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Bifurcation Analysis of the Dynamics of an Unbalanced Rubber Ellipsoid of Revolution

  • Elena N. Pivovarova [1] ; Alexander A. Kilin [1]
    1. [1] Udmurt State University

      Udmurt State University

      Rusia

  • Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 24, Nº 6, 2025
  • Idioma: inglés
  • Enlaces
  • Resumen

    • This paper addresses the problem of the rolling motion on a plane of an unbalanced ellipsoid of revolution with symmetrically truncated vertices. The problem is considered using the model of a rubber rolling. By the rubber model of rolling we mean the rolling of a body without slipping nor twisting at the point of contact. First integrals are presented and a reduction to quadratures is performed. Partial solutions of the resulting system are found and it is shown that they correspond either to the rolling motion of the body in a circle or to an equilibrium point. A gyroscopic function is analyzed to examine bifurcations of partial solutions and to carry out a classification of all types of bifurcation diagrams according to parameters.

  • Referencias bibliográficas
    • 1. Alves, J., Dias, J.: Design and control of a spherical mobile robot. Proceedings of the Institution of Mechanical Engineers, Part I: Journal...
    • 2. Arnold, V.I., Kozlov, V.V., Neishtadt, A.I.: Mathematical aspects of classical and celestial mechanics, vol. 3. Springer Science &...
    • 3. Balandin, D., Komarov, M., Osipov, G.: A motion control for a spherical robot with pendulum drive. J. Comput. Syst. Sci. Int. 52(4), 650–663...
    • 4. Balseiro, P., Sansonetto, N.: First integrals and symmetries of nonholonomic systems. Arch. Ration. Mech. Anal. 244(2), 343–389 (2022)
    • 5. Batista, M.: Integrability of the motion of a rolling disk of finite thickness on a rough plane. Int. J. Non-Linear Mech. 41(6–7), 850–859...
    • 6. Batista, M.: Steady motion of a rigid disk of finite thickness on a horizontal plane. Int. J. Non-Linear Mech. 41(4), 605–621 (2006)
    • 7. Bizyaev, I.A., Bolsinov, A., Borisov, A., Mamaev, I.: Topology and bifurcations in nonholonomic mechanics. Int. J. Bifurc. Chaos 25(10),...
    • 8. Bizyaev, I.A.,Mamaev, I.S.: Permanent rotations in nonholonomic mechanics. omnirotational ellipsoid. Regul. Chaotic Dyn. 27(6), 587–612...
    • 9. Bloch, A.M., Marsden, J.E., Zenkov, D.V.: Nonholonomic dynamics. Notices of the AMS 52(3), 320– 329 (2005)
    • 10. Bolsinov, A.V., Borisov, A.V., Mamaev, I.S.: Topology and stability of integrable systems. Russ. Math. Surv. 65(2), 259–318 (2010)
    • 11. Borisov, A., Mamaev, I.: The rolling motion of a rigid body on a plane and a sphere. Hierarchy of dynamics. Regul. Chaotic Dyn. 7(2),...
    • 12. Borisov, A., Mamaev, I.: Mathematical methods in the dynamics of vortex structures. Institute of Computer Sciences, Moscow-Izhevsk (2005)
    • 13. Borisov, A.V., Mamaev, I.S.: Conservation laws, hierarchy of dynamics and explicit integration of nonholonomic systems. Regul. Chaotic...
    • 14. Borisov, A.V., Mamaev, I.S., Bizyaev, I.A.: The hierarchy of dynamics of a rigid body rolling without slipping and spinning on a plane...
    • 15. Borisov, A.V., Mamaev, I.S., Kilin, A.A.: Dynamics of rolling disk. Regul. Chaotic Dyn. 8(2), 201–212 (2003)
    • 16. Cai, Y., Zhan, Q., Xi, X.: Path tracking control of a spherical mobile robot. Mech. Mach. Theory 51, 58–73 (2012)
    • 17. Chaplygin, S.A.: On a motion of a heavy body of revolution on a horizontal plane. Regul. Chaotic Dyn. 7(2), 119–130 (2002)
    • 18. Chen, W.H., Chen, C.P., Yu, W.S., Lin, C.H., Lin, P.C.: Design and implementation of an omnidirectional spherical robot omnicron. In:...
    • 19. Ehlers, K., Koiller, J.: Rubber rolling: Geometry and dynamics of 2-3-5 distributions. In: IUTAM Symposium on Hamiltonian Dynamics, Vortex...
    • 20. Ivanova, T.B., Pivovarova, E.N.: Dynamics and control of a spherical robot with an axisymmetric pendulum actuator. arXiv preprint (2015)....
    • 21. Ivochkin, M.Y.: Topological analysis of the motion of an ellipsoid on a smooth plane. Sbornik: Math. 199(6), 871 (2008)
    • 22. Karapetyan, A.: Families of permanent rotations of triaxial ellipsoid on rough horizontal plane and their branchings. Actual Problems...
    • 23. Karavaev, Y.L., Kilin, A.A.: Nonholonomic dynamics and control of a spherical robot with an internal omniwheel platform: Theory and experiments....
    • 24. Kharlamov, M.P., Ryabov, P.E., Savushkin, A.Y.: Topological atlas of the Kowalevski-Sokolov top. Regul. Chaotic Dyn. 21, 24–65 (2016)
    • 25. Kilin, A., Ivanova, T.: Bifurcation analysis and dynamical phenomena in the problem of the rolling motion of a dynamically symmetric spherical...
    • 26. Kilin, A.A., Artemova, E.M.: Bifurcation analysis of the problem of two vortices on a finite flat cylinder. Russ. J. Nonlinear Dyn. 20(1),...
    • 27. Kilin, A.A., Karavaev, Y.L.: The kinematic control model for a spherical robot with an unbalanced internal omniwheel platform. Russ. J....
    • 28. Kilin, A.A., Pivovarova, E.N.: The rolling motion of a truncated ball without slipping and spinning on a plane. Regul. Chaotic Dyn. 22(3),...
    • 29. Kilin, A.A., Pivovarova, E.N.: Integrable nonsmooth nonholonomic dynamics of a rubber wheel with sharp edges. Regul. Chaotic Dyn. 23,...
    • 30. Kilin, A.A., Pivovarova, E.N.: Qualitative analysis of the nonholonomic rolling of a rubber wheel with sharp edges. Regul. Chaotic Dyn....
    • 31. Kilin, A.A., Pivovarova, E.N.: Bifurcation analysis of the problem of a rubber ellipsoid of revolution rolling on a plane. Nonlinear Dyn....
    • 32. Kilin, A.A., Pivovarova, E.N., Ivanova, T.B.: Spherical robot of combined type: dynamics and control. Regul. Chaotic Dyn. 20(6), 716–728...
    • 33. Koiller, J., Ehlers, K.: Rubber rolling over a sphere. Regul. Chaotic Dyn. 12, 127–152 (2007)
    • 34. Kozlov, V.V.: On the integration theory of equations of nonholonomic mechanics. Regul. Chaotic Dyn. 7(2), 161–176 (2002)
    • 35. Magnus, K.: Kreisel: theorie und anwendungen. Springer-Verlag (2013)
    • 36. Pivovarova, E., Kilin, A.: Dynamics of a balanced truncated rubber ellipsoid of revolution. Russ. J. Nonlinear Dyn. 21 (2025). https://doi.org/10.20537/nd250802
    • 37. Ryabov, P.E., Sokolov, S.V.: Bifurcation diagram of the model of a lagrange top with a vibrating suspension point. Math. 11(3), 533 (2023)
    • 38. Zhan, Q., Cai, Y., Yan, C.: Design, analysis and experiments of an omni-directional spherical robot. In: 2011 IEEE International Conference...
    • 39. Zhao, B., Li, M., Yu, H., Hu, H., Sun, L.: Dynamics and motion control of a two pendulums driven spherical robot. In: 2010 IEEE/RSJ International...
    • 40. Zhao, B., Wang, P., Hu, H., Li, M., Sun, L.: Study on turning in place of a spherical robot based on stick-slip principle. In: 2009 IEEE...
    • 41. Zheng, M., Qiang, Z., Jinkun, L., Yao, C.: Control of a spherical robot: Path following based on nonholonomic kinematics and dynamics....
    • 42. Zobova, A.: Various friction models in two-sphere top dynamics. Mech. Solids 48, 134–139 (2013)
    • 43. Zobova, A., Karapetyan, A.: Analysis of the steady motions of the tippe top. J. Appl. Math. Mech. 73(6), 623–630 (2009)
    • 44. Zobova, A.A.: On conjugation of solutions to two integrable problems: Rolling of pointed body in the plane. Automat. Rem. Contr. 68(8),...
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