Calculation of derivative of complex modes using classical normal modes

In this paper, a method to calculate derivatives of eigenvectors of damped discrete linear dynamic system with respect to the system parameter is presented. Due to the non-proportional nature of the damping, the eigenvectors become complex, and as a consequence, the derivatives also become complex. The derivatives are calculated using small damping assumption, and the method avoids using the state-space approach. The results are obtained in terms of complex modes and frequencies of the second-order system, which in turn are related to the eigensolutions of the undamped system using perturbation method. Based on the derivatives, an expression for total change of the complex eigenvectors is obtained for a more general case when all the elements of mass, stiffness and damping matrices are varying. Application and accuracy of the derived expressions are demonstrated by considering numerical examples.