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      Wittrick-Williams algorithm proof of bracketing and convergence theorems for eigenvalues of constrained structures with positive and negative penalty parameters

      Ilanko, Sinniah; Williams, F.W.
      DOI
       10.1002/nme.2247
      Link
       www3.interscience.wiley.com
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      Citation
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      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.
      Permanent Research Commons link: https://hdl.handle.net/10289/2703
      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.
      Date
      2007
      Type
      Journal Article
      Publisher
      Wiley InterScience
      Collections
      • Science and Engineering Papers [3122]
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