Evaluating response of simple oscillators to a particular kind of time–frequency non-separable evolutionary stochastic processes

This article develops a Laplace-frequency method to derive the exact closed-form solution for the response evolutionary power spectral density (EPSD) of a simple oscillator subjected to an evolutionary stochastic process characterized by a particular kind of time–frequency (t,ω) non-separable EPSD: SF(t,ω)=t2λe−2γ(ω)tΦ(ω). The proposed Laplace-frequency method also gains insightful physics for the response EPSD. Although the transient mean square response can be easily evaluated by taking the integral of the response EPSD with respect to the frequency, a closed-form solution for the mean square response is not always possible. The correctness of the computed mean square response based on numerical integration has been verified by Monte Carlo simulations