Nonstationary Envelope in Random Vibration Theory

Abstract

In this paper, it is shown that the input process in the nonstationary case must be defined as a complex process, i.e., as the product of an analytic random stationary process by a deterministic shaping function. Defining the input in this manner, the complex nonstationary cross‐correlation matrix is evaluated, and the nonstationary spectral moments take on a self‐evident physical meaning as variances of the complex response and of its time derivatives. Using the complex process, the nonstationary envelope, too, becomes a natural consequence of the previous definition, i.e., the modulus of the complex response of linear systems subjected to such input. In the framework of complex processes, the mean rate threshold crossing for circular barriers and the first‐passage probability are evaluated using the one‐step memory Markov process.