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Continuous rolling motion of a disk on a vibrating plate

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Abstract

We studied the behavior of the continuous rolling motion (CRM) of a disk placed on a vibrating plate observed in the experiment using numerical simulations. Numerical simulations show that a rolling disk on a vibrating plate abruptly stops in case of pure rolling without slipping, whereas CRM occurs in the case of slipping. CRM occurs in two frequency bands separated by a gap. We use numerical simulations to determine the gap and the frequency domains for different values of the coefficient of sliding friction. The characteristics of rolling motion depend on the coefficient of slip friction and frequency of vibration.

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References

  1. Bendik, J.: The official Euler’s disk website. http://www.eulersdisk.com (2000). Accessed 28 Feb 2020

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

    Article  Google Scholar 

  3. Easwar, K., Rouyer, F., Menon, N.: Speeding to a stop: the finite-time singularity of a spinning disk. Phys. Rev. E 66(3), 045102(R) (2002)

    Article  Google Scholar 

  4. Caps, H., Dorbolo, S., Ponte, S., Croisier, H., Vandewalle, N.: Rolling and slipping motion of Euler’s disk. Phys. Rev. E 69, 056610 (2004)

    Article  MathSciNet  Google Scholar 

  5. McDonald, A.J., McDonald, K.T.: The rolling motion of a disk on a horizontal plane, Preprint Archive. Los Alamos National Laboratory. arXiv:Physics/0008227 (2000)

  6. Kessler, P., O’Reilly, O.M.: The ringing of Euler’s disk. Regul. Chaotic Dyn. 7, 49–60 (2002)

    Article  MathSciNet  Google Scholar 

  7. O’Reilly, O.M.: The dynamics of rolling disks and sliding disks. Nonlinear Dyn. 10(3), 287–305 (1996)

    Article  Google Scholar 

  8. Stanislavsky, A.A., Weron, K.: Nonlinear oscillations in the rolling motion of Euler’s disk. Physica D 156, 247–259 (2001)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  10. Le Saux, C., Leine, R.I., Glocker, C.: Dynamics of a rolling disk in the presence of dry friction. J. Nonlinear Sci. 15, 27–61 (2005)

    Article  MathSciNet  Google Scholar 

  11. Leine, R.I., Le Saux, C., Glocker, C.: Friction models for the rolling disk. In: Proceedings of the ENOC: conference. Eindhoven, CD-ROM (2005)

  12. Leine, R.I.: 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  Google Scholar 

  13. Ma, D., Liu, C., Zhao, Z., Zhang, H.: Rolling friction and energy dissipation in a spinning disc. Proc. R. Soc. A 470, 20140191 (2014)

    Article  Google Scholar 

  14. Dorbolo, S., Volfson, D., Tsimring, L., Kudrolli, L.: Dynamics of a bouncing dimer. Phys. Rev. Lett. 95, 044101 (2005)

    Article  Google Scholar 

  15. Kubo, Y., Inagaki, S., Ichikawa, M., Yoshikawa, K.: Mode bifurcation of a bouncing dumbbell with chirality. Phys. Rev. E 91, 052905 (2015)

    Article  MathSciNet  Google Scholar 

  16. Zhuravlev, V.G.: The model of dry friction in the problem of the rolling of rigid bodies. J. Appl. Math. Mech. 62, 705–710 (1998)

    Article  MathSciNet  Google Scholar 

  17. Leine, R.I., Glocker, C.: A set-valued force law for spatial Coulomb–Contensou friction. Eur. J. Mech. A Solids 22, 193–215 (2003)

    Article  MathSciNet  Google Scholar 

  18. Kireenkov, A.A.: Combined model of sliding and rolling friction in dynamics of bodies on a rough plane. Mech. Solids 43, 412–425 (2008)

    Article  Google Scholar 

  19. Kudra, G., Awrejcewicz, J.: Approximate modelling of resulting dry friction forces and rolling resistence for elliptic contact shape. Eur. J. Mech. A Solids 42, 358–375 (2013)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Educational Studies on Global-ICT-Teaching and Learning, Joetsu University of Education, Yamayashiki, Joetsu, Niigata, 943-8512, Japan

    Hiroshi Takano

  2. Faculty of Life and Medical Sciences, Doshisha University, Kyotanabe, Kyoto, 610-0394, Japan

    Satoshi Takatori

  3. Faculty of Biology-Oriented Science and Technology, Kindai University, Kinokawa, Wakayama, 649-6493, Japan

    Takatoshi Ichino

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  1. Hiroshi Takano
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  2. Satoshi Takatori
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  3. Takatoshi Ichino
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Correspondence to Hiroshi Takano.

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Takano, H., Takatori, S. & Ichino, T. Continuous rolling motion of a disk on a vibrating plate. Nonlinear Dyn 100, 2205–2214 (2020). https://doi.org/10.1007/s11071-020-05664-w

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  • DOI: https://doi.org/10.1007/s11071-020-05664-w

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