This paper deals with unsteady-state brush tyre models. Starting from tyre-road contact theory, we provide a full analytical solution to the partial differential equations (PDEs) describing the bristle deformation in the adhesion region of the contact patch. We show that the latter can be divided in two different regions, corresponding to two different domains for the solution of the governing PDEs of the system. In the case of constant sliding speed inputs, the steady-state solution coincides with the one provided by the classic steady-state brush theory. For a rectangular contact patch and parabolic pressure distribution, the time trend of the shear stresses is investigated. For the pure interactions (longitudinal, lateral and camber), some important conclusions are drawn about the relaxation length. Finally, an approach to derive simplified formulae for the tangential forces arising in the contact patch is introduced; the tyre formulae obtained by using the proposed approach are not based on the common slip definition, and can be employed when the rolling speed approaches zero. The outlined procedure is applied to the cases of linear tyre forces and parabolic pressure distribution.

本文涉及非稳态刷轮胎模型。从轮胎-路面接触理论出发，我们为描述接触面粘附区域刷毛变形的偏微分方程 (PDE) 提供了完整的解析解。我们表明后者可以分为两个不同的区域，对应于解决系统控制偏微分方程的两个不同域。在恒定滑动速度输入的情况下，稳态解与经典稳态电刷理论提供的解一致。对于矩形接触面和抛物线压力分布，研究了剪应力的时间趋势。对于纯相互作用（纵向、横向和弯度），得出了一些关于弛豫长度的重要结论。最后，介绍了一种推导接触面中产生的切向力的简化公式的方法；使用所提出的方法获得的轮胎公式不是基于常见的滑移定义，并且可以在滚动速度接近零时使用。概述的程序适用于线性轮胎力和抛物线压力分布的情况。