Regular and Chaotic Dynamics

, Volume 17, Issue 2, pp 142–149

# On a periodic motion of a rigid body carrying a material point in the presence of impacts with a horizontal plane

• Anatoly P. Markeev
Article

## Abstract

A material system consisting of an outer rigid body (a shell) and an inner body (a material point) is considered. The system moves in a uniform field of gravity over a fixed absolutely smooth horizontal plane. The central ellipsoid of inertia of the shell is an ellipsoid of rotation. The material point moves according to the harmonic law along a straight-line segment rigidly attached to the shell and lying on its axis of dynamical symmetry. During its motion, the shell may collide with the plane. The coefficient of restitution for an impact is supposed to be arbitrary. The periodic motion of the shell is found when its symmetry axis is situated along a fixed vertical, and the shell rotates around this vertical with an arbitrary constant angular velocity. The conditions for existence of this periodic motion are obtained, and its linear stability is studied.

## Keywords

rigid body dynamics collision periodic motion stability

## MSC2010 numbers

70E55 70F35 70K42 70E50

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