Solution Techniques for Large Eigenvalue Problems in Structural Dynamics
Lee, in-Won
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https://hdl.handle.net/2142/66899
Description
Title
Solution Techniques for Large Eigenvalue Problems in Structural Dynamics
Author(s)
Lee, in-Won
Issue Date
1981
Department of Study
Civil Engineering
Discipline
Civil Engineering
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Engineering, Civil
Language
eng
Abstract
This study deals with the determination of eigenvalues and eigenvectors of large algebraic systems. In particular, the methods developed are applicable to finding the natural frequencies and modes of vibration of large structural systems such as frames or continuous systems discretized in some way.
For distinct eigenvalues the method is an application of the modified Newton-Raphson method that turns out to be mode efficient than the standard competing schemes. It is also more efficient than the Robinson-Harris technique, which makes use of the ordinary Newton-Raphson method.
For close or multiple eigenvalues, the modified Newton-Raphson method is generalized to form an altogether new process. The entire set of close eigenvalues and their eigenvectors are found at the same time in a two-step procedure. The subspace of the approximate eigenvectors is first projected onto the subspace of the true eigenvectors. This part of the technique is rapid and efficient. If the eigenvalues are multiple, the results of the first stage indicate this fact and no further calculation is needed. If they are merely close, a single rotation in the newly found space solves the small eigenvalue problem and provides the final eigenvectors and eigenvalues of the close set. The method for performing the subspace projection can be expressed in terms of a simple extremum problem that generalizes the well-known extremum property of the eigenvectors.
Methods for the determination of initial approximations are presented. Theoretical convergence rates are derived and compared with actual experience in the solution of three example problems: (i) vibrations of a ten-story, ten-bay frame; (ii) Vibrations of a deep arch; and (iii) vibrations of square plates and of rectangular plates having aspect ratio close to unity.
The computational effort for these problems are compared for solution by the proposed method, by subspace iteration, and by the Robinson-Harris method (where applicable). In all the problems, the proposed method turned out to be the most efficient, the advantage increasing with the size of the problem. For instance, for comparable accuracy the subspace iteration method requires almost three times as much effort in obtaining the first four eigenvectors and eigenvalues as the proposed method.
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