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On the decomposition of generalized eigenproblems for the free vibration analysis of cyclically symmetric finite element models
Author(s) -
Koohestani K.
Publication year - 2009
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.2770
Subject(s) - finite element method , quadrilateral , mathematics , discretization , vibration , kronecker delta , matrix (chemical analysis) , block (permutation group theory) , symmetry (geometry) , block matrix , mathematical analysis , geometry , eigenvalues and eigenvectors , structural engineering , physics , engineering , quantum mechanics , composite material , materials science
Free vibration analysis is a major part of any dynamic analysis. Natural frequencies and related mode shapes may be obtained from free vibration analysis as the solutions of generalized eigenproblems. Although the eigensolutions of large‐scale structures require large computational efforts, these solutions may be achieved simply for symmetric structures. We present an efficient method for the decomposition of generalized eigenproblems related to finite element models with cyclic symmetry (having nodes at the axis of symmetry) into eigensubproblems with significantly smaller dimensions. This decomposition is obtained by block diagonalization of a matrix with a special pattern known as a block circulant, using the concept of the Kronecker product and similarity transformations. The proposed method is applied to three finite element models discretized by triangular and four‐node quadrilateral plate and shell elements, and its efficiency, accuracy and simplicity are evaluated. Copyright © 2009 John Wiley & Sons, Ltd.