Unsteady-state brush theory

Abstract

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.

Brush tyre models for large camber angles and steering speeds

“…which are identical to the formulae derived in [52]. It can be immediately observed that, in (29), both u − x (ξ ) and u − y (ξ ) are linear functions of σ x , σ y and ϕ and also u − x (ξ ) does not depend on the lateral slip and vice versa.…”

“…The micro-sliding speed v st (x, t) reads, in vector form, Pacejka [2], Guiggiani [21], Limebeer and Massaro [23], Romano et al [24], Romano et al [52]…”

“…Thus, we need to ascertain that Equations (24) and (27) are equivalent to the linear expressions predicted by classic 1DM for small values of ε ψ and ϕ. These formulae can be found in many books, although here we refer explicitly to the notation used in [52]. We start from Equation 24by expanding the trigonometric functions in Taylor series up to the second term to get…”

“…Recently, the authors have developed an exhaustive theory which captures the dynamics of the bristles in the contact patch [52]. A class of simplified formulae has also been introduced to deal with more complex scenarios and take into account the major effects connected with the tyre carcass dynamics.…”

Autonomous Vehicle Maneuvering at the Limit of Friction

“…With its roots in physical modeling, the brush model provides some insights into the tire traction forces. For further analytical insights of the tire behavior, unsteady-state versions of the brush model can be considered [66].…”