Envelope process statistics for linear dynamic system subject to nonstationary random vibrations

Envelope process statistic plays a central role in evaluation of threshold crossing of many dynamic physical systems subject to random inputs. Commonly used envelope definition in stationary condition deals with modulus of a complex process, whose real part is the original process itself and imaginary part is its Hilbert transformation. A simplified envelope process definition is commonly used in technical literature for nonstationary modulated situations. This is to overcome some physical and computational inconveniences that take place when the stationary envelope definition is directly extended to the nonstationary case. This simplification consists in replacing, in the imaginary part of the above defined complex process, the response to the Hilbert transform of a Modulated stationary Input (HMI) with the response to a Modulated Hilbert transform of a stationary Input (MHI). In this work the efficiency of this approximation has critically evaluated, showing that it is able to extend the un-symmetry of original state space and its Hilbert transformation envelope covariance matrix, that is true in stationary conditions. Moreover, it is also shown that above approximation will induce some errors in envelope statistics, which are not negligible when input modulation function has a very fast variation rate