Hysteresis Effect in the Nonlinear Stability of Towed Wheels

In this paper the dynamics of towed elastic wheels are studied with the help of the brush tyre model. To calculate the lateral deformation of the contact patch centre-line distributed time-delay is taken into account for the rolling parts, whereas parabolic limits are used to determine the deformation in case of side-slip. After linear stability analysis of the rectilinear motion the limit cycles of the non-smooth time-delayed system are calculated with the method of numerical collocation. With the help of bifurcation diagrams it is demonstrated how the periodic orbits develop from the linear stability boundary in a structure characteristic of piecewise-smooth systems. Moreover, it is shown that the contact memory effect and the dry friction yield bistable parameter ranges besides the linearly unstable domains. Namely, for one particular towing velocity a stable equilibrium corresponding to straight-line motion and a stable periodic orbit coexist resulting a hysteresis effect in the stability of the straight-line motion.