Reliability, Life Testing and the Prediction of Service Lives (eBook)

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

"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."