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Analysis of the periodic solutions of a smooth and discontinuous oscillator

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 pbkp_redictions and an excellent efficiency of the analysis for this particular system, which also suggests the analysis is applicable to strongly nonlinear systems. © 2013 The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg.