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About this book

This book highlights the mechanics of tire performance, offering detailed explanations of deriving basic equations for the fundamental properties of tires, and discussing ways to improve tire performance using these equations. It also compares the theory with practical measurements. The book commences with composite mechanics, which is the fundamental theory for belt and carcass tires, and covers classical, modified and discrete lamination theory. It then addresses the theory of tire shape and spring properties and the mechanics of tread pattern contact properties, as was well as the performance of various tires. This comprehensive book is a valuable resource for engineers involved in tire design and offers unique insights and examples of improvement of tire performances.

Table of Contents

Frontmatter

Chapter 1. Unidirectional Fiber-Reinforced Rubber

Abstract
The mystery why a tire, just one part of a vehicle, realizes many functions simultaneously can be explained by the fact that a tire has a composite structure made from composite material. Unidirectional fiber-reinforced rubber (UFRR) is the main composite material of a tire. The elastic and viscoelastic properties of UFRR are defined in principal directions, which are the direction of the fiber or cord and the direction perpendicular to the fiber or cord. Applying the rotation matrix to the properties of UFRR in the principal directions, the properties of UFRR can then be calculated in an arbitrary direction. Because UFRR has nonlinear properties with respect to the direction of the applied external force, the angle and width of UFRR used for a tire belt need to be carefully determined in tire design.
Yukio Nakajima

Chapter 2. Lamination Theory

Abstract
A laminate structure is used for the belt of radial tires and the ply of bias tires that consist of steel cords and rubber, or organic cords and rubber. Properties of the laminate, such as the extensional stiffness and bending stiffness of tire belt, can be estimated using classical lamination theory (CLT). Although interlaminar shear is not considered in CLT, the elastic properties predicted by CLT agree with measurements fairly well. The finite element method is used to predict the effect of the belt structure on the tire performance. Furthermore, the belt structure optimized for maneuverability or durability is obtained by combining the genetic algorithm (GA) with the finite element method.
Yukio Nakajima

Chapter 3. Modified Lamination Theory

Abstract
The belt failure of steel radial tires usually occurs at belt ends where the interlaminar shear stress increases drastically.
Yukio Nakajima

Chapter 4. Discrete Lamination Theory

Abstract
The properties of a composite laminate have been modeled using the micromechanics of Chap. 1, the CLT of Chap. 2 and the MLT of Chap. 3.
Yukio Nakajima

Chapter 5. Theory of Tire Shape

Abstract
The theory of tire shape has been well studied since the early 1900s because it has been easier to develop theory for the sidewall shape of a tire than to develop theory for other tire design elements, such as the crown shape, bead structure, belt structure, pattern and material. Three important theories have been established in the history of the development of theory of the sidewall shape: natural equilibrium theory for a bias tire, natural equilibrium theory for a radial tire and ultimate tire shape theory where the finite element method is combined with optimization technology. These theories and applications are discussed in this chapter.
Yukio Nakajima

Chapter 6. Spring Properties of Tires

Abstract
The spring properties of tires are the most fundamental characteristics of tires and are closely related to the four fundamental functions discussed in Sect. 1.​1.
Yukio Nakajima

Chapter 7. Mechanics of the Tread Pattern

Abstract
Various tread patterns have been used in tire design because tires are used on various road surfaces,
Yukio Nakajima

Chapter 8. Tire Vibration

Abstract
Vibration properties are fundamental properties of tires related to riding comfort and tire noise. The noise, vibration and ride harshness (NVH) of a vehicle are classified according to the riding comfort at a low frequency below 50 Hz and the interior noise in wide frequency range from 20 Hz to a few thousand Hertz.
Yukio Nakajima

Chapter 9. Contact Properties of Tires

Abstract
Tires are a vehicle’s only points of contact with the road. The contact properties of a tire defined by the contact pressure distribution and footprint shape of the contact patch are thus related to not only maneuverability but also riding comfort, noise, durability, rolling resistance, wear and traction/braking.
Yukio Nakajima

Chapter 10. Tire Noise

Abstract
Tire/road noise consists of noise due to tire surface vibration and noise related to aerodynamics. The former noise is usually explained by three elements, namely the external force applied to a tire, vibration properties of the tire and the acoustic field relating to the tire surface and road surface. The external force includes the tread impact associated with the lateral grooves and road roughness. The surface vibration of a tire is calculated by multiplying external forces by the tire’s vibration properties expressed by a transfer function. Tire/road radiation noise is then calculated by surface vibration via the Helmholtz equation [i.e., employing the boundary element method (BEM)]. The most important element for tire noise is the external force because other elements may not be controlled by tire design without deteriorating other tire performances. The external force due to lateral grooves is estimated by using a phenomenological model that uses the contact shape, pattern geometry and contact pressure or by conducting FEA. Meanwhile, the external force due to road roughness is estimated by measuring the spindle force of a tire rolling over a simple roughness and conducting contact analysis employing a Winker model with nonlinear contact stiffness or FEA. The vibration properties of a tire can be predicted by conducting FEA or using an elastic ring model. In pattern design, the phenomenological model is used as a design tool to determine the pattern geometry, and the pitch sequence is optimized by a GA. The Helmholtz resonator may be added to the circumferential grooves to reduce the pipe resonance noise. Furthermore, the special wheel or sound-absorbing material may be used to reduce the acoustic cavity noise.
Yukio Nakajima

Chapter 11. Cornering Properties of Tires

Abstract
The motion of a vehicle is determined by the force and moment generated by tires, which are the only points of contact between a vehicle and the road.
Yukio Nakajima

Chapter 12. Traction Performance of Tires

Abstract
The traction performance of a tire is important to the safety of a vehicle. Traction models of a tire have been proposed for different road conditions, such as dry, wet, snowy, icy and muddy conditions. Simple analytical models for tire traction in braking and driving have been developed by extending the Fiala model of cornering performance. One is a model where the contact pressure distribution does not change under braking and driving forces and the sliding point between the adhesion and sliding regions can be determined using a simple equation. Another is a model where the contact pressure distribution changes under braking and driving forces and the sliding point between adhesion and sliding regions can be determined by iterative calculation. The hydroplaning phenomenon is analyzed employing the equilibrium of the hydrodynamic pressure and tire load, the two-dimensional Reynolds equation for squeezing water out of the tread block and computational fluid dynamics (CFD) simulation where a tire is modeled by FEA and water is modeled using the finite volume method (FVM). The snow reaction is analyzed using an analytical model where the shear strength of snow is determined from the density of snow and conducting CFD simulation in an approach similar to that adopted for hydroplaning. The traction on ice is analyzed using a brush model with a friction model on ice where the friction coefficient is a function of the sliding velocity and other parameters of thermodynamics, and another brush model where the shear force distribution of a freely rolling tire is included. The traction on mud is analyzed using an analytical model where the tire deformation is classified as being in a rigid mode or elastic mode according to the difference between tire rigidity and load retention properties of mud, and CFD simulation is conducted in a manner similar to that adopted for hydroplaning. FEA has recently become popular in the design of the tire pattern for traction on wet, snowy and muddy road conditions.
Yukio Nakajima

Chapter 13. Rolling Resistance of Tires

Abstract
The rolling resistance (RR) of a tire depends on three factors: (i) the energy loss of rubber and reinforcements due to the cyclic deformation of the rolling tire, (ii) frictional energy in the contact area and (iii) the air resistance of the tire.
Yukio Nakajima

Chapter 14. Wear of Tires

Abstract
Wear is phenomenologically characterized by not only physical factors, such as fracture, but also chemical factors, such as oxidization.
Yukio Nakajima

Chapter 15. Standing Waves in Tires

Abstract
The phenomenon of standing waves in tires occurs in the sidewall and tire tread area when the vehicle speed exceeds a critical speed. Two approaches can be adopted in the study of standing waves: the analytical approach and FEA. The analytical approach, such as the adoption of a membrane model or elastic ring model, is further classified into the wave propagation approach and resonance approach. This chapter discusses the analytical approach and FEA for standing waves.
Yukio Nakajima

Chapter 16. Tire Properties for Wandering and Vehicle Pull

Abstract
Wandering is a phenomenon of handling pull in straight driving due to the road slope, wheel track unevenness, rain grooves or road roughness.
Yukio Nakajima

Backmatter

Additional information