The dynamics of the accelerating bicycle

The purpose of this paper is to study the dynamics of the accelerating bicycle under straight-running and cornering conditions. If the bicycle is cornering at constant acceleration and roll angle, it is shown that for low values of acceleration (and braking), it follows closely a logarithmic spiral shaped trajectory. The studies presented are facilitated by a novel control scheme that centres the machine's trajectory on an arbitrary point in the ground plane; the origin of the inertial reference frame is typical. We study the machine's dynamics at constant speed and then include forces of inertia that represent the relevant acceleration influences. This use of d'Alembert's principle allows one to interpret the stability of the accelerating machine in terms of constant-speed equilibria. The bicycle model employed is based on that originally developed by Whipple, and comprises two road wheels and two laterally-symmetric frame assemblies that are free to rotate relative to each other along an inclined steering axis. For the most part the front frame is treated as being flexible and the bicycle is fitted with force generating road tyres rather than road wheels that are constrained in pure-rolling.