Alexander Kuleshov

This paper gives the further development for the mathematical model of a derivative of a skateboard known as the snakeboard. As against to the model, proposed by Lewis et al. [13] and investigated by various methods in [1]-[13], our model takes into account an opportunity that platforms of a snakeboard can rotate independently from each other. This assumption has been made earlier only by Golubev [13]. Equations of motion of the model are derived in the Gibbs–Appell form. Analytical and numerical investigations of these equations are fulfilled assuming harmonic excitations of the rotor and platforms angles. The basic snakeboard gaits are analyzed and shown to result from certain resonances in the rotor and platforms angle frequencies. All the obtained theoretical results are confirmed by numerical experiments.

In this review we discuss methods of investigation of steady motions of nonholonomic mechanical systems. General conclusions are illustrated by examples from the rigid bodies dynamics on a absolutely rough horisontal plane.

The problem of the motion of a rotational symmetric rigid body along a perfectly rough surface is considered. The conditions of existence of a Chaplygin-type integral are obtained. It is shown, that these conditions are valid in the case of the motion of a nonhomogeneous dynamically symmetric sphere along a perfectly rough plane or along the internal surface of a sphere.