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?
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
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
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
βt
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
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
48. Versatility of Weibull Model
β −1
βt
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