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Acta Mechanica Sinica

, Volume 29, Issue 4, pp 575–582 | Cite as

Analysis of the periodic solutions of a smooth and discontinuous oscillator

  • Zhi-Xin Li
  • Qing-Jie CaoEmail author
  • Marian Wiercigroch
  • Alain Léger
Research Paper

Abstract

In this paper, the periodic solutions of the smooth and discontinuous (SD) oscillator, which is a strongly irrational nonlinear system are discussed for the system having a viscous damping and an external harmonic excitation. A four dimensional averaging method is employed by using the complete Jacobian elliptic integrals directly to obtain the perturbed primary responses which bifurcate from both the hyperbolic saddle and the non-hyperbolic centres of the unperturbed system. The stability of these periodic solutions is analysed by examining the four dimensional averaged equation using Lyapunov method. The results presented herein this paper are valid for both smooth (α > 0) and discontinuous (α = 0) stages providing the answer to the question why the averaging theorem spectacularly fails for the case of medium strength of external forcing in the Duffing system analysed by Holmes. Numerical calculations show a good agreement with the theoretical predictions and an excellent efficiency of the analysis for this particular system, which also suggests the analysis is applicable to strongly nonlinear systems.

Keywords

SD oscillator Averaging method Periodic solution Irrational nonlinearity Elliptic integral 

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Copyright information

© The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Zhi-Xin Li
    • 1
  • Qing-Jie Cao
    • 1
    Email author
  • Marian Wiercigroch
    • 2
  • Alain Léger
    • 3
  1. 1.Centre for Nonlinear Dynamics ResearchHarbin Institute of Technology, School of AstronauticsHarbinChina
  2. 2.Centre for Applied Dynamics Research, School of EngineeringUniversity of Aberdeen, King’s CollegeAberdeenScotland, UK
  3. 3.Laboratoire de Mécanique et d’ AcoustiqueCNRSMarseille Cedex 20France

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