Nonlinear analysis of a shimmying wheel with contact-force characteristics featuring higher-order discontinuities

In this study, the yaw dynamics of a towed caster wheel system is analysed via an in-plane, one degree-of-freedom mechanical model. The force and aligning torque generated by the elastic tyre are calculated by means of a semi-stationary tyre model, in which the piecewise-smooth characteristic of the tyre forces is also considered, resulting in a dynamical system with higher-order discontinuities. The focus of our analysis is the Hopf bifurcation affected by the non-smoothness of the system. The structure of the analysis is organised in a similar way as in case of smooth bifurcations. Firstly, the centre-manifold reduction is performed, then we compose the normal form of the bifurcation. Based on the Galerkin technique an approximate, semi-analytical method to calculate the limit cycles is introduced and compared with the method of collocation. The analysis provides a deeper insight into the development of the vibrations associated with wheel shimmy and demonstrate how the non-smoothness due to contact-friction influences the dynamic behaviour.