Calculation of eigenvector derivatives for structures with repeated eigenvalues

Abstract

In structural optimization and system identification, eigenvector derivatives provide important information for updating design/model parameters. When the current parameter values yield repeated eigenvalues, it has not been possible previously to calculate unique eigenvector derivatives. Recent work has provided a method for determining unique eigenvalue derivatives for this case, but methods for calculating the eigenvector sensitivities have been incomplete. In this work, a complete method for calculation of repeated-root eigenvector derivatives is shown for the real, symmetric structural eigenproblem. The derivation is completed by using information from the second derivative of the eigen problem and is limited to the case of distinct eigenvalue sensitivities. As an example, the repeated-root eigenvector sensitivities are calculated for a simple three degree-of-freedom beam grillage. Comparisons of linear approximations (using these derivatives) to the calculated eigenvectors support the accuracy of the formulation.