Eigensolutions of perturbed nearly defective matrices

Perturbation methods for approximating eigenvalues and eigenvectors of perturbed nearly defective (mistuned) systems have been discussed. It is shown that the mistuning causes Taylor expansions to be not uniformly valid even in small intervals of the perturbation parameter, subsequently rendering them useless for practical purposes. The problem is overcome by starting the perturbation expansion from an exactly defective (tuned) system ''close'' to the mistuned one. Asymptotic expansions are then obtained in terms of fractional powers of two parameters: the modification and the mistuning parameter. However, as the tuned system is unknown, an inverse problem has to be solved in order to determine it. An algorithm valid for the particular but frequent case of several couples of nearly coincident eigenvalues has been detailed for this problem; an outline of a more general case is also given. Illustrative examples are presented. (C) 1995 Academic Press Limited