Stability analysis of linear single-track models with transient tyre dynamics

The present paper investigates the behaviour of linear single-track vehicle models considering transient tyre dynamics. In particular, the tyre models covered in this paper include lumped-parameter approximations, like the famous single-contact point and two-regime formulations, as well as refined distributed brush-like representations formulated in terms of partial differential equations (PDEs). In the first case, the resulting description consists of a system of linear ordinary differential equations (ODEs), whereas, in the second case, the lateral vehicle dynamics are captured by an interconnection of ODEs and PDEs. Stability analyses are conducted concerning both variants. In particular, for single-track models equipped with lumped tyre approximations, sufficient conditions on the vehicle's parameters that ensure dynamical stability are explicitly derived. Concerning the more involved formulation that considers the exact distributed brush model, the exact criterion for stability is instead deduced analytically, and then a qualitative analysis is performed. Via a number of examples, it is revealed that, for a wide range of frequencies of the steering input, the vehicle's behaviour predicted by the two models is similar to that obtained using the classic single-track formulation. The findings contained in the paper may assist vehicle manufacturers in setting requirements on design parameters, and in developing control algorithms that account for transient tyre dynamics at different degrees of accuracy.