FACHHOCHSCHULE REGENSBURG UNIVERSITY OF APPLIED SCIENCES HOCHSCHULE FÜR TECHNIK WIRTSCHAFT SOZIALES SHORT COURSE Prof. Dr. Georg Rill © Brasil, August 2007 VEHICLE DYNAMICS Contents Contents I 1 Introduction 1 1.1 Terminology ................................... 1 1.1.1 Vehicle Dynamics ............................ 1 1.1.2 Driver .................................. 1 1.1.3 Vehicle .................................. 2 1.1.4 Load ................................... 3 1.1.5 Environment .............................. 3 1.2 Definitions .................................... 3 1.2.1 Reference frames ............................ 3 1.2.2 Toe-in, Toe-out ............................. 4 1.2.3 Wheel Camber ............................. 5 1.2.4 Design Position of Wheel Rotation Axis ............... 5 1.2.5 Steering Geometry ........................... 6 1.3 Driver ....................................... 8 1.4 Road ....................................... 9 2 TMeasy - An Easy to Use Tire Model 11 2.1 Introduction ................................... 11 2.1.1 Tire Development ............................ 11 2.1.2 Tire Composites ............................ 11 2.1.3 Tire Forces and Torques ........................ 12 2.1.4 Measuring Tire Forces and Torques ................. 13 2.1.5 Modeling Aspects ........................... 14 2.1.6 Typical Tire Characteristics ...................... 17 2.2 Contact Geometry ................................ 18 2.2.1 Basic Approach ............................. 18 2.2.2 Local Track Plane ............................ 21 2.2.3 Tire Deflection .............................. 22 2.2.4 Length of Contact Patch ........................ 24 2.2.5 Static Contact Point ........................... 25 2.2.6 Contact Point Velocity ......................... 27 2.2.7 Dynamic Rolling Radius ........................ 28 2.3 Steady State Forces and Torques ........................ 30 2.3.1 Wheel Load ............................... 30 I Contents 2.3.2 Tipping Torque ............................. 32 2.3.3 Rolling Resistance ........................... 33 2.3.4 Longitudinal Force and Longitudinal Slip .............. 34 2.3.5 Lateral Slip, Lateral Force and Self Aligning Torque ........ 37 2.3.6 Bore Torque ............................... 39 2.3.6.1 Modeling Aspects ...................... 39 2.3.6.2 Maximum Torque ...................... 40 2.3.6.3 Bore Slip ........................... 41 2.3.6.4 Model Realisation ...................... 42 2.3.7 Different Influences ........................... 42 2.3.7.1 Wheel Load .......................... 42 2.3.7.2 Friction ............................ 44 2.3.7.3 Camber ............................ 45 2.3.8 Combined Forces ............................ 48 2.3.8.1 Generalized Slip ....................... 48 2.3.8.2 Suitable Approximation .................. 50 2.3.8.3 Results ............................ 52 2.4 First Order Tire Dynamics ........................... 53 2.4.1 Simple Dynamic Extension ...................... 53 2.4.2 Enhanced Force Dynamics ....................... 54 2.4.2.1 Compliance Model ..................... 54 2.4.2.2 Relaxation Lengths ..................... 56 2.4.2.3 Performance at Stand Still ................. 57 2.4.3 Enhanced Torque Dynamics ...................... 57 2.4.3.1 Self Aligning Torque ..................... 57 2.4.3.2 Bore Torque .......................... 58 2.4.3.3 Parking Torque ........................ 60 3 Drive Train 63 3.1 Components ................................... 63 3.2 Engine ...................................... 64 3.3 Clutch ...................................... 64 3.4 Transmission ................................... 66 3.5 Drive Shafts, Half Shafts and Differentials .................. 68 3.5.1 Model Structure ............................. 68 3.5.2 Equation of Motion ........................... 70 3.5.3 Drive Shaft Torques .......................... 71 3.5.4 Locking Torques ............................ 72 3.6 Wheel Rotation ................................. 73 3.6.1 Driving and Braking Torques ..................... 73 3.6.2 Wheel Tire Dynamics .......................... 74 4 Suspension System 77 4.1 Purpose and Components ........................... 77 II Contents 4.2 Some Examples ................................. 78 4.2.1 Multi Purpose Systems ......................... 78 4.2.2 Specific Systems ............................ 79 4.3 Steering Systems ................................ 79 4.3.1 Components and Requirements ................... 79 4.3.2 Rack and Pinion Steering ....................... 80 4.3.3 Lever Arm Steering System ...................... 80 4.3.4 Drag Link Steering System ...................... 81 4.3.5 Bus Steer System ............................ 82 5 Force Elements 83 5.1 Standard Force Elements ............................ 83 5.1.1 Springs .................................. 83 5.1.2 Anti-Roll Bar .............................. 84 5.1.3 Damper ................................. 86 5.1.4 Rubber Elements ............................ 87 5.2 Dynamic Force Elements ............................ 88 5.2.1 Testing and Evaluating Procedures .................. 88 5.2.2 Simple Spring Damper Combination ................. 92 5.2.3 General Dynamic Force Model .................... 93 5.2.3.1 Hydro-Mount ........................ 95 6 Vertical Dynamics 99 6.1 Goals ....................................... 99 6.2 Basic Tuning ................................... 99 6.2.1 From complex to simple models ................... 99 6.2.2 Natural Frequency and Damping Rate ................ 102 6.2.3 Spring Rates ............................... 104 6.2.3.1 Minimum Spring Rates ................... 104 6.2.3.2 Nonlinear Springs ...................... 106 6.2.4 Influence of Damping ......................... 108 6.2.5 Optimal Damping ........................... 109 6.2.5.1 Avoiding Overshoots .................... 109 6.2.5.2 Disturbance Reaction Problem ............... 109 6.3 Sky Hook Damper ............................... 114 6.3.1 Modeling Aspects ........................... 114 6.3.2 Eigenfrequencies and Damping Ratios ................ 115 6.3.3 Technical Realization .......................... 116 6.4 Nonlinear Force Elements ........................... 117 6.4.1 Quarter Car Model ........................... 117 6.4.2 Results .................................. 119 7 Longitudinal Dynamics 121 7.1 Dynamic Wheel Loads ............................. 121 III Contents 7.1.1 Simple Vehicle Model ......................... 121 7.1.2 Influence of Grade ........................... 122 7.1.3 Aerodynamic Forces .......................... 123 7.2 Maximum Acceleration ............................. 124 7.2.1 Tilting Limits .............................. 124 7.2.2 Friction Limits ............................. 124 7.3 Driving and Braking .............................. 125 7.3.1 Single Axle Drive ............................ 125 7.3.2 Braking at Single Axle ......................... 126 7.3.3 Braking Stability ............................ 127 7.3.4 Optimal Distribution of Drive and Brake Forces .......... 128 7.3.5 Different Distributions of Brake Forces ................ 130 7.3.6 Anti-Lock-System ............................ 130 7.3.7 Braking on mu-Split .......................... 131 7.4 Drive and Brake Pitch ............................. 132 7.4.1 Vehicle Model .............................. 132 7.4.2 Equations of Motion .......................... 133 7.4.3 Equilibrium ............................... 135 7.4.4 Driving and Braking .......................... 135 7.4.5 Anti Dive and Anti Squat ....................... 137 8 Lateral Dynamics 139 8.1 Kinematic Approach .............................. 139 8.1.1 Kinematic Tire Model ......................... 139 8.1.2 Ackermann Geometry ......................... 139 8.1.3 Space Requirement ........................... 140 8.1.4 Vehicle Model with Trailer ....................... 142 8.1.4.1 Kinematics .......................... 142 8.1.4.2 Vehicle Motion ........................ 143 8.1.4.3 Entering a Curve ....................... 145 8.1.4.4 Trailer Motions ........................ 145 8.1.4.5 Course Calculations ..................... 146 8.2 Steady State Cornering ............................. 147 8.2.1 Cornering Resistance .......................... 147 8.2.2 Overturning Limit ........................... 149 8.2.3 Roll Support and Camber Compensation .............. 152 8.2.4 Roll Center and Roll Axis ....................... 154 8.2.5 Wheel Loads .............................. 155 8.3 Simple Handling Model ............................ 155 8.3.1 Modeling Concept ........................... 155 8.3.2 Kinematics ................................ 156 8.3.3 Tire Forces ................................ 157 8.3.4 Lateral Slips ............................... 157 8.3.5 Equations of Motion .......................... 158 IV Contents 8.3.6 Stability ................................. 159 8.3.6.1 Eigenvalues .......................... 159 8.3.6.2 Low Speed Approximation ................. 160 8.3.6.3 High Speed Approximation ................ 160 8.3.6.4 Critical Speed ........................ 161 8.3.7 Steady State Solution .......................... 162 8.3.7.1 Steering Tendency ...................... 162 8.3.7.2 Side Slip Angle ........................ 164 8.3.7.3 Slip Angles .......................... 165 8.3.8 Influence of Wheel Load on Cornering Stiffness .......... 165 8.4 Mechatronic Systems .............................. 167 8.4.1 Electronic Stability Control (ESC) ................... 167 8.4.2 Steer-by-Wire .............................. 168 9 Driving