Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 30840
An Iterative Updating Method for Damped Gyroscopic Systems

Authors: Yongxin Yuan


The problem of updating damped gyroscopic systems using measured modal data can be mathematically formulated as following two problems. Problem I: Given Ma ∈ Rn×n, Λ = diag{λ1, ··· , λp} ∈ Cp×p, X = [x1, ··· , xp] ∈ Cn×p, where p<n and both Λ and X are closed under complex conjugation in the sense that λ2j = λ¯2j−1 ∈ C, x2j = ¯x2j−1 ∈ Cn for j = 1, ··· , l, and λk ∈ R, xk ∈ Rn for k = 2l + 1, ··· , p, find real-valued symmetric matrices D,K and a real-valued skew-symmetric matrix G (that is, GT = −G) such that MaXΛ2 + (D + G)XΛ + KX = 0. Problem II: Given real-valued symmetric matrices Da, Ka ∈ Rn×n and a real-valued skew-symmetric matrix Ga, find (D, ˆ G, ˆ Kˆ ) ∈ SE such that Dˆ −Da2+Gˆ−Ga2+Kˆ −Ka2 = min(D,G,K)∈SE (D− Da2 + G − Ga2 + K − Ka2), where SE is the solution set of Problem I and · is the Frobenius norm. This paper presents an iterative algorithm to solve Problem I and Problem II. By using the proposed iterative method, a solution of Problem I can be obtained within finite iteration steps in the absence of roundoff errors, and the minimum Frobenius norm solution of Problem I can be obtained by choosing a special kind of initial matrices. Moreover, the optimal approximation solution (D, ˆ G, ˆ Kˆ ) of Problem II can be obtained by finding the minimum Frobenius norm solution of a changed Problem I. A numerical example shows that the introduced iterative algorithm is quite efficient.

Keywords: Model Updating, iterative algorithm, Optimal approximation, gyroscopic system, partially prescribed spectral data

Digital Object Identifier (DOI):

Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1143


[1] P. Lancaster, M. Tismenetsky. The Theory of Matrices. 2rd Edition, Academic Press, London, 1985.
[2] F. Tisseur, K. Meerbergen. The quadratic eigenvalue problem. SIAM Review, 43 (2001) 235-286.
[3] M. Baruch. Optimization procedure to correct stiffness and flexibility matrices using vibration tests. AIAA Journal, 16 (1978) 1208-1210.
[4] M. Baruch, I. Y. Bar-Itzhack. Optimal weighted orthogonalization of measured modes. AIAA Journal, 16 (1978) 346-351.
[5] A. Berman. Mass matrix correction using an incomplete set of measured modes. AIAA Journal, 17 (1979) 1147-1148.
[6] A. Berman, E. J. Nagy. Improvement of a large analytical model using test data. AIAA Journal, 21 (1983) 1168-1173.
[7] F. S. Wei. Stiffness matrix correction from incomplete test data. AIAA Journal, 18 (1980) 1274-1275.
[8] F. S. Wei. Mass and stiffness interaction effects in analytical model modification. AIAA Journal, 28 (1990) 1686-1688.
[9] F. S. Wei. Analytical dynamic model improvement using vibration test data. AIAA Journal, 28 (1990) 174-176.
[10] Y.B. Yang, Y. J. Chen, T. W. Hsu. Direct updating method for structural models based on orthogonality constraints. Mechanics of Advanced Materials and Structures, 16 (2009) 390-401.
[11] Y.B. Yang, Y. J. Chen. A new direct method for updating structural models based on measured modal data. Engineering Structures, 31 (2009) 32-42.
[12] Y. Yuan. A model updating method for undamped structural systems. Journal of Computational and Applied Mathematics, 219 (2008) 294- 301.
[13] M. I. Friswell, J. E. Mottershead. Finite Element Moodel Updating in Structural Dynamics. Dordrecht: Klumer Academic Publishers, 1995.
[14] M. I. Friswell, D. J. Inman, D. F. Pilkey. The direct updating of damping and stiffness matrices. AIAA Journal, 36 (1998) 491-493.
[15] D. F. Pilkey. Computation of a damping matrix for finite element model updating. Ph. D. Thesis, Dept. of Engineering Mechanics, Virginia Polytechnical Institute and State University, 1998.
[16] Y. C. Kuo, W. W. Lin, S. F. Xu. New methods for finite element moodel updating problems. AIAA Journal, 44 (2006) 1310-1316.
[17] D. L. Chu, M. Chu, W. W. Lin. Quadratic model updating with symmetry, positive definiteness, and no spill-over. SIAM Journal on Matrix Analysis And Applications, 31 (2009) 546-564.
[18] Y. Yuan. An inverse quadratic eigenvalue problem for damped structural systems. Mathematical Problems in Engineering, (2008), Article ID 730358, 9 pages.
[19] B. N. Datta, D. R. Sarkissian. Feedback control in distributed parameter gyroscopic systems: a solution of the partial eigenvalue assignment problem. Mechanical Systems and signal Processing, 16 (2002) 3-17.
[20] C. H. Guo. Numerical solution of a quadratic eigenvalue problem. Linear Algebra and its Applications, 385 (2004) 391-406.
[21] L. Meirovitch. A new method of solution of the eigenvalue problem for gyroscopic systems. AIAA Journal, 12 (1974) 1337-1342.
[22] R. H. Plaut. Alternative formulations for discrete gyroscopic eigenvalue problems. AIAA Journal, 14 (1976) 431-435.
[23] J. Qian, W. W. Lin. A numerical method for quadratic eigenvalue problems of gyroscopic systems. Journal of Sound and Vibration, 306 ( 2007) 284-296.
[24] D. R. Sarkissian. Theory and computations of partial eigenvalue and eigenstructure assignment problems in matrix second-order and distributed-parameter systems. Ph. D. thesis: Department of Mathematical Science, Northern Illinois University, 2001.
[25] A. Ben-Israel, T. N. E. Greville. Generalized Inverses. Theory and Applications (second ed). New York: Springer, 2003.