## Background

The Statistical life length analysis focus area is about estimating the remaining life length of components by using Bayesian statistics on failure data.Within for example Condition Based Maintenance it is often important to be able to estimate the remaining time until a component needs service, or calculate the probability that the component will fail before the next service occasion. The probability of failure after a certain operating time of a component is typically modelled as a Weibull distribution. If we can accurately estimate this distribution, we can answer a number of important questions, as for example what the risk of failure is, what the optimal service interval is, or what the expected cost of a suggested maintenance plan is. The latter can in turn be used to find an optimal maintenance plan.To estimate this distribution, some real data are needed. However, data on failures of components are always scarce, since the service intervals are placed with some margin to avoid failures, so there is usually only a small number of observed failures.Using Bayesian statistics it is possible to utilize the relatively small amounts of life length data from failed components, to make as good as possible predictions about the life length distribution, and then use this distribution to answer a number of useful questions.We have so far had two projects within this focus area:

## The project

We have developed and implemented methods for fully Bayesian analysis of life length data, suitable for a number of different maintenance scenarios, such as "Run till failure", "Interval based service", and "Inspection based service". Bayesian analysis involves both the scale and shape parameters of the Weibull distribution. Rather than producing just a point estimates of these parameters, a posterior distribution over them is calculated. This distribution is then used to find expected values and probability intervals for a number of useful entities, such as the risk of failure, the expected cost of a service interval, and the optimal service interval length.

## High Cycle Fatigue limit testing

In High Cycle Fatigue (HCF) limit testing the fatigue limit of a mechanical component is determined by applying cyclical stress of a certain amplitude and noting whether the component breaks or not. Since testing is time consuming and expensive, the number of test samples should be kept to a minimum. A common protocol for finding the fatigue limit distribution is the staircase method, in which thetesting amplitude is decreased or increased with a fixed step depending on whether the component in the previous test did break or not.We have in this project developed and implemented an alternative protocol, based on Bayesian experimental design, in which the amplitude of each test is selected to maximize the expected information gain of the test. Since the fatigue limit can be modelled by a Weibull distribution, the same methods as was implemented for Bayesian life length analysis can be used. Simulations have shown that with the proposed method the number of required test samples is significantly decreased as compared to with the staircase method.

An implementation of this test protocol can be made available for interested parties by a web interface.

Contact Anders Holst for more information.