Periodic Regimes of Motion of a Body with a Moving Internal Mass

The paper considers rectilinear motion of a system consisting of a body and a moving internal mass. The body moves in a resistive environment. The internal mass moves periodically with respect to the body. We investigate such periodic regimes of motion, that the velocity of the body is also periodic. We consider the problems of existence and uniqueness of the periodic regimes of motion and their stability. It is shown that a periodic regime of motion exists if the medium resistance is a monotonically decreasing unbounded function of the velocity and the relative velocity of the internal mass does not have jumps. A two-sided estimate of the initial velocity of the body for the periodic regime of motion is obtained. The uniqueness and exponential stability of the periodic regime of motion is proved for any monotonically decreasing law of resistance. In the special cases of linear and piecewise linear laws of the medium resistance, a periodic regime of motion is constructed and the rate of the exponential convergence of an arbitrary motion to this periodic regime is found. The general results are illustrated by simulations performed with the parameters of a physical prototype of a capsule robot.