dc.contributor.author Ilanko, Sinniah dc.contributor.author Williams, F.W. dc.date.accessioned 2009-07-22T23:58:21Z dc.date.available 2009-07-22T23:58:21Z dc.date.issued 2007 dc.identifier.citation Ilanko, S. & Williams, F. W. (2007). Wittrick-Williams algorithm proof of bracketing and convergence theorems for eigenvalues of constrained structures with positive and negative penalty parameters. International Journal for Numerical Methods in Engineering, 75(1), 83-102. en dc.identifier.uri https://hdl.handle.net/10289/2703 dc.description.abstract 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 pth 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 pth 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. en dc.language.iso en dc.publisher Wiley InterScience en_NZ dc.relation.uri http://www3.interscience.wiley.com/journal/117864974/abstract en dc.subject Wittrick-Williams algorithm en dc.subject eigenvalues en dc.subject Rayleigh-Ritz method en dc.subject penalty parameter en dc.title Wittrick-Williams algorithm proof of bracketing and convergence theorems for eigenvalues of constrained structures with positive and negative penalty parameters en dc.type Journal Article en dc.identifier.doi 10.1002/nme.2247 en dc.relation.isPartOf International Journal for Numerical Methods in Engineering en_NZ pubs.begin-page 83 en_NZ pubs.elements-id 34152 pubs.end-page 102 en_NZ pubs.issue 1 en_NZ pubs.volume 75 en_NZ
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