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Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 546364, 18 pages
http://dx.doi.org/10.1155/2012/546364
Research Article

Chaos in a Magnetic Pendulum Subjected to Tilted Excitation and Parametric Damping

1Center for Nonlinear Dynamics and Control, Department of Mechanical Engineering, Villanova University, 800 Lancaster Avenue, Villanova, PA 19085, USA
2Laboratory of Mechanics, University Hassan II, Casablanca, Morocco

Received 18 May 2012; Revised 3 August 2012; Accepted 3 August 2012

Academic Editor: Stefano Lenci

Copyright © 2012 C. A. Kitio Kwuimy et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. S. K. De and N. R. Aluru, “Complex oscillations and chaos in electrostatic microelectromechanical systems under superharmonic excitations,” Physical Review Letters, vol. 94, no. 20, 4 pages, 2005. View at Publisher · View at Google Scholar · View at Scopus
  2. B. E. DeMartini, H. E. Butterfield, J. Moehlis, and K. L. Turner, “Chaos for a microelectromechanical oscillator governed by the nonlinear mathieu equation,” Journal of Microelectromechanical Systems, vol. 16, no. 6, pp. 1314–1323, 2007. View at Publisher · View at Google Scholar · View at Scopus
  3. H. M. Ouakad and M. I. Younis, “The dynamic behavior of MEMS arch resonators actuated electrically,” International Journal of Non-Linear Mechanics, vol. 45, no. 7, pp. 704–713, 2010. View at Publisher · View at Google Scholar · View at Scopus
  4. I. Eker, “Experimental on-line identification of an electromechanical system,” ISA Transactions, vol. 43, no. 1, pp. 13–22, 2004. View at Scopus
  5. C. A. K. Kwuimy and P. Woafo, “Dynamics, chaos and synchronization of self-sustained electromechanical systems with clamped-free flexible arm,” Nonlinear Dynamics, vol. 53, no. 3, pp. 201–213, 2008. View at Publisher · View at Google Scholar · View at Scopus
  6. C. A. Kitio Kwuimy, B. Nana, and P. Woafo, “Experimental bifurcations and chaos in a modified self-sustained macro electromechanical system,” Journal of Sound and Vibration, vol. 329, no. 15, pp. 3137–3148, 2010. View at Publisher · View at Google Scholar · View at Scopus
  7. A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations, Wiley-Interscience, New York, NY, USA, 1979. View at Zentralblatt MATH
  8. B. R. Nana Nbendjo and P. Woafo, “Active control with delay of horseshoes chaos using piezoelectric absorber on a buckled beam under parametric excitation,” Chaos, Solitons & Fractals, vol. 32, no. 1, pp. 73–79, 2007. View at Publisher · View at Google Scholar · View at Scopus
  9. A. O. Belyakov, A. P. Seyranian, and A. Luongo, “Dynamics of the pendulum with periodically varying length,” Physica D, vol. 238, no. 16, pp. 1589–1597, 2009. View at Publisher · View at Google Scholar · View at Scopus
  10. G. Litak, M. Wiercigroch, B. W. Horton, and X. Xu, “Transient chaotic behaviour versus periodic motion of a parametric pendulum by recurrence plots,” ZAMM Zeitschrift für Angewandte Mathematik und Mechanik, vol. 90, no. 1, pp. 33–41, 2010. View at Publisher · View at Google Scholar · View at Scopus
  11. C. Hayashi, Nonlinear Oscillations in Physical Systems, McGraw-Hill Electrical and Electronic Engineering Series, McGraw-Hill, New York, NY, USA, 1964. View at Zentralblatt MATH
  12. A. Oksasoglu and D. Vavriv, “Interaction of low- and high-frequency oscillations in a nonlinear RLC circuit,” IEEE Transactions on Circuits and Systems I, vol. 41, no. 10, pp. 669–672, 1994. View at Publisher · View at Google Scholar · View at Scopus
  13. B. Nana and P. Woafo, “Synchronization in a ring of four mutually coupled van der Pol oscillators: theory and experiment,” Physical Review E, vol. 74, no. 4, Article ID 046213, 8 pages, 2006. View at Publisher · View at Google Scholar · View at Scopus
  14. S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, vol. 2 of Texts in Applied Mathematics, Springer, New York, NY, USA, 1990. View at Zentralblatt MATH
  15. O. Diallo and Y. Koné, “Melnikov analysis of chaos in a general epidemiological model,” Nonlinear Analysis: Real World Applications, vol. 8, no. 1, pp. 20–26, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. Z. Zhang, J. Peng, and J. Zhang, “Melnikov method to a bacteria-immunity model with bacterial quorum sensing mechanism,” Chaos, Solitons & Fractals, vol. 40, no. 1, pp. 414–420, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. G. Cicogna and F. Papoff, “Asymmetric duffing equation and the appearance of chaos,” Europhysics Letters, vol. 3, no. 9, pp. 963–967, 1978.
  18. S. Lenci and G. Rega, “Global optimal control and system-dependent solutions in the hardening Helmholtz-Duffing oscillator,” Chaos, Solitons & Fractals, vol. 21, no. 5, pp. 1031–1046, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. G. Litak, A. Syta, and M. Borowiec, “Suppression of chaos by weak resonant excitations in a non-linear oscillator with a non-symmetric potential,” Chaos, Solitons & Fractals, vol. 32, no. 2, pp. 694–701, 2007. View at Publisher · View at Google Scholar · View at Scopus
  20. H. Cao, J. M. Seoane, and M. A. F. Sanjuán, “Symmetry-breaking analysis for the general Helmholtz-Duffing oscillator,” Chaos, Solitons & Fractals, vol. 34, no. 2, pp. 197–212, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. B. P. Mann, “Energy criterion for potential well escapes in a bistable magnetic pendulum,” Journal of Sound and Vibration, vol. 323, no. 3–5, pp. 864–876, 2009. View at Publisher · View at Google Scholar · View at Scopus
  22. J. J. Thomsen, “Effective properties of mechanical systems under high-frequency excitation at multiple frequencies,” Journal of Sound and Vibration, vol. 311, no. 3–5, pp. 1249–1270, 2008. View at Publisher · View at Google Scholar · View at Scopus
  23. M. V. Bartuccelli, G. Gentile, and K. V. Georgiou, “On the dynamics of a vertically driven damped planar pendulum,” The Royal Society of London. Proceedings. Series A, vol. 457, no. 2016, pp. 3007–3022, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. J. M. Schmitt and P. V. Bayly, “Bifurcations in the mean angle of a horizontally shaken pendulum: analysis and experiment,” Nonlinear Dynamics, vol. 15, no. 1, pp. 1–14, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  25. H. Yabuno, M. Miura, and N. Aoshima, “Bifurcation in an inverted pendulum with tilted high-frequency excitation: analytical and experimental investigations on the symmetry-breaking of the bifurcation,” Journal of Sound and Vibration, vol. 273, no. 3, pp. 493–513, 2004. View at Publisher · View at Google Scholar · View at Scopus
  26. B. P. Mann and M. A. Koplow, “Symmetry breaking bifurcations of a parametrically excited pendulum,” Nonlinear Dynamics, vol. 46, no. 4, pp. 427–437, 2006. View at Publisher · View at Google Scholar · View at Scopus
  27. M. Belhaq and S. Mohamed Sah, “Fast parametrically excited van der Pol oscillator with time delay state feedback,” International Journal of Non-Linear Mechanics, vol. 43, no. 2, pp. 124–130, 2008. View at Publisher · View at Google Scholar · View at Scopus
  28. S. Sah and M. Belhaq, “Effect of vertical high-frequency parametric excitation on self-excited motion in a delayed van der Pol oscillator,” Chaos, Solitons & Fractals, vol. 37, no. 5, pp. 1489–1496, 2008. View at Publisher · View at Google Scholar · View at Scopus
  29. A. Fidlin and J. Juel Thomsen, “Non-trivial effects of high-frequency excitation for strongly damped mechanical systems,” International Journal of Non-Linear Mechanics, vol. 43, no. 7, pp. 569–578, 2008. View at Publisher · View at Google Scholar · View at Scopus
  30. F. C. Moon, J. Cusumano, and P. J. Holmes, “Evidence for homoclinic orbits as a precursor to chaos in a magnetic pendulum,” Physica D, vol. 24, no. 1–3, pp. 383–390, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  31. Y. Kraftmakher, “Experiments with a magnetically controlled pendulum,” European Journal of Physics, vol. 28, no. 5, pp. 1007–1020, 2007. View at Publisher · View at Google Scholar · View at Scopus
  32. S. M. Sah and M. Belhaq, “Control of a delayed limit cycle using the tilt angle of a fast excitation,” Journal of Vibration and Control, vol. 17, no. 2, pp. 175–182, 2011. View at Publisher · View at Google Scholar
  33. I. S. Gradstein and I. M. Rjikhik, Table of Integrals, Series and Products, Nauka, Moscow, Russia, 1971.
  34. http://www.gnu.org/software/gsl/manual/html_node/.