Efficiency of targeted energy transfers in coupled nonlinear oscillators associated with 1:1 resonance captures:Part II, analytical study
Efficiency of targeted energy transfers in coupled nonlinear oscillators associated with 1:1 resonance captures:Part II, analytical study
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DOI：
10.1016/j.jsv.2009.03.004
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发表时间：
2009-08
影响因子：
4.7
通讯作者：
T. Sapsis;A. Vakakis;O. Gendelman;L. Bergman;G. Kerschen;D. Quinn
中科院分区：
文献类型：
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作者：
T. Sapsis;A. Vakakis;O. Gendelman;L. Bergman;G. Kerschen;D. Quinn
研究了两自由度阻尼系统在1：1瞬态共振捕获条件下的目标能量传递问题。该系统由一个线性振子强耦合到一个基本上非线性的附件或非线性能量汇。在一篇配套论文中[Quinn等人，与1：1共振捕获相关联的耦合非线性振荡器中的目标能量转移的效率：Part I，JournalofSoundandVibration 311（2008）1228-1248]我们研究了该系统的哈密顿动力学的基本结构，并且表明对于足够小的粘性阻尼值，非线性阻尼跃迁强烈地受到哈密顿系统的周期和准周期轨道的基本拓扑结构的影响。在这项工作中，直接分析处理的管理强非线性阻尼运动方程进行通过慢/快分区的瞬态响应，以调查分析的最佳目标能量转移的参数区域。为此，我们确定的动态影响的能力的非线性附件被动地吸收和本地耗散宽带能量的线性振荡器的特征时间尺度。然后，我们证明了最佳的目标能量转移实现的初始能量接近的邻域的基础哈密顿系统的同宿轨道。我们研究分析瞬态轨道的同宿轨道的扰动导致的弱阻尼系统，并表明，这会产生一个额外的慢时间尺度的平均动力学，并导致最佳的有针对性的能量转移从线性振子的非线性能量汇在一个单一的“超慢”半周期。我们表明，在较高的能量，这种“超慢”的半周期被强非线性拍频，这导致显着的，但次优的目标能量转移效率所取代。最后，我们研究了在这个系统中的能量传递效率在很宽的系统参数范围内的数值目标，并验证了分析预测。
We study targeted energy transfer in a two degree-of-freedom damped system under the condition of 1:1 transient resonance capture. The system consists of a linear oscillator strongly coupled to an essentially nonlinear attachment or nonlinear energy sink. In a companion paper [Quinn et al., Efficiency of targeted energy transfers in coupled nonlinear oscillators associated with 1:1 resonance captures: part I, Journal of Sound and Vibration 311 (2008) 1228–1248] we studied the underlying structure of the Hamiltonian dynamics of this system, and showed that for sufficiently small values of viscous damping, nonlinear damped transitions are strongly influenced by the underlying topological structure of periodic and quasiperiodic orbits of the Hamiltonian system. In this work direct analytical treatment of the governing strongly nonlinear damped equations of motion is performed through slow/fast partitions of the transient responses, in order to investigate analytically the parameter region of optimal targeted energy transfer. To this end, we determine the characteristic time scales of the dynamics that influence the capacity of the nonlinear attachment to passively absorb and locally dissipate broadband energy from the linear oscillator. Then, we prove that optimal targeted energy transfer is realized for initial energies close to the neighborhood of a homoclinic orbit of the underlying Hamiltonian system. We study analytically transient orbits resulting as perturbations of the homoclinic orbit in the weakly damped system, and show that this yields an additional slow-time scale in the averaged dynamics, and leads to optimal targeted energy transfer from the linear oscillator to the nonlinear energy sink in a single “super-slow” half-cycle. We show that at higher energies, this “super-slow” half-cycle is replaced by strong nonlinear beats, which lead to significant but suboptimal targeted energy transfer efficiency. Finally, we investigate numerically targeted energy transfer efficiency in this system over a wide range of system parameters and verify the analytical predictions.