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The Supression of Chaos in Periodically Forced Oscillators

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Nonlinear Dynamics in Engineering Systems

Summary

A method for suppressing chaos in a class of periodically forced oscillators is presented. The method involves the manipulation of higher harmonics in the periodic excitation such that the homoclinic tangles which lie at the heart of the chaos are destroyed. The procedure is demonstrated via a particular example: an inverted pendulum with amplitude constraints subjected to periodic base excitation. The general theory is developed using a Melnikov type method (see [1–3] and is supported by some experimental evidence.

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References

  1. Melnikov, V.K.: “On the stability of the center for time periodic perturbations,” Trans. Moscow Math. Soc. 12 1 - 57, 1963.

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  2. Guckenheimer, J.; Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. 2nd printing, Springer Verlag, New York, Berlin, Heidelberg, Tokyo, 1986.

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  3. Wiggins, S.: Global Bifurcations and Chaos-Analytical Methods, Springer, Verlag, New York, Berlin, Heidelberg, Tokyo, 1988.

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  4. Moon, F.C.; Cusumano, J.; Holmes, P.J.: “Evidence for homoclinic orbits as a precursor to chaos in a magnetic pendulum,” Physica 24D 383 - 390 1987.

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  5. Moon, F.C.: Chaotic Vibrations, Wiley-Interscience, New York, 1987.

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  6. Shaw, S.W.; Rand, R.H.: “The transition to chaos in a simple mechanical system,” Int. J. of Nonlinear Mech. 24 41–56 1988.

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  7. Moore, D.B.; Shaw, S.W.: “The experimental response of an impacting pendulum system,” Int. J. of Nonlinear Mech., to appear.

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Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Michigan State University, East Lansing, MI, 48824, USA

    Steven W. Shaw

Authors
  1. Steven W. Shaw
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Editor information

Editors and Affiliations

  1. Institute B of Mechanics, University of Stuttgart, Pfaffenwaldring 9, D-7000, Stuttgart 80, Germany

    W. Schiehlen

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© 1990 Springer-Verlag Berlin Heidelberg

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Shaw, S.W. (1990). The Supression of Chaos in Periodically Forced Oscillators. In: Schiehlen, W. (eds) Nonlinear Dynamics in Engineering Systems. International Union of Theoretical and Applied Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-83578-0_36

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  • DOI: https://doi.org/10.1007/978-3-642-83578-0_36

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-83580-3

  • Online ISBN: 978-3-642-83578-0

  • eBook Packages: Springer Book Archive

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