## On the Calculation of Second and Higher Order Eigenvector Derivatives

### MI Friswell (University of Wales Swansea)

### AIAA Journal of Guidance, Control, and Dynamics, Vol. 18, No. 4, July-August 1995, pp. 919-921

### Abstract

This note extends Nelson's method for the calculation of the first order eigenvector derivatives, or sensitivities, to the second and higher order eigenvector derivatives. Most interest has focused on the use and computation of the first order eigensystem derivatives. Adelman and Haftka and Murthy and Haftka have given extensive surveys of the field. Interest in the second and higher order derivatives is increasing particularly to estimate the eigensystems of modified structures. For large design parameter changes the linear approximation inherent in the use of first order derivatives may be inadequate. Brandon discussed the significance of the second order derivatives and used the second order sensitivities to assess the error in predicting the effect of structural modifications using first order sensitivities. Chen *et al.* used the second order derivatives in the eigensolution reanalysis of modified structures. Chen *et al.* estimated bounds on the eigenvalues of modified systems using the second order derivatives of the eigenvalues and eigenvectors. Brandon and Chen *et al.* calculated the second order sensitivities by writing the sensitivities as the series in the eigenvectors. The series approach requires all the eigenvalues and their associated eigenvectors, although the series is often truncated to ease the computational burden. Another disadvantage is that errors in the calculation of any eigenvector will affect all the eigenvector derivatives. Tan *et al.* used an iterative method to calculate the second order derivatives of the eigenvalues and eigenvectors of a general matrix. Rudisill and Chu gave a direct method to calculate the second and higher order eigenvalue and eigenvector derivatives, although the sparse nature of the matrices in structural dynamics problems is not used. Jankovic calculated the second and higher order derivatives of the general eigenproblem. The method generates a full order, fully populated matrix that is then inverted. This is likely to be computationally expensive for structural dynamics applications with many degrees of freedom.
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