# A gradient based iterative algorithm for solving structural dynamics model updating problems

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## Abstract

The procedure of updating an existing but inaccurate model is an essential step toward establishing an effective model. Updating damping and stiffness matrices simultaneously with measured modal data can be mathematically formulated as following two problems. **Problem 1**: Let *M* _{ a }∈**SR** ^{ n×n } be the analytical mass matrix, and *Λ*=diag{*λ* _{1},…,*λ* _{ p }}∈**C** ^{ p×p }, *X*=[*x* _{1},…,*x* _{ p }]∈**C** ^{ n×p } be the measured eigenvalue and eigenvector matrices, where rank(*X*)=*p*, *p*<*n* and both *Λ* and *X* are closed under complex conjugation in the sense that \(\lambda_{2j} = \bar{\lambda}_{2j-1} \in\nobreak{\mathbf{C}} \), \(x_{2j} = \bar{x}_{2j-1} \in{\mathbf{C}}^{n} \) for *j*=1,…,*l*, and *λ* _{ k }∈**R**, *x* _{ k }∈**R** ^{ n } for *k*=2*l*+1,…,*p*. Find real-valued symmetric matrices *D* and *K* such that *M* _{ a } *XΛ* ^{2}+*DXΛ*+*KX*=0. **Problem 2**: Let *D* _{ a },*K* _{ a }∈**SR** ^{ n×n } be the analytical damping and stiffness matrices. Find \((\hat{D}, \hat{K}) \in\mathbf{S}_{\mathbf{E}}\) such that \(\| \hat{D}-D_{a} \|^{2}+\| \hat{K}-K_{a} \|^{2}= \min_{(D,K) \in \mathbf{S}_{\mathbf{E}}}(\| D-D_{a} \|^{2} +\|K-K_{a} \|^{2})\), where **S** _{ E } is the solution set of Problem 1 and ∥⋅∥ is the Frobenius norm. In this paper, a gradient based iterative (GI) algorithm is constructed to solve Problems 1 and 2. A sufficient condition for the convergence of the iterative method is derived and the range of the convergence factor is given to guarantee that the iterative solutions consistently converge to the unique minimum Frobenius norm symmetric solution of Problem 2 when a suitable initial symmetric matrix pair is chosen. The algorithm proposed requires less storage capacity than the existing numerical ones and is numerically reliable as only matrix manipulation is required. Two numerical examples show that the introduced iterative algorithm is quite efficient.

## Keywords

Model updating Iterative algorithm Damped structural system Partially prescribed spectral data Optimal approximation## Notes

### Acknowledgements

The authors would like to express their heartfelt thanks to the anonymous reviewers and Professor Alberto Carpinteri (Editor-in-Chief) for their constructive criticisms and helpful suggestions which substantially improved the quality of this paper.

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