Properties of random vibration in the time domain1675.2.3. Number of threshold crossings per unit time Let us consider a stationary and ergodic random vibration (t); and p(), the probability density function of the instantaneous values of ^(t). Let us seek to determine the number of times per unit time na the signal crosses a threshold chosen a priori with a positive slope. Let us set na the number of occasions per unit time that the signal crosses the interval a, a + da with a positive or negative arbitrary slope, da being an very small interval corresponding to the time increment dt. We have, on average, n; = -*2 [5-27]Let us set nQ the number of occasions per unit time that the signal crosses the threshold a = 0 with a positive slope (n^ gives an indication of the average frequency of the signal). Let us set finally 40 the derivative of the process 40 and b the value of 40 when i = a. Let us suppose that the time interval dt is sufficiently small that the variation of the signals between t and t + dt is linear. To a-40 cross the threshold a, the process must have a velocity 40 greater than dt The probability of crossing is related to the joint probability density p(l,l) between t and 't . Given a threshold a, the probability that: a < 4 0 a + da , and [5.28]b < l(t) < b + dbis thus, in a time unit, p(a,b) da db = PJa < f(t) < a + da, b < i(t) < b + db] Setting ta the time spent in the interval da: [5.29]da
