sthe Society shall not be responsible for statements or opinions advanced in papers or in discussion

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1 THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS 88-GT E. 47 St., New York, N.Y sthe Society shall not be responsible for statements or opinions advanced in papers or in discussion at meetings of the Society or of its Divisions or Sections, or printed ih its publications. M Discussion is printed only if the paper is published in an ASME Journal, Papers are available 1^L 0 from ASME for fifteen months after the meeting. Printed in USA. Copyright 1988 by ASME Weibull Analysis Techniques on a Desktop Computer J. L. BYERS Senior Aeronautical Engineer Naval Air Development Center Code 6052 Warminster, PA ABSTRACT This paper presents a summary of a task to provide individual Navy project engineers with analytical tools that enable them to perform Weibull failure and related analyses on a desktop computer. Similar tools have been available on mainframe computers for over a decade while only fragmented and incomplete tools existed for desktop computers. There is now available an integrated computer program that allows Navy project engineers and other Navy analysts to perform rigorous trade-off and what-if analyses in an interactive manner without having to send the problem off to their organizational central computer facility. The resulting computer codes exist in several forms to fit the various needs and computer configurations, such as: direct input of data, data file creation and update, and non-printing versions for those who have no printer available. Included in the codes are three Monte Carlo routines and several test plan generation codes. These codes have not been released to the general public as yet and are currently restricted to Navy units such as laboratories, Naval Aircraft Rework Facilities (NARF) and the Naval Air Systems Command Headquarters (NAVAIR). Public release is expected in mid-fy 89. INTRODUCTION For many years the United States Navy has been concerned about the best way to determine the number of spare parts to buy for its aircraft propulsion systems. The concern arises because there are three possible outcomes when spare parts are procured. One can buy too few, too many or exactly enough. The odds heavily favor the first two outcomes, which are both undesirable in almost every instance. Buying too few means some aircraft may not be able to fly when needed and buying too many is almost as bad since it wastes dollar resources for parts not needed. The Propulsion and Power Division of the Naval Air Systems Command (AIR 536) tasked the author to provide the analytical tools and methodology required to correlate known failure data with future parts usage to allow a more judicious parts ordering process. Constraints applied to this task included: -; the use of data from existing data systems - analytical tools developed should be simple and easy to use computer codes should be IBM PC compatible -, the methodology should not be incompatible with that used by industry -, techniques used should be usable even with a minimum of usage data such as early in the system life cycle. APPROACH After considering all of the constraints, it was soon evident that a statistical approach was required. The Weibull probability distribution function was considered since it is the almost singular method used for failure analysis in the aircraft propulsion industry. It was selected as the basis for this task because of the extra dimension of flexibility the Weibull distribution provided. As the slope or shape parameter (Beta) is varied, the Weibull distribution can approximate the exponential and normal distributions, two other widely used failure distributions (Figure 1). The fact that the Weibull distribution approximates other distributions as its shape parameter takes on different values is a clear indication that the Weibull distribution function is adaptable to a wider variety of failure applications than some other distribution functions. Also, since this probability distribution function is linear when plotted on the appropriate plotting paper, (Figure 2), it is possible to use a relatively small amount of data to make estimates for a generalized population of much larger size. With wider and wider use being made of the Weibull distribution methodology, it is rapidly becoming the basis for the analyses that are desired by Presented at the Gas Turbine and Aeroengine Congress Amsterdam, The Netherlands June 6-9, 1988

2 sparo parts forecasters as well as others interested in logistic support of aircraft propulsion systems and other industrial equipment. Weibull Si. p9 Olsteh'eian P P curve P - o.s normal ppol P ISO.. ---Expo,erm l 10any other 1111 [ curves ara pa5w OLD 0 BETA = ETA = 204, f P-to Figure 1. Weibull Slope or Shape Parameter..._._..._... _... _..._......<..;.,1.;1,...;...;,..;.j i i j<.i... ;...i..;.,.iii, IA Figure 2. Weibull Plotting Paper While it was quickly evident that Weibull distribution methods were widely used, it was also evident that only the most elemental analytical procedures were coded for use on desktop microcomputers. Very elaborate and highly capable analytical methods were found to be in use on large mainframe computers and these were generally treated as proprietary codes. Since the Navy wished to have this same general capability for use on desktop computers, it was decided to generate a new set of codes that could be integrated and manipulated by use of an executive routine. The overall program is named WEIBER, which is the acronym for the executive routine used : WEIBull Executive Routine. The balance of this paper will present a summary of the effort completed to arrive at an integrated Weibull analysis program for use on a desktop microcomputer. It is presumed that the reader has a fundamental knowledge of the Weibull probability distribution function and the analytical methods used. For those that require additional information, the references will provide ample guidance. there were six different analytical procedures and a list of historical Weibull slope parameters. These were: the Weibull parameter calculation routine, Weibayes characteristic life calculation, Weibayes life calculation, present risk analysis, future risk analysis, and a Monte Carlo failure analysis. Originally there were two versions of the Weibull parameter calculation: one that had the capability to give maximum likelihood values, and a second which provided the B10 and B50 lives as well as a table of instantaneous failure rate versus time. Today the program consists of a total of fourteen analytical codes. A third Weibull parameter calculation code was added that saves the data file of failures so that it can be updated and/or recalculated at a later time. Two more Monte Carlo failure analysis codes were added to optimize for speed of analysis and for maximum population or fleet size. New analytical codes were added to calculate confidence intervals for the Weibull slope (Beta), characteristic life (Eta), and time to first failure. In addition, reliability and confidence interval for reliability were added. Two codes were added to address test plan generation for substantiation testing of re-engineered parts or components. One test plan code provides requirements for a zero-failure test plan which consists of the number of parts to test and the number of hours or cycles that each part must successfully complete without failure. The second test plan code is for a non-zero failure test plan which presents the number of parts or components to test, the test hours or cycles for each, and the allowed number of failures for successful test completion. The current menu for the program codes (Figure 3) shows fifteen choices including the historical Weibull slope reference values. While there is space shown for eighteen codes in the current menu, this is not a limiting value. The menu can be presented on multiple screens and the codes can be spread out over any number of disks. As the overall program grows, it will become advantageous to have a hard disk KEIBULL PARAMETER CALCULATION. 16E CU) 2. NEIBULL PARAMETER CALCULATION AND SAVE INPUT DATA IPNA CO) 3. WEIBULL PARAMETER CALCULATION W/ MAX. LIKELIHOOD VALUES. (PWA CO) 4. CHARACTERISTIC LIFE CALCULATION. 5. NEIBAYES - WHEN WEIBULL PLOTS ARE IMPOSSIBLE 6, HISTORICAL VALUES OF BETA. 7. PRESENT RISK ANALYSIS. (SAVE DATA) B. FUTURE RISK ANALYSIS. (SAVE DATA) 9. CONFIDENCE INTERVAL CALCULATIONS FOR BETA & TIME TO FIRST FAILURE. 1B, RELIABILITY AND CONFIDENCE INTERVAL FOR RELIABILITY. 11. MEIBULL FAILURE ANALYSIS - RUNTY CARLO ANALYSIS, (SAVE DATA) I2. WEIBULL FAILURE ANALYSIS - SHORT PRINT-OUT. IRETREIVE DATA' 13.NEIBULL FAILURE ANALYSIS - FOR OVER IIBB ENGINES/PARTS. (RETREIVE DATA) 14.ZERO FAILURE TEST PLAN FOR SUBSTANTIATION TESTING. 15, NON-ZERO FAILURE TEST PLAN GENERATION, 16. FOR FUTURE USE - NOT YET AVAILABLE, 17,FOR FUTURE USE - NOT YET AVAILABLE, 18,FOR FUTURE USE - NOT YET AVAILABLE. TO CONTINUE CHOOSE OPTION NO. OR 90B TO QUIT,? THE PROGRAM The first problem was to decide what codes should be included in the program - at least in the initial version - so that it would be functional. Initially Figure 3. Weiber Program Menu 2

3 THE COMPUTER LANGUAGE The Basic language was chosen to write the codes for two reasons. First, it is the language with which the author is most familiar. Secondly, it was realized that frequent changes would be necessary if the codes were to be used with a maximum of flexibility. The codes were written in GW-Basic and developed on the highly PC- compatible ITT-XTRA. This made them extremely transportable since the vast majority of U.S. Navy desktop microcomputers are PC-compatible. The Basic language and PC -compatibility assures that necessary changes can be made at virtually any Naval organization in a minimum of time. Future versions are planned in compiled Basic and Pascal. WEIBULL PARAMETER CALCULATION There are three versions of the Weihull parameter calculation code currently in the program which can be use to calculate Beta and Eta. Two versions are derived from codes originally written by employees of Pratt and Whitney Aircraft and the other is derived from a code written by employees of the General Electric Company. As obtained from these sources, the codes were not suitable for integration into the WEIBER program. They were significantly revised to make them very user friendly. The General Electric Company derived code requires input of the number of units (parts) in the population, the time of failure for each failed unit, and the total number of failed units. As output, it furnishes Beta and Eta as well as the time when 10% and 50% of the population is expected to fail. These times are known as the B-10 and B-50 lives. With minor changes to the program, any population percentage desired can be substituted for the B-10 and B-50 values. The final output is a list of values for the instantaneous failure rate, or hazard rate, which is presented for selected values up to hours or cycles. Both Pratt and Whitney codes also required much modification to integrate them into WEIBER and to make them user friendly. The heart of each of the codes is identical, with the only substantial difference being the ability of one version to save a data file for later revision, addition of data, and/or recalculation. Both codes have the ability to calculate maximum likelihood values if desired. It is highly desirable to use the code that saves input as a data file if there is reason to believe that it will be necessary to run the same problem again. This is especially true when there are data changes and a large amount of input data. There are options available for creating, retrieving, verifying, and storing files. These file manipulation procedures were also developed by Pratt and Whitney personnel under Government contract for the U.S. Air Force. For one-time runs, either of the other two codes would be satisfactory, depending on the output data desired. CHARACTERISTIC LIFE AND WEIBAYES These are two separate codes that are complimentary. The characteristic life calculation code, Etacalc, is extremely useful when one has a good handle on the failure mode and hence has at least a range of values for the Weibull slope (Beta). All that is required is a population histogram, Beta and a count of the total number of failures. From this data one can determine the characteristic life Eta, even if there have been no failures. In fact it is a most useful method when there are few or no failures. The Weibayes code utilizes the calculated value of Eta (the Weibayes value) to in turn calculate the elapsed time or cycles (life) after which a specified proportion of a given population will have failed. These time periods are known as B lives, i.e., B-.1 or B - 50 for times when 0.1% or 50% of a population should have failed. PRESENT AND FUTURE RISK ANALYSIS These codes are used to calculate the numbers of failures to expect up to the present time or for a specified future time period, based on a given usage rate. There are obvious benefits in being able to calculate the number of expected failures, whether it is up to the present or for the future. These codes assume that one is interested in how many units will fail or should have failed without regard to the units being fixed, repaired, or replaced. In other words, the concern is for a given fixed number of units. The Present Risk code is useful in verifying that the values of Beta and Eta used are still valid for the failure mode under consideration. If the calculated number of failures does not closely match the actual number of failures, then the recalculation of Beta and Eta would be required. The Future Risk code is used when one is interested in the number of failures to expect over a reasonable time into the future. This may be for the purpose of confirming the number of spare parts required at a given location or to determine how personnel and/or equipment assets should be allocated. These risk analyses can be done for many reasons since the ability to forecast numbers of failures is always advantageous. NONTg CARLO FAILURE ANALYSIS There are three versions of Monte Carlo analyses in the Weiber program. These codes all allow for repair or replacement of failed units with subsequent return to service. The time period can be any length of time into the future and the usage rate can be varied as desired (flight hours or cycles per month). There is provision for specifying inspection time periods, at the end of which the units are considered to be either zero-timed by overhaul or at least certifiable for continued service through the next inspection time period. If there are to be no inspections (run to failure), then an inspection time is specified which is longer than that of the analysis. The time period of the analysis is determined by the usage rate multiplied by the months into the future. This means the inspection time is specified in terms of hours, cycles, A/B lights, etc., as is the time period of the analysis. Ilours or cycles are the most frequent choices but any age measurement units may be used if the data are available in that form or can be conveniently converted. The three versions of Monte Carlo analysis are generally categorized by the number of units in the population to be analyzed and the number of applicable failure modes. The first code, WEIBRISKI, is used on a relatively small population, 25 or less, with multiple failure modes in order to determine the dominant failure mode. This code prints out a maximum of 3

4 information, such as failure time for each unit in each failure mode for each iteration, and therefore would be too slow to use for a population larger than approximately twenty-five. Once the dominant failure mode is known, a second analysis would normally be run on the second code, SHRTWEIB. This code is used for a single failure mode and gives total failures by iteration and the average number of failures overall and per engine. The code is currently limited to 250 units or engines and has been optimized to run as fast as possible in an uncompiled mode. If the number of units in the population is very large (say up to 1,100) then it would be desirable to run the case of the single failure mode on the third code, BIGWEIBL. This code has been optimized for a maximum number of units as determined by the normal random access memory available in the basic language (64K); it therefore has a minimum of output data. Here the output is restricted to the number of failures per iteration (for all units), the average number of failures per unit for each iteration as well as the overall number of units. CONFIDENCE INTERVAL CODES in order to have a comfortable feeling with respect to the values of the Weibull parameters Beta and Eta, and the time-to-first-failure and reliability, (which are calculated using these parameters) it is desirable to calculate intervals of values based on the statistical sample data which is highly likely to contain the value of these parameters. These confidence intervals are usually calculated for levels of confidence of 90% or more and represent a measurement of the precision of the calculated results. Wide intervals generally indicate a lesser precision while narrow intervals generally reflect a higher precision. While confidence intervals do not guarantee acceptable levels of numerical answers, they should convey relative values of confidence and serve as an indication of precision or a lack thereof. In fact, many people feel that statistically based analyses should not be presented without presenting such data as confidence levels and confidence intervals. TEST PLAN GENERATION CODES The preceding codes generally indicate whether or not there is a problem with failures in a population of units in service. Many times the revealed problem demands a re-engineered part to allow a system to function with acceptable service. If so, it is desirable to use statistical methods to define substantiation test plans which will demonstrate that a newly engineered part has either eliminated a failure mode or greatly improved its failure characteristics such that it can yield acceptable service. The two test plan generation codes provide for the two most common test plan types used - zero-failure and non-zero-failure test plans. Test plans are generated based on the failure mode Weibull parameters of Beta and Eta and an initial estimate of test hours for each sample or unit to be tested. The test hours and resulting number of test articles define the test plan. In some cases, a second and more reasonable number of test hours must be tried to arrive at the most reasonable number of test articles. If the process does not converge quickly, an alternate method is available which consists of making an estimate of a reasonable number of test articles. If the process does not converge quickly, an alternate method is available which consists of making an estimate of a reasonable number of test articles and calculating the required number of test hours for each. In any case, the result is a specified number of test articles and the required test hours for each. In the case of the non-zero-failure test plan, an allowable number of failures is calculated which, if not exceeded, will still result in a successful substantiation test. WEIBER USAGE Each of the above described codes can be used independently as long as the required input data are available. The usual procedure is to determine the values of Beta and Eta from population data using one of the parameter calculation codes. If there is a lack of some data, it may be possible to estimate Beta based on historical failure mode data and calculate Eta using the Weibayes methodology. To give a good feeling of confidence, it is always recommended that sensitivities be run over a range of Beta values that is sure to include the real or actual value. Further analyses can then be run for risk analysis and/or failure predictions for a given population segment. Confidence intervals can also be calculated to get a further handle on probable accuracy. If a fix is called for in the failure, mode the test plan for substantiation can be generated. Any or all of these calculations can be made in an extremely short period of time by any project engineer with a minimum of computer experience. With the proliferation of desktop computers there is no reason for these analyses not to be done if failures are occurring. Even with the poorest quality of available data or a lack of data for an entire population, estimates can be made and sensitivity analyses can be run. This means that all project engineers can at least do worst case analyses to establish limits and guide decisions necessary regarding corrective actions to be taken. Each individual code has been checked for accuracy against the examples used in source documentation and each has been further demonstrated to yield correct answers using actual Navy fleet historical failure data. Currently the Weiber program is being used by the Naval Air Systems Command (NAVAIR); the Naval Aviation Maintenance Office ( NAMO) ; the Naval Air Development Center (NADC); the Naval Air Propulsion Center (NAPC); the Naval Air Rework Facilities (NARF) at Norfolk, VA., Jacksonville, FL, and San Diego, CA. The program has also been made available to personnel at the Naval Aviation Depot, MCAS at Cherry Point, NC and the Naval Sea Combat Systems Center at Norfolk,,VA. Further distribution within the Navy and other U.S. Government Agencies continues. 4

5 REFERENCES Abernethy, R. B., Breneman, J. E., Medlin, C. H., Reinman, G. L., 1983, "Weibull Analysis Handbook", report AFWAL-TR , Pratt and Whitney Aircraft, West Palm Beach,FL. Daub, W. J., Undated, "WeibulI Analysis For Aircraft Engine Components", General Electric Company. Caporal, P. M., 1985, An Introduction to the Weibull Distribution", General Electric Company. Monahan, M. H., Undated, "A Simplified Method of Weibull Analysis of Field or Test Data: Time Share", General Electric Company, AEG, Lynn, MA. Cyrus, J. D., 1986, "Engine Component Life Prediction Methodology for Conceptual Design Investigations", ASME Paper 86-GT-24 Zaretsky, E. V., 1985, "Fatigue Criterion to System Design, Life and Reliability", AIAA Paper AIAA Burkett, M. A., 1984, "Reliability Assessment from Small Sample Inspection Data for Gas Turbine Engine Components", SAE Paper