Advertisement

Regular and Chaotic Dynamics

, Volume 23, Issue 7–8, pp 887–907 | Cite as

Integrable Nonsmooth Nonholonomic Dynamics of a Rubber Wheel with Sharp Edges

  • Alexander A. KilinEmail author
  • Elena N. Pivovarova
Article
  • 1 Downloads

Abstract

This paper is concerned with the dynamics of a wheel with sharp edges moving on a horizontal plane without slipping and rotation about the vertical (nonholonomic rubber model). The wheel is a body of revolution and has the form of a ball symmetrically truncated on both sides. This problem is described by a system of differential equations with a discontinuous right-hand side. It is shown that this system is integrable and reduces to quadratures. Partial solutions are found which correspond to fixed points of the reduced system. A bifurcation analysis and a classification of possible types of the wheel’s motion depending on the system parameters are presented.

Keywords

integrable system system with a discontinuous right-hand side nonholonomic constraint bifurcation diagram body of revolution sharp edge wheel rubber model 

MSC2010 numbers

70E15 70E18 70E40 37Jxx 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    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.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Routh, E. J., A Treatise on the Dynamics of a System of Rigid Bodies: P. 2. The Advanced Part, 6th ed., New York: Macmillan, 1905; see also: New York: Dover, 1955 (reprint).zbMATHGoogle Scholar
  3. 3.
    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).zbMATHGoogle Scholar
  4. 4.
    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.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Astapov, I. S., On Rotational Stability of Celtic Stone, Vestn. Mosk. Univ. Ser. 1. Mat. Mekh., 1980, no. 2, pp. 97–100 (Russian).zbMATHGoogle Scholar
  6. 6.
    Karapetyan, A.V., On Realizing Nonholonomic Constraints by Viscous Friction Forces and Celtic Stones Stability, J. Appl. Math. Mech., 1981, vol. 45, no. 1, pp. 30–36; see also: Prikl. Mat. Mekh., 1981, vol. 45, no. 1, pp. 42–51.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Markeev, A.P., The Dynamics of a Rigid Body on an Absolutely Rough Plane, J. Appl. Math. Mech., 1983, vol. 47, no. 4, pp. 473–478; see also: Prikl. Mat. Mekh., 1983, vol. 47, no. 4, pp. 575–582.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Borisov, A.V., Kazakov, A.O., and Kuznetsov, S.P., Nonlinear Dynamics of the Rattleback: A Nonholonomic Model, Physics–Uspekhi, 2014, vol. 57, no. 5, pp. 453–460; see also: Uspekhi Fiz. Nauk, 2014, vol. 184, no. 5, pp. 493–500.Google Scholar
  9. 9.
    Borisov, A. V., Kilin, A.A., and Mamaev, I. S., New Effects in Dynamics of Rattlebacks, Dokl. Phys., 2006, vol. 51, no. 5, pp. 272–275; see also: Dokl. Akad. Nauk, 2006, vol. 408, no. 2, pp. 192–195.CrossRefzbMATHGoogle Scholar
  10. 10.
    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.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    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
  12. 12.
    Borisov, A. V., Kilin, A.A., and Karavaev, Yu. L., On the Retrograde Motion of a Rolling Disk, Physics-Uspekhi, 2017, vol. 60, no. 9, pp. 931–934; see also: Uspekhi Fiz. Nauk, 2017, vol. 60, no. 9, pp. 1003–1006.CrossRefGoogle Scholar
  13. 13.
    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.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    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.CrossRefzbMATHGoogle Scholar
  15. 15.
    Cendra, H. and Etchechoury, M., Rolling of a Symmetric Sphere on a Horizontal Plane without Sliding or Spinning, Rep. Math. Phys., 2006, vol. 57, no. 3, pp. 367–374.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    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
  17. 17.
    Pacejka, H., Tire and Vehicle Dynamics, 3rd ed., Oxford: Butterworth/Heinemann, 2012.Google Scholar
  18. 18.
    Neimark, Ju. I. and Fufaev, N.A., Dynamics of Nonholonomic Systems, Trans. Math. Monogr., vol. 33, Providence,R.I.: AMS, 2004.Google Scholar
  19. 19.
    Ivanova, T. B., Kilin, A.A., and Pivovarova, E. N., Controlled Motion of a Spherical Robot with Feedback: 1, J. Dyn. Control Syst., 2018, vol. 24, no. 3, pp. 497–510.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Ivanova, T. B., Kilin, A.A., and Pivovarova, E. N., Controlled Motion of a Spherical Robot of Pendulum Type on an Inclined Plane, Dokl. Phys., 2018, vol. 63, no. 7, pp. 302–306; see also: Dokl. Akad. Nauk, 2018, vol. 481, no. 3, pp. 258–263.CrossRefGoogle Scholar
  21. 21.
    Karavaev, Yu. L. and Kilin, A.A., Nonholonomic Dynamics and Control of a Spherical Robot with an Internal Omniwheel Platform: Theory and Experiments, Proc. Steklov Inst. Math., 2016, vol. 295, pp. 158–167; see also: Tr. Mat. Inst. Steklova, 2016, vol. 295, pp. 174–183.MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    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.MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Mobile Robots: Ball-Shaped Robot and Wheel Robot, A.V. Borisov, I. S.Mamaev, Yu. L. Karavaev (Eds.), Izhevsk: R&C Dynamics, Institute of Computer Science, 2013 (Russian).Google Scholar
  24. 24.
    Martynenko, Yu. G., Lenskii, A. V., and Kobrin, A. I., Decomposition of the Problem of Controlling a Mobile One-Wheel Robot with an Unperturbed Gyrostabilized Platform, Dokl. Phys., 2002, vol. 47, no. 10, pp. 772–774; see also: Dokl. Akad. Nauk, 2002, vol. 386, no. 6, pp. 767–769.MathSciNetCrossRefGoogle Scholar
  25. 25.
    Xu, Y. and Ou, Y., Control of Single Wheel Robots, Springer Tracts in Advanced Robotics, vol. 20, Berlin: Springer, 2005.Google Scholar
  26. 26.
    Moffatt, H. K., Euler’s Disk and Its Finite-Time Singularity, Nature, 2000, vol. 404, no. 6780, pp. 833–834.CrossRefGoogle Scholar
  27. 27.
    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
  28. 28.
    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
  29. 29.
    Kessler, P. and O’Reilly, O.M., The Ringing of Euler’s Disk, Regul. Chaotic Dyn., 2002, vol. 7, no. 1, pp. 49–60.MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    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.CrossRefzbMATHGoogle Scholar
  31. 31.
    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.Google Scholar
  32. 32.
    Cohen, C. M., The Tippe Top Revisited, Am. J. Phys., 1977, vol. 45, no. 1, pp. 12–17.CrossRefGoogle Scholar
  33. 33.
    Leine, R. I. and Gloker, Ch., A Set-Valued Force Law for Spatial Coulomb–Contensou Friction, Eur. J. Mech. A Solids, 2003, vol. 22, no. 2, pp. 193–216.MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Or, A.C., The Dynamics of a Tippe Top, SIAM J. Appl. Math., 1994, vol. 54, no. 3, pp. 597–609.MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    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.MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    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.MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    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.MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Kilin, A.A. and Pivovarova, E.N., The Rolling Motion of a Truncated Ball Without Slipping and Spinning on a Plane, Regul. Chaotic Dyn., 2017, vol. 22, no. 3, pp. 298–317.MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    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.CrossRefzbMATHGoogle Scholar
  40. 40.
    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.MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Borisov, A.V. and Mamaev, I. S., Isomorphism and Hamilton Representation of Some Nonholonomic Systems, Siberian Math. J., 2007, vol. 48, no. 1, pp. 26–36; see also: Sibirsk. Mat. Zh., 2007, vol. 48, no. 1, pp. 33–45.MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Borisov, A. V., Mamaev, I. S., and Bizyaev, I. A., The Jacobi Integral in Nonholonomic Mechanics, Regul. Chaotic Dyn., 2015, vol. 20, no. 3, pp. 383–400.MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Borisov, A. V., Fedorov, Yu. N., and Mamaev, I. S., Chaplygin Ball over a Fixed Sphere: An Explicit Integration, Regul. Chaotic Dyn., 2008, vol. 13, no. 6, pp. 557–571.MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Borisov, A. V., Kilin, A.A., and Mamaev, I. S., Hamiltonicity and Integrability of the Suslov Problem, Regul. Chaotic Dyn., 2011, vol. 16, nos. 1–2, pp. 104–116.MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Borisov, A. V., Kilin, A.A., and Mamaev, I. S., Generalized Chaplygin’s Transformation and Explicit Integration of a System with a Spherical Support, Regul. Chaotic Dyn., 2012, vol. 17, no. 2, pp. 170–190.MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Borisov, A.V. and Mamaev, I. S., Rigid Body Dynamics: Hamiltonian Methods, Integrability, Chaos, Izhevsk: R&C Dynamics, Institute of Computer Science, 2005 (Russian).zbMATHGoogle Scholar
  47. 47.
    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).CrossRefzbMATHGoogle Scholar
  48. 48.
    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
  49. 49.
    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.MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    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, pp. 21–60.MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    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.MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    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.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia

Personalised recommendations