ANALYSIS OF DAMPING EFFICIENCY OF NONLINEAR ENERGY SINK CELL
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摘要: 非线性能量汇(nonlinear energy sink, NES)具有减振频带宽、减振效果好等诸多优点. 但是NES没有线性刚度的特征导致其难以驱动大重量的NES振子, 从而难以应用于对大型工程结构的减振. 因此, 将NES以高效和便捷的方式应用于工程减振, 仍然是有待研究的问题. 将NES以胞元的形式装配于振动的主结构中, 通过多个NES胞元的共同作用, 是一种有前景的减振策略. 文章在偏心转子激励下, 探究了多个NES胞元对远大于单个NES自身重量的振动结构的减振效果, 分析了多个NES胞元耦合主结构组成的系统的整体响应特征. 建立了NES胞元减振系统的运动微分方程, 采用复化平均法(complexification-averaging, CxA)导出系统的慢不变流形及稳态响应满足的近似解析表达式, 通过慢变流形的扰动运动微分方程对稳态解进行稳定性分析, 再利用伪弧长法获得系统响应的近似解, 分析了NES胞元的减振规律及系统响应规律, 最后利用龙格−库塔(Runge-Kutta, R-K)法进行数值验证. 结果表明, 通过多个NES胞元共同作用, 能够有效控制较大重量的主系统振动, 而且减振效率随NES胞元个数和重量的增加而显著提高, 共振区的响应状态随着胞元个数的增加从稳定状态、强调制状态、稳定状态依次变化. 因此, 本研究有助于推动NES的工程应用.Abstract: Nonlinear energy sink (NES) possesses numerous advantages, such as a wide vibration reduction frequency bandwidth and excellent damping performance. However, NES has no linear stiffness, which makes it difficult to drive NES vibrator with large weight, so it is difficult to be applied to reduce the vibration of large engineering structures. Therefore, the efficient and convenient application of NES in engineering vibration mitigation remains a subject of ongoing research. Assembling NES in the form of cells within the vibrating primary structure, allowing multiple NES cells to act collectively, is a promising vibration reduction strategy. The damping effects of multiple NES cells on a vibrating structure, significantly heavier than a single NES, under the excitation of an eccentric rotor are explored in this paper. The overall response characteristics of a system composed of multiple coupled NES cells and the primary structure are analyzed. The governing motion differential equations are established, and the Complexification-Averaging (CxA) method is employed to derive approximate analytical expressions for steady-state response and the slow invariant manifold of the system. Stability analysis of steady-state solutions is conducted through the perturbed motion differential equations of the slow manifold. The pseudo-arc length method is then utilized to obtain approximate solutions for the response of the system. The vibration reduction effect of the NES cells and the characteristics of the response of the system are studied, and verification is performed using the Runge-Kutta (R-K) method. The results demonstrate that by the collective action of multiple NES cells, the vibration of the primary structure with large weight can be controlled effectively. The vibration reduction efficiency increases notably with the number and weight of NES cells. The response state in the resonance region transitions from a stable state to a strongly modulated state, and then back to a stable state as the number of cells increases. Hence, this study contributes to the engineering application of NES.
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图 2 非线性刚度对NES胞元减振效果影响的分析
Figure 2. Analysis of the influence of nonlinear stiffness on NES cell damping effect
图 3 频率岛上分支及其下方主曲线的时程图与相位图
Figure 3. Time domain response and phase diagram of the upper branch of the frequency island and main curve below the it
图 4 共振区的时程图与相位图 (续)
Figure 4. Time domain response and phase diagram of the resonance region (continued)
表 1 NES胞元重量占比
Table 1. NES cell weight ratio
Number of Cells βs/% βl/% 1 0.69 1.6 2 1.38 3.2 3 2.07 4.8 4 2.76 6.4 
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