Transient tyre models with a flexible carcass

The accuracy of transient tyre models may be largely improved by considering the flexibility of the tyre carcass. Several formulations, whereby the unsteady behaviour of the tyre is approximated using linear or nonlinear systems of ordinary differential equations (ODEs), are already available in the literature. However, when the tread behaviour is described using a distributed representation, that is, in terms of partial differential equations (PDEs), the inclusion of even the simplest model to represent the deformation of the tyre carcass leads to rather involved PDE or interconnected PDE-ODE systems, with nonlocal and boundary terms. Such descriptions require detailed analyses that have not been attempted so far. Therefore, this paper investigates the salient properties of the classic brush and LuGre-brush models considering the effect of a flexible carcass. For both formulations, the existence and uniqueness of the solution are discussed. For the standard version of the brush models, a closed-form solution is provided under the assumption of vanishing sliding, whereas the case of limited friction is explored only qualitatively. Concerning the LuGre-brush variant, the preliminary intuition gained from the analysis of the distributed representation is effectively used to develop approximated lumped formulations to be used in control-oriented applications.