HYBRID EXPANSION METHOD FOR FREQUENCY RESPONSES AND THEIR SENSITIVITIES, PART II: VISCOUSLY DAMPED SYSTEMS

Abstract

Frequency responses and their sensitivities have been broadly applied to finite element model updating, structural damage detection, dynamic optimization, vibration control and so on. In this paper, the modal acceleration method for the frequency responses and the double-modal acceleration method for their sensitivities, which have been discussed in the previous paper for undamped systems, are extended to viscously damped systems. The two methods are based on the hybrid expansion, power series expansion and modal superposition, of the inverse of the complex dynamic system matrix. Three steps are required to calculate the sensitivities using the proposed method. Firstly, frequency responses of a system excited by external forces are calculated by using the modal acceleration. Pseudo-force vector is then computed from the production of the sensitivity matrix and the frequency response vector. Finally, a second modal acceleration is applied to obtain the general frequency responses under the pseudo-forces, that is, the sensitivities. Two modal truncation schemes, middle-high-modal and low-high-modal truncation schemes, are presented according to the values of the exciting frequencies. The modal truncation errors of the modal acceleration method for frequency responses and the double-modal acceleration method for the sensitivities are also given to show the convergence of the proposed methods. Although the frequency responses and their sensitivities are discussed in this paper, the proposed methods are also valid for the frequency response functions, responses in time domain and their sensitivities. The results of a floating raft isolation system show that the proposed modal acceleration methods are efficient, especially for the sensitivity analysis. The modal truncation errors of the frequency responses and their sensitivities will reduce quickly when the two-modal acceleration methods are adopted.