Unlocking Weibull analysis
When products start failing, management management wants answers. Are they failing because because of manufacturing problems? Or is the design to blame? One of the most widely regarded methods for ferreting out the reason behind failures, as well as accurately predicting operational life, warranty claims and other product qualities is statistical analysis of a component’s or device’s failure data. Though there are many statistical distributions that could be used, including the eponential and lognormal, the Weibull distribution is particularly useful because it can characteri!e a wide range of data trends, including increasing, constant, and decreasing failure rates, a tas" its counterparts cannot handle. This characteristic also lets Weibull distributions mimic other statistical s tatistical distributions, which is why it is often an engineer’s first approimation for analy!ing failure d ata. What Weibull analysis can do
#any management decisions involving life$cycle costs and maintenance can be made more confidently from reliability estimates generated by Weibull analysis. %or eample, Weibull analysis can reveal the point at which a specific percentage of a population &such as a production run' will have failed, a valuable parameter for estimating when specific items should be serviced or replaced. Additionally, this analysis helps determine warranty periods that prevent ecessive replacement costs as well as customer dissatisfaction. Weibull analysis can be particularly helpful helpful in diagnosing the root cause of specific specific design failures, such as unanticipated or premature failures. Anomalies in Weibull plots are highlighted when items uncharacteristically fail compared to the rest of the population. (ngineers can then loo" for unusual circumstances that will help uncover the cause of these failures, which could include a bad production run, poor maintenance practices, or unique operating conditions, even when the design is good. )n addition to these factors, understanding the time and rate at which items fail contributes to other reliability analyses such as* ▶
%ailure modes, effects, and criticality analysis
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%ault tree analysis
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+eliability growth testing
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+eliability centered maintenance
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pares analysis
As with most analytical methods, the accuracy of a Weibull analysis depends on the quality of the data. %or valid Weibull analysis, and to interpret the results, there are several requirements for the data* ▶
)t must include item$specific failure data ×$to$failure' for the population being analy!ed.
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-ata for all items that did not fail must also be included.
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The analyst must "now all eperienced failure$mode root causes and be able to segregate them.
(ngineers often shy away from Weibull analysis because they believe it is too comple and esoteric. Although it’s true that an understanding of statistics is helpful, engineers can reap the benefits of a Weibull analysis without a strong statistical bac"ground. The Weibull distribution generally provides a good fit to data when the quality of that data is understood. alues for the resulting distribution parameters help eplain an item’s failure characteristics. These qualities can then influence cost$saving decisions made during design, development, and customer use. And when the distribution does not provide an acceptable fit, the qualities of the Weibull plot may still point the way to alternative distributions that might provide a better fit. Weibull terminology /ere are important terms in a two$parameter Weibull analysis* /a!ard +ate*
Probability Density Function (PDF):
0umulative -ensity %unction &0-%'*
Eta (η) represents the characteristic life of an item, defined as the time at which 63!" of the population has failed #he shape parameter, beta ($), is the slope of the best%&t line throu'h the data points on a eibull plot #he ariable t represents the time of interest when solin' these e*uations (+ee the 'raph asic eibull Plot)
This Weibull plot shows a best-fit line with a slope of beta going through four data points. The value for eta is derived by taking the point on the best-fit line that intersects with a line drawn from the y or CDF ais at !".#$% then finding the corresponding value on the or &'ge at failure( ais.
#he ha-ard rate describes how suriin' members of a population are failin' at a 'ien time #he ha-ard rate and eibull shape parameter, beta, hae a distinct relationship ▶ hen $ . /, the ha-ard rate decreases with time (re0ectin' infant mortality or failures soon after rollin' o1 the production line) ▶ hen $2/, the ha-ard rate is constant oer time ▶ hen $ /, the ha-ard rate increases with time (population wearout or products wearin' out at an increasin' rate as time passes) eibull distributions can also ta4e the form of other statistical
distributions dependin' on their $ alues ▶ hen $ . /, the eibull PDF is the same as the 'amma distribution ▶ hen $ 2 /, the PDF reduces to the e5ponential distribution with a failure rate, , e*ual to /7η ▶ hen $ 2 !, the eibull PDF becomes the 8aylei'h distribution (failure rate linearly increasin' with time) ▶ Data that fall into a normal distribution also 'enerate 'ood eibull plots, with $ 9 3 -ata types The accuracy of any analysis depends on the type, quantity, and quality of data being analy!ed. There are two categories of data used in a Weibull analysis* time$to$failure &TT%' and censored &or suspension' data. As the name implies, TT% data indicates how long an item lasts before failing. )t can be measured in hours, miles, or any other unit that defines a product’s life. )n some cases, the life of different parts of an item may be described by different metrics. %or eample, on airplanes, engine failures can be reported based on flight hours, while landing gear failures are trac"ed by the number of landing cycles. %ailures in different parts of an item must be treated separately for analysis and then combined to create a system$level life prediction. TT% data should also be associated with a specific failure mode for the part whenever possible. This data can come from various sources, including reliability growth testing, reliability qualification testing, and maintenance databases of field data. 0ensored data represents failure data recorded over an operating or test period. )t can be bro"en down into three categories, +ight$censored data includes test1operating times for items that did not fail &suspensions' and those that did. )nterval data includes all failures within a specific time interval, but the eact time$ to$failure is un"nown &e.g., warranty data'. %or left$censored data, the eact time an item failed is un"nown, other than the failure occurred before it was discovered. Although Weibull analysis can be done without considering individual root failure causes, "nowing and segregating failure modes lets engineers etract information about the item’s reliability. A decreasing failure rate &infant mortality or product failing soon after being made' is generally attributed to problems with manufacturing or part quality. )t indicates that reliability of the remaining products soon improves as defective items fail quic"ly and are weeded out. The rest of the items, considered defect free, either fail at a relatively constant rate or at an increasing rate as they wear out. 0onstant failure rates are common for electronic devices and comple systems due to the large number of different possible failures. #ost mechanical item failures, however, are caused by wearout.
2erforming Weibull analysis The traditional approach to performing two$parameter Weibull analysis is by plotting the data &usually done using commercial software'. (ach failure time is ran"ed based on the number of items that did and did not fail at that time, with the most popular ran"ings being mean and median. The data is then plotted manually or by software on Weibull probability 3paper,4 . 0ommercial software li"e Winmith, uper#ith, or 5uanterion’s 5uA+T$(+, simplifies analysis by automatically ran"ing and plotting failure d ata. Tools vary widely in price and can typically be downloaded off the internet. A best$fit line drawn through the data points lets engineers determine how well the underlying statistical distribution fits the data. )f the fit is valid, the best$fit line provides the item’s characteristic life and failure rate over time. )f the distribution does not provide an acceptable fit, the Weibull plot’s qualities can identify a more appropriate distribution or suggest a better interpretation of the data. %or eample, there may be other failure modes, a sudden change in the predominant cause of failures, or items were used in different environments. 2lot features such 3"nees4 &corners' or $hapes often indicate one or more of these problems may be responsible. )t is critical to understand that even though right$censored data is not plotted as part of a Weibull analysis, it significantly affects ran"ings that determine the Weibull plot’s shape. Omitting data on items that don’t fail can generate results that underestimate an item’s true reliability. &ee A Weibull plot using data with and without suspension.' This can prove costly in terms of more maintenance actions, inventory of spares, and an unnecessary redesign to improve reliability. Analysts should include all pertinent data to ensure accurate analysis results and interpretations.
)mitting suspension data produces results that underestimate the true reliability of the item being analy*ed. This plot% for eample% shows the mean time to failure +,TTF of /# hr for analysis without suspensions is less than that of a plot including suspensions +00 hr.
Assessing the ft +egardless of the technique used, an analyst must assess the assumed statistical distribution’s fit to a dataset. The failure data plot is particularly useful, as it not only allows for a simple chec" of whether a linear fit matches the data, but can also indicate potential solutions when the fit is not linear.
This graph of a #-parameter Weibull distribution provides a fairly good f it to the data. 1owever% an eperienced analyst would see that the curvature of the plotted data relative to the best-fit line suggests that either a "-parameter Weibull or a lognormal distribution may be more appropriate.
This is the same data used in the previous graph +#-parameter analysis for lognormal data but plotted with a lognormal distribution. 's epected% the linear fit is much better. The new r# value of /.223 confirms the results of the visual inspection +compared to an r# value of /.23# from previous graph% which means only /."$ of the variation from a perfect linear fit is not eplained by the lognormal distribution.
;ther statistical tests that *uantify the 'oodness%of%&t to particular distributions include the nderson% Darlin' tests Each has adanta'es and disadanta'es, dependin' on the *uantity of aailable data and the distribution bein' analy-ed ?nformation on these tests can be found in most reliability and statistics boo4s #o this point, a 'ood linear &t to a dataset has depended on selectin' an appropriate statistical distribution @oweer, this can be complicated
This plot clearly indicates a poor fit of the data to the #-parameter Weibull distribution.
poor !%parameter eibull plot) #he failures occurred in a lon'%term stora'e enironment and were not discoered until the deices were remoed for periodic testin' #his particular failure mode was deemed critical due to the lar'e number of failures Despite the fact most deices wor4ed when remoed for periodic testin', suspensions (those that passed the testin') were not recorded in the ori'inal dataset #his dataset, then, could not be
used to estimate the populationAs reliability Data from this testin' can still be used, howeer, to illustrate di1erent <erin' techni*ues that can improe a distributionAs &t to a dataset 6ecause the devices were in storage, failures were not discovered until they were tested. Actual failure times are un"nown, so interval &failure discovery' data must be used for analysis. There are also distinctive 3"nees4 in the plotted results, which can indicate several different types of failures, different operating environments, poorly manufactured lots, or parts made by different manufacturers. )n this eample, the reason for failures was "nown, storage environments were similar, and all parts were made by the same manufacturer. 0onsequently, further investigation was needed to identify the reason for the discrepancies. A review of serial numbers of failed parts did not reveal any indications of a manufacturing issue. /owever, it did indicate that failed parts were from two different versions of the device. Although the mechanical part was identical in both designs, data was filtered by version and an independent analysis performed on the two datasets. 2lotting 3ersion A4 data shows a noticeable improvement in the linear fit to the plot. The high beta value, 7.89, suggests a rapid wearout condition for the failing parts in this version.
This analysis% which looks at only data from items made with components from 4ersion ' of a production machine% has a much better fit than the previous plot.
> eibull plot for BCersion data (Fi'ure ) has a best%&t line with a smaller beta (3!) #hou'h used in two di1erent ersions of the same deice, the actual part, reason for failure, and stora'e enironment were identical, so there should not be a si-eable di1erence between beta alues for these two datasets
1ere5s the plot for data from items with components that came from 4ersion 6 of the same model production machine.
?t was subse*uently determined that the di1erence between the two ersions was the test circuitry used to dia'nose mechanical%part failures #hus, failure data for the two desi'ns were based on two di1erent test units >nalysts *uestioned whether the mechanical part or test circuitry was responsible for the failures
The 7-curve results because half the population is failing due to infant mortality while the remainder fails due to a strong wearout condition.
>nother problem arises when the eibull plot creates an +%shaped pro&le around the best%&t line ?t usually means a mi5ed dataset (+ee +%shaped eibull plot) +tandard <erin' techni*ues can separate the datasets to 'et accurate results #his re*uires identifyin' the reason why there are di1erent failures in the dataset ?t could be: ▶ Gore than one reason for the failures ▶ #he same reason for failure, but di1erent operatin' enironments ▶ #he same reason for failure, but di1erent part manufacturers or production lines ▶
This graph shows two distinct populations. The first +green% with a beta of /.8/9% shows infant mortality. The second +black% with a beta of 0.#% shows wearout. 6oth a visual inspection and the reported r s:uared values indicate an improved fit to individual datasets.
#he 'oal of a eibull analysis is to estimate the reliability of an item in a speci&c application or enironment 8esults are used to estimate reliability and the ade*uacy of a desi'n, and for deelopin' maintenance schedules and inentories of spares ;sing statistical analysis, it is easy to predict reliability. ince the y$ais coordinates on the Weibull plot refer to the percentage values of the 0-%, the reliability can be determined directly from the plotted dataset. Once the Weibull parameter values are "nown, reliability estimates can be calculated using the Weibull reliability function, +&t'. The reliability function can also be mathematically manipulated to solve for a reliability1life prediction for an item population. %or eample, one could calculate a 6<= life, which is equivalent to the time when the population’s reliability is >=. &ee 2redicting 6<= @ife.'
The hori*ontal line drawn where F+t ; /$ intersects the best-fit line at approimately 9" hours. This specific case% where /$ of the population has failed% is referred to as the 6/ or </ life.
+eliability predictions can also be performed on separated datasets, such as those for an item with competing failure modes, operated in different environments, produced by different manufacturers, etc. The prediction requires the use of a scaling function to address the impact of each failure mode distribution on the overall item’s reliability.
+esources ▶ The ew Weibull /andboo" B %ifth (dition, Culf 2ublishing 0o., D==7, by Abernethy, -r. +.6., &http*11tinyurl.com1cdrypo>' ▶ Applied @ife -ata Analysis, Eohn Wiley F ons, <>GD, )6 =H7<=>HIG7, by elson, W. &http*11tinyurl.com1c!Hcl8>' ▶ #echanical Analysis and Other peciali!ed Techniques for (nhancing +eliability B #AT(+, +eliability )nformation Analysis 0enter, D=7G$<$>88>=H$8>$>, by +ose, -., #ac-iarmid, A., @ein, 2., et al &http*11tinyurl.com1cqlyt7f '
