Analytical tyre models with two-regime dynamics: an iterative algorithm based on the implicit function theorem

Abstract This paper investigates the nonlinear transient dynamics of pneumatic tyres by extending the recently introduced two-regime modelling framework. Unlike classical single-contact-point models, which describe force generation via first-order relaxation dynamics, the two-regime approach directly models the evolution of tyre forces and aligning moment through a set of nonlinear ordinary differential equations derived from the underlying brush-type partial differential equations. A key contribution of this work is a general recursive methodology, based on the implicit function theorem, for reconstructing the slip surfaces from any analytical steady-state tyre model, including non-invertible formulations such as Pacejka’s Magic Formula. The stability properties of the proposed models are analysed, revealing the crucial role of the slip stiffness matrix in determining singular behaviours and transient instabilities. Simulations are conducted using an isotropic Magic Formula model for combined slips, demonstrating significant dynamical differences between the two-regime and single-contact-point formulations, including non-minimum phase effects and velocity-dependent relaxation phenomena. The results establish a direct connection between the steady-state and transient tyre characteristics, providing new insights into the nonlinear dynamics of rolling contact systems.