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The purpose of this publication is to provide illustrative examples of the more common mathematical calculations and statistical techniques utilized by reliability engineers in the practical performance of their daily activities. It is intended to be used as a companion to the RIAC “System Reliability Toolkit”, as the foundations of all of the techniques illustrated in this publication are discussed therein.
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As any practitioner in the field of reliability can attest, there exists a multitude of mathematical calculations and statistical techniques that can be used to assist the engineer in determining how system requirements should be allocated and how the system design should be modeled, assessed and re-designed in order to maximize the likelihood of satisfying the customer’s expectations or formal requirements. The purpose of this publication is to provide illustrative examples of the more common mathematical calculations and statistical techniques utilized by reliability engineers in the practical performance of their daily activities. It is intended to be used as a companion to the RIAC “System Reliability Toolkit”, as the foundations of all of the techniques illustrated in this publication are discussed therein.
The mathematical and statistical examples illustrated in this publication revolve around a single hypothetical case study, namely, a Flight Control System for a jet aircraft. In addition to providing a common thread throughout this publication, it is hoped that by basing all calculations on a single case study, the reader will come to understand that individual calculations do not exist in a vacuum. Rather, it will reinforce the notion that each phase of a project is linked to (and affects) other phases, and that conclusions drawn from a calculation are often used to influence design decisions, which may then require that previous analyses and calculations be revisited and updated.
Copyright © 2014 by Quanterion Solutions Incorporated. This handbook was developed by Quanterion Solutions Incorporated, in support of the prime contractor (Wyle Laboratories) in the operation of the Department of Defense Reliability Information Analysis Center (RIAC) under Contract HC1047-05-D-4005. The Government has a fully paid up perpetual license for free use of and access to this publication and its contents among all the DOD IACs in both hardcopy and electronic versions, without limitation on the number of users or servers. Subject to the rights of the Government, this document (hardcopy and electronic versions) and the content contained within it are protected by U.S. Copyright Law and may not be copied, automated, re-sold, or redistributed to multiple users without the express written permission. The copyrighted work may not be made available on a server for use by more than one person simultaneously without the express written permission. If automation of the technical content for other than personal use, or for multiple simultaneous user access to a copyrighted work is desired, please contact 877.808.0097 (toll free) or 315.732.0097 for licensing information.
| ISBN-10: | 1-933904-46-1 |
|---|---|
| ISBN-13: | 978-1-933904-46-7 |
| 1. Introduction | 1 |
| 1.1 CASE STUDY EXAMPLE | 1 |
| 1.2 STRUCTURE OF THE BOOK | 2 |
| 1.3 RELIABILITY ALLOCATION | 3 |
| 1.4 RELIABILITY MODELING | 3 |
| 1.5 RELIABILITY PREDICTION | 3 |
| 1.6 DATA ANALYSIS OVERVIEW | 3 |
| 1.7 DATA ANALYSIS TECHNIQUES | 4 |
| 1.8 RELIABILITY GROWTH | 4 |
| 1.9 RELIABILITY DEMONSTRATION TESTING | 4 |
| 1.10 DESIGN OF EXPERIMENTS | 5 |
| 1.11 STATISTICAL PROCESS CONTROL | 5 |
| 2. Allocation of Hardware Reliability Requirements | 7 |
| 2.1 CASE STUDY | 7 |
| 2.2 EQUAL APPORTIONMENT TECHNIQUE | 8 |
| 2.3 ARINC APPORTIONMENT TECHNIQUE | 8 |
| 2.4 FEASIBILITY OF OBJECTIVES TECHNIQUE | 10 |
| 2.5 SUMMARY | 12 |
| 3. Hardware Reliability Modeling | 13 |
| 3.1 CASE STUDY | 13 |
| 3.2 RELIABILITY BLOCK DIAGRAM | 14 |
| 3.3 RELIABILITY MATH MODEL | 15 |
| 3.4 SUMMARY | 16 |
| 4. Hardware Reliability Prediction | 17 |
| 4.1 CASE STUDY | 17 |
| 4.2 RELIABILITY PREDICTION PROCESS | 18 |
| 4.3 RELIABILITY PREDICTION – ACTIVE STICK | 18 |
| 4.4 RELIABILITY PREDICTION – ACTUATOR | 18 |
| 4.5 RELIABILITY PREDICTION – CONTROL SURFACE INTERFACE | 19 |
| 4.6 RELIABILITY PREDICTION – SIGNAL PROCESSING COMPUTER | 19 |
| 4.7 RELIABILITY PREDICTION – FLIGHT CONTROL SYSTEM | 21 |
| 4.8 SUMMARY | 21 |
| 5. Overview of Data Collection and Analysis | 23 |
| 5.1 CASE STUDY | 23 |
| 5.2 ORDER DATA | 24 |
| 5.3 QUICK TEST TO DETERMINE IF EXPONENTIAL DISTRIBUTION MIGHT APPLY | 24 |
| 5.4 RANKING DATA | 25 |
| 5.5 NORMAL DISTRIBUTION | 26 |
| 5.6 WEIBULL DISTRIBUTION | 27 |
| 5.7 FAILURE DENSITY, HAZARD RATE AND RELIABILITY – TIME INTERVAL DATA | 28 |
| 5.8 FAILURE DENSITY, HAZARD RATE AND RELIABILITY – FAILURE INTERVAL DATA | 30 |
| 5.9 LAPLACE STATISTIC | 32 |
| 6. Data Analysis Techniques | 33 |
| 6.1 CASE STUDY | 33 |
| 6.2 ORDER DATA | 34 |
| 6.3 WEIBULL ANALYSIS | 35 |
| 6.4 REGRESSION ANALYSIS | 37 |
| 6.5 ANALYSIS OF VARIANCE | 39 |
| 6.6 HYPOTHESIS TESTING | 42 |
| 6.7 CONFIDENCE INTERVALS | 46 |
| 6.8 SUMMARY | 47 |
| 7. Reliability Growth Testing | 49 |
| 7.1 CASE STUDY | 49 |
| 7.2 DUANE MODEL – RGT PLANNING | 51 |
| 7.3 CROW/AMSAA MODEL – RELIABILITY GROWTH TRACKING MODEL CONTINUOUS (RGTMC) | 53 |
| 7.4 SUMMARY | 58 |
| 8. Reliability Demonstration/Qualification Testing | 59 |
| 8.1 CASE STUDY | 59 |
| 8.2 OVERVIEW OF THE RDT DEFINITION PROCESS | 61 |
| 8.3 ZERO-FAILURE TEST | 61 |
| 8.4 FIXED-LENGTH RDT/RQT | 62 |
| 8.5 PROBABILITY-RATIO SEQUENTIAL TEST (PRST) | 63 |
| 8.6 RDT PERFORMANCE | 66 |
| 8.7 SUMMARY | 66 |
| 9. Design of Experiments | 67 |
| 9.1 CASE STUDY | 67 |
| 9.2 FULL-FACTORIAL DOE | 69 |
| 9.3 CONCLUSION OF FULL-FACTORIAL EXPERIMENT EXAMPLE | 73 |
| 9.4 FRACTIONAL-FACTORIAL DOE | 74 |
| 9.5 CONCLUSION OF FRACTIONAL-FACTORIAL EXPERIMENT EXAMPLE | 76 |
| 10. Statistical Process Control | 77 |
| 10.1 CASE STUDY | 77 |
| 10.2 X̅ AND R CONTROL CHARTS | 79 |
| 11. Summary and Conclusions | 87 |
| 11.1 CASE STUDY SUMMARY | 87 |
| 11.2 CONCLUSION | 92 |
