Bare and stretched string tyre models with distributed FrBD dynamics

This paper presents a novel class of string tyre models with FrBD friction dynamics. By modelling the distributed carcass and tread deformations with string-like equations, the resulting formulation leads to a system of semilinear parabolic partial differential equations (PDEs) that describe the evolution of the tyre states without explicitly distinguishing between stick and slip regimes. Rigorous stability and passivity analyses are also conducted using a Lyapunov-based approach, establishing boundedness of the distributed states and energy dissipation during rolling contact. The proposed Lyapunov function admits a clear physical interpretation as the total elastic energy stored in the tyre, enabling a direct link between mechanical energy storage and frictional dissipation due to slip losses. The steady-state and transient behaviours of the model are investigated both numerically and experimentally, revealing that the new formulation can satisfactorily reproduce nonlinear relaxation phenomena excited by step slip inputs. The resulting model provides a physically interpretable, mathematically well-posed, and computationally efficient basis for advanced vehicle dynamics simulations and control-oriented applications.