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Fundamentals of ReliabilityFundamentals of Reliability Engineering and Applications
Part 3 of 3
E. A. Elsayed©2011 ASQ & Presentation ElsayedPresented live on Dec 14th, 2010
http://reliabilitycalendar.org/The_Reliability Calendar/Short Courses/Shliability_Calendar/Short_Courses/Short_Courses.html
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ASQ Reliability DivisionASQ Reliability Division Short Course SeriesShort Course SeriesThe ASQ Reliability Division is pleased to present a regular series of short courses
featuring leading international practitioners, academics and consultantsacademics, and consultants.
The goal is to provide a forum for the basic andThe goal is to provide a forum for the basic and continuing education of reliability
professionals.
http://reliabilitycalendar.org/The_Reliability Calendar/Short Courses/Shliability_Calendar/Short_Courses/Short_Courses.html
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Fundamentals of Reliability Engineering and Applications
E. A. [email protected]
Rutgers University
December 14, 2010
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OutlinePart 1. Reliability Definitions
Reliability Definition…Time dependent characteristics
Failure Rate Availability MTTF and MTBF Time to First Failure Mean Residual Life Conclusions
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OutlinePart 2. Reliability Calculations
1. Use of failure data 2. Density functions 3. Reliability function 4. Hazard and failure rates
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OutlinePart 2. Reliability Calculations
1. Use of failure data 2. Density functions 3. Reliability function 4. Hazard and failure rates
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OutlinePart 3. Failure Time Distributions
1. Constant failure rate distributions 2. Increasing failure rate distributions 3. Decreasing failure rate distributions 4. Weibull Analysis – Why use Weibull?
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Empirical Estimate of F(t) and R(t)
When the exact failure times of units is known, weuse an empirical approach to estimate the reliabilitymetrics. The most common approach is the RankEstimator. Order the failure time observations (failuretimes) in an ascending order:
1 2 1 1 1... ...i i i n nt t t t t t t− + −≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤
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Empirical Estimate of F(t) and R(t)
is obtained by several methods
1. Uniform “naive” estimator
2. Mean rank estimator
3. Median rank estimator (Bernard)
4. Median rank estimator (Blom)
( )iF t
in
1i
n +0 30 4..
in−+
3 81 4
//
in−+
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[ ][ ]
[ ]
( ) 1 ( ) 1 exp
1- ( ) exp 1 exp
1 ( )1ln
1 ( )1 ln ln ln ln
1 ( )ln ln
λ
λ
λ
λ
λ
λ
= − = − −
= −
=−
=−
⇒ = +−
= += +
F t R t t
F t t
tF t
tF t
tF t
y ty a bx
Exponential Distribution
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Median Rank Calculations
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i t (i) t(i+1) F=(i-0.3)/(n+0.4) R=1-F f(t) h(t)0 0 70 0 1 0.0014 0.00141 70 150 0.067307692 0.9327 0.0013 0.00142 150 250 0.163461538 0.8365 0.001 0.00133 250 360 0.259615385 0.7404 0.0009 0.00134 360 485 0.355769231 0.6442 0.0008 0.00135 485 650 0.451923077 0.5481 0.0006 0.00126 650 855 0.548076923 0.4519 0.0005 0.00127 855 1130 0.644230769 0.3558 0.0004 0.00128 1130 1540 0.740384615 0.2596 0.0002 0.00129 1540 - 0.836538462 0.1635
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Failure Rate
10
0
0.0004
0.0008
0.0012
0.0016
0.002
0 1 2 3 4 5 6 7 8 9
Failu
re R
ate
Time
Failure Rate
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Probability Density Function
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0
0.0002
0.0004
0.0006
0.0008
0.001
0.0012
0.0014
0.0016
0 1 2 3 4 5 6 7 8 9
Prob
abili
ty D
ensi
ty F
unct
ion
Time
Probability Density Function
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Reliability Function
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0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5 6 7 8 9 10
Rel
iabi
lity
Time
Reliability Function
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Exponential Distribution: Another Example
Given failure data:
Plot the hazard rate, if constant then use the exponential distribution with f(t), R(t) and h(t) as defined before.
We use a software to demonstrate these steps.
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Plot of the Data
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Exponential Fit
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Exponential Analysis
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Weibull Model
• Definition
1
( ) exp 0, 0, 0t tf t tβ β
β β ηη η η
− = − > > ≥
( ) exp 1 ( )tR t F tβ
η
= − = −
1
( ) ( ) / ( ) tt f t R tβ
βλη η
−
= =
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Weibull Model Cont.
1/0
1(1 )tMTTF t e dtβη ηβ
∞ −= = Γ +∫
22 2 1(1 ) (1 )Var η
β β
= Γ + − Γ +
1/Median life ((ln 2) )βη=
• Statistical properties
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Versatility of Weibull Model
Hazard rate:
Time t
1β =
Constant Failure Rate Region
Haz
ard
Rat
e
0
Early Life Region
0 1β< <
Wear-Out Region
1β >
1
( ) ( ) / ( ) tt f t R tβ
βλη η
−
= =
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Weibull Model
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Weibull Analysis: Shape Parameter
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Weibull Analysis: Shape Parameter
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Weibull Analysis: Shape Parameter
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Normal Distribution
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Failure Data
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Table 1: Failure Data
Design A Design BSample # Cycles Sample # Cycles
1 726,044 11 529,0822 615,432 12 729,9573 508,077 13 650,5704 807,863 14 445,8345 755,223 15 343,2806 848,953 16 959,9037 384,558 17 730,0498 666,686 18 730,6409 515,201 19 973,224
10 483,331 20 258,006
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( ) 1 ( ) 1 exp
1- ( ) exp
log
ln (1- ( )) =
1 ln ln ln ln1 ( )
tF t R t
tF t
Taking the
tF t
tF t
β
β
β
η
η
η
β β η
= − = − −
= −
−
⇒ = −−
Linearization of the Weibull Model
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2
1 ln ln ln ln1 ( )
,
using Excel
tF t
This is an equation of straight liney a bxUse linear regression obtain a and b by solving
y n a b x
xy a x b x
or by
β β η⇒ = −−
= +
= +
= +∑ ∑∑ ∑ ∑
Linearization of the Weibull Model
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( ) 1 ( ) 1 exp
1 ln ln ln ln1 ( )
tF t R t
tF t
β
η
β β η
= − = − −
⇒ = −−
Calculations using Excel
• Weibull Plot
is linear function of ln(time).
• Estimate at ti using Bernard’s Formulaˆ ( )iF t
0.3ˆ ( )0.4i
iF tn−
=+
For n observed failure time data 1 2( , ,..., ,... )i nt t t t
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Linearization of the Weibull Model
Design A Cycles Rank
Median Ranks 1/(1-Median Rank) ln(ln(1/(1-Median Rank))) ln(Design A Cycles)
384,558 1 0.067307692 1.072164948 -2.663843085 12.8598499483,331 2 0.163461538 1.195402299 -1.72326315 13.088457508,077 3 0.259615385 1.350649351 -1.202023115 13.13838829515,201 4 0.355769231 1.552238806 -0.821666515 13.15231239615,432 5 0.451923077 1.824561404 -0.508595394 13.33007974666,686 6 0.548076923 2.212765957 -0.230365445 13.41007445726,044 7 0.644230769 2.810810811 0.032924962 13.4953659755,223 8 0.740384615 3.851851852 0.299032932 13.53476835807,863 9 0.836538462 6.117647059 0.593977217 13.60214777848,953 10 0.932692308 14.85714286 0.992688929 13.6517591
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Linearization of the Weibull Model
-3-2.5
-2-1.5
-1-0.5
00.5
11.5
12.8 13 13.2 13.4 13.6 13.8
ln(ln
(1/(1
-Med
ian
Ran
k)))
ln(Design A Cycles)
Line Fit Plot
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Linearization of the Weibull ModelSUMMARY OUTPUT
Regression StatisticsMultiple R 0.98538223R Square 0.97097815Adjusted R Square 0.96735041Standard Error 0.20147761Observations 10
ANOVAdf SS
Regression 1 10.86495309Residual 8 0.324745817Total 9 11.18969891
Coefficients Standard ErrorIntercept -57.1930531 3.464488033ln(Design A Cycles) 4.2524822 0.259929377
Beta (or Shape Parameter) =4.25Alpha (or Characteristic Life) =693,380
lnβ η
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Reliability Plot
.0000
.1000
.2000
.3000
.4000
.5000
.6000
.7000
.8000
.9000
1.0000
0 200,000 400,000 600,000 800,000 1,000,000 1,200,000 1,400,000
Surv
ival
Pro
babi
lity
Cycles
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Plots of the Data
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Test for Weibull Fit
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Parameters for Weibull
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Example 2: Input Data
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Example 2: Plots of the Data
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Example 2: Weibull Fit
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Example 2:Test for Weibull Fit
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Example 2: Parameters for Weibull
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Versatility of Weibull Model
Hazard rate:
Time t
1β =
Constant Failure Rate Region
Haz
ard
Rat
e
0
Early Life Region
0 1β< <
Wear-Out Region
1β >
1
( ) ( ) / ( ) tt f t R tβ
βλη η
−
= =
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( ) 1 ( ) 1 exp
1 ln ln ln ln1 ( )
tF t R t
tF t
β
η
β β η
= − = − −
⇒ = −−
Graphical Model Validation
• Weibull Plot
is linear function of ln(time).
• Estimate at ti using Bernard’s Formulaˆ ( )iF t
0.3ˆ ( )0.4i
iF tn−
=+
For n observed failure time data 1 2( , ,..., ,... )i nt t t t
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Example - Weibull Plot
• T~Weibull(1, 4000) Generate 50 data
10-5
100
105
0.01
0.02
0.05
0.10
0.25
0.50
0.75 0.90 0.96 0.99
Data
Prob
abilit
y
Weibull Probability Plot
0.632
η
βIf the straight line fits the data, Weibull distribution is a good model for the data
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