Dynamic response of non-classically damped structures via reduced-order complex modal analysis: Two novel truncation measures

The Caughey-O'Kelly condition, which defines a specific property of the damping matrix with respect to the mass and stiffness matrix, is violated in several cases of interest in earthquake engineering (e.g. seismic base isolation, supplemental energy dissipation devices, etc.). In these circumstances, non-classically damped systems are associated with complex-valued natural modes of vibration, and the complex modal superposition approach can be invoked as a generalization of the real-valued classical modal analysis to decouple the equations of motion and to compute the earthquake response. In analogy to the classical modal analysis, measures that quantify the error resulting from truncating up to a certain mode are desirable, especially for structures with hundreds or thousands of degrees of freedom. The concept of modal participating mass ratio, widely used in classical modal analysis, is no longer applicable to non-classically damped systems. The authors introduce here two novel measures related to a generalized modal mass ratio and to a modal dissipation ratio of each mode in the complex modal analysis framework. These two real-valued measures, whose sum is equal to one when all the modes are present, should be simultaneously considered to anticipate the importance of each mode in the complex modal superposition approach. Therefore, they are useful to determine the number of modes to retain in a reduced-order model. These two measures and the novel concepts are illustrated through some simple numerical examples of free vibration response and earthquake response of non-classically damped systems, including structures equipped with fluid viscous dampers and base-isolated buildings.