Calculating Derivatives of Repeated and Non-Repeated Eigenvalues without Explicit Use of their Eigenvectors

U Prells & MI Friswell (University of Wales Swansea)

AIAA Journal, Vol. 38, No. 8, August 2000, pp. 1426-1436

Abstract

The analysis of inverse problems in parametric model updating often require the sensitivities of eigenvalues. The calculation of these sensitivities is mathematically related to the derivatives of the eigenvalues with respect to the model parameters. A common method to calculate these derivatives is the Nelson method, which requires the eigenvectors. The method introduced in this paper is derived from the characteristic equation of the underlying general eigenvalue problem, and allows the derivatives of eigenvalues with respect to the model parameters to be calculated without explicit use of the eigenvectors. The method is extended for the case of repeated eigenvalues, which leads to restrictions on the parameterisation. For repeated eigenvalues of multiplicity 2 these restictions are formulated expicitly. Applications and limitations of the method are demonstrated by examples.

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