Nonholonomic mechanical systems on a plane with a variable slope

This paper considers such nonholonomic mechanical systems as Chaplygin skate, inhomogeneous Chaplygin sleigh and Chaplygin sphere moving in the gravity field along an oscillating plane with a slope varying with the periodic law. By explicit integration of the equations of motion, analytical expressions for the velocities and trajectories of the contact point for Chaplygin skate and Chaplygin sleigh are obtained. Numerical parameters of the periodic law for the inclination angle change are found, such that the velocity of Chaplygin skate will be unbounded, that is, an acceleration will take place. In the case of inhomogeneous Chaplygin sleigh, on the contrary, numerical parameters of the periodic law of the inclination angle change are found, for which the sleigh velocity is bounded and there is no drift of the sleigh. For similar numerical parameters and initial conditions, when the sleigh moves along a horizontal or inclined plane with the constant slope, the velocity and trajectory of the contact point are unbounded, that is, there is a drift of the sleigh. A similar problem is solved for the Chaplygin sphere; its trajectories are constructed on the basis of numerical integration. The results are illustrated graphically. The control of the slope of the plane, depending on the angular momentum of the sphere, is proposed for discussion. Regardless of the initial conditions, such control can almost always prevent the drift of the sphere in one of the directions.