Alexander Shamin

This paper addresses the problem of the motion of the Chaplygin sleigh, a rigid body moving with three points in contact with a horizontal plane. One of them is equipped with a knife edge along which there is no slipping. Special attention is given to the case where dry friction is present at one of the points of support without the knife edge. The equations of motion of the body are written, the normal reactions are calculated, and the behavior of the phase curves in the neighborhood of an equilibrium point, depending on the geometric and mass characteristics of the body, is investigated by the method of introducing a small parameter.

This paper addresses the problem of the motion of the Chaplygin sleigh, a rigid body with three legs in contact with a horizontal plane, one of which is equipped with a semicircular skate orthogonal to the horizontal plane. The problem is considered in a nonholonomic setting: assuming that the blade cannot slide in a direction perpendicular to its plane, but unlike the Chaplygin problem, there is a dry friction force in the skate that is directed along the skate, along which the blade plane and the reference plane intersect. It is also assumed that at the two other points of support there are dry friction forces.
The equations of motion of the Chaplygin sleigh are obtained, and a number of properties are proved. It is proved that the movement ceases in finite time. The possibility of realizing the nonnegativity of normal reactions is discussed. The case of static friction is studied when the blade velocity is $v=0$. A region of stagnation where the system rotates about a fixed vertical axis is found. On this set, the equations of motion are integrated and the law of variation of the angular velocity is found. Examples of trajectories of the sleigh are given. A qualitative description of the motion is obtained: the behavior of the phase curves in a neighborhood of the equilibrium point is investigated depending on the geometric and mass characteristics of the system.

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.