Ball on a viscoelastic plane

  • A. A. Zobova
  • D. V. Treschev


We consider dynamical problems arising in connection with the interaction of an absolutely rigid ball and a viscoelastic support plane. The support is a relatively stiff viscoelastic Kelvin-Voigt medium that coincides with the horizontal plane in the undeformed state. We also assume that under the deformation the support induces dry friction forces that are locally governed by the Coulomb law. We study the impact appearing when a ball falls on the plane. Another problem of our interest is the motion of a ball “along the plane.” A detailed analysis of various stages of the motion is presented. We also compare this model with classical models of interaction of solid bodies.


Angular Velocity Friction Force STEKLOV Institute Mass Center Slip Velocity 
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  1. 1.
    V. V. Andronov and V. Ph. Zhuravlev, Dry Friction in Problems of Mechanics (Regular and Chaotic Dynamics, Izhevsk, 2010) [in Russian].Google Scholar
  2. 2.
    L. A. Galin, Contact Problems: The Legacy of L.A. Galin, Ed. by G. M. L. Gladwell (Springer, Dordrecht, 2008).Google Scholar
  3. 3.
    I. G. Goryacheva, Mechanics of Frictional Interaction (Nauka, Moscow, 2001) [in Russian].Google Scholar
  4. 4.
    A. Yu. Ishlinsky, “Rolling friction,” Prikl. Mat. Mekh. 2(2), 245–260 (1938).Google Scholar
  5. 5.
    A. Yu. Ishlinsky, “Theory of resistance to rolling (rolling friction) and related phenomena,” in All-Union Conf. on Friction and Wear in Machines (Akad. Nauk SSSR, Moscow, 1940), Vol. 2, pp. 255–264 [in Russian].Google Scholar
  6. 6.
    A. Yu. Ishlinsky, “On partial slip in rolling contact,” Izv. Akad. Nauk SSSR, Otd. Tekhn. Nauk, No. 6, 3–15 (1956).Google Scholar
  7. 7.
    M. V. Ishkhanyan and A. V. Karapetyan, “Dynamics of a homogeneous ball on a horizontal plane with sliding, spinning, and rolling friction taken into account,” Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, No. 2, 3–14 (2010) [Mech. Solids 45, 155–165 (2010)].Google Scholar
  8. 8.
    A. V. Karapetyan, “A two-parameter friction model,” Prikl. Mat. Mekh. 73(4), 515–519 (2009) [J. Appl. Math. Mech. 73, 367–370 (2009)].MathSciNetGoogle Scholar
  9. 9.
    A. V. Karapetyan, “Modelling of frictional forces in the dynamics of a sphere on a plane,” Prikl. Mat. Mekh. 74(4), 531–535 (2010) [J. Appl. Math. Mech. 74, 380–383 (2010)].MathSciNetGoogle Scholar
  10. 10.
    A. A. Kireenkov, “Coupled models of sliding and rolling friction,” Dokl. Akad. Nauk 419(6), 759–762 (2008) [Dokl. Phys. 53, 233–236 (2008)].MathSciNetGoogle Scholar
  11. 11.
    A. S. Kuleshov, D. V. Treschev, T. B. Ivanova, and O. S. Naimushina, “A rigid cylinder on a viscoelastic plane,” Nelinein. Din. 7(3), 601–625 (2011).Google Scholar
  12. 12.
    T. Levi-Civita and U. Amaldi, Lezioni di meccanica razionale (N. Zanichelli, Bologna, 1951, 1952), Vol. 2.zbMATHGoogle Scholar
  13. 13.
    A. P. Markeev, Dynamics of a Body Touching a Rigid Surface (Regular and Chaotic Dynamics, Izhevsk, 2011) [in Russian].Google Scholar
  14. 14.
    Nonholonomic Dynamical Systems: Integrability, Chaos, and Strange Attractors, Ed. by A. V. Borisov and I. S. Mamaev (Inst. Comput. Res., Moscow, 2002) [in Russian].Google Scholar
  15. 15.
    Ju. I. Neimark and N. A. Fufaev, Dynamics of Nonholonomic Systems (Am. Math. Soc., Providence, RI, 1972), Transl. Math. Monogr. 33.zbMATHGoogle Scholar
  16. 16.
    E. J. Routh, A Treatise on the Dynamics of a System of Rigid Bodies (Macmillan, London, 1905), Parts 1, 2.zbMATHGoogle Scholar
  17. 17.
    O. S. Sentemova, “Multicomponent models of friction,” Vestn. Mosk. Univ., Ser. 1: Mat., Mekh., No. 6, 57–59 (2011) [Moscow Univ. Mech. Bull. 66 (6), 138–140 (2011)].Google Scholar
  18. 18.
    S. A. Chaplygin, Studies on the Dynamics of Nonholonomic Systems (Gostekhteorizdat, Moscow, 1949); 2nd ed. (URSS, Moscow, 2007) [in Russian].Google Scholar
  19. 19.
    F. Al-Bender and K. De Moerlooze, “Characterization and modeling of friction and wear: an overview,” Sustainable Constr. Des. 2(1), 19–28 (2011).Google Scholar
  20. 20.
    P. Contensou, “Couplage entre frottement de glissement et frottement de pivotement dans la théorie de la toupie,” in Kreiselprobleme/Gyrodynamics: Proc. Symp., Celerina, 1962 (Springer, Berlin, 1963), pp. 201–216.Google Scholar
  21. 21.
    Th. Erismann, “Theorie und Anwendungen des echten Kugelgetriebes,” Z. Angew. Math. Phys. 5(5), 355–388 (1954).MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    H. Hertz, “Ueber die Berührung fester elastischer Körper,” J. Reine Angew. Math. 92, 156–171 (1882).zbMATHGoogle Scholar
  23. 23.
    T. Pöschel, N. V. Brilliantov, and A. Zaikin, “Bistability and noise-enhanced velocity of rolling motion,” Europhys. Lett. 69, 371–377 (2005).CrossRefGoogle Scholar
  24. 24.
    T. Pöschel, T. Schwager, and N. V. Brilliantov, “Rolling friction of a hard cylinder on a viscous plane,” Eur. Phys. J. B 10, 169–174 (1999).CrossRefGoogle Scholar
  25. 25.
    T. Pöschel, T. Schwager, N. V. Brilliantov, and A. Zaikin, “Rolling friction and bistability of rolling motion,” in Powders and Grains 2005: Proc. 5th Int. Conf. on Micromechanics of Granular Media, Stuttgart, 2005 (Taylor & Francis, London, 2005), Vol. 2, pp. 1247–1253.Google Scholar

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© Pleiades Publishing, Ltd. 2013

Authors and Affiliations

  • A. A. Zobova
    • 1
  • D. V. Treschev
    • 2
  1. 1.Faculty of Mechanics and MathematicsMoscow State UniversityMoscowRussia
  2. 2.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia

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