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| DOI：10.7511/jslx20081001 |
| KeyWord:averaging equations,periodic solution,chaotic solution,Hopf bifurcation |
| WANG Lian-hua ZHAO Yue-yu HU Jian-hua JIN Yi-xin |
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| Abstract: |
| Two sets of averaging equations of the cases of primary resonance of the first or third symmetric mode of the suspended cable are derived in this study,where the one vs three internal resonance is considered. The equilibrium solution,the periodic solution and chaotic solution of averaging equations are examined in this paper.The Newton-Naphson method and the pseudo-arclength path-following algorithm are used to obtain the frequency-response curves of the two cases of primary resonances,and the equilibrium solution's stability is determine by examining the eigenvalues of the corresponding Jacobian matrix.The supercritical Hopf bifurcations are found in the frequency-response curves.Choosing these bifurcations as the initial points,the periodic solution branches for the two cases of primary resonance are obtained with the help of the shooting method and the pseudo-arclength path-following algorithm.Moreover,the Floquet theory is used to determine the periodic solution's stability.The numerical simulation is used to study the period-doubling bifurcations scenario leading to chaos.At last,the non-linear response of the two-degree-of-freedom(DOF) model is investigated by using the Runge-Kutta algorithm. |
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