This is a three parts lecture series. The parts will cover the basics and fundamentals of reliability engineering. Part 1 begins with introduction of reliability definition and other reliability characteristics and measurements. It will be followed by reliability calculation, estimation of failure rates and understanding of the implications of failure rates on system maintenance and replacements in Part 2. Then Part 3 will cover the most important and practical failure time distributions and how to obtain the parameters of the distributions and interpretations of these parameters. Hands-on computations of the failure rates and the estimation of the failure time distribution parameters will be conducted using standard Microsoft Excel. Part 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?

1. Fundamentals of Reliability
Fundamentals of Reliability
Engineering and Applications
Part 3 of 3
E. A. Elsayed
©2011 ASQ & Presentation Elsayed
Presented live on Dec 14th, 2010
http://reliabilitycalendar.org/The_Re
liability_Calendar/Short_Courses/Sh
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ort_Courses.html

2. ASQ Reliability Division
ASQ Reliability Division
Short Course Series
Short Course Series
The ASQ Reliability Division is pleased to
present a regular series of short courses
featuring leading international practitioners,
academics, and consultants.
academics and consultants
The goal is to provide a forum for the basic and
The goal is to provide a forum for the basic and
continuing education of reliability
professionals.
http://reliabilitycalendar.org/The_Re
liability_Calendar/Short_Courses/Sh
liability Calendar/Short Courses/Sh
ort_Courses.html

3. Fundamentals of Reliability Engineering and
Applications
E. A. Elsayed
elsayed@rci.rutgers.edu
Rutgers University
December 14, 2010
1

4. Outline
Part 1. Reliability Definitions
 Reliability Definition…Time dependent
characteristics
 Failure Rate
 Availability
 MTTF and MTBF
 Time to First Failure
 Mean Residual Life
 Conclusions
2

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

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

7. Outline
Part 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?
5

8. Empirical Estimate of F(t) and R(t)
When the exact failure times of units is known, we
use an empirical approach to estimate the reliability
metrics. The most common approach is the Rank
Estimator. Order the failure time observations (failure
times) in an ascending order:
t 1 ≤ t 2 ≤ ... ≤ t i −1 ≤ t i ≤ t i +1 ≤ ... ≤ t n −1 ≤ t n
6

9. Empirical Estimate of F(t) and R(t)
F (ti ) is obtained by several methods
i
1. Uniform “naive” estimator
n
i
2. Mean rank estimator n +1
i − 0.3
3. Median rank estimator (Bernard) n + 0. 4
i −3/8
4. Median rank estimator (Blom)
n +1/ 4
7

10. Exponential Distribution
F (t ) = R(t ) = exp [ −λt ]
1− 1−
1- F (t ) exp [ −λt ]
=
1
= exp [ λt ]
1 − F (t )
1
ln = λt
1 − F (t )
1
⇒ ln ln ln t + ln λ
=
1 − F (t )
= ln λ + ln t
y
y= a + bx 8

11. Median Rank Calculations
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 0.0014
1 70 150 0.067307692 0.9327 0.0013 0.0014
2 150 250 0.163461538 0.8365 0.001 0.0013
3 250 360 0.259615385 0.7404 0.0009 0.0013
4 360 485 0.355769231 0.6442 0.0008 0.0013
5 485 650 0.451923077 0.5481 0.0006 0.0012
6 650 855 0.548076923 0.4519 0.0005 0.0012
7 855 1130 0.644230769 0.3558 0.0004 0.0012
8 1130 1540 0.740384615 0.2596 0.0002 0.0012
9 1540 - 0.836538462 0.1635 9

12. Failure Rate
Failure Rate
0.002
0.0016
Failure Rate
0.0012
0.0008
0.0004
0
0 1 2 3 4 5 6 7 8 9
Time
10

13. Probability Density Function
Probability Density Function
0.0016
0.0014
0.0012
Probability Density Function
0.001
0.0008
0.0006
0.0004
0.0002
0
0 1 2 3 4 5 6 7 8 9
Time
11

14. Reliability Function
Reliability Function
1.2
1
0.8
Reliability
0.6
0.4
0.2
0
0 1 2 3 4 5 6 7 8 9 10
Time
12

15. 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

16. Input Data
14

17. Plot of the Data
15

18. Exponential Fit
16

19. Exponential Analysis

20. Weibull Model
• Definition
β −1
βt   t β 
= exp  −    β > 0, η > 0, t ≥ 0
η η 
f (t )
   η  
 
β −1
 t β
 βt
 =(t ) λ (t )
1− F = f=
η η 
R(t ) =−  
exp  (t ) / R(t )
 η 
 
  
18

21. Weibull Model Cont.
• Statistical properties
∞ 1
MTTF = η ∫0 t1/ β e−t dt = ηΓ(1 +
β
)
 
2
2 2 1  
Var η Γ(1 + ) −  Γ(1 + ) 
= 
 β  β  
 

 
Median life = η ((ln 2)1/ β )
19

22. Versatility of Weibull Model
β −1
βt
Hazard rate: =λ (t ) f=
η η 
(t ) / R(t )
 
Hazard Rate
Constant Failure Rate
Region
β >1
0 < β <1
Early Life Wear-Out
Region Region
β =1
0 Time t
20

23. Weibull Model
21

24. Weibull Analysis: Shape Parameter
22

25. Weibull Analysis: Shape Parameter
23

26. Weibull Analysis: Shape Parameter
24

27. Normal Distribution
25

28. Failure Data
Table 1: Failure Data
Design A Design B
Sample # Cycles Sample # Cycles
1 726,044 11 529,082
2 615,432 12 729,957
3 508,077 13 650,570
4 807,863 14 445,834
5 755,223 15 343,280
6 848,953 16 959,903
7 384,558 17 730,049
8 666,686 18 730,640
9 515,201 19 973,224
10 483,331 20 258,006
26

29. Linearization of the Weibull Model
  t β 
F (t ) =R (t ) =exp  −   
1− 1−
 η  
 
  t β 
1- F = exp  −   
(t )
 η  
 
Taking the log
  t β 
ln (1- F (t )) =  −   
 η  
 
1
⇒ ln ln = ln η
β ln t − β
1 − F (t ) 27

30. Linearization of the Weibull Model
1
⇒ ln ln = ln η
β ln t − β
1 − F (t )
This is an equation of straight line
y = a + bx
Use linear regression , obtain a and b by solving
∑= n a + b ∑ x
y
∑ xy a ∑ x + b ∑ x
= 2
or by using Excel 28

31. Calculations using Excel
• Weibull Plot
  t β 
F (t ) = R(t ) =exp  −   
1− 1−
 η  
 
1
⇒ ln ln = ln η
β ln t − β is linear function of ln(time).
1 − F (t )
ˆ
• Estimate F (ti ) at ti using Bernard’s Formula
For n observed failure time data (t1 , t2 ,..., ti ,...tn )
ˆ (t ) = i − 0.3
F i
n + 0.4 29

32. Linearization of the Weibull Model
Design A Median
Cycles Rank Ranks 1/(1-Median Rank) ln(ln(1/(1-Median Rank))) ln(Design A Cycles)
384,558 1 0.067307692 1.072164948 -2.663843085 12.8598499
483,331 2 0.163461538 1.195402299 -1.72326315 13.088457
508,077 3 0.259615385 1.350649351 -1.202023115 13.13838829
515,201 4 0.355769231 1.552238806 -0.821666515 13.15231239
615,432 5 0.451923077 1.824561404 -0.508595394 13.33007974
666,686 6 0.548076923 2.212765957 -0.230365445 13.41007445
726,044 7 0.644230769 2.810810811 0.032924962 13.4953659
755,223 8 0.740384615 3.851851852 0.299032932 13.53476835
807,863 9 0.836538462 6.117647059 0.593977217 13.60214777
848,953 10 0.932692308 14.85714286 0.992688929 13.6517591
30

33. Linearization of the Weibull Model
Line Fit Plot
1.5
1
ln(ln(1/(1-Median Rank)))
0.5
0
-0.5
-1
-1.5
-2
-2.5
-3
12.8 13 13.2 13.4 13.6 13.8
ln(Design A Cycles)
31

34. Linearization of the Weibull Model
SUMMARY OUTPUT
Regression Statistics
Multiple R 0.98538223
R Square 0.97097815
Adjusted R Square 0.96735041
Standard Error 0.20147761
Observations 10
ANOVA
df SS
Regression 1 10.86495309
Residual 8 0.324745817
Total 9 11.18969891
β ln η
Coefficients Standard Error
Intercept -57.1930531 3.464488033
ln(Design A Cycles) 4.2524822 0.259929377
Beta (or Shape Parameter) = 4.25
Alpha (or Characteristic Life) = 693,380
32

35. Reliability Plot
1.0000
.9000
.8000
Survival Probability
.7000
.6000
.5000
.4000
.3000
.2000
.1000
.0000
0 200,000 400,000 600,000 800,000 1,000,000 1,200,000 1,400,000
Cycles

36. Input Data

37. Plots of the Data

38. Weibull Fit

39. Test for Weibull Fit

40. Parameters for Weibull

41. Weibull Analysis

42. Example 2: Input Data

43. Example 2: Plots of the Data

44. Example 2: Weibull Fit

45. Example 2:Test for Weibull Fit

46. Example 2: Parameters for Weibull

47. Weibull Analysis

48. Versatility of Weibull Model
β −1
βt
Hazard rate: =λ (t ) f=
η η 
(t ) / R(t )
 
Hazard Rate
Constant Failure Rate
Region
β >1
0 < β <1
Early Life Wear-Out
Region Region
β =1
0 Time t
46

49. Graphical Model Validation
• Weibull Plot
  t β 
F (t ) = R(t ) =exp  −   
1− 1−
 η  
 
1
⇒ ln ln = ln η
β ln t − β is linear function of ln(time).
1 − F (t )
ˆ
• Estimate F (ti ) at ti using Bernard’s Formula
For n observed failure time data (t1 , t2 ,..., ti ,...tn )
ˆ (t ) = i − 0.3
F i
n + 0.4 47

50. Example - Weibull Plot
• T~Weibull(1, 4000) Generate 50 data
Weibull Probability Plot
0.99
0.96
0.90
0.75
0.50 0.632
Probability
0.25
If the straight line fits
0.10
β the data, Weibull
distribution is a good
0.05
model for the data
0.02
0.01
10
-5
10
0
η 10
5
48
Data