presented by: mark e. sims reliability s&t engineer aviation and missile research, development...
TRANSCRIPT
Presented by:
Mark E. SimsReliability S&T Engineer
Aviation and Missile Research, Development and Engineering Center
UNCLASSIFIED
Intro Reliability GrowthIntro Reliability Growth
"Approved for public release; distribution unlimited. Review completed by the AMRDEC Public Affairs Office 11 Oct 2013; PR0073."
2
Mil-HDBK-189 DefinitionMil-HDBK-189 Definition
Reliability GrowthThe 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
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
Design
TestFailure
Analysis
Identified Deficiencies
4
log-log paper graphing
Normal graphing
1 10 100 1000 100001
10
100
1000Reliability Growth Chart
Cumulative Duane
Instantaneous Duane
Test Hours
MT
BF
0 200 400 600 800 1000 1200 1400 16000
20
40
60
80
100
120Reliability Growth Chart
Cumulative Duane
Instantaneous Duane
Test Hours
MT
BF
..
..
.
. .. .
.
Graphs
Duane Postulate:The cumulative MTBF versus cumulative operating time is a straight line on log-log paper.
5
Continuous GrowthContinuous Growth
0 500 1000 1500 2000 2500 3000 3500 4000 45000.000
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
Reliability Growth Chart
Test Hours
Fa
ilure
Ra
te
Continuous means time.
0 500 1000 1500 2000 2500 3000 3500 4000 45000
100
200
300
400
500
600
700
Reliability Growth Chart
Test Hours
MT
BF
You can plot failure rate or MTBF against the total test hours.
6
Discrete GrowthDiscrete Growth
0 20 40 60 80 100 120 140 160 180 20060.0%
65.0%
70.0%
75.0%
80.0%
85.0%
90.0%
95.0%
Reliability Chart
Trials
Re
lia
bil
ity Discrete means trials.
7
Discrete GrowthDiscrete Growth
0 20 40 60 80 100 120 140 160 180 20060.0%
65.0%
70.0%
75.0%
80.0%
85.0%
90.0%
95.0%
Reliability Chart
Trials
Re
lia
bil
ity
Reliability Growth follows a Learning Curve approach.
Note: More rapid growth occurs earlier in the process then flattens out!
8
Why Reliability Growth?
9
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
Example
A System has 18 Failures in 177 Trials
10
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
6 30
7 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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Failure TrialTrials
Between Failures
1 6 6
2 7 1
3 14 7
4 16 2
5 26 10
6 30 4
7 38 8
8 39 1
9 51 12
There appears to be reliability growth.
Example
Failure TimeTrials
Between Failures
10 55 4
11 64 9
12 71 7
13 79 8
14 98 19
15 108 10
16 129 21
17 145 16
18 148 3
Less trials between failures.More trials between failures.
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0 200 400 600 800 1000 12000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Trials
Reliability
0.9254
Example
Applying Reliability Growth Methodology, we get the following curve:
.
13
0 200 400 600 800 1000 12000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Trials
Reliability
Note: Reliability without applying growth is1 – (18 / 177) = 0.8983
0.9254
Example
Applying Reliability Growth Methodology, we get the following curve:
.
14
Why Reliability Growth?Saves Assets
Reduces Test TimeSaves $$$$$$$
15
Duane ModelPower Law Formulation
for reliability growth
16
Duane PostulateDuane 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
17
Duane PostulateDuane Postulate
Slope, α
Time (or Trial), t
MT
BF
(Ln(t1), Ln(M1))
(Ln(t3), Ln(M3))
(Ln(t2), Ln(M2))
Times MTBFCum
t1 M1
t2 M2
t3 M3
18
Duane PostulateDuane Postulate
Linear relationship:y = αx + b
Has a linear log-log relationship!
19
Calculating αthe growth rate
Calculating αthe growth rate
20
Calculating α (the growth rate)
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.
21
We will determine α from these two readings.
Time (hrs)
Total Failures
First reading 500 5
Last reading 4000 20
Calculating α (the growth rate)
Calculating α (the growth rate)
22
First calculate the cumulative MTBF for each reading.
Time (hrs)
Total Failures
MTBF
First reading 500 5 100
Last reading 4000 20 200
Calculating α (the growth rate)
Calculating α (the growth rate)
23
Time (hrs)
Total Failures
MTBFLn(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.
Calculating α (the growth rate)
Calculating α (the growth rate)
24
Slope, α
x-axis
y-a
xis
( Ln(500) , Ln(100) )
( Ln(4000) , Ln(200) )
Time (hrs)
Total Failures
MTBFLn(Time) Ln(MTBF)
First reading 500 5 100 Ln(500) Ln(100)
Last reading 4000 20 200 Ln(4000) Ln(200)
Plot the logs of the readings.
Calculating α (the growth rate)
Calculating α (the growth rate)
25
α = 0.33
x-axis
y-a
xis
( 6.215 , 4.605 )
( 8.294 , 5.298 )
Time (hrs)
Total Failures
MTBFLn(Time) Ln(MTBF)
First reading 500 5 100 Ln(500) Ln(100)
Last reading 4000 20 200 Ln(4000) Ln(200)
Calculating α (the growth rate)
Calculating α (the growth rate)
26
α = 0.33
x-axis
y-a
xis
Growth is indicated when 0 < α < 1
Calculating α (the growth rate)
Calculating α (the growth rate)
27
Duane ParametersDuane Parameters
α = Growth parameterTI = Initial test timeMI = Initial MTBFMF = Final MTBFTtotal = Total time
• These parameters go into the Duane equation.
• If you know 4 of the parameters, you can calculate the other.
28
Sensitivity of αSensitivity of α
α = Growth parameterTI = Initial test timeMI = Initial MTBFMF = Final MTBFTtotal = Total time
What is the Total Test time if we are given these 4 parameters?
α .40
TI 100
MI 50
MF 150
TTotal ?
29
Sensitivity of αSensitivity of α
α = Growth parameterTI = Initial test timeMI = Initial MTBFMF = Final MTBFTtotal = Total time
How does changing the growth parameter α affect the total test time?
α .40
TI 100
MI 50
MF 150
Ttotal 435
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α .40 .27 .46 .64
TI 100 100 100 100
MI 50 50 50 50
MF 150 150 150 150
Ttotal 435
Sensitivity of αSensitivity of α
α = Growth parameterTI = Initial test timeMI = Initial MTBFMF = Final MTBFTtotal = Total time
How does changing the growth parameter α affect the total test time?
31
α .40 .27 .46 .64
TI 100 100 100 100
MI 50 50 50 50
MF 150 150 150 150
Ttotal 435 1823 285 113
Sensitivity of αSensitivity of α
α = Growth parameterTI = Initial test timeMI = Initial MTBFMF = Final MTBFTtotal = Total time
The α is very sensitive to the Total Time!
How does changing the growth parameter α affect the total test time?
32
Instantaneousvs
Cumulative
Instantaneousvs
Cumulative
Duane MTBF Equation
Finding the true estimate of a system’s MTBF using reliability growth.
33
FailureNumber
Failure Time
1 10
2 40
3 90
4 160
5 250
Inst vs. Cum MTBF
What is the true estimate of the MTBF at 250 hours?
34
FailureNumber
Failure Time
MTBFCum
1 10 10
2 40 20
3 90 30
4 160 40
5 250 50
Inst vs. Cum MTBF
Is the MTBF 50 at time 250?
35
FailureNumber
Failure Time
MTBFCum
Time Between Failures
1 10 10 10
2 40 20 30
3 90 30 50
4 160 40 70
5 250 50 90
Inst vs. Cum MTBF
Or would you say the MTBF is 90 at 250 hours?
36
FailureNumber
Failure Time
MTBFCum
Time Between Failures
MTBFInst
1 10 10 10 31
2 40 20 30 43
3 90 30 50 52
4 160 40 70 59
5 250 50 90 66
Inst vs. Cum MTBF
Applying a Reliability Growth Tracking Model from AMSAA or ReliaSoft’s RGA software tool will give these numbers.
37
FailureNumber
Failure Time
MTBFCum
Time Between Failures
MTBFInst
1 10 10 10 31
2 40 20 30 43
3 90 30 50 52
4 160 40 70 59
5 250 50 90 66
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
MTBFInst
MTBFCum
MT
BF
Time (or Test), t
On Log-Log Graph Paper
10 100 1000 10,000
10
100
Inst vs. Cum MTBF
39
This is how the graphs lookIn standard Cartesian coordinate
MTBFInst
MTBFCum
MT
BF
Time (or Test), t
300
100
500 1000 1500 2000
200
Inst vs. Cum MTBF
40
ExerciseExercise
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?
41
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?
ExerciseExercise
42
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?
ExerciseExercise
43
Reliability Growth FormulasReliability Growth Formulas
Failure Rate
MTBF
Reliability
44
M(t) = 1 / r(t)
MTBF is the reciprocal of the failure rate.
45
rI = Initial failure rate
tI = Initial time corresponding to rI
α = Growth rate parameter
Failure Rate FormulaFailure Rate Formula
Initial Conditions
46
MI = Initial MTBF
tI = Initial time corresponding to MI
α = Growth rate parameter
MTBF FormulaMTBF Formula
Initial Conditions
47
RI = Initial Reliability
NI = Initial number of trials corresponding to RI
α = Growth rate parameter
Reliability (Discrete)Reliability (Discrete)
Initial Conditions
48
Deriving r(t) FormulaDeriving r(t) Formula
r(t) is sometimes called the Hazard Rate.
49
Deriving r(t) FormulaDeriving r(t) Formula
K = Constant for Power Law Equation
First, start with the Duane Postulate.
50
Insert initial conditions MI at TI , and solve for K.
Deriving r(t) FormulaDeriving r(t) Formula
tI is the Initial Test Time.MI is the Initial MTBF at time tI.
51
Now substitute for K.
Deriving r(t) FormulaDeriving r(t) Formula
52
The failure rate, r, is the inverse of the MTBF, so r(t) = 1 / M(t).
Deriving r(t) FormulaDeriving r(t) Formula
53
Deriving r(t) FormulaDeriving r(t) Formula
Now we will simplify and take the derivative.
54
Deriving r(t) FormulaDeriving r(t) Formula
Now we will simplify and take the derivative.
55
MI = Initial MTBF
tI = Initial time corresponding to MI
α = Growth rate parameter
Deriving M(t) FormulaDeriving M(t) Formula
56
Deriving M(t) FormulaDeriving M(t) Formula
Recall MTBF = 1/r, so take the inverse of r(t).
57
The Sensitivity of Duane’s Initial Conditions TI and MI on the Total Test Time.
58
TI 100 150 200 250
α .40 .40 .40 .40
MI 50 50 50 50
MF 150 150 150 150
Ttotal 435 ??? ??? ???
α = Growth parameterTI = Initial test timeMI = Initial MTBFMF = Final MTBFTtotal = Total time
What if we increase the initial time for a planning curve?
Sensitivity of Initial TimeSensitivity of Initial Time
59
Sensitivity of Initial TimeSensitivity of Initial Time
What if we increase the initial time for a planning curve?
A higher initial time significantly increases Ttotal!
TI 100 150 200 250
α .40 .40 .40 .40
MI 50 50 50 50
MF 150 150 150 150
Ttotal 435 652 869 1087
Why?
60
250 500 750 1000
50
100
150
TI 100 250
α .40 .40
MI 50 50
MF 150 150
Ttotal 435 1087
Sensitivity of Initial Time
Time
MT
BF
TI
61
250 500 750 1000
50
100
150
TI 100 250
α .40 .40
MI 50 50
MF 150 150
Ttotal 435 1087
Growth is more rapid the smaller TI is!
Sensitivity of Initial TimeM
TB
F
Time
TI
62
MI 50 25 70 85
α .40 .40 .40 .40
TI 100 100 100 100
MF 150 150 150 150
Ttotal 435 ??? ??? ???
α = Growth parameterTI = Initial test timeMI = Initial MTBFMF = Final MTBFTtotal = Total time
What if we change the initial MTBF for a planning curve?
Sensitivity of Initial MTBF
63
What if we change the initial MTBF for a planning curve?
A higher initial MTBF significantly decreases Ttotal!
MI 50 25 70 85
α .40 .40 .40 .40
TI 100 100 100 100
MF 150 150 150 150
Ttotal 435 2459 187 115
Sensitivity of Initial MTBF
64
RI = Initial Reliability
NI = Initial number of trials corresponding to RI
α = Growth rate parameter
Deriving Reliability FormulaDeriving Reliability Formula
65
Deriving Reliability FormulaDeriving Reliability Formula
Rcum = Cumulative Reliability
F = Number of Failures
N = Number of Trials
r is the failure rate.
66
Deriving Reliability FormulaDeriving Reliability Formula
Recall failure rate formula.
67
Deriving Reliability FormulaDeriving Reliability Formula
Subtract from 1.
68
Deriving Reliability FormulaDeriving Reliability Formula
Make substitutions.
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
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
Inst / Cum Conversions
72
AMSAA-Crow ModelProjection Method
for reliability growth planning
73
- 50 100 150 200 250 300
0.84
0.86
0.88
0.90
0.92
0.94
0.96
0.98
1.00
RG PotentialRGP = 0.9747
TL
CAP4
CAP3
CAP2
CAP1
RG = 0.9639
RLUT = 0.9568
RDT3 = 0.9455
RDT2 = 0.9260
RDT1 = 0.8987
PM2-Discrete Reliability Growth Planning Curve
Idealized Curve DT1 DT2 DT3 LUT IOT Requirement
Trials
Rel
iab
ilit
y
RR = 0.9200
Discrete PM2 Growth Plan Example
041712-Sims-Reliability Growth (TE Class)
74
Continuous PM2 Growth Plan Example
-
500
1,0
00
1,5
00
2,0
00
2,5
00
3,0
00
3,5
00
4,0
00
4,5
00
0
100
200
300
400
500
600
700
CAP10CAP9CAP8CAP7CAP6CAP5
CAP4
CAP4
CAP3
CAP3
CAP2
CAP2
CAP1
CAP10
PM2 Continuous Reliability Growth Planning Curve
Idealized Curve Series3 Series5 Series7 Series9
Series11 Series12 Series13 Series14 Series15
Series16 Hypothetical Last Step IOT Series22 Requirement
Test Time (hours)
MT
BF
322
DT1
DT2
DT3
MG,DT = 581
LUTMG,0T = 523
MI = 190
500
MR = 200
MGP = 782
415
041712-Sims-Reliability Growth (TE Class)
75
Continuous Curve EquationContinuous 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
76
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
Discrete Curve EquationDiscrete Curve Equation
Discrete curve is plotted using this equation.
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
Management Strategy Factor
A-Mode: Failures that are not fixed.B-Mode: Failures that will have a fix.
Failure Rates
A-ModeB-Mode
A “fix” means a reliability improvement corrective action, not just a remove and replace of the same component.
79
Management Strategy Factor
λA = Failure rate of A-modesλB = Failure rate of B-modesλA + λB = Overall system failure rate
Failure Rates
A-ModeB-Mode
80
Management Strategy Factor
Failure Rates
A-ModeB-Mode
Example: What is the MS here?
Failure mode
Failure mode rate
Mode Type
1 0.027 B
2 0.015 B
3 0.033 B
4 0.001 A
5 0.013 B
81
Management Strategy Factor
Failure Rates
A-ModeB-Mode
Example: What is the MS here?
Failure mode
Failure mode rate
Mode Type
1 0.027 B
2 0.015 B
3 0.033 B
4 0.001 A
5 0.013 B
Total B-modes 0.088
Total System 0.089
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
Number of tests = 20
Successful tests = 18
Hardware Failure
Software Failure
What is the reliability?
X
X
μ, Fix Effectiveness Factor
84
Number of tests = 20
Successful tests = 18
X
X
Software Failure
Hardware Failure
μ, Fix Effectiveness Factor
85
Number of tests = 20
Successful tests = 18
X
X
What is the updated reliability?
μ1 = 100%
μ2 = 75%
Software Failure
Hardware Failure
μ, Fix Effectiveness Factor
86
Number of tests = 20
Successful tests = 18
Hardware
Software
X
X100% Fix
75% Fix
μ, Fix Effectiveness Factor
87
Failure mode
Failure mode rate
Mode Type
1 0.027 B
2 0.015 B
3 0.033 B
4 0.001 A
5 0.013 B
μ, Fix Effectiveness Factor
Another Example: Say the average μ is 0.75 (or 75%). What is the updated System Failure Rate?
λA = 0.001λB = 0.088λSystem = 0.089
88
Failure mode
Failure mode rate
Mode Type
1 0.027 B
2 0.015 B
3 0.033 B
4 0.001 A
5 0.013 B
μ, Fix Effectiveness Factor
Another Example: Say the average μ is 0.75 (or 75%). What is the updated System Failure Rate?
OriginalλA = 0.001λB = 0.088λSystem = 0.089
UpdatedλA = 0.001λB = 0.088 * (1- 0.75) = 0.022λSystem = 0.023
89
Shape Parameter, βShape Parameter, β
β = Shape parameter
TT = Total Test Time
MG = MTBF Goal
MGP = MTBF Growth Potential
MI = Initial MTBF
90
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
Shape Parameter, βShape Parameter, β
91
Growth PotentialGrowth Potential
MGP = MTBF Growth PotentialThe theoretical upper limit on MTBF
92
Growth PotentialGrowth Potential
MGP = MTBF Growth PotentialThe theoretical upper limit on MTBF
For example:MS = 0.95μ = 0.80MI = 190
93
RA = The portion of the system reliability not impacted by the correction action effort
PM2 Curve EquationPM2 Curve Equation
MS = Management Strategy. Fraction of failures to be addressed by corrective action.
Medium Risk Range 0.90 – 0.96.
RI = Initial Reliability
94
RB = The portion of the system reliability addressed by the correction action effort
PM2 Curve EquationPM2 Curve Equation
MS = Management Strategy. Fraction of failures to be addressed by corrective action.
Medium Risk Range 0.90 – 0.96.
RI = Initial Reliability
95
Management Strategy FactorManagement 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
n = Shape parameter of the beta distribution representing pseudo trials
PM2 Growth PlanPM2 Growth Plan
RGP = Reliability Growth PotentialRG = Reliability Goal (to meet requirement)NT = Total trials before going into IOT phase
97
Reliability Growth PotentialReliability Growth Potential
RGP = Reliability Growth PotentialThe theoretical upper limit on system reliability
98
Reliability Growth PotentialReliability Growth Potential
RGP = Reliability Growth PotentialThe theoretical upper limit on system reliability
For example:MS = 0.95μ = 0.80MI = 190
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
ASMSA-Crow/Duane EquationsASMSA-Crow/Duane Equations
1. Single Shot Systems Expected Failures:
NNR
NEF
1
)(1)(
2. Continuously Operating Systems Expected Failures:
TTM
TEF
1
)(1
)(