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Rlt1toc Page 1 of 9 RELIABILITY & LIFE TESTING HANDBOOK VOLUME 1 TABLE OF CONTENTS CHAPTER 1- OBJECTIVES, TYPES, SCHEDULING AND MANAGEMENT OF RELIABILITY AND LIFE TESTING ..................................................................................................... 1 1.1- OBJECTIVES OF RELIABILITY AND LIFE TESTING .................................................. 1 1.2- WHY RELIABILITY AND LIFE TESTING? .................................................................... 4 1.3- TEST TYPES, THEIR APPLICATIONS AND BENEFITS ............................................... 4 1.4- TEST PLANNING, SCHEDULING AND DOCUMENTATION ...................................... 7 CHAPTER 2- RELIABILITY AND LIFE DATATHEIR ACQUISITION AND PROCESSING ................................................ 21 2.1- OBJECTIVES ..................................................................................................................... 21 2.2- RELIABILITY AND LIFE DATA NEEDED .................................................................... 21 2.3- RELIABILITY AND LIFE DATA ACQUISITION FORMS ........................................... 22 2.4- FINAL FORMAT OF DATA ............................................................................................. 30 2.5- FINAL REDUCED DATA - RESULTS ............................................................................ 30 2.6- RELIABILITY DATA SOURCES ..................................................................................... 35 2.7- RELIABILITY DATA ANALYSIS, FEEDBACK, AND CORRECTIVE ACTION SYSTEM ......................................................................... 36 CHAPTER 3- FIVE VERY IMPORTANT ANALYTICAL FUNCTIONS IN RELIABILITY AND LIFE TESTING ........................................................................................................... 49 3.1- THE FIVE FUNCTIONS ................................ .………………………………………….. 49 3.2-THE FAILURE PROBABILITY DENSITY FUNCTION ...................................................50 3.3-THE RELIABILITY FUNCTION ........................................................................................58 3.4- THE CONDITIONAL RELIABILITY FUNCTION ...........................................................62 3.5- THE FAILURE RATE FUNCTION ....................................................................................64 3.6-THE MEAN LIFE FUNCTION ............................................................................................69 3.7-ADDITIONAL COVERAGE ................................................................................................73 CHAPTER 4- FAILURE FREQUENCY, FAILURE RATE AND RELIABILITY DETERMINATION FROM FIELD DATA ................................................................................................................... 81 4.1- SIMPLE DATA ANALYSIS ........................................................................................... 81 http://www.u.arizona.edu/~dimitri/Rlt1toc Rlt1toc Page 2 of 9 4.2- EARLY FAILURE CHARACTERISTICS ..................................................................... 88 4.3- USEFUL LIFE CHARACTERISTICS ............................................................................ 88 4.4- WEAROUT LIFE CHARACTERISTICS ....................................................................... 93 4.5- MORE SOPHISTICATED DATA ANALYSES ............................................................. 93 CHAPTER 5- CHI-SQUARE, STUDENT'S t AND F DISTRIBUTIONS ......................................................................................... 105 5.1- THE CHI-SQUARE DISTRIBUTION ............................................................................. 105 5.2- THE STUDENT'S t DISTRIBUTION .............................................................................. 116 5.3- THE F DISTRIBUTION ................................................................................................... 123 CHAPTER 6- THE EXPONENTIAL DISTRIBUTION ....................................................... 133 6.1-EXPONENTIAL DISTRIBUTION CHARACTERISTICS .............................................. 133 6.2- EXPONENTIAL RELIABILITY CHARACTERISTICS ................................................ 143 6.3- EXPONENTIAL FAILURE RATE AND MEAN-TIMEBETWEEN-FAILURES CHARACTERISTICS ............................................................. 148 6.4- DETERMINATION OF THE EXPONENTIAL FAILURE RATE AND MTBF FROM INDIVIDUAL TIME-TO-FAILURE DATA ................................................................... 150 6.5- DETERMINATION OF THE EXPONENTIAL FAILURE RATE AND MTBF BY PROBABILITY PLOTTING ............................................................................................ 151 6.6 -A BETTER ESTIMATE OF RELIABILITY ................................................................................................................. 156 CHAPTER 7- MTBF CONFIDENCE LIMITS, TEST TIME AND OC CURVES FOR THE EXPONENTIAL CASE .................................................................................................................. 173 7.1-CONFIDENCE LEVEL AND CONFIDENCE LIMITS .................................................. 173 7.2-DETERMINATION OF THE CONFIDENCE LIMITS ON m …………………………..174 7.3-ONE-SIDED CONFIDENCE LIMITS ON m IN A FAILURE TERMINATED TEST .................................................................................... 184 7.4-TWO-SIDED AND ONE-SIDED CONFIDENCE LIMITS ON m IN A TIME TERMINATED TEST ........................................................................ 186 http://www.u.arizona.edu/~dimitri/Rlt1toc Rlt1toc Page 3 of 9 7.5- DETERMINATION OF THE CONFIDENCE LIMITS ON THE RELIABILITY FOR THE EXPONENTIAL CASE .................................................. 192 7.6- MTBF AND RELIABILITY DETERMINATION WHEN NO FAILURES ARE OBSERVED IN A TEST ....................................................................... 194 7.7- COMPARISON OF THE LOWER, ONE-SIDED CONFIDENCE LIMIT ON m WHEN ONE FAILURE OCCURS IN A FAILURE TERMINATED TEST WITH THE CASE WHEN NO FAILURES OCCUR IN A TIME TERMINATED TEST ........................................................................................................ 196 7.8- EFFECT OF CONFIDENCE LEVEL, SAMPLE SIZE, AND OCCURRENCE OF FAILURES ON THE CONFIDENCE LIMITS ON m .................................................................................................................... 197 7.9- THE SKEWNESS OF THE DISTRIBUTION OF TIIE MEAN TIME BETWEEN FAILURES .............................................................................. 198 7.10-TEST DURATION AND SAMPLE SIZE FOR VARIOUS TEST TYPES AND CONFIDENCE LEVELS .................................................................. 198 7.11-EXPECTED WAITING TIMES FOR RELIABILITY TESTS IN THE EXPONENTIAL CASE ........................................................................................ 207 7.12-EXPECTED NUMBER OF FAILURES FOR RELIABILITY TESTS IN THE EXPONENTIAL CASE ........................................................................... 213 7.13-OPERATING CHARACTERISTIC CURVES FOR RELIABILITY TESTS IN THE EXPONENTIAL CASE ...................................................................................................... 217 7.14-BINOMIAL RELIABILITY DEMONSTRATION TEST WHEN THE TEST TIME IS DIFFERENT THAN THE MISSION TIME, FOR THE EXPONENTIAL CASE ............................................. 223 CHAPTER 8- TESTS OF COMPARISON OF THE MEAN LIFE FOR THE EXPONENTIAL CASE ............................................................... 255 8.1- INTRODUCTION ............................................................................................................ 255 8.2- TEST PROCEDURE ........................................................................................................ 256 8.3- CHOICE OF THE SIGNIFICANCE LEVEL ................................................................... 263 8.4- QUANTITATIVE BASES FOR SELECTING A MANUFACTURER…………………264 CHAPTER 9- THE NORMAL DISTRIBUTION .................................................................. 269 9.1- NORMAL DISTRIBUTION CHARACTERISTICS ...................................................... 269 http://www.u.arizona.edu/~dimitri/Rlt1toc Rlt1toc Page 4 of 9 9.2- COMPUTATIONAL METHODS FOR THE DETERMINATION OF THE PARAMETERS OF THE NORMAL DISTRIBUTION .......................................................................................... 273 9.3- DETERMINATION OF THE PARAMETERS OF THE NORMAL DISTRIBUTION BY PROBABILITY PLOTTING .......................................................................................... 282 9.4- NORMAL RELIABILITY CHARACTERISTICS ......................................................... 288 9.5- NORMAL FAILURE RATE CHARACTERISTICS ...................................................... 294 CHAPTER 10- CONFIDENCE INTERVAL ON THE MEAN LIFE AND ON THE RELIABILITY OF NORMALLY DISTRIBUTED DATA .................................................................................. 303 10.1-TWO-SIDED CONFIDENCE LIMITS ON THE MEAN LIFE WHEN T IS KNOWN, OR N 25 ..................................................................... 303 10.2-ONE-SIDED CONFIDENCE LIMITS ON THE MEAN LIFE WHEN T IS KNOWN, OR N 25 ..................................................................... 308 10.3- CONFIDENCE LIMITS ON T WHEN T IS UNKNOWN, OR N < 25 ............................................................................................... 311 10.4- RANGE OF LIFE ABOUT THE MEAN LIFE WITH A GIVEN PROBABILITY WHEN T IS KNOWN, OR N 25 ................................................... 318 10.5- RANGE OF LIFE ABOUT THE MEAN LIFE FOR A GIVEN PROBABILITY WHEN T IS UNKNOWN, OR N < 25 ............................................ 320 10.6- TOLERANCE LIMITS ON LIFE .................................................................................. 322 10.7- LOWER, ONE-SIDED CONFIDENCE LIMIT ON THE RELIABILITY FOR THE NORMAL pdf CASE ......................................................... 326 CHAPTER 11- THE LOGNORMAL DISTRIBUTION ....................................................... 333 11.1-LOGNORMAL DISTRIBUTION CHARACTERISTICS .............................................. 333 11.2- PROBABILITY PLOTTING OF THE LOGNORMAL DISTRIBUTION ........................................................................................................... 347 11.3- LOGNORMAL RELIABILITY CHARACTERISTICS ................................................ 353 11.4- LOGNORMAL FAILURE RATE CHARACTERISTICS ............................................. 357 11.5- APPLICATIONS OF THE LOGNORMAL DISTRIBUTION ............................................................................................................. 361 11.6- PHENOMENOLOGICAL CONSIDERATIONS FOR USING THE LOGNORMAL DISTRIBUTION ......................................................... 361 CHAPTER 12- THE WEIBULL DISTRIBUTION ............................................................... 373 12.1- WEIBULL DISTRIBUTION CHARACTERISTICS .................................................... 373 http://www.u.arizona.edu/~dimitri/Rlt1toc Rlt1toc Page 5 of 9 12.2- WEIBULL RELIABILITY CHARACTERISTICS ....................................................... 381 12.3-WEIBULL FAILURE RATE CHARACTERISTICS .................................................... 383 12.4- ESTIMATION OF THE PARAMETERS OF THE WEIBULL DISTRIBUTION BY PROBABILITY PLOTTING………………………… ............................................................................ 384 12.5- CONSTRUCTION OF THE WEIBULL PROBABILITY PAPER ................................................................................................ 413 12.6- PROBABILITY OF PASSING A RELIABILITY TEST .............................................. 415 12.7- APPLICATIONS OF THE WEIBULL DISTRIBUTION ............................................. 416 12.8- PHENOMENOLOGICAL CONSIDERATIONS FOR USING THE WEIBULL DISTRIBUTION ................................................................... 416 12.9- ANALYSIS OF PROBABILITY PLOTS AND CHOOSING THE RIGIIT PROBABILITY PLOTTING PAPER ........................................................................................................................... 416 12.10-THE LEAST SQUARES, MATCHING MOMENTS AND MAXIMUM LIKELIHOOD ESTIMATORS METHODS OF DETERMINING THE PARAMETERS OF THE TWO-PARAMETER WEIBULL DISTRIBUTION ........................................... 416 12.11-THE LINEAR ESTIMATION OF THE PARAMETERS OF THE WEIBULL DISTRIBUTION - RELATIONSHIP BETWEEN THE WEIBULL AND THE EXTREME VALUE DISTRIBUTION ............................................................................................ 443 12.12-THE GENERAL LINEAR MODEL OF THE ORDER STATISTICS ................................................................................................ 444 12.13-THE BEST LINEAR UNBIASED ESTIMATE (BLUE) OF AND ............................................................................................... 445 12.14-ESTIMATING THE WEIBULL PARAMETERS' BLUE’S ....................................................................................................................... 447 12.15-THE BEST LINEAR INVARIANT ESTIMATE (BLIE) OF AND .................................................................................................. 448 CHAPTER 13- CONFIDENCE LIMITS ON THE RELIABILITY WITH WEIBULL DISTRIBUTED TIMES TO FAILURE, AS WELL AS ON THE MEAN LIFE, MISSION DURATION, AND ................................................... 465 13.1- PROCEDURE ............................................................................................................... 465 13.2- MISSION DURATION FOR GIVEN RELIABILITY AND CONFIDENCE LEVEL ...................................................................................... 469 http://www.u.arizona.edu/~dimitri/Rlt1toc Rlt1toc Page 6 of 9 13.3- MISSION RANGE FOR A GIVEN CONFIDENCE LEVEL AND RELIABILITY ..................................................................................... 469 13.4- THE TRUE WEIBULL SLOPE, ............................................................................... 470 13.5-THE WEIBULL POPULATION MEAN AND ITS CONFIDENCE LIMITS ................................................................................................ 473 13.6- CONFIDENCE INTERVALS ON THE PARAMETERS OF THE WEIBULL DISTRIBUTION .......................................................................... 476 CHAPTER 14- RANKS NOT AVAILABLE IN TABLES ................................................... 517 14.1- COMPUTATION OF ANY RANK ................................................................................ 517 14.2- ALTERNATE METHODS FOR THE COMPUTATION OF THE MEDIAN RANK .................................................................................................. 519 CHAPTER 15- TESTS OF COMPARISON FOR THE WEIBULL DISTRIBUTION ............................................................................................. 525 15.1- APPLICABILITY ........................................................................................................... 525 I5.2- METHODOLOGY AT THE MEAN LIFE ..................................................................... 525 15.3- TEST OF COMPARISON AT THE MEAN LIFE USING A NOMOGRAPH ............................................................................................. 531 15.4- TEST OF COMPARISON AT THE MEAN LIFE WHEN THE ’S ARE UNEQUAL ......................................................................................... 535 15.5- TEST OF COMPARISON AT THE 10% FAILED LEVEL .......................................... 540 15.6- TEST OF COMPARISON AT THE 10% FAILED LEVEL USING A NOMOGRAPH .............................................................................................. 542 15.7- CONSTRUCTION OF THE NOMOGRAPH GIVEN IN FIG. 15.6a ...................................................................................................................... 545 15.8- EQUATION FOR CONSTRUCTING THE NOMOGRAPH GIVEN IN FIG. 15.13 ................................................................................................... 555 CHAPTER 16- THE GAMMA DISTRIBUTION ................................................................. 565 16.1- GAMMA DISTRIBUTION CHARACTERISTICS ....................................................... 565 16.2- ESTIMATION OF THE PARAMETERS OF THE GAMMA DISTRIBUTION BY PROBABILITY PLOTTING ..................................................... 571 16.3- GAMMA RELIABILITY CHARACTERISTICS ......................................................... 574 16.4- GAMMA DISTRIBUTION FAILURE RATE CHARACTERISTICS ................................................................................................... 586 16.5- APPLICATIONS OF THE GAMMA DISTRIBUTION ............................................... 588 16.6- PHENOMENOLOGICAL CONSIDERATIONS FOR USING THE GAMMA DISTRIBUTION ..................................................................... 588 http://www.u.arizona.edu/~dimitri/Rlt1toc Rlt1toc Page 7 of 9 CHAPTER 17- THE BETA DISTRIBUTION ....................................................................... 593 17.1-BETA DISTRIBUTION'S CHARACTERISTICS .......................................................... 593 17.2- BETA DISTRIBUTION'S RELIABILITY CHARACTERISTICS ................................................................................................... 598 17.3- BETA DISTRIBUTION'S FAILURE RATE CHARACTERISTICS ................................................................................................... 600 17.4- ESTIMATION OF THE PARAMETERS OF THE BETA DISTRIBUTION ........................................................................................................... 601 17.5- APPLICATIONS OF THE BETA DISTRIBUTION ..................................................... 607 CHAPTER 18-METHODS OF PARAMETER ESTIMATION .......................................... 613 18.1- INTRODUCTION .......................................................................................................... 613 18.2- IDENTIFICATION OF DATA OUTLIERS .................................................................. 616 18.3- LEAST SQUARES METHOD OF PARAMETER ESTIMATION ................................................................................................................ 633 18.4- MATCHING MOMENTS METHOD OF PARAMETER ESTIMATION ................................................................................................................ 647 18.5- MAXIMUM LIKELIHOOD METHOD OF PARAMETER ESTIMATION ............................................................................................................... 653 18.6- MODIFIED MOMENTS METHOD OF PARAMETER ESTIMATION ............................................................................................................... 658 CHAPTER 19- CHI-SQUARED GOODNESS-OF-FIT TEST ............................................ 679 19.1- WHEN AND HOW TO APPLY .................................................................................... 679 19.2- COMMENTS ON CHI-SQUARED GOODNESS-OF-FIT TEST .............................................................................................................................. 689 CHAPTER 20- KOLMOGOROV-SMIRNOV GOODNESSOF-FIT TEST .................................................................................................. 697 20.1- INTRODUCTION .......................................................................................................... 697 20.2- STEP-BY-STEP METHOD FOR THE STANDARD K-S GOODNESS-OF-FIT TEST WITH THE DISTRIBUTION PARAMETERS KNOWN ............................................................................................. 698 20.3- THE REFINED K-S GOODNESS-OF-FIT TEST FOR CONTINUOUS DISTRIBUTIONS WITH PARAMETERS KNOWN ............................................................................................. 709 20.4- MODIFIED K-S GOODNESS-OF-FIT TEST FOR CONTINUOUS DISTRIBUTIONS WITH PARAMETERS UNKNOWN ......................................................................................... 725 http://www.u.arizona.edu/~dimitri/Rlt1toc Rlt1toc Page 8 of 9 20.5- COMMENTS ON THE KOLMOGOROV - SMIRNOV GOODNESS-OF-FIT TEST ............................................................................................ 729 CHAPTER 21- ANDERSON - DARLING AND CRAMER VON MISES GOODNESS-OF-FIT TESTS ................................................. 739 21.1-INTRODUCTION ............................................................................................................ 739 21.2-THE STANDARD ANDERSON - DARLING AND CRAMER - VON MISES GOODNESS-OF-FIT TESTS FOR CONTINUOUS DISTRIBUTIONS WITH PARAMETERS KNOWN ............................................................................................... 740 21.3-STEP-BY-STEP METHOD FOR THE STANDARD ANDERSON - DARLING AND CRAMER - VON MISES GOODNESS-OF-FIT TESTS WITH THE PARAMETERS KNOWN ............................................................................................... 742 21.4-MODIFIED AD AND CvM GOODNESSOF-FIT TEST FOR CONTINUOUS DISTRIBUTIONS WITH PARAMETERS UNKNOWN .............................................................................. 772 21.5-COMMENTS ON THE ANDERSON DARLING AND CRAMER - VON MISES GOODNESS-OF-FIT TESTS .......................................................................................... 784 21.6-RECOMMENDATIONS FOR THE SELECTION OF GOODNESS-OF-FIT TESTS .................................................................................... 784 APPENDICES ............................................................................................................................ 789 APPENDIX A- RANK TABLES ............................................................................................... 790 APPENDIX B- STANDARDIZED NORMAL DISTRIBUTION'S AREA TABLES ............................................................................................... 804 APPENDIX C- STANDARDIZED NORMAL DISTRIBUTION'S ORDINATE VALUES, OR PROBABILITY DENSITIES ...................................................................................................... 812 APPENDIX D-CHI-SQUARE DISTRIBUTION'S PERCENTILE VALUES ........................................................................................................... 813 APPENDIX E- STUDENT'S t DISTRIBUTION'S PERCENTILE VALUES ........................................................................................................... 817 APPENDIX F- F DISTRIBUTION'S PERCENTAGE POINTS ............................................... 818 APPENDIX G- KOLMOGOROV-SMIRNOV GOODNESS-OF-FIT TEST CRITICAL VALUE TABLES ................................................................ 837 APPENDIX H-TWO-SIDED TOLERANCE FACTORS FOR NORMAL DISTRIBUTIONS ................................................................. 847 http://www.u.arizona.edu/~dimitri/Rlt1toc Rlt1toc Page 9 of 9 APPENDIX I- ONE-SIDED TOLERANCE FACTORS FOR NORMAL DISTRIBUTIONS ................................................................. 849 APPENDIX J- ONE-SIDED TOLERANCE FACTORS FOR NORMAL DISTRIBUTIONS WITH 50% PROBABILITY OF INCLUSION ........................................................... 851 APPENDIX K- COEFFICIENTS OF THE BLUES FOR THE EXTREME VALUE DISTRIBUTION ............................................................ 852 APPENDIX L- WEIGHTS FOR OBTAINING THE BEST LINEAR INVARIANT ESTIMATES OF THE PARAMETERS OF THE WEIBULL DISTRIBUTION ............................................................................ 856 APPENDIX M- BINOMIAL OR ATTRIBUTE RELIABILITY TESTING CURVES FOR CONFIDENCE LEVELS OF 50%, 80%, 90%, 95% and 99% .................................................................. 896 INDEX ....................................................................................................................................... 901 ABOUT THE AUTHOR ............................................................................................................ 914 http://www.u.arizona.edu/~dimitri/Rlt1toc