Road Vehicle Dynamics Table of Contents: Foreword 1.9.14 Translation and Rotational Systems 1.9.15 Angular Momentum and Moments Preface of Inertia 1.9.16 Geared Systems Chapter 1 Introduction 1.10 Lagrange’s Equation 1.1 General 1.10.1 Degrees of Freedom 1.2 Vehicle System Classification 1.10.2 Generalized Coordinates 1.3 Dynamic System 1.10.3 Constraints 1.4 Classification of Dynamic System 1.10.4 Principle of Virtual Work Models 1.10.5 D’ Alembert’s Principle 1.5 Constraints, Generalized Coordinates, 1.10.6 Generalized Force and Degrees of Freedom 1.10.7 Lagrange’s Equations of Motion 1.6 Discrete and Continuous Systems 1.10.8 Holonomic Systems 1.7 Vibration Analysis 1.10.9 Nonholonomic Systems 1.8 Elements of Vibrating Systems 1.10.10 Rayleigh’s Dissipation Function 1.8.1 Spring Elements 1.11 Summary 1.8.2 Potential Energy of Linear Springs 1.12 References 1.8.3 Equivalent Springs 1.8.3.1 Springs in Parallel Chapter 2 Analysis of Dynamic Systems 1.8.3.2 Springs in Series 2.1 Introduction 1.8.4 Mass or Inertia Elements 2.2 Classification of Vibrations 1.8.5 Damping Elements 2.3 Classification of Deterministic Data 1.8.5.1 Viscous Damping 2.3.1 Sinusoidal Periodic Data 1.8.5.2 Coulomb Damping 2.3.2 Complex Periodic Data 1.8.5.3 Structural or Hysteretic Damping 2.3.3 Almost Periodic Data 1.8.5.4 Combination of Damping Elements 2.3.4 Transient Nonperiodic Data 1.9 Review of Dynamics 2.4 Linear Dynamic Systems 1.9.1 Newton’s Laws of Motion 2.4.1 Linear Single-Degree-of-Freedom 1.9.2 Kinematics of Rigid Bodies System 1.9.3 Linear Momentum 2.4.2 Free Vibration of a Single-Degree- 1.9.4 Principle of Conservation of Linear of-Freedom System Momentum 2.4.3 Forced Vibration of a Single- 1.9.5 Angular Momentum Degree-of-Freedom System 1.9.6 Equations of Motion for a Rigid 2.4.4 Linear Multiple-Degrees-of- Body Freedom System 1.9.7 Angular Momentum of a Rigid Body 2.4.5 Eigenvalues and Eigenvectors: 1.9.8 Principle of Work and Energy Undamped System 1.9.9 Conservation of Energy 2.4.6 Eigenvalues and Eigenvectors: 1.9.10 Principle of Impulse and Momentum Damped System 1.9.11 Mechanical Systems 2.4.7 Forced Vibration Solution of a 1.9.12 Translational Systems Multiple-Degrees-of-Freedom 1.9.13 Rotational Systems System Road Vehicle Dynamics 2.5 Nonlinear Dynamic Systems 3.3.2 Rolling Resistance of the Tire with Toe-In 2.5.1 Exact Methods for Nonlinear Systems 3.3.3 Rolling Resistance of the Turning Wheel 2.5.2 Approximate Methods for Nonlinear 3.3.4 Longitudinal Adhesion Coefficient Systems 3.3.5 Theoretical Model of Tire Longitudinal 2.5.2.1 Iterative Method Force Under Driving and Braking 2.5.2.2 Ritz Averaging Method 3.4 Tire Lateral Dynamics 2.5.2.3 Perturbation Method 3.4.1 Tire Cornering Characteristics 2.5.2.4 Variation of Parameter Method 3.4.2 Mathematical Model of the Tire 2.5.3 Graphical Method Cornering Characteristic 2.5.3.1 Phase Plane Representation 3.4.2.1 Simplified Mathematical Model of 2.5.3.2 Phase Velocity the Tire Cornering Characteristic 2.5.3.3 Pell’s Method 3.4.2.2 Cornering Characteristic with 2.5.4 Multiple-Degrees-of-Freedom Systems Lateral Bending Deformation of the 2.6 Random Vibrations Tire Case 2.6.1 Probability Density Function 3.4.3 Rolling Properties of Tires 2.6.2 Autocorrelation Function 3.4.3.1 Cambered Tire Models 2.7 Gaussian Random Process 3.4.3.2 Cambered Tire Model with Roll 2.7.1 Fourier Analysis Elastic Deformation of the Tire 2.7.1.1 Fourier Series Carcass 2.7.1.2 Fourier Integral 3.5 Tire Mechanics Model Considering 2.7.2 Response of a Single-Degree-of- Longitudinal Slip and Cornering Freedom Vibrating System Characteristics 2.7.2.1 Impulse Response Method 3.5.1 C.G. Gim Theoretical Model 2.7.2.2 Frequency Response Method 3.5.2 K.H. Guo Tire Model 2.7.3 Power Spectral Density Function 3.5.2.1 Steady-State Simplified 2.7.4 Joint Probability Density Function Theoretical Tire Model 2.7.5 Cross-Correlation Function 3.5.2.2 Nonsteady-State Semi-Empirical 2.7.6 Application of Power Spectral Tire Mechanics Model Densities to Vehicle Dynamics 3.5.3 H.B. Pacejka Magic Formula Model 2.7.7 Response of a Single-Degree-of- 3.6 References Freedom System to Random Inputs 2.7.8 Response of Multiple- Degrees- of- Chapter 4 Ride Dynamics Freedom Systems to Random Inputs 4.1 Introduction 2.8 Summary 4.2 Vibration Environment in Road Vehicles 2.9 References 4.2.1 Vibration Sources from the Road 4.2.1.1 Power Spectral Density in Spatial Chapter 3 Tire Dynamics Frequency 3.1 Introduction 4.2.1.2 Power Spectral Density in 3.2 Vertical Dynamics of Tires Temporal Frequency 3.2.1 Vertical Stiffness and Damping 4.2.2 Vehicle Internal Vibration Sources Characteristics of Tires 4.2.2.1 Vibration Sources from the 3.2.2 Vertical Vibration Mechanics Models of Powerplant Tires 4.2.2.1.1 Coordinates and Powerplant 3.2.2.1 Point Contact Model of Tires Modes 3.2.2.2 Fixed Contact Patch Model of 4.2.2.1.2 Vibration Sources from Tires Engine Firing Pulsation 3.2.2.3 Time-Varying Contact Patch Model 4.2.2.1.3 Vibration Sources from of Tires Powerplant Inertia Forces 3.2.3 Enveloping Characteristics of Tires and Moments 3.3 Tire Longitudinal Dynamics 4.2.2.1.4 Powerplant Isolation Design 3.3.1 Tire Rolling Resistance 4.2.2.2 Vibration Sources from the Driveline Road Vehicle Dynamics 4.2.2.2.1 Driveline Imbalance Chapter 5 Vehicle Rollover Analysis 4.2.2.2.2 Gear Transmission Error 5.1 Introduction 4.2.2.2.3 Second Order Excitation 5.1.1 Rollover Scenario 4.2.2.2.4 Driveshaft Modes and 5.1.2 Importance of Rollover Driveline Modes 5.1.3 Research on Rollover 4.2.2.3 Vibration Sources from the 5.1.4 Scope of This Chapter Exhaust System 5.2 Rigid Vehicle Rollover Model 4.3 Vehicle Ride Models 5.2.1 Rigid Vehicle Model 4.3.1 Quarter Car Model 5.2.2 Steady-State Rollover on a Flat Road 4.3.1.1 Modeling for the Quarter Car 5.2.3 Tilt Table Ratio Model 5.2.4 Side Pull Ratio 4.3.1.2 Modal Analysis for the Quarter Car 5.3 Suspended Vehicle Rollover Model Model 5.3.1 Steady-State Rollover Model for a 4.3.1.3 Dynamic Analysis for the Quarter Suspended Vehicle Car Model 5.3.2 Contribution from the Tire Deflection 4.3.1.3.1 Transmissibility Between the 5.3.3 Contribution from the Suspension Body Response and Road Deflection Excitation 5.3.4 Parameters Influencing the Suspended 4.3.1.3.2 Transmissibility Between the Rollover Model Body Response and Vehicle 5.4 Dynamic Rollover Model Excitation 5.4.1 Rigid Dynamic Model 4.3.1.3.3 Dynamic Response at 5.4.2 Dynamic Rollover Model for a Random Input Dependent Suspension Vehicle 4.3.2 Bounce-Pitch Model 5.4.3 Dynamic Rollover Model for an 4.3.3 Other Modeling Independent Suspension Vehicle 4.4 Seat Evaluation and Modeling 5.4.4 Rollover Simulation Tools 4.4.1 Introduction 5.5 Dynamic Rollover Threshold 4.4.2 SEAT Value 5.5.1 Dynamic Stability Index 4.4.3 Seat Velocity 5.5.2 Rollover Prevention Energy Reserve 4.4.4 Linear Seat Modeling and 5.5.3 Rollover Prevention Metric Transmissibility 5.5.4 Critical Sliding Velocity 4.4.5 Nonlinear Seat Modeling and 5.6 Occupant in Rollover Transmissibility 5.6.1 Overview of the Occupant and 4.5 Discomfort Evaluation and Human Body Rollover Model 5.6.2 Testing of an Occupant Model 4.5.1 Discomfort and Subjective Evaluation 5.6.3 Simulation of Occupant Rollover 4.5.2 Objective Evaluation of Ride Discomfort 5.7 Safety and Rollover Control 4.5.2.1 Weighted Root-Mean-Square 5.7.1 Overview of Rollover Safety Method 5.7.2 Sensing of Rollover 4.5.2.2 Objective Evaluation by the 5.7.3 Rollover Safety Control Vibration Dose Value 5.8 Summary 4.5.3 Linear Human Body Modeling 5.9 References 4.5.4 Objective Evaluation by Nonlinear Seat-Human Body Modeling Chapter 6 Handling Dynamics 4.6 Active and Semi-Active Control 6.1 Introduction 4.6.1 Introduction 6.1.1 Tire Cornering Forces 4.6.2 Basic Control Concepts 6.1.2 Forces and Torques in the Tire Contact 4.6.3 Active Control Area 4.6.4 Semi-Active Control 6.2 The Simplest Handling Models-Two- 4.7 Summary Degrees-of-Freedom Yaw Plane Model 4.8 References 6.3 Steady-State Handling Characteristics Road Vehicle Dynamics 6.3.1 Yaw Velocity Gain and Understeer 7.2.1.2 Torque Calculations Gradient 7.2.2 Disk Brakes 6.3.1.1 Neutral Steer 7.2.3 Consideration of Temperature 6.3.1.2 Understeer 7.3 Load Transfer During Braking 6.3.1.3 Oversteer 7.3.1 Simple Braking on a Horizontal Road 6.3.2 Difference Between Slip Angles of 7.3.2 Effect of Aerodynamic and Other Forces the Front and Rear Wheels 7.3.2.1 Rolling Resistance 6.3.3 Ratio of Radius of Turn 7.3.2.2 Aerodynamic Drag 6.4 Dynamic Characteristics of Handling 7.3.2.3 Powertrain Resistance 6.4.1 Handling Damping and Natural 7.3.2.4 Load Transfer on a Horizontal Frequency Plane 6.4.2 Step Steer Input Response 7.3.3 Effect of Grade 6.4.3 Ramp Steer Input Response 7.4 Optimal Braking Performance 6.4.4 Impulse Input Excitation Response 7.4.1 Braking of a Single Axle 6.4.5 Frequency Response of Yaw Velocity 7.4.1.1 Braking of the Front Axle 6.4.6 Stability Analysis 7.4.1.2 Braking of the Rear Axle 6.4.7 Curvature Response 7.4.1.3 Safety Considerations 6.5 Chassis System Effects on Handling 7.4.2 Braking at Both Axles Characteristics 7.4.2.1 Front Lock-Up 6.5.1 Lateral Force Transfer Effects on 7.4.2.2 Rear Lock-Up Cornering 7.4.3 Achieving Optimal Braking Performance 6.5.2 Steering System 7.5 Considerations of Vehicle Safety 6.5.3 Camber Change Effect 7.5.1 Skid (Slip) Condition and Braking 6.5.4 Roll Steer Effect 7.5.2 Anti-Lock Braking System 6.5.5 Lateral Force Compliance Steer 7.6 Pitch Plane Models 6.5.6 Aligning Torque Effects 7.7 Recent Advances in Automotive Braking 6.5.7 Effect