Optimization of damping performance of a nonlinear energy sink system under random excitation
HU Hongxiang1，CHEN Lincong1,2
1.College of Civil Engineering, Huaqiao University, Xiamen 361021, China;
2.Key Laboratory for Intelligent Infrastructure and Monitoring of Fujian Province, Huaqiao University, Xiamen 361021, China
摘要作为一类具有非线性刚度的典型非线性被动控制装置，非线性能量阱（nonlinear energy sinks ,NES）以其质量轻、频率鲁棒性强等优势引起了工程领域的广泛关注。目前，针对耦合NES的结构系统动力学研究主要是确定性载荷情形，仅少数涉及随机激励情形的研究。本文研究了随机激励下耦合NES的单自由度结构随机振动的参数优化问题。首先应用加权残值法，将原系统等效为具有精确平稳解的随机动力学系统，理论解和蒙特卡洛解在误差允许范围内吻合，显示所提的数值方法有效。然后利用原系统的平稳响应概率密度函数（probability density function，PDF）的近似解析表达式来构造目标函数，提出了一种以主结构位移与速度响应量均方最小为目标的NES参数优化设计策略，讨论了非线性能量阱的阻尼系数、非线性刚度系数、质量比等参数对减振性能的影响。结果表明：增加NES的质量比与阻尼系数，可以实现较强的减振性能，非线性刚度值对NES的减振性能的影响规律与质量比取值相关。相关工作可为NES的设计与应用提供参考。
关键词：随机振动；非线性能量阱；参数优化；加权残值法
Abstract：As a typical nonlinear passive control device with nonlinear stiffness, nonlinear energy sinks (NES) have attracted the attention of various engineering fields for its light weight and strong frequency robustness. At present, the studies dealing with the dynamics of structural systems containing NES were almost limited to deterministic load cases in literature. There were only very few investigations available for the random excitation. In this paper, the parameter optimization of random vibration of single-degree-of-freedom (SDOF) structure coupled with NES excited by Gaussian white noise is investigated. First, the original system are equivalent to stochastic dynamic systems with exact stationary solutions by using the method of weighted residual .The theoretical solution and Monte Carlo solution agree within the allowable error range, which shows the effectiveness of the proposed numerical method. Subsequently, the approximate analytic expression of probability density function (PDF) of the stationary response of the original system is utilized to construct the objective function. A parameter optimization strategy of NES with the target of minimizing the mean-square (MS) of displacement and velocity response of main structure is proposed. The effects of parameters such as damping, nonlinear stiffness and mass ratio of nonlinear energy sink on damping performance are discussed. The results show that the increase of mass ratio and damping of NES can achieve strong damping performance. The influence law of nonlinear stiffness on the damping performance of NES is related to the value of mass ratio. It can provide a feasible reference for the design and application of NES.
Key words: random vibration; nonlinear energy sink; parameters optimization; method of weighted residual
胡宏祥1，陈林聪1,2. 随机激励下非线性能量阱系统减振性能优化研究[J]. 振动与冲击, 2022, 41(24): 27-32.
HU Hongxiang1，CHEN Lincong1,2. Optimization of damping performance of a nonlinear energy sink system under random excitation. JOURNAL OF VIBRATION AND SHOCK, 2022, 41(24): 27-32.
[1] 陆泽琦，陈立群. 非线性被动隔振的若干进展[J]. 力学学报, 2017, 49(03): 550-564.
Lu Zeqi, Chen Liqun. Some recent progresses in nonlinear passive isolations of vibrations[J]. Chinese Journal of Theoretical and Applied Mechanics, 2017, 49(03): 550-564.
[2] 鲁正，王自欣，吕西林. 非线性能量阱技术研究综述[J]. 振动与冲击, 2020, 39(04): 1-16.
Lu Zheng, Wang Zixin, LÜ Xilin. A review on nonlinear energy sink technology[J]. Journal of Vibration and Shock, 2020, 39(04): 1-16.
[3] Lee Y S, Kerschen G, Vakakis A F, et al. Complicated dynamics of a linear oscillator with a light, essentially nonlinear attachment[J]. Physica D: Nonlinear Phenomena. 2005, 204(1-2): 41-69.
[4] Sapsis T P, Vakakis A F, Gendelman O V, et al. Efficiency of targeted energy transfers in coupled nonlinear oscillators associated with 1:1 resonance captures: Part II, analytical study[J]. Journal of Sound and Vibration. 2009, 325(1-2): 297-320.
[5] Starosvetsky Y, Gendelman O V. Attractors of harmonically forced linear oscillator with attached nonlinear energy sink. II: Optimization of a nonlinear vibration absorber[J]. Nonlinear Dynamics. 2007, 51(1-2): 47-57.
[6] Zang J, Chen L. Complex dynamics of a harmonically excited structure coupled with a nonlinear energy sink[J]. Acta Mechanica Sinica. 2017, 33(4): 801-822.
[7] Gendelman O V, Gorlov D V, Manevitch L I, et al. Dynamics of coupled linear and essentially nonlinear oscillators with substantially different masses[J]. Journal of Sound and Vibration. 2005, 286(1-2): 1-19.
[8] Sigalov G, Gendelman O V, Al-Shudeifat M A, et al. Resonance captures and targeted energy transfers in an inertially-coupled rotational nonlinear energy sink[J]. Nonlinear Dynamics. 2012, 69(4): 1693-1704.
[9] Luongo A, Zulli D. Dynamic analysis of externally excited NES-controlled systems via a mixed Multiple Scale/Harmonic Balance algorithm[J]. Nonlinear Dynamics. 2012, 70(3): 2049-2061.
[10] 李爽，楼京俊，柴凯，等. 柔性铰链型非线性能量阱设计与系统局部分岔特性分析[J]. 振动与冲击. 2020, 39(24): 156-163.
Li Shuang, Lou Jingjun, Chai Kai, et al. Design of a nonlinear energy sink flexible hinges and local bifurcation analysis of the system[J]. Journal of Vibration and Shock. 2020, 39(24): 156-163.
[11] Starosvetsky Y, Gendelman O V. Response regimes of linear oscillator coupled to nonlinear energy sink with harmonic forcing and frequency detuning[J]. Journal of Sound and Vibration. 2008, 315(3): 746-765.
[12] Malatkar P, Nayfeh A H. Steady-State dynamics of a linear structure weakly coupled to an essentially nonlinear oscillator[J]. Nonlinear Dynamics. 2006, 47(1-3): 167-179.
[13] Starosvetsky Y, Gendelman O V. Response regimes in forced system with non-linear energy sink: quasi-periodic and random forcing[J]. Nonlinear Dynamics. 2011, 64(1-2): 177-195.
[14] Xue J, Zhang Y, Ding H, et al. Vibration reduction evaluation of a linear system with a nonlinear energy sink under a harmonic and random excitation[J]. Applied Mathematics and Mechanics, 2020, 41(1): 1-14.
[15] 薛继仁，陈立群，张业伟，等. 单自由度NES在高斯白噪声随机激励下的响应分析[J]. 振动与冲击, 2020, 39(12): 235-241.
Xue Jiren, Chen Liqun, Zhang Yewei, et al. Response analysis of single degree of freedom NES under random excitation of gaussian white noise[J]. Journal of Vibration and Shock, 2020, 39(12): 235-241.
[16] Xiong H, Kong X, Yang Z, et al. Response regimes of narrow-band stochastic excited linear oscillator coupled to nonlinear energy sink[J]. Chinese Journal of Aeronautics, 2015, 28(2): 457-468.
[17] Shiroky I B, Gendelman O V. Essentially nonlinear vibration absorber in a parametrically excited system[J]. Journal of Applied Mathematics and Mechanics, 2008, 88(7): 573-596.
[18] Yang K, Zhang Y, Ding H, et al. The transmissibility of nonlinear energy sink based on nonlinear output frequency-response functions[J]. Communications in Nonlinear Science and Numerical Simulation, 2017, 44: 184-192.
