Dynamics and stability of a rimless spoked wheel: a simple 2D system with impacts

We discuss the dynamics and stability of a rigid rimless spoked wheel, or regular polygon, confined to 'rolling' in a vertical plane uphill or downhill. The wheel has smooth inverted pendulum motions punctuated by repeated dissipative spoke impacts. It is a simple mechanical analogue to legged locomotion. The problem is completely soluble in closed form. We derive a return map for the full nonlinear system. The map has two asymptotically stable fixed points whose existence depends on slope angle: (1) standing still on two spokes and (2) limit cycle motion. For small slopes, only standing still exists; for intermediate slopes, the fixed points coexist; and, for large slopes, only limit cycles exist. We also completely define their basins of attraction. The rimless wheel's dynamical behaviour is analogous to two other dissipative systems, one smooth (a constantly forced, damped pendulum) and the other non-smooth (a 2D discrete skate).