# Rainflow Analysis of Switching Markov Loads

## Pär Johannesson

### Centre for Mathematical Sciences Mathematical Statistics Lund Institute of Technology 1999

##### ISBN 91-628-3784-2 LUTFMS--1012--1999

Abstract:
Rainflow cycles are often used in fatigue analysis of materials for describing the variability of applied loads. Therefore, an important characteristic of a random load process in the intensity of rainflow cycles, also called the expected rainflow matrix (RFM), which can be used for evaluation of the fatigue life. In this thesis mainly two problems are addressed: the first is computation of the expected RFM for a given load model; and the second is modelling of a load from its RFM. Special interest is given to switching random loads, which are random loads where the stochastic properties change over time, due to changes of the system dynamics. For a vehicle, the change of properties could reflect different driving conditions.
The first Markov model to be considered is a Markov chain (MC). In rainflow analysis only the local extremes, also called turning points, of the load are of interest. Hence, another very useful approach in applications is to apply a Markov chain of turning points (MCTP) model. Both switching Markov chains and switching Markov chains of turning points are considered as models for switching loads. The switching is modelled by a Markov chain, which leads to a so called hidden Markov model.
The problem of computing the expected RFM is solved for all the Markov models described above, including the switching loads, and the results are explicit matrix formulas. Rainflow inversion, which is the problem of computing a load model given an expected RFM, is solved for the MCTP model. A switching load gives rise to a mixed RFM. Methods for decomposition of a measured mixed RFM are derived, where estimates of the proportions and the switching frequencies of the subloads, as well as estimates of the models for the subloads can be obtained. By including side-information in the decomposition the accuracy of the estimates can be improved. The rainflow inversion and decomposition can be used for generating random load sequences from a measured RFM. Finally, the exact distribution of the number of interval crossings by a MC for a finite time horizon is computed. Crossings of intervals have a direct connection to rainflow cycles. Several examples are given in order to illustrate the different topics in the thesis.
Keywords:
Switching process, Markov model, hidden Markov model, random load, vehicle load, fatigue, rainflow matrix