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 Nelin. Dinam., 2019, Volume 15, Number 2, Pages 159–169 (Mi nd649)

Nonlinear physics and mechanics

On the Motion of the Chaplygin Sleigh on a Horizontal Plane with Dry Friction at Three Points of Contact

A. Yu. Shamin

Moscow State University, Vorobievy gory 1, Moscow, 119899 Russia

Abstract: 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.

Keywords: dry friction, Chaplygin sleigh

 Funding Agency Grant Number Russian Foundation for Basic Research 19-01-00140 This work was supported by the Russian Foundation for Basic Research (19-01-00140).

DOI: https://doi.org/10.20537/nd190205

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MSC: 70F25, 70F40
Accepted:23.04.2019

Citation: A. Yu. Shamin, “On the Motion of the Chaplygin Sleigh on a Horizontal Plane with Dry Friction at Three Points of Contact”, Nelin. Dinam., 15:2 (2019), 159–169

Citation in format AMSBIB
\Bibitem{Sha19} \by A. Yu. Shamin \paper On the Motion of the Chaplygin Sleigh on a Horizontal Plane with Dry Friction at Three Points of Contact \jour Nelin. Dinam. \yr 2019 \vol 15 \issue 2 \pages 159--169 \mathnet{http://mi.mathnet.ru/nd649} \crossref{https://doi.org/10.20537/nd190205}