MOTIONS OF A RIMLESS SPOKED WHEEL - A SIMPLE 3-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 i n which the wheel pivots on one 'support' spoke until another spoke co llides with the ground, followed by transfer of support to that spoke, and so an. We carry out three-dimensional (3D) numerical and analytic al stability studies of steady motions of this system. At any fixed, l arge enough slope, the system has a one-parameter family of stable ste ady rolling motions. We find analytic approximations for the minimum r equired slope at a given heading for stable rolling in three dimension s, for the case of many spokes and small slope. The rimless wheel shar es 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 oscil lators and 3D walking machines. In contrast to a rolling disk on a pat surface which has steady rolling motions that ave only neutrally stab le at best, the rimless wheel can have asymptotic stability. In the li mit as the number of spokes approaches infinity, the behavior of the r imless wheel approaches that of a rolling disk in an averaged sense an d becomes neutrally stable. Also, in this averaged sense, the piecewis e holonomic system (rimless wheel) approaches a non-holonomic system ( disk).