Rainflow Analysis of Switching Markov Loads
Pär Johannesson
Centre for Mathematical Sciences
Mathematical Statistics
Lund Institute of Technology
1999
ISBN 9162837842
LUTFMS10121999

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 sideinformation 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