Inverse problems for damped vibrating systems

Lancaster, Peter; Prells, Uwe

doi:10.1016/j.jsv.2004.05.003

Cited by 71 publications

(52 citation statements)

References 11 publications

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“…We want to stress that the very brief argument above gives rise to precisely the sufficient conditions discussed in [22] for the solvability of the QIEP. Furthermore, the matrix coefficients M , C and K of the particular solution Q(λ) to the QIEP can be expressed as…”

Section: Theorem 41 Assume That the Matrix λ Has Only Simple Eigenvmentioning

confidence: 94%

“…There is already a long list of studies on this subject. See, for example, [2,16,17,19,20,21,22,25,27] and the references contained therein. In this demonstration, we consider the special QIEP where the entire eigeninformation is given:…”

Section: Applicationsmentioning

confidence: 99%

“…If the coefficients of the matrix polynomial are self-adjoint, then additional properties such as the existence of a self-adjoint triple and the sign characteristic can be developed from the theory [11,Chapter 10]. As an example, we mention a case studied in [22] for a real symmetric quadratic λ-matrix Q(λ) under the assumption that all eigenvalues are simple and nonreal. Let X c = [x 1 , .…”

Section: Introductionmentioning

confidence: 99%

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Spectral decomposition of real symmetric quadratic $\lambda $-matrices and its applications

Chu

2009

Math. Comp.

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Abstract. Spectral decomposition provides a canonical representation of an operator over a vector space in terms of its eigenvalues and eigenfunctions. The canonical form often facilitates discussions which, otherwise, would be complicated and involved. Spectral decomposition is of fundamental importance in many applications. The well-known GLR theory generalizes the classical result of eigendecomposition to matrix polynomials of higher degrees, but its development is based on complex numbers. This paper modifies the GLR theory for the special application to real symmetric quadratic matrix polynomials, Q(λ) = Mλ 2 + Cλ + K, M nonsingular, subject to the specific restriction that all matrices in the representation be real-valued. It is shown that the existence of the real spectral decomposition can be characterized through the notion of real standard pair which, in turn, can be constructed from the spectral data. Applications to a variety of challenging inverse problems are discussed.

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Section: Theorem 41 Assume That the Matrix λ Has Only Simple Eigenvmentioning

confidence: 94%

Section: Applicationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Spectral decomposition of real symmetric quadratic $\lambda $-matrices and its applications

Chu

2009

Math. Comp.

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“…Recently, Chu, Kuo and Lin [13] constructed a physical solution for the SIQEP in which the matrices M , C and K are real and symmetric with the matrices M and K being positive definite and positive semidefinite, respectively. Lancaster and Prells [31] considered the SQIEP for the case when M , C and K are real symmetric matrices with both M and K being positive definite and C being positive semi-definite based on the complete information on simple and non-real eigenvalues and the associated eigenvectors. Kuo, Lin and Xu [30] presented a general solution for the SIQEP when the matrices M , C, and K are all real and symmetric with M being positive definite.…”

Section: Introductionmentioning

confidence: 99%

A smoothing Newton's method for the construction of a damped vibrating system from noisy test eigendata

Bai¹,

Ching²

2008

Numerical Linear Algebra App

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In this paper we consider an inverse problem for a damped vibration system from the noisy measured eigendata, where the mass, damping, and stiffness matrices are all symmetric positive definite matrices with the mass matrix being diagonal and the damping and stiffness matrices tridiagonal. To take into consideration the noise in the data, the problem is formulated as a convex optimization problem involving quadratic constraints on the unknown mass, damping, and stiffness parameters. Then we propose a smoothing Newtontype algorithm for the optimization problem, which improves a pre-existing estimate of a solution to the inverse problem. We show that the proposed method converges both globally and quadratically. Numerical examples are also given to demonstrate the efficiency of our method.

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“…In a very recent paper [4], Chu, Kuo and Lin gave special solutions of two types of IQEP, namely the standard IQEP and the monic IQEP. In the paper of [8], Lancaster and Prells solved symmetric M , C and K with M > 0, C ≥ 0 and K > 0 for a special IQEP that all eigenvalues are simple and non-real, and the corresponding eigenvector matrix X ∈ C n×n has a special structure X = X R (I − ιΘ), where ι = √ −1, X R is nonsingular and Θ is orthogonal.…”

Section: Introductionmentioning

confidence: 99%

Solutions of the Partially Described Inverse Quadratic Eigenvalue Problem

Kuo

Lin²,

2007

SIAM J. Matrix Anal. & Appl.

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In this paper, we consider to solve a general form of real and symmetric n × n matrices M , C, K with M being positive definite for an inverse quadratic eigenvalue problem (IQEP):a partially prescribed subset of k eigenvalues and eigenvectors (k ≤ n). Via appropriate choice of free variables in the general form of IQEP, for k = n: we solve (i) an IQEP with K semi-positive definite, (ii) an IQEP having additionally assigned n eigenvalues, (iii) an IQEP having additionally assigned r eigenpairs (r ≤ √ n) under closed complex conjugation; for k < n: we solve (i) a unique monic IQEP with k = n−1 which has an additionally assigned complex conjugate eigenpair, (ii) an IQEP having additionally assigned 2(n−k) complex eigenvalues with nonzero imaginary parts. Some numerical results are given to show the solvability of the above described IQEPs.

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Inverse problems for damped vibrating systems

Cited by 71 publications

References 11 publications

Spectral decomposition of real symmetric quadratic $\lambda $-matrices and its applications

Spectral decomposition of real symmetric quadratic $\lambda $-matrices and its applications

A smoothing Newton's method for the construction of a damped vibrating system from noisy test eigendata

Solutions of the Partially Described Inverse Quadratic Eigenvalue Problem

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