Rates of change of eigenvalues and eigenvectors in damped dynamic system

Rates of change of eigenvalues and eigenvectors of a damped linear discrete dynamic system with respect to the system parameters are presented. A nonproportional viscous damping model is assumed. Because of the nonproportional nature of the damping, the mode shapes and natural frequencies become complex, and as a consequence the sensitivities of eigenvalues and eigenvectors are also complex. The results are presented in terms of the complex modes and frequencies of the second-order system, and the use of rather undesirable state-space representation is avoided. The usefulness of the derived expressions is demonstrated by considering all example of a nonproportionally damped two-degree-of-freedom system.