Motion of a dynamically symmetric paraboloid on a perfectly rough plane

Abstract

We consider the classical problem of nonholonomic system dynamics-the problem of motion of a dynamically symmetric body bounded by a surface of rotation on a fixed perfectly rough plane. We prove that this problem can be completely solved in quadratures in the case when the moving body is a paraboloid of revolution. The qualitative description of motion of a paraboloid on the plane is given. The trajectory of the point of contact M on the surface of paraboloid is the curve consisting of periodically repeating waves and tangent to two parallels of the paraboloid. The trajectory of the point of contact on the supporting plane has the same pattern and it is situated between two concentric circles. During the motion of the paraboloid the point of contact M touches these two circles in turn. The steady motions of the paraboloid on a perfectly rough plane are found and their stability is investigated. It is proved that all the steady motionsof the paraboloid (permanent rotations and regular precessions) arestable.

Keywords:

rolling paraboloid of revolution, analysis of quadratures, steady motions, stability