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Wittrick–Williams algorithm proof of bracketing and convergence theorems for eigenvalues of constrained structures with positive and negative penalty parameters
Author(s) -
Ilanko S.,
Williams F. W.
Publication year - 2007
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.2247
Subject(s) - eigenvalues and eigenvectors , mathematics , inertial frame of reference , penalty method , convergence (economics) , mathematical proof , bounded function , mathematical analysis , rayleigh quotient , mathematical optimization , geometry , classical mechanics , physics , quantum mechanics , economics , economic growth
The well‐established Wittrick–Williams algorithm is used to derive novel and general proofs that show that the eigenvalues of systems with constraints can be bracketed by replacing the constraints by positive and negative pairs of either ordinary or inertial penalty parameters. It is also shown that convergence occurs from both above and below when the numerical values of these parameters are increased towards infinity. The proofs are applicable in many contexts but are derived in that of structural systems, for which the eigenvalues are either buckling load factors or the squares of natural frequencies of vibration; ordinary penalty parameters are stiffnesses of translational and rotational springs; and inertial penalty parameters are either masses or rotary inertias. The penalty parameters can be used to constrain a system or to impose constraints between systems. It is shown that the use of inertial penalty parameters has several advantages compared with using ordinary ones. Then the p th eigenvalue of a system with n constraints is bounded closely from above by the ( p + n )th eigenvalue of the system with very large positive inertial penalty parameters and from below by the p th eigenvalue, when large negative values are used instead. This work is expected to enhance the versatility of numerical eigenproblem methods, e.g. the Rayleigh–Ritz method. Copyright © 2007 John Wiley & Sons, Ltd.