Hands-free circular motions of a benchmark bicycle

Abstract

We write nonlinear equations of motion for an idealized benchmark bicycle in two different ways and verify their validity. We then present a complete description of hands-free circular motions. Three distinct families exist. (i) A handlebar-forward family, starting from capsize bifurcation off straight-line motion and ending in unstable static equilibrium, with the frame perfectly upright and the front wheel almost perpendicular. (ii) A handlebar-reversed family, starting again from capsize bifurcation but ending with the front wheel again steered straight, the bicycle spinning infinitely fast in small circles while lying flat in the ground plane. (iii) Lastly, a family joining a similar flat spinning motion (with handlebar forward), to a handlebar-reversed limit, circling in dynamic balance at infinite speed, with the frame near upright and the front wheel almost perpendicular; the transition between handlebar forward and reversed is through moderate-speed circular pivoting, with the rear wheel not rotating and the bicycle virtually upright. Small sections of two families are stable.