The Science of Vehicle Dynamics: Handling, Braking, and Ride of Road and Race Cars

Preface to the Third Edition
Preface to the Second Edition
Preface to the First Edition
Contents
About the Author
1 Introduction
1.1 Vehicle Definition
1.2 Vehicle Basic Scheme
References
2 Mechanics of the Wheel with Tire
2.1 The Tire as a Vehicle Component
2.2 Carcass Features
2.3 Contact Patch
2.4 Rim Position and Motion
2.4.1 Reference System
2.4.2 Rim Kinematics
2.5 Footprint Force
2.5.1 Perfectly Flat Road Surface
2.6 Global Mechanical Behavior
2.6.1 Tire Transient Behavior
2.6.2 Tire Steady-State Behavior
2.6.3 Simplifications Based on Tire Tests
2.7 Definition of Pure Rolling for Tires
2.7.1 Zero Longitudinal Force (Rolling Radius)
2.7.2 Zero Lateral Force
2.7.3 Zero Vertical Moment
2.7.4 Zero Lateral Force and Zero Vertical Moment
2.7.5 Pure Rolling Summary
2.7.6 Rolling Velocity and Rolling Yaw Rate
2.8 Definition of Tire Slips
2.8.1 Theoretical Slips
2.8.2 The Simple Case (No Camber)
2.8.3 From Slips to Velocities
2.8.4 (Not so) Practical Slips
2.8.5 Tire Slips are Rim Slips Indeed
2.8.6 Slip Angle
2.9 Grip Forces and Tire Slips
2.10 Tire Testing
2.10.1 Tests with Pure Longitudinal Slip
2.10.2 Tests with Pure Lateral Slip
2.11 Magic Formula
2.11.1 Magic Formula Properties
2.11.2 Fitting of Experimental Data
2.11.3 Vertical Load Dependence
2.11.4 Horizontal and Vertical Shifts
2.11.5 Camber Dependence
2.12 Mechanics of the Wheel with Tire
2.12.1 Braking/Driving
2.12.2 Cornering
2.12.3 Combined
2.12.4 Camber
2.12.5 Grip
2.12.6 Vertical Moment
2.13 Rolling Resistance
2.14 Driving Torque and Tractive Force
2.14.1 Tractive Force
2.15 Exercises
2.15.1 Pure Rolling
2.15.2 Theoretical and Practical Slips
2.15.3 Tire Translational Slips and Slip Angle
2.15.4 Tire Spin Slip and Camber Angle
2.15.5 Motorcycle Tire
2.15.6 Finding the Magic Formula Coefficients
2.16 Summary
2.17 List of Some Relevant Concepts
2.18 Key Symbols
References
3 Vehicle Model for Handling and Performance
3.1 Mathematical Framework
3.1.1 Vehicle Axis System
3.2 Vehicle Congruence (Kinematic) Equations
3.2.1 Velocity of upper GG, and Yaw Rate of the Vehicle
3.2.2 Yaw Angle of the Vehicle, and Trajectory of upper GG
3.2.3 Velocity Center upper CC
3.2.4 Fundamental Ratios betaβ and rhoρ
3.2.5 Acceleration of upper GG and Angular Acceleration of the Vehicle
3.2.6 Radius of Curvature of the Trajectory of upper GG
3.2.7 Radius of Curvature of the Trajectory of a Generic Point
3.2.8 Telemetry Data and Mathematical Channels
3.2.9 Acceleration Center upper KK
3.2.10 Inflection Circle
3.3 Tire Kinematics (Tire Slips)
3.3.1 Translational Slips
3.3.2 Spin Slips
3.4 Steering Geometry
3.4.1 Ackermann Steering Kinematics
3.4.2 Best Steering Geometry
3.4.3 Position of the Velocity Center and Relative Slip Angles
3.5 Vehicle Constitutive (Tire) Equations
3.6 Vehicle Equilibrium Equations
3.6.1 Inertial Terms
3.6.2 External Force and Moment
3.7 Forces Acting on the Vehicle
3.7.1 Weight
3.7.2 Aerodynamic Force
3.7.3 Road-Tire Friction Forces
3.7.4 Road-Tire Vertical Forces
3.8 Vehicle Equilibrium Equations (More Explicit Form)
3.9 Vertical Loads and Load Transfers
3.9.1 Longitudinal Load Transfer
3.9.2 Lateral Load Transfers
3.9.3 Vertical Load on Each Tire
3.10 Suspension First-Order Analysis
3.10.1 Suspension Reference Configuration
3.10.2 Suspension Internal Coordinates
3.10.3 Kinematic Camber Variation
3.10.4 Kinematic Track Width Variation
3.10.5 Vehicle Internal Coordinates
3.10.6 Definition of Roll and Vertical Stiffnesses
3.10.7 Suspension Internal Equilibrium
3.10.8 Effects of a Lateral Force
3.10.9 No-Roll Centers and No-Roll Axis
3.10.10 Suspension Jacking
3.10.11 Roll Moment
3.10.12 Roll Angles and Lateral Load Transfers
3.10.13 Explicit Expressions of the Lateral Load Transfers
3.10.14 Lateral Load Transfers with Rigid Tires
3.11 Dependent Suspensions (Solid Axle)
3.11.1 Unsprung Masses and Lateral Load Transfers
3.12 Linked Suspensions
3.13 Differential Mechanisms
3.13.1 Relative Angular Speeds
3.13.2 Torque Balance
3.13.3 Global Power Balance
3.13.4 Internal Power Balance
3.13.5 Internal Efficiency
3.13.6 Slow Wheel and Fast Wheel
3.13.7 Torque Split Relationship
3.13.8 Locking Coefficient
3.13.9 Rule of Thumb
3.13.10 A Simple Mathematical Model
3.13.11 Alternative Governing Equations
3.13.12 Open Differential
3.13.13 Limited-Slip Differentials (LSD)
3.13.14 Geared Differentials
3.13.15 Clutch-Pack Differentials
3.13.16 Spindle Axle
3.13.17 Differential-Tire Interaction
3.13.18 Informal Summary About the Differential Behavior
3.14 Vehicle Model for Handling and Performance
3.14.1 Equilibrium Equations
3.14.2 Roll Angles
3.14.3 Lateral Load Transfers
3.14.4 Total Vertical Loads
3.14.5 Static Camber and Camber Variations
3.14.6 Steer Angles
3.14.7 Tire Slips
3.14.8 Tire Constitutive Equations
3.14.9 Equations Governing the Differential Mechanisms
3.14.10 Summary
3.15 The Structure of This Vehicle Model
3.16 Three-Axle Vehicles
3.17 Exercises
3.17.1 Center of Curvature upper E Subscript upper GEG of the Trajectory of upper GG
3.17.2 Track Variation
3.17.3 Camber Variation
3.17.4 Power Loss in a Limited-Slip Differential
3.17.5 Differential-Tires Interaction
3.18 Summary
3.19 List of Some Relevant Concepts
3.20 Key Symbols
References
4 Braking Performance
4.1 Pure Braking
4.2 Vehicle Model for Braking Performance
4.3 Equilibrium Equations
4.3.1 Rigorous Moment Equation
4.4 Longitudinal Load Transfer
4.5 Maximum Deceleration
4.6 Brake Balance
4.7 All Possible Braking Combinations
4.8 Changing the Grip
4.9 Changing the Weight Distribution
4.10 A Numerical Example
4.11 Braking, Stopping, and Safe Distances
4.12 Braking Performance of Formula Cars
4.12.1 Equilibrium Equations
4.12.2 Vertical Loads
4.12.3 Maximum Deceleration
4.12.4 Brake Balance
4.12.5 Speed Independent Brake Balance
4.12.6 Practical Brake Balance
4.12.7 Speed Independent Practical Brake Balance
4.12.8 Sensitivities
4.12.9 Typical F1 Braking Performance
4.13 Exercises
4.13.1 Minimum Braking Distance
4.13.2 Braking with Aerodynamic Downforces
4.13.3 GP2 Brake Balance
4.13.4 Speed Independent Brake Balance
4.14 Summary
4.15 List of Some Relevant Concepts
4.16 Key Symbols
References
5 The Kinematics of Cornering
5.1 Planar Kinematics of a Rigid Body
5.1.1 Velocity Field and Velocity Center
5.1.2 Fixed and Moving Centrodes
5.1.3 Acceleration Field and Acceleration Center
5.1.4 Inflection Circle and Radii of Curvature
5.2 The Kinematics of a Turning Vehicle
5.2.1 Moving and Fixed Centrodes of a Turning Vehicle
5.2.2 Inflection Circle of a Turning Vehicle
5.2.3 Tracking the Curvatures of Front and Rear Midpoints
5.2.4 Evolutes
5.3 Exercises
5.3.1 Front and Rear Radii of Curvature
5.3.2 Drawing Centrodes
5.4 List of Some Relevant Concepts
5.5 Key Symbols
References
6 Map of Achievable Performance (MAP)
6.1 MAP Fundamental Idea
6.2 Input Achievable Regions
6.3 Achievable Performances on Input Regions
6.4 Output Achievable Regions
6.5 Achievable Performances on Output Regions
6.6 Mixed Input/Output Achievable Regions
6.7 Achievable Performances on Mixed I/O Regions
6.8 MAP from Constant Speed Tests
6.9 MAP from Constant Steer Tests
6.10 Concluding Remarks
6.11 List of Some Relevant Concepts
6.12 Key Symbols
7 Handling of Road Cars
7.1 Additional Simplifying Assumptions for Road Car Modeling
7.1.1 Negligible Vertical Aerodynamic Loads
7.1.2 Open Differential
7.1.3 Almost Constant Forward Speed
7.2 Mathematical Model for Road Car Handling
7.2.1 Global Equilibrium
7.2.2 Approximate Lateral Forces
7.2.3 Lateral Load Transfers and Vertical Loads
7.2.4 Roll Angles
7.2.5 Camber Angle Variations
7.2.6 Steer Angles
7.2.7 Tire Slips
7.2.8 Simplified Tire Slips
7.2.9 Tire Lateral Forces
7.3 Double Track Model
7.3.1 Governing Equations of the Double Track Model
7.3.2 Dynamical Equations of the Double Track Model
7.3.3 Alternative State Variables (β and ρ)
7.4 Vehicle in Steady-State Conditions
7.5 Single Track Model
7.5.1 From Double Track to Single Track
7.5.2 ``Forcing'' the Lateral Forces
7.5.3 Axle Characteristics
7.5.4 Governing Equations of the Single Track Model
7.5.5 Dynamical Equations of the Single Track Model
7.5.6 Alternative State Variables (β and ρ)
7.5.7 Inverse Congruence Equations
7.5.8 β1 and β2 as State Variables
7.5.9 Driving Force
7.5.10 The Role of the Steady-State Lateral Acceleration
7.5.11 Slopes of the Axle Characteristics
7.6 Double Track, or Single Track?
7.7 Steady-State Maps
7.7.1 Steady-State Gradients
7.7.2 Alternative Steady-State Gradients
7.7.3 Understeer and Oversteer
7.7.4 Handling Diagram
7.8 Map of Achievable Performance (MAP)
7.8.1 MAP Fundamentals
7.8.2 MAP Curvature ρ Versus Steer Angle δ
7.8.3 Other Possible MAPs
7.9 Weak Concepts in Classical Vehicle Dynamics
7.9.1 The Understeer Gradient
7.9.2 Popular Definitions of Understeer/Oversteer
7.10 Double Track Model in Transient Conditions
7.10.1 Equilibrium Points
7.10.2 Free Oscillations (No Driver Action)
7.10.3 Stability of the Equilibrium
7.10.4 Forced Oscillations (Driver Action)
7.11 Relationship Between Steady-State Data and Transient Behavior
7.11.1 Stability Derivatives from Steady-State Gradients
7.11.2 Equations of Motion
7.11.3 Estimation of the Control Derivatives
7.11.4 Objective Evaluation of Car Handling
7.12 Stability (Again)
7.13 New Understeer Gradient
7.14 The Nonlinear Single Track Model Revisited
7.14.1 Very Different Vehicles with Identical Handling
7.15 Linear Single Track Model
7.15.1 Governing Equations
7.15.2 Solution for Constant Forward Speed
7.15.3 Critical Speed
7.15.4 Transient Vehicle Behavior
7.15.5 Steady-State Behavior: Steering Pad
7.15.6 Lateral Wind Gust
7.15.7 Banked Road
7.16 Compliant Steering System
7.16.1 Governing Equations
7.16.2 Effects of Steer Compliance
7.16.3 There Is Something Unsafe
7.17 Road Vehicles with Locked or Limited Slip Differential
7.18 Exercises
7.18.1 Camber Variations
7.18.2 Ackermann Coefficient
7.18.3 Toe-In
7.18.4 Steering Angles
7.18.5 Axle Characteristics
7.18.6 Playing with Linear Differential Equations
7.18.7 Static Margin
7.18.8 Banked Road
7.18.9 Rear Steer
7.18.10 Wind Gust
7.19 Summary
7.20 List of Some Relevant Concepts
7.21 Key Symbols
References
8 Handling of Race Cars
8.1 Assumptions for Race Car Handling
8.1.1 Aerodynamic Downforces and Drag
8.1.2 Limited-Slip Differential
8.2 Vehicle Model for Race Car Handling
8.2.1 Equilibrium Equations
8.2.2 Lateral Forces for Dynamic Equilibrium
8.2.3 Gyroscopic Torques
8.2.4 Roll Angles
8.2.5 Vertical Loads on Each Wheel
8.2.6 Lateral Load Transfers
8.2.7 In-Plane Tire Forces
8.2.8 Tire Slips
8.2.9 Camber Angles
8.2.10 Steer Angles
8.2.11 Simple Model of a Limited-Slip Differential
8.2.12 Reducing the Number of Equations
8.3 Double Track Race Car Model
8.4 Kinematics of Race Cars when Cornering
8.5 Handling of Race Cars with Open Differential
8.5.1 Single Track Model for Race Cars
8.5.2 What About Understeer/Oversteer?
8.6 Steady-State Handling Analysis
8.6.1 Map of Achievable Performance (MAP)
8.6.2 MAPs from Real Cases
8.6.3 Power-Off and Power-On
8.7 Stability Derivatives and Control Derivatives
8.8 Stability Derivatives from Steady-State Gradients
8.8.1 Alternative Independent Variables
8.9 Comparison of Limited-Slip Differentials
8.9.1 Yaw Moment
8.9.2 Difference of Angular Speeds
8.9.3 Internal Power Loss
8.9.4 Longitudinal Forces
8.10 Exercises
8.10.1 Vehicle Kinematic Equations
8.10.2 Spin Slip Contributions
8.10.3 Acceleration Center and Acceleration of the Velocity Center
8.10.4 Aerodynamic Downforces
8.10.5 Roll Stiffnesses in Formula Cars
8.10.6 Lateral Load Transfers in Formula Cars
8.10.7 Centrifugal Force Not Applied at the Center of Mass
8.10.8 Global Aerodynamic Force
8.11 Summary
8.12 List of Some Relevant Concepts
8.13 Key Symbols
References
9 Handling with Roll Motion
9.1 Vehicle Position and Orientation
9.2 Yaw, Pitch and Roll
9.3 Angular Velocity
9.4 Angular Acceleration
9.5 Vehicle Lateral Velocity
9.5.1 Track Invariant Points
9.5.2 Vehicle Invariant Point (VIP)
9.5.3 Lateral Velocity and Acceleration
9.6 Three-Dimensional Vehicle Dynamics
9.6.1 Velocity and Acceleration of G
9.6.2 Rate of Change of the Angular Momentum
9.6.3 Completing the Torque Equation
9.6.4 Equilibrium Equations
9.6.5 Including the Unsprung Mass
9.7 Handling with Roll Motion
9.7.1 Equilibrium Equations
9.7.2 Load Transfers
9.7.3 Constitutive (Tire) Equations
9.7.4 Congruence (Kinematic) Equations
9.8 Steady-State and Transient Analysis
9.9 Exercise
9.9.1 Roll Motion and Camber Variation
9.10 Summary
9.11 List of Some Relevant Concepts
9.12 Key Symbols
References
10 Ride Comfort and Road Holding
10.1 Vehicle Models for Ride and Road Holding
10.2 Quarter (Full) Car Model
10.2.1 The Inerter as a Spring Softener
10.2.2 Quarter Car Natural Frequencies and Modes
10.3 Damper Tuning
10.3.1 Optimal Damper for Comfort
10.3.2 Optimal Damper for Road Holding
10.3.3 The Inerter as a Tool for Road Holding Tuning
10.4 More General Suspension Layouts
10.5 Road Profiles
10.6 Free Vibrations of Road Cars
10.6.1 Governing Equations
10.6.2 Proportional Viscous Damping
10.6.3 Vehicle with Proportional Viscous Damping
10.6.4 Principal Coordinates
10.6.5 Selection of Front and Rear Suspension Vertical Stiffnesses
10.7 Tuning of the Suspension Stiffnesses
10.7.1 Optimality of Proportional Damping
10.7.2 A Numerical Example
10.8 Non-Proportional Damping
10.9 Interconnected Suspensions
10.10 Exercises
10.10.1 Playing with η
10.10.2 Playing with ρ
10.11 Summary
10.12 List of Some Relevant Concepts
10.13 Key Symbols
References
11 Tire Models
11.1 Brush Model Definition
11.1.1 Roadway and Rim
11.1.2 Shape of the Contact Patch
11.1.3 Pressure Distribution and Vertical Load
11.1.4 Force–Couple Resultant
11.1.5 Elastic Compliance of the Tire Carcass
11.1.6 Friction
11.1.7 Constitutive Relationship
11.1.8 Kinematics
11.1.9 Brush Model Slips
11.1.10 Sliding Velocity of the Bristle Tips
11.1.11 Summary of Relevant Velocities
11.2 General Governing Equations of the Brush Model
11.2.1 Data for Numerical Examples
11.3 Brush Model Steady-State Behavior
11.3.1 Steady-State Governing Equations
11.3.2 Adhesion and Sliding Zones
11.3.3 Force–Couple Resultant
11.3.4 Examples of Tangential Stress Distributions
11.4 Adhesion Everywhere (Linear Behavior)
11.5 Translational Slip Only (bold italic sigma not equals bold 0σneq0, phi equals 0=0)
11.5.1 Rectangular Contact Patch
11.5.2 Elliptical Contact Patch
11.6 Wheel with Pure Spin Slip (bold italic sigma equals bold 0σ=0, phi not equals 0neq0)
11.7 Wheel with Both Translational and Spin Slips
11.7.1 Rectangular Contact Patch
11.7.2 Elliptical Contact Patch
11.8 Brush Model Transient Behavior
11.8.1 Transient Models with Carcass Compliance Only
11.8.2 Transient Model with Carcass and Tread Compliance
11.8.3 Model Comparison
11.8.4 Selection of Tests
11.8.5 Longitudinal Step Input
11.8.6 Lateral Step Input
11.9 Exercises
11.9.1 Braking or Driving?
11.9.2 Carcass Compliance
11.9.3 Brush Model: Local, Linear, Isotropic, Homogeneous
11.9.4 Anisotropic Brush Model
11.9.5 Carcass Compliance 2
11.9.6 Skating Versus Sliding
11.9.7 Skating Slip
11.9.8 Simplest Brush Model
11.9.9 Velocity Relationships
11.9.10 Slip Stiffness Reduction
11.9.11 Total Sliding
11.9.12 Spin Slip and Camber Angle
11.9.13 The Right Amount of Camber
11.9.14 Slip Stiffness
11.10 Summary
11.11 List of Some Relevant Concepts
11.12 Key Symbols
References
Appendix Index
Index