1 Introduction |
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1 | (6) |
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2 | (1) |
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3 | (3) |
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6 | (1) |
2 Mechanics of the Wheel with Tire |
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7 | (60) |
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2.1 The Tire as a Vehicle Component |
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9 | (1) |
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9 | (1) |
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10 | (2) |
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2.4 Rim Position and Motion |
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12 | (4) |
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13 | (1) |
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13 | (3) |
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16 | (4) |
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2.5.1 Perfectly Flat Road Surface |
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18 | (2) |
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2.6 Global Mechanical Behavior |
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20 | (3) |
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2.6.1 Tire Transient Behavior |
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20 | (1) |
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2.6.2 Tire Steady-State Behavior |
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20 | (1) |
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2.6.3 Simplifications Based on Tire Tests |
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21 | (2) |
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2.7 Rolling Resistance Moment |
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23 | (2) |
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2.8 Definition of Pure Rolling for Tires |
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25 | (8) |
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2.8.1 Zero Longitudinal Force |
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26 | (2) |
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28 | (1) |
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2.8.3 Zero Vertical Moment |
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28 | (1) |
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2.8.4 Zero Lateral Force and Zero Vertical Moment |
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28 | (1) |
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2.8.5 Pure Rolling Summary |
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29 | (2) |
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2.8.6 Rolling Velocity and Rolling Yaw Rate |
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31 | (2) |
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2.9 Definition of Tire Slips |
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33 | (5) |
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34 | (1) |
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2.9.2 The Simple Case (No Camber) |
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35 | (1) |
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2.9.3 From Slips to Velocities |
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35 | (1) |
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2.9.4 (Not So) Practical Slips |
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36 | (1) |
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2.9.5 Tire Slips Are Rim Slips Indeed |
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36 | (1) |
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37 | (1) |
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2.10 Grip Forces and Tire Slips |
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38 | (1) |
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39 | (6) |
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2.11.1 Tests with Pure Longitudinal Slip |
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41 | (1) |
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2.11.2 Tests with Pure Lateral Slip |
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42 | (3) |
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45 | (5) |
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2.12.1 Magic Formula Properties |
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46 | (1) |
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2.12.2 Fitting of Experimental Data |
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47 | (1) |
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2.12.3 Vertical Load Dependence |
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47 | (3) |
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2.12.4 Horizontal and Vertical Shifts |
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50 | (1) |
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50 | (1) |
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2.13 Mechanics of the Wheel with Tire |
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50 | (8) |
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51 | (1) |
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51 | (2) |
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53 | (2) |
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55 | (1) |
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56 | (1) |
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57 | (1) |
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58 | (9) |
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58 | (1) |
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2.14.2 Theoretical and Practical Slips |
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58 | (1) |
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2.14.3 Tire Translational Slips and Slip Angle |
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58 | (1) |
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2.14.4 Tire Spin Slip and Camber Angle |
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59 | (1) |
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59 | (1) |
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2.14.6 Finding the Magic Formula Coefficients |
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60 | (7) |
3 Vehicle Model for Handling and Performance |
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67 | (102) |
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3.1 Mathematical Framework |
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68 | (1) |
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3.1.1 Vehicle Axis System |
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68 | (1) |
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3.2 Vehicle Congruence (Kinematic) Equations |
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69 | (12) |
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3.2.1 Velocity of G, and Yaw Rate of the Vehicle |
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69 | (1) |
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3.2.2 Yaw Angle of the Vehicle, and Trajectory of G |
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70 | (2) |
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72 | (1) |
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3.2.4 Fundamental Ratios β and ρ |
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73 | (1) |
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3.2.5 Acceleration of G and Angular Acceleration of the Vehicle |
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73 | (3) |
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3.2.6 Radius of Curvature of the Trajectory of G |
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76 | (2) |
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3.2.7 Radius of Curvature of the Trajectory of a Generic Point |
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78 | (1) |
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3.2.8 Telemetry Data and Mathematical Channels |
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78 | (1) |
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3.2.9 Acceleration Center K |
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79 | (1) |
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80 | (1) |
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3.3 Tire Kinematics (Tire Slips) |
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81 | (4) |
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3.3.1 Translational Slips |
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84 | (1) |
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85 | (1) |
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3.4 Steering Geometry (Ackermann) |
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85 | (5) |
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3.4.1 Ackermann Steering Kinematics |
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87 | (2) |
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3.4.2 Best Steering Geometry |
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89 | (1) |
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3.4.3 Position of Velocity Center and Relative Slip Angles |
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89 | (1) |
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3.5 Vehicle Constitutive (Tire) Equations |
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90 | (1) |
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3.6 Vehicle Equilibrium Equations |
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91 | (2) |
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92 | (1) |
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3.6.2 External Force and Moment |
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92 | (1) |
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3.7 Forces Acting on the Vehicle |
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93 | (7) |
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93 | (1) |
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93 | (2) |
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3.7.3 Road-Tire Friction Forces |
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95 | (4) |
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3.7.4 Road-Tire Vertical Forces |
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99 | (1) |
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3.8 Vehicle Equilibrium Equations (More Explicit Form) |
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100 | (2) |
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3.9 Vertical Loads and Load Transfers |
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102 | (2) |
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3.9.1 Longitudinal Load Transfer |
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102 | (1) |
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3.9.2 Lateral Load Transfers |
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103 | (1) |
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3.9.3 Vertical Load on Each Tire |
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103 | (1) |
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3.10 Suspension First-Order Analysis |
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104 | (20) |
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3.10.1 Suspension Reference Configuration |
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105 | (1) |
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3.10.2 Suspension Internal Coordinates |
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106 | (1) |
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3.10.3 Kinematic Camber Variation |
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107 | (1) |
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3.10.4 Kinematic Track Width Variation |
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108 | (1) |
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3.10.5 Vehicle Internal Coordinates |
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109 | (1) |
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3.10.6 Definition of Roll and Vertical Stiffnesses |
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109 | (4) |
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3.10.7 Suspension Internal Equilibrium |
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113 | (1) |
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3.10.8 Effects of a Lateral Force |
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113 | (2) |
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3.10.9 No-Roll Centers and No-Roll Axis |
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115 | (3) |
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3.10.10 Suspension Jacking |
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118 | (1) |
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118 | (2) |
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3.10.12 Roll Angles and Lateral Load Transfers |
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120 | (2) |
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3.10.13 Explicit Expressions of the Lateral Load Transfers |
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122 | (2) |
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3.10.14 Lateral Load Transfers with Rigid Tires |
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124 | (1) |
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3.11 Sprung and Unsprung Masses |
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124 | (1) |
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3.12 Dependent Suspensions (Solid Axle) |
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125 | (3) |
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128 | (1) |
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3.14 Differential Mechanisms |
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128 | (22) |
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3.14.1 Relative Angular Speeds |
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130 | (1) |
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130 | (1) |
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3.14.3 Internal Efficiency and TBR |
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131 | (4) |
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3.14.4 Locking Coefficient |
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135 | (1) |
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136 | (2) |
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3.14.6 A Simple Mathematical Model |
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138 | (1) |
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3.14.7 Alternative Governing Equations |
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138 | (1) |
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139 | (1) |
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3.14.9 Limited-Slip Differentials |
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139 | (1) |
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3.14.10 Geared Differentials |
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140 | (1) |
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3.14.11 Clutch-Pack Differentials |
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141 | (3) |
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144 | (1) |
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3.14.13 Differential-Tire Interaction |
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144 | (6) |
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3.14.14 Informal Summary About the Differential Behavior |
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150 | (1) |
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3.15 Vehicle Model for Handling and Performance |
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150 | (7) |
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3.15.1 Equilibrium Equations |
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150 | (2) |
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152 | (1) |
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153 | (1) |
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153 | (1) |
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154 | (1) |
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3.15.6 Tire Constitutive Equations |
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155 | (1) |
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3.15.7 Differential Mechanism Equations |
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156 | (1) |
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156 | (1) |
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3.16 The Structure of This Vehicle Model |
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157 | (1) |
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157 | (3) |
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160 | (4) |
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3.18.1 Center of Curvature QG of the Trajectory of G |
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160 | (1) |
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160 | (1) |
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160 | (1) |
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3.18.4 Power Loss in a Self-locking Differential |
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161 | (1) |
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3.18.5 Differential-Tires Interaction |
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161 | (3) |
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164 | (1) |
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3.20 List of Some Relevant Concepts |
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165 | (1) |
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165 | (2) |
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167 | (2) |
4 Braking Performance |
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169 | (20) |
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170 | (1) |
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4.2 Vehicle Model for Braking Performance |
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170 | (1) |
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4.3 Equilibrium Equations |
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171 | (1) |
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4.4 Longitudinal Load Transfer |
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172 | (1) |
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172 | (1) |
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173 | (1) |
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4.7 All Possible Braking Combinations |
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173 | (2) |
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175 | (1) |
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4.9 Changing the Weight Distribution |
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176 | (1) |
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176 | (1) |
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4.11 Braking Performance of Formula Cars |
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177 | (6) |
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4.11.1 Equilibrium Equations |
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177 | (1) |
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178 | (1) |
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4.11.3 Maximum Deceleration |
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179 | (1) |
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180 | (1) |
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4.11.5 Speed Independent Brake Balance |
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181 | (1) |
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4.11.6 Typical Formula 1 Braking Performance |
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181 | (2) |
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4.12 Braking, Stopping, and Safe Distances |
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183 | (1) |
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183 | (4) |
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4.13.1 Minimum Braking Distance |
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183 | (2) |
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4.13.2 Braking with Aerodynamic Downforces |
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185 | (1) |
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185 | (1) |
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4.13.4 Speed Independent Brake Balance |
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186 | (1) |
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187 | (1) |
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4.15 List of Some Relevant Concepts |
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187 | (1) |
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188 | (1) |
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188 | (1) |
5 The Kinematics of Cornering |
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189 | (24) |
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5.1 Planar Kinematics of a Rigid Body |
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189 | (7) |
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5.1.1 Velocity Field and Velocity Center |
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190 | (2) |
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5.1.2 Acceleration Field and Acceleration Center |
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192 | (1) |
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5.1.3 Inflection Circle and Radii of Curvature |
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193 | (3) |
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5.2 The Kinematics of a Turning Vehicle |
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196 | (14) |
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5.2.1 Moving and Fixed Centrodes of a Turning Vehicle |
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197 | (4) |
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5.2.2 Inflection Circle of a Turning Vehicle |
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201 | (4) |
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5.2.3 Tracking the Curvatures of Front and Rear Midpoints |
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205 | (5) |
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210 | (1) |
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210 | (1) |
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5.3.1 Front and Rear Radii of Curvature |
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210 | (1) |
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211 | (1) |
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211 | (1) |
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212 | (1) |
6 Handling of Road Cars |
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213 | (110) |
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6.1 Additional Simplifying Assumptions for Road Car Modeling |
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214 | (1) |
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6.1.1 Negligible Vertical Aerodynamic Loads |
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214 | (1) |
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6.1.2 Almost Constant Forward Speed |
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214 | (1) |
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215 | (1) |
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6.2 Mathematical Model for Road Car Handling |
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215 | (12) |
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216 | (1) |
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6.2.2 Approximate Lateral Forces |
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217 | (1) |
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6.2.3 Lateral Load Transfers and Vertical Loads |
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218 | (2) |
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220 | (1) |
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6.2.5 Camber Angle Variations |
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220 | (2) |
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222 | (1) |
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223 | (1) |
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6.2.8 Simplified Tire Slips |
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224 | (2) |
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6.2.9 Tire Lateral Forces |
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226 | (1) |
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227 | (2) |
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6.3.1 Governing Equations of the Double Track Model |
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227 | (1) |
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6.3.2 Dynamical Equations of the Double Track Model |
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228 | (1) |
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6.3.3 Alternative State Variables (β and ρ) |
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228 | (1) |
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6.4 Vehicle in Steady-State Conditions |
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229 | (2) |
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231 | (21) |
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6.5.1 From Double to Single |
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231 | (3) |
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6.5.2 "Forcing" the Lateral Forces |
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234 | (1) |
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6.5.3 Axle Characteristics |
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235 | (9) |
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6.5.4 Governing Equations of the Single Track Model |
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244 | (2) |
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6.5.5 Dynamical Equations of the Single Track Model |
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246 | (1) |
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6.5.6 Alternative State Variables (β and ρ) |
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247 | (1) |
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6.5.7 Inverse Congruence Equations |
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248 | (1) |
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6.5.8 β1 and β2 as State Variables |
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248 | (2) |
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250 | (1) |
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6.5.10 The Role of the Steady-State Lateral Acceleration |
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251 | (1) |
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6.5.11 Slopes of the Axle Characteristics |
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252 | (1) |
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6.6 Double Track, or Single Track? |
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252 | (1) |
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253 | (8) |
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6.7.1 Steady-State Gradients |
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255 | (1) |
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6.7.2 Alternative Steady-State Gradients |
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256 | (1) |
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6.7.3 Understeer and Oversteer |
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256 | (3) |
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259 | (2) |
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6.8 Map of Achievable Performance (MAP) |
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261 | (13) |
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262 | (6) |
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6.8.2 MAP Curvature ρ Versus Steer Angle δ |
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268 | (5) |
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6.8.3 Other Possible MAPS |
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273 | (1) |
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6.9 Weak Concepts in Classical Vehicle Dynamics |
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274 | (2) |
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6.9.1 The Understeer Gradient |
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275 | (1) |
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6.9.2 Popular Definitions of Understeer/Oversteer |
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276 | (1) |
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6.10 Double Track Model in Transient Conditions |
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276 | (8) |
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6.10.1 Equilibrium Points |
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277 | (1) |
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6.10.2 Free Oscillations (No Driver Action) |
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277 | (4) |
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6.10.3 MAP for Transient Behavior |
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281 | (1) |
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6.10.4 Stability of the Equilibrium |
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282 | (1) |
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6.10.5 Forced Oscillations (Driver Action) |
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282 | (2) |
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6.11 Relationship Between Steady-State Data and Transient Behavior |
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284 | (6) |
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6.11.1 Stability Derivatives from Steady-State Gradients |
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285 | (2) |
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6.11.2 Equations of Motion |
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287 | (1) |
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6.11.3 Estimation of the Control Derivatives |
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288 | (1) |
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6.11.4 Objective Evaluation of Car Handling |
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288 | (2) |
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290 | (1) |
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6.13 New Understeer Gradient |
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291 | (1) |
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6.14 The Nonlinear Single Track Model Revisited |
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292 | (6) |
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6.14.1 Different Vehicles with Identical Handling |
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295 | (3) |
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6.15 Linear Single Track Model |
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298 | (14) |
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6.15.1 Governing Equations |
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299 | (2) |
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6.15.2 Solution for Constant Forward Speed |
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301 | (2) |
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303 | (1) |
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6.15.4 Transient Vehicle Behavior |
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303 | (3) |
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6.15.5 Steady-State Behavior: Steering Pad |
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306 | (1) |
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307 | (4) |
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311 | (1) |
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6.16 Compliant Steering System |
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312 | (3) |
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6.16.1 Governing Equations |
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313 | (1) |
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6.16.2 Effects of Steer Compliance |
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314 | (1) |
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6.17 Road Vehicles with Locked or Limited Slip Differential |
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315 | (1) |
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315 | (4) |
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315 | (1) |
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6.18.2 Ackermann Coefficient |
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315 | (1) |
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316 | (1) |
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316 | (1) |
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6.18.5 Axle Characteristics |
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316 | (1) |
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6.18.6 Playing with Linear Differential Equations |
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317 | (1) |
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317 | (1) |
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317 | (1) |
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318 | (1) |
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318 | (1) |
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319 | (1) |
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6.20 List of Some Relevant Concepts |
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320 | (1) |
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320 | (2) |
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322 | (1) |
7 Handling of Race Cars |
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323 | (54) |
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7.1 Assumptions for Race Car Handling |
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323 | (2) |
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7.1.1 Aerodynamic Downloads |
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324 | (1) |
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7.1.2 Limited-Slip Differential |
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324 | (1) |
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7.2 Vehicle Model for Race Car Handling |
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325 | (12) |
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7.2.1 Equilibrium Equations |
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326 | (2) |
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7.2.2 Lateral Forces for Dynamic Equilibrium |
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328 | (1) |
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328 | (1) |
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329 | (1) |
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330 | (1) |
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331 | (1) |
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7.2.7 Vertical Loads on Each Wheel |
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332 | (1) |
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7.2.8 Lateral Load Transfers |
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333 | (1) |
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334 | (1) |
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7.2.10 Behavior of the Limited-Slip Differential |
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334 | (1) |
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7.2.11 Reducing the Number of Equations |
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335 | (2) |
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7.3 Double Track Race Car Model |
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337 | (1) |
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337 | (1) |
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7.4 Basics for Steady-State Handling Analysis |
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338 | (1) |
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7.5 The Handling Diagram Becomes the Handling Surface |
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339 | (13) |
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7.5.1 Handling with Locked Differential (and No Wings) |
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339 | (13) |
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7.6 Handling of Formula Cars |
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352 | (11) |
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353 | (1) |
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7.6.2 Map of Achievable Performance (MAP) |
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354 | (9) |
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363 | (9) |
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7.7.1 Vehicle Kinematic Equations |
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363 | (4) |
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7.7.2 Spin Slip Contributions |
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367 | (1) |
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7.7.3 Acceleration Center K and Acceleration of the Velocity Center C |
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368 | (1) |
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7.7.4 Aerodynamic Downforces |
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368 | (1) |
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7.7.5 Roll Stiffnesses in Formula Cars |
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369 | (1) |
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7.7.6 Lateral Load Transfers in Formula Cars |
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370 | (1) |
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7.7.7 Centrifugal Force not Applied at the Center of Mass |
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371 | (1) |
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7.7.8 Global Aerodynamic Force |
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371 | (1) |
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372 | (1) |
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7.9 List of Some Relevant Concepts |
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373 | (1) |
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373 | (2) |
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375 | (2) |
8 Map of Achievable Performance (MAP) |
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377 | (16) |
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377 | (1) |
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378 | (6) |
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8.2.1 Input Achievable Region |
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378 | (4) |
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8.2.2 Output Achievable Regions |
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382 | (2) |
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8.2.3 Mixed I/O Achievable Regions |
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384 | (1) |
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8.3 Achievable Performances on Input Regions |
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384 | (2) |
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8.4 Achievable Performances on Output Regions |
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386 | (1) |
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8.5 Achievable Performances on Mixed I/O Regions |
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387 | (1) |
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8.6 MAP from Slowly Increasing Steer Tests |
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388 | (2) |
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8.7 MAP from Constant Steer Tests |
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390 | (2) |
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392 | (1) |
|
|
392 | (1) |
9 Handling with Roll Motion |
|
393 | (24) |
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9.1 Vehicle Position and Orientation |
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393 | (1) |
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394 | (3) |
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397 | (2) |
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399 | (1) |
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9.5 Vehicle Lateral Velocity |
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399 | (6) |
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9.5.1 Track Invariant Points |
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399 | (4) |
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9.5.2 Vehicle Invariant Point (VIP) |
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403 | (1) |
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9.5.3 Lateral Velocity and Acceleration |
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|
404 | (1) |
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9.6 Three-Dimensional Vehicle Dynamics |
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|
405 | (5) |
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9.6.1 Velocity and Acceleration of G |
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405 | (2) |
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9.6.2 Rate of Change of the Angular Momentum |
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407 | (1) |
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9.6.3 Completing the Torque Equation |
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408 | (1) |
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9.6.4 Equilibrium Equations |
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|
408 | (1) |
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9.6.5 Including the Unsprung Mass |
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409 | (1) |
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9.7 Handling with Roll Motion |
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410 | (2) |
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9.7.1 Equilibrium Equations |
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410 | (1) |
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|
410 | (1) |
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9.7.3 Constitutive (Tire) Equations |
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411 | (1) |
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9.7.4 Congruence (Kinematic) Equations |
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411 | (1) |
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9.8 Steady-State and Transient Analysis |
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412 | (1) |
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|
412 | (1) |
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9.9.1 Roll Motion and Camber Variation |
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412 | (1) |
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413 | (1) |
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9.11 List of Some Relevant Concepts |
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|
413 | (1) |
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|
414 | (1) |
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|
415 | (2) |
10 Ride Comfort and Road Holding |
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417 | (44) |
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10.1 Vehicle Models for Ride and Road Holding |
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418 | (4) |
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422 | (8) |
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10.2.1 The Inerter as a Spring Softener |
|
|
426 | (1) |
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10.2.2 Quarter Car Natural Frequencies and Modes |
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426 | (4) |
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|
430 | (5) |
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10.3.1 Optimal Damper for Comfort |
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430 | (2) |
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10.3.2 Optimal Damper for Road Holding |
|
|
432 | (1) |
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10.3.3 The Inerter as a Tool for Road Holding Tuning |
|
|
433 | (2) |
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10.4 More General Suspension Layouts |
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|
435 | (1) |
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436 | (1) |
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10.6 Free Vibrations of Road Cars |
|
|
437 | (11) |
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10.6.1 Governing Equations |
|
|
438 | (2) |
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10.6.2 Proportional Viscous Damping |
|
|
440 | (1) |
|
10.6.3 Vehicle with Proportional Viscous Damping |
|
|
441 | (2) |
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10.6.4 Principal Coordinates |
|
|
443 | (2) |
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10.6.5 Selection of Front and Rear Suspension Vertical Stiffnesses |
|
|
445 | (3) |
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10.7 Tuning of Suspension Stiffnesses |
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|
448 | (4) |
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10.7.1 Optimality of Proportional Damping |
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|
449 | (1) |
|
10.7.2 A Numerical Example |
|
|
450 | (2) |
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10.8 Non-proportional Damping |
|
|
452 | (1) |
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10.9 Interconnected Suspensions |
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|
453 | (3) |
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|
456 | (1) |
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|
456 | (1) |
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|
456 | (1) |
|
|
457 | (1) |
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10.12 List of Some Relevant Concepts |
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|
457 | (1) |
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|
457 | (2) |
|
|
459 | (2) |
11 Tire Models |
|
461 | (78) |
|
11.1 Brush Model Definition |
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|
461 | (14) |
|
|
462 | (1) |
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11.1.2 Shape of the Contact Patch |
|
|
463 | (1) |
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11.1.3 Pressure Distribution and Vertical Load |
|
|
464 | (2) |
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11.1.4 Force-Couple Resultant |
|
|
466 | (1) |
|
11.1.5 Elastic Compliance of the Tire Carcass |
|
|
467 | (1) |
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|
468 | (1) |
|
11.1.7 Constitutive Relationship |
|
|
469 | (1) |
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|
470 | (2) |
|
|
472 | (1) |
|
11.1.10 Sliding Velocity of the Bristle Tips |
|
|
473 | (1) |
|
11.1.11 Summary of Relevant Velocities |
|
|
474 | (1) |
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11.2 General Governing Equations of the Brush Model |
|
|
475 | (3) |
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11.2.1 Data for Numerical Examples |
|
|
478 | (1) |
|
11.3 Brush Model Steady-State Behavior |
|
|
478 | (10) |
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11.3.1 Steady-State Governing Equations |
|
|
479 | (1) |
|
11.3.2 Adhesion and Sliding Zones |
|
|
479 | (4) |
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11.3.3 Force-Couple Resultant |
|
|
483 | (1) |
|
11.3.4 Examples of Tangential Stress Distributions |
|
|
484 | (4) |
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11.4 Adhesion Everywhere (Linear Behavior) |
|
|
488 | (3) |
|
11.5 Translational Slip Only (σ not equal to 0, φ = 0) |
|
|
491 | (19) |
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11.5.1 Rectangular Contact Patch |
|
|
498 | (9) |
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11.5.2 Elliptical Contact Patch |
|
|
507 | (3) |
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11.6 Wheel with Pure Spin Slip (σ = 0, φ not equal to 0) |
|
|
510 | (3) |
|
11.7 Wheel with Both Translational and Spin Slips |
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|
513 | (6) |
|
11.7.1 Rectangular Contact Patch |
|
|
513 | (2) |
|
11.7.2 Elliptical Contact Patch |
|
|
515 | (4) |
|
11.8 Brush Model Transient Behavior |
|
|
519 | (13) |
|
11.8.1 Transient Models with Carcass Compliance Only |
|
|
521 | (4) |
|
11.8.2 Transient Model with Carcass and Tread Compliance |
|
|
525 | (2) |
|
|
527 | (2) |
|
11.8.4 Selection of Tests |
|
|
529 | (1) |
|
11.8.5 Longitudinal Step Input |
|
|
529 | (2) |
|
11.8.6 Lateral Step Input |
|
|
531 | (1) |
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|
532 | (4) |
|
11.9.1 Braking or Driving? |
|
|
532 | (1) |
|
11.9.2 Carcass Compliance |
|
|
532 | (1) |
|
11.9.3 Brush Model: Local, Linear, Isotropic, Homogeneous |
|
|
532 | (1) |
|
11.9.4 Anisotropic Brush Model |
|
|
532 | (1) |
|
11.9.5 Carcass Compliance 2 |
|
|
533 | (1) |
|
11.9.6 Skating Versus Sliding |
|
|
533 | (1) |
|
|
533 | (1) |
|
11.9.8 Simplest Brush Model |
|
|
534 | (1) |
|
11.9.9 Velocity Relationships |
|
|
534 | (1) |
|
11.9.10 Slip Stiffness Reduction |
|
|
534 | (1) |
|
|
535 | (1) |
|
11.9.12 Spin Slip and Camber Angle |
|
|
535 | (1) |
|
11.9.13 The Right Amount of Camber |
|
|
535 | (1) |
|
|
536 | (1) |
|
|
536 | (1) |
|
11.11 List of Some Relevant Concepts |
|
|
536 | (1) |
|
|
537 | (1) |
|
|
538 | |
Correction to: The Science of Vehicle Dynamics |
|
C1 | |
Index |
|
539 | |