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Nonlinear Dynamics

, Volume 95, Issue 1, pp 699–714 | Cite as

The Chaplygin sleigh with friction moving due to periodic oscillations of an internal mass

  • Ivan A. Bizyaev
  • Alexey V. Borisov
  • Sergey P. Kuznetsov
Original Paper

Abstract

For a Chaplygin sleigh moving in the presence of weak friction, we present and investigate two mechanisms of arising acceleration due to oscillations of an internal mass. In certain parameter regions, the mechanism induced by small oscillations determines acceleration which is on average one-directional. The role of friction is that the velocity reached in the process of the acceleration is stabilized at a certain level. The second mechanism is due to the effect of the developing oscillatory parametric instability in the motion of the sleigh. It occurs when the internal oscillating particle is comparable in mass with the main platform and the oscillations are of a sufficiently large amplitude. In the nonholonomic model the magnitude of the parametric oscillations and the level of mean energy achieved by the system turn out to be bounded if the line of the oscillations of the moving particle is displaced from the center of mass; the observed sustained motion is in many cases associated with a chaotic attractor. Then, the motion of the sleigh appears to be similar to the process of two-dimensional random walk on the plane.

Keywords

Nonholonomic mechanics Chaplygin sleigh Parametric oscillator Strange attractor Lyapunov exponent Chaotic dynamics 

Notes

Acknowledgements

This work was supported by Grant No. 15-12-20035 of the Russian Science Foundation.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

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Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Udmurt State UniversityIzhevskRussian Federation
  2. 2.Kotel’nikov’s Institute of Radio-Engineering and Electronics of RASSaratovRussian Federation

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