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 Regul. Chaotic Dyn., 2014, Volume 19, Issue 2, Pages 198–213 (Mi rcd126)

The Dynamics of Nonholonomic Systems Consisting of a Spherical Shell with a Moving Rigid Body Inside

Ivan A. Bizyaeva, Alexey V. Borisovbcd, Ivan S. Mamaeva

a Udmurt State University, ul. Universitetskaya 1, Izhevsk, 426034 Russia
b Moscow Institute of Physics and Technology, Institutskii per. 9, Dolgoprudny, Moscow Region, 141700 Russia
c A. A.Blagonravov Mechanical Engineering Research Institute of RAS, ul. Bardina 4, Moscow, 117334 Russia
d National Research Nuclear University “MEPhI”, Kashirskoe sh. 31, Moscow, 115409 Russia

Abstract: In this paper we investigate two systems consisting of a spherical shell rolling without slipping on a plane and a moving rigid body fixed inside the shell by means of two different mechanisms. In the former case the rigid body is attached to 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 a nonholonomic hinge. 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. We also consider the problem of free motion of a bundle of two bodies connected by means of a nonholonomic hinge. For this system, integrable cases and various tensor invariants are found.

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

 Funding Agency Grant Number Russian Foundation for Basic Research 13-01-12462-ofi_m Ministry of Education and Science of the Russian Federation MD-2324.2013.1NSh-2964.2014.11746 The work of A. V.Borisov was carried out within the framework of the state assignment to the Udmurt State University “Regular and Chaotic Dynamics”. The work of I.S.Mamaev was supported by the RFBR grants 13-01-12462-ofi m. The work of I. A.Bizyaev was supported by the Grant of the President of the Russian Federation for Support of Young Doctors of Science MD- 2324.2013.1, and by the Grant of the President of the Russian Federation for Support of Leading Scientific Schools NSh-2964.2014.1.

DOI: https://doi.org/10.1134/S156035471402004X

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MSC: 70E18, 37J60, 37J35
Accepted:31.10.2013
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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”, Regul. Chaotic Dyn., 19:2 (2014), 198–213

Citation in format AMSBIB
\Bibitem{BizBorMam14} \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 Regul. Chaotic Dyn. \yr 2014 \vol 19 \issue 2 \pages 198--213 \mathnet{http://mi.mathnet.ru/rcd126} \crossref{https://doi.org/10.1134/S156035471402004X} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3189257} \zmath{https://zbmath.org/?q=an:1308.70003} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000334198000004} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
2. A. V. Borisov, I. S. Mamaev, “Equations of motion of non-holonomic systems”, Russian Math. Surveys, 70:6 (2015), 1167–1169
3. A. V. Borisov, I. S. Mamaev, I. A. Bizyaev, “The Jacobi Integral in Nonholonomic Mechanics”, Regul. Chaotic Dyn., 20:3 (2015), 383–400
4. Yury L. Karavaev, Alexander A. Kilin, “The Dynamics and Control of a Spherical Robot with an Internal Omniwheel Platform”, Regul. Chaotic Dyn., 20:2 (2015), 134–152
5. Alexander V. Sakharov, “Rotation of the Body with Movable Internal Masses Around the Center of Mass on a Rough Plane”, Regul. Chaotic Dyn., 20:4 (2015), 428–440
6. Alexander P. Ivanov, “On the Control of a Robot Ball Using Two Omniwheels”, Regul. Chaotic Dyn., 20:4 (2015), 441–448
7. Rasoul Akbarzadeh, Ghorbanali Haghighatdoost, “The Topology of Liouville Foliation for the Borisov–Mamaev–Sokolov Integrable Case on the Lie Algebra $so(4)$”, Regul. Chaotic Dyn., 20:3 (2015), 317–344
8. Alexey V. Borisov, Alexander A. Kilin, Ivan S. Mamaev, “Dynamics and Control of an Omniwheel Vehicle”, Regul. Chaotic Dyn., 20:2 (2015), 153–172
9. Ivan A. Bizyaev, Alexey V. Borisov, Alexey O. Kazakov, “Dynamics of the Suslov Problem in a Gravitational Field: Reversal and Strange Attractors”, Regul. Chaotic Dyn., 20:5 (2015), 605–626
10. V. V. Kozlov, “The dynamics of systems with servoconstraints. II”, Regular and Chaotic Dinamics, 20:4 (2015), 401–427
11. I. A. Bizyaev, A. V. Borisov, I. S. Mamaev, “Hamiltonization of elementary nonholonomic systems”, Russ. J. Math. Phys., 22:4 (2015), 444–453
12. A. V. Borisov, I. S. Mamaev, “A new integrable system of nonholonomic mechanics”, Dokl. Phys., 60:6 (2015), 269–271
13. Yu. L. Karavaev, A. A. Kilin, “Dinamika sferorobota s vnutrennei omnikolesnoi platformoi”, Nelineinaya dinam., 11:1 (2015), 187–204
14. Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev, “Dynamics of the Chaplygin Sleigh on a Cylinder”, Regul. Chaotic Dyn., 21:1 (2016), 136–146
15. Alexey V. Borisov, Ivan S. Mamaev, “Adiabatic Invariants, Diffusion and Acceleration in Rigid Body Dynamics”, Regul. Chaotic Dyn., 21:2 (2016), 232–248
16. Rasoul Akbarzadeh, “Topological Analysis Corresponding to the Borisov–Mamaev–Sokolov Integrable System on the Lie Algebra $so(4)$”, Regul. Chaotic Dyn., 21:1 (2016), 1–17
17. Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev, “The Hojman Construction and Hamiltonization of Nonholonomic Systems”, SIGMA, 12 (2016), 012, 19 pp.
18. I. A. Bizyaev, A. V. Borisov, I. S. Mamaev, “The Hess–Appelrot system and its nonholonomic analogs”, Proc. Steklov Inst. Math., 294 (2016), 252–275
19. Yu. L. Karavaev, A. A. Kilin, “Nonholonomic dynamics and control of a spherical robot with an internal omniwheel platform: theory and experiments”, Proc. Steklov Inst. Math., 295 (2016), 158–167
20. V. Kozlov, “The phenomenon of reversal in the Euler–Poincaré–Suslov nonholonomic systems”, J. Dyn. Control Syst., 22:4 (2016), 713–724
21. 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
22. A. V. Borisov, I. S. Mamaev, I. A. Bizyaev, “Dynamical systems with non-integrable constraints, vakonomic mechanics, sub-Riemannian geometry, and non-holonomic mechanics”, Russian Math. Surveys, 72:5 (2017), 783–840
23. Evgeny V. Vetchanin, Ivan S. Mamaev, “Dynamics of Two Point Vortices in an External Compressible Shear Flow”, Regul. Chaotic Dyn., 22:8 (2017), 893–908
24. Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev, “The Chaplygin Sleigh with Parametric Excitation: Chaotic Dynamics and Nonholonomic Acceleration”, Regul. Chaotic Dyn., 22:8 (2017), 955–975
25. M. Roozegar, M. J. Mahjoob, “Modelling and control of a non-holonomic pendulum-driven spherical robot moving on an inclined plane: simulation and experimental results”, IET Contr. Theory Appl., 11:4 (2017), 541–549
26. T. B. Ivanova, A. A. Kilin, E. N. Pivovarova, “Controlled motion of a spherical robot with feedback. I”, J. Dyn. Control Syst., 24:3 (2018), 497–510
27. S. V. Semendyaev, “Solid system with two massive eccentrics on a rough plane: rotational case”, IFAC-PapersOnLine, 51:2 (2018), 884–889
28. A. A. Kilin, T. B. Ivanova, E. N. Pivovarova, “Control of the Rolling Motion of a Spherical Robot on An Inclined Plane”, Dokl. Phys., 63:10 (2018), 435–440