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This paper is mainly concerned with a model updating problem for damped structural systems, which can be described as the following inverse quadratic eigenvalue problem (IQEP) and an optimal approximate problem (OAP). Problem IQEP: Given matrices Λ = diag { λ 1 , ... , λ p } ∈ C p × p and X = [ x 1 , ... , x p ] ∈ C n × p with p ≤ n , λ i ≠ λ j for i ≠ j , i , j = 1 , ... , p , rank (X) = p and both Λ and X being closed under complex conjugation, find real symmetric matrices M, D and K such that M X Λ 2 + D X Λ + K X = 0. Problem OAP: Given real symmetric matrices M a , D a and K a , find (M ^ , D ^ , K ^) ∈ S E such that ‖ M ^ − M a ‖ N 2 + ‖ D ^ − D a ‖ N 2 + ‖ K ^ − K a ‖ N 2 = min (M , D , K) ∈ S E (‖ M − M a ‖ N 2 + ‖ D − D a ‖ N 2 + ‖ K − K a ‖ N 2) , where S E is the solution set of Problem IQEP and ‖ ⋅ ‖ N is a weighted Frobenius norm. The representation of the general solution to Problem IQEP and the explicit formula for the optimal approximate solution of Problem OAP are given by applying the singular value decomposition. A given numerical example shows that the updated model can be in good agreement with the modal test data. [ABSTRACT FROM AUTHOR]

The finite element model errors mainly come from the complex parts of the geometry, boundary conditions and stress state of the structure. Therefore, the problem for updating mass and stiffness matrices can be reduced to an inverse problem for symmetric matrices with submatrix constraints (IP-MUP): Let Λ = d i a g (λ 1 , ... , λ p) ∈ R p × p and Φ = [ ϕ 1 , ... , ϕ p ] ∈ R n × p be the measured eigenvalue and eigenvector matrices with rank(Φ) = p. Find n × n symmetric matrices M and K such that K Φ = M Φ Λ , Φ ⊤ M Φ = I p , s. t. M (r) = M 0 , K (r) = K 0 , where M(r) and K(r) are the r × r leading principal submatrices of M and K, respectively. We then consider an optimal approximation problem (OAP): Given n × n symmetric matrices Ma and Ka. Find (M ˆ , K ˆ) ∈ S E such that ∥ K ˆ − K a ∥ 2 + ∥ M ˆ − M a ∥ 2 = min (M , K) ∈ S E (∥ K − K a ∥ 2 + ∥ M − M a ∥ 2) , where S E is the solution set of Problem IP-MUP. In this paper, the solvability condition for Problem IP-MUP is established, and the expression of the general solution of Problem IP-MUP is derived. Also, we show that the optimal approximation solution (M ˆ , K ˆ) is unique and derive an explicit formula for it. [ABSTRACT FROM AUTHOR]

Model updating for damped gyroscopic systems can be mathematically formulated as the following inverse quadratic eigenvalue problem (IQEP) and an optimal approximation problem (OAP). Problem IQEP: Given Λ = diag { λ 1 , ... , λ p } ∈ C p × p , X = [ x 1 , ... , x p ] ∈ C n × p , where p

Letandwe first solve the minimum Frobenius norm residual problem (Problem LSP):with unknown Toeplitz matricesXandY. We then consider a best approximation problem: given Toeplitz matricesand, findsuch thatwhereis the solution set of Problem LSP. We show that the best approximation solutionis unique and derive an explicit formula for it. [ABSTRACT FROM AUTHOR]

An efficient numerical method for updating finite element models with incomplete modal measured data using acceleration and displacement feedback technique is developed. By the method, the required control gain matrices of acceleration and displacement feedback are determined, and thus the optimal approximation mass matrix and stiffness matrix which satisfy the required orthogonality relation and eigenvalue equation are found under the Frobenius norm sense. The procedure of this method is very simple and easy to realize and the solution of this problem have a compact expression. The numerical examples show that the modal measured data are better incorporated into the updated model. [ABSTRACT FROM AUTHOR]

In this paper, the partial quadratic pole assignment problem with time delay is considered. We propose a multi-step method for solving this problem in which the unwanted eigenvalues are moved to desired values and all other eigenpairs remain unchanged. Our method is easy to realize and need not use the receptance matrix or solve the Sylvester equation. Numerical examples show that the proposed method is effective. [ABSTRACT FROM AUTHOR]

Abstract: Linear undamped gyroscopic systems are defined by three real matrices, , and ; the mass, stiffness, and gyroscopic matrices, respectively. In this paper an inverse problem is considered: given complete information about eigenvalues and eigenvectors, and , where the diagonal elements of are all purely imaginary, is of full row rank , and both and are closed under complex conjugation in the sense that for , find and such that . The solvability condition for the inverse problem and a solution to the problem are presented, and the results of the inverse problem are applied to develop a method for model updating. [Copyright &y& Elsevier]

Abstract: In this paper, we first give the representation of the general solution of the following inverse monic quadratic eigenvalue problem (IMQEP): given matrices , for , , , , and both and are closed under complex conjugation in the sense that , for , and , for , find real-valued symmetric matrices and such that . Then we consider a best approximation problem: given , find such that , where is a weighted Frobenius norm and is the solution set of IMQEP. We show that the best approximation solution is unique and derive an explicit formula for it. [ABSTRACT FROM AUTHOR]

Abstract: The matrix model updating problem (MMUP), considered in this paper, concerns updating a symmetric second-order finite element model so that the updated model can reproduce a given set of measured eigenvalues and eigenvectors by replacing the corresponding ones from the original model, and preserves the symmetry of the original model. The MMUP can be mathematically formulated as following two problems. Problem 1: Given , where and both and X are closed under complex conjugation in the sense that , for , and , for , find real-valued symmetric matrices D, K and a real-valued skew-symmetric matrix (that is, ) such that . Problem 2: Given real-valued symmetric matrices and a real-valued skew-symmetric matrix , find such that where is the solution set of and is the Frobenius norm. We provide the representation of the general solution of and show that the optimal approximation solution is unique and derive an explicit formula for it. [Copyright &y& Elsevier]

Abstract: Let be a nontrivial involution, i.e., . We say that is -symmetric if . The set of all -symmetric matrices is denoted by . In this paper, we first give the solvability condition for the following inverse eigenproblem (IEP): given a set of vectors in and a set of complex numbers , find a matrix such that and are, respectively, the eigenvalues and eigenvectors of . We then consider the following approximation problem: Given an matrix , find such that , where is the solution set of IEP and is the Frobenius norm. We provide an explicit formula for the best approximation solution by means of the canonical correlation decomposition. [Copyright &y& Elsevier]

Abstract: In this paper we develop an efficient numerical method for the finite element model updating of damped gyroscopic systems. This model updating of damped gyroscopic systems is proposed to incorporate the measured modal data into the finite element model to produce an adjusted finite element model on the damping and gyroscopic matrices that closely match the experimental modal data. [Copyright &y& Elsevier]

Abstract: In this paper, we first give the representation of the general solution of the following inverse eigenvalue problem (IEP): Given and a diagonal matrix find real-valued symmetric -diagonal matrices M and K such that We then consider an optimal approximation problem: Given real-valued symmetric -diagonal matrices , find such that where is the solution set of IEP. We show that the optimal approximation solution is unique and derive an explicit formula for it. [Copyright &y& Elsevier]

Abstract: In this paper, the following two problems are considered: Problem I Given a full column rank matrix , a diagonal matrix and matrices , find matrices such that , where and are, respectively, the leading principal submatrices of and . Problem II Given matrices with , find , such that , where is the solution set of Problem I. By applying the theory and methods of the algebraic inverse eigenvalue problems, the solvability condition and the general solution to Problem I are derived. The expression of the solution to Problem II is presented. [Copyright &y& Elsevier]

Abstract: Updating an existing but inaccurate structural dynamics model with measured data can be mathematically reduced to the problem of the best approximation to a given matrix pencil in the Frobenius norm under a given spectral constraint and a submatrix pencil constraint. In this paper, a direct method and the associated mathematical theories for solving this problem are proposed. It is shown that the best approximation solution is unique and an explicit expression of the solution is derived. Numerical examples show that the proposed method is quite accurate and efficient. [Copyright &y& Elsevier]