MTTF is greater than MTBF is estimated by dividing total operating time on all units by the number of failures observed. They are different parameters, although equal when there are no suspensions. Otherwise they can be vastly different. MTBF is used with repairable systems. See tbe Glossary, Appendix A. This is trne for all Weibulls. This was the case for the compressor inlet air seal rivets in the following example. The flare part of the rivet was missing from some of the rivets during inspection. After thorough failure analysis, a redesign was started. An accelerated laboratOly test using the old rivets established a baseline.
The results are shown in Table Rivet serial numbers 3, 6, and 7 arc considered nonrepresentative failures as they were produced by two other failure modes. For instrnetive purposes only, the three suspensions arc ignored Dr. That leaves five failure points for flarc failure. Table See Table Each failure is plotted at its time-to-failure I and an estimate ofF t , the percentage of the total population failing before it. The true percentage values are unknown. In the next section a Weibull plot of this data is produced ignoring suspensions. This is wrong. Always include suspensions. It is done here to illustrate the effect of suspensions.
The correct plot with suspensions will be produced in Section 2. Median ranks are reconnnended as most accurate and therefore, best practice, but there are other plotting positions discussed in Chapter 5. In Section 5. With reasonably sizcd samples there is little difference in results with different plotting positions. Median rank tables are provided in Appendix I. Enter the tables for a sample of five and find the median ranks shown in Table for the five flare failure times shown in the middle column. These same median rank plotting positions are used with all types of probability paper i.
Note that if two data points have the same time to failure on the X-axis, they are plotted at different median rank values on the Y-axis, each point gets its own individual verticalloeation. Thc median rank values of F t , probability of failure, are found by setting the beta probability equal to 0.
See page I-I for the Exccl function. For the lowest data point the median rank estimate, the probability plotting position is For the middle data point the probability plotting position is For the highest data point the probability plotting position is The median is more typical in these cases. For example, the majority of the population makes far less than the average mean income, bnt half the population makes more or less than the median income.
Most life data distribntions are skewed and therefore, the median plays an important role. Measure the slope of the line with a ruler by taking the ratio of rise over run measured with a ruler to estimate beta. Note, make sure you are using scale Weibull probability paper. Weibull plots generated by computers are rarely scale.
Most of the illustrations in this book are not I: 1. See the problem section at the end of this chapter for I: I Weibull paper. Select a staIting point and measure one inch in the horizontal direction run. Then, measure vertically, rise , to the line intersect. Analytical methods for establishing the line using regression analysis and maximum likelihood estimates will be discussed in Chapter 5.
Two parameters define the Weibull distribution. The first is J3, the slope or shape parameter, and the other is the characteristic life, eta denoted by YJ. YJ is the In Figure , the Weibull's suggestion has become the world standard for quoting bearing life. It is not clear where the B notation came from. The author believes it may be from the Gelman "Brucheinleitzeit" fracture initiation time. Others believe "B" stands for bearing. Aircraft industries use much lower B lives for design requirements.
In aerospace, BI life is used for benign failures, BO. I for serious failures and BO. OI for catastrophic failures. B lives may be read directly from the Weibull plot or determined more accurately from the Weibull equation. For example, the Bl life for the rivet data in Figure would be 9. These data cannot be ignored even though the suspensions are never plotted.
Times on suspended units must be included in the analysis. Suspensions are not plotted but the rank order and the median ranks must be adjusted for suspensions. Leonard Johnson is credited for the method of adjustment employed herein [Johnson ]. Auth's simple fonnu1a gives exactly the same adjustment to the rank order i. The Auth adjusted rank value then is used in Bernard's equation to calculate an adjusted median rank.
The procedure is to rank the Table data with the suspensions and use the Auth equation below, to determine the adjusted ranks, aecounting for the presenee of the suspensions. This eorreets the previous Weibull, Table , and Figure , which excluded the three suspensions. I Median Rank 9. Benard's approximation is given in the next section. The results in Table arc plotted in Figure The major effect of the suspensions is to increase I]. Beta is minimally affected. Note: suspended items do not affect rank numbers until after they occur earlier failure times have unadjusted rank numbers.
Also, if two items fail at the same age, they are assigned sequential rank order numbers. The median ranks are converted to percentages to plot on Weibull paper. For example, considering the first failure in Table with an adjusted rank of 1. Total Test Time minutes 2. The slope, beta, hardly changed but the characteristic life, eta, increased.
- Wes Fulton?
- The New Weibull Handbook.
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This effect generally is tme: adding suspensions has little effect on beta, but increases eta. Figure is the correct Weibnll plot as it includes the suspensions. Thus if suspensions are ignored, results will be pessimistic. These are the steps to plot data sets with suspensions: I. First rank the times, both failures and suspensions, from earliest too latest. Calculate the adjusted ranks for the failures suspensions are not plotted. Use Benard's approximation to calculate the adjusted median ranks.
Plot the failure times x versus the adjusted median ranks y on standard I x I Weibull paper. Incidentally, the same plotting positions may be used to plot the data on other probability paper such as Log Normal. Estimate eta by reading the B Estimate beta as the ratio of the rise over run measured with a ruler. Figure 2. The Weibull plot provides clues about the failure mechanism, since different slopes, betas, imply different classes of failure modes. The bathtub curve, Figure 2. The "hazard rate" is the instantaneous failure rate. Personally we hope the "failure rate" during the design lifc is zero from all causes.
The Bathtub Curve [or Human Mortality 2. Components with infant mortality are like red wine: the longer you leave it alone, the better it gets. The Weibull Hazard Function 1 W e i b lbeta 5. These failure modes are ageless. An old part is as good as a new one if its failure mode is random. Therefore, we might suspect: Maintenance cn-ors, hUan errors, abusive events Failurcs duc to nature, foreign objeet damage, lightning strikes on transformers, woodpeeker attacks on power poles Mixtmes of data from 3 or morc failure modes assuming they have different betas Systems or component Wcibulls mixtures Intervals between failures.
Here again, overhauls are not appropriate. Of those that survive to time t , a constant percentage fails in the next unit of time. This is known as a constant hazard rate, the instantaneons failure rate. The period for overhaul is read off the Weibull plot at the appropriate B life. Ifthe failure produces a safety hazard, the recommended useful life should be velY low, B.
If the failure mode is benign, the recommended age for overhaul or part replacement may be much higher BI-BIO. If the failure mode is wear out, betas greater than one, and the cost of an unplanned failure is higher than the cost of planned replacement, there will be an optimal replacement interval.
Replacing the part at the optimal replacement time will minimize cost. See Section 4. Thomas Jefferson at 71 years ofage writing to John Adams 78 on July 5, Both died on the Fourth ofJuly , exactly fifty years after the signing ofthe Declaration ofIndependence. Steep betas within the design life are a source of concern. There is risk that the entire fleet will fail quickly as they age into the steep Weibull. On the other hand, if the Weibull characteristic life is well beyond the design life, there is a negligible probability of failure before overhaul or retirement.
In this case, steep betas are a source of happiness. Most steep Weibulls have a safe period before the onset of failure within which the probability of failure is negligible. The steeper the slope, beta, the smaller variation in the times to failure and the more predictable the results. A vertical Weibull, with a beta of infinity implies perfect design, quality control and production. For example, the latest powder metallurgy may provide cleaner metal with fewer inclusions. Turbine blades made from purer metal should have steeper betas than blades made with dirtier metals. Some manufacturers now use steep betas to specify and control the quality of vendor parts.
Typical failure modes with old age, rapid wear out include: stTess corrosion, material properties, brittle materials like ceramics, some fmills of erosion. All wear out modes have increasing failure rates with age and therefore, decreasing reliability. Overhauls and inspections may be cost effective to replace parts that produce significant failures.
Airlines remove compressor and turbine disks at their B. Ollife to reduce the risk of non-contained turbine failures in flight. Nuclear steam generator tubes are inspected remotely for the presence of cracks from stress corrosion, fatigue and flaws. Pumps have bearing and seal failures. Buried electric cables short out with age.
Railroads have roller bearing failures that cause derailments. Pacemaker batteries wear out. Dental implants fall out. All of these arc Weibull wear out failure modes that require inspections and cOl1'ective action. If one is well in advance of the other, failures from the second mode will never occur unless the first failure mode is eliminated. The part will always fail from the first mode if the lines are widely separated. The first mode is said to "cover" the second. In humans, cancer may precede and therefore, "cover" heart disease.
All parts and components have multiple failure modes but often one mode covers the others to the extent we are unaware ofthem. Existence of these unknown failure modes makes warranties and guarantees risky because development tests may not uncover the hidden modes. In human mortality the same concept applies. If a fatal illness is eliminated by a breakthrough in medicine, it may have little effect on the maximum life span because other fatal illnesses are "uncovered".
During the last century the average life span has more than doubled but the probability of surviving for years has hardly changed because of uncovering other failure modes. For old systems that have operated many multiples of their design life, there is no sign that all the failure modes are ever solved or even found. For this reason, systems that involve safety must be exposed to accelerated testing well beyond their design life to uncover unknown failure modes.
The manufacturer must test well beyond the age of the customer's fleet leaders to assure that catastrophic modes are eliminated. Accelerated life tests are often used. However, these tests may be too costly except for extreme safety requirements like aircraft, automobiles, surgical implants, and nuclear systems. The author has reviewed data on complex systems that shows the incidence of the first failure for each mode plotted against age using the Crow-AMSAA model.
This is a linear function on log-log paper and never reaches a plateau. These results imply that there are always unknown failure modes that will occur in the future. There are always Weibulls beyond the Weibulls we know about! However, understanding the construction provides a better understanding of its use.
All probability papers have scales that transform the cnmulative probability distribution into a lincar scale. If data is plotted on the transformed scale and it conforms to a straight line, that snpports the supposition that the distribution is appropriate. If the data is plotted on several probability papers, say normal, log normal and Weibull; the plot with the best fit is usually the most appropriate distribution. Of course, whcnever possible, knowledge about the physics of failure should support the choice of distribution. The fraction of parts that have not failed up to time t is 1 - F t.
This is the reliability at time t, denoted by R t. As shown in Tables , and , Weibull paper can be constructed as follows.
Construction of Abscissa t In t t Illr o units I 2 0. It will have a one-to-one relationship for establishing the slope of the Weibull. The Weibull parameter p is estimated by simply measuring the slope of the line on 1-to-1 Weibull paper. Of course, the scales can be made in any relationship. For example, assume the range of the failure data, the X parameter, is less than one log cycle. Multiply the X scale by three, to produce 1-to-3 paper. On 1-to-3 paper the slope is multiplied by tluee to estimate p. These considerations were important in past decades when most Weibulls were hand plotted.
Today with good software available, hand plotting is rare. SSW automatically provides optimal scales for each data set, but these may be changed with the "Zoom" option. SSW also provides l-to-1 scales as an option. Sample 1-to-1 scale Weibull paper is included at the end of this chapter. Standard Weibull probability paper is located at the end of this chapter.
The standard method and best practice for small and moderate size samples , median rank regression X onto Y curve fitting using the times-to-failure as the dependent variable 2. Median rank regression curve fitting using the median ranks as the dependent variable Y onto X , not recommended , 3. Maximum likelihood estimation MLE for very large samples, over failures. Maximum likelihood estimation with Reduced Bias Adjustment for small and large samples, or less failures.
Grouped or interval data analysis for coarse and inspection data may use the "inspection option", 6. Interval analysis of destructive inspection or NDE nondestructive evaluation data, regression is Yon X may use the "Probit method, " 7. This method is used for most engineering Weibull analysis throughout the world witl, heritage directly from Waloddi Weibull. If you could only use one method this would be the best choice.
The last three methods are altemative methods used for special situations. All methods will be compared in detail in Chapter 5. This chapter was devoted to good Weibulls based on good data using the standard method. In Chapter 3, we will attack dirty interval data, uncertainties, and some of the methods needed to analyze these deficient data sets. Chapters 5 and 8 include additional methods for dirty data. If beta is known from prior experience, a method called Weibayes may be employed. For small samples, say 20 or less, Weibayes is much more accurate than Weibull and is best practice if beta is known.
Weibayes is treated in Chapter 6. Today personal computers and friendly software have eliminated these cumbersome chores and provide solutions that were impossible by hand such as maximum likelihood solutions. Therefore, it may seem strange that we presented hand calculation methods for Weibull analysis in this chapter.
The author recommends every reader plot at least one or two Weibulls by hand to really understand the method. Two such problems are provided in this section. There is standard scale Weibull Probability following the problems for your use. After those problems, the student is encouraged to use good friendly software for as many of the remaining problems as possible, herein and elsewhere.
Problem Ten fatigue specimens were put on test. They were all tested to failure. The failure times in hours were as follows: , 85, , , , , , , , and Rank the data and note that identical failure times are assigned seqnential rank order numbers. Look up the median ranks in Appendix I. What is the slope and what class of failure mode is it? What is the eharaeteristie life?
What is the B1 life? What percent of the samples are expected to fail before hours? What is the reliability at hours? Problem Plot a Weibull with Suspensions. There have been five failures out of eight parts in service. The time in hours on the parts is as follows: Serial Time 9. Construct a Weibull plot. What are c. Hints: Rank the data from the first failure or suspension to the last. Enter in the table. Remember suspensions have no effeet on adjusted ranks or median ranks until after they occur.
Form Non. SloCked Re Form Non-Stocked Rev. What is the hardest part of Weibull analysis? Air conditioning compressor failure data is perhaps as dirty as dirty can be. In most cases the only data on failed compressors is the date shipped from the compressor manufacturer to the air conditioner manufacturer and the date returned under warranty. Operating time is highly seasonal and is not reported. How would you determine if this data is good enough to do Weibull analysis? The slope parameter is p. It is sometimes called what other name? What is another name for a Weibull distribution with e.
What is the standard method for doing Weibull analysis? What class of failure, [infant mortality, random, or wear out], and what range of expect for the following data sets? Woodpecker attacks on wooden electric power poles? Alzheimer's disease? Chicken Pox or Whooping cough?
Bearings from spalling or skidding balls? Fatigue cracking? Turbine vanes may fail early from hot spots from plugged fuel nozzles, a quality problem, or much later from stress corrosion cracking? What two classes? Solutions to these problems are in Appendix K. No peeking!! The answer is it does not make any difference if you do it properly as the two Weibulls are consistent.
For simplicity, assume the disk contains 4 blades. Plot the two Weibulllines on Weibull probability paper. Check on solution: Compare the first time-to-failure for one of four blades to the first failure of the system. The median rank Appendix I for first of four failures is The Bl5. Chapter 3 is devoted to "bad" Weibull plots.
Bad Weibull plots are often informative, In contrast to other distributions, the bad plot can provide significant information for the engineer who learns how to interpret results. Therefore, always plot life data on Weibull probability paper as the first step in the analysis. The objective of this chapter is to present the methods for interpreting bad Weibull plots, the analysis of dirty data, small sample risks, goodness of fit, and measures of uncertainty.
Insensibly one begins to twist facts to suit theories, instead of theories to suit facts. However, in cases of safety and extraordinary financial loss, there may be no alternative to employing Weibulls with only 1,2 or 3 failures. The analyst cannot request more "crash and burn" type failures. There is an urgent need to provide a source of direction rather than waiting for more failures. It is, therefore, plUdent to consider the associated uncertainties of these small sample applications.
It is remarkable that Weibull engineering analysis may be useful with an order of magnitude fewer samples than other commonly used statistical distributions. By comparison, if success-failure data is analyzed with the binomial distribution, a minimum sample size might be three hundred.
If dimensions and performance measurements are normally distributed, a minimum sample would be thirty. It is fair to ask why Weibull engineering analysis is possible with only a few failures. First, with service data, failures tend plot in the lower left hand comer of the Weibull because of the multitude of suspensions. This is the area of engineering interest, precisely where the data appears.
This is the neighborhood ofBO. The second reason is that the data set includes both failures and successes suspensions. Although there may be only three failures, there may be thousands of suspensions. Suspensions are not weighted as much as failures but they do count. With regression the failure data in the lower left COmer is over weighted. See Chapter 5. Although this degrades mathematical rigor, it increases the accuracy in the area of interest. Weibull probability paper is unique in that the lower area is expanded. In other words the area of small probabilities is magnified for engineering purposes.
Therefore, for engineering predictions of Bl life or BO. For statistical purposes, it is true that larger samples are needed to accurately estimate the Weibull parameters, eta and beta, and to determine which distribution is most appropriate, log normal or Weibull. For these statistical purposes a minimum of twenty one failures are needed, but thirty would be better, particularly for estimating beta.
The Weibull analyst can quantify the statistical uncertainly of his small sample by using SSWL software. The improvement in uncertainty with increasing the sample size is illustrated in Figures , and For example, in Figure for sample size three, the B5 life varies from about to in repeated samples from the same parent Weibulljust due to statistical uncertainty.
Chapter 7 provides a detailed discussion of confidence intervals, There are methods for reducing the small sample uncertainty. Obviously, the sample size may be increased, if this is not too expensive. A more cost effective method is to employ prior experience with the subject failure mode. This method, called "Weibayes," is described in Chapter 6 and depends on assuming the slope, f3. The approach is based on engineering experience and a WeibulllibraIy to select f3. It will reduce unceliainty by factors of two or three for small samples. Sample Size 3. The experienced analyst will have a sense of fit by studying the plot.
However, small samples make it difficult to gauge goodness of fit. The author prefers the simple correlation coefficient. The correlation coefficient, "1'," measures the strength of a linear relationship between two variables so it is ideal for our purpose. As life data probability plots always havc positive slopes, they always have positive cOlTelation coefficients. The closer l' is to one the better the fit.
Calculation of the correlation coefficient is illustrated in Appendix B. This section will introduce goodness of fit based on "r. New research and other measures of goodness of fit will be described in Appendix D. Mathematicians object to using the correlation coefficient on probability plots because using median ranks artificially increases the observed cOlTelation. To overcome this objection, the author employed Monte Carlo simulation programmed by Wes Fulton to approximate the distribution of the correlation coefficient from ideal Weibulls based on median rank plotting positions.
If your correlation coefficient, "1''', is largcr than the CCC, you have a good fit. Sce Figure The CCC is found by ranking the "1''' values for the correlation coefficient from 10, Monte Carlo simulation trials and choosing the th value. If your "1''' is larger than the CCC, the "10 p", you have a good fit. See [Edgeman ], and [Gan and Koehler ] for related studies with other plotting positions.
The correlation coefficient squared is the coefficient of determination, "1"". Statisticians prefer 1" to I' as a goodness of fit measure. CCC'is the tenth percentile of 1". Figure provides critical values CCC' of r' based on ten million trials and up to a million failures for the 2 and 3 parameter Weibull and log-nonnal distributions.
Notice the distributions have different CCC'. In other words, the three parameter Weibull 1" must be much higher than the r' for the other two distributions to be an equally good fit. For example, consider a sample of 10 failures whose , is 0. On the plot this would be a good fit for the two parameter Weibull and log normal, but a bad fit for the three parameter Weibull. Critical Correlation Coefficientl lOP 0. This is the most accurate indicator of goodness of fit.
To compare the fit of one distribution with another, say a Weibull with a log nonnal, we need a moderate size sample, say 21 or more failures, and the Pvc. The distribution with the highest Pve among competing distributions is the best statistical choice. The Pve offers many benefits as a goodness of fit indicator. Chi Chao Liu  concluded for regression that the P-value provides an excellent indication of goodness-of-fit. However, there is capability to show 1'2 or 1'2 - CCC2. Prior knowledge of the physics of failure and past experience should weigh heavily in choosing the best distribution.
There can be real concerns about the validity of the data. Often the snspicion will be that the data is non-homogenous; several heats of metal were used, several vendors produced the units, the service environment was not unifonn, the testing procedure was poorly defined, the data collection was poor, there is a mixturc of failure modes, and on and on.
These concems may be sufficient to negate the analysis. The solution to these worries is in the data. What does the data say? If the data fit the distribution well it implies these concerns are negligible, but if the fit is poor they may be significant. Remember thc basic premise of statistics is that no two things, parts, tests, twins or data points are exactly alike. Statistical methods allow us to deal with uncertainty.
In other words a good fit may be a cure for insomnia. Carl [Tamm] of Delphi using the author's CCC values discovered that these critical correlation coefficients are approximately Weibull distributed and generated the results in shown in previous editions of the Handbook, a Weibull plot showing the CCC up to failures. By extending the coverage to a million failurcs it is now obvious that the straight line Weibull is inadequate as curvature is now obvious.
The more recent simulations by Carl, Todd Marquart, and Paul Barringer have produced an updated plot shown in Figure 3. It may havc failed by a different failure mode or not failed at all. A bolt that fails in the bolt head is a suspension in a pull test for thread failures. An "early suspension" is one that was suspended before the first failure time. A "late suspension" is suspended after the last failure. Suspensions between failures are called "internal" or Hrandom" suspensions.
As a rule, suspensions increase the characteristic life, eta, but have little effect on the slope, beta. Early suspensions have negligible effect on the Weibull plot. Late suspensions have more significant effects, and may reduce tlle slope, beta. The presence of many late suspensions usually indicates a batch problem; newer units are failing, older units are not failing.
Internal suspensions are more difficult statistically, particularly for confidence interval estimates. A bolt that fails in the bolt head is a suspension in a pull test for thread failures. An "early suspension" is one that was suspended before the first failure time. A "late suspension" is suspended after the last failure. As a rule, suspensions increase the characteristic life, eta, but have little effect on the slope, beta.
Early suspensions have negligible effect on the Weibull plot. Late suspensions have more significant effects, and may reduce the slope, beta. The presence of many late suspensions usually indicates a batch problem-, newer units are failing, older units are not failing. Internal suspensions are more difficult statistically, particularly for confidence interval estimates. Figure shows the effect of 90 late and early suspensions with 10 failures.
Internal suspensions will tend to shift the Weibull line somewhere between the early and late suspensions shifts. The SSW software provides this capability. Confidence bounds are discussed in Chapter 7.
Effect of Suspensions on the Weibull Line Statisticians always prefer complete samples i. For example, all in-service and warranty data sets have suspensions. Fortunately, the Weibull and log normal distributions handle suspensions well. For unknown reasons materials curves for low cycle fatigue and creep often have a suspect first point.
Deleting this point is very attractive and may reduce the weight of the design by a significant amount. This will be a critical, perhaps dangerous, decision. If not, concern about the presence of this value will disappear. With a suspect point you should investigate the engineering aspects of data recording, test records, instrument calibrations, etc. This is the proper thing to do. Statistics can be of some help in this investigation but should not be used to justify rejecting an outlier without supporting engineering. Wes Fulton then modeled the test for outliers using simulation.
The outlier test is conducted as follows: The null hypothesis is that the suspect point is not an outlier. However, if it is an outlier it can have a large effect on the parameter estimates. Thus, the suspect point is deleted from the set, and the parameters and the p-value calculated. Data is precious and should not be rejected without sufficient evidence. In the SuperSmith Weibull software select the calculator option for an outlier test.
See Chapter 11 for Case Study The message from the data is the origin may be in the wrong place. For a curved plot time may not start at zero.
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For example, it may be Dr. A bearing failure due to spalling or unbalance cannot occur without bearing rotation inducing enough damage to fail the bearing. Time starts when failures are possible. Figure shows the same data with the origin shifted to 24 million revolutions. There is a guaranteed failure free period the first 24 million revolutions , within which the probability of failure is zero.
Notice the significant improvement in the goodness of fit. Note that because beta is so steep the plot looks good even though there is extreme curvature. Steep betas hide patterns that are disclosed when the X axis is expanded. The model indicates it is physically impossible to fail the plate at a low level of stress. Figure shows the effect of the t 0 shift. There are many possible reasons for an origin shift. The manufacturer may have put time or mileage on the system as part of production acceptance, but reported that the units are "zero time" at delivery.
The purpose of production acceptance is to eliminate the infant mortality failures. Electronic components often are subjected to burn-in or environmental stress screening for the same purpose. In these cases the units have aged before being delivered as "zero time" systems. For material properties, where the Weibull ordinate is stress or strain, it may be impossible for fracture or creep or other properties to produce failure near the origin on the scale. Spare parts like rubber, chemicals and ball bearings may age in storage and use part of their life on the shelf, requiring a negative t 0.
For these reasons and others, the Weibull plot may be curved and needs an origin shift, from zero to t 0. Anytime beta is above 5 or 6 the data should be scrutinized carefully as it will appear to be a good fit but can have curvature or outliers. When the tp correction is applied to the data, the resulting plot will follow more of a straight line if the correction is appropriate. Figure shows the fracture data in Figure with the tp correction. Note that the Weibull ordinate scale and the characteristic life are now in the tp domain. To convert back to real time, add tp back.
Of course this produces a curved tg Weibull line when plotted in the real time domain but the real time domain is easier to explain without adding or subtracting data. The three parameter con-elation coefficient with t 0 will always show a higher con-elation coefficient better fit than the two parameter simply because it is a more complex model.
Similarly, a quadratic curve fit will have a higher conelation coefficient than a linear fit. The following four criteria should always be met before using the three parameter Weibull: 1. The Weibull plot should show curvature. There should be a physical explanation of why failures cannot occur before tg. For example, bearings cannot spall instantaneously. Many rotations are required to produce the damage. A larger sample size , at least 21 failures should be available.
It takes much data to estimate three parameters than two parameters. If there is prior knowledge from earlier Weibulls that the third parameter is appropriate, a smaller sample size, say eight to ten may be acceptable. The correlation coefficient "P" value should be greater than the "P" value for the next best distribution. Concave upward suggests a negative tg, and is more difficult to explain physically. Some parts have been aged before installation. Shelf life, burn-in for electronics, and production are examples.
Another possibility is the classic mixture of two failure modes, the BiWeibull. There are several ways to estimate tg. A curve may be "eyeballed" through the data and extrapolated down to the horizontal time scale. The intersection will be an approximate tg. If the earliest portion of the data is missing, tg may compensate for the missing data, although this may not always be successful.
The cable data was grouped by vintage year and the aging scale was in months. There was no data recorded prior to Note the tg corrections in Figure Please do not expect this kind of close agreement, but it shows the value of good data in large quantities. We thank Florida Power for permission j Dr. The computer will iterate on tp until the correlation coefficient is maximized. Left suspensions are earlier than the first failure and should be deleted before calculating to; otherwise to cannot exceed the earliest left suspension. Note that the deletion of left suspensions will have a small effect on the plot line.
In summary, concave downward plots indicate the origin needs to be shifted to the right, subtracting to from each time-to-failure to get a straight line fit in the to time domain. Concave upward plots indicate the origin has to be shifted to the left and 1 q must be added to each time-to-failure to get a straight line fit.
The plot in "as recorded" time scale is easier to understand. See Case Studies Log normal data plotted on Weibull paper is concave downward, very much like the three parameter Weibull. The same data plotted on log normal probability paper follows a straight line. If x is log normally distributed, the distribution of x will be skewed to the right and log x will have the familiar bell shaped normal distribution.
Figure shows this effect with the probability density function plotted. The three-parameter Weibull plot with positive t zero correction looks just like the log normal plot with small samples, so these are competitive distributions. The author has witnessed many arguments over which distribution best fits the data.
With bearing failures the arguments went on for two decades until the industry arbitrarily selected the three parameter Weibull as the standard. Technically however, the log normal is sometimes a better choice even with bearing data and this is true with the data in Figure If the distribution of log x is normal, x is log normally distributed. All the standard normal statistics apply in the logarithmic domain. The log normal parameters are the mean of log x, mu , and the standard deviation of log x, sigma.
Both mu and sigma are expressed in logarithms. S74 1. With log normal data the antilog of the mean of log x will be less than the mean of x. The antilog of the mean of log x approximates the median value of x. Sigma is also a logarithm and therefore, its antilog is a multiplying factor. The equations are given in Appendix G. Log Normal Probability Density Functions Physically, the log normal models a process where the time-to-failure results from a multiplication of effects. Progressive deterioration will be log normal. For example, a crack grows rapidly with high stress because the stress increases progressively as the crack grows.
If so, the growth rate will be log normal. Vibration tends to loosen bolts and fasteners which increases the vibration. In a gas turbine, loss of blade tip clearance will cause an increase in fuel consumption. If the rate of blade tip loss accelerates the increase in fuel consumption will be log normal. The log normal has many applications such as materials properties, personal incomes, bank deposits, growth rate of cracks, and the distribution of flaw sizes. If T lab has a Weibull distribution, T s will also have a Weibull distribution.
However, if the K factors are significant, the in- service times will be lognormally distributed because of the Central Limit Theorem: sums of samples from any shape distributions tend to be normally distributed and the Ln of T s is such a sum. Note the multiplication in this model produces the log normal times-to-failure. Data should be plotted on Weibull and log normal probability paper to compare the goodness of fit. The three parameter Weibull and the two parameter log normal will usually provide con-elation coefficients within plus or minus one per cent of each other.
Do not be deceived by this into thinking the two distributions are equivalent. The software provides solutions for two and three parameter Weibulls and log nonnals, so comparisons are easily made with Pve. S indicates reference may be downloaded from ban-ingerl. With heavy censoring e. The log normal distribution should be the first choice if there is good prior information and more than twenty failures.
For example, many material characteristics employ the log normal distribution. Times-to-repair and crack growth-to-nipture are often log normal. Knowledge that the physics of failure indicates progressive deterioration would also be a clue that the data may be log normal. Some semiconductor chip failures are log normal. The distribution of the Weibull slope parameter, beta, is approximately log normal, while eta is more normally distributed. This is not always true comparing the Weibull three parameter with the log normal.
See Case Study Considering that distribution analysis is not creditable with small samples, 20 or fewer failures and that the Weibull 2 parameter distribution is more conservative than the log normal, best practice is always use the Weibull 2 parameter for small samples. Examples are given illustrating the following: a. Failures are mostly low-time parts with high time parts unaffected, suggesting a batch problem.
Serial numbers of failed parts are close together, also suggesting a batch problem Dr. The data has a "dogleg" bend or cusps when plotted on Weibull paper, probably caused by a mixture of failure modes. The first or last point appears suspect as an outlier, indicating data problems or perhaps evidence of a different failure mode.
Gas turbine engines are tested before being shipped to the customer, and since there were over of these engines in the field with no problems, what was going wrong? Upon examining the failed oil pumps it was found that they contained oversized parts. Something had changed in the manufacturing process that created this problem, a batch problem.
The oversized parts caused an interference with the gears in the pump that resulted in failure. This was traced to a machining operation and corrected. Low-time failures can suggest wear out by having a slope greater than one, but more often, they will show infant mortality, slopes less than one. Low-time failures provide a clue to a production or assembly process change, especially when there are many successful high-time units in the field.
Overhaul and scheduled maintenance also may produce these "batch" effects. Times since overhaul or maintenance may provide a clue. The presence of many late suspensions is also a clue that a batch problem exists. In the next chapter methods for using failure forecasting to confirm the presence of a batch problem will be presented. There is summary of batch clues and analysis at the end of Chapter 8. Appendixes F and J contain advanced material on detecting batch effects. For example, if low-time units have no failures, mid-time units have failures, and high-time units have no failures, a batch problem is strongly suggested.
Something may have changed in the manufacturing process for a short period and then changed back. Closeness of serial numbers of the failed parts suggests a batch problem. Figure is a prime example of a process change that happened midstream in production. Bearings were failing in new augmentor pumps. The failures occurred at to hours. At least units had more time than the highest time failure. These failures were traced to a process change incorporated as a cost reduction for manufacturing bearing cages. Augmentor Pump Bearing Failure 3. This was the case for a compressor start bleed system binding problem.
Upon examination of the data, 10 out of 19 failures had occurred at one base. The base was located on the ocean and the salt air was the factor. The data were Figure A categorized into separate Weibull plots with this Figure B engineering knowledge. The first Weibull had a slope of 0. More attention to maintenance resolved the problem. Dogleg Weibulis are caused by mixtures of more than one failure mode. These are competitive failure modes, competing to produce failure. For instance, fuel pump failures could be due to bearings, housing cracks, leaks, etc. If these different failure modes are plotted on one Weibull plot, several dogleg bends will result.
When this occurs, a close examination of the failed parts is the best way to separate the data into different failure modes. If this is done correctly, separate good Weibulis will result. With small samples it is hard to distinguish curves from cusps, see Figures A and B. There can be mixtures of modes and populations, perhaps batches and competing failure modes.
A steep slope followed by a shallow slope usually indicates a batch problem. A steep slope followed by a gentle curve to the right indicates there are some "perpetual survivors" that are not subject to the failure mode. For example, there may be defects in some parts but not all the parts, a batch problem. The Classic Bi-Weibull Many hydro mechanical components show infant mortality from production and quality problems followed by wear out later in life as competing failure modes. This is called the "Classic Bi-Weibull," a shallow slope followed by a steep slope.
The author recommends engineering analysis to separate or categorize the data by failure mode. Many analysts assume that all failures earlier than the comer belong to the first Weibull and all the later failures belong to the second. This is rarely true, but statistically it may be approximately true.
This approach will be illustrated with an example. However they experienced early failures after overhaul indicating a quality problem in the overhaul process. Many hydro mechanical parts experience this combination of a few failures early in life followed by wear out. The corrective action is to identify and eliminate the quality problem. The failed units were not available for inspection to categorize them into separate failure modes. Therefore a statistical separation based on the likelihood ratio test was employed to estimate the parameters of the two Weibulls using SSW.
This test will be treated in Chapter 7. The corner showing the strongest evidence of two sets is selected if there is strong evidence supporting two sets versus one set. To estimate the four parameters at least 21 failures are needed for creditable results. For mixtures of three failure modes at least failures should be available. However, engineering separation of data mixtures is always preferred to statistical methods.
Thus Weibulls for a system or component with many modes mixed together will tend toward a ft of one. These Weibulls should not be employed if there is any way to categorize the data into separate, more accurate failure modes. Using a Weibull plot with mixtures of many failure modes is equivalent to assuming the exponential distribution applies.
Webinar: Weibull Engineering Intro Using Weibull Plotting Software by Wes Fulton
The exponential results are often misleading and yet this is common practice. MIL-HDBK use the exponential assumption for modeling electronic systems whereas field failure data will indicate infant mortality, or wearout failures. See case Study The steep plot often hides bad Weibull data. All the messages from the data such as curves, outliers, doglegs tend to disappear.
Apparently good Weibulls may have poor fits. An example is Figure Here, at first glance the plots appear to be good fits, but there is curvature and perhaps an outlier. If SSW is used, a simple solution to amplify the data problems is to use the "zoom" option to magnify the curvature portion. This will make the problems with the data more obvious. With large samples a flat spot in the middle of the plot can indicate a log normal. The Weibull 3P is a continuous curve. There may or may not be a batch problem. It is my intent to share experience.
Waloddi Weibull preached simplicity; plot the data, and look at it, and the author strongly agrees. Waloddi said these methods may have "some utility. This handbook may appear to be complex because the "menu" of possible data sets in industry is infinite. Many of you have probably browsed through Chapter 3, leaving some sections under the heading, "if I ever need this material, I know where it is. Use what applies in your application; ignore the rest. Chapter 10 provides a summary of the methods for handling the good, the bad and the dirty data.
Further, it now includes a useful logic diagram that will take you step-by-step through your analysis. Plot on Weibull paper. What value is needed to straighten the Weibull? Hint: "Eyeball" a curve through the data and read an approximate t 0 where it intersects the bottom scale. Will the value found in "a" be added or subtracted from the failure values? Repeat with SSW using the t 0 options to find a more accurate answer. Study Figures through Assume you have no prior knowledge. If you are in a Weibull Workshop talk it over with the "experts" sitting near you. Use the information provided on the plot.
List your comments and the actions you would take in analyzing each data set. Garbage Broken Units. Broken Units. What Kind of Data? Laboratory Connector Failures Figure Cable Wire Fatigue Samples Dr. Yield Strength - Steel Dr. What is This? And This? J GumbeL There are similar vignettes of other great statisticians that I admire at the end of chapters that relate to their contributions.
These are not required reading. Dorian Shainin was one of the earliest pioneers for Weibull analysis and engineering statistics.
He inspired the author to do a doctorate in statistics although Dorian thought it was a waste of time. His "Random Balance" preceded Taguchi and was equally controversial. His humor was endless. He claimed he drank carrot juice and vodka; "I can get just as high but I can see better. ASQ has now named an award after Dorian. His greatest contribution was getting engineers and managers excited about using statistics. His talent here is unsurpassed, actually never tied.
Gumbel spent much of his life studying the statistics of extreme values, rare events. He and Waloddi Weibull did a sabbatical together at Columbia University and became good friends. It is employed for predicting maximum and minimum values, flood levels, wind gusts, the size of inclusions in metal. The Weibull and the Gumbel minimum are related like the normal and the log normal through a logarithmic transformation.
Section 8. Responsible managers demand a forecast of the number of failures expected to occur in the future. How many failures will there be next month, the next six months, the next year?
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What will the costs be? Managers need failure forecasts to set priorities and allocate resources for corrective action. This is risk analysis, a prediction of the magnitude of the problem, a clear view of the future. Further, the uncertainty will increase as the time span for the forecast increases. With this information a failure forecast can be produced. The techniques used to produce the failure forecast vary from simple calculations to complicated analysis requiring Monte Carlo simulation.
These will be explained in the following sections. The batch failure mode does not apply to all the units in the population. Detection of batch problems is the principal reason for calculating the expected failures now. To calculate Dr. Earlier editions of the handbook evaluated the Now Risk as probability of failure by time tj summed over the number of units, N, including failures, r, and suspensions, s. New research has shown this formula to be slightly biased.
There are 5 suspensions at ages of and hours, and 4 at and hours, respectively. The first question is, "What is the expected number of failures from time zero to now for this population? Table shows the results. If the expected-failures-now is much larger than the observed number of failures, the Weibull may not apply to the entire population. This indicates a "batch" problem, i. Batch problems are very common. If you have a batch problem you have the wrong data set and your first priority is to identify the units in the batch.
See Chapter S for more details on batch problems and how to analyze them. The failure forecast will be wrong if you use the wrong data set. Appendix F describes the Aggregated Cumulative Hazard method for analyzing batch problems. It works for complete samples which the now risk method cannot handle.
If a Poisson lower confidence bound on the "Now Risk" is greater than the observed number of failures a batch problem is indicated with high confidence. The Now Risk detection capability is available with rank regression, but not with maximum likelihood estimates MLE. SSWcan place lower and upper bounds on "Now Risk. Let us use the next year as an example. Given the 18 pumps at risk in Table , the expected number of failures over the next 12 months can be predicted. Yearly usage of each pump will be hours. The failure forecast for one year sums the predicted failures of the hour pumps running to hours, plus the predicted failures of the hour pumps running to hours, plus the predicted failures of the hour pumps running to hours, etc.
The forecast only involves the unfailed pumps, the living suspensions, pumps still "at risk". All future failures will occur on today's survivors. If pump i has accumulated t j hours to date without failure , and will accumulate u additional hours in a future period, the failure forecast for the 18 pumps at risk is given by Equation F tj is the probability of pump i failing in the first tj hours of service, assuming it follows a Weibull failure distribution, it is the 12 month usage, hours.
Note that the summation is over the 18 unfailed, suspended pumps. Unfailed pumps are still "at risk" because they have survived up to now. In this example, failed pumps are dead, no longer "at risk. Therefore, the failure prediction is 2. Without replacement the fleet size will decrease with time.
With replacement the fleet size is constant and the failure of the replacements increases the number of failures. The replacement unit is usually assumed to be a new, zero-timed, addition to the fleet of the same design, but may be a redesign with an improved Weibull or a repaired part with a worse Weibull. With new units, if the forecast interval is B50 life or less, the forecast with or without replacement is about the same since the replacement units do not age enough to fail.
In cases where the failure forecast is greater than the B50 life, the chance of more than one failure over the interval starts to become significant, i. In this case, the expected number of failures may be calculated by adding a new zero time unit whenever the failure forecast increases by a unit amount. Production will increase the fleet size and the failure prediction. The production rate may be constant, or it may be seasonal varying by the month of the year, or it may be sampled from a distribution.
The calculations for all these variables are complex and require software like WSW. These equations assume failed units are not replaced. F t may be calculated from the Weibull, normal or log normal. Without renewal the future risk relates to a decreasing fleet size. The future risk with renewal may be estimated by adding a new production unit to the fleet whenever the future risk increases by a unit amount. SSWprovides upper and lower bounds on the future risk with and without renewal.
Several case studies will illustrate the ideas developed in the previous sections. The first two examples, Sections 4. The asterisk indicates advance material which may be skipped on first reading. In Chapter 11 case studies Case studies 5. Advanced material may be omitted on first reading. The fleet originally contained bearings in service with ages up to hours. Failures are also shown.
Figure shows the bearing cage Weibull failure distribution. This was much less than the B10 design life of hours, so a redesign was undertaken immediately. Additionally, management wanted to know how many failures would be observed before this redesign entered the field. Bearing Cage Data from Table The risk questions are: 1. How many failures could be expected before the units reach hours if all the suspensions were new, zero time parts?
Calculate the number of units that will fail by hours, assuming failed units are not replaced. Enter the x-axis of the Weibull plot Figure at hours and read the y-axis. Approximately 1.
That is, after the entire population of Dr. WSW more precisely calculates How many failures could be expected in the next year? How many in the next five years? Using the methodology explained in Section 4. Thus, about Note that the 12 month forecast is How many failures could be expected over a hour period if we replace every bearing at hours? From the answer to Question 1, the probability of a bearing failure by hours is 0.
Therefore, if it is assumed that each hour replacement makes the bearing "good as new" relative to cage fracture, there is a total expectation offailure for each bearing by hours of approximately 0. So, if all bearings ran to hours with hour replacements, 0. See Chapter 7. Bearing Cage Failures 4.
Suspensions are listed in Table The high incidence at air base D prompted a 12 month failure forecast. Weibull analysis of the failures at air base D Figure shows a rapid wear out characteristic. All Bases Except Base D Chapter 7 presents methods to statistically show that they are significantly different. Barron Test Coordinator DaimlerChrysler Corporation "An excellent overview of Weibull analyses that leaves the student with a basic understanding and the resources for further learning about the topic. Brought me up-to-speed quickly. Techniques learned are invaluable.
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