Mechanical Sciences (Apr 2022)
Analysis of divergent bifurcations in the dynamics of wheeled vehicles
Abstract
This paper presents the bifurcation approach to analyze divergent loss of stability of the symmetric solution of a nonlinear dynamic model in Lyapunov's critical case of a single zero root. Under such a condition, material birth-annihilation bifurcations of multiple stationary states take place. Moreover, the equilibrium surface of stationary states in a small neighborhood of the symmetric solution is a generalized Whitney fold. In the simplest case of a fold peculiarity, the corresponding bifurcation manifold locally coincides with the discriminant manifold of a third-degree polynomial that determines the manifold of stationary states in a small neighborhood of the symmetric solution. An algorithm to construct the corresponding polynomial is introduced. Through the algorithm, the bifurcation manifold is determined, and the conditions for safe/unsafe loss of stability of the symmetric solution are derived analytically. The proposed approach to analyze divergent loss of stability is implemented for a nonlinear bicycle model of a two-axle wheeled vehicle. It represents a further development of Pevzner–Pacejka's well-known graph-analytical method. The paper determines the critical values of constructive parameters that are responsible for safe/unsafe loss of stability of the vehicle's straight-line motion, and it discusses strategies for the bifurcation approach to analyze divergent loss of stability.