Skip to main content
SpringerLink
Log in

On the loss of contact of the Euler disk

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. Bendik, J.: The official Euler’s disk website. http://www.eulerdisk.com (2000). Accessed 14 June 2013

  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)

    Article  Google Scholar 

  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. Korteweg, D.: Extrait d’une lettre a M. Appel. Rendiconti del circolo matematico di Palermo. 14, 7–8 (1900)

    Article  MATH  Google Scholar 

  5. Borisov, A.V., Mamaev, I.S., Kilin, A.A.: Dynamic of rolling disk. Regul. Chaotic Dyn. 8(2), 201–212 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  8. Moffatt, H.K.: Euler’s disk and its finite-time singularity. Nature 404, 833–834 (2000)

    Article  Google Scholar 

  9. van den Engh, G., Nelson, P., Roach, J.: Numismatic gyrations. Nature 408, 540 (2000)

    Article  Google Scholar 

  10. Bildsten, L.: Viscous dissipation for Euler’s disk. Phys. Rev. E. 66(2), 056309 (2002)

    Article  MathSciNet  Google Scholar 

  11. Villanueva, R., Epstein, M.: Vibrations of Euler’s disk. Phys. Rev. E. 71(7), 066609 (2005)

    Article  Google Scholar 

  12. Kessler, P., O’Reilly, O.M.: The Ringing of Euler’s disk. Reg. Chaotic Dyn. 7(1), 49–60 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  13. O’Reilly, O.M.: The dynamics of rolling disks and sliding disks. Nonlinear Dyn. 10, 287–305 (1996)

    Article  MathSciNet  Google Scholar 

  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. 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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  17. Stanislavsky, A.A., Weron, K.: Nonlinear oscillations in the rolling motion of Euler’s disk. Physica D. 156(10), 247259 (2001)

    MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  19. Leine R.L.: Measurements of the finite-time singularity of the Euler disk. In: 7th EUROMECH Solid Mechanics Conference, Lisbon, Portugal (2009)

  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)

    Article  MATH  Google Scholar 

  21. Saje, M., Zupan, D.: The rattling of Euler’s disk. Multidiscip. Model. Mater. Struct. 2(1), 49–66 (2006)

    Article  Google Scholar 

  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)

  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

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  25. Batista, M.: The nearly horizontally rolling of a thick disk on a rough plane. Regul. Chaotic Dyn. 13(4), 344354 (2008)

  26. Batista, M.: Self-induced jumping of a rigid body of revolution on a smooth horizontal surface. NonLinear Mech. 43, 26–35 (2008)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

  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. Torrence, C., Compo, G. P.: A practical guide to wavelet analysis. Bull. Am. Meteorol. Soc. 79(1), 61–78 (1998)

Download references

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.

Author information

Authors and Affiliations

  1. Laboratory of Nonlinear Analysis and the Design of New Types of Vehicles, Institute of Computer Science, Udmurt State University, Universitetskaya 1, Izhevsk, 426034, Russia

    Alexey V. Borisov & Ivan S. Mamaev

  2. Deparment of Mechatronic Systems, Kalashnikov Izhevsk State Technical University, Studencheskaya st. 7, Izhevsk, 426069, Russia

    Yury L. Karavaev

Authors
  1. Alexey V. Borisov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ivan S. Mamaev
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yury L. Karavaev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yury L. Karavaev.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Borisov, A.V., Mamaev, I.S. & Karavaev, Y.L. On the loss of contact of the Euler disk. Nonlinear Dyn 79, 2287–2294 (2015). https://doi.org/10.1007/s11071-014-1811-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-014-1811-5

Keywords

Search

Navigation