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

2019, Vol. 15, no. 2, pp.  159-169

Author(s): Shamin A. Y.

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
Citation: Shamin A. Y., On the Motion of the Chaplygin Sleigh on a Horizontal Plane with Dry Friction at Three Points of Contact, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 2, pp.  159-169
DOI:10.20537/nd190205

## References

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[2] Neimark, Ju. I. and Fufaev, N. A., Dynamics of Nonholonomic Systems, Trans. Math. Monogr., 33, AMS, Providence, R.I., 1972, 518 pp.
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[4] Karapetyan, A. V. and Shamin, A. Yu., “On the Movement of the Chaplygin Sleigh on a Horizontal Plane with Dry Friction”, J. Appl. Math. Mech., 83:2 (2019), 251–256; Prikl. Mat. Mekh., 83:2 (2019), 228–233 (Russian)
[5] Ivanov, A. P., Fundamentals of the Theory of Systems with Friction, R&C Dynamics, Institute of Computer Science, Moscow – Izhevsk, 2011, 304 pp. (Russian)
[6] Sumbatov, A. S. and Yunin, E. K., Selected Problems of Mechanics of Systems with Dry Friction, Fizmatlit, Moscow, 2013, 200 pp. (Russian)
[7] Kuleshov, A. S., Treschev, D. V., Ivanova, T. B., and Naimushina, O. S., “A Rigid Cylinder on a Viscoelastic Plane”, Nelin. Dinam., 7:3 (2011), 601–625 (Russian)