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 Regul. Chaotic Dyn., 2018, Volume 23, Issue 7-8, Pages 803–820 (Mi rcd368)

Comparing Dynamics Initiated by an Attached Oscillating Particle for the Nonholonomic Model of a Chaplygin Sleigh and for a Model with Strong Transverse and Weak Longitudinal Viscous Friction Applied at a Fixed Point on the Body

Alexey V. Borisovab, Sergey P. Kuznetsovbc

a Moscow Institute of Physics and Technology, Institutskii per. 9, Dolgoprudnyi, 141700 Russia
b Udmurt State University, ul. Universitetskaya 1, Izhevsk, 426034 Russia
c Kotel’nikov’s Institute of Radio-Engineering and Electronics of RAS, Saratov Branch, ul. Zelenaya 38, Saratov, 410019 Russia

Abstract: This paper addresses the problem of a rigid body moving on a plane (a platform) whose motion is initiated by oscillations of a point mass relative to the body in the presence of the viscous friction force applied at a fixed point of the platform and having in one direction a small (or even zero) value and a large value in the transverse direction. This problem is analogous to that of a Chaplygin sleigh when the nonholonomic constraint prohibiting motions of the fixed point on the platform across the direction prescribed on it is replaced by viscous friction. We present numerical results which confirm correspondence between the phenomenology of complex dynamics of the model with a nonholonomic constraint and a system with viscous friction — phase portraits of attractors, bifurcation diagram, and Lyapunov exponents. In particular, we show the possibility of the platform’s motion being accelerated by oscillations of the internal mass, although, in contrast to the nonholonomic model, the effect of acceleration tends to saturation. We also show the possibility of chaotic dynamics related to strange attractors of equations for generalized velocities, which is accompanied by a two-dimensional random walk of the platform in a laboratory reference system. The results obtained may be of interest to applications in the context of the problem of developing robotic mechanisms for motion in a fluid under the action of the motions of internal masses.

Keywords: Chaplygin sleigh, friction, parametric oscillator, strange attractor, Lyapunov exponents, chaotic dynamics, fish-like robot

 Funding Agency Grant Number Russian Foundation for Basic Research 18-08-00999-a18-29-10051-mk Russian Science Foundation 15-12-20035 The work of A.V. Borisov (Introduction and formulation of the equations of motion (Section 1)) was supported by the RFBR under grants No. 18-08-00999-a and 18-29-10051-mk. Numerical simulation and analysis of the results obtained (Sections 2–6) were carried out by A.V. Borisov and S.P. Kuznetsov within the framework of the RSF grant No. 15-12-20035.

DOI: https://doi.org/10.1134/S1560354718070018

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Bibliographic databases:

MSC: 37C10, 34D45, 37E30, 34C60, 37J60
Accepted:28.11.2018
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Citation: Alexey V. Borisov, Sergey P. Kuznetsov, “Comparing Dynamics Initiated by an Attached Oscillating Particle for the Nonholonomic Model of a Chaplygin Sleigh and for a Model with Strong Transverse and Weak Longitudinal Viscous Friction Applied at a Fixed Point on the Body”, Regul. Chaotic Dyn., 23:7-8 (2018), 803–820

Citation in format AMSBIB
\Bibitem{BorKuz18} \by Alexey V. Borisov, Sergey P. Kuznetsov \paper Comparing Dynamics Initiated by an Attached Oscillating Particle for the Nonholonomic Model of a Chaplygin Sleigh and for a Model with Strong Transverse and Weak Longitudinal Viscous Friction Applied at a Fixed Point on the Body \jour Regul. Chaotic Dyn. \yr 2018 \vol 23 \issue 7-8 \pages 803--820 \mathnet{http://mi.mathnet.ru/rcd368} \crossref{https://doi.org/10.1134/S1560354718070018} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000458183900001} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85061226885}