Complex modes in linear stochastic systems

Linear dynamic systems must generally be expected to ex-hibit non-proportional damping. Non-proportionally damped linear systems do not possess classical normal modes but possess complex modes. In this paper we analyze the complex modes arising in multiple degree-of-freedom non-proportionally damped discrete linear stochastic systems. Second-order statistics of the complex eigenvalues and eigen-vectors are presented by assuming the randomness of the sys-tem is small so that the first-order perturbation approach is valid. The proposed method is illustrated by considering nu-merical example based on a linear array of damped spring-mass oscillators. It is shown that the approach can predict the statistics of eigenvalues and eigenvectors with good accuracy when compared with independent Monte Carlo simulations.