fundamentals of reliability engineering and applications · fundamentals of reliability engineering...
TRANSCRIPT
![Page 1: Fundamentals of Reliability Engineering and Applications · Fundamentals of Reliability Engineering and Applications E. A. Elsayed elsayed@rci.rutgers.edu Rutgers University Quality](https://reader030.vdocument.in/reader030/viewer/2022033119/5e34e3803a9e3a54a735be59/html5/thumbnails/1.jpg)
1
Fundamentals of Reliability Engineering and Applications
E. A. [email protected]
Rutgers University
Quality Control & Reliability Engineering (QCRE)IIE
February 21, 2012
![Page 2: Fundamentals of Reliability Engineering and Applications · Fundamentals of Reliability Engineering and Applications E. A. Elsayed elsayed@rci.rutgers.edu Rutgers University Quality](https://reader030.vdocument.in/reader030/viewer/2022033119/5e34e3803a9e3a54a735be59/html5/thumbnails/2.jpg)
OutlinePart 1. Reliability Definitions
Reliability Definition…Time dependent characteristics
Failure Rate Availability MTTF and MTBF Time to First Failure Mean Residual Life Conclusions
2
![Page 3: Fundamentals of Reliability Engineering and Applications · Fundamentals of Reliability Engineering and Applications E. A. Elsayed elsayed@rci.rutgers.edu Rutgers University Quality](https://reader030.vdocument.in/reader030/viewer/2022033119/5e34e3803a9e3a54a735be59/html5/thumbnails/3.jpg)
OutlinePart 2. Reliability Calculations
1. Use of failure data 2. Density functions 3. Reliability function 4. Hazard and failure rates
3
![Page 4: Fundamentals of Reliability Engineering and Applications · Fundamentals of Reliability Engineering and Applications E. A. Elsayed elsayed@rci.rutgers.edu Rutgers University Quality](https://reader030.vdocument.in/reader030/viewer/2022033119/5e34e3803a9e3a54a735be59/html5/thumbnails/4.jpg)
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?
4
![Page 5: Fundamentals of Reliability Engineering and Applications · Fundamentals of Reliability Engineering and Applications E. A. Elsayed elsayed@rci.rutgers.edu Rutgers University Quality](https://reader030.vdocument.in/reader030/viewer/2022033119/5e34e3803a9e3a54a735be59/html5/thumbnails/5.jpg)
OutlinePart 2. Reliability Calculations
1. Use of failure data a) Interval data (no censoring)b) Exact failure times are known
2. Density functions 3. Reliability function 4. Hazard and failure rates
5
![Page 6: Fundamentals of Reliability Engineering and Applications · Fundamentals of Reliability Engineering and Applications E. A. Elsayed elsayed@rci.rutgers.edu Rutgers University Quality](https://reader030.vdocument.in/reader030/viewer/2022033119/5e34e3803a9e3a54a735be59/html5/thumbnails/6.jpg)
6
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 surviving components at time t , then we define:
Failure density function
Failure rate function
Reliability function6
0
( )ˆ( ) fn tf tn t
( )ˆ( ) , ( )f
s
n th tn t t
0
( )ˆ ( ) ( ) sr
n tR t P T tn
![Page 7: Fundamentals of Reliability Engineering and Applications · Fundamentals of Reliability Engineering and Applications E. A. Elsayed elsayed@rci.rutgers.edu Rutgers University Quality](https://reader030.vdocument.in/reader030/viewer/2022033119/5e34e3803a9e3a54a735be59/html5/thumbnails/7.jpg)
7
Basic Definitions Cont’dThe 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 theinterval
0-10001001-20002001-30003001-40004001-50005001-60006001-7000
1004020151087
Total 200
7
![Page 8: Fundamentals of Reliability Engineering and Applications · Fundamentals of Reliability Engineering and Applications E. A. Elsayed elsayed@rci.rutgers.edu Rutgers University Quality](https://reader030.vdocument.in/reader030/viewer/2022033119/5e34e3803a9e3a54a735be59/html5/thumbnails/8.jpg)
8
Calculations
Time Interval
Failure Density( )f t x 410
Hazard rate ( )h t x 410
0-1000
1001-2000
2001-3000
……
6001-7000
3
1 0 0 5 .02 0 0 1 0
3
4 0 2 .02 0 0 1 0
3
2 0 1 .02 0 0 1 0
…….. 3
7 0 .3 52 0 0 1 0
3
1 0 0 5 .02 0 0 1 0
3
4 0 4 .01 0 0 1 0
3
2 0 3 .3 36 0 1 0
……
3
7 1 07 1 0
Time Interval (Hours)
Failures in theinterval
0-10001001-20002001-30003001-40004001-50005001-60006001-7000
1004020151087
Total 200
8
![Page 9: Fundamentals of Reliability Engineering and Applications · Fundamentals of Reliability Engineering and Applications E. A. Elsayed elsayed@rci.rutgers.edu Rutgers University Quality](https://reader030.vdocument.in/reader030/viewer/2022033119/5e34e3803a9e3a54a735be59/html5/thumbnails/9.jpg)
9
Failure Density vs. Time
1 2 3 4 5 6 7 x 103
Time in hours9
×10-
4
![Page 10: Fundamentals of Reliability Engineering and Applications · Fundamentals of Reliability Engineering and Applications E. A. Elsayed elsayed@rci.rutgers.edu Rutgers University Quality](https://reader030.vdocument.in/reader030/viewer/2022033119/5e34e3803a9e3a54a735be59/html5/thumbnails/10.jpg)
10
Hazard Rate vs. Time
1 2 3 4 5 6 7 × 103
Time in Hours
10
×10-
4
![Page 11: Fundamentals of Reliability Engineering and Applications · Fundamentals of Reliability Engineering and Applications E. A. Elsayed elsayed@rci.rutgers.edu Rutgers University Quality](https://reader030.vdocument.in/reader030/viewer/2022033119/5e34e3803a9e3a54a735be59/html5/thumbnails/11.jpg)
11
Reliability Calculations
Time Interval Reliability ( )R t 0-1000 1001-2000 2001-3000 …… 6001-7000
5/5=1.0 2.0/4.0=0.5 1/3.33=0.33 …… 0.35/10=.035
Time Interval(Hours)
Failures in theinterval
0-10001001-20002001-30003001-40004001-50005001-60006001-7000
1004020151087
Total 200
11
![Page 12: Fundamentals of Reliability Engineering and Applications · Fundamentals of Reliability Engineering and Applications E. A. Elsayed elsayed@rci.rutgers.edu Rutgers University Quality](https://reader030.vdocument.in/reader030/viewer/2022033119/5e34e3803a9e3a54a735be59/html5/thumbnails/12.jpg)
12
Reliability vs. Time
1 2 3 4 5 6 7 x 103
Time in hours
12
![Page 13: Fundamentals of Reliability Engineering and Applications · Fundamentals of Reliability Engineering and Applications E. A. Elsayed elsayed@rci.rutgers.edu Rutgers University Quality](https://reader030.vdocument.in/reader030/viewer/2022033119/5e34e3803a9e3a54a735be59/html5/thumbnails/13.jpg)
13
Exponential Distribution
Definition
( ) exp( )f t t
( ) exp( ) 1 ( )R t t F t
( ) 0, 0h t t
(t)
Time
13
![Page 14: Fundamentals of Reliability Engineering and Applications · Fundamentals of Reliability Engineering and Applications E. A. Elsayed elsayed@rci.rutgers.edu Rutgers University Quality](https://reader030.vdocument.in/reader030/viewer/2022033119/5e34e3803a9e3a54a735be59/html5/thumbnails/14.jpg)
14
Exponential Model Cont’d
1
MTTF
2
1Variance
12Median life ︵ln ︶
Statistical Properties
14
6 Fa ilu res/h r5 10MTTF=200,000 hrs or 20 years
Median life =138,626 hrs or 14 years
6 Failures/hr5 10Standard deviation of MTTF is200,000 hrs or 20 years
![Page 15: Fundamentals of Reliability Engineering and Applications · Fundamentals of Reliability Engineering and Applications E. A. Elsayed elsayed@rci.rutgers.edu Rutgers University Quality](https://reader030.vdocument.in/reader030/viewer/2022033119/5e34e3803a9e3a54a735be59/html5/thumbnails/15.jpg)
15
Exponential Model Cont’d
1
MTTF
Statistical Properties
15
6 Failures/hr5 10MTTF=200,000 hrs or 20 years
It is important to note that the MTTF= (1/failure rate) is only applicable for the constant failure rate case (failure time follow exponential distribution.
![Page 16: Fundamentals of Reliability Engineering and Applications · Fundamentals of Reliability Engineering and Applications E. A. Elsayed elsayed@rci.rutgers.edu Rutgers University Quality](https://reader030.vdocument.in/reader030/viewer/2022033119/5e34e3803a9e3a54a735be59/html5/thumbnails/16.jpg)
16
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
![Page 17: Fundamentals of Reliability Engineering and Applications · Fundamentals of Reliability Engineering and Applications E. A. Elsayed elsayed@rci.rutgers.edu Rutgers University Quality](https://reader030.vdocument.in/reader030/viewer/2022033119/5e34e3803a9e3a54a735be59/html5/thumbnails/17.jpg)
17
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
n0 30 4..
in
3 81 4
//
in
![Page 18: Fundamentals of Reliability Engineering and Applications · Fundamentals of Reliability Engineering and Applications E. A. Elsayed elsayed@rci.rutgers.edu Rutgers University Quality](https://reader030.vdocument.in/reader030/viewer/2022033119/5e34e3803a9e3a54a735be59/html5/thumbnails/18.jpg)
18
Empirical Estimate of F(t) and R(t)
Assume that we use the mean rank estimator
18
1
ˆ ( )11ˆ( ) 0,1, 2,...,
1
i
i i i
iF tnn iR t t t t i n
n
Since f (t ) is the derivative of F(t ), then
11
ˆ ˆ( ) ( )ˆ ( ).( 1)1ˆ ( )
.( 1)
i ii i i i
i
ii
F t F tf t t t tt n
f tt n
![Page 19: Fundamentals of Reliability Engineering and Applications · Fundamentals of Reliability Engineering and Applications E. A. Elsayed elsayed@rci.rutgers.edu Rutgers University Quality](https://reader030.vdocument.in/reader030/viewer/2022033119/5e34e3803a9e3a54a735be59/html5/thumbnails/19.jpg)
19
Empirical Estimate of F(t) and R(t)
19
1ˆ( ).( 1 )
ˆ ˆ( ) ln ( ( )
ii
i i
tt n i
H t R t
Example:
Recorded failure times for a sample of 9 units are observed at t=70, 150, 250, 360, 485, 650, 855, 1130, 1540. Determine F(t), R(t), f(t), ,H(t)︵ ︶t
![Page 20: Fundamentals of Reliability Engineering and Applications · Fundamentals of Reliability Engineering and Applications E. A. Elsayed elsayed@rci.rutgers.edu Rutgers University Quality](https://reader030.vdocument.in/reader030/viewer/2022033119/5e34e3803a9e3a54a735be59/html5/thumbnails/20.jpg)
20
Calculations
20
i t (i) t(i+1) F=i/10 R=(10-i)/10 f=0.1/t =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
![Page 21: Fundamentals of Reliability Engineering and Applications · Fundamentals of Reliability Engineering and Applications E. A. Elsayed elsayed@rci.rutgers.edu Rutgers University Quality](https://reader030.vdocument.in/reader030/viewer/2022033119/5e34e3803a9e3a54a735be59/html5/thumbnails/21.jpg)
21
Reliability Function
21
0
0.2
0.4
0.6
0.8
1
1.2
0 200 400 600 800 1000 1200
Reliability
Time
![Page 22: Fundamentals of Reliability Engineering and Applications · Fundamentals of Reliability Engineering and Applications E. A. Elsayed elsayed@rci.rutgers.edu Rutgers University Quality](https://reader030.vdocument.in/reader030/viewer/2022033119/5e34e3803a9e3a54a735be59/html5/thumbnails/22.jpg)
22
Probability Density Function
22
0.000000
0.000200
0.000400
0.000600
0.000800
0.001000
0.001200
0.001400
0.001600
0 200 400 600 800 1000 1200
Density Function
Time
![Page 23: Fundamentals of Reliability Engineering and Applications · Fundamentals of Reliability Engineering and Applications E. A. Elsayed elsayed@rci.rutgers.edu Rutgers University Quality](https://reader030.vdocument.in/reader030/viewer/2022033119/5e34e3803a9e3a54a735be59/html5/thumbnails/23.jpg)
23
Constant Failure Rate
23
0.000100
0.000300
0.000500
0.000700
0.000900
0.001100
0.001300
0.001500
0.001700
0.001900
0 200 400 600 800 1000 1200
Failure Rate
Time
![Page 24: Fundamentals of Reliability Engineering and Applications · Fundamentals of Reliability Engineering and Applications E. A. Elsayed elsayed@rci.rutgers.edu Rutgers University Quality](https://reader030.vdocument.in/reader030/viewer/2022033119/5e34e3803a9e3a54a735be59/html5/thumbnails/24.jpg)
24
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.
24
![Page 25: Fundamentals of Reliability Engineering and Applications · Fundamentals of Reliability Engineering and Applications E. A. Elsayed elsayed@rci.rutgers.edu Rutgers University Quality](https://reader030.vdocument.in/reader030/viewer/2022033119/5e34e3803a9e3a54a735be59/html5/thumbnails/25.jpg)
25
Input Data
25
![Page 26: Fundamentals of Reliability Engineering and Applications · Fundamentals of Reliability Engineering and Applications E. A. Elsayed elsayed@rci.rutgers.edu Rutgers University Quality](https://reader030.vdocument.in/reader030/viewer/2022033119/5e34e3803a9e3a54a735be59/html5/thumbnails/26.jpg)
26
Plot of the Data
26
![Page 27: Fundamentals of Reliability Engineering and Applications · Fundamentals of Reliability Engineering and Applications E. A. Elsayed elsayed@rci.rutgers.edu Rutgers University Quality](https://reader030.vdocument.in/reader030/viewer/2022033119/5e34e3803a9e3a54a735be59/html5/thumbnails/27.jpg)
27
Exponential Fit
27
![Page 28: Fundamentals of Reliability Engineering and Applications · Fundamentals of Reliability Engineering and Applications E. A. Elsayed elsayed@rci.rutgers.edu Rutgers University Quality](https://reader030.vdocument.in/reader030/viewer/2022033119/5e34e3803a9e3a54a735be59/html5/thumbnails/28.jpg)
Exponential Analysis
![Page 29: Fundamentals of Reliability Engineering and Applications · Fundamentals of Reliability Engineering and Applications E. A. Elsayed elsayed@rci.rutgers.edu Rutgers University Quality](https://reader030.vdocument.in/reader030/viewer/2022033119/5e34e3803a9e3a54a735be59/html5/thumbnails/29.jpg)
Summary
In this part, we presented the three most importantrelationships 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.
29