Very Short Introduction (Plus an Example) of Weibull Engineering Basics

By Wes Fulton, CEO, Fulton Findings LLC

(originally written on 24 JUL 2010, last edited on 8 JAN 2012)

… pictures below represent some of the many uses for Weibull Engineering

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The simple concepts in statistics can appear complicated to beginners because many books on the subject use long and strange words. However, this introduction hopefully uses words that are generally shorter and more familiar.

Perhaps the only object without variability is a good digital copy of a digital original. Practically every other product and service has variability. For example, a bearing is not going to be the same as another bearing even though they may have the same part number and may have been manufactured one after the other. These bearings could be very similar, but are not exactly the same. One expected difference between them is how long they operate successfully. Some tiny amount of variability in the type of usage can make a big difference in operating life capability. Along with lifetime variability, there are other areas where variability effects are important such as money markets, quality satisfaction levels, disease cure rates, satellite reliability, maintenance scheduling, warranty analysis, and so on with an almost unending list of additional areas. Understanding variability and making decisions about variability is straightforward with a proper variability model. Such variability models are called “distributions”. From the correct probability distribution you can estimate the expected probability of getting a particular result in test or in customer usage. Picking the appropriate model for measurement variability is the goal of “applied statistics”.

Most distributions are presented in either probability density function (“PDF”) form or cumulative distribution function (“CDF”) form. PDF and CDF are just two different ways to describe the same distribution … but more about that later.

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Six different “extreme-value” distributions were discovered by E.J. Gumbel in the 1920’s for modeling the occurrence of rare events like flooding and wind gusts and power surges. One of the six possible extreme-value distributions is now called the Weibull distribution. It is one of the most widely-used solutions for modeling how things vary, especially for lifetime data (age-to-failure data) and reliability. The name Weibull is usually pronounced in English-speaking countries as “Waee-bull”, but is no doubt pronounced differently in Sweden where Wallodi Weibull was born. The subject technique was promoted in the beginning by the technically-gifted Waloddi Weibull. He started investigating variability models (as well as many other things) during the 1930’s eventually writing over 60 papers plus a book titled “Fatigue Testing and Analysis of Results”. The book was published in 1961 by Pergamon Press [1].

Weibull distribution methods have been updated continually with an explosion of use since the 1950’s and with new applications being added almost continually. Now, there are computer programs available for Weibull modeling along with Weibull classes to teach application. A main reference book for this is the handbook by Dr. Bob Abernethy [2]. This was the first book written specifically for Weibull Engineering (now titled “The New Weibull Handbook©”). The first version of the handbook was published in the early 1980’s. It has since been updated continually by Dr. Abernethy adding the latest methods and software solutions to make it the de-facto world standard. Dr. Abernethy, whose doctorate is in statistics, also invented the engines that powered the fastest manned aircraft during the cold war between the United States and Russia (the SR-71 Blackbird spy plane). His technical expertise plus statistics expertise plus simple writing style come together to provide good reading, explaining things clearly for practical solutions to real issues. So, reference 2 below is especially recommended for further reading.

The Weibull distribution is simple and yet more versatile than some of the other commonly used models for data variability. It can exactly duplicate distributions like exponential and Rayleigh, and by embracing additional distributions such as normal, lognormal, and “type one” extreme-value (also called Gumbel after E. J. Gumbel) we get into Weibull Engineering. Weibull Engineering has a wider scope than just the analysis of fatigue testing results. It also includes root-cause detection, event forecasting, spare parts projection, test planning, optimum-replacement interval, accelerated testing analysis, design comparison, process reliability, manufacturing control, cost management, as well as others. It currently enjoys wide popularity with many people in the fields of design, development, finance, fabrication, maintenance, operations, quality, reliability, safety, and testing.

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Before some details of the Weibull distribution get presented, the well-worn “normal” distribution should be briefly described. The normal distribution is the king of distributions, used to model many things … but not everything. Waloddi Weibull had an early paper on the Weibull distribution rejected because it was not on the normal distribution! The normal distribution probability density function (PDF) has only one basic shape (Figure 1 below) which may be either wider-and-shorter or thinner-and-taller, but it always takes a bell-like shape. It is sometimes called the “bell curve”, and sometimes called the “Gaussian” distribution in honor of Johann Carl Friedrich Gauss. For example, the normal distribution applies when modeling variability in such cases as error measurements, student test scores, performance variability, X-bar quality control charts, and miss-distance for a single axis in a machining operation. There is something called “the central limit theorem” which explains why the normal distribution fits so well often when different effects are mixed in the data. The normal distribution model has 2 parameters, meaning that it only requires two numbers for its description. Greek letters provide a quick naming convention for these. To completely describe the normal distribution, only the mean value (“Mu”, pronounced like “mew”) and the standard deviation value (“Sigma”) are required. However, product life and reliability usually need something different.

Life-data measurements exhibit variability not closely symmetrical around a central value. Symmetry around a central value is required for using the normal distribution effectively. But if used incorrectly, the normal distribution can produce a negative estimate for life. So the normal distribution is not generally the right choice for modeling lifetime, age-to-failure, data.

The Weibull distribution works well in modeling lifetime data. The Weibull PDF can take many shapes (Figure 2 below) and can fit to non-symmetrical data. Also, the simple 1-parameter and 2-parameter versions of the Weibull distribution will not produce a negative value for life. That’s a nice feature! The 2-parameter version of Weibull is standard, and in some ways it is even simpler than the 2-parameter normal distribution. The math is straightforward for the cumulative Weibull distribution, but the cumulative normal distribution must resort to higher-math integral approximations. The standard 2-parameter Weibull has characteristic value (“Eta” pronounced like “ey-tah”) and slope value (“Beta” pronounced like “bey-tah”) for its two parameters. “Shape parameter” is another name for Weibull slope, since the Weibull PDF shape changes with different Beta values. Other references may use different parameter names than used here, however Eta and Beta are used for the Weibull 2-parameter names in Dr. Abernethy’s handbook and also in the international standard, “IEC 61649, Edition 2, Weibull Analysis” [3].

The figures above represent probability density function (PDF) shapes. All PDF’s, Weibull or normal or whatever, are identical in one respect, i.e. the area underneath each PDF curve is exactly one (1) or 100%. So the PDF represents 100% of where to expect a similar measurement. The amount of area under the PDF curve, to the left of any point along the PDF plot horizontal measurement scale, is the cumulative distribution function (CDF) form. The CDF is simply another way to express the same model. The CDF representation is generally more useful than the PDF when answering most questions about variability such as … “How long can a product go with only a specific proportion of failures expected?” A Weibull CDF plot is displayed below in Figure 3 for the worked-out example of Weibull Engineering analysis following a little later in this introduction.

The Weibull fit to the data can be better than the normal distribution for some specific applications simply due to its multi-shape capability. The data itself selects the most appropriate Weibull shape for best solution. Weibull often fits better and works better as a model for small samples (a small sample is taken here to be twenty or less measured occurrence values). With a larger sample size, more than 20 occurrences, the more complex 3-parameter version of Weibull may be used in modeling data variability. The Weibull distribution third parameter (t0 … pronounced like “tee-zeeroh”) represents a time shift for occurrence age variability. The 3-parameter Weibull works well in cases where either there is a delay in the onset of the occurrence mechanism (a failure free period), or the aging process starts before the item officially begins operation (prior deterioration). With very small sample sizes even down to zero occurrences the Weibull 1-parameter model, also called “Weibayes”, is the most accurate solution provided there is sufficient and appropriate historical data to help. For Weibayes, with a good estimate of the Weibull slope (Beta) already provided from prior experience, the solution requires only finding the Weibull characteristic value (Eta) to get a simpler and more accurate answer.

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Knowing the root cause of an occurrence mechanism provides tremendous help in determining corrective action. Sometimes the Weibull solution can suggest the type of root cause given that the data is generated from a single root cause. With only one root cause being analyzed the Weibull slope is usually above 1.0 (“wear out”) or below 1.0 (“infant mortality”). Mixing different root causes together can complicate the analysis even though there are reasonable solutions available for that (as long as there are only two or three mechanisms and a larger quantity of data). Mixing many different root cause mechanisms together in the same data set usually produces a Weibull solution with Beta slope near 1.0 and with a reasonable goodness of fit to the data. However, with many mechanisms mixed together there is additional randomization and loss of resolution needed to suggest appropriate corrective action. A major recommendation is to try and focus on one root cause at a time if possible.

Once the basic Weibull model is determined, it can provide the foundation for forecasting such as “Abernethy Risk”. An additional piece of information needed to forecast is the expected usage rate such as the amount of aging experienced per item each month. For example, distance travelled (miles or kilometers) per vehicle per month would be the applicable usage rate for an automotive application. Flight hours per engine per month might be applicable for aircraft as well as patients per doctor per month for hospitals. Forecasting may be one of the most useful aspects of Weibull Engineering. Such calculated expectations provide the basis for spare parts requirements and warranty programs.

Weibull Engineering is not limited to the analysis of life data. It is in demand for evaluating process reliability, instrument calibration intervals, economic variability, and quality control. Specifically, Weibull should be used for quality control monitoring instead of the classic normal distribution where Weibull is more appropriate. One additional appropriate area of application for Weibull Engineering is with regard to “six sigma” type quality control efforts such as controlling plating thickness variability, rotating shaft wobble, progressive deterioration, and contamination level monitoring. These last few applications are not modeled well with the normal distribution as the measurements cannot be negative. The standard Weibull and the lognormal distributions are usually more accurate for these applications, since they only predict positive values.

WORKED-OUT EXAMPLE

The Issue - Your organization uses hundreds of batteries all of a similar design. These batteries go into a data recording module that cannot be monitored externally with sensors, as the module is buried within a small medical device implanted into cancer patients. The batteries are seldom required and only operate a few seconds at a time, but the life requirement is in hours to minimize chance of failure. When a battery dies, there is loss of data which requires an excessive amount of money in re-testing and re-analysis to reproduce the desired data. This costly loss of data is happening too often and your management does not like it. Testing on a new sample of seven batteries provides the following failure data with regard to operating hours since new:

130 hours of battery life before failure (Battery #1)

165 hours of battery life before failure (Battery #2)

234 hours of battery life before failure (Battery #3)

252 hours of battery life before failure (Battery #4)

253 hours of battery life before failure (Battery #5)

295 hours of battery life before failure (Battery #6)

389 hours of battery life before failure (Battery #7)

The Goal – Your management wants you to find out how long any particular battery of the same design should be used before replacement if the chance of failure is limited to only 2 percent (%). Plus, you want to be able to defend your decision to management by being conservative in your estimate of life capability.

The Analysis – The battery test data is plotted on Weibull CDF scaling. Then a straight line (on this scaling) is fit to the data. NOTE: The CDF scaling is done in such a way that a straight line going from lower left to upper right on this scaling is a solution. This plot models the variability of battery life capability. The Weibull CDF plot of the test results (Figure 3 below) shows good fit of the Weibull distribution to the data with p-value estimate (pve%) of 79.51 for goodness-of-fit. A pve% value can possibly go from 0 to 100, and the nominal value is 50 for data sampled from the same distribution used for the plot. Therefore, the pve% of 79.51 for the Weibull distribution fit to this battery data is above average, i.e. good. A pve% of 10 or higher is usually acceptable. The fit line has Weibull slope (Beta) of 3.028, which is above one. Any Weibull slope above one indicates “wear out” as the type of occurrence mechanism. Wear out means that older items fail at a relatively faster rate than newer items. The wear out indication for the batteries allows for a useful planned replacement interval. A nominal estimate of 75 hours life comes by reading the horizontal value from the straight data fit line where it crosses 2% occurrence on the vertical scale.

However, this nominal result does not provide a conservative estimate. A lower estimate line, called a “confidence line”, was added to the plot for a conservative lower estimate of life capability. The curved confidence line on the plot is a 90% lower estimate of life capability. When read from the lower confidence line, the conservative estimate of life capability is 30 hours (horizontal scale) limiting failure probability to only 2% (vertical scale). Note that 2% failure occurrence probability equates to 98% reliability. With additional battery test data, the lower confidence line should get closer to the nominal line. This may give an even higher estimate of life for the same 98% reliability.

Adding one cost factor for planned replacement (lower cost) and a second cost factor for emergency replacement due to failure (higher cost) would allow a more detailed “optimum replacement” study at a later time to achieve minimum operational cost.

The Result – You make a recommendation for a pre-emptive planned replacement of each battery after 30 hours of operation based upon your Weibull analysis. Later you have more testing accomplished, and these results (consistent with before) reduce the uncertainty by adding the latest data to original data thus bringing the conservative lower bound higher and closer to nominal. That initial conservative estimate of 30 hours life is eventually raised to a very comfortable higher operational value. There are very few problems with this battery after your corrective action is implemented. Management likes you. You are promoted, you are happier, and the Weibull estimate for your own life expectancy increases.

CONCLUSIONS

More probability emphasis is coming to such fields as aerospace, energy production, food production, financial markets, medicine, military, oil refining, physics, public safety, and transportation. Weibull Engineering and similar probability-centered analysis, such as Reliability Centered Maintenance (RCM), leads the way in providing useful answers to difficult questions. For more information visit http://www.WeibullNews.com on the web, and check out the following references.

REFERENCES:

1. Weibull, Waloddi, “Fatigue Testing and Analysis of Results”, Pergamon Press, 1961 (Weibull’s only book)

2. Abernethy, Robert B., “The New Weibull Handbook©”, self-published (first complete self-study reference for Weibull Engineering)

3. IEC 61649, Edition 2, “Weibull Analysis” (the official international standard)

4. Click Here for SAE’s “Introduction to Weibull Engineering” Fast Track