悬索在外激励作用下的1∶3内共振分析(II)∶数值结果  

On the one vs three internal resonances of suspended cables subjected to external excitation (II)∶numerical result

作　　者：王连华[1] 赵跃宇[1] 胡建华[1] 金怡新[1] 

机构地区：[1]湖南大学土木工程学院,长沙湖南410082

出　　处：《计算力学学报》2008年第1期104-111,共8页Chinese Journal of Computational Mechanics

基　　金：国家自然科学基金(1027204110502020)资助项目

摘　　要：本研究的第一部分已经推导了悬索在第一阶面内对称模态主共振和第三阶面内对称模态主共振下的平均方程,其中考虑了这两阶模态之间1∶3内共振。本文对平均方程的稳态解、周期解以及混沌解进行了研究。利用Newton-Naphson方法和拟弧长的延拓算法确定了主共振情况下的幅频响应曲线,通过利用Jacobian矩阵的特征值判断幅频响应曲线中解的稳定性。在这些幅频响应曲线中,都存在超临界Hopf分叉,导致平均方程的周期解。以这些超临界Hopf分叉为起点,利用打靶法和拟弧长的延拓算法确定了两种主共振情况下的周期解分支,同时通过利用Floquet理论判断这些周期解的稳定性。然后利用数值结果研究了两种主共振情况下的周期解经过倍周期分叉通向混沌的过程。最后利用Runge-Kutta法研究了悬索两自由度离散模型的非线性响应。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 stud response of the two- Y d the period-doubling bifurcations scenario leading to chaos. At last, the non-linear egree-of-freedom （DOF） model is investigated by using the Runge-Kutta algorithm.

关 键 词：平均方程 周期解 混沌解 HOPF分叉 

分 类 号：O322[理学—一般力学与力学基础]