Abstract
We first give the representation of the general solution of the following inverse quadratic eigenvalue problem (IQEP): given , , and both and are closed under complex conjugation in the sense that , for , and for , find real-valued symmetric -diagonal matrices , and such that . We then consider an optimal approximation problem: given real-valued symmetric -diagonal matrices , find such that , where is the solution set of IQEP. We show that the optimal approximation solution is unique and derive an explicit formula for it.
1. Introduction
Throughout this paper, we will adopt the following notation. and denote the set of all complex and real matrices, respectively. denotes the set of all symmetric matrices in . and stand for the transpose and the Moore-Penrose generalized inverse of a real matrix . represents the identity matrix of size ; denotes the conjugate of the complex number . For , an inner product in is defined by , then is a Hilbert space. The matrix norm induced by the inner product is the Frobenius norm. Given two matrices and , the Kronecker product of and is defined by , and the stretching function is defined by where , is the th column vector of Furthermore, for a matrix let and stand for the two orthogonal projectors and
Using finite element techniques, vibrating structures such as beams, buildings, bridges, highways, and large space structures can be discretized to matrix second-order models (referred to as analytical models). A matrix second-order model of the free motion of a vibrating system is a system of differential equations of the formwhere and are the analytical mass, damping, and stiffness matrices. The system represented by (1.1) is called damped structural system. It is well known that all solutions of (1.1) can be obtained via the algebraic equationComplex numbers and nonzero complex vectors for which this relation holds are, respectively, the eigenvalues and eigenvectors of the system. It is known that (1.2) has finite eigenvalues over the complex field, provided that the leading matrix coefficient is nonsingular.
Due to the complexity of the structure, the finite element model is only an approximation to the practical structure. On the other hand, a part of the natural frequencies (eigenvalues) and corresponding mode shapes (eigenvectors) of the structure can be obtained experimentally by performing vibration tests [1]. Generally speaking, very often natural frequencies and mode shapes of an analytical model described by (1.2) do not match very well with experimentally measured frequencies and mode shapes. Thus, engineers would like to improve the analytical model of the structure such that the updated model predicts the observed dynamic behavior. Then, the updated model may be considered to be a better dynamic representation of the structure. This model can be used with greater confidence for the analysis of the structure under different boundary conditions or with physical structural changes.
For undamped systems (i.e., ), various techniques for updating mass and stiffness matrices using measured response data have been discussed by Baruch [2], Baruch and Bar-Itzhack [3], Berman [4], Berman and Nagy [5], and Wei [6, 7]. For damped structural systems, the theory and computation were first proposed by Friswell et al. [8, 9]; they applied the ideas in [2, 3] to minimize changes between the analytical and updated model subject to the spectral constraints. Kuo et al. [10] have recently proposed a direct method to close the weaknesses in [8] which seems more efficient and reliable. All these existing methods can reproduce the given set of measured data while updated matrices symmetry, but the connectivity of the original finite element model is not necessarily preserved, causing the addition of unwanted load paths.
The purpose of the work presented in this paper is to develop a new method for finite element model updating problems which preserves the connectivity of the original model. Assume that and are real-valued symmetric -diagonal matrices. Thus, the problem of updating mass, damping, and stiffness matrices simultaneously can be mathematically formulated as follows.
Problem IQEP. Given matrices , and both and are closed under complex conjugation in the sense that , for , and , for , find real-valued symmetric -diagonal matrices and such that Problem II. Let be the solution set of IQEP. Find such that
The paper is organized as follows. In Section 2, using the Kronecker product and stretching function of matrices, we give an explicit representation of the solution set of Problem IQEP. In Section 3, we show that there exists a unique solution in Problem II and present the expression of the unique solution Finally, in Section 4, a numerical algorithm to acquire the optimal approximation solution under the Frobenius norm sense is described, and a numerical example is provided.
2. The Solution of Problem IQEP
To begin with, we introduce a lemma [11]. Lemma 2.1. If then has a solution if and only if . In this case, the general solution of the equation can be described as , where is an arbitrary vector.
Let be the set of all real-valued symmetric -diagonal matrices, then is a linear subspace of , and the dimension of is
Define as where and is the th column vector of the identity matrix It is easy to verify that forms an orthonormal basis of the subspace that is, Now, if are -diagonal matrices, then can be expressed as where the real numbers are yet to be determined.
Define a matrix as where It is easy to verify that is a unitary matrix, that is, Using this transformation matrix, we have where and are, respectively, the real part and the imaginary part of the complex number ; and , are, respectively, the real part and the imaginary part of the complex vector for .
It follows from (2.5) and (2.6) that (1.3) can be equivalently written asSubstituting (2.3) into (2.7), we haveWhen setting We see that (2.8) is equivalent toIt follows from Lemma 2.1 that (2.10) with unknown vector has a solution if and only ifUsing Lemma 2.1 again, we know that (2.11) with respect to has a solution if and only ifwhere It follows from Lemma 2.1 that (2.12) with respect to is always solvable, and the general solution to the equation iswhere and is an arbitrary vector. Substituting (2.13) into (2.11) and applying Lemma 2.1, we obtainwhere is an arbitrary vector. Inserting (2.14) and (2.13) into (2.10) yields where is an arbitrary vector.
As a summary of the above discussion, we have proved the following result.
Theorem 2.2. Suppose that , and both and are closed under complex conjugation. The real matrices and are given by (2.5) and (2.6). Let be given as in (2.1), (2.9). Write Then the solution set of problem IQEP can be expressed as where are, respectively, given by (2.15), (2.14), and (2.13) with being arbitrary vectors.
3. The Solution of Problem II
It follows from Theorem 2.2 that the set is always nonempty. It is easy to verify that is a closed convex subset of . From the best approximation theorem (see [12]), we know there exists a unique solution in such that (1.4) holds.
We now focus our attention on seeking the unique solution in . For the real-valued symmetric -diagonal matrices and , it is easily seen that and can be expressed as the linear combinations of the orthonormal basis , that is, where are uniquely determined by the elements of and . Let Then, for any triple of matrices in (2.16), by the relations of (2.2) and (3.1), we see thatSubstituting (2.13), (2.14), and (2.15) into the relation of , we havewhere Therefore, Clearly, if and only ifNote that Therefore, implies that where Substituting (3.7) into yields where
Clearly, is equivalent toUpon substituting (3.7) and (3.9) into (2.13), (2.14), and (2.15), we obtainwhere is given by (3.8).
By now, we have proved the following result.
Theorem 3.1. Let the real-valued symmetric -diagonal matrices and be given. Then, Problem II has a unique solution, and the unique solution of Problem II can be expressed as where and are given by (3.10).
4. A Numerical Example
Based on Theorems 2.2 and 3.1, we can describe an algorithm for solving problem IQEP and Problem II as follows.
Example 4.1. Consider a five-DOF system
modelled analytically with mass, damping, and stiffness matrices given by
That is, are symmetric
3-diagonal matrices. The measured eigenvalue and eigenvector matrices and are given by
Using Algorithm 1, we obtain the
unique solution of Problem II as follows:
We define the residual asand the numerical results shown in the following table.
Therefore, the
prescribed eigenvalues (the diagonal elements of the matrix ) and
eigenvectors (the column vectors of the matrix ) are embedded
in the new model and the updated
matrices are also
symmetric 3-diagonal matrices, which implies that the structural connectivity
information of the analytical model is preserved.
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