reliability engg
TRANSCRIPT
-
8/11/2019 Reliability engg
1/63
1
Fundamentals of Reliability Engineering
and Applications
Anurag Dixit
Department of Mechanical Engineering
Department of Mechanical Engineering
School of Engineering
Gautam Buddha University,
Greater Noida
July-2014
-
8/11/2019 Reliability engg
2/63
2
Reliability Engineering
Outline
Reliability definition
Reliability estimation
System reliability calculations
2
-
8/11/2019 Reliability engg
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
-
8/11/2019 Reliability engg
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)
-
8/11/2019 Reliability engg
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
-
8/11/2019 Reliability engg
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
-
8/11/2019 Reliability engg
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
-
8/11/2019 Reliability engg
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
-
8/11/2019 Reliability engg
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
-
8/11/2019 Reliability engg
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
-
8/11/2019 Reliability engg
11/63
11
Mean Time Between Failure: MTBF
11
-
8/11/2019 Reliability engg
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
-
8/11/2019 Reliability engg
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 n0identical 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
-
8/11/2019 Reliability engg
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
-
8/11/2019 Reliability engg
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
-
8/11/2019 Reliability engg
16/63
16
Failure Density vs. Time
1 2 3 4 5 6 7 x 103
Time in hours
16
10-4
-
8/11/2019 Reliability engg
17/63
17
Hazard Rate vs. Time
1 2 3 4 5 6 7 103
Time in Hours
17
10-4
-
8/11/2019 Reliability engg
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
-
8/11/2019 Reliability engg
19/63
19
Reliability vs. Time
1 2 3 4 5 6 7 x 103
Time in hours
19
-
8/11/2019 Reliability engg
20/63
20
Exponential Distribution
Definition
( ) exp( )f t t
( ) exp( ) 1 ( )R t t F t
( ) 0, 0t t
(t)
Time
20
-
8/11/2019 Reliability engg
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
-
8/11/2019 Reliability engg
22/63
22
Empirical Estimate of F(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
-
8/11/2019 Reliability engg
23/63
23
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
i
n
1
i
n
0 3
0 4
.
.
i
n
3 8
1 4
/
/
i
n
-
8/11/2019 Reliability engg
24/63
24
Empirical Estimate of F(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 of F(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
-
8/11/2019 Reliability engg
25/63
25
Empirical Estimate of F(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
-
8/11/2019 Reliability engg
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
-
8/11/2019 Reliability engg
27/63
27
Reliability Function
27
0
0.2
0.4
0.6
0.8
1
1.2
0 200 400 600 800 1000 1200
Reliability
Time
-
8/11/2019 Reliability engg
28/63
28
Probability Density Function
28
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
DensityFunction
Time
-
8/11/2019 Reliability engg
29/63
29
Failure Rate
Constant
29
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
-
8/11/2019 Reliability engg
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
-
8/11/2019 Reliability engg
31/63
31
Input Data
31
-
8/11/2019 Reliability engg
32/63
32
Plot of the Data
32
-
8/11/2019 Reliability engg
33/63
33
Exponential Fit
33
E ti l A l i
-
8/11/2019 Reliability engg
34/63
Exponential Analysis
-
8/11/2019 Reliability engg
35/63
35
Go Beyond Constant Failure Rate
- Weibull Distribution (Model) and
Others
35
-
8/11/2019 Reliability engg
36/63
36
The General Failure Curve
Time t
1
Early Life
Region
2
Constant Failure Rate
Region
3
Wear Out
Region
Failu
reRate
0
ABC
Module
36
-
8/11/2019 Reliability engg
37/63
37
Related Topics (1)
Time t
1
Early Life
Region
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
-
8/11/2019 Reliability engg
38/63
38
21
MotivationSimple 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
-
8/11/2019 Reliability engg
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
-
8/11/2019 Reliability engg
40/63
40
Related Topics (2)
Time t
3
Wear Out
Region
Haza
rdRate
0
Maintenance:An important assumption for
effective maintenance is that
component has an
increasing failure rate.
Why?
40
-
8/11/2019 Reliability engg
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
-
8/11/2019 Reliability engg
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
-
8/11/2019 Reliability engg
43/63
43
Weibull Model
43
-
8/11/2019 Reliability engg
44/63
44
Weibull Analysis: Shape Parameter
44
-
8/11/2019 Reliability engg
45/63
45
Weibull Analysis: Shape Parameter
45
-
8/11/2019 Reliability engg
46/63
46
Weibull Analysis: Shape Parameter
46
-
8/11/2019 Reliability engg
47/63
47
Normal Distribution
47
-
8/11/2019 Reliability engg
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
-
8/11/2019 Reliability engg
49/63
Input Data
-
8/11/2019 Reliability engg
50/63
Plots of the Data
-
8/11/2019 Reliability engg
51/63
Test for Weibull Fit
-
8/11/2019 Reliability engg
52/63
Test for Weibull Fit
Parameters for Weibull
-
8/11/2019 Reliability engg
53/63
Parameters for Weibull
-
8/11/2019 Reliability engg
54/63
E l 2 I t D t
-
8/11/2019 Reliability engg
55/63
Example 2: Input Data
-
8/11/2019 Reliability engg
56/63
Example 2: Plots of the Data
-
8/11/2019 Reliability engg
57/63
Example 2:Test for Weibull Fit
-
8/11/2019 Reliability engg
58/63
Example 2:Test for Weibull Fit
Example 2: Parameters for Weibull
-
8/11/2019 Reliability engg
59/63
Example 2: Parameters for Weibull
Weibull Analysis
-
8/11/2019 Reliability engg
60/63
Weibull Analysis
-
8/11/2019 Reliability engg
61/63
61
Versatility of Weibull Model
Hazard rate:
Time t
1
Constant Failure Rate
Region
HazardR
ate
0
Early Life
Region
0 1
Wear Out
Region
1
1
( ) ( ) / ( ) tt f t R t
61
-
8/11/2019 Reliability engg
62/63
62
( ) 1 ( ) 1 exp
1 ln ln ln ln
1 ( )
tF t R t
tF t
Graphical Model Validation
Weibull Plot
is linear function of ln(time).
Estimate attiusing Bernards Formula
( )iF t
0.3 ( )
0.4i
iF t
n
For nobserved failure time data 1 2( , ,..., ,... )i nt t t t
62
-
8/11/2019 Reliability engg
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 fits
the data, Weibull
distribution is a good
model for the data