Unsteady-state brush theory

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.

Automated summary

This paper focuses on unsteady-state brush tyre models, providing a full analytical solution to the partial differential equations describing bristle deformation in the adhesion region of the contact patch. It reveals that the solution can be divided into two different regions, with investigations into the time trend of shear stresses for a rectangular contact patch and parabolic pressure distribution. Additionally, important conclusions are drawn about the relaxation length for pure interactions and a simplified approach to derive tangential forces in the contact patch is introduced.