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.