Analysis of the periodic solutions of a smooth and discontinuous oscillator  

作　　者：Zhi-Xin Li Qing-Jie Cao Marian Wiercigroch Alain Léger 

机构地区：[1]Centre for Nonlinear Dynamics Research, Harbin Institute of Technology, School of Astronautics [2]Centre for Applied Dynamics Research, School of Engineering, University of Aberdeen, King's College [3]Laboratoire de Mécanique et d' Acoustique,CNRS, 31, Chemin Joseph Aiguier, 13402 Marseille Cedex 20, France

出　　处：《Acta Mechanica Sinica》2013年第4期575-582,共8页力学学报（英文版）

基　　金：supported by the National Natural Science Foundation of China(11072065)

摘　　要：In this paper, the periodic solutions of the smooth and discontinuous （SD） oscillator, which is a strongly irra- tional 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 un- perturbed system. The stability of these periodic solutions is analysed by examining the four dimensional averaged equa- tion using Lyapunov method. The results presented herein this paper are valid for both smooth （e 〉 0） and discontin- uous （ce = 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.In this paper, the periodic solutions of the smooth and discontinuous （SD） oscillator, which is a strongly irra- tional 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 un- perturbed system. The stability of these periodic solutions is analysed by examining the four dimensional averaged equa- tion using Lyapunov method. The results presented herein this paper are valid for both smooth （e 〉 0） and discontin- uous （ce = 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.

关 键 词：SD oscillator ~ Averaging method. Periodic so-lution ~ Irrational nonlinearity ~ Elliptic integral 

分 类 号：TN752[电子电信—电路与系统]