# 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.

## References

- 1.Tisseur F, Meerbergen K (2001) The quadratic eigenvalue problem. SIAM Rev 43:235–286 MathSciNetADSCrossRefMATHGoogle Scholar
- 2.Ding F, Chen T (2005) Gradient based iterative algorithms for solving a class of matrix equations. IEEE Trans Autom Control 50:1216–1221 MathSciNetCrossRefGoogle Scholar
- 3.Ding F, Chen T (2006) On iterative solutions of general coupled matrix equations. SIAM J Control Optim 44:2269–2284 MathSciNetCrossRefMATHGoogle Scholar
- 4.Ding F, Liu PX, Ding J (2008) Iterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle. Appl Math Comput 197:41–50 MathSciNetCrossRefMATHGoogle Scholar
- 5.Yuan Y, Liu H (2012) An iterative updating method for undamped structural systems. Meccanica 47:699–706 MathSciNetCrossRefGoogle Scholar
- 6.Baruch M (1978) Optimization procedure to correct stiffness and flexibility matrices using vibration tests. AIAA J 16:1208–1210 ADSCrossRefMATHGoogle Scholar
- 7.Baruch M, Bar-Itzhack IY (1978) Optimal weighted orthogonalization of measured modes. AIAA J 16:346–351 ADSCrossRefGoogle Scholar
- 8.Berman A (1979) Mass matrix correction using an incomplete set of measured modes. AIAA J 17:1147–1148 ADSCrossRefGoogle Scholar
- 9.Berman A, Nagy EJ (1983) Improvement of a large analytical model using test data. AIAA J 21:1168–1173 ADSCrossRefGoogle Scholar
- 10.Wei FS (1980) Stiffness matrix correction from incomplete test data. AIAA J 18:1274–1275 ADSCrossRefMATHGoogle Scholar
- 11.Wei FS (1990) Mass and stiffness interaction effects in analytical model modification. AIAA J 28:1686–1688 ADSCrossRefGoogle Scholar
- 12.Wei FS (1990) Analytical dynamic model improvement using vibration test data. AIAA J 28:174–176 ADSGoogle Scholar
- 13.Yang YB, Chen YJ (2009) A new direct method for updating structural models based on measured modal data. Eng Struct 31:32–42 CrossRefGoogle Scholar
- 14.Yuan Y (2008) A model updating method for undamped structural systems. J Comput Appl Math 219:294–301 MathSciNetADSCrossRefMATHGoogle Scholar
- 15.Yuan Y (2009) A symmetric inverse eigenvalue problem in structural dynamic model updating. Appl Math Comput 213:516–521 MathSciNetCrossRefMATHGoogle Scholar
- 16.Friswell MI, Mottershead JE (1995) Finite element model updating in structural dynamics. Kluwer Academic, Dordrecht CrossRefMATHGoogle Scholar
- 17.Friswell MI, Inman DJ, Pilkey DF (1998) The direct updating of damping and stiffness matrices. AIAA J 36:491–493 ADSCrossRefGoogle Scholar
- 18.Pilkey DF (1998) Computation of a damping matrix for finite element model updating. PhD Thesis, Dept of Engineering Mechanics, Virginia Polytechnical Institute and State University Google Scholar
- 19.Kuo YC, Lin WW, Xu SF (2006) New methods for finite element model updating problems. AIAA J 44:1310–1316 ADSCrossRefGoogle Scholar
- 20.Chu DL, Chu M, Lin WW (2009) Quadratic model updating with symmetry, positive definiteness, and no spill-over. SIAM J Matrix Anal Appl 31:546–564 MathSciNetCrossRefMATHGoogle Scholar
- 21.Yuan Y (2008) An inverse quadratic eigenvalue problem for damped structural systems. In: Mathematical problems in engineering. Article ID 730358, 9 pp Google Scholar
- 22.Yuan Y, Dai H (2011) On a class of inverse quadratic eigenvalue problem. J Comput Appl Math 235:2662–2669 MathSciNetCrossRefMATHGoogle Scholar
- 23.Ding J, Ding F (2008) The residual based extended least squares identification method for dual-rate systems. Comput Math Appl 56:1479–1487 MathSciNetCrossRefMATHGoogle Scholar
- 24.Han LL, Ding F (2009) Multi-innovation stochastic gradient algorithms for multi-input multi-output systems. Digit Signal Process 19:545–554 CrossRefGoogle Scholar
- 25.Dehghan M, Hajarian M (2010) An iterative method for solving the generalized coupled Sylvester matrix equations over generalized bisymmetric matrices. Appl Math Model 34:639–654 MathSciNetCrossRefMATHGoogle Scholar
- 26.Ding F, Chen T (2005) Iterative least squares solutions of coupled Sylvester matrix equations. Syst Control Lett 54:95–107 MathSciNetCrossRefMATHGoogle Scholar
- 27.Ben-Israel A, Greville TNE (2003) Generalized inverses. Theory and applications, 2nd edn. Springer, New York MATHGoogle Scholar
- 28.Zimmerman D, Widengren M (1990) Correcting finite element models using a symmetric eigenstructure assignment technique. AIAA J 28:1670–1676 ADSCrossRefGoogle Scholar
- 29.Boisvert R, Pozo R, Remington K, Barrett R, Dongarra JJ. http://math.nist.gov/MatrixMarket