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A Smooth and Discontinuous Oscillator pp 103–120Cite as

Extended Averaging Method

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Part of the book series: Springer Tracts in Mechanical Engineering ((STME))

Abstract

This chapter aims at studying the periodic solutions and the Hopf bifurcations of the SD oscillator using the so-called averaging method. This will be done in the case where the system has a viscous damping and an external harmonic excitation. A four dimensional averaging method is introduced by using the complete Jacobian elliptic integrals, directly to obtain the perturbed primary responses which bifurcate from both the hyperbolic saddle point and the non-hyperbolic centers of the unperturbed system. The stability of these periodic solutions is obtained by examining the four dimensional averaged equation using Lyapunov method.

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Notes

  1. 1.

    This chapter relies on a common work with Dr. Zhixin Li, Center for Nonlinear Dynamics Research, School of Astronautics, Harbin Institute of Technology, Harbin 150001 Chian, and Professor Ruilan Tian, Center for Nonlinear Dynamics Research, Shijiazhang Tiedao University, Shijiazhuang 050043 China.

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Author information

Authors and Affiliations

  1. School of Astronautics, Harbin Institute of Technology, Harbin, China

    Qingjie Cao

  2. Laboratoire de Mécanique et d’Acoustique, CNRS, Marseille Cedex13, France

    Alain Léger

Authors
  1. Qingjie Cao
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  2. Alain Léger
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Correspondence to Qingjie Cao .

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Cao, Q., Léger, A. (2017). Extended Averaging Method. In: A Smooth and Discontinuous Oscillator. Springer Tracts in Mechanical Engineering. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53094-8_8

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  • DOI: https://doi.org/10.1007/978-3-662-53094-8_8

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-53092-4

  • Online ISBN: 978-3-662-53094-8

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