Zero Crossings, Peaks, and Other Statistical Measures of Random Responses

Several statistical properties of a stationary random process that are of interest in random vibration applications are reviewed. In the first three sections, the zero crossings, threshold crossings, distribution of peaks, and a simple model of fatigue damage are obtained directly from calculations based on the joint probability density p(x,ẋ). It is assumed that p(x,ẋ) is available, e.g., from the Fokker-Planck equation. In the fourth and fifth sections, it is alternatively assumed that x(t) is the response of a particular nonlinear oscillator to random excitation. It is shown that certain statistical results can be obtained by utilizing the oscillator properties. Two illustrative examples are described in the sixth section. Finally, in the seventh section, the directions for future research on higher-order statistics are sketched.