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ISSN:2164-6376 (print)
ISSN:2164-6414 (online)
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com

Mathematical Modelling and Simulation of the Bifurcational Wobblestone Dynamics

Discontinuity, Nonlinearity, and Complexity 3(2) (2014) 123--132 | DOI:10.5890/DNC.2014.06.002

Jan Awrejcewicz; Grzegorz Kudra

Department of Automation, Biomechanic and Mechatronics, Lodz University of Technology, Lodz Poland

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Abstract

The Celtic stone, sometimes also called wobbles tone or rattleback usually is a semi-ellipsoidal solid with a special mass distribution. Most celts lied on aflat and horizontal base, set in rotational motion around a vertical axis can rotate in only one direction. In this work the dynamics of the celt is simulated numerically, but the solid is forced untypically, i.e. it is situated on a harmonically vibrating base. Essential part of the model are approximate functions describing the contact forces, i.e. dry friction forces and rolling resistance. They are based on previous works of the authors, but some modifications of friction model are made, which can be described as a generalization of the earlier used Padé approximants. Periodic, quasiperiodic and chaotic dynamics of a harmonically forced rattleback is found and presented by the use of Poincaré maps and bifurcation diagrams.

Acknowledgments

This paper was financially supported by the National Science Centre of Poland under the grant MAESTRO 2, No. 2012/04/A/ST8/00738, for years 2013-2016.

References

  1. [1]  Kalker, J.J. (1982), A fast algorithm for the simplified theory of rolling contact, Vehicle Systems Dynamics, 11(1), 1-13.
  2. [2]  Kalker, J.J. (1990), Three-dimensional Elastic Bodies in Rolling Contact, Kluwer, Dordrecht.
  3. [3]  Contensou, P. (1962), Couplage entre frottement de glissementet de pivotementdans la téorie de la toupe, Kreiselprobleme Gyrodynamics: IUTAM Symp., Calerina, 201-216.
  4. [4]  Goyal, S., Ruina, A. and Papadopoulos, J. (1991), Planar sliding with dry friction. Part 1. Limit surface and moment function, Wear 143, 307-330.
  5. [5]  Goyal, S., Ruina, A. and Papadopoulos, J. (1991), Planar sliding with dry friction. Part 2. Dynamics of motion, Wear 143, 331-352.
  6. [6]  Howe, R. D. and Cutkosky, M. R. (1996), Practical force-motion models for sliding manipulation, The International Journal of Robotics Research, 15(6), 557-572.
  7. [7]  Zhuravlev, V.P. (1998), The model of dry friction in the problem of the rolling of rigid bodies, Journal of Applied Mathematics and Mechanics, 62(5), 705-710.
  8. [8]  Zhuravlev, V.P. and Kireenkov, A. A. (2005), Padé expansions in the two-dimensional model of Coulomb friction, Mechanics of Solids, 40(2), 1-10.
  9. [9]  Kireenkov, A.A. (2008), Combined model of sliding and rolling friction in dynamics of bodies on a rough plane, Mechanics of Solids, 43(3), 412-425.
  10. [10]  Leine, R.I. andGlocker, Ch. (2003), A set-valued force law for spatial Coulomb-Contensou friction, European Journal of Mechanics - A/Solids, 22(2), 193-216.
  11. [11]  Möller, M., Leine, R.I. and Glocker, Ch. (2009), An efficient approximation of orthotropic set-valued force laws of normal cone type, Proceedings of the 7th EUROMECH Solid Mechanics Conference, Lisbon, Portugal.
  12. [12]  Walker, GT. (1896), On a dynamical top, The Quarterly Journal of Pure and Applied Mathematics, 28, 175-184.
  13. [13]  Bondi, S.H. (1986), The rigid body dynamics of unidirectional spin, Proceedings of the Royal Society of London. SeriesA, 405, 265-279.
  14. [14]  Magnus, K. (1974), Zur Theorie der keltischen Wackelsteine, Zeitschrift fürangewandte Mathematik und Mechanik, 5, 54-55.
  15. [15]  Caughey, T.K. (1980), A mathematical model of the 'Rattleback', International Journal of Non-Linear Mechanics, 15 (4-5), 293-302.
  16. [16]  Kane, T.R. and Levinson, D.A. (1982), Realistic mathematical modeling of the rattleback, International Journal of Non-Linear Mechanics, 17(3), 175-186.
  17. [17]  Lindberg, J. and Longman, R.W. (1983), On the dynamic behavior of the wobblestone, Acta Mechanica, 49(1-2), 81-94.
  18. [18]  Garcia, A. and Hubbard, M. (1988), Spin reversal of the rattleback: theory and experiment, Proceedings of the Royal Society of London. Series A, 148, 165-197.
  19. [19]  Markeev, A.P. (2002), On the dynamics of a solid on an absolutely rough plane, Regular and Chaotic Dynamics, 7(2), 153-160.
  20. [20]  Borisov, A.V., Kilin, A.A. and Mamaev, I.S. (2006), New effects of rattlebacks, Doklady Physics, 51(5), 272-275.
  21. [21]  Zhuravlev, V.Ph. and Klimov, D.M. (2008), Global motion of the celt, Mechanics of Solids, 43(3), 320-327.
  22. [22]  Kudra, G. andAwrejcewicz, J. (2011), Tangens hyperbolicus approximations of the spatial model of friction coupled with rolling resistance, International Journal of Bifurcation and Chaos, 21(10), 2905-2017.
  23. [23]  Awrejcewicz, J. and Kudra, G. (2012), Celtic stone dynamics revisited using dry friction and rolling resistance, Shock and Vibration, 19, 1-9.
  24. [24]  Kudra, G. and Awrejcewicz, J. (2013), Approximate modelling of resulting dry friction forces and rolling resistance for elliptic contact shape. European Journal of Mechanics - A/Solids, 42, 358-375.

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