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Normal modes of a double pendulum at low energy levels

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Abstract

This study is concerned with a double pendulum and its regular behaviour associated with low energy levels and the influence of the associated initial conditions on the frequency of normal modes. The case of nonlinear oscillations described by the exact equations of motion is examined. A global qualitative insight is provided via energy diagrams and Poincaré maps. Then, the case of linear oscillations, their normal modes and associated frequencies is analysed. Further, quantitative insights via two approaches (Lindstedt–Poincaré method and harmonic balancing) are also achieved to determine analytically the influence of initial amplitudes on the existence and frequency of nonlinear normal modes. These results are compared with the one corresponding to the linear normal modes as well as with the corresponding numerical solutions of the exact equations of motion.

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References

  1. Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, New York (1979)

    MATH  Google Scholar 

  2. Kovacic, I., Rand, R.H.: About a class of nonlinear oscillators with amplitude-independent frequency. Nonlinear Dyn. 74, 455–465 (2013)

    Article  MathSciNet  Google Scholar 

  3. Poschel, J.: A lecture on the classical KAM theorem. Proc. Symp. Pure Math. 69, 707–732 (2001)

    Article  MathSciNet  Google Scholar 

  4. Richter, P.H., Scholz, H.J.: Chaos in classical mechanics: the double pendulum. In: Schuster, P. (ed.) Stochastic Phenomena and Chaotic Behaviour in Complex Systems. Springer Series in Synergetics, vol. 21, pp. 86–97. Springer, Berlin (1984)

    Chapter  Google Scholar 

  5. Shinbrot, T., Grebogi, C., Wisdom, J., Yorke, J.A.: Chaos in a double pendulum. Am. J. Phys. 60, 491–499 (1992)

    Article  Google Scholar 

  6. Levien, R.B., Tan, S.M.: Double pendulum: an experiment in chaos. Am. J. Phys. 61, 1038–1044 (1993)

    Article  Google Scholar 

  7. Stachowiak, T., Okada, T.: A numerical analysis of chaos in the double pendulum. Chaos Solitons Fract. 29, 417–422 (2006)

    Article  Google Scholar 

  8. Rafat, M.Z., Wheatland, M.S., Bedding, T.R.: Dynamics of a double pendulum with distributed mass. Am. J. Phys. 77, 216–223 (2009)

    Article  Google Scholar 

  9. Jyotirmoy, R., Mallik, A.K., Bhattacharjee, J.K.: Role of initial conditions in the dynamics of a double pendulum at low energies. Nonlinear Dyn. 73, 993–1004 (2013)

    Article  MathSciNet  Google Scholar 

  10. Kovacic, I., Radomirovic, D.: Mechanical Vibrations: Fundamentals with Solved Examples. Willey, London (2017)

    Google Scholar 

  11. Rosenberg, R.: The normal modes of nonlinear n-degree-of-freedom systems. J. Appl. Mech. 2, 7–14 (1962)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors acknowledge support of the Ministry of Education and Science of Serbia, Grant III41007.

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Authors and Affiliations

  1. Centre for Vibro-Acoustic Systems and Signal Processing, Faculty of Technical Sciences, University of Novi Sad, Novi Sad, Serbia

    Ivana Kovacic & Miodrag Zukovic

  2. Faculty of Agriculture, University of Novi Sad, Novi Sad, Serbia

    Dragi Radomirovic

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  1. Ivana Kovacic
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  2. Miodrag Zukovic
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  3. Dragi Radomirovic
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Correspondence to Ivana Kovacic.

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Kovacic, I., Zukovic, M. & Radomirovic, D. Normal modes of a double pendulum at low energy levels. Nonlinear Dyn 99, 1893–1908 (2020). https://doi.org/10.1007/s11071-019-05424-5

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