Symmetric Inverse Eigenvalue Vibration Problem and Its Application

Starek, Ladislav; Inman, Dan

doi:10.1006/mssp.2000.1349

Cited by 20 publications

(9 citation statements)

References 7 publications

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“…The book by Gladwell (1986b) and his follow-up review (Gladwell 1996) cover a broad scope of practices and references of IEPs for small mechanical systems. Applications of IEPs to model updating problems and fault detection problems for machine and structure diagnostics are discussed by Starek and Inman (2001). Other individual articles such as Barcilon (1979), Dai (1995), Gladwell (1984), Gladwell and Gbadeyan (1985), Gladwell (1986aGladwell ( , 1997Gladwell ( , 1999, Ram and Caldwell (1992) and Ram and Gladwell (1994) represent some typical applications to vibrating rods and beams.…”

Section: Applied Mechanicsmentioning

confidence: 99%

Structured inverse eigenvalue problems

Chu¹,

Golub²

2002

Acta Numerica

192

101

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An inverse eigenvalue problem concerns the reconstruction of a structured matrix from prescribed spectral data. Such an inverse problem arises in many applications where parameters of a certain physical system are to be determined from the knowledge or expectation of its dynamical behaviour. Spectral information is entailed because the dynamical behaviour is often governed by the underlying natural frequencies and normal modes. Structural stipulation is designated because the physical system is often subject to some feasibility constraints. The spectral data involved may consist of complete or only partial information on eigenvalues or eigenvectors. The structure embodied by the matrices can take many forms. The objective of an inverse eigenvalue problem is to construct a matrix that maintains both the specific structure as well as the given spectral property. In this expository paper the emphasis is to provide an overview of the vast scope of this intriguing problem, treating some of its many applications, its mathematical properties, and a variety of numerical techniques.

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Section: Applied Mechanicsmentioning

confidence: 99%

Structured inverse eigenvalue problems

Chu¹,

Golub²

2002

Acta Numerica

192

101

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“…In a large and complicated physical system [11,12,14], it is often impractical and impossible to obtain the entire spectral information. Thus, it is more sensible to consider a partially described IQEP (PD-IQEP) where only a prescribed partial set of eigenpairs is given.…”

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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“…For instance, the QIEP studied by Ram and Ehlay in [25] involves only symmetric tridiagonal matrix coefficients where two sets of eigenvalues are given. The QIEP studied by Starek and Inman in [27] is associated with nonproportional underdamped systems. Lancaster and Prells [23] considered the QIEP with symmetric and positive semi-definite damping C where complete information on eigenvalues and eigenvectors is given and all eigenvalues are simple and non-real.…”

Section: Introduction the Time-invariant Second Order Differential Smentioning

confidence: 99%

Structured Quadratic Inverse Eigenvalue Problem, I. Serially Linked Systems

Chu¹,

Buono²,

Yu³

2007

SIAM J. Sci. Comput.

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Abstract. Quadratic pencils arising from applications are often inherently structured. Factors contributing to the structure include the connectivity of elements within the underlying physical system and the mandatory nonnegativity of physical parameters. For physical feasibility, structural constraints must be respected. Consequently, they impose additional challenges on the inverse eigenvalue problems which intend to construct a structured quadratic pencil from prescribed eigeninformation. Knowledge of whether a structured quadratic inverse eigenvalue problem is solvable is interesting in both theory and applications. However, the issue of solvability is problem dependent and has to be addressed structure by structure. This paper considers one particular structure where the elements of the physical system, if modeled as a mass-spring system, are serially linked. The discussion recasts both undamped or damped problems in a framework of inequality systems that can be adapted for numerical computation. Some open questions are described.

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Symmetric Inverse Eigenvalue Vibration Problem and Its Application

Cited by 20 publications

References 7 publications

Structured inverse eigenvalue problems

Structured inverse eigenvalue problems

Solutions of the Partially Described Inverse Quadratic Eigenvalue Problem

Structured Quadratic Inverse Eigenvalue Problem, I. Serially Linked Systems

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