Skip to main content
SpringerLink
Log in

Approximate analytical solutions for oscillatory and rotational motion of a parametric pendulum

  • Original Article
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

In this paper, the authors have studied dynamic responses of a parametric pendulum by means of analytical methods. The fundamental resonance structure was determined by looking at the undamped case. The two typical responses, oscillations and rotations, were investigated by applying perturbation methods. The primary resonance boundaries for oscillations and pure rotations were computed, and the approximate analytical solutions for small oscillations and period-one rotations were obtained. The solution for the rotations has been derived for the first time. Comparisons between the analytical and numerical results show good agreements.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Xu, X., Wiercigroch, M., Cartmell, M.P.: Rotating orbits of a parametrically-excited pendulum. Chaos Solitons Fractals 23, 1537–1548. (2005)

    Article  Google Scholar 

  2. Bishop, S.R., Sofroniou, A., Shi, P.: Symmetry-breaking in the response of the parametrically excited pendulum model. Chaos Solitons Fractals 25, 257–264 (2005)

    Article  Google Scholar 

  3. Clifford, M.J., Bishop, S.R.: Locating oscillatroy orbits of the parametrically-excited pendulum. J. Aust. Math. Soc. Ser. B – Appl. Math. 37, 309–319. (1996)

    MathSciNet  Google Scholar 

  4. Clifford, M.J., Bishop, S.R.: Approximating the excape zone for the parametrically excited pendulum. J. Sound Vib. 172(4), 572–576 (1994)

    Article  MathSciNet  Google Scholar 

  5. Szemplí'nska-Stupnicka, W., Tyrkiel, E.: The oscillation-rotation attractors in the forced pendulum and their peculiar properties. Int. J. Bifurcation and Chaos 12, 159–168 (2002)

    Article  Google Scholar 

  6. Szemplí'nska-Stupnicka, W., Tyrkiel, E.: Common features of the onset of the persistent chaos in nonlinear oscillators: a phenomenological approach. Nonlinear Dyn. 27, 271–293 (2002)

    Article  Google Scholar 

  7. Szemplí'nska-Stupnicka, W., Zubrzycki, A., Tyrkiel, E.: Properties of chaoteic and regular boundary crisis in dissipative driven nonlinear sscillators. Nonlinear Dyn. 19, 19–36 (1999)

    Article  Google Scholar 

  8. Koch, B.P., Leven, R.W., Pompe, B., Wilke, C.: Experimental evidence for chaotic behavior of a parametrically forced pendulum. Phys. Lett. A 96(5) 219–224 (1983)

    Article  Google Scholar 

  9. Leven, R.W., Pompe, B., Wilke, C., Koch, B.P.: Experiments on periodic and chaotic motion of a parametrically driven pendulum. Physica D 16, 371–384 (1985)

    Article  MathSciNet  Google Scholar 

  10. Yabuno, H., Miura, M., Aoshima, N.: Bifurcation in an inverted pendulum with tilted high-frequency excitation: analytical and experimental investigations on the symmetry-breaking of the bifurcation. J. Sound Vib. 273, 493–513 (2004)

    Article  Google Scholar 

  11. Koch, B.P., Leven, R.W.: Subharmonic and homoclinic bifurcations in a parametrically forced pendulum. Physica D 16, 1–13 (1984)

    Article  MathSciNet  Google Scholar 

  12. Miles, J.: On resonant rotation of a weakly damped pendulum. J. Sound Vib. 280, 401–406 (2005)

    Article  MathSciNet  Google Scholar 

  13. Nusse, H.E., Yorke, A.Y.: Dynamics: Numerical Explorations, Springer-Verlag, New York, Inc. (1997)

  14. Xu, X.: Nonlinear dynamics of parametric pendulum for wave energy extraction, PhD thesis, University of Aberdeen. (2005)

Download references

Author information

Authors and Affiliations

  1. Centre for Applied Dynamics Research, Department of Engineering, University of Aberdeen, Aberdeen, AB24 3UE, Scotland, UK

    Xu Xu & M. Wiercigroch

Authors
  1. Xu Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Wiercigroch
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Wiercigroch.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Xu, X., Wiercigroch, M. Approximate analytical solutions for oscillatory and rotational motion of a parametric pendulum. Nonlinear Dyn 47, 311–320 (2007). https://doi.org/10.1007/s11071-006-9074-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-006-9074-4

Keywords

Search

Navigation