Improved complex mode theory and truncation and acceleration of complex mode superposition

Abstract

The traditional complex mode theory is substantially improved. By means of the conjugacy of the complex characteristic value and vector, the formulae to compute the accuracy solution to the time-domain dynamic response for a symmetrical system with generalized linear viscous damping are established. Based on the formulation constructed, the truncation of the complex mode superposition is investigated. By employing the complex mode expansion of the system flexibility matrix discovered in this work, an accelerating method for the complex mode superposition after truncation is proposed. According to such improved complex mode theory, the system dynamical parameters, for example, the natural frequency and the complex vibration shape, are all the functions with respect to the mass, stiffness, and damping. However, the influence of the damping on the natural frequency may always be relatively slight. The numerical simulating investigation shows that the precise dynamic response can be obtained simply and conveniently in line with the formulae established. Meanwhile, the truncating and accelerating methods are effective for both the narrow or wide band external excitation and can provide the approximate solution to the dynamic response with better precision even when the frequency of the excitation is higher. Under certain conditions, the accelerating approach can be used to boost the accuracy of the truncated response.