On the non-linear dynamics of vehicle shimmy coupled with handling motions: modelling, bifurcation analysis, and validations

A novel vehicle dynamic model considering the coupling mechanism between the front wheel shimmy and the vehicle lateral handling motions is established to investigate the interacting effect between the steering shimmy instability and the general motion instability, that focuses especially on the influence of the tyre force saturation induced by lateral-longitudinal coupling when different degrees and allocation strategy of braking force are applied. The typical 3-degree-of-freedom front wheel shimmy structure of dependent suspension is dynamically modelled by using the Lagrange method, alongside the equations for plane vehicle handling motions introduced. The stretched-string-based tyre kinematic equations, that describe the transient tyre-road contact status, are proposed by analytically deriving the coupling relations between the angular shimmy variables and vehicle motion status. A theoretically approximated linear tyre model, based on the coupling between vertical, longitudinal, and lateral directional forces, is analytically derived for stability analysis. Meanwhile, a complete PAC-2002 tyre model, with experimentally identified parameters, is employed for simulation verification. The stability chart of linear stability analysis shows the variation of the system bifurcation curves and stable parameter region under different parametric configurations, which gives that the degrees and allocations of braking force largely affect the stable parameter domain and the global stability. The variations of the system characteristic roots indicate that both Hopf and S-N bifurcation could happen due to shimmy oscillations and destabilization of motion, respectively, and their co-existence at certain parameter domains make the dynamic behaviour further unpredictable and complex. Numerical simulation verifies the dynamical characteristics of shimmy, and further analysis shows that three types of vehicle motion instability which coexisted with shimmy instability could be triggered by the coupling effect. In addition, the variation law of limit-cycle amplitude under varying vehicle driving inputs is summarized, and finally, a preliminary experiment is conducted for validations.