ANALYSIS ON VIBRATION SUPPRESSION RESPONSE OF NONLINEAR ENERGY SINK WITH COMBINED NONLINEAR DAMPING
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摘要: 非线性能量阱是一种振动能量吸收装置, 其在结构振动抑制中具有十分重要的作用. 文章对具有组合非线性阻尼非线性能量阱的系统进行振动抑制相关的分析. 首先对具有组合非线性阻尼非线性能量阱的系统进行理论模型的描述, 对系统模型的运动方程利用复变量平均法进行推导, 得到系统的慢变方程. 其次对系统的慢变方程运用多尺度法进行强调制响应的分析, 通过对系统进行慢不变流形和相轨迹的研究, 描述系统强调制响应发生的条件基础. 此外, 还利用一维映射对系统进行分析, 揭示外激励幅值对强调制响应存在时频率失谐系数取值区间的影响规律. 最后利用能量谱、时间响应和庞加莱映射对耦合组合非线性阻尼非线性能量阱系统进行了振动抑制的相关研究, 揭示组合非线性阻尼的非线性能量阱不同阻尼比、阻尼和刚度对其振动抑制效果的影响规律, 得出组合非线性阻尼非线性能量阱和主结构响应存在一致性的现象, 并验证所提出的组合非线性阻尼非线性能量阱模型具有较好的振动抑制能力.Abstract: Nonlinear energy sink is a kind of vibration energy absorption device, which plays an important role in vibration suppression of structure. In this paper, the correlation analysis of vibration suppression for a system with combined nonlinear damping nonlinear energy sink is carried out. Firstly, the theoretical model of the system with combined nonlinear damping nonlinear energy sink is described. The motion equations of the system model are derived by using the complex variable average method, and the slow variable equations of the system are obtained. Secondly, the slow variable equations of the system are analyzed by using the multi-scale method. By studying the slow invariant manifold and phase trajectories of the system, the condition basis of the strongly modulated response of the system is described. In addition, the influence law of the external excitation amplitude on the frequency detuning coefficient interval in the presence of the strongly modulated response is revealed by analyzing the system with one-dimensional mapping. Finally, the energy spectrum, time response and Poincare mapping are applied to study the vibration suppression of the system with combined nonlinear damping nonlinear energy sink, the influence law of different damping ratio, damping and stiffness of nonlinear energy sink on its vibration suppression effect is revealed. Meanwhile, it is found that the response of the nonlinear energy sink with combined nonlinear damping is consistent with that of the main structure. In addition, it is verified that the nonlinear energy sink with combined nonlinear damping proposed in this study has good vibration suppression ability.
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图 1 具有组合非线性阻尼NES系统的理论模型
Figure 1. Theoretical model of a system with combined nonlinear damping NES
图 2 当k23 = 4/3, λ221 = λ223 = 0.2时SIM图, 带箭头的点线表示跳跃轨迹, 虚线表示不稳定部分, 实线表示稳定部分.
Figure 2. SIM when k23 = 4/3, λ221 = λ223 = 0.2, the dotted line with arrow represents the jump trajectory, the dotted line stands for the unstable part, and the solid line denotes the stable part
图 3 当λ221 = 0.2, λ223 = 0.2, k23 = 4/3, A = 3时系统的相轨迹图
Figure 3. Phase trajectories of the system when λ221 = 0.2, λ223 = 0.2, k23 = 4/3, A = 3
图 4 当A = 3时, δ从−8.7 ~ 31.4变化的系统一维映射图
Figure 4. One dimensional mapping of system as δ changes from −8.7 to 31.4 when A = 3
图 5 当k23 = 4/3时, 不同阻尼比及不同阻尼NES的主结构能量谱图
Figure 5. The energy spectrum of the main structure of NES with different damping ratios and different damping when k23 = 4/3
图 6 不同阻尼NES的时间响应和庞加莱映射图, 其中红点表示系统达到稳定状态,
xN, x1表示NES和主结构的位移, vN, v1代表NES和主结构的速度
Figure 6. Time response and poincare mapping of NES with different damping where the red dot indicates that the system reaches steady state, xN, x1 represent the displacement of NES and the main structure, and vN, v1 stand for the velocity of NES and main structure
图 7 当λ223 = 0.3, λ221 = 0.03时, 不同刚度NES的主结构能量谱对比图
Figure 7. Comparison of energy spectrum of main structure with different stiffness NES when λ223 = 0.3, λ221 = 0.03
图 8 不同刚度和不同频率的时间响应和庞加莱映射图
Figure 8. Time response and Poincare mapping with different stiffness and different frequencies
8 不同刚度和不同频率的时间响应和庞加莱映射图(续)
8. Time response and Poincare mapping with different stiffness and different frequencies (continued)
图 9 具有不同NES主结构的能量谱对比图
Figure 9. Comparison of energy spectrum of main structures with different NES
表 1 当A变化时SMR存在的δ区间
Table 1. δ interval where SMR exists when A changes
A δ interval where SMR exists A = 2 δ ∈ [−7.0, 18.6] A = 5 δ ∈ [−10.8, 56.3] A = 7 δ ∈ [−15.3, 80.5] 
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