用加速度和位移反馈修正无阻尼振动系统的一种迭代法

Abstract

Abstract: In this paper, an iterative method for simultaneously updating mass and stiffness matrices is presented. By the method, the required control gain matrices of acceleration and displacement feedback are determined, and thus the optimal approximation mass and stiffness matrices which satisfies the required eigenvalue equation are found under the Frobenius norm sense and the large number of unmeasured and unknown eigeninformation of the original model is preserved. Two numerical examples show that the proposed method is reliable and attractive.

Key words: Model updating, Iterative algorithm, Partially prescribed spectral information, No spill-over, Undamped structural system, Optimal approximation

CLC Number:

O241

Zhang Qian, Yuan Yongxin. An Iterative Updating Method for Undamped Vibration Systems Using Acceleration and Displacement Feedback[J].Acta mathematica scientia,Series A, 2018, 38(2): 358-371.

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References

[1] Friswell M I, Mottershead J E. Finite Element Model Updating in Structural Dynamics. Dordrecht:Kluwer Academic Publishers, 1995
[2] Baruch M. Optimization procedure to correct stiffness and flexibility matrices using vibration tests. AIAA Journal, 1978, 16:1208-1210
[3] Baruch M, Bar-Itzhack I Y. Optimal weighted orthogonalization of measured modes. AIAA Journal, 1978, 16:346-351
[4] Berman A, Nagy E J. Improvement of a large analytical model using test data. AIAA Journal, 1983, 21:1168-1173
[5] Wei F S. Structural dynamic model improvement using vibration test data. AIAA Journal, 1990, 28:175-177
[6] Wei F S. Mass and stiffness interaction effects in analytical model modification. AIAA Journal, 1990, 28:1686-1688
[7] Dai H. Optimal approximation of real symmetric matrix pencil under spectral restriction. Numer Math J Chin Univ, 1990, 12:177-187
[8] Carvalho J, Datta B N, Gupta A, Lagadapati M. A direct method for model updating with incomplete measured data and without spurious modes. Mechanical Systems and Signal Processing, 2007, 21:2715-2731
[9] Mao X B, Dai H. A quadratic inverse eigenvalue problem in damped structural model updating. Applied Mathematical Modelling, 2016, 40:6412-6423
[10] Yuan Y, Dai H. On a class of inverse quadratic eigenvalue problem. Journal of Computational and Applied Mathematics, 2011, 235:2662-2669
[11] Bai Z J, Chu D, Sun D. A dual optimization approach to inverse quadratic eigenvalue problems with partial eigenstructure. SIAM Journal on Scientific Computing, 2007, 29:2531-2561
[12] Kuo Y C, Lin W W, Xu S F. New model updating method for quadratic eigenvalue problems using a symmetric eigenstructure assignment. AIAA Journal, 2005, 43:2593-2598
[13] Carvalho J B, Datta B N, Lin W W, Wang C S. Symmetry preserving eigenvalue embedding in finite-element model updating of vibrating structures. Journal of Sound and Vibration, 2006, 290:839-864
[14] Saad Y. Projection and deflation method for partial pole assignment in linear state feedback. IEEE Trans Automatic Control, 1988, 33:290-297
[15] Xu S F, Qian J. Orthogonal basis selection method for robust partial eigenvalue assignment problem in second-order control systems. Journal of Sound and Vibration, 2008, 317:1-19
[16] Cai Y F, Qian J, Xu S F. Robust partial pole assignment problem for high order control systems. Automatica, 2012, 48:1462-1466
[17] Cai Y F, Xu S F. A new eigenvalue embedding approach for finite element model updating. Taiwanese Journal of Mathematics, 2010, 14:911-932
[18] Brahma S, Datta B. An optimization approach for minimum norm and robust partial quadratic eigenvalue assignment problems for vibrating structures. Journal of Sound and Vibration, 2009, 324:471-489
[19] Datta B N, Lin W W, Wang J N. Robust partial pole assignment for vibrating systems with aerodynamic effects. IEEE Trans Automatic Control, 2006, 51:1979-1984
[20] Datta B N. Finite element model updating, eigenstructure assignment and eigenvalue embedding for vibrating structures. Mechanical Systems and Signal Processing, 2002, 16:83-96
[21] Bai Z J, Datta B N, Wang J. Robust and minimum norm partial quadratic eigenvalue assignment in vibrating systems:A new optimization approach. Mechanical Systems and Signal Processing, 2010, 24:766-783
[22] Cai Y F, Qian J, Xu S F. The formulation and numerical method for partial quadratic eigenvalue assignment problems. Numer Linear Algebra Appl, 2011, 18:637-652
[23] Zhang J F, Ouyang H, Yang J. Partial eigenstructure assignment for undamped vibration systems using acceleration and displacement feedback. Journal of Sound and Vibration, 2014, 333:1-12
[24] Yuan Y X, Zhao W H, Liu H. Analytical dynamic model modification using acceleration and displacement feedback. Applied Mathematical Modelling, 2016, 40:9584-9593
[25] Lancaster P, Tismenetsky M. The Theory of Matrices. London:Academic Press, 1985
[26] Aubin J P. Applied Functional Analysis. New York:John Wiley & Sons, 1979
[27] Ben-Israel A, Greville T N E. Generalized Inverses. Theory and Applications. New York:Springer, 2003
[28] Mitra S K. A pair of simultaneous linear matrix equations A₁XB₁=C₁, A₂XB₂=C₂ and a matrix programming problem. Linear Algebra and Its Applications, 1990, 131:107-123

An Iterative Updating Method for Undamped Vibration Systems Using Acceleration and Displacement Feedback

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