Regular and Chaotic Dynamics

, Volume 22, Issue 3, pp 298–317 | Cite as

The rolling motion of a truncated ball without slipping and spinning on a plane

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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 

MSC2010 numbers

70E15 70E18 70E40 37Jxx 

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References

  1. 1.
    Ehlers, K.M. and Koiller, J, Rubber Rolling: Geometry and Dynamics of 2 - 3 - 5 Distributions, in Proc. IUTAM Symposium 2006 on Hamiltonian Dynamics, Vortex Structures, Turbulence (Moscow, Russia, 25–30 August 2006), pp. 469–480.Google Scholar
  2. 2.
    Borisov, A.V. and Mamaev, I. S, Conservation Laws, Hierarchy of Dynamics and Explicit Integration of Nonholonomic Systems, Regul. Chaotic Dyn., 2008, vol. 13, no. 5, pp. 443–490.CrossRefMATHGoogle Scholar
  3. 3.
    Borisov, A. V., Mamaev, I. S., and Bizyaev, I. A, The Hierarchy of Dynamics of a Rigid Body Rolling without Slipping and Spinning on a Plane and a Sphere, Regul. Chaotic Dyn., 2013, vol. 18, no. 3, pp. 277–328.MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Borisov, A. V., Mamaev, I. S., and Treschev, D.V, Rolling of a Rigid Body without Slipping and Spinning: Kinematics and Dynamics, J. Appl. Nonlinear Dyn., 2013, vol. 2, no. 2, pp. 161–173.CrossRefMATHGoogle Scholar
  5. 5.
    Borisov, A.V., Kazakov, A.O., and Pivovarova, E.N, Regular and Chaotic Dynamics in the Rubber Model of a Chaplygin Top, Regul. Chaotic Dyn., 2016, vol. 21, nos. 7–8, pp. 885–901.MathSciNetCrossRefGoogle Scholar
  6. 6.
    Ciocci, M.C., Malengier, B., Langerock, B., and Grimonprez, B, Towards a Prototype of a Spherical Tippe Top, J. Appl. Math., 2012, Art. 268537, 34 pp.MathSciNetMATHGoogle Scholar
  7. 7.
    Cohen, C. M, The Tippe Top Revisited, Am. J. Phys., 1977, vol. 45, no. 1, pp. 12–17.CrossRefGoogle Scholar
  8. 8.
    Or, A.C, The Dynamics of a Tippe Top, SIAM J. Appl. Math., 1994, vol. 54, no. 3, pp. 597–609.MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Rauch-Wojciechowski, S., Sköldstam, M., and Glad, T, Mathematical Analysis of the Tippe Top, Regul. Chaotic Dyn., 2005, vol. 10, no. 4, pp. 333–362.MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Zobova, A.A. and Karapetyan, A.V, Analysis of the Steady Motions of the Tippe Top, J. Appl. Math. Mech., 2009, vol. 73, no. 6, pp. 623–630; see also: Prikl. Mat. Mekh., 2009 vol. 73, no. 6, pp. 867–877.MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Moffatt, H.K., Euler’s Disk and Its Finite-Time Singularity, Nature, 2000, vol. 404, no. 6780, pp. 833–834.CrossRefGoogle Scholar
  12. 12.
    Petrie, D., Hunt, J. L., and Gray, C. G, Does the Euler Disk Slip during Its Motion?, Amer. J. Phys., 2002, vol. 70, no. 10, pp. 1025–1028.CrossRefGoogle Scholar
  13. 13.
    Borisov, A.V., Mamaev, I. S., and Karavaev, Yu. L, On the Loss of Contact of the Euler Disk, Nonlinear Dynam., 2015, vol. 79, no. 4, pp. 2287–2294.MathSciNetCrossRefGoogle Scholar
  14. 14.
    Kessler, P. and O’Reilly, O.M, The Ringing of Euler’s Disk, Regul. Chaotic Dyn., 2002, vol. 7, no. 1, pp. 49–60.MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Batista, M, The Nearly Horizontally Rolling of a Thick Disk on a Rough Plane, Regul. Chaotic Dyn., 2008, vol. 13, no. 4, pp. 344–354.MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Batista, M, Integrability of the Motion of a Rolling Disk of Finite Thickness on a Rough Plane, Internat. J. Non-Linear Mech., 2006, vol. 41, pp. 850–859.CrossRefMATHGoogle Scholar
  17. 17.
    Mushtari, Kh. M., Über das Abrollen eines schweren starren Rotationskörpers auf einer unbeweglichen horizontalen Ebene, Mat. Sb., 1932, vol. 39, nos. 1–2, pp. 105–126 (Russian).MATHGoogle Scholar
  18. 18.
    Zobova, A.A, On the Conjugation of Solutions of Two Integrable Problems: Rolling of a Pointed Body on a Plane, Autom. Remote Control, 2007, vol. 68, no. 8, pp. 1438–1443; see also: Avtomat. i Telemekh., 2007 no. 8, pp. 156–162.MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Halme, A., Schonberg, T., and Wang, Y, Motion Control of a Spherical Mobile Robot, in Proc. of the 4th Internat. Workshop on Advanced Motion Control (Mie, Japan, 1996): Vol. 1, pp. 259–264.CrossRefGoogle Scholar
  20. 20.
    Borisov, A. V., Kilin, A.A., and Mamaev, I. S, How to Control Chaplygin’s Sphere Using Rotors, Regul. Chaotic Dyn., 2012, vol. 17, nos. 3–4, pp. 258–272.MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Borisov, A. V., Kilin, A.A., and Mamaev, I. S, How to Control Chaplygin’s Sphere Using Rotors: 2, Regul. Chaotic Dyn., 2013, vol. 18, nos. 1–2, pp. 144–158.MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Ivanova, T. B. and Pivovarova, E. N, Comments on the Paper by A.V. Borisov, A.A.Kilin,I. S. Mamaev “How To Control the Chaplygin Ball Using Rotors: 2”, Regul. Chaotic Dyn., 2014, vol. 19, no. 1, pp. 140–143.MATHGoogle Scholar
  23. 23.
    Svinin, M., Morinaga, A., and Yamamoto, M, On the Dynamic Model and Motion Planning for a Class of Spherical Rolling Robots, in Proc. of the IEEE Internat. Conf. on Robotics and Automation (ICRA, 14–18 May, 2012), pp. 3226–3231.Google Scholar
  24. 24.
    Bizyaev, I.A., Borisov, A.V., and Mamaev, I. S, The Dynamics of Nonholonomic Systems Consisting of a Spherical Shell with a Moving Rigid Body Inside, Regul. Chaotic Dyn., 2014, vol. 19, no. 2, pp. 198–213.MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Bhattacharya, S. and Agrawal, S.K, Design,Experiments and Motion Planning of a Spherical Rolling Robot, in Proc. of the IEEE Internat. Conf. on Robotics and Automation (San Francisco,Calif., USA, 2000): Vol. 2, pp. 1207–1212.Google Scholar
  26. 26.
    Alves, J. and Dias, J, Design and Control of a Spherical Mobile Robot, J. Syst. Control Eng., 2003, vol. 217, pp. 457–467.Google Scholar
  27. 27.
    Karavaev, Yu. L. and Kilin, A.A, The Dynamics and Control of a Spherical Robot with an Internal Omniwheel Platform, Regul. Chaotic Dyn., 2015, vol. 20, no. 2, pp. 134–152.MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Kilin, A.A., Pivovarova, E.N., and Ivanova, T.B, Spherical Robot of Combined Type: Dynamics and Control, Regul. Chaotic Dyn., 2015, vol. 20, no. 6, pp. 716–728.MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Svinin, M., Bai, Y., and Yamamoto, M., Dynamic Model and Motion Planning for a Pendulum-Actuated Spherical Rolling Robot, in Proc. of the 2015 IEEE Internat. Conf. on Robotics and Automation (ICRA), pp. 656–661.Google Scholar
  30. 30.
    Pivovarova, E.N. and Klekovkin, A. V, Influence of Rolling Friction on the ControlledMotion of a Robot Wheel, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2015, vol. 25, no. 4, pp. 583–592 (Russian).CrossRefGoogle Scholar
  31. 31.
    Borisov, A. V. and Mamaev, I. S, The Rolling Motion of a Rigid Body on a Plane and a Sphere: Hierarchy of Dynamics, Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 177–200.MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Chaplygin, S.A, On a Motion of a Heavy Body of Revolution on a Horizontal Plane, Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 119–130.MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Borisov, A.V. and Mamaev, I. S., Rigid Body Dynamics: Hamiltonian Methods, Integrability, Chaos, Izhevsk: R&C Dynamics, Institute of Computer Science, 2005 (Russian).MATHGoogle Scholar
  34. 34.
    Pivovarova, E.N. and Ivanova, T. B, Stability Analysis of Periodic Solutions in the Problem of the Rolling of a Ball with a Pendulum, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2012, no. 4, pp. 146–155 (Russian).CrossRefMATHGoogle Scholar
  35. 35.
    Borisov, A. V., Mamaev, I. S., and Ivanova, T. B, Stability of a Liquid Self-Gravitating Elliptic Cylinder with Intrinsic Rotation, Nelin. Dinam., 2010, vol. 6, no. 4, pp. 807–822 (Russian).CrossRefGoogle Scholar
  36. 36.
    Borisov, A. V., Mamaev, I. S., and Kilin, A.A, Dynamics of Rolling Disk, Regul. Chaotic Dyn., 2003, vol. 8, no. 2, pp. 201–212.MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Cushman, R., Hermans, J., and Kemppainen, D., The Rolling Disc, in Nonlinear Dynamical Systems and Chaos, H. W. Broer, S. A. van Gils, I. Hoveijn, F. Takens (Eds.), Progr. Nonlinear Differential Equations Appl., vol. 19, Basel: Birkhäuser, 1996.Google Scholar
  38. 38.
    Kuleshov, A. S, The Steady Rolling of a Disc on a Rough Plane, J. Appl. Math. Mech., 2001, vol. 65, no. 1, pp. 171–173; see also: Prikl. Mat. Mekh., 2001 vol. 65, no. 1, pp. 173–175.CrossRefMATHGoogle Scholar
  39. 39.
    O’Reilly, O.M, The Dynamics of Rolling Disks and Sliding Disks, Nonlinear Dynam., 1996, vol. 10, no. 3, pp. 287–305.CrossRefGoogle Scholar
  40. 40.
    Kilin, A.A, The Dynamics of Chaplygin Ball: The Qualitative and Computer Analysis, Regul. Chaotic Dyn., 2001, vol. 6, no. 3, pp. 291–306.MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Borisov, A. V., Kilin, A.A., and Mamaev, I. S, The Problem of Drift and Recurrence for the Rolling Chaplygin Ball, Regul. Chaotic Dyn., 2013, vol. 18, no. 6, pp. 832–859.MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Bolsinov, A.V., Borisov, A. V., and Mamaev, I. S, Topology and Stability of Integrable Systems, Russian Math. Surveys, 2010, vol. 65, no. 2, pp. 259–318; see also: Uspekhi Mat. Nauk, 2010 vol. 65, no. 2, pp. 71–132.MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Kozlov, V.V. and Kolesnikov, N.N, On Theorems of Dynamics, J. Appl. Math. Mech., 1978, vol. 42, no. 1, pp. 26–31; see also: Prikl. Mat. Mekh., 1978 vol. 42, no. 1, pp. 28–33.MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Moshchuk, N.K., A Qualitative Analysis of the Motion of a Heavy Solid of Revolution on an Absolutely Rough Plane, J. Appl. Math. Mech., 1988, vol. 52, no. 2, pp. 159–165; see also: Prikl. Mat. Mekh., 1988 vol. 52, no. 2, pp. 203–210.MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Fedorov, Yu.N., On Disk Rolling on Absolutely Rough Surface, Izv. Akad. Nauk SSSR. Mekh. Tverd. Tela, 1987, no. 4, pp. 67–75 (Russian).Google Scholar
  46. 46.
    Cushman, R. H. and Duistermaat, J. J, Nearly Flat Falling Motions of the Rolling Disk, Regul. Chaotic Dyn., 2006, vol. 11, no. 1, pp. 31–60.MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Afonin, A.A. and Kozlov, V.V, Problem on Falling of Disk Moving on Horizontal Plane, Izv. Ross. Akad. Nauk. Mekh. Tverd. Tela, 1997, no. 1, pp. 7–13 (Russian).Google Scholar
  48. 48.
    Kozlov, V.V, Motion of a Disk on an Inclined Plane, Izv. Ross. Akad. Nauk. Mekh. Tverd. Tela, 1996, no. 5, pp. 29–35 (Russian).Google Scholar
  49. 49.
    Borisov, A. V., Kilin, A.A., and Karavaev, Yu. L, On the Retrograde Motion of a Rolling Disk, Physics–Uspekhi, 2017 (accepted).Google Scholar
  50. 50.
    Jalali, M. A., Sarebangholi, M. S., and Alam, M.-R., Terminal Retrograde Turn of Rolling Rings, Phys. Rev. E, 2015, vol. 92, no. 3, 032913, 5 pp.CrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Udmurt State UniversityIzhevskRussia
  2. 2.Institute of Mathematics and Mechanics of the Ural Branch of RASEkaterinburgRussia

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