IN AD-A143 E103

1 IN AD-A143 E103i%UfAFWAL-Tf-83-2073-\, 2'1D"-1Dr. R. 0. AbernethyJ. E. 3renemanC. H. Medlin FlainmanPratt & Whitney AircraftGovernment Products Technolojis Box 2C31 West Palm Beach, Florida 3-4Z2 November 1S33 Final Report for Period 1 July 1- 2 to 31 AL;zt 1::3 Approved for Pub!c , G;,,,,4 ,,z ,.. 1 ELECTE_; JUL181984 "A ero Prc:!.v*.-;!n Ln"z :7,,c '.4,A~r Fron CT~- 7 tReproduced FromBest Available CopyALL*,-. v., t 4: _.,WU1fAJ *N , : .;i:I'):.;, /', .: ::::,.: .., ..NOTICEWhen Government drawings, specifications, or other data are used for any purpose other than incomnection -with a definitely related Government procurement operation. the United States Governmentthereby incurs no responsibility nor any obligation whosoever; and the fact that the government mayhave formulated, furnished, or in any ,-ay suppiivd the said drawings, specifications, or other data. isnot to be regarded by implication or otherwise as in any manner licensing the holder or any otherperson or corporation, or conveying any rights or permission to manufacture use, or sell any patentedinvention that may in any way be related report has been reviewed by the Office of Public Affairs (ASD/PA) and is releasable to theNational Technical Information Service (NTIS).

2 At NTIS, it will be available to the general foreign technical report has been reviewed and is approved for A. N iITT /ACK RICHENSP roject Engineer Chief, Components BranchT MTurbine Engine DivisionS FOR THE COMMANDERH. I. BUSHD irectorTurbine Engine DivisionAero Propulsion Laboratory"I1f your address has changed, if you wish to be removed f m omr mailing ILqt. or if the addresseeis no longer employed by your organization please notify OH 45433 to help us maintain a current mailing of this report should not be returned umless retu*rn is required by security obligations, or notice on a specific " * a i * SECURITY CLASSIFICATION Of THIS PAGE (Whent Dat. Entered)REPOT DCUMNTATON AGEREAD INSTRUCTIONSREPRTDOUMNTAIO PGEBEFORECOMPLETINGFORM1. REPORT NUMBER ACCESSION NO. 3. RECIPIENT'S CATALOG NUMBER* ~~~ (J79 A~4q/D _____4. TITLE (and ) S. TYPE Of REPORT & PERIOD COVEREDF inal_____WEIBULL ANALYSIS HANDBOOK .I ulv 1982 to 31 August I1J83S. PERFORMING ORG. REPORT NUMBER_____ P&WA/GlIDl FR- 175797.)

3 AUTNOR(s) G. CONTRACT OR GRANT R. B. Abernethy; .1. E. Breneman; 3fl-2C24C. H. Medlin, G'. L. Reinman9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. ELEMENT, PROJECT, TASKU~nited Technologies Corporation IAREA & WORK UNIT N4 UMBERSP ratt & Whitney Aircraft GDroup 30660649 Chwernment Pri4 NIIVIJ Divis~ Box 2%91. West Palm Beach. _ _ __ _ _ _11. CON4 TROLLING OFFICE NMAE AND ADDRESS I2. REPORT DATEy Aero Propulsion LaihOFat(Iy (AFWAL/PO1TC) Nov'ember 1983* ~~Air Force Wright. Aeronautiald Labo~ratories (AFSC) .NMUROPAEWrgtPatterson Air Force Base, Ohio 4543:3 228t MNITORING AGENCY NAME & #4 ODRESS(ff Re'Htern~t from Controlling Office) IS. SECURITY CLASS. (ofthi rep. toi)5 UnclassifiedIS&. DECLASSIFICATION/DOWNGRADINGSCHEDULE16. DIST RIBUTION STATEMENT (l this Rope"Approved for public release, distribution unlimited17. DISTRIBUTION STATEMENT (of th abstract eatershi 20. if l'nU bronikofp~eft)IS. SUPPLEMENTARY NOTES19. REY WORDS (Coothbue an roeres. *fee it MWd Ides!4', IV block member)Weiluall.))

4 Weihlmyes, Risk Analysis, Reliability Testing, Thorndike (Charts, Fatigue Testing. LifeTesting. Expornential D~istribution' RI~~2. ABSTRACT (Gintms~ame n eese. Old Un . i daj) ~ handbook is intended to provide instructions or. how to do Weibull analysis. It will provide an]understanding of Weibull analysis that is common between the military and industry. The handbookU. contains seven chapters plus an appendix. The chapters are written containing a minimum ofmathematics with proofs given in the 1473 90 ITIOw OF I NOV 6SI 08 11 SOLET19 UNCLASSIFIED8/0 0102-014-4" d1 SECURITY CLASSIFICATION OF THIS PAGE (When. Date FAIMrn4 FOREWORDA dvanced Weibull methods have been developed at Pratt & Whitney Aircraft in a jointeffort between the Governmetit Products Division and the Commercial Products these methods have been used in aircraft engine projects in both Divisions. theadvanced technologies have never been published, even though they have been presented andused by the LT. S. Air Force (WPAFB), U.))

5 S. Navy (NAVAIR) and several authors would like to acknowledge the contribution to this work made by several otherPratt & Whitney Aircraft employees: D. E. Andress, F. E. l)auser, J1. W. Grdenick,J. H. Isiminger, B. J. Kracunas, R. Morin, M. E. Obernesser, M. A. Proschan, andB. G. key Air Force personnel that encouraged publication were: Gary Adams, Dr. TomCurran, Jim Day, Bill Troha and Don Zabierek (the USAF Program Manager), all at W1' following members of the Ameriean Institute of Aeronautics and AstronauticsSystems Effectiveness and Safety Committee provided valuable constructive reviews for whichthe authors are indebed: M. Berssenbrugge, R. Cosgrove, P. Dick, T. P. Enright, J. F. Kent, , T. Prasinos, B. F. Shelley, and K. L authors would be pleased to review constructive comments for future ~n ForTTS G;RA&-lOU'D@ .1:~ TABi U 'ouxicedif V a I cIlWILAvn 'n1)t1!VtY codesS~LU4a1 TABLE OF CONTENTSC(haptrr Page .I INTI'IDOI))('i'ION TO W' Il'II,.I, ANALYSIS .. O hje 'tiiv.

6 Background .. Examples ..F Scope .. Advantages of Weibull Analysis .. Aging Time or Cycles .. Failure Distribution .. Risk Predictions .. Engineering Changes and Maintenance Plan Evaluation .. Mathematical Models .. _.; .. 1. Weibulls with Cusps or Curves .. System Weibulls .. No-Failure Weibulls .. It) Small Failure Sample Weibulls .. Changing Weibulis Establishing the Weibull Line .. Summary ..13"2 PERFORMING A WEIBULL ANALYSIS .. Foreword .. Weibull Paper and Its Construction .. Failure Data Analysis -Exemple .. Stspended Test Items -Nonfailures .. : .. Weibull Curve Interpretation .. Data Inconsistencies and Muhtimode Failures .. L[w-Time Failures .. Close Serial Numbers .. 11)gleg Curved Weibulls .. Problems .. 353 WEIBULL RISK AND FOREiCAST ANALYSIS .. Foreword .. Risk Analysis Defi~iition .. Forecasting Techniques .. Calculating Risk .. Present Risk .. Future Risk When Failed Units Are Not Fixed .. Future Risk When Failed Units Are Repaired.]

7 The Use of Simulation in Risk Analysis .. Case Studieb .. Case 93tudy 1: Bearing Cage Fracture .. Case Study 2: Bleed System Failures .. Case Study 3: System Risk Analysis Utilizing a SimulationM odel .. Problems .. TABLE OF CONTENTS (Continued)4 WEIIIAYIS WtIEN WEIIIJLIS ARE IMPOSSIIlIE .. Foreword .. W eibayes M ethod .. Weibayes -No Failures .. Weibest -No .. Unknown Failure Times .. Weibayes Worries and Concerns .. Examples of Problems/Analytical Solutions .. Problem s .. 755 SUBSTANTIATION AND RELIABILITY TESTING .. 77I Foreword .. Zero-Failure Test Plans for Substantiation 'resting .. Zero-Failure Test Plans for Reliability Testing .. Total Test Time .. Advantages and Limitations of the Zero-Failure Test Plans ., Non-Zero-Failure Test Plans .. Designing Test Plans .. Recommended Method oor Solving Equations .. Problems .. 926 CASE HISTORIES WITH WEIBULL APPLICATIONS .. FOREWORD .. Example 1: Turbopump Bearing Failures .. Initial Analysis -7 Small Sample.

8 Two Months Later -Batch Identified .. 5 :lisk Prediction ..-- -- Four Months Later -Final Weibull Plot .. Example 2: Main Gearbox Housing Cracks .. Information Available for Analysis .. Risk Anal)'is .. Determining the Fix .. How Good 'Were the Forecasts? .. Example 3" Opportunistic Maintenance Screening Intervals Structuring the Problem . Finding the Optimum Interva .. Example 4: Support Cost Model .. Role of the W eibuil .. Example 5: Vane and Case Field Cracks .. Resolving the Questions .. 107 Concluding Remarks .. 107vi-rrrr~r -.**.,.*,* * .g r .w . >f -i, . TABLE OF CONTENTS (Continued)Chapter Page%; 7 CONFIDFNCE LIMITS AND OTHER ASPECTS OF THE 1W EIBU I.[ .. Foreword .. Confidence Intervals .. Confidence Intervals for 1 and 'p .. Confidence Intervals for Reliability .. Confidence Intervals About a Failure Time .. Confidence Bands on the Weibull Line .. 114h Weibull "Thorndike" Charts .. Shifting W eibulls .. Weibull Goodness of Fit.]

9 Comparing the Weibull to Other Distributions .. PROBLEM S .. 113 APPENDIX A -Glossary .. 141 APPENDIX B -Median Ranks, 5"; Ranks, and 95', Ranks .. 143 APPENDIX C -Rank Regression (Weibull Plot) Method of WeibullAnalysis .. 1551. Method ..1552. Example and Step-by-Step Procedure .. 155 APPENDIX D -Maximum Likelihood Method of Weibull Analysis .1591. Foreword .. 1592. The Likelihood Function .. 1593. Maximizing the Likelihood Function .. 1604. Example .. 161AP'iNDIX E -Weibayes Methods .. 16:1I. Foreword 163i 2. Derivation of the Weibayes Equation ..163 APP NDIX F -Monte Carlo Simulation Study -Acruracy ofWeib 11 Analysis Methods .. 1661. Foreword .. 1662. ] sBeta (Fo) Estimates .. 1683. " Life Estimates .. 1684. Eta (qL) Estimates .. 17:684. Risk Forecasts .. Risk Forecast Accuracy .. Risk Forpcast Precision .. 1786. M onte Carlo Simulation .. 179vii .. "-TABLE OF CONTENTS (Continued)('hooter Pa e'eA1P' )IX (G Rank IMetrehosidn vt h~d vs Maximmn ILikv-!i~ "M th d of \ ei u l m.)

10 , i ..A1'PENI)IX H -- \Veilull Parameter ;,inat ion Computer Pr,)gjamrns * AP'PENDIX i -- W eiuill (;raphs .. 'PPENDIX .J -- ANSWERS TO IP .)ILEMS .. 199I. Chapter I A nsw er.; ..1992. C hapler 2 A nswers .. 1993. Chapter :1 Aaswers .. 2064. C hapive 4 A nsw ers .. 25. Chaipter .r Answers .. Chapter 6 Answer .. 2227. Chapter 7 An:swer .. 222 IiViliia im LIST OF ILLUSTRATIONSAMU" lPerforming a Weilull Analysis .. Understanding the Results .41.:" 11) Louver Craicking .. To Correction to Curved Weibull .. Mixing Failure Modes, MFP .. Mixing Failure Modes, EEC .. Small Sample Beta Estimates Are Too Steep .. Construction of Weibull Paper .. Rivet Failures .. Rivet Failures With Suspensions .. Failure Dintribution Charecteristic .. M ain Oil Pumps .. Weibull Plot for Augmentor Pump .. Compressor Start Bleed System .. Compressor Start Bleed System, Excluding Ocean Base .. Compressor Start Bleed System, Ocean Base .. A Curved Weibull Needs to Correction .. Plotting to Correction.