Stationary and Transient Response Envelopes

An envelope is introduced by using the Hilbert transform to define a complex conjugate to the excitation and response processes of a linear structure. Timelimited stationary excitation is treated in detail, and the complex correlation function is shown to follow from its stationary equivalent by use of a suitable differential operator. Simple expressions are derived for the case of rational spectral density, and a parametric study of the influence of the frequency content is carried out. It is found that envelope crossings can be predicted by use of stationary measures for the mean crossing frequency and the bandwidth of the response combined with the nonstationary intensity. This is computationally important as these parameters are often available in closed form. Finally, the envelope is used to study the firstpassage probability.