Keandalan 2.pptx
-
Upload
febriantoutomo -
Category
Documents
-
view
267 -
download
5
Transcript of Keandalan 2.pptx
-
7/27/2019 Keandalan 2.pptx
1/63
1
Fundamentals of Reliability Engineering
and Applications
Dr. E. A. Elsayed
Department of Industrial and Systems Engineering
Rutgers University
Systems Engineering DepartmentKing Fahd University of Petroleum and Minerals
KFUPM, Dhahran, Saudi Arabia
April 20, 2009
-
7/27/2019 Keandalan 2.pptx
2/63
2
Reliability Engineering
Outline
Reliability definition
Reliability estimation
System reliability calculations
2
-
7/27/2019 Keandalan 2.pptx
3/63
3
Reliability Importance
One of the most important characteristics of a product, itis a measure of its performance with time (Transatlanticand Transpacific cables)
Products recalls are common (only after time elapses). InOctober 2006, the Sony Corporation recalled up to 9.6million of its personal computer batteries
Products are discontinued because of fatal accidents(Pinto, Concord)
Medical devices and organs (reliability of artificial organs)
3
-
7/27/2019 Keandalan 2.pptx
4/63
4
Reliability Importance
Business data
4
Warranty costs measured in million dollars for several large
American manufacturers in 2006 and 2005.
(www.warrantyweek.com)
-
7/27/2019 Keandalan 2.pptx
5/63
Maximum Reliability level
Reliability
WithRepairs
Time
NoRepairs
Some Initial Thoughts
Repairable and Non-Repairable
Another measure of reliability is availability (probability
that the system provides its functions when needed).
5
-
7/27/2019 Keandalan 2.pptx
6/63
Some Initial Thoughts
Warranty
Will you buy additional warranty? Burn in and removal of early failures.
(Lemon Law).
Time
Failu
reRate
Early FailuresConstantFailure Rate
IncreasingFailureRate
6
-
7/27/2019 Keandalan 2.pptx
7/63
7
Reliability Definitions
Reliabilityis a time dependent characteristic.
It can only be determined after an elapsed time but
can be predicted at any time.
It is the probability that a product or service will
operate properly for a specified period of time (design
life) under the design operating conditions withoutfailure.
7
-
7/27/2019 Keandalan 2.pptx
8/63
8
Other Measures of Reliability
Availabilityis used for repairable systems
It is the probability that the system is operationalat any random time t.
It can also be specified as a proportion of timethat the system is available for use in a given
interval (0,T).
8
-
7/27/2019 Keandalan 2.pptx
9/63
9
Other Measures of Reliability
Mean Time To Failure (MTTF):It is the averagetime that elapses until a failure occurs.
It does not provide information about the distribution
of the TTF, hence we need to estimate the varianceof the TTF.
Mean Time Between Failure (MTBF):It is theaverage time between successive failures.
It is used for repairable systems.
9
-
7/27/2019 Keandalan 2.pptx
10/63
10
Mean Time to Failure: MTTF
1
1
n
i
i
MTTF tn
0 0( ) ( )MTTF tf t dt R t dt
Time t
R(t)
1
0
1
22 is better than 1?
10
-
7/27/2019 Keandalan 2.pptx
11/63
11
Mean Time Between Failure: MTBF
11
-
7/27/2019 Keandalan 2.pptx
12/63
12
Other Measures of Reliability
Mean Residual Life (MRL):It is the expected remaininglife, T-t, given that the product, component, or a system
has survived to time t.
Failure Rate (FITs failures in 109hours):The failure rate in
a time interval [ ] is the probability that a failure per
unit time occurs in the interval given that no failure has
occurred prior to the beginning of the interval.
Hazard Function:It is the limit of the failure rate as the
length of the interval approaches zero.
1 2t t
1( ) [ | ] ( )
( ) tL t E T t T t f d t
R t
12
-
7/27/2019 Keandalan 2.pptx
13/63
13
Basic Calculations
0
1
0 0
0
( ), ( )
( ) ( ) ( ) , ( ) ( )
( )
n
i
fi
f sr
s
t
n tMTTF f tn n t
n t n tt R t P T t
n t t n
Suppose n0 identical units are subjected to atest. During the interval (t, t+t), we observed
nf(t) failed components. Let ns(t) be the
surviving components at time t, then the MTTF,
failure density, hazard rate, and reliability attime t are:
13
-
7/27/2019 Keandalan 2.pptx
14/63
14
Basic Definitions Contd
The unreliability F(t) is
( ) 1 ( )F t R t
Example: 200 light bulbs were tested and the failures in1000-hour intervals are
Time Interval (Hours) Failures in the
interval
0-1000
1001-2000
2001-3000
3001-4000
4001-5000
5001-6000
6001-7000
100
40
20
15
10
8
7
Total 200
14
-
7/27/2019 Keandalan 2.pptx
15/63
15
Calculations
Time
Interval
Failure Density
( )f t x 410
Hazard rate
( )h t x 410
0-1000
1001-2000
2001-3000
6001-7000
3
1005.0
200 10
3
402.0
200 10
3
201.0
200 10
..
3
70.35
200 10
3
1005.0
200 10
3
404.0
100 10
3
203.33
60 10
3
710
7 10
Time Interval
(Hours)
Failures
in the
interval
0-10001001-2000
2001-3000
3001-4000
4001-5000
5001-6000
6001-7000
10040
20
15
10
8
7
Total 200
15
-
7/27/2019 Keandalan 2.pptx
16/63
16
Failure Density vs. Time
1 2 3 4 5 6 7 x 103
Time in hours
16
10-4
-
7/27/2019 Keandalan 2.pptx
17/63
17
Hazard Rate vs. Time
1 2 3 4 5 6 7 103
Time in Hours
17
10-4
-
7/27/2019 Keandalan 2.pptx
18/63
18
Calculations
Time Interval Reliability ( )R t
0-1000
1001-2000
2001-3000
6001-7000
200/200=1.0
100/200=0.5
60/200=0.33
0.35/10=.035
Time Interval
(Hours)
Failures
in the
interval
0-1000
1001-20002001-3000
3001-4000
4001-5000
5001-6000
6001-7000
100
4020
15
10
8
7
Total 200
18
-
7/27/2019 Keandalan 2.pptx
19/63
19
Reliability vs. Time
1 2 3 4 5 6 7 x 103
Time in hours
19
-
7/27/2019 Keandalan 2.pptx
20/63
20
Exponential Distribution
Definition
( ) exp( )f t t
( ) exp( ) 1 ( )R t t F t
( ) 0, 0t t
(t)
Time
20
-
7/27/2019 Keandalan 2.pptx
21/63
21
Exponential Model Contd
1MTTF
2
1Variance
12Median life (ln )
Statistical Properties
21
6Failures/hr5 10
MTTF=200,000 hrs or 20 years
Median life =138,626 hrs or 14
years
-
7/27/2019 Keandalan 2.pptx
22/63
22
Empirical Estimate ofF(t)and R(t)
When the exact failure times of units is known, weuse 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:
1 2 1 1 1... ...
i i i n n t t t t t t t
-
7/27/2019 Keandalan 2.pptx
23/63
23
Empirical Estimate ofF(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
i
n
1
i
n
0 3
0 4
.
.
i
n
3 8
1 4
/
/
i
n
-
7/27/2019 Keandalan 2.pptx
24/63
24
Empirical Estimate ofF(t)and R(t)
Assume that we use the mean rank estimator
24
1
( )1
1 ( ) 0,1,2,...,
1
i
i i i
iF t
n
n iR t t t t i n
n
Since f(t) is the derivative ofF(t), then
11
( ) ( )( )
.( 1)
1 ( )
.( 1)
i ii i i i
i
i
i
F t F tf t t t tt n
f tt n
-
7/27/2019 Keandalan 2.pptx
25/63
25
Empirical Estimate ofF(t)and R(t)
25
1
( ) .( 1 )
( ) ln ( ( )
i
i
i i
t t 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
-
7/27/2019 Keandalan 2.pptx
26/63
26
Calculations
26
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
-
7/27/2019 Keandalan 2.pptx
27/63
27
Reliability Function
27
Reliability
Time
-
7/27/2019 Keandalan 2.pptx
28/63
28
Probability Density Function
28
y = 0.0014e-0.002xR = 0.9949
DensityFunction
Time
-
7/27/2019 Keandalan 2.pptx
29/63
29
Failure Rate
Constant
29
Failure Rate
Time
-
7/27/2019 Keandalan 2.pptx
30/63
30
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) asdefined before.
We use a software to demonstrate these steps.
30
-
7/27/2019 Keandalan 2.pptx
31/63
31
Input Data
31
-
7/27/2019 Keandalan 2.pptx
32/63
32
Plot of the Data
32
-
7/27/2019 Keandalan 2.pptx
33/63
33
Exponential Fit
33
E ti l A l i
-
7/27/2019 Keandalan 2.pptx
34/63
Exponential Analysis
-
7/27/2019 Keandalan 2.pptx
35/63
35
Go Beyond Constant Failure Rate
- Weibull Distribution (Model) and
Others
35
-
7/27/2019 Keandalan 2.pptx
36/63
36
The General Failure Curve
Time t
1
Early LifeRegion
2
Constant Failure RateRegion3
Wear-OutRegion
Failu
reRate
0
ABC
Module
36
-
7/27/2019 Keandalan 2.pptx
37/63
37
Related Topics (1)
Time t
1
Early LifeRegion
Failur
eRate
0
Burn-in:
According to MIL-STD-883C,
burn-in is a test performed to
screen or eliminate marginalcomponents with inherent
defects or defects resulting
from manufacturing process.
37
-
7/27/2019 Keandalan 2.pptx
38/63
38
21
Motivation Simple Example
Suppose the life times (in hours) of severalunits are: 1 2 3 5 10 15 22 28
1 2 3 5 10 15 22 28 10.75 hours8
MTTF
3-2=1 5-2=3 10-2=8 15-2=13 22-2=20 28-2=26
1 3 8 13 20 26(after 2 hours) 11.83 hours >
6MRL MTTF
After 2 hours of burn-in
-
7/27/2019 Keandalan 2.pptx
39/63
39
Motivation - Use of Burn-in
Improve reliability using cull eliminator
1
2
MTTF=5000 hours
Company
Company
After burn-inBefore burn-in
39
-
7/27/2019 Keandalan 2.pptx
40/63
40
Related Topics (2)
Time t
3
Wear-OutRegion
Haza
rdRate
0
Maintenance:An important assumption for
effective maintenance is that
component has an
increasing failure rate.
Why?
40
-
7/27/2019 Keandalan 2.pptx
41/63
41
Weibull Model
Definition
1
( ) exp 0, 0, 0t t
f t t
( ) exp 1 ( )
t
R t F t
1
( ) ( ) / ( )t
t f t R t
41
-
7/27/2019 Keandalan 2.pptx
42/63
42
Weibull Model Cont.
1/
0
1(1 )tMTTF t e dt
2
2 2 1(1 ) (1 )Var
1/Median life ((ln2) )
Statistical properties
42
-
7/27/2019 Keandalan 2.pptx
43/63
43
Weibull Model
43
-
7/27/2019 Keandalan 2.pptx
44/63
44
Weibull Analysis: Shape Parameter
44
-
7/27/2019 Keandalan 2.pptx
45/63
45
Weibull Analysis: Shape Parameter
45
-
7/27/2019 Keandalan 2.pptx
46/63
46
Weibull Analysis: Shape Parameter
46
-
7/27/2019 Keandalan 2.pptx
47/63
47
Normal Distribution
47
-
7/27/2019 Keandalan 2.pptx
48/63
Weibull Model
1
( ) ( ) .
t
h t
( )1( ) ( )
tt
f t e
0
( )1( ) ( )
t
F t e d
( )
( ) 1
t
F t e
( )
( )t
R t e
I t D t
-
7/27/2019 Keandalan 2.pptx
49/63
Input Data
-
7/27/2019 Keandalan 2.pptx
50/63
Plots of the Data
Weibull Fit
-
7/27/2019 Keandalan 2.pptx
51/63
Weibull Fit
Test for Weibull Fit
-
7/27/2019 Keandalan 2.pptx
52/63
Test for Weibull Fit
Parameters for Weibull
-
7/27/2019 Keandalan 2.pptx
53/63
Parameters for Weibull
Weibull Analysis
-
7/27/2019 Keandalan 2.pptx
54/63
Weibull Analysis
E l 2 I t D t
-
7/27/2019 Keandalan 2.pptx
55/63
Example 2: Input Data
-
7/27/2019 Keandalan 2.pptx
56/63
Example 2: Plots of the Data
Example 2: Weibull Fit
-
7/27/2019 Keandalan 2.pptx
57/63
Example 2: Weibull Fit
Example 2:Test for Weibull Fit
-
7/27/2019 Keandalan 2.pptx
58/63
Example 2:Test for Weibull Fit
Example 2: Parameters for Weibull
-
7/27/2019 Keandalan 2.pptx
59/63
Example 2: Parameters for Weibull
Weibull Analysis
-
7/27/2019 Keandalan 2.pptx
60/63
Weibull Analysis
-
7/27/2019 Keandalan 2.pptx
61/63
61
Versatility of Weibull Model
Hazard rate:
Time t
1
Constant Failure RateRegion
HazardR
ate
0
Early LifeRegion
0 1
Wear-OutRegion
1
1
( ) ( ) / ( ) tt f t R t
61
-
7/27/2019 Keandalan 2.pptx
62/63
62
( ) 1 ( ) 1 exp
1ln ln ln ln
1 ( )
tF t R t
tF t
Graphical Model Validation
Weibull Plot
is linear function ofln(time).
Estimate attiusing Bernards Formula
( )iF t
0.3 ( )
0.4i
iF t
n
Forn observed failure time data 1 2( , ,..., ,... )i nt t t t
62
-
7/27/2019 Keandalan 2.pptx
63/63
Example - Weibull Plot
T~Weibull(1, 4000) Generate 50 data
-5 0 5
0.01
0.02
0.05
0.10
0.25
0.50
0.75
0.90
0.96
0.99
Probability
Weibull Probability Plot
0.632
If the straight line fitsthe data, Weibulldistribution is a good
model for the data