On Translational Rectilinear Motion of a Rigid Body Carrying a Movable Inner Mass

Abstract

We consider the motion of the mechanical system consisting of the case (a rigid body) and the inner mass (a material point). The inner mass circulates inside the rigid body on a circle centered at the center of mass of the rigid body. We suppose that the absolute value of the velocity of circular motion of the inner mass is constant. The rigid body moves translationally and rectilinearly on a flat horizontal surface with forces of viscous friction and dry Coulomb friction on it. The inner mass moves in the vertical plane.We perform the full qualitative investigation of the dynamics of this system. We prove that there always exists a unique motion of the rigid body with periodic velocity. We study all possible types of such a periodic motion. We establish that for any initial velocity, the rigid body either reaches the periodic mode of motion in a finite time or asymptotically approaches to it depending on the parameters of the problem.