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Presented by: Mark E. Sims Reliability S&T Engineer Aviation and Missile Research, Development and Engineering Center UNCLASSIFIED Intro Reliability Growth.

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Presentation on theme: "Presented by: Mark E. Sims Reliability S&T Engineer Aviation and Missile Research, Development and Engineering Center UNCLASSIFIED Intro Reliability Growth."— Presentation transcript:

1 Presented by: Mark E. Sims Reliability S&T Engineer Aviation and Missile Research, Development and Engineering Center UNCLASSIFIED Intro Reliability Growth "Approved for public release; distribution unlimited. Review completed by the AMRDEC Public Affairs Office 11 Oct 2013; PR0073."

2 2 Mil-HDBK-189 Definition Reliability Growth The positive improvement in a reliability parameter over a period of time due to changes in product design or the manufacturing process. MIL-HDBK-189 is a Department of Army Handbook for Reliability Growth Management

3 3 J.T. Duane was an engineer at the Aerospace Electronics Department of the General Electric Company. He published a paper in 1964 that applied a “learning curve approach” to reliability monitoring. He observed that the cumulative MTBF versus cumulative operating time followed a straight line when plotted on log-log paper. The learning (i.e., growing) is accomplished through a “test, analyze, and fix” (TAAF) process. Beginnings

4 4 log-log paper graphing Normal graphing Graphs Duane Postulate: The cumulative MTBF versus cumulative operating time is a straight line on log-log paper.

5 5 Continuous Growth Continuous means time. You can plot failure rate or MTBF against the total test hours.

6 6 Discrete Growth Discrete means trials.

7 7 Discrete Growth Reliability Growth follows a Learning Curve approach. Note: More rapid growth occurs earlier in the process then flattens out!

8 8 Why Reliability Growth?

9 Example A System has 18 Failures in 177 Trials

10 10Example A system has 18 failures in 177 trials. The failures are listed the tables below. FailureTrial FailureTrial

11 11 FailureTrial Trials Between Failures There appears to be reliability growth. Example FailureTime Trials Between Failures Less trials between failures. More trials between failures.

12 Example Applying Reliability Growth Methodology, we get the following curve:.

13 Example Applying Reliability Growth Methodology, we get the following curve:.

14 14 Why Reliability Growth? Saves Assets Reduces Test Time Saves $$$$$$$

15 15 Duane Model Power Law Formulation for reliability growth

16 16 Duane Postulate During Reliability Growth, Graphing the log of time (or tests) against its corresponding log of MTBF Will be a straight line with slope α. MTBF Cum = Cumulative Mean-Time-Between-Failure t = Time K = Constant for Power Law Equation α = Growth parameter

17 17 Duane Postulate Slope, α Time (or Trial), t MTBF (Ln(t 1 ), Ln(M 1 )) (Ln(t 3 ), Ln(M 3 )) (Ln(t 2 ), Ln(M 2 )) TimesMTBF Cum t1t1 M1M1 t2t2 M2M2 t3t3 M3M3

18 18 Duane Postulate Linear relationship: y = αx + b Has a linear log-log relationship!

19 19 Calculating α the growth rate

20 20 Calculating α (the growth rate) Time (hrs) Total Failures First reading5005 Last reading We will determine α from these two readings.

21 21 We will determine α from these two readings. Time (hrs) Total Failures First reading5005 Last reading Calculating α (the growth rate)

22 22 First calculate the cumulative MTBF for each reading. Time (hrs) Total Failures MTBF First reading Last reading Calculating α (the growth rate)

23 23 Time (hrs) Total Failures MTBF Ln(Time)Ln(MTBF) First reading Ln(500)Ln(100) Last reading Ln(4000)Ln(200) Take logs of the readings. Calculating α (the growth rate)

24 24 Slope, α x-axis y-axis ( Ln(500), Ln(100) ) ( Ln(4000), Ln(200) ) Time (hrs) Total Failures MTBF Ln(Time)Ln(MTBF) First reading Ln(500)Ln(100) Last reading Ln(4000)Ln(200) Plot the logs of the readings. Calculating α (the growth rate)

25 25 α = 0.33 x-axis y-axis ( 6.215, ) ( 8.294, ) Time (hrs) Total Failures MTBF Ln(Time)Ln(MTBF) First reading Ln(500)Ln(100) Last reading Ln(4000)Ln(200) Calculating α (the growth rate)

26 26 α = 0.33 x-axis y-axis Growth is indicated when 0 < α < 1 Calculating α (the growth rate)

27 27 Duane Parameters α = Growth parameter T I = Initial test time M I = Initial MTBF M F = Final MTBF T total = Total time These parameters go into the Duane equation. If you know 4 of the parameters, you can calculate the other.

28 28 Sensitivity of α α = Growth parameter T I = Initial test time M I = Initial MTBF M F = Final MTBF T total = Total time What is the Total Test time if we are given these 4 parameters? α.40 TITI 100 MIMI 50 MFMF 150 T Total ?

29 29 Sensitivity of α α = Growth parameter T I = Initial test time M I = Initial MTBF M F = Final MTBF T total = Total time How does changing the growth parameter α affect the total test time? α.40 TITI 100 MIMI 50 MFMF 150 T total 435

30 30 α TITI 100 MIMI 50 MFMF 150 T total 435 Sensitivity of α α = Growth parameter T I = Initial test time M I = Initial MTBF M F = Final MTBF T total = Total time How does changing the growth parameter α affect the total test time?

31 31 α TITI 100 MIMI 50 MFMF 150 T total Sensitivity of α α = Growth parameter T I = Initial test time M I = Initial MTBF M F = Final MTBF T total = Total time The α is very sensitive to the Total Time! How does changing the growth parameter α affect the total test time?

32 32 Instantaneous vs Cumulative Duane MTBF Equation Finding the true estimate of a system’s MTBF using reliability growth.

33 33 Failure Number Failure Time Inst vs. Cum MTBF What is the true estimate of the MTBF at 250 hours?

34 34 Failure Number Failure Time MTBF Cum Inst vs. Cum MTBF Is the MTBF 50 at time 250?

35 35 Failure Number Failure Time MTBF Cum Time Between Failures Inst vs. Cum MTBF Or would you say the MTBF is 90 at 250 hours?

36 36 Failure Number Failure Time MTBF Cum Time Between Failures MTBF Inst Inst vs. Cum MTBF Applying a Reliability Growth Tracking Model from AMSAA or ReliaSoft’s RGA software tool will give these numbers.

37 37 Failure Number Failure Time MTBF Cum Time Between Failures MTBF Inst Inst vs. Cum MTBF Applying a Reliability Growth Tracking Model from AMSAA or ReliaSoft’s RGA software tool to get these numbers. So, 66 is the true MTBF at 250 operating hours, if reliability growth is occurring.

38 38 MTBF Time (or Test), t On Log-Log Graph Paper , Inst vs. Cum MTBF

39 39 This is how the graphs look In standard Cartesian coordinate MTBF Time (or Test), t Inst vs. Cum MTBF

40 40ExerciseExercise 10 system failures occurred after 500 hours of reliability growth testing, with a calculated growth parameter of What is the system’s instantaneous MTBF?

41 41 10 system failures occurred after 500 hours of reliability growth testing, with a calculated growth parameter of What is the system’s instantaneous MTBF?ExerciseExercise

42 42 10 system failures occurred after 500 hours of reliability growth testing, with a calculated growth parameter of What is the system’s instantaneous MTBF?ExerciseExercise

43 43 Reliability Growth Formulas Failure Rate MTBF Reliability

44 44 M(t) = 1 / r(t) MTBF is the reciprocal of the failure rate.

45 45 r I = Initial failure rate t I = Initial time corresponding to r I α = Growth rate parameter Failure Rate Formula Initial Conditions

46 46 M I = Initial MTBF t I = Initial time corresponding to M I α = Growth rate parameter MTBF Formula Initial Conditions

47 47 R I = Initial Reliability N I = Initial number of trials corresponding to R I α = Growth rate parameter Reliability (Discrete) Initial Conditions

48 48 Deriving r(t) Formula r(t) is sometimes called the Hazard Rate.

49 49 Deriving r(t) Formula K = Constant for Power Law Equation First, start with the Duane Postulate.

50 50 Insert initial conditions M I at T I, and solve for K. Deriving r(t) Formula t I is the Initial Test Time. M I is the Initial MTBF at time t I. t I is the Initial Test Time. M I is the Initial MTBF at time t I.

51 51 Now substitute for K. Deriving r(t) Formula

52 52 The failure rate, r, is the inverse of the MTBF, so r(t) = 1 / M(t). Deriving r(t) Formula

53 53 Deriving r(t) Formula Now we will simplify and take the derivative.

54 54 Deriving r(t) Formula Now we will simplify and take the derivative.

55 55 M I = Initial MTBF t I = Initial time corresponding to M I α = Growth rate parameter Deriving M(t) Formula

56 56 Deriving M(t) Formula Recall MTBF = 1/r, so take the inverse of r(t).

57 57 The Sensitivity of Duane’s Initial Conditions T I and M I on the Total Test Time.

58 58 TITI α.40 MIMI 50 MFMF 150 T total 435??? α = Growth parameter T I = Initial test time M I = Initial MTBF M F = Final MTBF T total = Total time What if we increase the initial time for a planning curve? Sensitivity of Initial Time

59 59 Sensitivity of Initial Time What if we increase the initial time for a planning curve? A higher initial time significantly increases T total ! TITI α.40 MIMI 50 MFMF 150 T total Why?

60 TITI α.40 MIMI 50 MFMF 150 T total Sensitivity of Initial Time Time MTBF TITI

61 TITI α.40 MIMI 50 MFMF 150 T total Growth is more rapid the smaller T I is! Sensitivity of Initial Time MTBF Time TITI

62 62 MIMI α.40 TITI 100 MFMF 150 T total 435??? α = Growth parameter T I = Initial test time M I = Initial MTBF M F = Final MTBF T total = Total time What if we change the initial MTBF for a planning curve? Sensitivity of Initial MTBF

63 63 What if we change the initial MTBF for a planning curve? A higher initial MTBF significantly decreases T total ! MIMI α.40 TITI 100 MFMF 150 T total Sensitivity of Initial MTBF

64 64 R I = Initial Reliability N I = Initial number of trials corresponding to R I α = Growth rate parameter Deriving Reliability Formula

65 65 Deriving Reliability Formula R cum = Cumulative Reliability F = Number of Failures N = Number of Trials r is the failure rate.

66 66 Deriving Reliability Formula Recall failure rate formula.

67 67 Deriving Reliability Formula Subtract from 1.

68 68 Deriving Reliability Formula Make substitutions.

69 69 Initially, System A has 3 failures after 100 firings. If you expect a growth rate of 0.25, what would be the expected reliability after 1000 flight tests? Exercise

70 70 Initially, System A has 3 failures after 100 firings. If you expect a growth rate of 0.25, what would be the expected reliability after 1000 flight tests? Exercise

71 71 Inst / Cum Conversions

72 72 AMSAA-Crow Model Projection Method for reliability growth planning

73 73 R R = Discrete PM2 Growth Plan Example Sims-Reliability Growth (TE Class)

74 74 Continuous PM2 Growth Plan Example M I = M R = 200 M GP = Sims-Reliability Growth (TE Class)

75 75 Continuous Curve Equation Continuous curve is plotted using this equation. MTBF(T) = System Mean-Time-Between-Failures at time T MTBF I = Initial MTBF MS = Management Strategy µ = Average Fix Effectiveness Factor (FEF) β = Shape parameter

76 76 R(N) = System Reliability at trial N. R A = The portion of the system reliability not impacted by the correction action effort R B = The portion of the system reliability addressed by the correction action effort MS = Management Strategy µ = Average Fix Effectiveness Factor (FEF) n = Shape parameter of the beta distribution representing pseudo trials Discrete Curve Equation Discrete curve is plotted using this equation.

77 77 Management Strategy Factor Management Strategy (MS) is the fraction of the overall system failure rate to be address by the corrective action plan. λ = Failure rate. For various reasons (prohibitive cost, improbability of reoccurrence), some failure modes will not have a corrective action.

78 78 Management Strategy Factor A-Mode: Failures that are not fixed. B-Mode: Failures that will have a fix. A “fix” means a reliability improvement corrective action, not just a remove and replace of the same component.

79 79 Management Strategy Factor λ A = Failure rate of A-modes λ B = Failure rate of B-modes λ A + λ B = Overall system failure rate

80 80 Management Strategy Factor Example: What is the MS here? Failure mode Failure mode rate Mode Type B B B A B

81 81 Management Strategy Factor Example: What is the MS here? Failure mode Failure mode rate Mode Type B B B A B Total B-modes Total System 0.089

82 82 μ, Fix Effectiveness Factor Mil-HDBK-189 Definition: Fix Effectiveness Factor, μ = A fraction representing the reduction in an individual initial mode failure rate due to implementation of a corrective action. Essentially Fix Effectiveness Factors discount failures. A couple examples will follow.

83 83 Number of tests = 20 Successful tests = 18 Hardware Failure Software Failure What is the reliability? X X μ, Fix Effectiveness Factor

84 84 Number of tests = 20 Successful tests = 18 X X Software Failure Hardware Failure μ, Fix Effectiveness Factor

85 85 Number of tests = 20 Successful tests = 18 X X What is the updated reliability? μ 1 = 100% μ 2 = 75% Software Failure Hardware Failure μ, Fix Effectiveness Factor

86 86 Number of tests = 20 Successful tests = 18 Hardware Software X X 100% Fix 75% Fix μ, Fix Effectiveness Factor

87 87 Failure mode Failure mode rate Mode Type B B B A B μ, Fix Effectiveness Factor Another Example: Say the average μ is 0.75 (or 75%). What is the updated System Failure Rate? λ A = λ B = λ System = 0.089

88 88 Failure mode Failure mode rate Mode Type B B B A B μ, Fix Effectiveness Factor Another Example: Say the average μ is 0.75 (or 75%). What is the updated System Failure Rate? Original λ A = λ B = λ System = Updated λ A = λ B = * ( ) = λ System = 0.023

89 89 Shape Parameter, β β = Shape parameter T T = Total Test Time M G = MTBF Goal M GP = MTBF Growth Potential M I = Initial MTBF

90 90 η = Shape parameter of the beta distribution representing pseudo trials N T = Total Number of Trials R G = Reliability Goal R GP = Reliability Growth Potential R I = Initial Reliability Shape Parameter, β

91 91 Growth Potential M GP = MTBF Growth Potential The theoretical upper limit on MTBF

92 92 Growth Potential M GP = MTBF Growth Potential The theoretical upper limit on MTBF For example: MS = 0.95 μ = 0.80 M I = 190 For example: MS = 0.95 μ = 0.80 M I = 190

93 93 R A = The portion of the system reliability not impacted by the correction action effort PM2 Curve Equation MS = Management Strategy. Fraction of failures to be addressed by corrective action. Medium Risk Range 0.90 – R I = Initial Reliability

94 94 R B = The portion of the system reliability addressed by the correction action effort PM2 Curve Equation MS = Management Strategy. Fraction of failures to be addressed by corrective action. Medium Risk Range 0.90 – R I = Initial Reliability

95 95 Management Strategy Factor A-Mode: Failures that are not fixed. B-Mode: Failures that will have a fix. λ = Failure rate. Fraction of failures to be addressed by the corrective action plan.

96 96 n = Shape parameter of the beta distribution representing pseudo trials PM2 Growth Plan R GP = Reliability Growth Potential R G = Reliability Goal (to meet requirement) N T = Total trials before going into IOT phase

97 97 Reliability Growth Potential R GP = Reliability Growth Potential The theoretical upper limit on system reliability

98 98 Reliability Growth Potential R GP = Reliability Growth Potential The theoretical upper limit on system reliability For example: MS = 0.95 μ = 0.80 M I = 190 For example: MS = 0.95 μ = 0.80 M I = 190

99 99 Summary Reliability Growth applies a “Learning Curve” Approach System must undergo Test-Analyze-And-Fix for reliability to grow. Initial Conditions are sensitive to a growth plan.

100 100 ASMSA-Crow/Duane Equations 1. Single Shot Systems Expected Failures: 2. Continuously Operating Systems Expected Failures:


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