On the Motion of a Vibrating Robot on a Horizontal Plane with Anisotropic Friction

This paper is concerned with a mechanical system consisting of a rigid body (outer body) placed on a horizontal rough plane and of an internal moving mass moving in a circle lying in a vertical plane, so that the radius vector of the point has a constant angular velocity. The interaction of the outer body and the horizontal plane is modeled by the Coulomb – Amonton law of dry friction with anisotropy (the friction coefficient depends on the direction of the body’s motion). The equation of the body’s motion is a differential equation with a discontinuous right-hand side. Based on the theory of A. F. Filippov, it is proved that, for this equation, the existence and right-hand uniqueness of the solution takes place, and that there exists a continuous dependence on initial conditions. Some general properties of the solutions are established and possible periodic regimes and their features are considered depending on the parameters of the problem. In particular, the existence of a periodic regime is proved in the case where the motion occurs without sticking of the outer body, and conditions for the existence of such a regime are shown. An analysis is made of the final dynamics of how the system reaches a periodic regime in the case where the outer body sticks twice within a period of revolution of the internal mass. This periodic regime exhibits sticking in the so-called upper and lower deceleration zones. The outer body comes twice to a stop and is at rest in these zones for some time and then continues its motion. This paper gives a complete description of the solution pattern for such motions. It is shown that, in the parameter space of the system where such a regime exists, the solution reaches this regime in finite time. All qualitatively different solutions in this case are described. In particular, special attention is devoted to the terminal motion, namely, to the solution of the system during the last period of motion of the internal mass before reaching the periodic regime. Existence regions of such solutions are found and the boundaries of the regions of initial conditions determining the qualitatively different dynamics of the system are established.