Closed-form solutions for the evolutionary frequency response function of linear systems subjected to separable or non-separable non-stationary stochastic excitations

In the seismic engineering, in order to reproduce the typical characteristics of real earthquakes ground-motion time history, the so-called uniformly modulated and the fully non-stationary random processes have been introduced. The first process is constructed as the product of a stationary zero-mean Gaussian random process by a deterministic function of time; for this reason it is also called separable non-stationary stochastic process. However, this process catches only the time-varying intensity of the accelerograms. To consider simultaneously both the amplitude and frequency changes, time-frequency varying deterministic functions have been introduced in the characterization of the input process. The latter process is referred as fully or non-separable non-stationary stochastic process. The evolutionary frequency response function plays a central role in the evaluation of the statistics of the response of linear structural systems subjected to both separable or non-separable stochastic excitations. In fact, by means of this function, it is possible to evaluate in explicit form the evolutionary power spectral density of the response and consequently the non-geometric spectral moments, which are required in the prediction of the safety of structural systems subjected to non-stationary random excitations. In this paper a method to evaluate in closed-form the evolutionary frequency response function of classically damped linear structural systems subjected to both separable and non-separable non-stationary excitations is presented. In order to evidence the flexibility of the proposed procedure the evolutionary frequency response function is evaluated by very handy explicit closed-form solutions for the most adopted time varying and time-frequency varying modulating functions.