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Reliability Engineering Probabilistic Models and Maintenance Methods

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MECHANICAL ENGINEERING A Series of Textbooks and Reference Books

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Founding Editor L. L. Faulkner Columbus Division, Battelle Memorial Institute and Department of Mechanical Engineering The Ohio State University Columbus, Ohio

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

Spring Designer’s Handbook, Harold Carlson Computer-Aided Graphics and Design, Daniel L. Ryan Lubrication Fundamentals, J. George Wills Solar Engineering for Domestic Buildings, William A. Himmelman Applied Engineering Mechanics: Statics and Dynamics, G. Boothroyd and C. Poli Centrifugal Pump Clinic, Igor J. Karassik Computer-Aided Kinetics for Machine Design, Daniel L. Ryan Plastics Products Design Handbook, Part A: Materials and Components; Part B: Processes and Design for Processes, edited by Edward Miller Turbomachinery: Basic Theory and Applications, Earl Logan, Jr. Vibrations of Shells and Plates, Werner Soedel Flat and Corrugated Diaphragm Design Handbook, Mario Di Giovanni Practical Stress Analysis in Engineering Design, Alexander Blake An Introduction to the Design and Behavior of Bolted Joints, John H. Bickford Optimal Engineering Design: Principles and Applications, James N. Siddall Spring Manufacturing Handbook, Harold Carlson Industrial Noise Control: Fundamentals and Applications, edited by Lewis H. Bell Gears and Their Vibration: A Basic Approach to Understanding Gear Noise, J. Derek Smith Chains for Power Transmission and Material Handling: Design and Applications Handbook, American Chain Association Corrosion and Corrosion Protection Handbook, edited by Philip A. Schweitzer Gear Drive Systems: Design and Application, Peter Lynwander Controlling In-Plant Airborne Contaminants: Systems Design and Calculations, John D. Constance CAD/CAM Systems Planning and Implementation, Charles S. Knox Probabilistic Engineering Design: Principles and Applications, James N. Siddall

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Traction Drives: Selection and Application, Frederick W. Heilich III and Eugene E. Shube Finite Element Methods: An Introduction, Ronald L. Huston and Chris E. Passerello Mechanical Fastening of Plastics: An Engineering Handbook, Brayton Lincoln, Kenneth J. Gomes, and James F. Braden Lubrication in Practice: Second Edition, edited by W. S. Robertson Principles of Automated Drafting, Daniel L. Ryan Practical Seal Design, edited by Leonard J. Martini Engineering Documentation for CAD/CAM Applications, Charles S. Knox Design Dimensioning with Computer Graphics Applications, Jerome C. Lange Mechanism Analysis: Simplified Graphical and Analytical Techniques, Lyndon O. Barton CAD/CAM Systems: Justification, Implementation, Productivity Measurement, Edward J. Preston, George W. Crawford, and Mark E. Coticchia Steam Plant Calculations Manual, V. Ganapathy Design Assurance for Engineers and Managers, John A. Burgess Heat Transfer Fluids and Systems for Process and Energy Applications, Jasbir Singh Potential Flows: Computer Graphic Solutions, Robert H. Kirchhoff Computer-Aided Graphics and Design: Second Edition, Daniel L. Ryan Electronically Controlled Proportional Valves: Selection and Application, Michael J. Tonyan, edited by Tobi Goldoftas Pressure Gauge Handbook, AMETEK, U.S. Gauge Division, edited by Philip W. Harland Fabric Filtration for Combustion Sources: Fundamentals and Basic Technology, R. P. Donovan Design of Mechanical Joints, Alexander Blake CAD/CAM Dictionary, Edward J. Preston, George W. Crawford, and Mark E. Coticchia Machinery Adhesives for Locking, Retaining, and Sealing, Girard S. Haviland Couplings and Joints: Design, Selection, and Application, Jon R. Mancuso Shaft Alignment Handbook, John Piotrowski BASIC Programs for Steam Plant Engineers: Boilers, Combustion, Fluid Flow, and Heat Transfer, V. Ganapathy Solving Mechanical Design Problems with Computer Graphics, Jerome C. Lange Plastics Gearing: Selection and Application, Clifford E. Adams Clutches and Brakes: Design and Selection, William C. Orthwein Transducers in Mechanical and Electronic Design, Harry L. Trietley Metallurgical Applications of Shock-Wave and High-Strain-Rate Phenomena, edited by Lawrence E. Murr, Karl P. Staudhammer, and Marc A. Meyers Magnesium Products Design, Robert S. Busk How to Integrate CAD/CAM Systems: Management and Technology, William D. Engelke

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Cam Design and Manufacture: Second Edition; with cam design software for the IBM PC and compatibles, disk included, Preben W. Jensen Solid-State AC Motor Controls: Selection and Application, Sylvester Campbell Fundamentals of Robotics, David D. Ardayfio Belt Selection and Application for Engineers, edited by Wallace D. Erickson Developing Three-Dimensional CAD Software with the IBM PC, C. Stan Wei Organizing Data for CIM Applications, Charles S. Knox, with contributions by Thomas C. Boos, Ross S. Culverhouse, and Paul F. Muchnicki Computer-Aided Simulation in Railway Dynamics, by Rao V. Dukkipati and Joseph R. Amyot Fiber-Reinforced Composites: Materials, Manufacturing, and Design, P. K. Mallick Photoelectric Sensors and Controls: Selection and Application, Scott M. Juds Finite Element Analysis with Personal Computers, Edward R. Champion, Jr. and J. Michael Ensminger Ultrasonics: Fundamentals, Technology, Applications: Second Edition, Revised and Expanded, Dale Ensminger Applied Finite Element Modeling: Practical Problem Solving for Engineers, Jeffrey M. Steele Measurement and Instrumentation in Engineering: Principles and Basic Laboratory Experiments, Francis S. Tse and Ivan E. Morse Centrifugal Pump Clinic: Second Edition, Revised and Expanded, Igor J. Karassik Practical Stress Analysis in Engineering Design: Second Edition, Revised and Expanded, Alexander Blake An Introduction to the Design and Behavior of Bolted Joints: Second Edition, Revised and Expanded, John H. Bickford High Vacuum Technology: A Practical Guide, Marsbed H. Hablanian Pressure Sensors: Selection and Application, Duane Tandeske Zinc Handbook: Properties, Processing, and Use in Design, Frank Porter Thermal Fatigue of Metals, Andrzej Weronski and Tadeusz Hejwowski Classical and Modern Mechanisms for Engineers and Inventors, Preben W. Jensen Handbook of Electronic Package Design, edited by Michael Pecht Shock-Wave and High-Strain-Rate Phenomena in Materials, edited by Marc A. Meyers, Lawrence E. Murr, and Karl P. Staudhammer Industrial Refrigeration: Principles, Design and Applications, P. C. Koelet Applied Combustion, Eugene L. Keating Engine Oils and Automotive Lubrication, edited by Wilfried J. Bartz Mechanism Analysis: Simplified and Graphical Techniques, Second Edition, Revised and Expanded, Lyndon O. Barton Fundamental Fluid Mechanics for the Practicing Engineer, James W. Murdock Fiber-Reinforced Composites: Materials, Manufacturing, and Design, Second Edition, Revised and Expanded, P. K. Mallick Numerical Methods for Engineering Applications, Edward R. Champion, Jr.

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Turbomachinery: Basic Theory and Applications, Second Edition, Revised and Expanded, Earl Logan, Jr. Vibrations of Shells and Plates: Second Edition, Revised and Expanded, Werner Soedel Steam Plant Calculations Manual: Second Edition, Revised and Expanded, V. Ganapathy Industrial Noise Control: Fundamentals and Applications, Second Edition, Revised and Expanded, Lewis H. Bell and Douglas H. Bell Finite Elements: Their Design and Performance, Richard H. MacNeal Mechanical Properties of Polymers and Composites: Second Edition, Revised and Expanded, Lawrence E. Nielsen and Robert F. Landel Mechanical Wear Prediction and Prevention, Raymond G. Bayer Mechanical Power Transmission Components, edited by David W. South and Jon R. Mancuso Handbook of Turbomachinery, edited by Earl Logan, Jr. Engineering Documentation Control Practices and Procedures, Ray E. Monahan Refractory Linings Thermomechanical Design and Applications, Charles A. Schacht Geometric Dimensioning and Tolerancing: Applications and Techniques for Use in Design, Manufacturing, and Inspection, James D. Meadows An Introduction to the Design and Behavior of Bolted Joints: Third Edition, Revised and Expanded, John H. Bickford Shaft Alignment Handbook: Second Edition, Revised and Expanded, John Piotrowski Computer-Aided Design of Polymer-Matrix Composite Structures, edited by Suong Van Hoa Friction Science and Technology, Peter J. Blau Introduction to Plastics and Composites: Mechanical Properties and Engineering Applications, Edward Miller Practical Fracture Mechanics in Design, Alexander Blake Pump Characteristics and Applications, Michael W. Volk Optical Principles and Technology for Engineers, James E. Stewart Optimizing the Shape of Mechanical Elements and Structures, A. A. Seireg and Jorge Rodriguez Kinematics and Dynamics of Machinery, Vladimír Stejskal and Michael Valásek Shaft Seals for Dynamic Applications, Les Horve Reliability-Based Mechanical Design, edited by Thomas A. Cruse Mechanical Fastening, Joining, and Assembly, James A. Speck Turbomachinery Fluid Dynamics and Heat Transfer, edited by Chunill Hah High-Vacuum Technology: A Practical Guide, Second Edition, Revised and Expanded, Marsbed H. Hablanian Geometric Dimensioning and Tolerancing: Workbook and Answerbook, James D. Meadows Handbook of Materials Selection for Engineering Applications, edited by G. T. Murray

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Handbook of Thermoplastic Piping System Design, Thomas Sixsmith and Reinhard Hanselka Practical Guide to Finite Elements: A Solid Mechanics Approach, Steven M. Lepi Applied Computational Fluid Dynamics, edited by Vijay K. Garg Fluid Sealing Technology, Heinz K. Muller and Bernard S. Nau Friction and Lubrication in Mechanical Design, A. A. Seireg Influence Functions and Matrices, Yuri A. Melnikov Mechanical Analysis of Electronic Packaging Systems, Stephen A. McKeown Couplings and Joints: Design, Selection, and Application, Second Edition, Revised and Expanded, Jon R. Mancuso Thermodynamics: Processes and Applications, Earl Logan, Jr. Gear Noise and Vibration, J. Derek Smith Practical Fluid Mechanics for Engineering Applications, John J. Bloomer Handbook of Hydraulic Fluid Technology, edited by George E. Totten Heat Exchanger Design Handbook, T. Kuppan Designing for Product Sound Quality, Richard H. Lyon Probability Applications in Mechanical Design, Franklin E. Fisher and Joy R. Fisher Nickel Alloys, edited by Ulrich Heubner Rotating Machinery Vibration: Problem Analysis and Troubleshooting, Maurice L. Adams, Jr. Formulas for Dynamic Analysis, Ronald L. Huston and C. Q. Liu Handbook of Machinery Dynamics, Lynn L. Faulkner and Earl Logan, Jr. Rapid Prototyping Technology: Selection and Application, Kenneth G. Cooper Reciprocating Machinery Dynamics: Design and Analysis, Abdulla S. Rangwala Maintenance Excellence: Optimizing Equipment Life-Cycle Decisions, edited by John D. Campbell and Andrew K. S. Jardine Practical Guide to Industrial Boiler Systems, Ralph L. Vandagriff Lubrication Fundamentals: Second Edition, Revised and Expanded, D. M. Pirro and A. A. Wessol Mechanical Life Cycle Handbook: Good Environmental Design and Manufacturing, edited by Mahendra S. Hundal Micromachining of Engineering Materials, edited by Joseph McGeough Control Strategies for Dynamic Systems: Design and Implementation, John H. Lumkes, Jr. Practical Guide to Pressure Vessel Manufacturing, Sunil Pullarcot Nondestructive Evaluation: Theory, Techniques, and Applications, edited by Peter J. Shull Diesel Engine Engineering: Thermodynamics, Dynamics, Design, and Control, Andrei Makartchouk Handbook of Machine Tool Analysis, Ioan D. Marinescu, Constantin Ispas, and Dan Boboc

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Implementing Concurrent Engineering in Small Companies, Susan Carlson Skalak Practical Guide to the Packaging of Electronics: Thermal and Mechanical Design and Analysis, Ali Jamnia Bearing Design in Machinery: Engineering Tribology and Lubrication, Avraham Harnoy Mechanical Reliability Improvement: Probability and Statistics for Experimental Testing, R. E. Little Industrial Boilers and Heat Recovery Steam Generators: Design, Applications, and Calculations, V. Ganapathy The CAD Guidebook: A Basic Manual for Understanding and Improving Computer-Aided Design, Stephen J. Schoonmaker Industrial Noise Control and Acoustics, Randall F. Barron Mechanical Properties of Engineered Materials, Wolé Soboyejo Reliability Verification, Testing, and Analysis in Engineering Design, Gary S. Wasserman Fundamental Mechanics of Fluids: Third Edition, I. G. Currie Intermediate Heat Transfer, Kau-Fui Vincent Wong HVAC Water Chillers and Cooling Towers: Fundamentals, Application, and Operation, Herbert W. Stanford III Gear Noise and Vibration: Second Edition, Revised and Expanded, J. Derek Smith Handbook of Turbomachinery: Second Edition, Revised and Expanded, edited by Earl Logan, Jr. and Ramendra Roy Piping and Pipeline Engineering: Design, Construction, Maintenance, Integrity, and Repair, George A. Antaki Turbomachinery: Design and Theory, Rama S. R. Gorla and Aijaz Ahmed Khan Target Costing: Market-Driven Product Design, M. Bradford Clifton, Henry M. B. Bird, Robert E. Albano, and Wesley P. Townsend Fluidized Bed Combustion, Simeon N. Oka Theory of Dimensioning: An Introduction to Parameterizing Geometric Models, Vijay Srinivasan Handbook of Mechanical Alloy Design, edited by George E. Totten, Lin Xie, and Kiyoshi Funatani Structural Analysis of Polymeric Composite Materials, Mark E. Tuttle Modeling and Simulation for Material Selection and Mechanical Design, edited by George E. Totten, Lin Xie, and Kiyoshi Funatani Handbook of Pneumatic Conveying Engineering, David Mills, Mark G. Jones, and Vijay K. Agarwal Clutches and Brakes: Design and Selection, Second Edition, William C. Orthwein Fundamentals of Fluid Film Lubrication: Second Edition, Bernard J. Hamrock, Steven R. Schmid, and Bo O. Jacobson Handbook of Lead-Free Solder Technology for Microelectronic Assemblies, edited by Karl J. Puttlitz and Kathleen A. Stalter Vehicle Stability, Dean Karnopp

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Mechanical Wear Fundamentals and Testing: Second Edition, Revised and Expanded, Raymond G. Bayer Liquid Pipeline Hydraulics, E. Shashi Menon Solid Fuels Combustion and Gasification, Marcio L. de Souza-Santos Mechanical Tolerance Stackup and Analysis, Bryan R. Fischer Engineering Design for Wear, Raymond G. Bayer Vibrations of Shells and Plates: Third Edition, Revised and Expanded, Werner Soedel Refractories Handbook, edited by Charles A. Schacht Practical Engineering Failure Analysis, Hani M. Tawancy, Anwar Ul-Hamid, and Nureddin M. Abbas Mechanical Alloying and Milling, C. Suryanarayana Mechanical Vibration: Analysis, Uncertainties, and Control, Second Edition, Revised and Expanded, Haym Benaroya Design of Automatic Machinery, Stephen J. Derby Practical Fracture Mechanics in Design: Second Edition, Revised and Expanded, Arun Shukla Practical Guide to Designed Experiments, Paul D. Funkenbusch Gigacycle Fatigue in Mechanical Practive, Claude Bathias and Paul C. Paris Selection of Engineering Materials and Adhesives, Lawrence W. Fisher Boundary Methods: Elements, Contours, and Nodes, Subrata Mukherjee and Yu Xie Mukherjee Rotordynamics, Agnieszka Muszynska Pump Characteristics and Applications: Second Edition, Michael Volk Reliability Engineering: Probabilistic Models and Maintenance Methods, Joel A. Nachlas

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Reliability Engineering Probabilistic Models and Maintenance Methods

Joel A. Nachlas

Boca Raton London New York Singapore

A CRC title, part of the Taylor & Francis imprint, a member of the Taylor & Francis Group, the academic division of T&F Informa plc.

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CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2005 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20110713 International Standard Book Number-13: 978-1-4200-5618-1 (eBook - PDF) 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. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www. copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. 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.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

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Dedicated to the memory of Betty Nachlas

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Preface The motivation for the preparation of this book is my wish to create an integrated introductory resource for the study of reliability evaluation and maintenance planning. The focus across all of the topics treated is the support of design activities that lead to the production of dependable and efficient equipment. The orientation of the topical development is that probability models provide an effective vehicle for portraying and evaluating the variability that is inherent in the performance and longevity of equipment. The book is intended to support either an introductory course in reliability theory and preventive maintenance planning or a sequence of courses that address these topics. Fairly comprehensive coverage of the basic models and of various methods of analysis is provided. An understanding 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.

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The background required of the reader is a sound understanding of probability. 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. One caveat concerning the coverage here is that the treatment is strictly limited to hardware. Reliability and maintenance models have been developed for applications to software, humans, and services systems. No criticism of those efforts is intended, but the focus here is simply hardware. The organization of the text is reasonably straightforward. The elementary concepts of reliability theory are presented sequentially in Chapters 1 through 6. Following this, commonly used statistical methods for evaluating component reliability are described in Chapters 7 and 8. Chapters 9 through 13 treat repairable systems and thus maintenance planning models. Here again, the presentation is sequential in that simple failure models precede those that include preventive actions, and the renewal cases are treated before the more realistic nonrenewal cases. In the final chapter, four interesting special topics, including warranties, are discussed. It is worth noting that three appendices that address aspects of numerical computation are provided. 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 has been the support and encouragement of my wife Beverley. An important practical component of my success has been the support of Virginia Tech, particularly in providing computing resources and time during my sabbatical. I also wish to express my profound gratitude to 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.

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About the Author Joel A. Nachlas serves on the faculty of the Grado Department of Industrial and Systems Engineering at Virginia Tech. He has taught at Virginia Tech since 1974 and acts as the coordinator for the department’s graduate program in Operations Research. For the past 12 years, he has also taught Reliability Theory regularly at the Ecole Superiore d’Ingenieures de NiceSophia Antipolis. Dr. Nachlas earned his B.E.S. from the Johns Hopkins University in 1970. He earned his M.S. in 1972. and his Ph.D. in 1976, both from the University of Pittsburgh. His research interests are in the applications of probability and statistics to problems in reliability and quality control. His work in microelectronics reliability has been performed in collaboration with and under the support of the IBM Corp., INTELSAT, and the Bull Corp. He is the coauthor of over 50 refereed articles, has served in numerous editorial and referee capacities, and has lectured on reliability and maintenance topics throughout North America and Europe. Dr. Nachlas’ collaborative study of the deterioration of microelectronic communications circuits with J.L. Stevenson vii

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of INTELSAT earned them the 1991 P.K. McElroy Award. In addition, his work on the use of nested renewal functions to study opportunistic maintenance policies earned him the 2004 William A. Golomski Award. Since 1986, Dr. Nachlas has served on the Management Committee for the Annual Reliability and Maintainability Symposium. Since 2002, he has been Proceedings Editor for the annual Reliability and Maintainability Symposium (RAMS). He is a member of INFORMS, IIE, and SRE, and is a Fellow of the ASQ. During most of his tenure at Virginia Tech, Dr. Nachlas has also served as head coach of the men’s lacrosse team, and in 2001 he was selected by U.S. Lacrosse as the men’s division intercollegiate associates national coach of the year.

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Contents Chapter 1 Introduction ........................................................ 1 Chapter 2 System Structures .............................................. 7 2.1 2.2

2.3 2.4 2.5

Status Functions ............................................................. 7 System Structures and Status Functions ................... 10 2.2.1 Series Systems................................................... 10 2.2.2 Parallel System.................................................. 11 2.2.3 k-out-of-n Systems............................................. 14 2.2.4 Equivalent Structures....................................... 15 Modules of Systems....................................................... 21 Multistate Components and Systems .......................... 22 Exercises ........................................................................ 24

Chapter 3 Reliability of System Structures ..................... 27 3.1 3.2

3.3 3.4 3.5 3.6 3.7

Probability Elements .................................................... 27 Reliability of System Structures .................................. 28 3.2.1 Series Systems................................................... 28 3.2.2 Parallel Systems ................................................ 29 3.2.3 k-out-of-n Systems............................................. 30 3.2.4 Equivalent Structures....................................... 30 Modules .......................................................................... 38 Reliability Importance .................................................. 39 Reliability Allocation..................................................... 41 Conclusion...................................................................... 43 Exercises ........................................................................ 43

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Chapter 4 Reliability over Time ........................................ 45

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4.1 4.2

4.3 4.4

Reliability Measures ..................................................... 45 Life Distributions .......................................................... 52 4.2.1 The Exponential Distribution........................... 52 4.2.2 The Weibull Distribution .................................. 54 4.2.3 The Normal Distribution .................................. 58 4.2.4 The Lognormal Distribution............................. 61 4.2.5 The Gamma Distribution.................................. 61 4.2.6 Other Distributions ........................................... 62 System Level Models .................................................... 64 Exercises ........................................................................ 69

Chapter 5 Failure Processes .............................................. 73 5.1

5.2

5.3

5.4

Mechanical Failure Models........................................... 74 5.1.1 Stress-Strength Interference ............................ 74 5.1.2 Shock and Cumulative Damage ....................... 78 Electronic Failure Models............................................. 84 5.2.1 The Arrhenius Model ........................................ 84 5.2.2 The Eyring Model.............................................. 85 5.2.3 Power Law Model .............................................. 85 5.2.4 A Defect Model .................................................. 85 Other Failure Models .................................................... 86 5.3.1 A Diffusion Process Model................................ 86 5.3.2 Proportional Hazards ........................................ 88 5.3.3 Competing Risks................................................ 89 Exercises ........................................................................ 92

Chapter 6 Age Acceleration ............................................... 95 6.1 6.2 6.3 6.4

Age Acceleration for Electronic Devices ...................... 97 Age Acceleration for Mechanical Devices .................. 101 Step Stress Strategies................................................. 103 Exercises ...................................................................... 104

Chapter 7 Nonparametric Statistical Methods .............. 105 7.1 7.2

Data Set Notation and Censoring.............................. 106 Estimates Based on Order Statistics......................... 108

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7.3 7.4 7.5

7.6

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Estimates and Confidence Intervals.......................... 110 Tolerance Bounds ........................................................ 114 TTT Transforms .......................................................... 116 7.5.1 Theoretical Construction................................. 117 7.5.2 Application to Complete Data Sets................ 120 7.5.3 Application to Censored Data Sets ................ 127 Exercises ...................................................................... 132

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Chapter 8 Parametric Statistical Methods..................... 135 8.1 8.2 8.3 8.4

8.5

Graphical Methods ...................................................... 136 Method of Moments .................................................... 143 Method of Maximum Likelihood ................................ 147 Special Topics .............................................................. 157 8.4.1 Method of Moments with Censored Data...... 158 8.4.2 Data Analysis under Step-Stress Testing...... 161 8.4.3 Data Analysis with Group Censoring ............ 165 Exercises ...................................................................... 168

Chapter 9 Repairable Systems I — Renewal and Instantaneous Repair ..................................... 173 9.1 9.2 9.3 9.4 9.5

Renewal Processes....................................................... 174 Classification of Distributions and Bounds on Renewal Measures ................................................. 185 Residual Life Distribution .......................................... 191 Conclusion.................................................................... 193 Exercises ...................................................................... 194

Chapter 10 Repairable Systems II — Nonrenewal and Instantaneous Repair ........................... 197 10.1 Minimal Repair Models .............................................. 198 10.2 Imperfect Repair Models ............................................ 202 10.3 Equivalent Age Models ............................................... 207 10.3.1 The Kijima Models .......................................... 207 10.3.2 The Quasi-Renewal Process............................ 216 10.4 Conclusion.................................................................... 220 10.5 Exercises ...................................................................... 220

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Chapter 11 Availability Analysis .................................... 221 11.1 Availability Measures ................................................. 224 11.2 Example Computations............................................... 228 11.2.1 The Exponential Case ..................................... 228 11.2.2 A Numerical Case ........................................... 231 11.3 System-Level Availability ........................................... 233 11.4 The Nonrenewal Cases ............................................... 240 11.5 Markov Models ............................................................ 243 11.6 Exercises ...................................................................... 250

Chapter 12 Preventive Maintenance .............................. 251 12.1 Replacement Policies................................................... 252 12.1.1 The Elementary Models.................................. 252 12.1.2 Availability Model for Age Replacement ....... 259 12.1.3 Availability Model for Block Replacement..................................................... 262 12.1.4 Availability Model for Opportunistic Age Replacement ............................................. 263 12.1.4.1 A Failure Model ................................ 270 12.1.4.2 An Opportunistic Failure Replacement Policy........................... 273 12.1.4.3 A Partial Opportunistic Age Replacement Policy........................... 278 12.1.4.4 A Full Opportunistic Age Replacement Policy........................... 281 12.1.4.5 Analysis of the Opportunistic Replacement Models......................... 282 12.2 Nonrenewal Models..................................................... 284 12.2.1 Imperfect PM Models ...................................... 286 12.2.2 Models Based on the Quasi-Renewal Process.............................................................. 289 12.2.3 Models Based on the Kijima Model ............... 291 12.4 Conclusion.................................................................... 293 12.5 Acknowledgment ......................................................... 294 12.6 Exercises ...................................................................... 294

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Chapter 13 Predictive Maintenance ............................... 297 13.1 13.2 13.3 13.4 13.5

System Deterioration .................................................. 298 Inspection Scheduling ................................................. 300 More Complete Policy Analysis .................................. 301 Conclusion.................................................................... 306 Exercises ...................................................................... 306

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Chapter 14 Special Topics................................................ 307 14.1 Warranties ................................................................... 307 14.1.1 Full Replacement Warranties......................... 309 14.1.2 Pro Rata Warranties ....................................... 312 14.2 Reliability Growth....................................................... 314 14.3 Dependent Components .............................................. 316 14.4 Bivariate Reliability.................................................... 319 14.4.1 Collapsible Models........................................... 320 14.4.2 Bivariate Models ............................................. 321 14.4.2.1 Stochastic Functions ........................ 322 14.4.2.2 Correlation Models ........................... 325 14.4.2.3 Probability Analysis ......................... 326 14.4.2.4 Failure and Renewal Models ........... 331 14.5 Acknowledgment ......................................................... 338 14.6 Exercises ...................................................................... 339 Appendix A

Numerical Approximations........................ 341

A.1 Normal Distribution Function.................................... 341 A.2 Gamma Function......................................................... 344 A.3 Psi (Digamma) Function............................................. 344 Appendix B

Numerical Evaluation of Weibull Renewal Functions..................................... 347

Appendix C

Laplace Transform for the Key Renewal Theorem ............................... 359

Bibliography ......................................................................... 361 Index...................................................................................... 367