Computing Transient Response of Dynamic Systems in the Frequency Domain

Frequency-domain methods are usually more efficient computationally than time-domain methods to compute the responses of linear dynamic systems. However, a common drawback of frequency-domain methods is that computed responses are merely the steady-state response, not the total response. It appears that no analytical solution has been derived for the transient response of a multiple-degree-of-freedom (MDOF) dynamic system to arbitrary periodic loading. An analytical contribution of this article is deriving a closed-form solution for the transient responses of linear MDOF systems to arbitrary periodic excitations. Together with the fast Fourier transform (FFT) algorithm, a very efficient numerical method is developed to compute the transient and total responses of MDOF systems, suitable for both damped and undamped systems. In the newly developed method, the computational time required for obtaining the transient response is much less than that for the steady-state response. Three numerical examples are provided in this paper to verify the correctness, and demonstrate the effectiveness as well, of the newly developed method.