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On the Chaotic Dynamics of a Spherical Pendulum with a Harmonically Vibrating Suspension

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Abstract

The equations of motion for a lightly damped spherical pendulum are considered. The suspension point is harmonically excited in both vertical and horizontal directions. The equations are approximated in the neighborhood of resonance by including the third order terms in the amplitude. The stability of equilibrium points of the modulation equations in a four-dimensional space is studied. The periodic orbits of the spherical pendulum without base excitations are revisited via the Jacobian elliptic integral to highlight the role played by homoclinic orbits. The homoclinic intersections of the stable and unstable manifolds of the perturbed spherical pendulum are investigated. The physical parameters leading to chaotic solutions in terms of the spherical angles are derived from the vanishing Melnikov–Holmes–Marsden (MHM) integral. The existence of real zeros of the MHM integral implies the possible chaotic motion of the harmonically forced spherical pendulum as a result from the transverse intersection between the stable and unstable manifolds of the weakly disturbed spherical pendulum within the regions of investigated parameters. The chaotic motion of the modulation equations is simulated via the 4th-order Runge–Kutta algorithms for certain cases to verify the analysis.

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References

  1. Chernous'ko, F. L., ‘Optimum translation of a pendulum,’ Journal of Applied Mathematics and Mechanics 39, 1975, 775–787.

    Article  MATH  MathSciNet  Google Scholar 

  2. Kral, R., Kreuzer, E., and Wilmers, C., ‘Nonlinear oscillations of a crane ship,’ Zeitschrift fur Angewandte Mathematik und Mechanik 76(S4), 1996, 5–8.

    Google Scholar 

  3. Markeyev, A. P., ‘The dynamics of a spherical pendulum with a vibrating suspension,’ Journal of Applied Mathematics and Mechanics 63, 1999, 205–211.

    MathSciNet  Google Scholar 

  4. Ghigliazza, R. M. and Holmes, P., ‘On the dynamics of cranes, or spherical pendula with moving supports,’ International Journal of Non-Linear Mechanics 37, 2002, 1211–1221.

    Article  Google Scholar 

  5. Abdel-Rahman, E. M., Nayfeh, A. H., and Masoud, Z. N., ‘Dynamics and control of cranes: A review,’ Journal of Vibration and Control 9, 2003, 863–908.

    Google Scholar 

  6. Miles, J. W., ‘Resonantly forced waves in a circular cylinder,’ Journal of Fluid Mechanics 149, 1984, 15–31.

    MATH  MathSciNet  Google Scholar 

  7. Miles, J. W. and Henderson, D., ‘Parametrically forced surface waves,’ Annul Review of Fluid Mechanics 32, 1990, 345–365.

    Google Scholar 

  8. Miles, J. W., ‘Stability of forced oscillations of a spherical pendulum,’ Quarterly of Applied Mathematics 20, 1962, 21 -32.

    Google Scholar 

  9. Lorenz, E. N., ‘Deterministic nonperiodic flow,’ Journal of Atmospheric Sciences 20, 1963,130–141.

    Google Scholar 

  10. Miles, J. W., ‘Resonant motion of a spherical pendulum,’ Physica D 11, 1984, 309–323.

    Article  MATH  MathSciNet  Google Scholar 

  11. Miles, J. W., ‘Internal resonances of a detuned spherical pendulum,’ Journal of Applied Mathematics and Physics 36, 1985, 609–615.

    MATH  MathSciNet  Google Scholar 

  12. Tritton, D. J., ‘Ordered and chaotic motion of a forced spherical pendulum,’ European Journal of Physics 7, 1986, 162–169.

    Article  Google Scholar 

  13. Bryant, P. J., ‘Breakdown to chaotic motion of a forced, damped, spherical pendulum,’ Physica D 64, 1993, 324–339.

    Article  MATH  MathSciNet  Google Scholar 

  14. Miles, J. W. and Zou, Q. P., ‘Parametric excitation of a detuned spherical pendulum,’ Journal of Sound and Vibration 164, 1993, 237–250.

    Article  MathSciNet  Google Scholar 

  15. Kana, D. D., and Fox, D. J., ‘Distinguishing the transition to chaos in a spherical pendulum,’ Chaos 5, 1995, 298–310.

    Article  Google Scholar 

  16. Tritton, D. J. and Groves, M., ‘Lyapunov exponents for the Miles’ spherical pendulum equations,’ Physica D 126, 1999, 83–98.

    Article  MathSciNet  Google Scholar 

  17. Chen, Y. S. and Leung, A. Y. T., Bifurcation and Chaos in Engineering, Springer-Verlag: London, 1998.

    Google Scholar 

  18. Guckenheimer, J. and Holmes, P., Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983.

    Google Scholar 

  19. Thompson, J. M. T. and Stewart, H. B., Nonlinear Dynamics and Chaos: Geometrical Methods for Engineers and Scientists, Chichester [West Sussex]; Wiley, New York, 1986.

  20. Melnikov, V. K., ‘On the stability of the centre for time-periodic perturbations,’ Transactions of the Moscow Mathematical Society 12, 1963, 1–57.

    MATH  Google Scholar 

  21. Kuang, J. L., Tan, S. H., Arichandran, K., and Leung, A. Y. T., ‘Chaotic dynamics of an asymmetrical gyrostat,’ International Journal of Non-Linear Mechanics 36, 2001, 1213–1233.

    Article  Google Scholar 

  22. Kuang, J. L., Tan, S. H., and Leung, A. Y. T., ‘On Melnikov's method in study of chaotic motion of a gyrostat,’ International Journal of Control 75, 2002, 328–351.

    Article  MathSciNet  Google Scholar 

  23. Wiggins, S. and Shaw, S. W., ‘Chaos and three-dimensional horseshoe in slowly varying oscillators,’ Journal of Applied Mechanics 55, 1988, 959–968.

    MathSciNet  Google Scholar 

  24. Kuang, J. L. and Leung, A. Y. T., ‘Homoclinic orbits of the Kovalevskaya top with perturbations,’ Zeitschrift fur Angewandte Mathematik und Mechanik 85, 2005, 277–302.

    MathSciNet  Google Scholar 

  25. Wittenburg, J., Dynamics of Systems of Rigid Bodies, Teubner, Stuttgart, 1977.

  26. Chin, C., Nayfeh, A. H., and Abdel-Rahman, E., ‘Nonlinear dynamics of a boom crane,’ Journal of Vibration and Control 7, 2001, 199–220.

    Google Scholar 

  27. Nayfeh, A. H. and Mook, D. T., Nonlinear Oscillators, Wiley, New York, 1979.

    Google Scholar 

  28. Nayfeh, A. H., Nonlinear Interactions, Wiley, New York, 2000.

    Google Scholar 

  29. Hagedorn, P., Non-linear Oscillations (translated and edited by Wolfram Stadler) Oxford University Press, Oxford, Clarendon Press, New York, 1982.

  30. Routh, E. J., A Treatise on the Dynamics of a System of Rigid Bodies, Part 2: The advanced part, Sixth Edition (London: Macmillan, 1905), pp. 186–202.

  31. Golubev, V. V., Lectures on Integration of the Equations of Motion of a Rigid Body About a Fixed Point. Translated from Russian by J. Shorr-Kon and published for the National Science Foundation by the Israel Program for Scientific Translations. Washington D. C., Office of Technical Services, U. S. Department of Commerce (1960).

  32. Lawden, D. F., Elliptic Functions and Applications, Springer-Verlag, New York, 1980.

    Google Scholar 

  33. Holmes, P. J. and Marsden, J. E., ‘A partial differential equation with infinitely many periodic orbits: Chaotic oscillations of a forced beam,’ Archive for Rational Mechanics and Analysis 76, 1981, 135–165.

    Article  MathSciNet  Google Scholar 

  34. Nayfeh, T. A., Asrar, W., and Nayfeh, A. H., ‘Three-mode interactions in harmonically excited systems with quadratic nonlinearities,’ Nonlinear Dynamics 3, 1992, 385–410.

    Google Scholar 

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Authors and Affiliations

  1. Faculty of Science and Engineering, City University of Hong Kong, Hong Kong SAR, P. R. China

    A. Y. T. Leung & J. L. Kuang

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  1. A. Y. T. Leung
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  2. J. L. Kuang
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Correspondence to J. L. Kuang.

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Leung, A.Y.T., Kuang, J.L. On the Chaotic Dynamics of a Spherical Pendulum with a Harmonically Vibrating Suspension. Nonlinear Dyn 43, 213–238 (2006). https://doi.org/10.1007/s11071-006-7426-8

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  • DOI: https://doi.org/10.1007/s11071-006-7426-8

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