Selecting maintenance tasks using Weibull Analysis

The data, distribution and what this means for maintenance

In my career I have often found that Weibull analysis has been treated as if it is far too difficult and expensive to conduct. This might have been true when computing power and digital technology for gathering and processing data was uncommon. Today, the old reasons for dismissing Weibull analysis have largely evaporated.

The Weibull distribution is a continuous data distribution in the exponential distribution family. It enables us to determine whether components are failing and by which pattern of failure. This blog will outline the data required to conduct a Weibull Analysis, how to plot the Weibull distribution and what this means for maintenance.

Why the Weibull distribution?

Weibull was invented seventy years ago, and has been used extensively for life or failure analysis. Other distributions can be used such as the exponential or lognormal, however, these are normally used for specialised situations.

What data do we need?

For each part that we want to carry out the analysis on, we need its failure and change event data. We also need to know the operating age of the components at the time they’re changed.

The Weibull process is enhanced if we can use age data for parts that have been subject to scheduled replacement, when the removed part is still working. These functional component changes are termed ‘censored’ or ‘suspended’ data. Because suspended parts have not achieved their failure age at removal it adds evidence to the analysis because of their survival and makes the results more accurate.

How to define age

If machinery operates or is exposed 24/7, age could be elapsed calendar time. If not it is best to use operating hours, or some kind of cyclic count.

For example, the life of a medium voltage air circuit breaker’s main contactors is proportional to the number of on-off cycles, as electric arcing dissipates contactor material. The age of the breaker contactors is expressed in switching cycles.

As well as components that have been subject to planned removals, the age of the currently fitted parts should also be included in the data as ‘suspended’ events.

Fitting age data to a Weibull distribution

There are a number of well-known estimation methods for fitting age data to a Weibull distribution. Ranked Regression (using the median ranks on the y values) Maximum Likelihood Estimation (MLE), and the Method of Moments. The latter only takes into account the failure events, and so is not as useful.

A useful web site using excel can be found at www.real-statistics.com. Years ago Reliability engineers used specially printed log-log (probability plot) graph paper to manually plot the failure ages and fit a regression line by eye, before reading off the parameters that represented the Weibull distribution. The ‘goodness of fit’ of the regression line was seen by eye. Today specialised software is used to calculate the estimates.

Usually people with an engineering background prefer the ranked regression method, because the probability chart allows the data and the fit to be visualised. It is easy to see whether the data suffers from mixed failure modes, or whether it is appropriate to use the location parameter from the visual line fit.

Those from a statistical background usually prefer the MLE method as in general terms it is more accurate. My experience suggests that there are circumstances where you may prefer either. The MLE method has an inherent bias, which is worse when the shape parameter is < 1 and the number of data points is < 15. The dispersion of data is greater the lower the shape. In these circumstances I generally use the ranked regression method. MLE has separate methods for bias correction.

Parameters

The estimations can yield 1, 2 or 3 parameters that describe a Weibull distribution. The Parameters are the shape, scale and location. Most often a 2 parameter Weibull is used with the shape and scale, the location projects the distribution left or right on the x-axis. If deterioration begins and there is a probability of failure after manufacturing with components in stock before they are fitted to an operating machine. Shape and scale are often referred to using Greek letters beta and eta respectively.

Let’s plot!

We will take three sets of simulated data, to represent the three fundamental patterns of failure we introduced above, premature, random and wear-out. The number of simulated failure events (500) is higher than would normally be encountered in the field, so we can show these examples clearly. The first three charts show a histogram of the event age data for the three cases:

We now pass the three data sets into the Ranked Regression Weibull estimator, and see the probability plots (checked by eye) are a reasonable fit for the Weibull distribution.

There are two other views of the Weibull distribution, the Probability distribution function (PDF) and the Cumulative Distribution Function (CDF). You will note that each chart above provides the estimates of the shape and scale for each case. The shape and scale parameters are needed to plot the PDF and CDF below.

The PDF plots should be very similar to the histograms above. The premature PDF seems to be confined to the bottom left-hand corner of the chart, this is because the distribution is so dispersed, stretching out the values in x-axis.

The Cumulative Distribution Function (CDF) is useful as it can tell us the probability of failure of the components at a specific age. The Weibull Scale parameter is always equivalent to the age or cycles equal to the probability of 0.63. Probability is a number that is always between 0 and 1, and so is easily mentally converted to percentages. The Scale is the age at which 63% of the components are likely to have failed.

A convention often used by reliability engineers is to quote reliability as “B-xx”, so using the scale factor, this is equivalent to ‘B-63’. Each case was generated from a Weibull distribution with a scale of 1000, but to ensure each of the charts were readable any sample ages above 6000 cycles were not included in the data set. This is why the estimated scale of the premature data set is 843.

Interestingly, if we look at the B-20 figures for each of the cases, where 20% of the components are likely to have failed, the difference between the three cases with similar scale parameters is strikingly different. This illustrates why premature failure is so bad:

So what does this mean for maintenance?

If we reflect on our previous blog where we align failure patterns with maintenance task prerequisites, we can now link Weibull analysis and the Weibull shape parameter.

Just a cautionary note, there are other pre-requisites and conditions that need to be present for the various maintenance task types to be applicable, we can discuss these in a later blog.

Weibull analysis is not only extremely useful, but viable given the advancement in digital technology for gathering and processing the appropriate data. The shape analysis informs us what maintenance interventions and task types are applicable to the three basic patterns of failure. Thus, Weibull analysis helps justify our maintenance regime.

What is your experience using Weibull? Have you thought it too difficult or expensive to conduct or have you embraced it? Please let us know your experiences and stories.

The previous blogs have discussed defined maintenance, and classified maintenance types, applicability and other attributes. You can find them here..

In the next blog we are going to explore and demonstrate how a new emerging database technology can be used to better store, retrieve process and more importantly reason on the data we hold. We will look at the traditional methods and information systems that we use in maintenance and show how we can transform, innovate and improve effectiveness using new technology.

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OXMT: Using our experience in Formula 1®, aerospace and the resources sector, we help businesses maintain high-performing, reliable and efficient equipment. Optimise maintenance processes in capital intensive industries with IronMan®

The Maintenance Guru

23 years in operating and maintaining complex plant, 17 years in leading the development of predictive maintenance. Involved with international standards