Computation of frequency responses and their sensitivities for undamped systems

This paper presents a numerical method for calculating the frequency response and its sensitivity of an undamped system in a frequency interval. The Sturm sequence number is first used to adaptively determine the number of the lowest modes that need to be calculated. The corresponding modes can be computed by the Lanczos or subspace iteration methods. The complementary portion of contribution of these computed modes is then transformed into the solution to a new system by using the mass orthogonality. The solution of the new system is approximated by the partial sum of the convergent power series of the excitation frequencies, and the number of items therein can be adaptively determined by utilizing only the highest excitation frequency. The sensitivity expression of the frequency response is also established. The resulting expressions of the frequency response and its sensitivity are valid for the entire range of excitation frequencies of interest. By changing only the excitation frequency, we can obtain frequency responses and their sensitivities. This computational methodology is illustrated by its applications to two examples. The results show that the proposed method can remarkably reduce the CPU time required by the direct method.