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AN IMPROVED STATISTICAL METHOD for MEASURING RELIABILITY with SPECIAL REFERENCE to ROCKET ENGINES By• Robert B. Abernethy Subm

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AN IMPROVED STATISTICAL METHOD for MEASURING RELIABILITY with SPECIAL REFERENCE to ROCKET ENGINES By• Robert B. Abernethy Subm AN IMPROVED STATISTICAL METHOD FOR MEASURING RELIABILITY WITH SPECIAL REFERENCE TO ROCKET ENGINES by• Robert B. Abernethy Submitted as a thesis for the Ph.D. Degree in the Faculty of Science.at the Imperial College of Science and Technology in the University of London. July 1965 - 2 ABSTRACT The current procedure employed in industry to estimate rocket engine reliability is based entirely on discrete, success-failure, variables. The precision of this method is inadequate at high levels of reliability; large changes in reliability cannot be detected. A new method is developed based on treating the performance parameters as continuous, measured variables and the mechanical • characteristics as discrete variables. The new method is more precise, more accurate, and has wider applica- tion to the complex problems of estimating vehicle and mission reliability. Procedures are provided for frequen- tist, likelihood and Bayesian reliability estimates. Estimates of the proportion of a normal distribution exceeding or satisfying a limit, or limits, are treated in detail and tables of these estimates are tabulated. Monte Carlo simulation is used to verify analytical results. Computer programs in Extended Mercury-Atlas Autocode and Fortran IV languages are included with typical com- puter output for the estimates developed. 3 ACKNOWLEDGEMENT The author is deeply indebted to Professor G. A. Barnard and Professor E. H. Lloyd, his advisors, for their guidance and counsel; to Dr. D. J. Farlie for his many constructive comments; to Mr. T. E. Roughley for his help with computer services; and to his wife for her encouragement and patience. INDEX Part I Introduction Chapter 1 OBJECTIVE 11 a. Requirements for Reliability Estimates b. Rocket Engine Project Life Span c. Development Program Data d. Objective e. Organization THE CURRENT METHOD 20 a. Reliability Growth b. High Failure Rate Components Dangerous Test Objectives .Variable Firing Duration' e. Ground Test Versus Mission Populations • Trimming Propellant Mixture Ratio and Thrust Controls Documentation Criticism THE PROPOSED METHOD 34 a. ---Fighiti kck-ite...n...0-1-tc_QI Akoctrt( • Overall Reliability Estimates ▪ Independence and Normality .Data Description INDEX Part II Estimates of the Proportion of a Normal Distribution Defined by a Fixed Limit or Limits Chapter 4 INTRODUCTION 43 a. Required Estimates b. Current Estimates c. Point Estimates Interval Estimates Likelihood Estimates DISTRIBUTION OF .k = X-L a. Sampling Distribution Tables Simulation .Expectation of k • Variance of k Comparison Higher Moments • Median TABLE III THE DISTRIBUTION OF (k) .The, Sampling Distribution b. Transformations c. Expectation of 1-1 d. Bias Chapter 7 THE DISTRIBUTION OF h, AN UNBIASED ESTIMATE OF 11 96 a. An Iterative Method b. Exact Method c.. Interval Estimates of h d. Sides and Limits e. Examples LIKELIHOOD ESTIMATES 109 Likelihood Function b. Maximum Likelihood Estimate Bayesian Estimates Double Limits THE DISTRIBUTION OF h', KNOWN VARIANCE a. The Estimate hi . The Distribution of h' c. Interval Estimates of hi 10 A RELATED PROBLEM-VARIABLE LIMITS 132 The Variable Limit b. The Distribution c. References d. Discussion INDEX Part-III THE SIMPLF MODEL ABSTRACT 136 Chapter 11 FUNCTIONAL RELIABILITY 137 a. Definition b. Functional Performance Limits c. Weighting Factors d. Estimates 12 PERFORMANCE RELIABILITIES 3_44 Specific Impulse ▪ Mixture*Ratio and Known Variance OVERALL RELIABILITY. 150 Point Estimates Interval Estimate Overall Reliability Distribution • Simulation and DiscrqtigecLResuls Confidence Interval Approxim4tions EfficienCy .Summary Chapter 14 LIKELIHOOD AND BAYESIAN ESTIMATES 174 a. Likelihood Function b. Maximum Likelihood Estimate c. Bayesian Confidence Interval d. Simple Method e. Complex Method f. Choice of Estimates Conclusion Part. IV COMPLEX MODELS ABSTRACT 190 Chapter 15 MULTIFIRING SAMPLE SPACE AND TRIMMING 191 a. Complex Versus Simple Models b. The Complex Model c. Engine-to-Engine Variation d. Run-to-Run Variation e. Noncentral t ar Small Compared to aE a Not Small Compared to Trimming Stability of Variance Ratio j. Approximate Degrees of Freedom k. Current Method 16 PROBLEMS FOR FUTURE RESEARCH 218 a. Multifiring-Multi-engine Mission Space b. Variable Number of Firings.Per Engine Assembly c. True. Versus Indicated Reliabilities d. Likelihood-Bayesian Estimates - 10 - APPENDIX I REFERENCES 228 II NOTATION 232 VOLUME II COMPUTER PROGRAMS AND RESULTS 5.1 Percentiles of the Noncentral t/sqrtN 7.1 Iterated Bias 7.2 Simulation Normal Double Limits 8.1 Normal Reliability-Likelihood Contours 13.1 Binomial Trinormal Simulation 13.1A Binomial Trinormal SimulatiOn with Variance, Efficiency 13.2 Overall Reliability Confidence Bound (Fortran) 14.1 Overall Reliability Likelihood Functions 14.2 Bayesian Confidence Bound Based on Chi- Squared Prior Distributions 15.1 Degrees of FreedoM Estimatet - 11 - CHAPTER I OBJECTIVE 1.a. Requirements for Reliability Estimates The first American space vehicles and military missiles were both costly and unreliable. The most unreliable component systems were propulsion and elec- tronics and as a result the electronics and rocket engine industries were the first to be required by contract to monitor and report estimates of reliability. The stati- stical methods developed are reminiscent of early quality control methods; they are based on discrete random variables and generally suffer from poor precision. Although these methods were satisfactory at low levels of reliability, better methods are required at the current higher levels of reliability (Abernethy 1963 (b) ). In addition to production control, there are other requirements for rocket engine reliability estimates. The design of vehicle engine failure detection systems and abort systems and the use of redundant engines are based on engine reliability estimates. (Abernethy 1963 (a)). Many advanced missions involve multiple launches; the number of vehicles needed to complete the mission is a random variable that is dependent on engine reliability. -12 - The optimum single engine thrust size for multi-engine vehicles is a function of total thrust required, reliability and cost: the smaller the engine, the more development tests can be made for a fixed cost and therefore, the higher reliability, but in production the cost per pound thrust favors larger engines. It follows that there is an optimum size single engine for each vehicle as a func- tion of reliability, total cost and total thrust required. (Muiready et al ) 1.b. Rocket Engine Project Life Span The lifespan of a rocket engine project may be divided into three overlapping periods: design, develop- ment and production. THE LIFESPAN OF A ROCKET ENGINE PROJECT PRODUCTION ADVANCE MODEL 4 DEVELOPMENT DESIGN ---- --,.-PRODUCTION FLIGHT TEST DEVELOPMENT' DESIGN 0 1 2 3 4 YEARS -- FIGURE. 1.1 . -13 - In the design period there are no full scale engine test data available for reliability estimation. In the same period with electronic systems it is common practice to make system reliability estimates based on component test data, but with rocket engines analogous methods have failed; the mechanical and .thRrmodynamic interactions between components'are significant and the engine environment of vibration, cryogenic and combustion temperatures, and transient shock is virtually impossible to simulate on a component test stand. Therefore, rocket engine reliability estimates in the design period will be largely guesses. Historically the best guess has been the reliability of the last engine produced by the same engineering team. The production-flight test period is characterized by reasonably large quantities of valid data from flight tests and from production engine ground test. The statistical problems of making estimates in this period are trivial and the need for estimates is less than during the development program. The development period is the critical period for reliability estimates for two r.easons: the need for reliability estimates is most acute at this time because the vehicle design and mission analysis is done concurrently and the quantity of reliability data -14 - available is small and it is of poor quality. 1.c. Development Program Data Development period data consists of the results of full scale tests of experimental engines constructed of the following types of components: a. Experimental: untested new designs b. Obsolete: inferior designs that have been replaced in the bill of materials c. Overage: parts with test exposure exceeding ten or fifteen times the design life d. Damaged or repaired parts e. Green run parts - untested parts f. Standard bill7of-materials parts. Flight engines consist of bill-of-materials parts only. Reliability estimates for flight engines based on devel- opment data must utilize data from experimental, non- homogeneous engines as it is the only data available. Further, the test programs and test objectives of experimental engines test conflict with the objectives of reliability demonstration. A good development program will uncover as many weaknesses and failure modes as possible; failure should often be a test objective, either directly or indirectly. The tests that are particularly -15 - dangerous must be excluded from the reliability evaluation as non-representative of a mission firing. The test firing duration may be l'ess than or greater than a mission firing and therefore a system of weighting test data by test duration is required; a successful short duration test should not count as much as a successful
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