In this study, the numerical bifurcation analysis of a shimmying wheel is performed
with a non-smooth, time-delayed model of the tyre-ground contact. This model is capable
of reproducing the bistable behaviour often observed in experiments: a stable equilibrium
and a stable periodic orbit coexisting for the same set of system parameters, that
the simpler quasi-steady tyre models fail to capture. In the bistable parameter domain,
there also exists an unstable periodic orbit within the separatrix between the domains
of attractions of the two stable steady-state solutions. Although this solution never
appears in a real-life system, one may still gain valuable information from tracing
it as it gives an indication about the level of perturbation that would drive the
system from one stable solution to the other. However, the complexity of the laws
governing partial sticking and sliding in the tyre-ground contact makes the numerical
bifurcation analysis with the traditional, collocation-based techniques infeasible.
Instead, this study is based on numerical simulations and the technique of control-based
continuation (CBC) to track the stable and unstable periodic solutions of the system
allowing for the assessment of the accuracy of the non-smooth, delayed tyre model
in replicating the dynamics observed in experiments. In the meantime, the physics-based
model provides an insight into the relationship between the sticking and sliding regions
appearing in the tyre-ground contact and the global dynamics of the system.
