On the computation of non geometric spectral moments for non stationary filtered input processes

The spectral moments have been introduced in the literature for the case of structural systems subjected to stationary Gaussian input processes and were defined as the geometric moments of the one-sided power spectral density function of the response. It has been proved that the spectral moments play an important role for the safety prediction of structural systems. Extension of the definition of spectral moments for non stationary Gaussian input processes has been provided by means of the geometric interpretation applied to the one-sided evolutionary power spectral density function of the response. In the case of white input process, such a geometric approach leads to divergences and needs adjustments in order to be exploited for the safety assessment in the non stationary case. The exact formulation for the spectral moments has been proved to be based rather on a non geometric time domain approach aiming at the evaluation of the covariances of the pre-envelope response process. The geometric and the non geometric approaches lead to the same spectral moments for the stationary case only. In this paper, a procedure for the evaluation of the non geometric spectral moments is presented with particular attention to the case of non stationary filtered Gaussian white noise which is both amplitude and frequency modulated. The presented procedure is based on the formulation of the pre-envelope covariance differential equations whose solution is obtained by means of an expression of the time dependent transition matrix of the system constituted by the structure and the time varying frequency filter.