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**Intro Reliability Growth**

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

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**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

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**Identified Deficiencies**

Beginnings 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. Identified Deficiencies Design Failure Analysis Test

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**Graphs . . . . . . . . . . log-log paper graphing Duane Postulate:**

The cumulative MTBF versus cumulative operating time is a straight line on log-log paper. . . . . . . Normal graphing . . . .

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**Continuous Growth Continuous means time.**

You can plot failure rate or MTBF against the total test hours.

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Discrete Growth Discrete means trials.

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**Discrete Growth Reliability Growth follows a Learning Curve approach.**

Note: More rapid growth occurs earlier in the process then flattens out!

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**Why Reliability Growth?**

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**Example A System has 18 Failures in 177 Trials 1 2 3 4 5 6 7 8 9 10 11**

12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177

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Example A system has 18 failures in 177 trials. The failures are listed the tables below. Failure Trial 1 6 2 7 3 14 4 16 5 26 30 38 8 39 9 51 Failure Trial 10 55 11 64 12 71 13 79 14 98 15 108 16 129 17 145 18 148

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**Trials Between Failures Trials Between Failures**

Example There appears to be reliability growth. Failure Trial Trials Between Failures 1 6 2 7 3 14 4 16 5 26 10 30 38 8 39 9 51 12 Failure Time Trials Between Failures 10 55 4 11 64 9 12 71 7 13 79 8 14 98 19 15 108 16 129 21 17 145 18 148 3 More trials between failures. Less trials between failures.

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Example Applying Reliability Growth Methodology, we get the following curve: . 0.9254

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Example Applying Reliability Growth Methodology, we get the following curve: . 0.9254

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**Why Reliability Growth?**

Saves Assets Reduces Test Time Saves $$$$$$$

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**for reliability growth**

Duane Model Power Law Formulation for reliability growth

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**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. MTBFCum = Cumulative Mean-Time-Between-Failure t = Time K = Constant for Power Law Equation α = Growth parameter

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**Duane Postulate MTBF Time (or Trial), t Times MTBFCum t1 M1 t2 M2 t3**

Slope, α Time (or Trial), t MTBF (Ln(t3), Ln(M3)) (Ln(t2), Ln(M2)) (Ln(t1), Ln(M1))

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**Duane Postulate Linear relationship: y = αx + b**

Has a linear log-log relationship!

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**Calculating α the growth rate**

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**Calculating α (the growth rate)**

Time (hrs) Total Failures First reading 500 5 Last reading 4000 20 We will determine α from these two readings.

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**Calculating α (the growth rate)**

Time (hrs) Total Failures First reading 500 5 Last reading 4000 20 We will determine α from these two readings.

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**Calculating α (the growth rate)**

Time (hrs) Total Failures MTBF First reading 500 5 100 Last reading 4000 20 200 First calculate the cumulative MTBF for each reading.

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**Calculating α (the growth rate)**

Time (hrs) Total Failures MTBF Ln(Time) Ln(MTBF) First reading 500 5 100 Ln(500) Ln(100) Last reading 4000 20 200 Ln(4000) Ln(200) Take logs of the readings.

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**Calculating α (the growth rate)**

Time (hrs) Total Failures MTBF Ln(Time) Ln(MTBF) First reading 500 5 100 Ln(500) Ln(100) Last reading 4000 20 200 Ln(4000) Ln(200) Slope, α x-axis y-axis ( Ln(4000) , Ln(200) ) Plot the logs of the readings. ( Ln(500) , Ln(100) )

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**Calculating α (the growth rate)**

Time (hrs) Total Failures MTBF Ln(Time) Ln(MTBF) First reading 500 5 100 Ln(500) Ln(100) Last reading 4000 20 200 Ln(4000) Ln(200) α = 0.33 x-axis y-axis ( , ) ( , )

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**Calculating α (the growth rate)**

Growth is indicated when 0 < α < 1 α = 0.33 x-axis y-axis

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**Duane Parameters α = Growth parameter TI = Initial test time**

MI = Initial MTBF MF = Final MTBF Ttotal = Total time These parameters go into the Duane equation. If you know 4 of the parameters, you can calculate the other.

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**Sensitivity of α α .40 TI 100 MI 50 MF 150 TTotal ?**

What is the Total Test time if we are given these 4 parameters? α .40 TI 100 MI 50 MF 150 TTotal ? α = Growth parameter TI = Initial test time MI = Initial MTBF MF = Final MTBF Ttotal = Total time

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**Sensitivity of α α .40 TI 100 MI 50 MF 150 Ttotal 435**

How does changing the growth parameter α affect the total test time? α .40 TI 100 MI 50 MF 150 Ttotal 435 α = Growth parameter TI = Initial test time MI = Initial MTBF MF = Final MTBF Ttotal = Total time

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**Sensitivity of α α .40 .27 .46 .64 TI 100 MI 50 MF 150 Ttotal 435**

How does changing the growth parameter α affect the total test time? α .40 .27 .46 .64 TI 100 MI 50 MF 150 Ttotal 435 α = Growth parameter TI = Initial test time MI = Initial MTBF MF = Final MTBF Ttotal = Total time

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**Sensitivity of α α .40 .27 .46 .64 TI 100 MI 50 MF 150 Ttotal 435 1823**

How does changing the growth parameter α affect the total test time? α .40 .27 .46 .64 TI 100 MI 50 MF 150 Ttotal 435 1823 285 113 The α is very sensitive to the Total Time! α = Growth parameter TI = Initial test time MI = Initial MTBF MF = Final MTBF Ttotal = Total time

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**Instantaneous vs Cumulative**

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

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**Inst vs. Cum MTBF Failure Number Failure Time 1 10 2 40 3 90 4 160 5**

250 What is the true estimate of the MTBF at 250 hours?

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**Inst vs. Cum MTBF Failure Number Failure Time MTBFCum 1 10 2 40 20 3**

90 30 4 160 5 250 50 Is the MTBF 50 at time 250?

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**Inst vs. Cum MTBF Failure Number Failure Time MTBFCum**

Time Between Failures 1 10 2 40 20 30 3 90 50 4 160 70 5 250 Or would you say the MTBF is 90 at 250 hours?

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**Inst vs. Cum MTBF Failure Number Failure Time MTBFCum**

Time Between Failures MTBFInst 1 10 31 2 40 20 30 43 3 90 50 52 4 160 70 59 5 250 66 Applying a Reliability Growth Tracking Model from AMSAA or ReliaSoft’s RGA software tool will give these numbers.

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**Inst vs. Cum MTBF Failure Number Failure Time MTBFCum**

Time Between Failures MTBFInst 1 10 31 2 40 20 30 43 3 90 50 52 4 160 70 59 5 250 66 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.

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

MTBFInst MTBF 100 MTBFCum 10 10 100 1000 10,000 Time (or Test), t

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**This is how the graphs look In standard Cartesian coordinate**

Inst vs. Cum MTBF This is how the graphs look In standard Cartesian coordinate 300 MTBF MTBFInst 200 100 MTBFCum 500 1000 1500 2000 Time (or Test), t

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Exercise 10 system failures occurred after 500 hours of reliability growth testing, with a calculated growth parameter of 0.40. What is the system’s instantaneous MTBF?

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Exercise 10 system failures occurred after 500 hours of reliability growth testing, with a calculated growth parameter of 0.40. What is the system’s instantaneous MTBF?

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Exercise 10 system failures occurred after 500 hours of reliability growth testing, with a calculated growth parameter of 0.40. What is the system’s instantaneous MTBF?

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**Reliability Growth Formulas**

Failure Rate MTBF Reliability

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

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**Failure Rate Formula rI = Initial failure rate**

tI = Initial time corresponding to rI α = Growth rate parameter Initial Conditions

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**MTBF Formula MI = Initial MTBF tI = Initial time corresponding to MI**

α = Growth rate parameter Initial Conditions

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**Reliability (Discrete)**

RI = Initial Reliability NI = Initial number of trials corresponding to RI α = Growth rate parameter Initial Conditions

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

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**Deriving r(t) Formula First, start with the Duane Postulate.**

K = Constant for Power Law Equation

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**Deriving r(t) Formula tI is the Initial Test Time.**

Insert initial conditions MI at TI , and solve for K. tI is the Initial Test Time. MI is the Initial MTBF at time tI.

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Deriving r(t) Formula Now substitute for K.

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

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

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

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**Deriving M(t) Formula MI = Initial MTBF**

tI = Initial time corresponding to MI α = Growth rate parameter

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

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**The Sensitivity of Duane’s Initial Conditions TI and MI on the Total Test Time.**

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**Sensitivity of Initial Time**

What if we increase the initial time for a planning curve? TI 100 150 200 250 α .40 MI 50 MF Ttotal 435 ??? α = Growth parameter TI = Initial test time MI = Initial MTBF MF = Final MTBF Ttotal = Total time

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**Sensitivity of Initial Time**

What if we increase the initial time for a planning curve? TI 100 150 200 250 α .40 MI 50 MF Ttotal 435 652 869 1087 A higher initial time significantly increases Ttotal! Why?

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**Sensitivity of Initial Time**

100 250 α .40 MI 50 MF 150 Ttotal 435 1087 150 MTBF 100 50 TI 250 500 750 1000 Time

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**Sensitivity of Initial Time**

100 250 α .40 MI 50 MF 150 Ttotal 435 1087 Growth is more rapid the smaller TI is! 150 MTBF 100 50 TI 250 500 750 1000 Time

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**Sensitivity of Initial MTBF**

What if we change the initial MTBF for a planning curve? MI 50 25 70 85 α .40 TI 100 MF 150 Ttotal 435 ??? α = Growth parameter TI = Initial test time MI = Initial MTBF MF = Final MTBF Ttotal = Total time

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**Sensitivity of Initial MTBF**

What if we change the initial MTBF for a planning curve? MI 50 25 70 85 α .40 TI 100 MF 150 Ttotal 435 2459 187 115 A higher initial MTBF significantly decreases Ttotal!

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**Deriving Reliability Formula**

RI = Initial Reliability NI = Initial number of trials corresponding to RI α = Growth rate parameter

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**Deriving Reliability Formula**

r is the failure rate. Rcum = Cumulative Reliability F = Number of Failures N = Number of Trials

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**Deriving Reliability Formula**

Recall failure rate formula.

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**Deriving Reliability Formula**

Subtract from 1.

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**Deriving Reliability Formula**

Make substitutions.

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**Exercise 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?

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**Exercise 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?

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Inst / Cum Conversions

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**for reliability growth planning**

AMSAA-Crow Model Projection Method for reliability growth planning

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**Discrete PM2 Growth Plan Example**

RR = Sims-Reliability Growth (TE Class)

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**Continuous PM2 Growth Plan Example**

MGP = 782 500 415 MI = 190 MR = 200 Sims-Reliability Growth (TE Class)

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**Continuous Curve Equation**

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

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**Discrete Curve Equation**

Discrete curve is plotted using this equation. R(N) = System Reliability at trial N. RA = The portion of the system reliability not impacted by the correction action effort RB = 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

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**Management Strategy Factor**

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

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**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.

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**Management Strategy Factor**

λA = Failure rate of A-modes λB = Failure rate of B-modes λA + λB = Overall system failure rate

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**Management Strategy Factor**

Example: What is the MS here? Failure mode Failure mode rate Mode Type 1 0.027 B 2 0.015 3 0.033 4 0.001 A 5 0.013

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**Management Strategy Factor**

Example: What is the MS here? Failure mode Failure mode rate Mode Type 1 0.027 B 2 0.015 3 0.033 4 0.001 A 5 0.013 Total B-modes 0.088 Total System 0.089

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**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.

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**μ, Fix Effectiveness Factor**

Number of tests = 20 Successful tests = 18 What is the reliability? Software Failure X Hardware Failure X

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**μ, Fix Effectiveness Factor**

Number of tests = 20 Successful tests = 18 Software Failure X Hardware Failure X

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**μ, Fix Effectiveness Factor**

Number of tests = 20 Successful tests = 18 What is the updated reliability? Software Failure X μ1 = 100% Hardware Failure X μ2 = 75%

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**μ, Fix Effectiveness Factor**

Number of tests = 20 Successful tests = 18 Software X 100% Fix Hardware X 75% Fix

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**μ, Fix Effectiveness Factor**

Another Example: Say the average μ is 0.75 (or 75%). What is the updated System Failure Rate? Failure mode Failure mode rate Mode Type 1 0.027 B 2 0.015 3 0.033 4 0.001 A 5 0.013 λA = λB = λSystem =

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**μ, Fix Effectiveness Factor**

Another Example: Say the average μ is 0.75 (or 75%). What is the updated System Failure Rate? Failure mode Failure mode rate Mode Type 1 0.027 B 2 0.015 3 0.033 4 0.001 A 5 0.013 Original λA = λB = λSystem = Updated λA = λB = * ( ) = λSystem =

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**Shape Parameter, β β = Shape parameter TT = Total Test Time**

MG = MTBF Goal MGP = MTBF Growth Potential MI = Initial MTBF

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Shape Parameter, β η = Shape parameter of the beta distribution representing pseudo trials NT = Total Number of Trials RG = Reliability Goal RGP = Reliability Growth Potential RI = Initial Reliability

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**MGP = MTBF Growth Potential The theoretical upper limit on MTBF**

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**MGP = MTBF Growth Potential The theoretical upper limit on MTBF**

For example: MS = 0.95 μ = 0.80 MI = 190

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**PM2 Curve Equation MS = Management Strategy.**

RA = The portion of the system reliability not impacted by the correction action effort MS = Management Strategy. Fraction of failures to be addressed by corrective action. Medium Risk Range 0.90 – 0.96. RI = Initial Reliability

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**PM2 Curve Equation MS = Management Strategy.**

RB = The portion of the system reliability addressed by the correction action effort MS = Management Strategy. Fraction of failures to be addressed by corrective action. Medium Risk Range 0.90 – 0.96. RI = Initial Reliability

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**Management Strategy Factor**

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

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**PM2 Growth Plan RGP = Reliability Growth Potential**

n = Shape parameter of the beta distribution representing pseudo trials RGP = Reliability Growth Potential RG = Reliability Goal (to meet requirement) NT = Total trials before going into IOT phase

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**Reliability Growth Potential**

RGP = Reliability Growth Potential The theoretical upper limit on system reliability

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**Reliability Growth Potential**

RGP = Reliability Growth Potential The theoretical upper limit on system reliability For example: MS = 0.95 μ = 0.80 MI = 190

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**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.

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**ASMSA-Crow/Duane Equations**

1. Single Shot Systems Expected Failures: 2. Continuously Operating Systems Expected Failures:

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