A gradient based iterative algorithm for solving structural dynamics model updating problems
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A gradient based iterative algorithm for solving structural dynamics model updating problems Yongxin Yuan · Hao Liu
Received: 28 December 2011 / Accepted: 6 April 2013 © Springer Science+Business Media Dordrecht 2013
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 Ma ∈ SRn×n be the analytical mass matrix, and Λ = diag{λ1 , . . . , λp } ∈ Cp×p , X = [x1 , . . . , xp ] ∈ Cn×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 λ2j = λ¯ 2j −1 ∈ C, x2j = x¯2j −1 ∈ Cn for j = 1, . . . , l, and λk ∈ R, xk ∈ Rn for k = 2l + 1, . . . , p. Find real-valued symmetric matrices D and K such that Ma XΛ2 + DXΛ + KX = 0. Problem 2: Let Da , Ka ∈ SRn×n be the analytical damping and stiffness matrices. Find ˆ K) ˆ ∈ SE such that Dˆ − Da 2 + Kˆ − Ka 2 = (D, min(D,K)∈SE (D − Da 2 + K − Ka 2 ), where SE is the solution set of Problem 1 and · is the Frobenius norm. In this paper, a gradient based iterative (GI)
Research supported by the National Natural Science Foundation of China (No. 10926130). Y. Yuan () School of Mathematics and Physics, Jiangsu University of Science and Technology, Zhenjiang 212003, P.R. China e-mail: [email protected] H. Liu Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, P.R. China e-mail: [email protected]
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
1 Introduction Using finite element techniques, the equation of motion of an n-degree-of-freedom damped linear system in free vibration can be written in the form ¨ + Da q(t) ˙ + Ka q(t) = 0. Ma q(t)
(1)
Here, q(t) is the displacement vector and Ma , Da and Ka are the analytical mass, damping and stiffness matrices, respectively. In many practical applications, the matrix Ma is often symmetric positive definite and Ka is symmetric positive semi-definite. The damping matrix Da is hard to determine in practice, however, very
Meccanica
often, for the sake of computational convenience and other practical considerations, it is assumed to be symmetric. If a fundamental solution to (1) i
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