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Improved first-order approximation of eigenvalues and eigenvectors

Improved first-order approximation of eigenvalues and eigenvectors
Improved first-order approximation of eigenvalues and eigenvectors
A method based on reduced basis approximation concepts is presented for improved first-order approximation of eigenvalues and eigenvectors of modified structural dynamic systems. The terms of a local approximation based on Taylor or matrix power series are used as basis vectors for approximating the perturbed eigenparameters.
For each eigenmode, a reduced eigensystem is generated by using the baseline eigenvector and the first-order approximation term as Ritz vectors. The solution of the reduced eigensystem leads to two possible estimates of each perturbed eigenvalue and eigenvector. Criteria for selection of the best approximation are presented. The
zero- and first-order Rayleigh quotient approximation can be directly recovered as special cases of the present method. Results are presented for approximate dynamic reanalysis of a 25-bar planar truss structure. It is shown that high-quality approximation of the perturbed eigenparameters can be obtained for moderate perturbations in the stiffness and mass matrices. For very large local perturbations of the structure, including deletion of some structural members, it is shown that the present method yields reasonable-quality approximations.
0001-1452
1721-1727
Nair, Presanth B.
c19c2a6e-825f-4cb2-b7ca-16cfdc6eac8a
Keane, Andrew J.
26d7fa33-5415-4910-89d8-fb3620413def
Langley, Robin S.
a1935f7e-ec85-4930-a392-cd3a28071606
Nair, Presanth B.
c19c2a6e-825f-4cb2-b7ca-16cfdc6eac8a
Keane, Andrew J.
26d7fa33-5415-4910-89d8-fb3620413def
Langley, Robin S.
a1935f7e-ec85-4930-a392-cd3a28071606

Nair, Presanth B., Keane, Andrew J. and Langley, Robin S. (1998) Improved first-order approximation of eigenvalues and eigenvectors. AIAA Journal, 36 (9), 1721-1727.

Record type: Article

Abstract

A method based on reduced basis approximation concepts is presented for improved first-order approximation of eigenvalues and eigenvectors of modified structural dynamic systems. The terms of a local approximation based on Taylor or matrix power series are used as basis vectors for approximating the perturbed eigenparameters.
For each eigenmode, a reduced eigensystem is generated by using the baseline eigenvector and the first-order approximation term as Ritz vectors. The solution of the reduced eigensystem leads to two possible estimates of each perturbed eigenvalue and eigenvector. Criteria for selection of the best approximation are presented. The
zero- and first-order Rayleigh quotient approximation can be directly recovered as special cases of the present method. Results are presented for approximate dynamic reanalysis of a 25-bar planar truss structure. It is shown that high-quality approximation of the perturbed eigenparameters can be obtained for moderate perturbations in the stiffness and mass matrices. For very large local perturbations of the structure, including deletion of some structural members, it is shown that the present method yields reasonable-quality approximations.

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Published date: 1998

Identifiers

Local EPrints ID: 21198
URI: https://eprints.soton.ac.uk/id/eprint/21198
ISSN: 0001-1452
PURE UUID: 49c106ff-9413-4544-ae54-1011832406e3

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Date deposited: 09 Nov 2006
Last modified: 17 Jul 2017 16:27

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Contributors

Author: Presanth B. Nair
Author: Andrew J. Keane
Author: Robin S. Langley

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