Bicycle dynamic modeling and stability analysis with a toroidal-shaped tire geometry

Traditional bicycle modeling often uses line contacts to simplify wheel-ground constraints. However, the geometric shape of the tire can greatly alter the constraint nature, thereby affecting relative equilibria and their stability in dynamics. This work develops a nonlinear dynamic model of the bicycle system with a toroidal-shaped tire geometry to establish a realistic wheel-ground constraint. Such tire geometry significantly increases the complexity of the dynamic equations. Therefore, calculation of the nontrivial equilibria that involves a set of strongly nonlinear algebraic equations is more difficult. To solve the issue, we establish the balance equation in a non-inertial frame to decouple the complex coupling terms of the high-dimensional nonlinear system, which simplifies the calculation of the equilibria. Furthermore, this work proposes a method to facilitate stability analysis of the relative equilibria, where linearization of the multi-dimensional dynamic equations is performed first, and then explicit dimensionality reduction is followed. Finally, we investigate the effects of tire geometric parameters on the equilibrium stability and find that the solution of the equilibria and their stability properties change significantly with the tire geometric shape.