Fundamentals of ReliabilityFundamentals of Reliability Engineering and Applications

Part 3 of 3

E. A. Elsayed©2011 ASQ & Presentation ElsayedPresented live on Dec 14th, 2010

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Fundamentals of Reliability Engineering and Applications

E. A. Elsayedelsayed@rci.rutgers.edu

Rutgers University

December 14, 2010

OutlinePart 1. Reliability Definitions

Reliability Definition…Time dependent characteristics

Failure Rate Availability MTTF and MTBF Time to First Failure Mean Residual Life Conclusions

2

OutlinePart 2. Reliability Calculations

1. Use of failure data 2. Density functions 3. Reliability function 4. Hazard and failure rates

3

OutlinePart 2. Reliability Calculations

1. Use of failure data 2. Density functions 3. Reliability function 4. Hazard and failure rates

4

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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6

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−+

8

[ ][ ]

[ ]

( ) 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

8

Median Rank Calculations

9

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

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

Probability Density Function

11

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

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

13

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.

13

14

Input Data

14

15

Plot of the Data

15

16

Exponential Fit

16

Exponential Analysis

18

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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19

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

19

20

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

−

= =

20

21

Weibull Model

21

22

Weibull Analysis: Shape Parameter

22

23

Weibull Analysis: Shape Parameter

23

24

Weibull Analysis: Shape Parameter

24

25

Normal Distribution

25

26

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

2828

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

29

( ) 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

29

3030

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

3131

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

3232

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

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

Input Data

Plots of the Data

Weibull Fit

Test for Weibull Fit

Parameters for Weibull

Weibull Analysis

Example 2: Input Data

Example 2: Plots of the Data

Example 2: Weibull Fit

Example 2:Test for Weibull Fit

Example 2: Parameters for Weibull

Weibull Analysis

46

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

−

= =

46

47

( ) 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

47

48

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

48