带有自由转动附着质量的悬挂绳的非线性动力学

Filip Novkoski, Jules Fillette, Chi-Tuong Pham, Eric Falcon
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摘要

我们的研究表明,在一根悬挂的柔性弦的下端添加一个可自由转动的附着质量后,其自然共振频率会发生显著变化(变化幅度为一个数量级)。然后,这个铰接系统会表现出复杂的非线性动力学特性,如弯曲振荡,类似于挥杆松弛时的振荡,从而极大地改变了系统共振,而系统共振是由枢轴质量的长度控制的。利用远程非侵入式磁参量增强技术对该动力学进行了实验研究。为此,用一根柔性绳索将永久磁铁悬挂在垂直振荡的导电板上。实验报告了谐波不稳定性和周期加倍不稳定性,并使用希尔方程对其进行了建模,从而得出了能准确描述实验观察到的舌状不稳定性曲线的解析解。
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Nonlinear dynamics of a hanging string with a freely pivoting attached mass
We show that the natural resonant frequency of a suspended flexible string is significantly modified (by one order of magnitude) by adding a freely pivoting attached mass at its lower end. This articulated system then exhibits complex nonlinear dynamics such as bending oscillations, similar to those of a swing becoming slack, thereby strongly modifying the system resonance that is found to be controlled by the length of the pivoting mass. The dynamics is experimentally studied using a remote and noninvasive magnetic parametric forcing. To do so, a permanent magnet is suspended by a flexible string above a vertically oscillating conductive plate. Harmonic and period-doubling instabilities are experimentally reported and are modeled using the Hill equation, leading to analytical solutions that accurately describe the experimentally observed tonguelike instability curves.
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