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Nelin. Dinam., 2013, Volume 9, Number 3, Pages 547–566 (Mi nd405)  

This article is cited in 1 scientific paper (total in 1 paper)

The dynamics of nonholonomic systems consisting of a spherical shell with a moving rigid body inside

Ivan A. Bizyaevab, Alexey V. Borisovabcd, Ivan S. Mamaevabcd

a Institute of Computer Science
b Laboratory of nonlinear analysis and the design of new types of vehicles, Udmurt State University, Universitetskaya 1, Izhevsk, 426034 Russia
c A. A. Blagonravov Mechanical Engineering Research Institute of RAS, Bardina str. 4, Moscow, 117334, Russia
d Institute of Mathematics and Mechanics of the Ural Branch of RAS, S. Kovalevskaja str. 16, Ekaterinburg, 620990, Russia

Abstract: In this paper we investigate two systems consisting of a spherical shell rolling on a plane without slipping and a moving rigid body fixed inside the shell by means of two different mechanisms. In the former case the rigid body is fixed at the center of the ball on a spherical hinge. We show an isomorphism between the equations of motion for the inner body with those for the ball moving on a smooth plane. In the latter case the rigid body is fixed by means of the nonholonomic hinge. The equations of motion for this system have been obtained and new integrable cases found. A special feature of the set of tensor invariants of this system is that it leads to the Euler–Jacobi–Lie theorem, which is a new integration mechanism in nonholonomic mechanics.

Keywords: nonholonomic constraint, tensor invariants, isomorphism, nonholonomic hinge.

Full text: PDF file (440 kB)
References: PDF file   HTML file
UDC: 531.38
MSC: 70E18, 37J60, 37J35
Received: 04.09.2013
Revised: 31.10.2013

Citation: Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev, “The dynamics of nonholonomic systems consisting of a spherical shell with a moving rigid body inside”, Nelin. Dinam., 9:3 (2013), 547–566

Citation in format AMSBIB
\Bibitem{BizBorMam13}
\by Ivan~A.~Bizyaev, Alexey~V.~Borisov, Ivan~S.~Mamaev
\paper The dynamics of nonholonomic systems consisting of a spherical shell with a moving rigid body inside
\jour Nelin. Dinam.
\yr 2013
\vol 9
\issue 3
\pages 547--566
\mathnet{http://mi.mathnet.ru/nd405}


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    This publication is cited in the following articles:
    1. I. A. Bizyaev, A. V. Bolsinov, A. V. Borisov, I. S. Mamaev, “Topology and bifurcations in nonholonomic mechanics”, International Journal of Bifurcation and Chaos, 25:10 (2015), 15300–21  mathnet  crossref  isi
  • Нелинейная динамика
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