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A direct method for the simultaneous updating of finite element mass, damping and stiffness matrices
Linear and Multilinear Algebra ( IF 1.1 ) Pub Date : 2022-06-25 , DOI: 10.1080/03081087.2022.2092047
Jiajie Luo 1 , Lina Liu 1 , Sisi Li 1 , Yongxin Yuan 1
Affiliation  

This paper is mainly concerned with a model updating problem for damped structural systems, which can be described as the following inverse quadratic eigenvalue problem (IQEP) and an optimal approximate problem (OAP). Problem IQEP: Given matrices Λ=diag{λ1,,λp}Cp×p and X=[x1,,xp]Cn×p with pn, λiλjλiλj for ij, i,j=1,,p, rank(X)=p and both Λ and X being closed under complex conjugation, find real symmetric matrices M, D and K such that MXΛ2+DXΛ+KX=0. Problem OAP: Given real symmetric matrices Ma,Da and Ka, find (M^,D^,K^)SE such that M^MaN2+D^DaN2+K^KaN2=min(M,D,K)SE(MMaN2+DDaN2+KKaN2), where SE is the solution set of Problem IQEP and N is a weighted Frobenius norm. The representation of the general solution to Problem IQEP and the explicit formula for the optimal approximate solution of Problem OAP are given by applying the singular value decomposition. A given numerical example shows that the updated model can be in good agreement with the modal test data.



中文翻译:

一种同时更新有限元质量、阻尼和刚度矩阵的直接方法

本文主要关注阻尼结构系统的模型更新问题,可以描述为以下逆二次特征值问题(IQEP)和最优近似问题(OAP)。问题 IQEP:给定矩阵Λ=诊断{λ1,……,λp}εCp×pX=[X1,……,Xp]εCn×ppn,λ iλ jλλj为了j,,j=1,……,p,X=p并且 Λ 和X在复共轭下都是闭合的,找到实对称矩阵MDK使得中号XΛ2+DXΛ+KX=0问题OAP:给定实对称矩阵中号A,DAKA, 寻找中号^,D^,K^εS这样中号^-中号A2+D^-DA2+K^-KA2=分钟中号,D,KεS中号-中号A2+D-DA2+K-KA2, 在哪里S是问题 IQEP 的解集,是加权弗罗贝尼乌斯范数。通过应用奇异值分解,给出了问题 IQEP 的通解的表示和问题 OAP 的最优近似解的显式公式。给定的数值例子表明,更新后的模型与模态试验数据能够很好地吻合。

更新日期:2022-06-25
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