Constructing the physical parameters of a damped vibrating system from eigendata

In this paper we consider the inverse problem for a discrete damped mass–spring system where the mass, damping, and stiffness matrices are all symmetric tridiagonal. We first show that the model can be constructed from two real eigenvalues and three real eigenvectors or two complex conjugate eigenpa...

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Vydáno v:Linear algebra and its applications Ročník 428; číslo 2; s. 625 - 656
Hlavní autor: Bai, Zheng-Jian
Médium: Journal Article
Jazyk:angličtina
Vydáno: Elsevier Inc 15.01.2008
Témata:
Ill-posedness
Inverse problem
Regularization
Vibration system
Vibration system
Regularization
Ill-posedness
Inverse problem
ISSN:0024-3795
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Shrnutí:In this paper we consider the inverse problem for a discrete damped mass–spring system where the mass, damping, and stiffness matrices are all symmetric tridiagonal. We first show that the model can be constructed from two real eigenvalues and three real eigenvectors or two complex conjugate eigenpairs and a real eigenvector. Then, we study the general under-determined and over-determined problems. In particular, we provide the sufficient and necessary conditions on the given two real or complex conjugate eigenpairs so that the under-determined problem has a physical solution. However, for large model order, the construction from these data may be sensitive to perturbations. To reduce the sensitivity, we propose the minimum norm solution over the under-determined noisy data and the least squares solution to the over-determined measured data. We also discuss the physical realizability of the required model by the positivity-constrained regularization method for the ill-posed under-determined problem and the least squares optimization problems with positivity-constraints for the ill-posed over-determined problem. Finally, we give simple numerical examples to illustrate the effectiveness of our methods.
ISSN:0024-3795
DOI:10.1016/j.laa.2007.08.014