Feedback linearization of the shimmying wheel

The classical shimmying wheel is the simplest mechanical model to describe vibration problems in the rolling dynamics of aircraft nose gears, motorcycles , automotive systems and tractor-trailer systems. Such systems can exhibit undesirable unstable rolling motion which can often lead to disastrous results. The classical shimmy model includes a rigid wheel towed by a horizontal caster attached to a robust cart of constant speed by an elastic king pin. In the presence of dry friction, the single contact point between the wheel and the ground may have a zero velocity, the wheel rolls, and the system is non-holonomic. If the Coulomb friction cannot provide the necessary friction forces for rolling, the wheel slips, and the system is holonomic. There is a non-smooth change between the two kinds of dynamics. Apart of the smooth and symmetric geometrical nonlinearities of the two dynamics, the switches between them provide another strong and important nonlinearity. The bifurcation analysis of the rolling system proves the existence of a subcritical Hopf bifurcation in the system. Numerical simulation also shows unstable limit cycle for certain parameter values. There exists also an at-tractor outside the unstable limit cycle which belongs partly to the rolling system, and partly to the sliding one. Energy is introduced into the system via the constant speed of towing, while energy is dissipated at the wheel-ground contact point in case of sliding. It is difficult to identify the above attractor even with careful numerical simulation. The attractor may look chaotic, and some semi-analytical