Saunders Reliability, Life Testing and the Prediction of Service Lives

For Engineers and Scientists
1. Auflage 2010
ISBN: 978-0-387-48538-6
Verlag: Springer US
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 308 Seiten

Reihe: Springer Series in Statistics

This book is intended for students and practitioners who have had a calculus-based statistics course and who have an interest in safety considerations such as reliability, strength, and duration-of-load or service life. Many persons studying statistical science will be employed professionally where the problems encountered are obscure, what should be analyzed is not clear, the appropriate assumptions are equivocal, and data are scant. In this book there is no disclosure with many of the data sets what type of investigation should be made or what assumptions are to be used.

Autoren/Hrsg.

Saunders, Sam C.

Inhaltsverzeichnis

1;Preface;6
2;Acknowledgements;7
2.1;Vörtrekkers;7
3;Glossary;8
4;Admonitions;9
5;Table of Contents;11
6;CHAPTER 1 Requisites;14
6.1;1.1. Why Reliability Is Important;14
6.2;1.2. Valuable Concepts;16
6.2.1;1.2.1. Concepts from Probability;16
6.2.2;1.2.2. Concepts from Statistics;19
7;CHAPTER 2 Elements of Reliability;23
7.1;2.1. Properties of Life Distributions;23
7.2;2.2. Useful Parametric Life Distributions;27
7.2.1;2.2.1. The Epstein (Exponential) Distribution;27
7.2.2;2.2.2. The Gamma Distribution;28
7.2.3;2.2.3. The Pareto Distribution;28
7.2.4;2.2.4. The Gaussian or Normal Distribution;29
7.2.5;2.2.5. Transformations to Normality;30
7.2.5.1;The Truncated Normal Distribution;30
7.2.5.2;The Log-Normal Distribution;31
7.2.5.3;The Xi-Normal Family;32
7.2.6;2.2.6. The Fatigue-Life Distribution;33
7.2.7;2.2.7. The Inverse-Gaussian Distribution;33
7.2.8;2.2.8. The Extreme-Value Distribution of Minima;34
7.2.9;2.2.9. Some Other Distributions;35
8;CHAPTER 3 Partitions and Selection;39
8.1;3.1. Binomial Coefficients and Sterling Numbers;39
8.1.1;3.1.1. Lagrange Coefficients;41
8.2;3.2. Lotteries and Coupon Collecting;43
8.2.1;3.2.1. Lotteries;43
8.2.2;3.2.2. Coupon Collecting;44
8.3;3.3. Occupancy and Allocations;47
8.3.1;3.3.1. Occupancy;47
8.3.1.1;Multiple Occupancy;49
8.3.2;3.3.2. Allocations;51
8.4;3.4. Related Concepts;52
8.4.1;3.4.1. The Sum of Epstein Waiting Times;52
8.4.2;3.4.2. Interpolation and Numerical Integration;53
9;CHAPTER 4 Coherent Systems;57
9.1;4.1. Functional Representation;57
9.2;4.2. Event-Tree Depiction;63
9.2.1;4.2.1. Associated Random Variables;64
9.3;4.3. Evaluation of Reliability;66
9.3.1;4.3.1. System Life;68
9.4;4.4. Use of Association to Bound Reliability;73
9.5;4.5. Shape of the Reliability Function;76
9.6;4.6. Diagnostics and Importance of System Components;79
9.6.1;4.6.1. Importance;79
9.6.2;4.6.2. Diagnostics Using Reliability;79
9.7;4.7. Hazard Rates and P.lya Frequency Functions;81
9.8;4.8. Closure Properties;82
9.8.1;4.8.1. Further Closure Properties;84
10;CHAPTER 5 Applicable Life Distributions;88
10.1;5.1. The Gaussian or Normal Distribution;88
10.2;5.2. Epstein's Distribution;90
10.2.1;5.2.1. The Erlang-k Distribution;91
10.3;5.3. The Galton and Fatigue-Life Distributions;91
10.3.1;5.3.1. The Log-Normal Distribution;91
10.3.2;5.3.2. The Fatigue-Life Distribution;92
10.4;5.4. Discovery and Rediscovery;93
10.5;5.5. Extreme Value Theory and Association;95
10.5.1;5.5.1. Gumbel's Theory;95
10.5.2;5.5.2. Maximum Loads and Association;97
11;CHAPTER 6 Philosophy, Science, and Sense;102
11.1;6.1. Likelihood without Priors;102
11.2;6.2. Likelihood for Complete Samples;105
11.3;6.3. Properties of the Likelihood;107
11.3.1;6.3.1. The Likelihood Depends upon the Model;107
11.3.2;6.3.2. Relative Likelihoods Are Not Probabilities on T;108
11.3.3;6.3.3. Likelihoods Invariant under Transformations;109
11.3.4;6.3.4. Likelihoods on Simple Parameter Spaces;109
11.3.5;6.3.5. Bayes' Theorem and Its Application;111
11.4;6.4. Types of Censoring of Data;114
11.4.1;6.4.1. Estimation for Type I Censoring;115
11.4.2;6.4.2. Estimation for Type II Censoring;115
11.4.3;6.4.3. Estimation for Type III Random Censoring;116
11.4.4;6.4.4. Transformation to the Standard Weibull;117
11.5;6.5. Generation of Ordered Observations;118
11.6;6.6. A Parametric Model of Censoring;121
11.7;6.7. The Empirical Cumulative Distribution;124
12;CHAPTER 7 Nonparametric Life Estimators;127
12.1;7.1. The Empiric Survival Distribution;127
12.1.1;7.1.1. Life-Table Methods;127
12.1.1.1;The Reduced Sample Method;128
12.1.1.2;The Actuarial Method;128
12.1.2;7.1.2. The Kaplan-Meier Estimator;128
12.2;7.2. Expectation and Bias of the K-M Estimator;130
12.2.1;Proportional Hazards;133
12.3;7.3. The Variance and Mean-Square Error;135
12.4;7.4. The Nelson-Aalen Estimator;137
12.4.1;7.4.1. Extensions and Generalizations;138
13;CHAPTER 8 Weibull Analysis;141
13.1;8.1. Distribution of Failure Times for Systems;141
13.2;8.2. Estimation for the Weibull Distribution;141
13.2.1;8.2.1. Right-Censored Estimation;142
13.2.2;8.2.2. Left-Censored Estimation;142
13.3;8.3. Competing Risks;143
13.3.1;8.3.1. The Bathtub-Shaped Hazard;143
13.4;8.4. Analysis of Censored Data;144
13.4.1;8.4.1. Estimation under Independent Competing Risks;144
13.4.2;8.4.2. Observing Both Time and Cause of Failure;145
13.4.3;8.4.3. Estimation with Dependent Failure Modes;147
13.4.4;8.4.4. Estimation under Random Censoring on Both Sides;148
13.4.5;8.4.5. Censoring for the Reciprocal Weibull;150
13.5;8.5. Change Points and Multiple Failure Mechanisms;152
13.5.1;8.5.1. A Known Change Point;153
13.5.2;8.5.2. A Change Point at an Unknown Location;157
13.5.3;8.5.3. Conclusions;159
14;CHAPTER 9 Examine Data, Diagnose and Consult;161
14.1;9.1. Scientific Idealism;161
14.2;9.2. Consultation and Diagnosis;162
14.3;9.3. Datasets in Service-Life Prediction;164
14.4;9.4. Data, Consulting, and Modeling;170
15;CHAPTER 10 Cumulative Damage Distributions;173
15.1;10.1. The Past as Prologue;173
15.2;10.2. The Fatigue-Life Distribution;175
15.3;10.3. The Mixed Class of Cumulative Damage Distributions;177
15.4;10.4. Elementary Derivation of Means and Variances;179
15.5;10.5. Behavior of the Hazard Rate;181
15.6;10.6. Mixed Variate Relationships;185
15.7;10.7. Estimation for Wald's Distributions;189
15.7.1;10.7.1. Estimation for Complete Samples;189
15.7.1.1;Estimation of a When ß Is Known;190
15.7.1.2;Estimation of ß When a Is Known;190
15.7.1.3;Unbised Estimation;191
15.7.2;10.7.2. Estimation for Incomplete Wald Samples;193
15.8;10.8. Estimation for the FL-Distribution;195
15.8.1;10.8.1. Complete Samples;195
15.8.2;10.8.2. Incomplete Samples of Fatigue-life Distribution;197
15.9;10.9. Estimation for Tweedie's Distribution;200
15.10;10.10. Cases of Misidentification;202
15.10.1;10.10.1. When the FL-Distribution Is Unknown;202
15.10.2;10.10.2. When the CD-Distributions Are Unknown;202
15.10.3;10.10.3. Weibull Distribution Contrasted with the FL-Distribution;203
15.10.4;10.10.4. Galton Distribution Mistaken for FL-Distribution;204
16;CHAPTER 11 Analysis of Dispersion;207
16.1;11.1. Applicability;207
16.2;11.2. Schrödinger's Distribution;208
16.3;11.3. Sample Distributions under Consonance;208
16.3.1;11.3.1. And Student's Distribution?;217
16.4;11.4. Classifications for Dispersion Analysis;219
16.4.1;11.4.1. A Single Classification;220
16.4.2;11.4.2. A Two-Way Classification for Multiplicative Effects;221
16.4.2.1;No Row or Column Effects;221
16.4.2.2;No Column Effects;222
16.4.2.3;No Row Effects;224
16.4.2.4;When Does Consonance Occur?;224
17;CHAPTER 12 Damage Processes;227
17.1;12.1. The Poisson Process;227
17.1.1;12.1.1. The Superposition of Poisson Processes;229
17.1.2;12.1.2. The Decomposition of Poisson Processes;229
17.2;12.2. Damage Due to Intermittant Shocks;229
17.3;12.3. Renewal Processes;232
17.3.1;12.3.1. Renewal Function for the Wald Distribution;234
17.3.2;12.3.2. Negligible Replacement Times for Units in Service;236
17.3.3;12.3.3. Tauberian Theorems for the Laplace Transform;236
17.4;12.4. Shock Models with Varying Intensity;237
17.4.1;12.4.1. The Marshall-Olkin Distribution;238
17.4.2;12.4.2. The Bivariate Poisson;240
17.5;12.5. Stationary Renewal Processes;240
17.6;12.6. The Miner-Palmgren Rule and Additive Damage;242
17.6.1;12.6.1. Miner's Rule as an Expectation;243
17.6.2;12.6.2. How Applicable Is This Theory?;244
17.7;12.7. Other Cumulative Damage Processes;245
17.7.1;12.7.1. Deterioration of Polymer Coatings;245
17.7.2;12.7.2. Varying Duty Cycles;246
17.8;12.8. When Linear Cumulative Damage Fails;248
17.8.1;12.8.1. Load-Order Effects in Crack Propagation;249
18;CHAPTER 13 Service Life of Structures;253
18.1;13.1. Wear under Spectral Loading;253
18.2;13.2. Multivariate Fatigue Life;254
18.2.1;13.2.1. Two-Component Load Sharing;255
18.2.2;13.2.2. The Multivariate Fatigue-Life Distribution;256
18.3;13.3. Correlations between Component Damage;261
18.3.1;13.3.1. Covariance and Association;262
18.4;13.4. Implementation;266
18.4.1;13.4.1. Estimation for Small Censored Samples;267
18.4.2;13.4.2. Relating Cumulative-Damage Parameters to the Exposure;268
19;CHAPTER 14 Strength and Durability;271
19.1;14.1. Range of Applicability;271
19.1.1;14.1.1. Introduction;271
19.1.2;14.1.2. Reliability Analysis of Strength;272
19.1.2.1;Static Strength for a Column;272
19.1.3;14.1.3. Strength of an Airframe Subsystem;274
19.2;14.2. Accelerated Tests for Strength;275
19.2.1;14.2.1. Determination of the Part of Least Accord;277
19.3;14.3. Danger of Extrapolation from Tests;279
19.3.1;14.3.1. Relating Parameters to the Exposure;281
19.3.1.1;The Pagett Models Using the Wald Distribution;281
19.4;14.4. Fracture Mechanics and Stochastic Damage;283
20;CHAPTER 15 Maintenance of Systems;286
20.1;15.1. Introduction;286
20.2;15.2. Availability;286
20.2.1;15.2.1. Application of Tauberian Properties;288
20.2.2;15.2.2. System Availability;289
20.2.2.1;Systems with Spares;290
20.3;15.3. Age Replacement with Renewal;290
20.3.1;15.3.1. A Single Machine with Repair;292
20.4;15.4. The Inversion of Transforms;294
20.5;15.5. Problems in Scheduled Maintenance;297
20.5.1;15.5.1. A Problem with Unscheduled Fleet Maintenance;298
20.5.2;15.5.2. A Problem with Scheduled Fleet Maintenance;299
21;CHAPTER 16 Mathematical Appendix;302
21.1;16.1. Integration;302
21.1.1;16.1.1. Stieltjes Integrals;302
21.2;16.2. Probability and Measure;304
21.3;16.3. Distribution Transforms;306
21.4;16.4. A Compendium of Discrete Distributions;310
21.5;16.5. A Compendium of Continuous Distributions;311
22;Bibliography;312
23;Index;318

Leseproben

"CHAPTER 5 Applicable Life Distributions (p. 75-76)

Often a number of parametric distributions can be used to summarize a given sample of life-length data. Sometimes several of them can do it quite well. For example, if we take the Data-Set VII in Chapter 9 (101 observations of the fatigue-life of aluminum coupons) we find there are several unimodal, skewed to the left, two-parameter life distributions that will fit it adequately in the region ofcentral tendency. These include the Galton, Weibull, Gamma, and fatiguelife distributions; certainly there are others.

How does one decide which of these distributions is most appropriate? In certain instances it makes little difference which of these families of distributions is adopted for use. But if the life of airframe components, made of the same material as that tested, must be predicted under many different loading conditions, all at some fraction of the maximum stress applied during the test, great differences arise among the families in their realistic predictive capability when the service-life is extrapolated from test data. Obtaining fatigue-life data at unrealistically high stress levels is necessitated by having to complete the testing within a small fraction of the design life.

After all, time is money. This is called an accelerated test since the stress level is beyond that encountered in service. What is desired is a method to calculate a safe-life for critical components when the maximum stress in service is, say, one-hundredth of that imposed in the test. That is, we must have a statistical model in which the parameters of the life distribution are constructs of the physical factors, such as the stress regime and the type of material (both of which are known to be of primary importance) so that if these physical factors are changed the appropriate modifications to the distribution of service life are possible, with valid predictions over the range of applicable service-life conditions. This is especially true whenever public health and safety are at risk.

5.1. The Gaussian or Normal Distribution

Under what conditions should the normal distribution be used? It is applied so universally and so uncritically that, simultaneously, it is the most used, and misused, distribution in statistics. The Central Limit Theorem (the limit theorem which is central to so much of statistical theory) is given by the classical LindebergFeller normal convergence criterion."