Closed-form expressions for eigenvalue and eigenvectors of stochastic symmetric matrices using the probability transformation method基于概率变换法的随机对称矩阵特征值与特征向量闭式表达
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Laudani R, Falsone G, 2024. Closed-form expressions for eigenvalue and eigenvectors of stochastic symmetric matrices using the probability transformation method. Probabilistic Engineering Mechanics, 78: 103706.DOI: 10.1016/j.probengmech.2024.103706
本研究采用概率变换法 (probability transformation method, PTM) 推导了随机实对称矩阵特征对的闭式概率密度函数 (probability density function, PDF)。特别地，概率变换法可以由系统不确定性的联合概率密度函数 (joint PDF, JPDF) 直接估计特征对的概率密度函数。通过一些数值应用研究了线性随机系统的随机性对自振频率和振型的影响。即使所研究结构样本很简单，但这只与作者使用的 Mathematica 软件有关，其在一定程度上限制了高维问题的精度。然而从理论角度看并不受此限制，问题的维度对方法精度没有影响。与蒙特卡罗模拟对比的分析结果证实了所提随机程序的有效性。关键词: 随机特征值问题, 概率密度函数, 精确随机解法This work shows the use of the Probability Transformation Method (PTM) for deriving a closed-form probability density function (PDF) of the eigenpair of stochastic real-valued symmetric matrices. In particular, the PTM allows the direct evaluation of the eigenpair PDF starting from the joint PDF (JPDF) of the system’s uncertainties. The impact of the linear stochastic systems’ randomness in the natural frequencies and mode shape is investigated through some numerical applications. Even if the structural samples investigated are intentionally simple, that aspect is only linked to the authors’ use of the Mathematica software that, in some ways, limits the resolution for high dimensional problems. From a theoretical perspective, though, this is not a restriction, and the problem’s dimension has no impact on the method’s accuracy. The obtained analytical results compared with Monte Carlo simulations have confirmed the goodness of the proposed stochastic procedure.Keywords: Random eigenvalue problem; Probability density function; Exact stochastic solution method.Fig. 1. Mean zero-crossing rates with n = 4
图 2: 例 1 自振频率的概率密度函数: (a) 随机情形 (1); (b) 随机情形 (2)Fig. 2. Example 1, PDF of the natural frequencies ω_1, ω_2 and ω_3: (a) Random scenario (1); (b) Random scenario (2)
图 3: 例 1 自振频率的概率密度函数: (a) 随机情形 (3); (b) 随机情形 (4); (c) 随机情形 (5); (d) 随机情形 (6)Fig. 3. Example 1, PDF of the natural frequencies ω_1, ω_2 and ω_3: (a) Random scenario (3); (b) Random scenario (4); (c) Random scenario (5); (d) Random scenario (6)
图 4: 例 1 随机情形 (1) 下随机频率的联合概率密度函数: (a) 第 1, 2 阶自振频率的联合概率密度函数; (b) 第 1, 3 阶自振频率的联合概率密度函数; (c) 第 2, 3 阶自振频率的联合概率密度函数Fig. 4. Example 1, JPDF of the random frequencies for the random scenario (1): (a) p_{ω_1,ω_2} (ω_1, ω_2); (b) p_{ω_1,ω_3} (ω_1, ω_3); (c) p_{ω_2,ω_3} (ω_2, ω_3)
图 5: 例 1 特征向量分量的概率密度函数: (a,b) 随机情形 (1); (c,d) 随机情形 (2)Fig. 5. Example 1, PDF of the eigenvector components, ψ_{i,2} and ψ_{i,3}: (a,b) Random scenario (1); (c,d) Random scenario (2)
图 6: 例 1 特征向量分量的概率密度函数: (a,b) 随机情形 (3); (c,d) 随机情形 (4)Fig. 6. Example 1, PDF of the eigenvector components, ψ_{i,2} and ψ_{i,3}: (a,b) Random scenario (3); (c,d) Random scenario (4)
图 7: 例 1 特征向量分量的概率密度函数: (a,b) 随机情形 (5); (c,d) 随机情形 (6)Fig. 7. Example 1, PDF of the eigenvector components, ψ_{i,2} and ψ_{i,3}: (a,b) Random scenario (5); (c,d) Random scenario (6)
图 8: 例 1 随机情形 (2) 下框架振型及其相关置信区间表示Fig. 8. Example 1, representation of the frame mode’s shapes with its related confidence interval for the random scenario (2)
Fig. 9. Example 2, PDF of the natural frequencies ω_1, ω_2 and ω_3
Fig. 10. Example 2, representation of the frame mode’s shapes with its related confidence interval
作者信息 | Authors
Rossella Laudani, 通讯作者 (Corresp.)意大利墨西拿大学 (University of Messina) 工程系Email: rlaudani@unime.it
意大利墨西拿大学 (University of Messina) 工程系
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)
