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- 1. Predicting Product Predicting Product Life Using Reliability Life Using Reliability y Analysis Methods Steven Wachs ©2011 ASQ & Presentation Steven Presented live on Nov 09th ~ 11th, 2012http://reliabilitycalendar.org/The_Reliability_Calendar/Short_Courses/Shliability Calendar/Short Courses/Short_Courses.html
- 2. ASQ Reliability Division ASQ Reliability Division Short Course Series Short Course Series One of the monthly webinars One of the monthly webinars on topics of interest to reliability engineers. To view recorded webinar (available to ASQ Reliability Division members only) visit asq.org/reliability ) / To sign up for the free and available to anyone live webinars visit reliabilitycalendar.org and select English Webinars to find links to register for upcoming eventshttp://reliabilitycalendar.org/The_Reliability_Calendar/Short_Courses/Shliability Calendar/Short Courses/Short_Courses.html
- 3. Predicting Product LifeUsing Reliability Analysis Methods Steven Wachs Principal Statistician Integral Concepts, Inc. www.integral-concepts.com 248-884-2276 www.integral-concepts.com 1 ©2012 Copyright
- 4. Agenda Reliability Concepts & Reliability Data Probability, Statistics, and Distributions Assessing & Selecting Models Estimating Reliability Statistics Systems Reliability Reliability Test Planning Accelerated Life Testing Warranty Analysis www.integral-concepts.com 2 ©2012 Copyright
- 5. MotivationIntense Global CompetitionCustomer ExpectationsCustomer LoyaltyProduct Liability www.integral-concepts.com 3 ©2012 Copyright
- 6. Defining Reliability Reliability is the probability that a material, component, or system will perform its intended function under defined operating conditions for a specified period of time. www.integral-concepts.com 4 ©2012 Copyright
- 7. Ambiguity in Definition What is the intended function? What are the defined operating conditions? How should time be defined?We must clearly define these characteristics when defining reliability for a specific application www.integral-concepts.com 5 ©2012 Copyright
- 8. Reliability Data Life Data Time-To-Failure (TTF) Data Time-Between-Failure (TBF) Data Survival Data Event-time Data Degradation Data www.integral-concepts.com 6 ©2012 Copyright
- 9. Unique Aspects of Reliability Data Presence of Censoring Reliability Models based on positive random variables (e.g. exponential, lognormal, Weibull, gamma) Interpolation and extrapolation often required www.integral-concepts.com 7 ©2012 Copyright
- 10. Repairable vs. Non-repairable The focus of this course is non- repairable components or systems (characterized by time to failure) Repairable systems are characterized by time between failure www.integral-concepts.com 8 ©2012 Copyright
- 11. The Bathtub Curve www.integral-concepts.com 9 ©2012 Copyright
- 12. The Reliability FunctionR(to ) = P(T > to )where T =“time”to failure www.integral-concepts.com 10 ©2012 Copyright
- 13. Censored Data When exact failure times are not known Provides useful information for estimation of reliability (Do NOT drop from analysis) Types of Censoring – Right Censoring – Left Censoring – Interval Censoring www.integral-concepts.com 11 ©2012 Copyright
- 14. Types of Censoring Left Censoring Right Censoring Interval Censoring www.integral-concepts.com 12 ©2012 Copyright
- 15. Other Censoring Ideas Competing Risks • Impact on Reliability estimates • Alternatives (if extreme censoring exists) – Use Accelerated Testing Conditions – Use Degradation Data www.integral-concepts.com 13 ©2012 Copyright
- 16. Agenda Reliability Concepts & Reliability Data Probability, Statistics, and Distributions Assessing & Selecting Models Estimating Reliability Statistics Systems Reliability Reliability Test Planning Accelerated Life Testing Warranty Analysis www.integral-concepts.com 14 ©2012 Copyright
- 17. Describing Time to Failure www.integral-concepts.com 15 ©2012 Copyright
- 18. Integrating the PDF B c fxdx d PX B PX c, d www.integral-concepts.com 16 ©2012 Copyright
- 19. Reliability Distribution Plot 0.0008 0.0007 0.0006 0.0005 Density 0.0004 0.0003 0.0002 0.0001 R(1500) 0.0000 0 1500 X www.integral-concepts.com 17 ©2012 Copyright
- 20. Failure Probability (Cumulative) Distribution Plot 0.0008 F(500) 0.0007 0.0006 0.0005 Density 0.0004 0.0003 0.0002 0.0001 0.0000 0 500 X www.integral-concepts.com 18 ©2012 Copyright
- 21. The CDF fxdx t Ft PX , t www.integral-concepts.com 19 ©2012 Copyright 19
- 22. CDF/Reliability Relationship www.integral-concepts.com 20 ©2012 Copyright
- 23. Hazard Function (Rate)The propensity to fail in the next instant given that it hasn’t failed up to that time (“instantaneous failure rate function”) ft ftht 1Ft Rt www.integral-concepts.com 21 ©2012 Copyright
- 24. www.integral-concepts.com 22©2012 Copyright
- 25. Mean Time to Failure (MTTF) The Expected Value or the Mean of the Time to Failure Random Variable The average time to failure (often significantly larger than the median time to failure) The MTTF can be misleading as often as much as 70% of the population will fail before the MTTF MTTF ET 0 tftdt 0 Rtdt www.integral-concepts.com 23 ©2012 Copyright
- 26. B-Life (or Quantile) The time at which a specified proportion of the population is expected to fail www.integral-concepts.com 24 ©2012 Copyright
- 27. Advantages of Parametric Models May be described concisely with a few parameters Allows extrapolation (in time) Provide “smooth” estimates of failure time distributions www.integral-concepts.com 25 ©2012 Copyright
- 28. Common Distributions in Reliability Weibull Exponential Lognormal Gamma Binomial Loglogistic Etc. www.integral-concepts.com 26 ©2012 Copyright
- 29. Exponential Distribution The simplest model used in reliability analysis (and sometimes misused) Described by a single parameter, l which is the hazard rate (inverse of MTTF) Key property: the hazard rate is constant (the only distribution with this property) www.integral-concepts.com 27 ©2012 Copyright
- 30. Exponential Distribution• pdf: f(t) = le-lt• cdf: F(t) = 1 - e-lt• Reliability: R(t) = e-lt• Hazard rate: h(t) = l• MTTF = 1/l = q• Quantile: F-1(p) = (1/l)[-ln(1-p)] www.integral-concepts.com 28 ©2012 Copyright
- 31. Exponential Distribution ExampleLight bulb lifetime may be described by an exponential distribution. The MTTF = 12,000 hrs.Find:A. Hazard RateB. Proportion failing by 12,000 hrsC. Proportion failing by 24,000 hrs www.integral-concepts.com 29 ©2012 Copyright
- 32. Exponential Distribution Example Solution A. l = 1/12,000 = 0.000083 = 83 failures per million hrs 12,000 B. F12, 000 1 e 12,000 1 1 e 0. 632 24,000 C. F24, 000 1 e 12,000 1 1 0. 865 e2 www.integral-concepts.com 30 ©2012 Copyright
- 33. Exponential Distribution Guidelines Constant hazard rate implies that the probability that a unit will fail in the next instant does not depend on the unit’s age Reasonable for many electronic components that do not wear out Usually inappropriate for modeling TTF of mechanical components that are subject to fatigue, corrosion, or wear www.integral-concepts.com 31 ©2012 Copyright
- 34. The Weibull Distribution The most common model in reliability analysis Described by 2 parameters: h = “scale” parameter b = “shape” parameter Flexible model that can effectively model a wide variety of failure distributions www.integral-concepts.com 32 ©2012 Copyright
- 35. The Weibull Distribution (some functions) 1 t pdf: ft t e t cdf: Ft 1 e t Reliability: Rt e 1 Hazard rate: ht t www.integral-concepts.com 33 ©2012 Copyright
- 36. The Weibull Shape ParameterFailure Rate ? www.integral-concepts.com 34 ©2012 Copyright
- 37. The Weibull Scale Parameter ? www.integral-concepts.com 35 ©2012 Copyright
- 38. Weibull Characteristics h is also referred to as the 63.2nd percentile To see this: set t = h in F(t) Ft F 1 e 1 1e 1 e 0. 632 1 • The value of b is irrelevant when t = h www.integral-concepts.com 36 ©2012 Copyright
- 39. Conditional Reliability An application of conditional probability Needed to estimate reliability when “burn- in” is used or to estimate reliability after a warranty period. Rtt 0 Rt|t 0 Rt 0 www.integral-concepts.com 37 ©2012 Copyright
- 40. Agenda Reliability Concepts & Reliability Data Probability, Statistics, and Distributions Assessing & Selecting Models Estimating Reliability Statistics Systems Reliability Reliability Test Planning Accelerated Life Testing Warranty Analysis www.integral-concepts.com 38 ©2012 Copyright
- 41. Selecting Models Given time-to-failure data (failures and censored data), which distribution best describes the data? Graphical Methods (probability plotting) and/or Statistical Methods may be utilized www.integral-concepts.com 39 ©2012 Copyright
- 42. Probability Plots Graphical method to assess “fit” Fit is determined by how well the plotted points align along a straight line Plotted variables are “transformed” so that y is a linear function of x – X axis: Plot observed failure times – Y axis: Plot estimated cumulative probabilities (p) www.integral-concepts.com 40 ©2012 Copyright
- 43. Probability Plotting www.integral-concepts.com 41 ©2012 Copyright
- 44. Constructing Probability Plots• X-Axis – Observed/Transformed Failure Times• Y-Axis – Estimated/Transformed Cumulative Probabilities• Transformed quantities for plot depend on the distribution www.integral-concepts.com 42 ©2012 Copyright
- 45. Constructing Probability Plots www.integral-concepts.com 43 ©2012 Copyright
- 46. Linearizing the CDF - Example Consider the Weibull distribution. Recall that the Weibull cdf is: t Ft 1 e • We need to transform F(t) to achieve a linear function www.integral-concepts.com 44 ©2012 Copyright
- 47. Linearizing the CDF - Example t 1 Ft e t 1 1Ft e ln 1Ft 1 t ln ln 1Ft 1 lnt ln By setting: y ln ln 1Ft 1 x lnt C ln we have: y x C www.integral-concepts.com 45 ©2012 Copyright
- 48. Graphical Estimation 2.75 b = 2.0/2.75 = 0.732.0 h www.integral-concepts.com 46 ©2012 Copyright
- 49. Selecting Models Multiple Distributions may adequately describe the time-to-failure data Sensitivity Analysis is recommended to assess how reliability predictions vary with alternative viable models Confidence Intervals on reliability estimates do not include model uncertainty www.integral-concepts.com 47 ©2012 Copyright
- 50. Selecting a Distribution www.integral-concepts.com 48 ©2012 Copyright
- 51. Handling Multiple Failure Modes Multiple Failure Modes should be modeled separately (if data exists) Failure rates of the various failure modes are typically different Overall Reliability may be predicted using system reliability concepts (series model) www.integral-concepts.com 49 ©2012 Copyright
- 52. Handling Multiple Failure Modes ReliaSoft Weibull++ 7 - www.ReliaSoft.com Probability - W eibull 99.000 Probability-Weibull Data 1 Weibull-2P 90.000 ML E SRM MED FM F=40/S=0 Data Points Probability Line 50.000 U n re l i a b i l i ty , F (t) 10.000 5.000 Steven Wachs integral Concepts, Inc. 10/28/2011 1:05:27 PM 1.000 0.010 0.100 1.000 10.000 100.000 1000.000 10000.000 Time, ( t) b h www.integral-concepts.com 50 ©2012 Copyright
- 53. Handling Multiple Failure Modes ReliaSoft Weibull++ 7 - www.ReliaSoft.com Probability - W eibull 99.000 Probability-Weibull Data 1 Weibull-CFM 90.000 ML E SRM MED FM CFM 1 Points CFM 2 Points CFM 1 L ine CFM 2 L ine Probability Line 50.000 U n re l i a b i l i ty , F (t) 10.000 5.000 Steven Wachs integral Concepts, Inc. 10/28/2011 1:03:08 PM 1.000 0.010 0.100 1.000 10.000 100.000 1000.000 10000.000 Time, ( t) b h b h www.integral-concepts.com 51 ©2012 Copyright
- 54. Agenda Reliability Concepts & Reliability Data Probability, Statistics, and Distributions Assessing & Selecting Models Estimating Reliability Statistics Systems Reliability Reliability Test Planning Accelerated Life Testing Warranty Analysis www.integral-concepts.com 52 ©2012 Copyright
- 55. Reliability Estimation From the time-to-failure distribution we can estimate quantities like: • Reliability at various times • Time at which x% are expected to fail • Failure (hazard) rates • Mean time to failure Confidence Intervals or Bounds should be included to account for estimation uncertainty www.integral-concepts.com 53 ©2012 Copyright
- 56. Reliability Estimation Estimation Methods • Maximum Likelihood Estimation (MLE): nice statistical properties, handles censored data well, biased estimates for small sample sizes • Rank Regression: Unbiased estimates but poorer precision and does not handle censored data as well as MLE www.integral-concepts.com 54 ©2012 Copyright
- 57. Properties of Estimators Bias – The extent to which the estimator differs on average from the true value. (An unbiased estimator equals the true value on average) Precision – The amount of variability in the estimates. www.integral-concepts.com 55 ©2012 Copyright
- 58. Properties of Estimators www.integral-concepts.com 56 ©2012 Copyright
- 59. Estimation Methods Maximum Likelihood Estimation – Generally preferred by statisticians (minimum variance) although the estimates tend to be biased – ML method finds parameter values which maximize the likelihood function (the joint probability of observing all of the data). – The maximization of the likelihood function usually must be done numerically (rather than analytically). www.integral-concepts.com 57 ©2012 Copyright
- 60. MLE Example (Weibull) • Given failure time data, we need to estimate h, b. i1 fx i fx 1 fx 2 . . . fx n nL L e n x 1 xi i i1 • We maximize likelihood function by taking derivatives with respect to each parameter www.integral-concepts.com 58 ©2012 Copyright
- 61. Effect of Censored Data on the Likelihood Function • With no censoring, the likelihood function is: i1 fx i fx 1 fx 2 . . . fx n n L • Censored observations cannot use the pdf since the failure time is unknown www.integral-concepts.com 59 ©2012 Copyright
- 62. Effect of Censored Data on the Likelihood Function • Suppose we have a right-censored observation at time = 1500? • What function indicates the probability of this occurring? www.integral-concepts.com 60 ©2012 Copyright
- 63. Effect of Censored Data on the Likelihood Function • Suppose we have a right-censored observation at time = 1500? • What function indicates the probability of this occurring? • R(1500) gives the probability that a unit fails at time 1500 or later. www.integral-concepts.com 61 ©2012 Copyright
- 64. Effect of Censored Data on the Likelihood Function Distribution Plot 0.0008 0.0007 0.0006 0.0005 Density 0.0004 0.0003 0.0002 0.0001 R(1500) 0.0000 0 1500 X www.integral-concepts.com 62 ©2012 Copyright
- 65. Effect of Censored Data on the Likelihood Function • Suppose we have a left-censored observation at time = 500? • What function indicates the probability of this occurring? www.integral-concepts.com 63 ©2012 Copyright
- 66. Effect of Censored Data on the Likelihood Function • Suppose we have a left-censored observation at time = 500? • What function indicates the probability of this occurring? • F(500) gives the probability that a unit fails at time 500 or earlier. www.integral-concepts.com 64 ©2012 Copyright
- 67. Effect of Censored Data on the Likelihood Function Distribution Plot 0.0008 F(500) 0.0007 0.0006 0.0005 Density 0.0004 0.0003 0.0002 0.0001 0.0000 0 500 X www.integral-concepts.com 65 ©2012 Copyright
- 68. Effect of Censored Data on the Likelihood Function • Suppose we have an interval censored condition where the failure occurred between 1000 and 1300. • What function indicates the probability of this occurring? www.integral-concepts.com 66 ©2012 Copyright
- 69. Effect of Censored Data on the Likelihood Function • Suppose we have an interval censored condition where the failure occurred between 1000 and 1300. • What function indicates the probability of this occurring? • F(1300)-F(1000) gives the probability that a unit fails between 1000 and 1300 www.integral-concepts.com 67 ©2012 Copyright
- 70. Effect of Censored Data on the Likelihood Function Distribution Plot 0.0008 0.0007 0.0006 0.0005 F(1300)-F(1000) Density 0.0004 0.0003 0.0002 0.0001 0.0000 0 1000 1300 X www.integral-concepts.com 68 ©2012 Copyright
- 71. Estimation Methods• Rank Regression – Determines best fit line on the probability plot by using least squares regression – Fitted line is used to estimate parameters www.integral-concepts.com 69 ©2012 Copyright
- 72. Failure Probability Plot www.integral-concepts.com 70 ©2012 Copyright
- 73. Reliability Estimation www.integral-concepts.com 71 ©2012 Copyright
- 74. Estimating with Multiple Failure Modes Failure Time Failure Model Failure Time Failure Model 63 linkage 791 motor 116 linkage 808 motor 237 linkage 823 motor 249 linkage 841 motor 297 linkage 869 motor 384 linkage 874 linkage 386 linkage 878 motor 420 linkage 981 motor 467 linkage 991 motor 485 motor 999 motor 522 linkage 1005 motor 541 linkage 1007 motor 592 linkage 1046 motor 595 linkage 1084 motor 601 linkage 1086 motor 624 linkage 1190 motor 655 motor 1299 motor 662 linkage 1481 linkage 702 linkage 1502 motor 721 linkage 1581 motor www.integral-concepts.com 72 ©2012 Copyright
- 75. Linkage Failure ModeDistribution Analysis: Failure TimeVariable: Failure TimeFailure Mode: fm = linkageCensoring Information CountUncensored value 20Right censored value 20Estimation Method: Maximum LikelihoodDistribution: WeibullParameter Estimates Standard 95.0% Normal CIParameter Estimate Error Lower UpperShape 1.34641 0.264909 0.915592 1.97994Scale 1325.81 240.466 929.169 1891.76 www.integral-concepts.com 73 ©2012 Copyright
- 76. Motor Failure ModeDistribution Analysis: Failure TimeVariable: Failure TimeFailure Mode: fm = motorCensoring Information CountUncensored value 20Right censored value 20Estimation Method: Maximum LikelihoodDistribution: WeibullParameter Estimates Standard 95.0% Normal CIParameter Estimate Error Lower UpperShape 4.17342 0.634609 3.09784 5.62245Scale 1154.46 62.7168 1037.86 1284.17 www.integral-concepts.com 74 ©2012 Copyright
- 77. Multiple Failure Modes Probability Plot for Failure Time Complete Data - ML Estimates F ailure M ode = linkage Failure Mode = linkage Failure Mode = motor S hape S cale Weibull - 95% CI Weibull - 95% CI 1.34641 1325.81 F ailure M ode = motor 95 95 S hape S cale 4.17342 1154.46 80 80 50 50 Percent 20 Percent 20 5 5 2 2 1 1 10 100 1000 10000 500 1000 2000 Failure Time Failure Time www.integral-concepts.com 75 ©2012 Copyright
- 78. Multiple Failure Modes Survival Plot for Failure Time Complete Data - ML Estimates F ailure M ode = linkage Failure Mode = linkage Failure Mode = motor S hape S cale Weibull - 95% CI Weibull - 95% CI 1.34641 1325.81 F ailure M ode = motor 100 100 S hape S cale 4.17342 1154.46 80 80 60 60 Percent Percent 40 40 20 20 0 0 0 1500 3000 4500 500 1000 1500 Failure Time Failure Time www.integral-concepts.com 76 ©2012 Copyright
- 79. Multiple Failure Modes Survival Plot for Failure Time Multiple Distributions - 95% CI Complete Data - ML Estimates 100 80 60 Percent 40 20 0 0 200 400 600 800 1000 1200 1400 1600 Failure Time www.integral-concepts.com 77 ©2012 Copyright
- 80. Confidence Intervals • An interval (l, u) around the point estimate that contains the true value with high probability • The interval is said to be a P% confidence interval if P percent of the intervals we might calculate from replicated studies contain the true parameter value www.integral-concepts.com 78 ©2012 Copyright
- 81. Improving Precision of Estimates More Data (Failures) = Better Precision (tighter confidence intervals) Can make more assumptions (assume distribution parameters) Reduce confidence level (not a real solution) www.integral-concepts.com 79 ©2012 Copyright
- 82. Agenda Reliability Concepts & Reliability Data Probability, Statistics, and Distributions Assessing & Selecting Models Estimating Reliability Statistics Systems Reliability Reliability Test Planning Accelerated Life Testing Warranty Analysis www.integral-concepts.com 80 ©2012 Copyright
- 83. System Reliability A System may be thought of as a collection of components or subsystems System Reliability Depends on: a. Component reliability b. Configuration (redundancy) c. Time www.integral-concepts.com 81 ©2012 Copyright
- 84. A Series System i1 R it R 1 tR 2 tR 3 tR 4 t 4R s t Example: If the component reliabilities are 0.9, 0.9. 0.8, 0.8 at 1 year Then the System reliability at 1 year is: 0.9*0.9*0.8*0.8 = 0.52 www.integral-concepts.com 82 ©2012 Copyright
- 85. Series Model for Multiple Failure Modes n Rt R it R A tR BtR C tR Dt i1 www.integral-concepts.com 83 ©2012 Copyright
- 86. A Parallel System (Redundant Components) R s t 1 F s t 1 F 1 tF 2 tF 3 t 1 1 R 1 t1 R 2 t1 R 3 t www.integral-concepts.com 84 ©2012 Copyright
- 87. A Parallel System (Redundant Components) Example: If the component reliabilities are 0.9, 0.9. 0.9 at 1 year Then the System reliability at 1 year is: 1 - (.1)*(.1)*(.1) = 0.999 www.integral-concepts.com 85 ©2012 Copyright
- 88. k-out-of-n Parallel Systems System consists of n components in which k of the n components must function in order for the system to function For example, if 2 of 4 engines are required to fly, then the system will not fail if: – All 4 engines operate – Any 3 operate – Any 2 operate www.integral-concepts.com 86 ©2012 Copyright
- 89. k-out-of-n Parallel SystemsIf all components have the same reliability, R(t):The probabilities of all possible combinations leading to success aresummed www.integral-concepts.com 87 ©2012 Copyright
- 90. k-out-of-n Parallel Systems Example Suppose a system consists of 6 identical pumps. For the system to function, at least 4 of the 6 pumps must operate. If the reliability of each pump at 3 years in service is 0.90, what is the system reliability at 3 years? www.integral-concepts.com 88 ©2012 Copyright
- 91. Effect of k on System Reliability As k increases, system reliability decreases If k = 1 Pure Parallel System If k = n Series System www.integral-concepts.com 89 ©2012 Copyright
- 92. Effect of k on System Reliability System Reliability vs k (k-out-of-6, R = 0.90) 1.0 0.9 0.8 k Reliability Reliability 1 1.0000 2 0.9999 0.7 3 0.9987 4 0.9842 5 0.8857 0.6 6 0.5314 0.5 1 2 3 4 5 6 k www.integral-concepts.com 90 ©2012 Copyright
- 93. k-out-of-n Parallel Systems When the components in the k-out-of-n parallel configuration do not share the same reliability function, all possible combinations must be computed Example follows www.integral-concepts.com 91 ©2012 Copyright
- 94. k-out-of-n System ExampleThree generators areconfigured in parallel. At 0.90least two of thegenerators mustfunction in order for the 0.8 2/3system to function. At 5 7years: R1 = 0.90, R2 =0.87, R3 = 0.80. What is 0.80the System Reliability at5 years? www.integral-concepts.com 92 ©2012 Copyright
- 95. k-out-of-n System Example Here, k = 2, n = 3 The following combinations of events lead to a reliable system at 5 years in service: – generator 1,2 operate and generator 3 fails – generator 1,3 operate and generator 2 fails – generator 2,3 operate and generator 1 fails – All three generators operate www.integral-concepts.com 93 ©2012 Copyright
- 96. k-out-of-n System Example R1 = 0.90 R2 = 0.87 R3 = 0.80 www.integral-concepts.com 94 ©2012 Copyright
- 97. Reliability Block Diagrams Used to Model System and Estimate System Reliability www.integral-concepts.com 95 ©2012 Copyright
- 98. Reliability Allocation ProblemsGiven a reliability target for the system, howshould subsystem and/or component levelreliability requirements be established so that thesystem objective is met?Typical Goals a. Maximize the System Reliability for a given cost b. Minimize the Cost for a given System Reliability Improve component reliability or add redundancies? www.integral-concepts.com 96 ©2012 Copyright
- 99. Agenda Reliability Concepts & Reliability Data Probability, Statistics, and Distributions Assessing & Selecting Models Estimating Reliability Statistics Systems Reliability Reliability Test Planning Accelerated Life Testing Warranty Analysis www.integral-concepts.com 97 ©2012 Copyright
- 100. Reliability Test Planning Estimation Test Plans Determine sample size needed to estimate reliability characteristics with a specified precision Planning information such as assumed distribution parameters, testing time, and censoring scheme is required Failures during testing are required www.integral-concepts.com 98 ©2012 Copyright
- 101. Sample Sizes for DesiredPrecision We select sample size to achieve the desired precision in our estimates Larger sample size Greater precision Greater precision Smaller confidence intervals www.integral-concepts.com 99 ©2012 Copyright
- 102. Sample Size CalculationsCalculation Depends On: Distribution used to model the failure data Level of precision desired Confidence level Presence of censoring Length of test (for Type I censoring) Failure proportion (for Type II censoring) www.integral-concepts.com 100 ©2012 Copyright
- 103. Estimation Test Plan Type I right-censored data (Single Censoring) Estimated parameter: 50th percentile Calculated planning estimate = 124.883 Target Confidence Level = 95% Planning distribution: Weibull Scale = 150, Shape = 2 Actual Censoring Sample Confidence Time Precision Size Level 100 62.435 8 96.2010 www.integral-concepts.com 101 ©2012 Copyright
- 104. Reliability Test Planning Demonstration Test Plans Determine sample size (or testing time) needed to demonstrate reliability characteristics (e.g. lower bound on reliability) Planning information such as assumed distribution and parameter is required Failures during testing are not required www.integral-concepts.com 102 ©2012 Copyright
- 105. Reliability Demonstration Evaluates the following hypothesis H0: The reliability is less than or equal to a specified value H1: The reliability is greater than a specified value www.integral-concepts.com 103 ©2012 Copyright
- 106. Types of Test Plans Zero-Failure Test Plans – Test demonstrates reliability if zero failures are observed during test – Useful for highly reliable items M-Failure Test Plans – Test demonstrates reliability if no more than m failures occur – Permit verification of test design assumptions www.integral-concepts.com 104 ©2012 Copyright
- 107. Planning Information Assumptions needed: – Distribution – Shape Parameter (for Weibull) – Scale Parameter (for other distributions such as lognormal, loglogistic, logistic, extreme value) – Assumptions based on expert opinions, prior studies, similar products – Sensitivity analysis is recommended www.integral-concepts.com 105 ©2012 Copyright
- 108. Computing Test Time or Sample Size We specify either the sample size or the testing time allocated for each unit (the other quantity is computed) Demonstration Test Plan consists of: – The maximum number of failures allowed – The sample size – The testing time for each unit www.integral-concepts.com 106 ©2012 Copyright
- 109. Example: Demonstration Test Plan Reliability Goal: 1st percentile > 80,000 mi TTF estimated by Weibull w/ b = 2.5 Can test for 120,000 miles How many units are needed for zero-failure and 1-failure test plans? www.integral-concepts.com 107 ©2012 Copyright
- 110. Example: Demonstration Test PlanDemonstration Test PlansReliability Test PlanDistribution: Weibull, Shape = 2.5Percentile Goal = 80000,Target Confidence Level = 95% ActualFailure Testing Sample Confidence Test Time Size Level 0 120000 108 94.9768 www.integral-concepts.com 108 ©2012 Copyright
- 111. Example: Demonstration Test PlanDemonstration Test PlansReliability Test PlanDistribution: Weibull, Shape = 2.5Percentile Goal = 80000,Target Confidence Level = 95% ActualFailure Testing Sample Confidence Test Time Size Level 1 120000 172 95.0241 www.integral-concepts.com 109 ©2012 Copyright
- 112. Example: Demonstration Test Plan Suppose we can only test 50 units?Reliability Test PlanDistribution: Weibull, Shape = 2.5Percentile Goal = 80000,Actual Confidence Level = 95%Failure Sample Testing Test Size Time 0 50 163392 www.integral-concepts.com 110 ©2012 Copyright
- 113. Probability of Passing (POP) Likelihood of Passing for Weibull Model Maximum Failures = 0, Target Alpha = 0.05 Time = 120000, N = 108, Actual alpha = 0.0502316 100 80 60 Percent 40 20 0 2 4 6 8 10 Ratio of Improvement www.integral-concepts.com 111 ©2012 Copyright
- 114. Probability of Passing (POP) Likelihood of Passing for Weibull Model Maximum Failures = 1, Target Alpha = 0.05 Time = 120000, N = 172, Actual alpha = 0.0497587 100 80 60 Percent 40 20 0 2 4 6 8 10 Ratio of Improvement www.integral-concepts.com 112 ©2012 Copyright
- 115. Demonstration Test Plan (1st Percentile)Reliability Test PlanDistribution: Weibull, Shape = 2.5Percentile Goal = 80000, Target Confidence Level = 95% ActualFailure Testing Sample Confidence Test Time Size Level 0 120000 108 94.9768 1 120000 172 95.0241 2 120000 228 94.9669 3 120000 281 94.9567 www.integral-concepts.com 113 ©2012 Copyright
- 116. Demonstration Test Plans Test Units vs Test Time 772.775 0 Failures 1 Failures 2 Failures 3 Failures 639.853 506.932 N u m b e r o f T e s t U n i ts 374.010 241.088 Steven Wachs integral Concepts, Inc. 10/28/2011 2:25:16 PM 108.167 80000.000 88000.000 96000.000 104000.000 112000.000 120000.000 Test Time www.integral-concepts.com 114 ©2012 Copyright
- 117. Agenda Reliability Concepts & Reliability Data Probability, Statistics, and Distributions Assessing & Selecting Models Estimating Reliability Statistics Systems Reliability Reliability Test Planning Accelerated Life Testing Warranty Analysis www.integral-concepts.com 115 ©2012 Copyright
- 118. Introduction to ALT Purpose:To estimate reliability on a timely basis Inducefailures sooner by testing at accelerated stress conditions Extrapolateresults obtained at accelerated conditions to use conditions (using acceleration models) Focus on one or a small number of failure modes www.integral-concepts.com 116 ©2012 Copyright
- 119. ALT Models (2 parts) www.integral-concepts.com 117 ©2012 Copyright
- 120. Life is a Function of Time and Stress www.integral-concepts.com 118 ©2012 Copyright
- 121. Life-Stress Relationship ReliaSoft AL TA 7 - www.ReliaSoft.com Lif e vs Stress 100000.000 Life Data 1 Ey ring Weibull 323 F=30 | S=0 Eta L ine 1% 99% 393 Stress Lev el Points Eta Point Imposed Pdf 408 Stress Lev el Points Eta Point Imposed Pdf 423 Stress Lev el Points Eta Point Imposed Pdf L i fe 10000.000 Steven Wachs integral Concepts, Inc. 7/7/2011 3:07:17 PM 1000.000 300.000 328.000 356.000 384.000 412.000 440.000 Temperature Beta=4.2918; A=-11.0878; B=1454.0864 www.integral-concepts.com 119 ©2012 Copyright
- 122. Accelerated Stress Testing Combination of Statistical Modeling and understanding of Physics of Failure Care must be taken in designing tests to yield useful information ALT models should be refined based on correlation to actual results obtained at normal use conditions www.integral-concepts.com 120 ©2012 Copyright
- 123. Accelerated Life Testing - Topics Purpose and Key Concepts Accelerated Life Test Models One, Two, and Multiple Stress Models ALT Test Planning Accelerated Degradation Models Pitfalls, Guidelines, and Examples www.integral-concepts.com 121 ©2012 Copyright
- 124. Introduction to ALT Purpose: To estimate reliability on a timely basis Induce failures sooner by testing at accelerated conditions Extrapolate results obtained at accelerated conditions to use conditions (using acceleration models) Focus on one or a small number of failure modes www.integral-concepts.com 122 ©2012 Copyright
- 125. Types of Accelerated Testing Accelerated Life Testing – Units tested until failure – Accelerating factor(s) are used to shorten the time to failure Accelerated Degradation Testing – Accelerating factor(s) are used to promote degradation – Amount of degradation observed during test – Degradation data used to predict actual time to failure at stressed conditions www.integral-concepts.com 123 ©2012 Copyright
- 126. Accelerating Methods1. Increase Usage Rate – Increase usage rate from normal usage rate – ex. Car door hinges have median lifetime of 44,000 cycles (15 years at 8 cycles per day) – Increasing rate to 5000 cycles per day will reduce median lifetime to 9 days. – Assumes TTF is independent of usage rate – Need to avoid unintended “stress” (e.g. temp) caused by higher usage rate www.integral-concepts.com 124 ©2012 Copyright
- 127. Accelerating Methods2. Test Under Stress Conditions • Test at higher levels of one or multiple stress factors • Common stress factors – temperature – thermal cycling – voltage – pressure – mechanical load – humidity www.integral-concepts.com 125 ©2012 Copyright
- 128. Types of Stress Loading www.integral-concepts.com 126 ©2012 Copyright
- 129. Accelerated Life Test Models ALT Models have 2 parts1. Stochastic Part • failure time distribution at each level of stress • Use distribution fitting to fit appropriate models (Weibull, lognormal, etc.) at each level of stress2. Structural Part • Life-stress relationship • Use regression models to relate the stress variable to the Time To Failure Distribution www.integral-concepts.com 127 ©2012 Copyright
- 130. ALT Models have 2 parts www.integral-concepts.com 128 ©2012 Copyright
- 131. Acceleration Models Acceleration models relate accelerating factors (e.g. temp, voltage) to the TTF distribution. Model depends on acceleration method (usage or stress) and the type of stress Physical models are based on physical or chemical theory that describes the failure causing process www.integral-concepts.com 129 ©2012 Copyright
- 132. Life-Stress Models Increased stress promotes earlier failures and life is predicted as a function of time and stress Common stress factors include: – Temperature, Load, Pressure, Voltage, Current, Thermal cycling, etc. The models assume stress levels are positive. For temperature, use absolute temperature (Kelvin) instead of Celsius or Farenheit www.integral-concepts.com 130 ©2012 Copyright
- 133. Acceleration Factor Quantifies the degree to which a given stress accelerates failure times AF = Life at Use Condition / Life at Stress Condition Acceleration factor increases with stress www.integral-concepts.com 131 ©2012 Copyright
- 134. Acceleration Factor ReliaSoft ALTA 7 - www.ReliaSoft.com Acc el erati on Factor vs Stres s 10.000 Acceleration Factor Data 1 Arrhenius Weibull 323 F= | S= 30 0 AF Line 8.000 A c c e le r a t io n F a c t o r 6.000 4.000 2.000 Steven Wachs integral Concepts, Inc. 8/17/2011 9:47:29 PM 0.000 300.000 340.000 380.000 420.000 460.000 500.000 Temp erat u re Beta=4.2916; B=1861.6187; C=58.9848 www.integral-concepts.com 132 ©2012 Copyright
- 135. Arrhenius Model (Temp Acceleration) Commonly used for products which fail as a result of material degradation at elevated temperatures Based on a kinetic model that describes the effect of temperature on the rate of a simple chemical reaction. www.integral-concepts.com 133 ©2012 Copyright
- 136. Arrhenius Relationship Rate = rate of a chemical reaction (rate is inversely proportional to life) tempK = absolute temperature in the Kelvin scale = temp in deg C + 273.15 kB = Boltzmann’s constant = 8.6171x10-5= 1/11605 electron volts per deg C Ea = activation energy in electron volts g = a constant (Ea and g are product or material characteristics) www.integral-concepts.com 134 ©2012 Copyright
- 137. www.integral-concepts.com 135©2012 Copyright
- 138. Arrhenius Model (ALTA Formulation) Rate = rate of a chemical reaction (rate is inversely proportional to life) T = absolute temperature in Kelvin kB = Boltzmann’s constant = 8.6171x10-5= 1/11605 electron volts per deg C Ea = activation energy in electron volts C = a constant www.integral-concepts.com 136 ©2012 Copyright
- 139. Arrhenius Model (ALTA Formulation) Let: Then: www.integral-concepts.com 137 ©2012 Copyright
- 140. Arrhenius-Weibull ModelThe Weibull PDF Scale Parameter b, B, and C are estimated from the data (MLE) (the PDF is a function of time and temperature) www.integral-concepts.com 138 ©2012 Copyright
- 141. www.integral-concepts.com 139©2012 Copyright
- 142. Inverse Power Law Model Supports a variety of stress variables such as voltage, temperature, load, etc. Assumes that the product life is proportional to the inverse power of the stress induced www.integral-concepts.com 140 ©2012 Copyright
- 143. Inverse Power Law Relationship where: T(V) = TTF at a given voltage V = Voltage A = constant (product characteristic) a = constant (product characteristic) (Voltage is the acceleration variable here) www.integral-concepts.com 141 ©2012 Copyright
- 144. Inverse Power Model (ALTA Formulation)Taking logs of both sides, we have:If failure time and stress are on log scales, this is a linear relationship www.integral-concepts.com 142 ©2012 Copyright
- 145. Other Models Some 2-stress and multiple stress models will be mentioned later Many specific models have been developed (for certain materials, failure modes, and applications) although most may be modeled with general formulations. www.integral-concepts.com 143 ©2012 Copyright
- 146. Guidelines for ALT Models Acceleration Factor(s) should be chosen to accelerate failure modes The amount of extrapolation between test stresses and use condition should be minimized Different failure modes may be accelerated at different rates (best to focus on one or two modes) The available data will generally provide little power to detect model lack of fit. An understanding of the physics is important. www.integral-concepts.com 144 ©2012 Copyright
- 147. Guidelines for ALT Models Sensitivity analysis should be performed to assess the impact of changing model assumptions ALT should be planned and conducted by teams including personnel knowledgeable about the product, its use environment, the physical/chemical/mechanical aspects of the failure mode, and the statistical aspects of the design and analysis of reliability tests ALT results should be correlated with longer term tests or field data www.integral-concepts.com 145 ©2012 Copyright
- 148. Strategy for Analyzing ALT Data1. Examine the data graphically2. Generate multiple probability plots3. Fit an overall model4. Perform residual analysis5. Assess reasonableness of the model6. Utilize model for predictions (with uncertainly quantified) www.integral-concepts.com 146 ©2012 Copyright
- 149. Example – Analyzing ALT Data ALT of mylar-polyurethane insulation used in high performance electromagnets* Insulation has a characteristic dielectric strength which may degrade over time When applied voltage exceeds dielectric strength a short circuit will occur Accelerating variable is voltage*From Meeker & Escobar (1998) www.integral-concepts.com 147 ©2012 Copyright
- 150. Example – Analyzing ALT DataTime to Failure (Minutes) of Mylar-Polyurethane Insulation Voltage Stress (kV/mm) 219.0 157.1 122.4 100.3 15.0 49.0 188.0 606.0 16.0 99.0 297.0 1012.0 36.0 154.5 405.0 2520.0 50.0 180.0 744.0 2610.0 55.0 291.0 1218.0 3988.0 95.0 447.0 1340.0 4100.0 122.0 510.0 1715.0 5025.0 129.0 600.0 3382.0 6842.0 625.0 1656.0 700.0 1721.0 www.integral-concepts.com 148 ©2012 Copyright
- 151. Example – Analyzing ALT Data• TTF data collected at four stress (voltage) levels• Normal operating voltage level is 50 kV/mm• Fit appropriate model• Find 95% confidence interval for the B10 life www.integral-concepts.com 149 ©2012 Copyright
- 152. Graphical Analysis www.integral-concepts.com 150 ©2012 Copyright
- 153. Multiple Probability Plots www.integral-concepts.com 151 ©2012 Copyright
- 154. Finds the best fitting stochastic model given a specified structural model www.integral-concepts.com 152 ©2012 Copyright
- 155. Fitting the Model Model: Inverse Power LawStd. = scale parameter for Distribution: LognormalLognormal distribution Analysis: MLE Std: 1.049793128The location parameteris a function of Voltage K: 1.149419255E-012per the IPL model n: 4.289109625 LK Value: -271.4247009 Fail Susp: 36 0 www.integral-concepts.com 153 ©2012 Copyright
- 156. ReliaSoft AL TA 7 - www.ReliaSoft.com Probabi l i ty - Lognormal 99.000 Probability Data 1 Inverse Power Law Lognormal 100.3 F= | S= 8 0 Stress Level Points Stress Level Line 122.4 F= | S= 8 0 Stress Level Points Stress Level Line 157.1 F= | S= 10 0 Stress Level Points Stress Level Line 219 F= | S= 10 0 U n r e lia b ilit y Stress Level Points Stress Level Line 50 50.000 Use Level Line 10.000 5.000 Steven Wachs integral Concepts, Inc. 8/19/2011 3:15:43 PM 1.000 10.000 100.000 1000.000 10000.000 100000.000 TimeStd=1.0498; K=1.1494E-12; n=4.2891 www.integral-concepts.com 154 ©2012 Copyright
- 157. ReliaSoft AL TA 7 - www.ReliaSoft.com Us e Level Probabi l i ty Lognormal 99.000 Use Level CB@90% 2-Sided Data 1 Inverse Power Law Lognormal 50 F= | S= 36 0 Data Points Use Level Line Top CB-II Bottom CB-II U n r e lia b ilit y 50.000 10.000 5.000 Steven Wachs integral Concepts, Inc. 8/19/2011 3:22:01 PM 1.000 1000.000 10000.000 100000.000 1000000.000 1.000E+7 TimeStd=1.0498; K=1.1494E-12; n=4.2891 www.integral-concepts.com 155 ©2012 Copyright
- 158. ReliaSoft AL TA 7 - www.ReliaSoft.com R el i abi l i ty vs Ti me 1.000 Reliability CB@90% 2-Sided [R] Data 1 Inverse Power Law Lognormal 50 F= | S= 36 0 0.800 Data Points Reliability Line Top CB-II Bottom CB-II 0.600 R e lia b ilit y 0.400 0.200 Steven Wachs integral Concepts, Inc. 8/19/2011 3:25:11 PM 0.000 0.000 60000.000 120000.000 180000.000 240000.000 300000.000 TimeStd=1.0498; K=1.1494E-12; n=4.2891 www.integral-concepts.com 156 ©2012 Copyright
- 159. ReliaSoft AL TA 7 - www.ReliaSoft.com Unrel i abi l i ty vs Ti me 1.000 Unreliability Data 1 Inverse Power Law Lognormal 50 F= | S= 36 0 Data Points 0.800 Unreliability Line 0.600 U n r e lia b ilit y 0.400 0.200 Steven Wachs integral Concepts, Inc. 8/19/2011 3:19:32 PM 0.000 0.000 60000.000 120000.000 180000.000 240000.000 300000.000 TimeStd=1.0498; K=1.1494E-12; n=4.2891 www.integral-concepts.com 157 ©2012 Copyright
- 160. ReliaSoft AL TA 7 - www.ReliaSoft.com Fai l ure R ate vs Ti me 5.000E-5 Failure Rate Data 1 Inverse Power Law Lognormal 50 F= | S= 36 0 Failure Rate Line 4.000E-5 3.000E-5 F a ilu r e R a t e 2.000E-5 1.000E-5 Steven Wachs integral Concepts, Inc. 8/19/2011 3:52:01 PM 0.000 0.000 100000.000 200000.000 300000.000 400000.000 500000.000 TimeStd=1.0498; K=1.1494E-12; n=4.2891 www.integral-concepts.com 158 ©2012 Copyright
- 161. ReliaSoft AL TA 7 - www.ReliaSoft.com Li fe vs Stres s 100000.000 Life Data 1 Inverse Power Law Lognormal 50 F= | S= 36 0 Median Line 100.3 Stress Level Points 10000.000 Median Point Imposed Pdf 122.4 Stress Level Points Median Point Imposed Pdf 157.1 Stress Level Points Median Point Imposed Pdf 219 Stress Level Points L ife 1000.000 Median Point Imposed Pdf 100.000 Steven Wachs integral Concepts, Inc. 8/19/2011 4:15:19 PM 10.000 10.000 100.000 1000.000 V olt ag eStd=1.0498; K=1.1494E-12; n=4.2891 www.integral-concepts.com 159 ©2012 Copyright
- 162. ReliaSoft AL TA 7 - www.ReliaSoft.com Acc el erati on Factor vs Stres s 600.000 Acceleration Factor Data 1 Inverse Power Law Lognormal 50 F= | S= 36 0 AF Line 480.000 A c c e le r a t io n F a c t o r 360.000 240.000 120.000 Steven Wachs integral Concepts, Inc. 8/19/2011 4:17:07 PM 0.000 10.000 68.000 126.000 184.000 242.000 300.000 V olt ag eStd=1.0498; K=1.1494E-12; n=4.2891 www.integral-concepts.com 160 ©2012 Copyright

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