Laplace Domain Approach for Computing Transient Response of Simple Oscillators to Stationary Excitation

Evaluating the mean-square response at the transient state of a single degree of freedom (SDOF) system to stationary random excitation has been extensively studied in the past, but the explicit closed-form solution has not been available unless the excitation was considered to be a physically unrealizable white noise process. In this paper, explicit closed-form solutions for the mean-square response are derived for arbitrary input power spectral density (psd) functions. While all existing solution methods for evaluating the mean-square response have always been conducted in the time and/or frequency domains, the proposed method is operated in the complex plane (Laplace domain) based on pole-residue formulations. Not only is the proposed approach is much more efficient than other existing approaches, but also meaningful physical and mathematical insights can be gained in its solution procedure. To demonstrate the procedure, this paper considers the excitation process characterized by a white and nonwhite psd, respectively, and the corresponding closed-form solutions for the transient mean-square response of SDOF systems are derived. The correctness of the closed-form solution for the nonwhite psd is verified by Monte Carlo simulations.