The stability of the rotation of a spheroid about a horizontal axis with impacts on a horizontal plane

The motion of a uniform spheroid over a fixed absolutely smooth horizontal plane with impacts on the plane when an impact is absolutely elastic is considered. In the unperturbed motion, the spheroid rotates about a horizontally positioned axis of symmetry with a constant angular velocity and its centre of gravity in the intervals between the impacts moves along a fixed vertical. The stability of this motion with respect to the angle of inclination of the axis of symmetry of the spheroid from the vertical is investigated in a strict nonlinear formulation. Analytical expressions are obtained for the stability and instability conditions in terms of the problem parameters and geometric illustrations of the results obtained are constructed.