Advertisement

Nonlinear Dynamics

, Volume 79, Issue 4, pp 2287–2294 | Cite as

On the loss of contact of the Euler disk

  • Alexey V. Borisov
  • Ivan S. Mamaev
  • Yury L. Karavaev
Original Paper

Abstract

This paper is an experimental investigation of a round uniform disk rolling on a horizontal surface. Two methods for experimentally determining the loss of contact of the rolling disk from the horizontal surface before its stop are proposed. Results of experiments for disks having different masses and manufactured from different materials are presented. Causes of “microlosses of contact” detected in the processes of motion are discussed.

Keywords

Euler’s disk Loss of contact Experiment 

Notes

Acknowledgments

The authors thank A. Ruina, A.P. Ivanov, and D.V.Treshev for useful discussions, and S.R. Gallyamov and S.A. Trefilov for technical consultation.

References

  1. 1.
    Bendik, J.: The official Euler’s disk website. http://www.eulerdisk.com (2000). Accessed 14 June 2013
  2. 2.
    Appel, P.: Sur l’integration des equations du mouvement d’un corps pesant de redolution roulant par une arete circulaire sur up plan horizontal; cas parculier du cerceau. Rendiconti del circolo matematico di Palermo. 14, 1–6 (1900)CrossRefGoogle Scholar
  3. 3.
    Chaplygin, S.A.: On motion of heavy rigid body of revolution on horizontal plane. Proc. Phys. Sci. Sect. Soc. Amat. Nat. Sci. 9(1), 10–16 (1897)Google Scholar
  4. 4.
    Korteweg, D.: Extrait d’une lettre a M. Appel. Rendiconti del circolo matematico di Palermo. 14, 7–8 (1900)CrossRefzbMATHGoogle Scholar
  5. 5.
    Borisov, A.V., Mamaev, I.S., Kilin, A.A.: Dynamic of rolling disk. Regul. Chaotic Dyn. 8(2), 201–212 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Borisov, A.V., Mamaev, I.S.: The rolling motion of a rigid body on a plane and a sphere. Hierarchy of dynamics. Reg. Chaotic Dyn. 7(2), 177–200 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    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 and a sphere. Reg. Chaotic Dyn. 18(3), 277–328 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Moffatt, H.K.: Euler’s disk and its finite-time singularity. Nature 404, 833–834 (2000)CrossRefGoogle Scholar
  9. 9.
    van den Engh, G., Nelson, P., Roach, J.: Numismatic gyrations. Nature 408, 540 (2000)CrossRefGoogle Scholar
  10. 10.
    Bildsten, L.: Viscous dissipation for Euler’s disk. Phys. Rev. E. 66(2), 056309 (2002)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Villanueva, R., Epstein, M.: Vibrations of Euler’s disk. Phys. Rev. E. 71(7), 066609 (2005)CrossRefGoogle Scholar
  12. 12.
    Kessler, P., O’Reilly, O.M.: The Ringing of Euler’s disk. Reg. Chaotic Dyn. 7(1), 49–60 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    O’Reilly, O.M.: The dynamics of rolling disks and sliding disks. Nonlinear Dyn. 10, 287–305 (1996)CrossRefMathSciNetGoogle Scholar
  14. 14.
    McDonald A.J., McDonald K.T.: The rolling motion of a disk on a horizontal plane. Preprint Archive, Los Alamos National Laboratory. arXiv: physics/008227(2000)
  15. 15.
    Le, Saux C., Leine, R.L., Glocker, C.: Dynamics of a rolling disk in the presence of dry friction. J. Nonlinear Sci. 15, 27–61 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Caps, H., Dorbolo, S., Ponte, S., Croisier, H., Vandewalle, N.: Rolling and slipping motion of Euler’s disk. Phys. Rev. E. 69(6), 056610 (2004)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Stanislavsky, A.A., Weron, K.: Nonlinear oscillations in the rolling motion of Euler’s disk. Physica D. 156(10), 247259 (2001)MathSciNetGoogle Scholar
  18. 18.
    Easwar, K., Rouyer, F., Menon, N.: Speeding to a stop: the finite-time singularity of a spinning disk. Phys. Rev. E. 66(3), 045102 (2002)CrossRefGoogle Scholar
  19. 19.
    Leine R.L.: Measurements of the finite-time singularity of the Euler disk. In: 7th EUROMECH Solid Mechanics Conference, Lisbon, Portugal (2009)Google Scholar
  20. 20.
    Leine, R.L.: Experimental and theoretical investigation of the energy dissipation of a rolling disk during its final stage of motion. Arch. Appl. Mech. 79, 1063–1082 (2009)CrossRefzbMATHGoogle Scholar
  21. 21.
    Saje, M., Zupan, D.: The rattling of Euler’s disk. Multidiscip. Model. Mater. Struct. 2(1), 49–66 (2006)CrossRefGoogle Scholar
  22. 22.
    Petrie, D., Hunt, J.L., Gray, C.G.: Does the Euler disk slip during its motion? Am. J. Phys. 70(10), 1025–1028 (2002)Google Scholar
  23. 23.
    Mitsui, T., Aihara, K., Terayama, C., Kobayashi, H., Shimomura, Y.: Can a spinning egg really jump? Proc. R. Soc. A. 462, 2897–2905 (2006). doi: 10.1098/rspa.2006.1718 CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Branicki, M., Shimomura, Y.: Dynamics of an axisymmetric body spinning on a horizontal surface. IV. Stability of steady spin states and the ’rising egg’ phenomenon for convex axisymmetric bodies. Proc. R. Soc. A. 462, 3253–3275 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Batista, M.: The nearly horizontally rolling of a thick disk on a rough plane. Regul. Chaotic Dyn. 13(4), 344354 (2008)Google Scholar
  26. 26.
    Batista, M.: Self-induced jumping of a rigid body of revolution on a smooth horizontal surface. NonLinear Mech. 43, 26–35 (2008)CrossRefzbMATHGoogle Scholar
  27. 27.
    Ivanov, A.P.: On detachment conditions in the problem on the motion of a rigid body on a rough plane. Reg. Chaotic Dyn. 13(4), 355–368 (2008)CrossRefzbMATHGoogle Scholar
  28. 28.
    Go, D.B., Pohlman, D.A.: A mathematical model of the modified Paschen’s curve for breakdown in microscale gaps. J. Appl. Phys. 107, 103303 (2010)Google Scholar
  29. 29.
    Albert J., Levit W., Levit L.: Electrical breakdown and ESD phenomena for devices with nanometer-to-micron gaps. In: Proc. SPIE 4980, Reliability, Testing, and Characterization of MEMS/MOEMS II, 87. (2003). doi: 10.1117/12.478191
  30. 30.
    Torrence, C., Compo, G. P.: A practical guide to wavelet analysis. Bull. Am. Meteorol. Soc. 79(1), 61–78 (1998)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Alexey V. Borisov
    • 1
  • Ivan S. Mamaev
    • 1
  • Yury L. Karavaev
    • 2
  1. 1.Laboratory of Nonlinear Analysis and the Design of New Types of Vehicles, Institute of Computer ScienceUdmurt State UniversityIzhevskRussia
  2. 2.Deparment of Mechatronic SystemsKalashnikov Izhevsk State Technical UniversityIzhevskRussia

Personalised recommendations