Motions of a rimless spoked wheel: a simple three-dimensional system with impacts

This paper discusses the mechanics of a rigid rimless spoked wheel, or regular polygon, 'rolling' downhill. By 'rolling', we mean motions in which the wheel pivots on one 'support' spoke until another spoke collides with the ground, followed by transfer of support to that spoke, and so on. We carry out three-dimensional (3D) numerical and analytical stability studies of steady motions of this system. At any fixed, large enough slope, the system has a one-parameter family of stable steady rolling motions. We find analytic approximations for the minimum required slope at a given heading for stable rolling in three dimensions, for the case of many spokes and small slope. The rimless wheel shares some qualitative features with passive-dynamic walking machines; it is a passive 3D system with intermittent impacts and periodic motions. In terms of complexity, it lies between one-dimensional impact oscillators and 3D walking machines. In contrast to a rolling disk on a flat surface which has steady rolling motions that are only neutrally stable at best, the rimless wheel can have asymptotic stability. In the limit as the number of spokes approaches infinity, the behavior of the rimless wheel approaches that of a rolling disk in an averaged sense and becomes neutrally stable. Also, in this averaged sense, the piecewise holonomic system (rimless wheel) approaches a non-holonomic system (disk).