An efficient algebraic method for the computation of natural frequency and mode shape sensitivities—Part II. Multiple natural frequencies

Abstract

In this paper an efficient algorithm whose stability is proved is derived for the computation of eigenpair derivatives for the real symmetric eigenvalue problem with multiple eigenvalues. The eigenpair derivatives can be obtained by solving algebraic equations with side conditions in the case of multiple eigenvalues, as well as distinct ones. For the eigenvalue problem with multiple natural frequencies, the adjacent eigenvectors are used in the algebraic equation as side conditions, they lie adjacent to the m (multiplicity of multiple eigenvalue) distinct eigenvalues, which appear when a design parameter varies. As an example to demonstrate the efficiency of the proposed method in the case of multiple eigenvalues, a cantilever beam is considered. The results of the proposed method for calculating the eigenpair derivatives are compared with those of Dailey's method which finds the exact eigenvector derivatives. The design parameter of the cantilever beam is its height.