Accurate methods for frequency responses and their sensitivities of proportionally damped systems

A modal acceleration method for frequency responses and a double modal acceleration method for their sensitivities are derived for proportionally damped systems in this paper. These two methods are based on the power series expansion and modal superposition of the dynamic flexible matrix. Three steps are required to calculate the sensitivities of the frequency responses by using the proposed methods. Two modal truncation schemes, middle–high-modal truncation and low–high-modal truncation, are presented according to the values of the excited frequencies. General stiffness and mass matrices, which are the combinations of stiffness and damping matrices, mass and damping matrices, respectively, are introduced before the power series expansion is applied. This makes the modal truncated errors of frequency responses and their sensitivities reduce quickly with an increase of the number of the power series item. The modal truncated errors of the frequency responses and their sensitivities are also presented to examine the advantages and convergence of the two modal acceleration methods. It can be seen that the methods for the frequency responses and their sensitivities of undamped systems can be looked as a particular case of the present 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 two-dimensional frame show that the proposed acceleration methods are efficient.