The motion of a randomly perturbed Chaplygin sledge

• Published in: Journal of applied mathematics and mechanics 1994, Vol.58 (5), p.831-839
• Main Author: Moshchuk, N.K.
• Format: Article
• Language: English
• ISSN: 0021-8928; 0021-8928
• DOI: 10.1016/0021-8928(94)90008-6

The motion of a Chaplygin sledge [1] on an inclined plane is considered when sliding and rotational friction are present, together with random “white noise” perturbations produced by, for example, translational vibrations of the base. Stochastic equations of motion are set up and the problem of their statistical analysis is considered. In the case when the plane is horizontal, and the friction and perturbations are small, the analysis is carried out by an averaging method. All finite kinetic energy distributions of the sledge are found. It is shown that a limiting steady-state mode with a γ-distribution is established. The motion of a skate (which is a special case of a Chaplygin sledge) on an inclined plane when there is no rotational friction is briefly considered. It is shown that when the sliding friction is small the skate will “on average” slowly slip downwards, i.e. the mathematical expectation of the coordinate changes slightly, whereas the “root mean square” slippage will be significant, i.e. the dispersion of the coordinate varies strongly. In the case of a Chaplygin sledge moving along an inclined plane with arbitrary coefficients of sliding and rotational friction the analysis is performed using the method of orthogonal expansions. Numerical results are presented.