Complex modes in damped linear dynamic systems

This paper is concerned about real and complex vibration modes in structural dynamics. In 1965, Caughey and O'Kelly provided a necessary and sufficient condition for classifying those systems that have a complete set of normal modes. However, there remains the challenge for researchers to develop a suitable theory for the study of complex modes. In this paper, a fundamental vibration model (D-Λ_k model) is advanced, which can adequately describe both proportionally and nonproportionally damped systems. By defining the concept of strongly/weakly complex system eigenvectors, a general analysis of modes and mode shapes is performed. This proposed method yields a simple criterion which characterizes the same class of systems specified by Caughey. At the same time it also is useful to study individual complex modes with the aid of a general D-Λ_k model.