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

• The response of linear systems under fully non-stationary input is analyzed.
• The spectral analysis is performed by means of the evolutionary power spectral density.
• The evolutionary frequency response is handled by closed form solutions.
• The spectral characteristics of the response are evaluated by handy formulas.
• Numerical results demonstrate both the accuracy and consistency of the proposed model.

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.