Fractional-order filter approximations for efficient stochastic response determination of wind-excited linear structural systems风激线性结构系统高效随机响应分析的分数阶滤波近似
![]()
Roncallo L, Mavromatis I, Kougioumtzoglou IA, Tubino F, 2024. Fractional-order filter approximations for efficient stochastic response determination of wind-excited linear structural systems. Probabilistic Engineering Mechanics, 78: 103696.DOI: 10.1016/j.probengmech.2024.103696
提出了湍流风随机激励模型的一类分数阶滤波近似方法。具体地，通过最小化原功率谱密度和近似功率谱密度的频域误差来确定未知滤波参数。结果表明，与标准整数阶滤波的极限情形相比，在相同数量待优化参数下，所提含有理阶导数项的分数阶滤波可提高精度。此外，所提滤波近似方法几乎可以零计算成本分析线性结构系统的稳态响应矩。该方法对滤波状态变量方程采用复模态分析处理，并依靠 Chachy 留数定理分析相应随机振动积分。与蒙特卡罗模拟数据估计值进行对比，结果表明其精度很高。关键词: 风工程, 随机动力学, 分数阶导数, 滤波近似, 随机振动积分A fractional-order filter approximation is developed for a wind turbulence stochastic excitation model. Specifically, the unknown filter parameters are determined by minimizing the error in the frequency domain between the original and the approximate power spectral densities. It is shown that compared to the limiting case of a standard integer-order filter, and for the same number of parameters to be optimized, the determined fractional-order filter with derivative elements of rational order yields enhanced accuracy. Further, the developed filter approximation enables the analytical calculation of stationary response moments of linear structural systems at practically zero computational cost. This is done by employing a complex modal analysis treatment of the filter state-variable equations, and by relying on Cauchy's residue theorem for evaluating analytically the related random vibration integrals. Comparisons with estimates based on Monte Carlo simulation data demonstrate a quite high degree of accuracy.
Keywords: Wind engineering; Stochastic dynamics; Fractional derivative; Filter approximation; Random vibration integral.![]()
Fig. 1. Complex domain of integration
Fig. 2. Integer-order (β = 1) and fractional-order (β = 5/6) filter approximations of the excitation PSD; Comparisons with the original PSD: (a) Linear scale; (b) Logarithmic scale
Fig. 3. Eigenvalues of matrix D of Eq. (39); Comparisons between numerical calculation (blue) and approximate analytical estimates (red)
Fig. 4. Fractional-order (β = 5/6) filter approximation of the excitation PSD; Comparisons between the original PSD, Eq. (46) using the numerically calculated complete set of n = 12 eigenvalues, and truncated Eq. (46) using n = 5 terms based on the approximate analytical expressions of Eqs. (52), (53), (54): a) linear scale, and b) logarithmic scale.
图 5: 自振频率与阻尼比参数值为 π/5 与 5% 的线性振子响应功率谱密度Fig. 5. Response PSD of a linear oscillator with parameter values ω_0 = π/5 and ξ = 5%; Comparisons between the integer-order (β = 1) filter approximation, the fractional-order (β = 5/6) filter approximation, and the exact result: (a) Linear scale; (b) Logarithmic scale
图 6: 自振频率与阻尼比参数值为 2π 与 0.2% 的线性振子响应功率谱密度Fig. 6. Response PSD of a linear oscillator with parameter values ω_0 = 2π and ξ = 0.2%; Comparisons between the integer-order (β = 1) filter approximation, the fractional-order (β = 5/6) filter approximation, and the exact result: (a) Linear scale; (b) Logarithmic scale
图 7: 自振频率与阻尼比参数值为 π/5 与 5% 的线性振子位移响应方差Fig. 7. Response displacement variance of a linear oscillator with parameter values ω_0 = π/5 and ξ = 5%; Comparisons between the integer-order (β = 1) filter approximation, the fractional-order (β = 5/6) filter approximation, the exact result obtained by numerical integration in the frequency domain, and MCS-based estimates (10,000 realizations)
Fig. 8. Eigenvalues of matrix -D of Eq. (39)
作者信息 | Authors
Luca Roncallo, 通讯作者 (Corresp.)意大利热那亚大学 (University of Genoa) 理工学院Email: luca.roncallo@edu.unige.it
美国哥伦比亚大学 (Columbia University) 土木工程与工程力学系
Ioannis A. Kougioumtzoglou美国哥伦比亚大学 (Columbia University) 土木工程与工程力学系
意大利热那亚大学 (University of Genoa) 理工学院
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)
