Extrapolation of Rainflow Matrices

An important area in mechanical engineering is fatigue of materials. The load process is an essential parameter in fatigue design, and is often summarized in the so-called rainflow cycle matrix. Extrapolation of a load measurement to longer distances is needed, and hence one is interested in estimating the limiting rainflow matrix (i.e. the ergodic distribution). An interesting fact is that counting rainflow cycles is equivalent to counting crossings of intervals. The main result is an asymptotic expression for the intensity of interval upcrossings as the upper and lower levels tend to ±∞, where it is assumed that the point processes of upcrossings of the upper and lower levels converge to two independent Poisson processes. The asymptotic expression only involves the intensity of upcrossings of the upper and lower levels. When estimating the limiting rainflow matrix from a load measurement, the asymptotic result is used where it is valid, i.e. for large interval crossings, and kernel smoothing is used elsewhere. To use the asymptotic result, one needs to extrapolate the level upcrossings for high and for low levels, where it is suggested to use peak over threshold methods. An algorithm for the estimation is given in detail. The usefulness and validity of the asymptotic result is illustrated by several examples, both from simulated processes and measured signals from an automobile.