Stationary and Transient Response Envelopes

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Abstract
An envelope is introduced by using the Hilbert transform to define a complex conjugate to the excitation and response processes of a linear structure. Time‐limited 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 band‐width of the response combined with the non‐stationary intensity. This is computationally important as these parameters are often available in closed form. Finally, the envelope is used to study the first‐passage probability.

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