents

11.8.3 Model Comparison 527

11.8.4 Selection of Tests 529

11.8.5 Longitudinal Step Input 529

11.8.6 Lateral Step Input 531

11.9 Exercises 532

11.9.1 Braking or Driving? 532

11.9.2 Carcass Compliance 532

11.9.3 Brush Model: Local, Linear, Isotropic,Homogeneous 532

11.9.4 Anisotropic Brush Model 532

11.9.5 Carcass Compliance 2 533

11.9.6 Skating Versus Sliding 533

11.9.7 Skating Slip 533

11.9.8 Simplest Brush Model 534

11.9.9 Velocity Relationships 534

11.9.10 Slip Stiffness Reduction 534

11.9.11 Total Sliding 535

11.9.12 Spin Slip and Camber Angle 535

11.9.13 The Right Amount of Camber 535

11.9.14 Slip Stiffness 536

11.10 Summary 536



References 538

Index 539

the science of vehicle dynamics - gbv

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