Nonlinear Dynamics

, Volume 46, Issue 4, pp 427–437

Symmetry breaking bifurcations of a parametrically excited pendulum

  • B. P. Mann
  • M. A. Koplow
Original Article

Abstract

This paper examines the bifurcation behavior of a planar pendulum subjected to high-frequency parametric excitation along a tilted angle. Parametric nonlinear identification is performed on the experimental system via an optimization approach that utilizes a developed approximate analytical solution. Experimental and theoretical efforts then consider the influence of a subtle tilt angle in the applied parametric excitation by contrasting the predicted and observed mean angle bifurcations with the bifurcations due to excitation applied in either the vertical or horizontal direction. Results show that small deviations from either a perfectly vertical or horizontal excitation will result in symmetry breaking bifurcations as opposed to pitchfork bifurcations.

Keywords

Parametric excitation Pendulum Symmetry breaking bifurcations 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Acheson, D.J.: A pendulum theorem. Proc. Royal Soc. Lond. A 443, 239–245 (1993)MATHCrossRefGoogle Scholar
  2. 2.
    Bartuccelli, M.V., Gentile, G., Georgiou, K.V.: On the dynamics of a vertically driven damped planar pendulum. Proc. Royal Soc. Lond. A 457, 3007–3022 (2001)MATHMathSciNetGoogle Scholar
  3. 3.
    Thomsen, J.J.: Some general effects of strong high-frequency excitation: stiffening, biasing, and smoothening. J. Sound Vib. 253(4), 807–831 (2002)CrossRefGoogle Scholar
  4. 4.
    Butcher, E.A., Sinha, S.C.: Symbolic computation of secondary bifurcations in a parametrically excited simple pendulum. Int. J. Bifurcation Chaos 8(3), 627–637(1998)MATHCrossRefGoogle Scholar
  5. 5.
    Scmitt, J.M., Bayly, P.V.: Bifurcations in the mean angle of a horizontally shaken pendulum: Analysis and experiment. Nonlinear Dyn. 15, 1–14 (1999)CrossRefGoogle Scholar
  6. 6.
    Clifford, M.J., Bishop, S.R.: Inverted oscillations of a driven pendulum. Proc. Royal Soc. Lond. A 454, 2811–2817 (1997)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Balachandran, B., Nayfeh, A.H.: Applied Nonlinear Dynamics. Wiley, New York (1995)Google Scholar
  8. 8.
    Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, New York (1979)Google Scholar
  9. 9.
    Coleman, T., Li, Y.: An interior, trust region approach for nonlinear minimization subject to bounds. SIAM J. Optim. 6, 418–445 (1996)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Virgin, L.N.: Introduction to Experimental Nonlinear Dynamics. Cambridge University Press, Cambridge, UK (2000)Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • B. P. Mann
    • 1
  • M. A. Koplow
    • 2
  1. 1.Nonlinear Dynamics Laboratory, Department of Mechanical and Aerospace EngineeringUniversity of Missouri-ColumbiaMissouri-ColumbiaUSA
  2. 2.Nonlinear Dynamics Laboratory, Department of Mechanical and Aerospace EngineeringUniversity of Florida-GainesvilleFlorida-GainesvilleUSA

Personalised recommendations