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2022 | Book

Advanced Brush Tyre Modelling


About this book

This book provides an in-depth understanding of tyre dynamics. In the first three chapters, new concepts are introduced in respect to the classic theory. Analytical results are derived step-by-step to guide the reader through the governing equations of the model. Transient phenomena are explained in a relatively simple way according to the brush theory. The analysis is extended to cover the case of large camber angles, which is often disregarded. The principal mechanisms behind the transient generation of tyre forces and moment are discussed in general terms, and exact solutions to the rolling contact problem are given for simple contact geometries.
The last part of the book is dedicated to the approximated theories of transient generation of tyre forces and moment. Two different models, suitable for use in vehicle dynamics applications, are presented and compared.

Table of Contents

Chapter 1. The Tyre as a Nonlinear System
At a high level of description, the tyre may be thought of as a nonlinear dynamical system, which produces certain outputs, often referred to as tyre characteristics, when subjected to opportune inputs. This interpretation allows defining some fundamental quantities that contribute to determining both the steady-state and the transient response of the tyre. Amongst these, the slip variables play the most important role. In steady-state conditions, the tyre characteristics may be described as real analytic functions of the slips, which may be defined in three main different ways. The Jacobians of the steady-state characteristics with respect to the slip variables are often called matrices of generalised slip stiffnesses. The local properties of the steady-state tyre characteristics may be deduced from the entries of these matrices.
Luigi Romano
Chapter 2. Tyre-Road Contact Mechanics Equations
The brush theory constitutes the simplest, full-physical approach to model the tyre-road interaction. It describes the tyre-wheel system as a rigid body equipped with bristles, which undergo tangential deformations in their attempt to stick to the ground. This chapter introduces the governing equations of the brush models. These may be divided into four separate sets: the friction model, the constitutive relationships, the tyre-road kinematic relationships and the equilibrium equations. The fundamental assumptions behind the proposed formulation are outlined, and the boundary and initial conditions are stated in a general manner.
Luigi Romano
Chapter 3. Steady-State Brush Theory
In steady-state conditions, explicit expressions for tyre characteristics may be derived using the theoretical framework provided by the brush theory. This chapter is, thus, dedicated to addressing the stationary problem from both the local and global perspectives. The fundamental concepts of critical slip and spin are introduced with respect to an isotropic tyre, and the deformation of the bristles inside the contact patch is investigated for different operating conditions of the tyre. Analytical functions describing the tyre forces and moment acting inside the contact patch are obtained for the particular case of a rectangular contact patch. The analysis is qualitative in nature.
Luigi Romano
Chapter 4. Unsteady-State Brush Theory
Many transient phenomena concerning the tyre-road interaction are effectively explained within the theoretical framework of the brush theory. The analysis in vanishing sliding conditions is relatively simple and may be conducted with respect to any time-varying slip input. The case of limited friction available inside the contact patch is rather involving. In this context, the investigations proposed in this chapter are limited to small spin slips under the assumption of a thin tyre. The situation further complicates when considering a flexible tyre carcass, but it may be still approached using some intuition from Chap. 3. A rather general formulation of the transient problem is proposed, which allows to gain some preliminary insights about the relaxation behaviour of the tyre.
Luigi Romano
Chapter 5. Theory for Large Spin Slips
Large camber angles excite a two-dimensional velocity field inside the contact patch. This phenomenon causes the camber and turn spin slips to induce different effects on both the transient and steady-state tyre behaviours. As a consequence, the total spin slip is not sufficient alone to describe the tyre characteristics, and the two contributions must be regarded separately. The present analysis addresses the general problem under the assumption of a rigid carcass. First, it is shown how to solve the governing PDEs of the brush model in vanishing sliding conditions. Closed-form solutions for the bristle deflection are then derived explicitly for rectangular and elliptical contact geometries. The analysis is extended qualitatively to the situation of limited friction available inside the contact patch.
Luigi Romano
Chapter 6. Theories of Transient Generation of Tyre Forces and Moment
The knowledge gained from the analysis of the transient brush theory may be effectively used to derive simplified, pragmatic tyre models to be used in vehicle dynamics simulations. Two different formulations have been proposed in the literature. The first is the so-called single contact point. It consists of a simple class of models, which usually find good agreement with experimental data and are based on a rather intuitive approximation. However, they systematically neglect the dynamics of the bristles inside the contact patch. The second formulation presented in this chapter is the two-regime approach, which also captures the transient effect due to the bristles. It relies on more sophisticated mathematical tools, and, albeit being relatively easy to implement, rarely admit closed-form representations. The two different models are compared together against the exact description introduced in Chap. 4.
Luigi Romano
Advanced Brush Tyre Modelling
Dr. Luigi Romano
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