HYBRID EXPANSION METHOD FOR FREQUENCY RESPONSES AND THEIR SENSITIVITIES, PART I: UNDAMPED SYSTEMS

Frequency responses and their sensitivities have been broadly applied to the areas of finite element model updating, structural damage detection, structural dynamic optimization and so on. A modal acceleration method for the frequency responses and a double-modal acceleration method for their sensitivities of undamped systems are derived in this paper. The two methods are based on the hybrid expansion, power series expansion and modal superposition, of the dynamic flexible 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 modal acceleration. A pseudo-force vector is then computed from the product of the sensitivity matrix and the frequency response vector. Finally, a second-modal acceleration is applied to obtain the general frequency responses, that is, the sensitivities, under the pseudo-forces. Two modal truncation schemes, middle–high–modal and low–high–modal truncation schemes, are presented according to the values of the excited frequencies. The modal truncated errors of the frequency responses and their sensitivities will be reduced quickly when the two-modal acceleration methods are adopted. Although only 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 two-dimensional frame show that the proposed modal acceleration methods are efficient, especially for the sensitivities.