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An algorithm for exact eigenvalue calculations for rotationally periodic structures
Author(s) -
Williams F. W.
Publication year - 1986
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.1620230407
Subject(s) - eigenvalues and eigenvectors , stiffness , stiffness matrix , mathematics , algorithm , matrix (chemical analysis) , vibration , cover (algebra) , mathematical analysis , direct stiffness method , geometry , structural engineering , engineering , physics , mechanical engineering , materials science , quantum mechanics , composite material
An existing algorithm ensures that no eigenvalues are missed when using the stiffness matrix method of structural analysis, where the eigenvalues are the natural frequencies of undamped free vibration analyses or the critical load factors of buckling problems. The algorithm permits efficient multi‐level substructuring and gives ‘exact’ results when the member equations used are those obtained by solving appropriate differential equations. The present paper extends this algorithm to cover rotationally periodic (i.e. cyclically symmetric) three‐dimensional structures which are analysed by using complex arithmetic to obtain a stiffness matrix which involves only one of the rotationally repeating portions of the structure. Nodes and members are allowed to coincide with the axis of rotational periodicity and the resulting modes are classified. Rigid body freedoms are accounted for empirically, and the ‘exact’ member equations and efficient multi‐level substructuring of the earlier algorithm can be used when assembling the stiffness matrix of the repeating portion.