Effect of External Excitations on a Nonlinear System with Time Delay |
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Authors: | Email author" target="_blank">J?C?JiEmail author Colin?H?Hansen Xinye?Li |
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Institution: | (1) School of Mechanical Engineering, The University of Adelaide, SA 5005, Australia;(2) School of Mechanical Engineering, Hebei University of Technology, Tianjin, 300130, P.R. China |
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Abstract: | The trivial equilibrium of a two-degree-of-freedom autonomous system may become unstable via a Hopf bifurcation of multiplicity two and give rise to oscillatory bifurcating solutions, due to presence of a time delay in the linear and nonlinear terms. The effect of external excitations on the dynamic behaviour of the corresponding non-autonomous system, after the Hopf bifurcation, is investigated based on the behaviour of solutions to the four-dimensional system of ordinary differential equations. The interaction between the Hopf bifurcating solutions and the high level excitations may induce a non-resonant or secondary resonance response, depending on the ratio of the frequency of bifurcating periodic motion to the frequency of external excitation. The first-order approximate periodic solutions for the non-resonant and super-harmonic resonance response are found to be in good agreement with those obtained by direct numerical integration of the delay differential equation. It is found that the non-resonant response may be either periodic or quasi-periodic. It is shown that the super-harmonic resonance response may exhibit periodic and quasi-periodic motions as well as a co-existence of two or three stable motions. |
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Keywords: | coexistence of stable motions Hopf bifurcation non-resonances quasi-periodic motions super-harmonic resonances time delay two degree-of-freedom nonlinear system |
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