Rubber Technology

1 UNIDIRECTIONAL FIBER-REINFORCED RUBBER

1.1 Composite Materials Used for Tires

1.2 Stress/Strain Relationship

1.3 Mechanics of a Composite

1.3.1 Plane Stress

1.3.2 Transformation of Strain Between Two Coordinate Systems

1.3.3 Constitutive Equation (Hook’s Law)

1.3.4 Representation of the Stiffness Matrix Using Invariants

1.3.5 Properties of a Composite in an Arbitrary Direction

1.4 Micromechanics

1.4.1 Parallel and Series Models

1.4.2 Modified Micromechanics

1.4.3 Upper and Lower Bounds of the Modulus of Composites

1.4.4 Halpin–Tsai Model

1.5 Micromechanics of Unidirectionally Cord-Reinforced Rubber (UDCRR)

1.5.1 Models for UDCRR

1.5.2 Comparison of the Micromechanics Model for Fiber-Reinforced Rubber .

1.6 Mechanics of UDCRR Under an FRR Approximation

1.6.1 Approximate Equations for UDCRR

1.6.2 Properties of UDCRR in an Arbitrary Direction

1.6.3 Particular Angle for UDCRR

1.6.4 Comparison of Micromechanics and Experimental Results .

1.7 Viscoelastic Properties of a UDCRR Plate

1.7.1 Studies on the Viscoelastic Properties of a UDCRR Plate

1.7.2 Analytical Damping Model

1.7.3 Finite Element Model for Viscoelastic Properties

1.8 Mechanics of Short-Fiber-Reinforced Rubber (SFRR)

1.8.1 Micromechanics of SFRR

1.8.2 Modulus of SFRR in an Arbitrary Direction

1.8.3 Viscoelastic Properties of SFRR

Problems

Appendix: Viscoelasticity

2 LAMINATION THEORY

2.1 CLT

2.1.1 Coordinate System for Laminates and Representation of the Laminate Configuration

2.1.2 CLT

2.2 Properties of a Symmetric Laminate

2.2.1 Constitutive Equation of a Symmetric Laminate

2.2.2 In-Plane Stiffness of a Symmetric Laminate

2.2.3 Bending Properties of a Symmetric Laminate

2.3 Properties of a Bias Laminate

2.3.1 Stiffness of a Bias Laminate

2.3.2 In-Plane and Out-of-Plane Coupling Deformation of a Bias Laminate

2.3.3 FRR Approximation for a Bias Laminate

2.3.4 Comparison of CLT and Experimental Results for a Bias Laminate

2.3.5 Viscoelastic Properties of a Bias Laminate

2.4 Optimization of the Belt Structure of a Tire

2.4.1 Computer-Aided Composite Design

2.4.2 Optimization of the Belt Construction Through Mathematical Programming

2.4.3 Optimization of the Belt Construction Using a GA

Problems

3 MODIFIED LAMINATION THEORY

3.1 Introduction

3.2 MLT of a Two-Ply Laminate Without Out-of-Plane Coupling Deformations (Symmetric Four-Ply Laminate)

3.2.1 Fundamental Equations

3.2.2 Analysis of a Bias Laminate Under Uniform Stress and Displacement

3.2.3 Analysis of FRR

3.3 MLT of a Two-Ply Laminate Including Transverse Stress Without Out-of-Plane Coupling Deformation (Symmetric

Four-Ply Laminate)

3.3.1 Fundamental Equations of the MLT of a Two-Ply Laminate Including Transverse Stress Without Out-of-Plane Coupling Deformation

3.3.2 Comparison of CLT and MLT and a Parameter Study on the Interlaminar Shear Strain of UDCRR

3.4 MLT of a Two-Ply Laminate with Coupling Deformation

3.4.1 MLT for a Two-Ply Laminate with Coupling Deformation

3.4.2 Fundamental Equations of MLT for a Two-Ply Laminate with Coupling Deformation.

3.4.3 Bias Belt Under Uniaxial Uniform Displacement

3.5 MLT for In-plane Bending

3.5.1 Fundamental Equations

3.5.2 Comparison Between Theory and Experiment.

3.6 MLT of Three-Ply with Coupling Deformation

3.6.1 Fundamental Equations

3.6.2 In-plane Bending Properties of the Folded Belt of a Tire

3.7 MLT of Out-of-Plane Torsional Rigidity of a Bias Belt

3.7.1 Fundamental Equations

3.7.2 Comparison Between Theory and Experiment.

3.8 MLT of the Buckling of a Two-Ply Bias Belt Under an In-plane Bending Moment

3.8.1 Buckling of a Two-Ply Bias Belt Under an In-plane Bending Moment

3.8.2 Fundamental Equations for the Buckling of a Tire Belt Under an In-plane Bending Moment

3.8.3 Buckling Analysis of Passenger-Car Tires Under an In-plane Bending Moment

3.8.4 Simplified Equation for Buckling Analysis of Passenger-Car Tires Under an in-Plane Bending Moment

3.9 MLT of the Buckling of a Two-Ply Bias Belt Under a Compressive Force

3.9.1 MLT of the Buckling of a Two-Ply Bias Belt Under a Compressive Force

3.9.2 Buckling Analysis of Passenger-Car Tires Under a Compressive Force

Problems

Appendix: Beam Theory for a Tire Belt Under Buckling Caused by an In-plane Bending Moment

4 DISCRETE LAMINATION THEORY

4.1 DLT of a Two-Ply Bias Belt with Out-of-Plane Coupling Deformation Under an Extensional Load

4.1.1 Fundamental Equations for DLT

4.1.2 Displacements in DLT

4.1.3 Strain Energy of Parts of a Two-Ply Bias Laminate

4.1.4 Stationary Condition of Total Strain Energy

4.1.5 Solution of the Differential Equation for Two-Ply Bias Laminate

4.1.6 Determination of Integral Constants by Boundary Conditions

4.1.7 Equivalent Young’s Modulus for a Two-Ply Bias Laminate

4.1.8 Interlaminar Shear Stress and Interfacial Shear Stress

4.1.9 Analysis of a Two-Ply Bias Laminate Using DLT

4.2 DLT of a Two-Ply Bias Belt Without Out-of-Plane Coupling Deformation Under a Bending Moment

4.2.1 Displacements of DLT

4.2.2 Strain Energies of the Cord and Rubber

4.2.3 Stationary Condition and Natural Boundary Conditions

4.2.4 Solution to the Differential Equation for a Two-Ply Bias Laminate

4.2.5 Analysis of a Two-Ply Bias Laminate Using DLT

4.3 FEA Using a Discrete Model of a Two-Ply Bias Laminate Without Out-of-Plane Coupling Deformation

Problems .

Appendix 1: Parameters in Equations for a Two-Ply Bias Belt Under an Extensional Load

Appendix 2: Parameters in Equations for a Two-Ply Bias Belt Under a Bending Moment

5 Theory of Tire Shape

5.1 Studies on Tire Shape

5.2 Theory of the Natural Equilibrium Shape Based on Netting Theory

5.2.1 Fundamental Equations for the Natural Equilibrium Shape Based on Netting Theory

5.2.2 Natural Equilibrium Shape with the Cord Path of Pantograph Deformation

5.2.3 Natural Equilibrium Shapes of Bias Tires

5.2.4 Natural Equilibrium Shape of Radial Tires Without a Belt

5.2.5 Natural Equilibrium Shape with the Cord Path of the Geodesic Line

5.2.6 Natural Equilibrium Shapes with Other Cord Paths

5.2.7 Natural Equilibrium Shape of Bias Tires Under a Centrifugal Force

5.3 Effects of the Tire Shape on Tire Properties

5.3.1 Effect of the Tire Shape on Cord Tension

5.3.2 Effect of Tire Shape on Bead Tension

5.3.3 Effect of Tire Shape on the Interlaminar Shear Stress of Bias Tires

5.4 Theory of the Natural Equilibrium Shape for Belted Radial Tires

5.4.1 Fundamental Equations for Belted Radial Tires Having a Uniform Partition of Pressure

5.4.2 Cord Lengths of Belted Radial Tires

5.5 General Theory for the Shape of Belted Tires

5.5.1 General Equation of the Natural Equilibrium Shape for Belted Tires

5.5.2 Tire Shape of Belted Radial Tires with Partitioned Pressure in Both Crown and Bead Areas

5.6 Nonequilibrium Tire Shape

5.6.1 Application of the Nonequilibrium Tire Shape to Passenger-Car Tires

5.6.2 Application of the Nonequilibrium Tire Shape to Truck/Bus Tires

5.7 Ultimate Theory of the Tire Sidewall Shape

5.7.1 Theory of Optimization

5.7.2 Application and Validation of GUTT

Problems .

Appendix: Equation of the Tire Shape for the Partitioned Tire Pressure of the Belt Given by Eq. (5.108)

6 SPRING PROPERTIES OF TIRES

6.1 Tire Spring of a Simple Tire Model

6.1.1 Spring Properties of Tires

6.1.2 Radial Fundamental Spring Rate

6.1.3 Lateral Fundamental Spring Rate

6.1.4 Circumferential Fundamental Spring Rate

6.1.5 Contribution of Structural and Tensile Stiffness to the Vertical Spring Rate .

6.2 Tire Spring Rates of the Rigid Ring Model. .

6.2.1 Torsional Spring Rate

6.2.2 Lateral Spring Rate

6.2.3 Eccentric Spring Rate of the Rigid Ring Model

6.2.4 In-Plane Rotational Spring Rate

6.2.5 Fore–Aft Spring Rate of the Rigid Ring Model

6.2.6 Measurement Procedure for Fundamental Spring Rates and Tire Spring Rates

6.3 Tire Spring Rates of the Flexural Ring Model

6.3.1 Lateral Spring Rate of a Tire with a Flexural Ring

6.3.2 Torsional Spring Rate of a Tire with a Flexural Ring

6.4 Fundamental Spring Rates Based on the Equilibrium Shape of Belted Radial Tires

6.4.1 Theory of Tire Shape

6.4.2 Lateral Fundamental Spring Rate

6.4.3 Circumferential Fundamental Spring Rate

6.4.4 Radial Fundamental Spring Rate

6.5 Modification of Yamazaki’s Model 364

6.5.1 Modification of Yamazaki’s Model

6.5.2 Contribution of Bending and Extensional Deformation of the Sidewall Material to Fundamental Spring Rates

6.6 Line Spring Rate

6.7 Visualization of the Spring Rate

Problems

7 MECHANICS OF THE TREAD PATTERN

7.1 Shear Spring Rate of the Tire Block Pattern

7.1.1 Fundamental Equations for an Analytical Approach

7.1.2 Calculation of the Block Rigidity of a Practical Block Pattern

7.1.3 Comparison of Prediction and Measurement Results

7.1.4 FEA Approaches for Block Rigidity

7.2 Compressive Modulus of a Bonded Rubber Block

7.2.1 Studies on the Compressive Modulus of a Bonded Rubber Block

7.2.2 Two-Dimensional Rectangular Block

7.2.3 Rectangular Solid Block

7.3 Properties of the Block in Contact with the Road

7.3.1 Properties of the Block Under a Compressive Force

7.3.2 Properties of a Block Under Both Compressive and Shear Forces

7.4 Pressure Dependence of the Friction Coefficient on a Dry Surface

7.4.1 Hertz Theory

7.4.2 JKR Theory

7.4.3 Archard Theory (Multiple-Contact Theory)

7.4.4 Greenwood and Williamson’s Theory (Statistics-Based Model)

7.4.5 Pressure Dependence of the Friction Coefficient Determined by Local Slip

7.5 Pressure Distribution of a Block and the Frictional Force

7.5.1 Theory and Experiment

7.5.2 Shape of the Block Surface that Makes the Pressure Distribution Uniform

7.6 Contact Properties of a Block Pattern with Sipes

7.6.1 Studies on Studless Tires

7.6.2 FEA Prediction of the Tread Block for Studless Tires

7.6.3 Comparison of FEA and Experimental Results for Blocks with Two- and Three-Dimensional Sipes

7.7 Other FEA Studies on the Block Pattern

7.7.1 Improvement of the Radial Run-out and the Difference of Block Rigidity Between the Small and Large Pitches

7.7.2 Pattern Effect on Hydroplaning and Wear

Problems

8 TIRE VIBRATION

8.1 Vibration Properties of Tires

8.1.1 Fundamental Frequencies of Unloaded Tires

8.1.2 Natural Frequencies of Tires with One Point of Contact

8.1.3 Calculation of Fundamental Frequencies

8.2 Elastic Ring Model Without a Tread Spring

8.2.1 Fundamental Equations

8.2.2 Comparison Between Experiment and Calculation

8.2.3 Analysis of the Contribution of Each Parameter to Natural Frequencies

8.3 Frequency Response Function of Tires

8.3.1 Equation of Motion in the Mixed Modal and Physical Coordinate System

8.3.2 Frequency Response Function of the Tire–Wheel System

8.3.3 Transfer Function Under the Axle–Free Condition Determined by Two Frequency Response Functions

8.4 Tire Models Rolling Over Cleats (Ride Harshness)

8.4.1 Envelope Properties

8.4.2 Tire Vibration Model of the Enveloping Response

Appendix 1: Representations of Tire Mode Shapes

Appendix 2: Elements of the Matrix of Eq. (8.155)

Appendix 3: Elements of the Matrix of Eq. (8.156)

9 CONTACT PROPERTIES OF TIRES

9.1 Studies on the Contact Properties of Tires

9.2 Contact Analysis of Tires Using an Elastic Ring Model

9.2.1 Fundamental Equations

9.2.2 Contact Analysis of Tires Using the Elastic Ring Model

9.2.3 Boundary Conditions and Solutions for the Contact Analysis of Tires Obtained Using the Elastic Ring Model 9.2.4 Comparison Between Calculation and Experiment

9.3 Contact Analysis of Tires Using the Elastic Ring Model with a Fourier Series

9.3.1 Elastic Ring Model and Tire Deformation Model Near the Contact Patch

9.3.2 Governing Equation for Contact Analysis

9.3.3 Comparison Between Calculation and Measurement

9.4 Optimization of the Tire Crown Shape Allowing the Free Control of the Contact Pressure Distribution of Tires

9.4.1 Optimization Procedure for the Tire Crown Shape

9.4.2 Objective Function and Constraints for Passenger-Car Tires

9.4.3 Validation of the Optimized Crown Shape

Appendix: Explicit Expression of Eq. (9.58)

10 TIRE NOISE

10.1 Background of Studies on Tire Noise

10.2 Classification of Tire Noise

10.3 Mechanism of Tire/Road Noise

10.3.1 Mechanism of Tire/Road Noise

10.3.2 External Forces Acting on Tires.

10.3.3 Stick–Slip and Stick–Snap Processes

10.3.4 External Forces Acting on Tires Due to Road Roughness

10.3.5 Source of Tire/Road Noise

10.3.6 Transfer Function of Tires

10.3.7 Characteristics of the Acoustic Field

10.3.8 Air Pumping Noise

10.3.9 Speed Dependence of Tire Noise

10.3.10 Tire Vibration and Tire Noise Due to the Nonuniformity of Tires

10.4 Ways of Improving Tire/Road Noise Through Pattern Design

10.5 Models of Tire Noise

10.6 Models for Vehicle Interior Noise

10.6.1 Tire/Wheel/Suspension Model

10.6.2 Contribution of Structure-Borne and Air-Borne Noise to Interior Noise

10.7 Phenomenological Model for Pattern Pitch Noise

10.7.1 Spectrum Analysis of Pitch Noise

10.7.2 Theory on the Magic Angle (Rectangular Footprint)

10.7.3 Theory on the Magic Angle (Hexagonal Footprint)

10.7.4 Theory on the Magic Shape

10.7.5 Tire Noise Prediction for a Practical Pattern and a Comparison with Measurements

10.7.6 Other Studies on the Effect of the Tire Pattern on Tire Noise

10.8 Optimization of the Tire Pitch Sequence

10.8.1 Difficulty in Optimizing the Tire Pitch Sequence

10.8.2 Parameters for Optimization of the Pitch Sequence

10.8.3 GAs

10.8.4 GAs with Growth

10.8.5 Experiment and Discussion

10.8.6 Tire Noise Quality Characterized Using Psychoacoustic Parameters638

10.9 Pipe Resonance Noise

10.9.1 Pipe Resonance Noise and Resonance Frequencies of Simple Sub-resonators

10.9.2 Tire Noise for a Simple Resonator

10.9.3 Other Technology Used to Reduce Pipe Resonance Noise.

10.10 Acoustic Cavity Noise of Tires

10.10.1 Characteristics of Acoustic Cavity Noise

10.10.2 Fundamental Equation of Acoustic Cavity Noise

10.10.3 Ways to Improve the Acoustic Cavity Resonance of a Tire

10.11 Horn Effect

10.11.1 Studies on the Horn Effect

10.11.2 Ray Theory for the Horn Effect

10.11.3 Comparison of Ray Theory and Measurements

10.11.4 Other Experimental Research on the Horn Effect

10.12 Models of the External Force Acting on a Tire and the Interaction Between the Tire and the Road Roughness . .

10.12.1 Representation of Road Roughness

10.12.2 Hybrid Model

10.12.3 Analytical Models

10.12.4 Two- or Three-Dimensional FEA

10.13 Tire Noise Prediction

10.13.1 Procedure of Predicting Tire Noise Using an Analytical Model for Tire Vibration and a BEM for Noise Radiation

10.13.2 Procedure of Tire Noise Prediction Employing FEA and a BEM

10.13.3 Comparison of Analytical Models and the WFEM.

10.13.4 SEA .

10.13.5 Hybrid Models: TRIAS and SPERoN

10.13.6 Other Studies on Tire Noise

10.13.7 Effect of Tire Rolling on the Natural Frequencies of Tires

11 CORNERING PROPERTIES OF TIRES

11.1 Tire Models for Cornering Properties

11.1.1 Solid Tire Model with a Brush Model for Pure Slipping

11.1.2 String Model

11.1.3 Beam Model and Fiala Model

11.1.4 Comparison of the Force and Moment for Various Tire Models

11.2 Cornering Properties with a Large Slip Angle

11.2.1 Lateral Deformation of the Tread Ring Due to a Side Force

11.2.2 Torsional Deformation by Self-aligning Torque Around the Z-Axis

11.2.3 Kinetic Friction Coefficient of Rubber Changing with Sliding Speed

11.2.4 Shape of the Contact Pressure Distribution

11.2.5 Tire Model for Cornering Properties with a Large Slip Angle

11.3 Cornering Properties with a Small Slip Angle During Driving/Braking

11.3.1 Fundamental Equations

11.3.2 Example Calculations

11.4 Cornering Properties for a Large Slip Angle During Driving/Braking

11.4.1 Friction Ellipse Model

11.4.2 Sakai’s Model

11.5 Neo-Fiala Model

11.5.1 Neo-Fiala Model for a Small Slip Angle

11.5.2 Neo-Fiala Model for a Combined Slip

11.6 Dynamic Cornering Properties

11.6.1 Tire Model for Dynamic Cornering Properties at Low Speed

11.6.2 Tire Model for Dynamic Cornering Propertiesof Tires at High Speed

11.6.3 Dynamic Cornering Properties Obtained Using the Neo-Fiala Model

11.7 Thermomechanical Tire Model

11.8 Finite Element Model for Cornering Properties

11.8.1 FEA of the Tire Model

11.8.2 FEA of the Vehicle/Tire System

11.9 Vehicle Dynamics and Tire Properties

11.9.1 Effects of Nonlinear Cornering Properties of Tires on Cornering Performance

11.9.2 TPC Specification System

11.9.3 Vehicle Dynamics and Wheel Alignment

12 TRACTION PERFORMANCE OF TIRES

12.1 Traction Performances on Dry and Wet Surfaces

12.1.1 Tire Traction Model Without Changing the Pressure Distribution Under Braking and Driving

12.1.2 Tire Traction Model Including the Change in Pressure Distribution Under a Braking or Driving Condition

12.1.3 Transient Tire Model in Braking and Driving Conditions

12.1.4 Effects of an ABS on the Braking Performance of Tires

12.2 Hydroplaning

12.2.1 Three-Zone Concept of Hydroplaning

12.2.2 Models for Complete Hydroplaning

12.2.3 Models for Partial Hydroplaning

12.2.4 Hydroplaning Prediction Using Computational Mechanics

12.3 Traction on Snow

12.3.1 Difficulty in Improving Tire Performance for a Wide Range of Road Conditions

12.3.2 Models of Tire Traction on a Snow-Covered Road

12.3.3 Analytical Model of Tire Traction on a Snow-Covered Road

12.3.4 FEA of Tire Traction on Snow

12.4 Traction on Ice

12.4.1 Studies on Friction on Ice

12.4.2 Studies on Tire Traction on Ice

12.4.3 Friction Coefficient of a Rubber Block on Ice

12.4.4 Analytical Studies on the Friction Coefficient of Tire on Ice

12.4.5 Analytical Model for Braking and Driving Forces of a Tire on Ice

12.4.6 FEA of a Largely Deformed Block with Three-Dimensional Sipes

12.5 Logic Tree of Traction on Snow and Ice

12.6 Traction on Soil

12.6.1 Studies on Tire Traction on Soil

12.6.2 Tire Deformation and Shear Stress on Soil

12.6.3 Traction on Soil

12.6.4 Fundamental Equations of Semi-empirical Theory

12.6.5 FEA of Tire Traction on Soil

13 ROLLING RESISTANCE OF TIRES

13.1 RR of Tires

13.1.1 RR of Tires

13.1.2 Energy Loss of Tires

13.1.3 Calculation of the Strain Energy Loss of Tires

13.1.4 Models of the RR of Tires

13.1.5 Simple Model of the Energy Loss of Tread Rubber

13.1.6 Deformation Index

13.1.7 Effect of Drum Curvature on the RR of Tires

13.1.8 Effect of Road Texture on the RR

13.1.9 Method of Evaluating the RR of Tires

13.2 RR in Cornering and Driving/Braking

13.2.1 RR in Cornering

13.2.2 RR with a Fore–aft Force in a Steady State

13.3 Transient RR

13.3.1 Transient RR

13.3.2 Tire Temperature in the Steady State

13.3.3 RR and Tire Temperature in the Transient State

13.3.4 Transient RR in the Case of a Speed Change Within a Short Time

13.4 RR of Tires and Fuel Economy

13.4.1 Relation of RR and Fuel Economy

13.4.2 Effect of Reducing the RR of an Eco-Tire on Fuel Consumption

13.4.3 Ways of Improving Fuel Economy Other than Reducing RR

13.5 Numerical Prediction of RR

13.5.1 Prediction of RR by FEA

13.5.2 Modules Used to Predict RR

13.5.3 Consideration of the Nonlinearity of the Loss Modulus

13.6 Technology That Reduces RR

13.6.1 Logic Tree for RR

13.6.2 Tire Designs Obtained Using Optimization Technology to Decrease RR

13.6.3 Tire Pattern

13.6.4 Inflation Pressure

13.6.5 Other Tire Design Parameters Used to Decrease RR

13.7 Future Tires

13.7.1 From Tire Design to Mobility Design

13.7.2 Tire Size of a Next-Generation Low-RR Tire

14 WEAR OF TIRES

14.1 Wear of Tires

14.1.1 Wear of Tires and Rubber

14.1.2 Wear Energy and Tread Wear

14.1.3 Factors Related to Tire Wear

14.2 Wear at a Small Slip Angle and Slip Ratio

14.2.1 Wear Energy in Cornering

14.2.2 Wear Energy for a Fore–aft Force and Wear Energy for Both Cornering and a Fore–aft Force

14.2.3 Effect of the Hysteresis Loss of Rubber on Wear Energy

14.3 Wear Life in the Case of Uniform Wear

14.3.1 Fundamental Equations

14.3.2 Comparison Between Calculation and Experiment.

14.4 Composite Mechanics and Wear

14.5 Wear and Irregular Wear for a Simple External Force

14.6 Wear in a Combined Slip

14.6.1 Models for Wear Energy in a Combined Slip

14.6.2 Calculation of Wear Energy in a Combined Slip.

14.7 Expansion of the Brush Model for Wear .

14.7.1 Models for Lateral Slip in the Sliding Region

14.7.2 One-Element Model

14.7.3 Three-Element Model

14.7.4 Comparison Between Calculation and Measurement

14.8 Progression of Irregular Wear

14.8.1 Progression of Step-Down Wear and River Wear

14.8.2 Wear Model of Dual Tires (i.e., Circumferential Force Acting on Freely Rolling Dual Tires)

14.8.3 Model of the Progression of Step-Down Wear

14.9 Effect of Vehicle Alignment on Tire Wear

14.9.1 External Force of Tires in a Vehicle Model

14.9.2 Optimized Toe for Tire Wear

14.9.3 Comparison Between Calculation and Measurement

14.10 Diagonal Wear and Polygonal Wear

14.10.1 Diagonal Wear and Polygonal Wear of Tires

14.10.2 Model for Polygonal Wear

14.10.3 Comparison of the Calculation and Measurement of the Polygonal Wear of a Truck/Bus Tire

14.11 Indoor Wear Evaluation

14.11.1 Methods of Indoor Wear Evaluation.

14.11.2 Basic Steps in Indoor Wear Simulation

14.11.3 Characterization of External Forces Applied to a Tire

14.11.4 Indoor Wear Drum Test

14.11.5 Wear Energy Machine

14.11.6 Wear Prediction by FEA

14.11.7 Wear Abradability

14.12 Ways of Improving Wear and Irregular Wear

14.12.1 Contact Pressure Distribution and Irregular Wear

14.12.2 Rib for Absorbing Braking Force Causing Irregular Wear (i.e., A Braking Control Rib)

14.12.3 Groove for Defending Against a Side Force

14.12.4 Three-Dimensional Dome-Shaped Block

14.12.5 Surface Shape of the Block

14.12.6 Three-Dimensional Sipes

14.12.7 Waved Belt Construction

14.12.8 Other Improvements

15 STANDING WAVES IN TIRES

15.1 Studies on Standing Waves in Tires

15.2 Simple Explanation of a Standing Wave

15.3 One-Dimensional Model of a Standing Wave in a Bias Tire

15.3.1 Membrane Theory

15.3.2 Critical Speed of the Standing Wave Including the Effect of a Centripetal Force

15.3.3 Consumption of Energy by a Standing Wave

15.3.4 Comparison Between Calculation and Experiment.

15.4 One-Dimensional Model of a Standing Wave of a Radial Tire (i.e., a Beam with Tension on an Elastic Foundation)

15.4.1 Wave Propagation Approach

15.4.2 Resonance Approach

15.5 Prediction of the Standing Wave in Tires Employing FEA

16 TIRE PROPERTIES FOR WANDERING AND VEHICLE PULL

16.1 Wandering Due to Ruts

16.1.1 Wandering Due to Ruts

16.1.2 Theory of Wandering on a Rut Based on Tire Mechanics

16.1.3 Theory of Wandering on a Rutted Road Based on Vehicle Dynamics

16.2 Rain Groove Wandering

16.2.1 Studies on Rain Groove Wandering

16.2.2 Theory on the Rain Groove Wandering of a Tire with Circumferential Grooves

16.2.3 Analytical Theory on the Rain Groove Wandering of a Practical Tire Pattern

16.2.4 Prediction of Rain Groove Wandering Employing FEA

16.3 Vehicle Pull

16.3.1 Force and Moment of Tires at a Small Slip Angle

16.3.2 Vehicle Pull and Tire Mechanics

16.3.3 Effect of the Caster Trail on Vehicle Pull

References

Solutions

Index