Dynamics and bifurcations of a coupled column-pendulum oscillator
Dynamics and bifurcations of a coupled column-pendulum oscillator
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耦合柱摆振荡器的动力学和分岔
DOI：
10.1006/jsvi.1995.0207
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发表时间：
1995
期刊：
影响因子：
--
通讯作者：
A. Ertas
中科院分区：
文献类型：
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作者：
G. Mustafa;A. Ertas
摘要本文研究了末端质量摆布置的大型柔性柱的动力学问题。该系统是带有尖端附件的柔性结构的减振装置的概念化。平均系统的分岔图表明，系统通过两条不同的路径失去稳定性：一条导致鞍点分叉，另一条导致Hopf分叉，这表明存在一个不变环面。在强迫幅值的变化下，这些分叉合并。这一现象具有重要的全球影响，因为与Hopf分叉相关的周期调制往往具有无穷大的周期，这是同宿轨道存在的一个强烈指标。该系统还具有孤立解(所谓的“Isolas”)，它们形成远离零的孤立环路。当强迫幅值变化时，等值线出现、消失或与规则溶液分支合并。响应曲线表明，柱子的振幅呈现饱和状态，摆起了减振作用。然而，也有一个发生能量反向流动的频率范围，在这个频率范围内，钟摆的幅度降低，而柱子的幅度却很大。实验动力学表明，周期运动产生准周期响应，证实了环面的存在。在准周期区域内，存在包含错综复杂的锁模周期响应网络的窗口。力幅的增加会导致环面破裂，这一现象类似于流体动力学中的湍流开始。
Abstract This study deals with the dynamics of a large flexible column with a tip mass-pendulum arrangement. The system is a conceptualization of a vibration-absorbing device for flexible structures with tip appendages. The bifurcation diagrams of the averaged system indicate that the system loses stability via two distinct routes; one leading to a saddle-node bifurcation, and the other to the Hopf bifurcation, indicating the existence of an invariant torus. Under the change of forcing amplitude, these bifurcations coalesce. This phenomenon has important global ramifications, in the sense that the periodic modulations associated with the Hopf bifurcation tend to have an infinite period, a strong indicator of existence of homoclinic orbits. The system also possesses isolated solutions (the so-called “isolas”) that form isolated loops bounded away from zero. As the forcing amplitude is varied, the isolas appear, disappear or coalesce with the regular solution branches. The response curves indicate that the column amplitude shows saturation and the pendulum acts as a vibration absorber. However, there is also a frequency range over which a reverse flow of energy occurs, where the pendulum shows reduced amplitude at the cost of large amplitudes of the column. The experimental dynamics shows that the periodic motion gives rise to a quasi-periodic response, confirming the existence of tori. Within the quasi-periodic region, there are windows containing intricate webs of mode-locked periodic responses. An increase in the force amplitude causes the tori to break up, a phenomenon similar to the onset of turbulence in hydrodynamics.