RELIABILITY ENGINEERING Probabilistic Models and Maintenance Methods Second Edition RELIABILITY ENGINEERING Probabilistic Models and Maintenance Methods Second Edition JOEL A. NACHLAS Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Group, an informa business CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2017 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U. Government works Printed on acid-free paper Version Date: 20161019 International Standard Book Number-13: 978-1-4987-5247-3 (Pack - Book and Ebook) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use.
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Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.com and the CRC Press Web site at http://www.com Dedicated to the memory of Betty Nachlas Contents Preface .2 System Structures and Status Functions .3 k-out-of-n Systems .3 Modules of Systems .4 Multistate Components and Systems. 19 3 Reliability of System Structures .2 Reliability of System Structures .3 k-out-of-n Systems. 37 4 Reliability over Time .3 System-Level Models.
58 vii viii Contents 5 Failure Processes .1 Mechanical Failure Models .1 Stress–Strength Interference .2 Shock and Cumulative Damage .2 Electronic Failure Models .3 Power Law Model .3 Other Failure Models.1 Diffusion Process Model.1 Age Acceleration for Electronic Devices .2 Age Acceleration for Mechanical Devices .3 Step Stress Strategies. 93 7 Nonparametric Statistical Methods .1 Data Set Notation and Censoring .2 Estimates Based on Order Statistics .3 Estimates and Confidence Intervals .4 Kaplan–Meier Estimates .1 Continuous Monitoring of Test Unit Status .2 Periodic Monitoring of Test Unit Status .2 Application to Complete Data Sets.3 Application to Censored Data Sets.7 Nelson Cumulative Hazard Estimation Method. 124 8 Parametric Statistical Methods .2 Method of Moments .2 Confidence Intervals for the Estimates .3 Method of Maximum Likelihood .4 Maximum Likelihood Method with Data Censoring .1 Method of Moments with Censored Data .2 Data Analysis under Step Stress Testing. 167 9 Repairable Systems I: Renewal and Instantaneous Repair .2 Classification of Distributions and Bounds on Renewal Measures .3 Residual Life Distribution.
190 10 Repairable Systems II: Nonrenewal and Instantaneous Repair .1 Minimal Repair Models .2 Imperfect Repair Models .3 Equivalent Age Models .2 Quasi-Renewal Process .3 System-Level Availability .1 Availability under Imperfect Repair .2 Availability Analysis for the Quasi-Renewal Model .2 Availability Model for Age Replacement .3 Availability Model for Block Replacement.4 Availability Model for Opportunistic Age Replacement .2 Opportunistic Failure Replacement Policy .3 Partial Opportunistic Age Replacement Policy .4 Full Opportunistic Age Replacement Policy .5 Analysis of the Opportunistic Replacement Models.1 Imperfect PM Models .2 Models Based on the Quasi-Renewal Process .3 Models Based on the Kijima Model .3 More Complete Policy Analysis .4 Models and Analysis Based on Continuous Process Monitoring .1 Observable Degradation Processes .2 Unobservable Degradation Processes .1 Time Series Methods .2 Conditional Probability Methods .1 Statistical Analysis of Repairable System Data.1 Data from a Single System .2 Data from Multiple Identical Systems .1 Full Replacement Warranties .2 Pro Rata Warranties.4 Failure and Renewal Models. 341 Contents xi Appendix A: Numerical Approximations .343 Appendix B: Numerical Evaluation of the Weibull Renewal Functions. 347 Appendix C: Laplace Transform for the Key Renewal Theorem. 353 Appendix D: Probability Tables.
365 Preface The motivation for the preparation of a second edition was my wish to expand the treatment of several topics while maintaining an integrated introductory resource for the study of reliability evaluation and maintenance planning. The focus across all of the topics treated is the use of analytical methods to support the design of dependable and efficient equipment and the planning for the servicing of that equipment. The orientation of the topical develop- ment is that probability models provide an effective vehicle for portraying and evaluating the variability that is inherent in the performance and lon- gevity of equipment. The book is intended to support either an introductory graduate course in reliability theory and preventive maintenance planning or a sequence of courses that address these topics.
A fairly comprehensive coverage of the basic models and of various methods of analysis is provided. An under- standing of the topics discussed should permit the reader to comprehend the literature describing new and advanced models and methods. Notwithstanding the emphasis upon initial study, the text should also serve well as a resource for practicing engineers. Engineers who are involved in the design process should find a coherent explanation of the reliability and maintenance issues that will influence the success of the devices they create.
Similarly, engineers responsible for the analysis and verification of product reliability or for the planning of maintenance support of fielded equipment should find the material presented here to be relevant and easy to access and use. In preparing this second edition, the treatment of statistical methods for evaluating reliability has been expanded substantially. Several methods for constructing confidence intervals as part of the parametric estimation effort are described and methods for treating data derived from operating repair- able devices have also been added. In addition, the analysis of nonstation- ary models of repairable equipment maintenance has been updated and expanded.
These expansions along with numerous other minor improve- ments to the text should make this book an even more useful resource for both students and practitioners. The background required of the reader is a sound understanding of prob- ability. This subsumes capability with calculus. More specifically, the reader should have an understanding of distribution theory, Laplace transforms, convolutions, stochastic processes, and Markov processes.
It is also worth mentioning that the use of the methods discussed in this book often involves substantial computational effort, so facility with numerical methods and access to efficient mathematical software is desirable. xiii xiv Preface One caveat concerning the coverage here is that the treatment is strictly limited to hardware. Reliability and maintenance models have been devel- oped for applications to software, humans, and services systems. No criti- cism of those efforts is intended but the focus here is simply hardware.
The organization of the text is reasonably straightforward. The elemen- tary concepts of reliability theory are presented sequentially in Chapters 1 through 6. Following this, the commonly used statistical methods for eval- uating component reliability are described in Chapters 7 and 8. Chapters 9 through 13 treat repairable systems and maintenance planning models.
Here again the presentation is sequential in that simple failure models pre- cede those that include preventive actions and the renewal cases are treated before the more realistic nonrenewal cases. In the final chapter, four inter- esting special topics, including warranties, are discussed. It is worth noting that four appendices that address aspects of numerical computation are pro- vided. These should be quite useful to the reader.
Naturally, many people have contributed to the preparation of this text. The principal factor in the completion of this book was the support and encouragement of my wife Beverley. An important practical component of my success was the support of Virginia Tech, especially during sabbaticals when progress with writing is so much easier. I acknowledge the significant computational capability provided to me by the Mathematica software.
Many of the analyses included in this text would have been much more taxing or even impossible without the strength and efficiency the Wolfram software provides. I also wish to extend my thanks directly to three of my students, each of whom contributed to my efforts. Edvin Beqari stimulated my increased inter- est in and analysis of the diffusion models of degradation. He also directed much of my analysis of that topic.
Elliott Mitchell-Colgan helped to expand the sets of exercises included at the end of the chapters. Paul D’Agostino invested very many hours in verifying a majority of the complicated numeri- cal analyses used for examples or for exercise solutions. I express my profound gratitude to all of my graduate students who have taught me so much about these topics over the years. May we all continue to learn and grow and to enjoy the study of this important subject.
Nachlas received his BES from Johns Hopkins University in 1970, his MS in 1972 and his PhD in 1976, both from the University of Pittsburgh. He served on the faculty of the Grado Department of Industrial and Systems Engineering at Virginia Tech for 41 years and retired in March 2016. His research interests are in the applications of probability and statistics to prob- lems in reliability and quality control. In addition to his normal teaching activities during his time at Virginia Tech, he served as the coordinator for the department’s graduate program in operations research and for their dual master’s degree that is operated with École des Mines de Nantes in France.
From 1992 to 2011, he regularly taught reliability theory at the École Polytechnique de l’Université Nice Sophia Antipolis. He is the coauthor of more than 50 refereed articles, has served in numerous editorial and referee capacities, and has lectured on reliability and maintenance topics through- out North America and Europe. xv 1 Introduction Although we rarely think of it, reliability and maintenance are part of our everyday lives. The equipment, manufactured products, and fabricated infrastructure that contribute substantively to the quality of our lives have finite longevity.
Most of us recognize this fact, but we do not always fully perceive the implications of finite system life for our efficiency and safety. Many, but not all, of us also appreciate the fact that our automobiles require regular service, but we do not generally think about the fact that roads and bridges, smoke alarms, electricity generation and transmission devices, and many other machines and facilities we use also require regular maintenance. We are fortunate to live at a time in which advances in the understanding of materials and energy have resulted in the creation of an enormous variety of sophisticated products and systems, many of which (1) were inconceiv- able 100 or 200 or even 20 years ago; (2) contribute regularly to our comfort, health, happiness, efficiency, or success; (3) are relatively inexpensive; and (4) require little or no special training on our part. Naturally, our reliance on these devices and systems is continually increasing and we rarely think about failure and the consequences of failure.
Occasionally, we observe a catastrophic failure. Fatigue failures of the fuse- lage of aircraft [1], the loss of an engine by a commercial jet [1], the Three Mile Island [1] and Chernobyl [1] nuclear reactor accidents, and the Challenger [2] and Discovery [3] space shuttle accidents are all widely known examples of catastrophic equipment failures.