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Dynamics of a rolling and sliding disk in a plane. Asymptotic solutions, stability and numerical simulations

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Abstract

We present a qualitative analysis of the dynamics of a rolling and sliding disk in a horizontal plane. It is based on using three classes of asymptotic solutions: straight-line rolling, spinning about a vertical diameter and tumbling solutions. Their linear stability analysis is given and it is complemented with computer simulations of solutions starting in the vicinity of the asymptotic solutions. The results on asymptotic solutions and their linear stability apply also to an annulus and to a hoop.

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References

  1. Appell, P., Sur l’intégration deséquations du mouvement d’un corps pesant de révolution roulant par une arête circulaire sur un plan horizontal: cas particulier du cerceau, Rend. Circ. Mat. Palermo, 1900, vol. 14, no. 1, pp. 1–6.

    Article  MATH  Google Scholar 

  2. Borisov, A. V., Mamaev, I. S., and Kilin, A.A., Dynamics of Rolling Disk, Regul. Chaotic Dyn., 2003, vol. 8, no. 2, pp. 201–212.

    Article  MathSciNet  MATH  Google Scholar 

  3. Borisov, A.V., Mamaev, I. S., and Karavaev, Yu. L., On the Loss of Contact of the Euler Disk, Nonlinear Dynam., 2015, vol. 79, no. 4, pp. 2287–2294.

    Article  MathSciNet  Google Scholar 

  4. Chaplygin, S.A., On a Motion of a Heavy Body of Revolution on a Horizontal Plane, Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 119–130; see also: Collected Works: Vol. 1, Moscow: Gostekhizdat, 1948, pp. 51–57.

    Article  MathSciNet  MATH  Google Scholar 

  5. Cohen, C. M., The Tippe Top Revisited, Am. J. Phys., 1977, vol. 45, no. 1, pp. 12–17.

    Article  Google Scholar 

  6. Contensou, P., Couplage entre frottement de pivotement et frottement de pivotement dans la théorie de latoupie, in Kreiselprobleme Gyrodynamics: IUTAM Symp. Celerina, Berlin: Springer, 1963, pp. 201–216.

    Google Scholar 

  7. Demidovich, B.P., Lectures on the Mathematical Stability Theory, Moscow: Nauka, 1967 (Russian).

    MATH  Google Scholar 

  8. Ebenfeld, S. and Scheck, F., A New Analysis of the Tippe Top: Asymptotic States and Liapunov Stability, Ann. Physics, 1995, vol. 243, no. 2, pp. 195–217.

    Article  MathSciNet  MATH  Google Scholar 

  9. Gallop, E. G., On the Rise of a Spinning Top, Proc. Camb. Phylos. Soc., 1904, vol. 19, no. 3, pp. 356–373.

    MATH  Google Scholar 

  10. Gantmacher, F.R., The Theory of Matrices: Vol. 2, New York: Chelsea, 1959.

    MATH  Google Scholar 

  11. Ivanov, A.P., On Detachment Conditions in the Problem on the Motion of a Rigid Body on a Rough Plane, Regul. Chaotic Dyn., 2008, vol. 13, no. 4, pp. 355–368.

    Article  MathSciNet  MATH  Google Scholar 

  12. Karapetian, A.V., On the Regular Precession of a Body of Revolution on a Horizontal Plane with Friction, J. Appl. Math. Mech., 1982, vol. 46, no. 4, pp. 450–453; see also: Prikl. Mat. Mekh., 1982, vol. 46, no. 4, pp. 568–572.

    Article  MATH  Google Scholar 

  13. Karapetian, A.V., Global Qualitative Analysis of Tippe Top Dynamics, Mech. Solids, 2008, vol. 43, no. 3, pp. 342–348; see also: Izv. Akad. Nauk SSSR. Mekh. Tverd. Tela, 2008, no. 3, pp. 33–41.

    Article  Google Scholar 

  14. Karapetian, A.V., A Two-Parameter Friction Model, J. Appl. Math. Mech., 2009, vol. 73, no. 4, pp. 367–370; see also: Prikl. Mat. Mekh., 2009, vol. 73, no. 4, pp. 515–519.

    Article  MathSciNet  Google Scholar 

  15. Korteweg, D. J., Extrait d’une lettre à M. Appell, Rend. Circ. Mat. Palermo, 1900, vol. 14, no. 1, pp. 7–8.

    Article  MATH  Google Scholar 

  16. Leine, R. I., Le Saux, C., and Glocker, Ch., Friction Models for the Rolling Disk, in Proc. of the ENOC Conf. (Eindhoven, Netherlands, 7–12 August, 2005), 10 pp.

  17. Le Saux, C., Leine, R. I., and Glocker, Ch., Dynamics of a Rolling Disk in the Presence of Dry Friction, J. Nonlinear Sci., 2005, vol. 15, no. 1, pp. 27–61.

    Article  MathSciNet  MATH  Google Scholar 

  18. Markeev, A.P., Dynamics of a Body Being Contiguous to a Rigid Surface, Moscow: Nauka, 1992 (Russian).

    MATH  Google Scholar 

  19. McDonald, A. J. and McDonald, K.T., The Rolling Motion of a Disk on a Horizontal Plane, arXiv:physics/0008227 (2000), 21 pp.

    Google Scholar 

  20. Moshchuk, N.K., Motion of a Heavy Rigid Body on a Horizontal Plane with Viscous Friction, J. Appl. Math. Mech., 1985, vol. 49, no. 1, pp. 49–53; see also: Prikl. Mat. Mekh., 1985, vol. 49, no. 1, pp. 66–71.

    Article  MATH  Google Scholar 

  21. Munitsyna, M. A., The Motions of a Spheroid on a Horizontal Plane with Viscous Friction, J. Appl. Math. Mech., 2012, vol. 76, no. 2, pp. 154–161; see also: Prikl. Mat. Mekh., 2012, vol. 76, no. 2, pp. 214–223.

    Article  MathSciNet  MATH  Google Scholar 

  22. O’Reilly, O.M., The Dynamics of Rolling Disks and Sliding Disks, Nonlinear Dynam., 1996, vol. 10, no. 3, pp. 287–305.

    Article  MathSciNet  Google Scholar 

  23. Press, W. H., Teukolsky, S.A., Vetterling, W. T., and Flannery, B.P., Numerical Recipes: The Art of Scientific Computing, 3rd ed., Cambridge: Cambridge Univ. Press, 2007.

    MATH  Google Scholar 

  24. Rauch-Wojciechowski, S. and Rutstam, N., Dynamics of the Tippe Top: Properties of Numerical Solutions Versus the Dynamical Equations, Regul. Chaotic Dyn., 2013, vol. 18, no. 4, pp. 453–467.

    Article  MathSciNet  MATH  Google Scholar 

  25. Routh, E. J., The Advanced Part of a Treatise on the Dynamics of a System of Rigid Bodies: Being Part II of a Treatise on the Whole Subject, 6th ed., New York: Dover, 1955.

    MATH  Google Scholar 

  26. Zhuravlev, V. F., On a Model of Dry Friction in the Problem of the Rolling of Rigid Bodies, J. Appl. Math. Mech., 1998, vol. 62, no. 5, pp. 705–710; see also: Prikl. Mat. Mekh., 1998, vol. 62, no. 5, pp. 762–767.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institute of Physics, University of Zielona Góra, Licealna 9, PL-65–417, Zielona Góra, Poland

    Maria Przybylska

  2. Department of Mathematics, Linköping University, 581 83, Linköping, Sweden

    Stefan Rauch-Wojciechowski

Authors
  1. Maria Przybylska
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  2. Stefan Rauch-Wojciechowski
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Correspondence to Maria Przybylska.

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Przybylska, M., Rauch-Wojciechowski, S. Dynamics of a rolling and sliding disk in a plane. Asymptotic solutions, stability and numerical simulations. Regul. Chaot. Dyn. 21, 204–231 (2016). https://doi.org/10.1134/S1560354716020052

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  • DOI: https://doi.org/10.1134/S1560354716020052

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