Analysis of Reliability and Quality Control: Fracture Mechanics 1

Title Page......Page 5
Contents......Page 7
Preface......Page 11
 1.1. Introduction......Page 15
  1.1.1. The importance of true physical acceleration life models (accelerated tests = true acceleration or acceleration)......Page 17
  1.1.2. Expression for linear acceleration relationships......Page 18
 1.2. Fundamental expression of the calculation of reliability......Page 19
 1.3. Continuous uniform distribution......Page 23
  1.3.2. Distribution function......Page 24
 1.4. Discrete uniform distribution (discrete U)......Page 26
  1.5.1. Discrete triangular distribution version......Page 27
  1.5.3. Links with uniform distribution......Page 28
 1.6. Beta distribution......Page 29
  1.6.1. Function of probability density......Page 30
  1.6.2. Distribution function of cumulative probability......Page 32
  1.6.3. Estimation of the parameters (p, q) of the beta distribution......Page 33
  1.7.1. Arithmetic mean......Page 34
  1.7.2. Reliability......Page 36
  1.7.3. Stabilization and normalization of variance error......Page 37
 1.9. The Gumbel distribution......Page 42
  1.9.1. Random variable according to the Gumbel distribution (CRV, E1 Maximum)......Page 43
  1.9.2. Random variable according to the Gumbel distribution (CRV E1 Minimum)......Page 44
 1.10. The Frechet distribution (E2 Max)......Page 45
 1.11. The Weibull distribution (with three parameters)......Page 46
 1.12. The Weibull distribution (with two parameters)......Page 49
  1.12.1. Description and common formulae for the Weibull distribution and its derivatives......Page 51
  1.12.2. Areas where the extreme value distribution model can be used......Page 53
  1.12.3. Risk model......Page 54
  1.12.4. Products of damage......Page 55
 1.13. The Birnbaum–Saunders distribution......Page 56
  1.13.1. Derivation and use of the Birnbaum–Saunders model......Page 57
  1.14.1. Probability density function......Page 59
  1.14.3. Cumulative risk function......Page 62
  1.14.5. Inverse survival function......Page 63
 1.15. Rayleigh distribution......Page 64
 1.16. The Rice distribution (from the Rayleigh distribution)......Page 66
 1.17. The Tukey-lambda distribution......Page 67
 1.18. Student’s (t) distribution......Page 69
  1.19.1. Probability distribution function of chi-square law (χ2)......Page 71
  1.19.2. Probability distribution function of chi-square law (χ2)......Page 72
 1.20. Exponential distribution......Page 73
  1.20.1. Example of applying mechanics to component lifespan......Page 77
  1.21.2. Probability density function......Page 80
  1.21.3. Cumulated distribution probability function......Page 81
 1.22. Bernoulli distribution......Page 82
 1.23. Binomial distribution......Page 85
 1.25. Geometrical distribution......Page 89
  1.25.1. Hypergeometric distribution (the Pascal distribution) versus binomial distribution......Page 90
 1.26. Hypergeometric distribution (the Pascal distribution)......Page 92
 1.27. Poisson distribution......Page 94
 1.28. Gamma distribution......Page 95
 1.31. Erlang distribution (characteristic of gamma distribution, Γ)......Page 99
 1.32. Logistic distribution......Page 103
  1.33.1. Mathematical–statistical characteristics of log-logistic distribution......Page 105
 1.34. Fisher distribution (F-distribution or Fisher–Snedecor)......Page 106
 1.35. Analysis of component lifespan (or survival)......Page 109
 1.36. Partial conclusion of Chapter 1......Page 110
 1.37. Bibliography......Page 111
 2.1. Introduction to assessment and statistical tests......Page 113
  2.1.1. Estimation of parameters of a distribution Overview:......Page 114
  2.1.2. Estimation by confidence interval......Page 116
  2.1.3. Properties of an estimator with and without bias......Page 117
 2.3. Method of maximum likelihood......Page 120
  2.3.1. Estimation of maximum likelihood......Page 121
  2.3.2. Probability equation of reliability-censored data......Page 122
  2.3.3. Punctual estimation of exponential law......Page 123
  2.3.4. Estimation of the Weibull distribution......Page 124
  2.3.5. Punctual estimation of normal distribution......Page 125
 2.4. Moving least-squares method......Page 127
  2.4.1. General criterion: the LSC......Page 128
  2.4.2. Examples of nonlinear models......Page 132
  2.4.3. Example of a more complex process......Page 136
 2.5. Conformity tests: adjustment and adequacy tests......Page 137
  2.5.1. Model of the hypothesis test for adequacy and adjustment......Page 139
  2.5.2. Kolmogorov–Smirnov Test (KS 1930 and 1936)......Page 140
  2.5.4. Simulated test (2nd application)......Page 145
  2.5.5. Example 1......Page 146
  2.5.6. Example 2 (Weibull or not?)......Page 149
  2.5.7. Cramer–Von Mises (CVM) test......Page 153
  2.5.8. The Anderson–Darling test......Page 154
  2.5.10. Adequacy test of chi-square (χ2)......Page 159
 2.6. Accelerated testing method......Page 165
  2.6.3. Example of the Weibull model......Page 166
  2.6.5. Example of the extreme value distribution model (E-MIN)......Page 167
  2.6.6. Example of the study on the Weibull distribution......Page 168
  2.6.7. Example of the BOX–COX model......Page 170
 2.7. Trend tests......Page 171
  2.7.1. A unilateral test......Page 172
  2.7.4. Homogenous Poisson Process (HPP)......Page 174
 2.8. Duane model power law......Page 178
 2.9. Chi-Square test for the correlation quantity......Page 180
  2.9.1. Estimations and χ2 test to determine the confidence interval......Page 181
  2.9.2. t_test of normal mean......Page 184
 2.10. Chebyshev’s inequality......Page 185
 2.11. Estimation of parameters......Page 187
 2.12. Gaussian distribution: estimation and confidence interval......Page 188
  2.12.4. Estimation of the Gaussian mean of unknown variance......Page 189
 2.13. Kaplan–Meier estimator......Page 192
  2.13.1. Empirical model using the Kaplan–Meier approach......Page 193
  2.13.2. General expression of the KM estimator......Page 194
 2.14. Case study of an interpolation using the bi-dimensional spline function......Page 195
 2.15. Conclusion......Page 197
 2.16. Bibliography......Page 198
 3.1. Introduction to errors and uncertainty1......Page 201
 3.2. Definition of uncertainties and errors as in the ISO norm......Page 203
 3.3. Definition of errors and uncertainty in metrology......Page 205
  3.3.1. Difference between error and uncertainty......Page 206
 3.4. Global error and its uncertainty......Page 216
 3.5. Definitions of simplified equations of measurement uncertainty......Page 218
  3.5.1. Expansion factor k and range of relative uncertainty......Page 220
  3.5.2. Determination of type A and B uncertainties according to GUM......Page 222
 3.6. Principal of uncertainty calculations of type A and type B......Page 243
  3.6.1. Standard and expanded uncertainties......Page 245
  3.6.3. Error on repeated measurements: composed uncertainty......Page 246
 3.7. Study of the basics with the help of the GUMic software package: quasi-linear model......Page 253
 3.9. Bibliography......Page 259
Glossary......Page 263
Index......Page 271