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 Regul. Chaotic Dyn., 2017, Volume 22, Issue 3, Pages 298–317 (Mi rcd258)

The Rolling Motion of a Truncated Ball Without Slipping and Spinning on a Plane

Alexander A. Kilinab, Elena N. Pivovarovab

a Institute of Mathematics and Mechanics of the Ural Branch of RAS, ul. S. Kovalevskoi 16, Ekaterinburg, 620990 Russia
b Udmurt State University, ul. Universitetskaya 1, Izhevsk, 426034 Russia

Abstract: This paper is concerned with the dynamics of a top in the form of a truncated ball as it moves without slipping and spinning on a horizontal plane about a vertical. Such a system is described by differential equations with a discontinuous right-hand side. Equations describing the system dynamics are obtained and a reduction to quadratures is performed. A bifurcation analysis of the system is made and all possible types of the top’s motion depending on the system parameters and initial conditions are defined. The system dynamics in absolute space is examined. It is shown that, except for some special cases, the trajectories of motion are bounded.

Keywords: integrable system, system with discontinuity, nonholonomic constraint, bifurcation diagram, absolute dynamics

 Funding Agency Grant Number Russian Foundation for Basic Research 15-08-09093-a15-38-20879 mol_a_ved Russian Science Foundation 15-12-20035 The work of A.A. Kilin (Sections 1, 2, and 4) is supported by the RFBR grants 15-08-09093-a and 15-38-20879 mol_a_ved. The work of E.N. Pivovarova (Section 3) is carried out within the framework of the RSF grant no. 15-12-20035.

DOI: https://doi.org/10.1134/S156035471703008X

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Bibliographic databases:

Document Type: Article
MSC: 70E15, 70E18, 70E40, 37Jxx
Language: English

Citation: Alexander A. Kilin, Elena N. Pivovarova, “The Rolling Motion of a Truncated Ball Without Slipping and Spinning on a Plane”, Regul. Chaotic Dyn., 22:3 (2017), 298–317

Citation in format AMSBIB
\Bibitem{KilPiv17} \by Alexander A. Kilin, Elena N. Pivovarova \paper The Rolling Motion of a Truncated Ball Without Slipping and Spinning on a Plane \jour Regul. Chaotic Dyn. \yr 2017 \vol 22 \issue 3 \pages 298--317 \mathnet{http://mi.mathnet.ru/rcd258} \crossref{https://doi.org/10.1134/S156035471703008X} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3658427} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000402746300008} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85020210423} 

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This publication is cited in the following articles:
1. Yu. L. Karavaev, A. V. Klekovkin, A. A. Kilin, “Dinamicheskaya model treniya kacheniya sfericheskikh tel po ploskosti bez proskalzyvaniya”, Nelineinaya dinam., 13:4 (2017), 599–609
2. Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev, “An Invariant Measure and the Probability of a Fall in the Problem of an Inhomogeneous Disk Rolling on a Plane”, Regul. Chaotic Dyn., 23:6 (2018), 665–684
3. Alexander A. Kilin, Elena N. Pivovarova, “Integrable Nonsmooth Nonholonomic Dynamics of a Rubber Wheel with Sharp Edges”, Regul. Chaotic Dyn., 23:7-8 (2018), 887–907
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