Derivatives of Complex Eigenvectors using Nelson's Method

MI Friswell (University of Wales Swansea) & S Adhikari (Cambridge University)

AIAA Journal, Vol. 38, No. 12, December 2000, pp. 2355-2357


The calculation of derivatives of natural frequencies and mode shapes with respect to model parameters is vital for design optimisation, model updating, fault detection and many other applications. The methods to calculate these derivatives are well established for undamped structures. Fox and Kapoor calculated the derivative of the eigenvectors by expressing these derivatives as a linear combination of the undamped eigenvectors. Nelson introduced the approach, extended in this paper, where only the eigenvector of interest was required. Adhikari1 extended the method of Fox and Kapoor to systems with non-proportional damping using the mass, damping and stiffness matrices directly. Lee et al. calculated the eigenvector derivatives of self-adjoint systems using a similar approach to Nelson. This technical note extends Nelson's method to non-proportionally damped systems with complex modes. This method has the great advantage that only the eigenvector of interest is required. The method proposed by Adhikari1 obtained the eigenvector derivative as a linear combination of all the eigenvectors. For large scale structures, with many degrees of freedom, obtaining all the eigenvectors is a computationally expensive task. Both self-adjoint and non self-adjoint systems are considered.

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