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

1. Fundamentals of Reliability
Fundamentals of Reliability
Engineering and Applications
Part 2 of 3
E. A. Elsayed
©2011 ASQ & Presentation Elsayed
Presented live on Dec 07th, 2010
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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.
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liability_Calendar/Short_Courses/Sh
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ort_Courses.html

3. Fundamentals of Reliability Engineering and
Applications
E. A. Elsayed
elsayed@rci.rutgers.edu
elsayed@rci rutgers edu
Rutgers University
December 7, 2010
,
1

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

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

6. Outline
Part 3. Failure Time Distributions
1. Constant failure rate distributions
1 C t t f il t di t ib ti
2. Increasing failure rate distributions
3.
3 Decreasing failure rate distributions
4. Weibull Analysis – Why use Weibull?
4

7. Outline
Part 2. Reliability Calculations
1. Use f f il
1 U of failure d t
data
a) Interval data (no censoring)
b) Exact failure times are known
2. Density functions
3. Reliability function
y
4. Hazard and failure rates
5

8. Basic Calculations
Suppose n0 identical units are subjected to a
test. During the interval (t, t +∆t ), we observed
nf (t ) failed components Let ns (t ) be the
components.
surviving components at time t , then we define:
Failure density function f ˆ (t )  n f (t )
n 0 t
ˆ (t )  nf (t ) ,
h
Failure rate function ns (t )t
Reliability function Rˆ (t )  P (T  t )  n s (t )
r
n0 6

9. Basic Definitions Cont’d
The unreliability F(t) is
1
F t   R t
︵
︶
︵
︶
Example: 200 light bulbs were tested and the failures in
1000-hour intervals are
Time Interval (Hours) Failures in the
interval
0-1000 100
1001-2000 40
2001-3000
2001 3000 20
3001-4000 15
4001-5000 10
5001-6000 8
6001-7000 7
Total 200
7

10. Calculations
Time Failure Density Hazard rate
Interval
I t l f (t ) x 104 h(t ) x 104
100 100
Time Interval Failures 0-1000 200  103
 5 .0
200  103
 5 .0
(Hours) in the
interval
0-1000 100
1001-2000
1001 2000
40 40
 2 .0
0  4 .0
0
1001-2000 40 200  103 100  103
2001-3000 20
3001-4000 15 2001 3000
2001-3000
20
 1 .0
20
 3 .3 3
4001-5000 10 200  103 60  103
5001-6000 8
6001-7000 7 …… …….. ……
Total 200
7 7
6001-7000 200  103
 0 .3 5
7  103
 10
8

11. Failure Density vs. Time
×10-4
1 2 3 4 5 6 7 x 103
Time i h
Ti in hours
9

12. Hazard Rate vs. Time
×10-4
-
1 2 3 4 5 6 7 × 103
Time in Hours
10

13. Reliability Calculations
Time Interval Failures
(Hours) in the Time Interval Reliability R (t )
interval 0-1000
0 1000 5/5=1.0
5/5=1 0
0-1000 100
1001-2000 40 1001-2000 /
2.0/4.0=0.5
2001-3000
2001 3000 20
3001-4000 15
4001-5000 10 2001-3000 1/3.33=0.33
5001-6000
5001 6000 8
6001-7000 7
…… ……
Total 200
6001-7000 0.35/10=.035
11

14. Reliability vs. Time
y
1 2 3 4 5 6 7 x 103
Time in hours
12

15. Exponential Distribution
Definition (t)
h (t )     0, t  0 Time
f (t )   exp( t )
R (t )  exp( t )  1  F (t )
p(
13

16. Exponential Model Cont’d
Cont d
Statistical Properties
5 1 0 6 F a il u r e s / h r
1    
M TTF  MTTF=200,000 hrs or 20 years

5 1 0  6 F a il u r e s / h r
1  
V a ri a n c e 
2 Standard deviation of MTTF is
200,000 hrs or 20 years
M e d i a n lif e ln 1 Median life =138,626 hrs or 14
 2

︵
︶
years
14

17. Exponential Model Cont’d
Cont d
Statistical Properties
5 1 0 6 F a il u r e s / h r
1    
M TTF  MTTF=200,000 hrs or 20 years

It is important to note that the MTTF= 1/failure
MTTF
is only applicable for the constant failure rate
case (failure time follow exponential distribution.
( p
15

18. 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
pp
Estimator. Order the failure time observations (failure
times) in an ascending order:
... ...
t1  t2   ti 1  ti  ti  1   t n 1  t n
16

19. Empirical Estimate of F(t) and R(t)
p () ()
F (ti ) is obtained by several methods
y
i
1.
1 Uniform “naive” estimator
naive
n
i
2.
2 Mean rank estimator n1 .
i  0 .3
3. Median
3 M di rank estimator (B
k ti t (Bernard)
d) n 0 4
/
i 3 / 8
4. Median rank estimator (Blom)
n 1 4
17

20. Empirical Estimate of F(t) and R(t)
Assume that we use the mean rank estimator
ˆ (t )  i
F i
n 1
ˆ (t )  n  1  i
R i ti  t  ti 1 i  0,1, 2,..., n
n 1
Since f (t ) is the derivative of F(t ), then
F (ti 1 )  F (ti )
ˆ ˆ
f (ti ) 
ˆ ti  ti 1  ti
ti .(n  1)
(
1
f (ti ) 
ˆ
ti .(n  1)
18

21. Empirical Estimate of F(t) and R(t)
1
 (t ) 
ˆ
ti .(n  1  i )
i
(
H (ti )   ln ( R(ti )
ˆ ˆ
Example:
Recorded failure times for a sample of 9 units are
observed at t=70 150 250 360 485 650 855
=70, 150, 250, 360, 485, 650, 855,
1130, 1540. Determine F(t), R(t), f(t),  t ,H(t)
︵
︶
19

22. Calculations
i t (i) t(i+1) F=i/10 R=(10-i)/10
F i/10 R (10 i)/10 f 0.1/t
f=0.1/t  1/(t.(10 i))
=1/(t.(10‐i)) H(t)
0 0 70 0 1 0.001429 0.001429 0
1 70 150 0.1 0.9 0.001250 0.001389 0.10536052
2 150 250 0.2 0.8 0.001000 0.001250 0.22314355
3 250 360 0.3 0.7 0.000909 0.001299 0.35667494
4 360 485 0.4 0.6 0.000800 0.001333 0.51082562
5 485 650 0.5 0.5 0.000606 0.001212 0.69314718
6 650 855 0.6 0.4 0.000488 0.001220 0.91629073
7 855 1130 0.7 0.3 0.000364 0.001212 1.2039728
8 1130 1540 0.8 0.2 0.000244 0.001220 1.60943791
9 1540 - 0.9 0.1 2.30258509
20

23. Reliability Function
y
1.2
1
Reliability
R li bilit 0.8
0.6
0.4
0.2
0
0 200 400 600 800 1000 1200
Time
21

24. Probability Density Function
0.001600
0.001400
0.001200
0.001000
Density
0.000800
Function
0.000600
0.000400
0.000200
0.000000
0 000000
0 200 400 600 800 1000 1200
Time
22

25. Constant Failure Rate
0.001900
0.001700
0.001500
0 001500
0.001300
0.001100
Failure Rate
0.000900
0.000700
0.000500
0.000300
0.000100
0 200 400 600 800 1000 1200
Time
23

26. Exponential Distribution: Another Example
Given failure d t
Gi f il 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.
24

27. Input Data
25

28. Plot of the Data
26

29. Exponential Fit
27

30. Exponential Analysis

31. Summary
In this part, we presented the three most important
relationships in reliability engineering.
We estimated obtained estimate functions for failure
rate, reliability and failure time. We obtained these
function for interval time and exact failure times
times.
29