Closed-form expressions for eigenvalue and eigenvectors of stochastic symmetric matrices using the probability transformation method

IF 3 3区工程技术 Q2 ENGINEERING, MECHANICAL

Probabilistic Engineering Mechanics Pub Date : 2024-10-01 DOI:10.1016/j.probengmech.2024.103706

Rossella Laudani, Giovanni Falsone

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引用次数: 0

Abstract

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.

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使用概率变换法的随机对称矩阵特征值和特征向量闭式表达式

这项工作展示了如何利用概率变换法（PTM）推导出随机实值对称矩阵特征对的闭式概率密度函数（PDF）。特别是，PTM 可以从系统不确定性的联合概率密度函数（JPDF）出发，直接评估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固有频率和模态振型的影响。尽管所研究的结构样本有意简单化，但这只是与作者使用的 Mathematica 软件有关，该软件在某些方面限制了高维问题的分辨率。不过，从理论角度来看，这并不是一个限制，问题的维度对方法的准确性没有影响。所获得的分析结果与蒙特卡罗模拟结果相比较，证实了所提出的随机程序的优越性。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Probabilistic Engineering Mechanics 工程技术-工程：机械

CiteScore

3.80

自引率

15.40%

发文量

审稿时长

13.5 months

期刊介绍： This journal provides a forum for scholarly work dealing primarily with probabilistic and statistical approaches to contemporary solid/structural and fluid mechanics problems encountered in diverse technical disciplines such as aerospace, civil, marine, mechanical, and nuclear engineering. The journal aims to maintain a healthy balance between general solution techniques and problem-specific results, encouraging a fruitful exchange of ideas among disparate engineering specialities.