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Nonlinear Dynamics

, Volume 97, Issue 2, pp 1635–1648 | Cite as

Nonholonomic rolling of a ball on the surface of a rotating cone

  • Alexey V. BorisovEmail author
  • Tatiana B. Ivanova
  • Alexander A. Kilin
  • Ivan S. Mamaev
Original paper
  • 57 Downloads

Abstract

This paper investigates the rolling without slipping of a homogeneous heavy ball on the surface of a rotating cone in two settings: without dissipation in a nonholonomic setting and with rolling friction torque which is proportional to the angular velocity of the ball. In the nonholonomic setting, the resulting system of five differential equations on the level set of first integrals is reduced to quadratures. A bifurcation analysis of the above system is carried out to determine the possible types of motion. In the second case, it is shown that there are not only trajectories emanating from the lower point of the cone (its vertex), but also trajectories to the vertex of the cone (fall). An analysis of the dependence of the type of terminal motion of the center of mass of the ball on initial conditions is carried out.

Keywords

Rotating surface Cone Nonholonomic constraint Rolling friction Jacobi integral Bifurcation analysis 

Notes

Funding

The work of A. V. Borisov and T. B. Ivanova (Sect. 2) was supported by Grant 15-12-20035 of the Russian Science Foundation. The work of I. S. Mamaev and A. A. Kilin (Sect. 3) was carried out at MIPT within the framework of Project 5-100 for state support for leading universities of the Russian Federation.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    Borisov, A.V., Mamaev, I.S., Kilin, A.A.: The rolling motion of a ball on a surface: new integrals and hierarchy of dynamics. Regul. Chaotic Dyn. 7(2), 201–219 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Borisov, A.V., Mamaev, I.S., Bizyaev, I.A.: The Jacobi integral in nonholonomic mechanics. Regul. Chaotic Dyn. 20(3), 383–400 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Borisov, A.V., Ivanova, T.B., Karavaev, Y.L., Mamaev, I.S.: Theoretical and experimental investigations of the rolling of a ball on a rotating plane (turntable). Eur. J. Phys. 39(6), 065001 (2018)CrossRefGoogle Scholar
  4. 4.
    Earnshaw, S.: Dynamics, or An Elementary Treatise on Motion, 3rd edn. Deighton, Cambridge (1844)Google Scholar
  5. 5.
    Routh, E.J.: Dynamics of a System of Rigid Bodies. MacMillan, London (1891)zbMATHGoogle Scholar
  6. 6.
    Weltner, K.: Movement of spheres on rotating discs—a new method to measure coefficients of rolling friction by the central drift. Mech. Res. Commun. 10(4), 223–232 (1983)CrossRefzbMATHGoogle Scholar
  7. 7.
    Zengel, K.: The electromagnetic analogy of a ball on a rotating conical turntable. Am. J. Phys. 85(12), 901–907 (2017)CrossRefGoogle Scholar
  8. 8.
    Gary, D.: White: on trajectories of rolling marbles in cones and other funnels. Am. J. Phys. 81(12), 890–898 (2013)CrossRefGoogle Scholar
  9. 9.
    English, L.Q., Mareno, A.: Trajectories of rolling marbles on various funnels. Am. J. Phys. 80(11), 996–1000 (2012)CrossRefGoogle Scholar
  10. 10.
    Moeckel, R.: Embedding the Kepler problem as a surface of revolution. Regul. Chaotic Dyn. 23(6), 695–703 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Contensou, P.: Couplage entre frottement de glissement et frottement de pivotement dans la theorie de la toupie. Kreiselprobleme Gydrodynamics: IUTAM Symp. Celerina, pp. 201–216. Springer, Berlin (1963)Google Scholar
  12. 12.
    Sokirko, A.V., Belopolskii, A.A., Matytsyn, A.V., Kossalkowski, D.A.: Behavior of a ball on the surface of a rotating disk. Am. J. Phys. 62(2), 151–156 (1994)CrossRefGoogle Scholar
  13. 13.
    Ehrlich, R., Tuszynski, J.: Ball on a rotating turntable: comparison of theory and experiment. Am. J. Phys. 63(4), 351–359 (1995)CrossRefGoogle Scholar
  14. 14.
    Lewis, A.D., Murray, R.M.: Murrayy Variational principles for constrained systems: theory and experiment. Int. J. Non-Linear Mech. 30(6), 793–815 (1995)CrossRefzbMATHGoogle Scholar
  15. 15.
    Borisov, A.V., Mamaev, I.S.: Conservation laws, hierarchy of dynamics and explicit integration of nonholonomic systems. Regul. Chaotic Dyn. 13(5), 443–490 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Borisov, A.V., Kilin, A.A., Mamaev, I.S.: A nonholonomic model of the paul trap. Regul. Chaotic Dyn. 23(3), 339–354 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Fasso, F., Sansonetto, N.: Conservation of ‘moving’ energy in nonholonomic systems with affine constraints and integrability of spheres on rotating surfaces. J. Nonlinear Sci. 26(2), 519–544 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Arnol’d, V.I., Kozlov, V.V., Neishtadt, A.I.: Mathematical Aspects of Classical and Celestial Mechanics. Encyclopaedia of Mathematical Sciences, vol. 3, 3rd edn. Springer, Berlin (2006)CrossRefGoogle Scholar
  19. 19.
    Bolsinov, A.V., Borisov, A.V., Mamaev, I.S.: Hamiltonization of nonholonomic systems in the neighborhood of invariant manifolds. Regul. Chaotic Dyn. 16(5), 443–464 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Borisov, A.V., Kilin, A.A., Mamaev, I.S.: On a nonholonomic dynamical problem. Math. Notes. 79(5–6), 734–740 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Vitolo, R., Broer, H., Simo, C.: Routes to chaos in the Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms. Nonlinearity 23(8), 1919–1947 (2010)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Moscow Institute of Physics and TechnologyDolgoprudnyiRussia
  2. 2.Udmurt State UniversityIzhevskRussia

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