The controlled motion of a bicycle

The motion of a vertically positioned bicycle is considered when a horizontal control force, which may be both internal and external in relation to the bicycle, is applied to its pedal. Tangential forces of dry friction obeying the Euler-Coulomb law act at points of contact of the wheels with the horizontal support plane. The constraint at the points of contact of the wheels with the support is assumed to be unilateral. The problem of determining the acceleration of the centre of mass of the bicycle and the realized motions of its wheels (with or without slip, and with or without detachment) with different values of the design parameters and control force is solved. Cases of non-uniqueness of motion - the Painlevé paradox - are found.