Nonstationary response statistics of structures with hysteretic damping to evolutionary stochastic excitation演变随机激励下滞回阻尼结构的非平稳响应统计
Cao QY, Hu SLJ, Li HJ, 2024. Nonstationary response statistics of structures with hysteretic damping to evolutionary stochastic excitation. Probabilistic Engineering Mechanics, 77: 103659.DOI: 10.1016/j.probengmech.2024.103659
结构阻尼通常建模为线性滞回阻尼 (linear hysteretic damping, LHD),因此其对应的运动方程 (equation of motion, EOM) 是一个包含响应位移 Hilbert 变换的积分微分方程。由于系统本质上是非因果的,计算其在演变随机激励下的非平稳响应统计特性变得困难。本文提出了一种高效求解方法,用于获取各类非平稳响应统计的闭式解,包括演变功率谱 (evolutionary power spectrum, EPS)、相关函数和均方值。该新方法利用因果化时间概念,引入 “因果化” 脉冲响应函数 (impulse response function, IRF),通过极点-残差方法计算因果响应统计。该方法需要从系统频响函数 (frequency response function, FRF) 获得传递函数 (transfer function, TF) 的极点-残差形式,而频响函数可以从运动方程中获得。随后,将因果响应统计值移回原始时间,获得所需响应统计。为获得传递函数的极点-残差形式,需要两个步骤: (1) 对振子的频响函数进行 Fourier 逆变换以获得离散脉冲响应函数,(2) 使用 Prony 方法将该离散脉冲响应函数分解,以获得与传递函数相关的极点残差。对滞回阻尼和混合黏滞阻尼振子在白噪声、调制白噪声和调制 Kanai-Tajimi 模型随机激励下的算例进行蒙特卡罗模拟,验证了该方法的正确性。关键词: 滞回阻尼, 极点, 残差, 响应统计, 演变随机激励, 非因果The damping of a structure has often been modeled as linear hysteretic damping (LHD), so its corresponding equation of motion (EOM) is an integro-differential equation that involves the Hilbert transform of response displacement. As a result, the system is non-causal in nature, and it is challenging to compute its nonstationary response statistics under evolutionary stochastic excitation. This article develops an efficient solution method to obtain closed-form solutions for various nonstationary response statistics, including the evolutionary power spectrum (EPS), correlation function and mean square values. The novel solution method utilizes the concept of causalization time to introduce a “causalized” impulse response function (IRF), by which causal response statistics are computed based on a pole-residue approach. This approach requires obtaining a pole-residue form of the transfer function (TF) from the frequency response function (FRF) of the system, which is readily obtained from the EOM. Subsequently, the desired response statistics are obtained by shifting the causal response statistics back to the original time. To obtain the pole-residue form of the TF, two steps are necessary: (1) taking the inverse Fourier transform of the FRF of the oscillator to obtain a discrete IRF and (2) using the Prony-SS method to decompose this discrete IRF to obtain the pole residues associated with the TF. The correctness of the proposed method is numerically verified by Monte Carlo simulations through examples of hysteretic damping and mixed viscous-hysteretic damping oscillators that are subjected to white noise, modulated white noise and modulated Kanai–Tajimi model random excitations.
Keywords: Hysteretic damping; PoleResidue; Response statistics;Evolutionary stochastic excitation; Non-causal.Fig. 1. Computing the causal IRF h^c(t)
图 2: 纯滞回阻尼的脉冲响应函数与时间移位下因果脉冲响应函数Fig. 2. h(t) and h^c(t) by time shift for Example 1
Fig. 3. Normalized singular values of IRF
图 4: 纯滞回阻尼的原因果脉冲响应函数与 10, 50, 163, 180 项下重构的脉冲响应函数对比Fig. 4. Comparison between original h^c(t) and reconstructed IRFs with 10, 50, 163 and 180 terms in Example 1
图 5: 白噪声激励下滞回阻尼单自由度系统的均方响应计算Fig. 5. Computed E[X^2(t)] for the SDOF system with hysteretic damping to white noise excitation
Fig. 6. Computed mean square response E[X^2(t)] with different t_c values
图 7: 调制白噪声激励下滞回阻尼单自由度系统的演变功率谱密度计算Fig. 7. Computed S_X(t,ω) for the SDOF system with hysteretic damping to the modulated white noise excitation
图 8: 调制白噪声激励下滞回阻尼单自由度系统的自相关响应计算Fig. 8. Computed E[X(t_1)X(t_2)] for the SDOF system with hysteretic damping to the modulated white noise excitation
图 9: 左 y 轴: 调制函数; 右 y 轴: 调制白噪声激励下滞回阻尼单自由度系统的均方响应计算Fig. 9. Left y-axis: Modulating function; Right y-axis: Computed E[X^2(t)] for the SDOF system with hysteretic damping to the modulated white noise excitation
图 10: 参数为 h_0 = 0.5, ν_0 = 4π, B = 1 的 Kanai-Tajimi 功率谱密度模型Fig. 10. Kanai-Tajimi model S_F(ω) with h_0 = 0.5, ν_0 = 4π and B = 1
图 11: 调制 Kanai-Tajimi 激励下滞回阻尼单自由度系统的演变功率谱密度计算Fig. 11. Computed S_X(t,ω) for the SDOF system with hysteretic damping to the modulated Kanai–Tajimi excitation
图 12: 调制 Kanai-Tajimi 激励下滞回阻尼单自由度系统的自相关响应计算Fig. 12. Computed E[X(t_1)X(t_2)] for the SDOF system with hysteretic damping to the modulated Kanai–Tajimi excitation
图 13: 左 y 轴: 调制函数; 右 y 轴: 调制 Kanai-Tajimi 激励下滞回阻尼单自由度系统的均方响应计算Fig. 13. Left y-axis: Modulating function; Right y-axis: Computed E[X^2(t)] for the SDOF system with hysteretic damping to the modulated Kanai–Tajimi excitation
图 14: 混合黏滞阻尼的脉冲响应函数与时间移位下因果脉冲响应函数Fig. 14. h(t) and h^c(t) by time shift for Example 2
图 15: 混合黏滞阻尼的原因果脉冲响应函数及其重构的对比Fig. 15. Comparison between original h^c(t) and reconstructed one for Example 2
图 16: 白噪声激励下黏滞阻尼单自由度系统的均方响应计算Fig. 16. Computed E[X^2(t)] for the SDOF system with viscous-hysteretic damping to the white noise excitation
图 17: 左 y 轴: 调制函数; 右 y 轴: 调制白噪声激励下黏滞阻尼单自由度系统的均方响应计算Fig. 17. Left y-axis: Modulating function; Right y-axis: Computed E[X^2(t)] for the SDOF system with viscous-hysteretic damping to the modulated white noise excitation
图 18: 左 y 轴: 调制函数; 右 y 轴: 调制 Kanai-Tajimi 激励下黏滞阻尼单自由度系统的均方响应计算Fig. 18. Left y-axis: Modulating function; Right y-axis: Computed of E[X^2(t)] for the SDOF system with viscous-hysteretic damping to the modulated Kanai–Tajimi excitation
作者信息 | Authors
美国布朗大学 (Brown University) 应用数学系
S. L. James Hu, 通讯作者 (Corresp.)美国罗德岛大学 (University of Rhode Island) 海洋工程系Email: jameshu@uri.edu
中国海洋大学 (Ocean University of China) 山东省海洋工程重点实验室
律梦泽 M.Z. Lyu | 编辑 (Ed)
P.D. Spanos | 审校 (Rev)
陈建兵 J.B. Chen | 审校 (Rev)
彭勇波 Y.B. Peng | 审校 (Rev)