可靠性基础课程之二 by ASQ
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Time Interval (Hours) 0-1000 1001-2000 2001 3000 2001-3000 3001-4000 4001-5000 5001-6000 6001-7000 Total
t F
t R
1
Failures in the interval 100 40 20 15 10 8 7 200
Example: Recorded failure times for a sample of 9 units are observed at t=70, =70 150, 150 250, 250 360, 360 485, 485 650, 650 855, 855 1130, 1540. Determine F(t), R(t), f(t), ︵ ︶,H(t)
e c n a i r a V
2
Standard deviation of MTTF is 200,000 hrs or 20 years
︵ 2 ︶ n l
1
Median life =138,626 hrs or 14 years
14
e f i l n a i d e M
Exponential Model Cont’d Cont d
t
19
Calculations
i 0 1 2 3 4 5 6 7 8 9 t (i) 0 70 150 250 360 485 650 855 1130 1540 t(i+1) F F=i/10 i/10 R R=(10-i)/10 (10 i)/10 70 150 250 360 485 650 855 1130 1540 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 f=0.1/ f 0.1/t 0.001429 0.001250 0.001000 0.000909 0.000800 0.000606 0.000488 0.000364 0.000244 =1/( 1/(t.(10‐i)) 0.001429 0.001389 0.001250 0.001299 0.001333 0.001212 0.001220 0.001212 0.001220 H(t) 0 0.10536052 0.22314355 0.35667494 0.51082562 0.69314718 0.91629073 1.2039728 1.60943791 2.30258509
Fundamentals of Reliability Engineering and Applications
E. A. Elsayed elsayed@rci rutgers edu elsayed@rci.rutgers.edu Rutgers University
December 7, , 2010
3
Part 3.
Outline Failure Time Distributions
1. Constant 1 C t t failure f il rate t distributions di t ib ti 2. Increasing failure rate distributions 3 Decreasing failure rate distributions 3. 4. Weibull Analysis – Why use Weibull?
7
Calculationsቤተ መጻሕፍቲ ባይዱ
Time I t Interval l
Time Interval (Hours) 0-1000 1001-2000 2001-3000 3001-4000 4001-5000 5001-6000 6001-7000 Total Failures in the interval 100 40 20 15 10 8 7 200
15
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 pp is the Rank Estimator. Order the failure time observations (failure times) in an ascending order:
1
2
3
4
5
6
7
x 103
Ti Time i in h hours
9
Hazard Rate vs. Time
-4 ×10-
1
2
3
4
5
6
7
× 103
Time in Hours
10
Reliability Calculations
Time Interval (Hours) 0-1000 1001-2000 2001 3000 2001-3000 3001-4000 4001-5000 5001 6000 5001-6000 6001-7000 Total Failures in the interval 100 40 20 15 10 8 7 200
2
Part 2.
Outline Reliability Calculations
1. Use 1 U of f failure f il d data t 2. Density functions 3 Reliability function 3. 4. Hazard and failure rates
Failure Density Hazard rate f (t ) x 104 h(t ) x 104
100 5 .0 200 103 100 5 .0 200 103
0-1000
1001 2000 1001-2000 2001-3000 2001 3000 …… 6001-7000
Exponential Model Cont’d Cont d
Statistical Properties
r h / s e r u l i a F 6 0 1
1
5
F T T M
MTTF=200,000 hrs or 20 years
1
r h / s e r u l i a F 6 0 1 5
1
Part 1.
Outline Reliability Definitions
Reliability Definition…Time Definition Time dependent characteristics Failure Rate Availability MTTF and MTBF Time to First Failure Mean Residual Life C Conclusions l i
in
2 Mean rank estimator 2.
1
in in
3 Median 3. M di rank k estimator ti t (B (Bernard) d) 4. Median rank estimator (Blom)
0 3 0 4 . .
/ /
3 8 1 4
17
Empirical Estimate of F(t) and R(t)
Time Interval 0 1000 0-1000 1001-2000 2001-3000 …… 6001-7000
Reliability R (t ) 5/5=1 0 5/5=1.0 / 2.0/4.0=0.5 1/3.33=0.33 …… 0.35/10=.035
11
Reliability y vs. Time
5
Basic Calculations
Suppose n0 identical units are subjected to a test. During the interval (t, t +∆t ), we observed nf (t ) failed components components. Let ns (t ) be the surviving components at time t , then we define: Failure density function Failure rate function Reliability function
4
Part 2.
Outline Reliability Calculations
1 Use 1. U of f failure f il d data t
a) b)
Interval data (no censoring) Exact failure times are known
2. Density functions 3. Reliability y function 4. Hazard and failure rates
t
1
t
t
t
t
t
t
2
1
1
1
. . .
. . .
i
i
i
n
n
16
Empirical p Estimate of F(t) ( ) and R(t) ()
F (ti )
is obtained by y several methods
in
1 Uniform “naive” 1. naive estimator
n f (t ) ˆ f (t ) n 0 t
nf (t ) ˆ h (t ) , ns (t )t
n s (t ) ˆ R (t ) Pr (T t ) n0
6
Basic Definitions Cont’d
The unreliability F(t) is
︵︶ ︵︶ Example: 200 light bulbs were tested and the failures in 1000-hour intervals are
Assume that we use the mean rank estimator i ˆ F (ti ) n 1 n 1 i ˆ R (ti ) ti t ti 1 i 0,1, 2,..., n n 1 Since f (t ) is the derivative of F(t ), then
40 2 .0 0 200 103
40 4 .0 0 100 103 20 3 .3 3 60 103
20 1 .0 200 103
……..
7 0 .3 5 200 103
……
7 10 7 103
8
Failure Density vs. Time
×10-4
1
2
3
4
5
6
7
x 103
Time in hours
12
Exponential Distribution
Definition
h (t )
(t)
0, t 0
Time
f (t ) exp( t ) R (t ) exp( p( t ) 1 F (t )
13
ˆ (t ) F ˆ (t ) F i 1 i ˆ (t ) f i ti .( (n 1) ˆ (t ) f i 1 ti .(n 1) ti ti 1 ti
18
Empirical Estimate of F(t) and R(t)
ˆ (t ) 1 i ti .( (n 1 i ) ˆ (t ) ln ( R ˆ (t ) H i i
Statistical Properties
r h / s e r u l i a F 6 0 1 5
1
F T T M
MTTF=200,000 hrs or 20 years
It is important to note that the MTTF= MTTF 1/failure is only applicable for the constant failure rate case ( (failure time follow exponential p distribution.
t F
t R
1
Failures in the interval 100 40 20 15 10 8 7 200
Example: Recorded failure times for a sample of 9 units are observed at t=70, =70 150, 150 250, 250 360, 360 485, 485 650, 650 855, 855 1130, 1540. Determine F(t), R(t), f(t), ︵ ︶,H(t)
e c n a i r a V
2
Standard deviation of MTTF is 200,000 hrs or 20 years
︵ 2 ︶ n l
1
Median life =138,626 hrs or 14 years
14
e f i l n a i d e M
Exponential Model Cont’d Cont d
t
19
Calculations
i 0 1 2 3 4 5 6 7 8 9 t (i) 0 70 150 250 360 485 650 855 1130 1540 t(i+1) F F=i/10 i/10 R R=(10-i)/10 (10 i)/10 70 150 250 360 485 650 855 1130 1540 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 f=0.1/ f 0.1/t 0.001429 0.001250 0.001000 0.000909 0.000800 0.000606 0.000488 0.000364 0.000244 =1/( 1/(t.(10‐i)) 0.001429 0.001389 0.001250 0.001299 0.001333 0.001212 0.001220 0.001212 0.001220 H(t) 0 0.10536052 0.22314355 0.35667494 0.51082562 0.69314718 0.91629073 1.2039728 1.60943791 2.30258509
Fundamentals of Reliability Engineering and Applications
E. A. Elsayed elsayed@rci rutgers edu elsayed@rci.rutgers.edu Rutgers University
December 7, , 2010
3
Part 3.
Outline Failure Time Distributions
1. Constant 1 C t t failure f il rate t distributions di t ib ti 2. Increasing failure rate distributions 3 Decreasing failure rate distributions 3. 4. Weibull Analysis – Why use Weibull?
7
Calculationsቤተ መጻሕፍቲ ባይዱ
Time I t Interval l
Time Interval (Hours) 0-1000 1001-2000 2001-3000 3001-4000 4001-5000 5001-6000 6001-7000 Total Failures in the interval 100 40 20 15 10 8 7 200
15
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 pp is the Rank Estimator. Order the failure time observations (failure times) in an ascending order:
1
2
3
4
5
6
7
x 103
Ti Time i in h hours
9
Hazard Rate vs. Time
-4 ×10-
1
2
3
4
5
6
7
× 103
Time in Hours
10
Reliability Calculations
Time Interval (Hours) 0-1000 1001-2000 2001 3000 2001-3000 3001-4000 4001-5000 5001 6000 5001-6000 6001-7000 Total Failures in the interval 100 40 20 15 10 8 7 200
2
Part 2.
Outline Reliability Calculations
1. Use 1 U of f failure f il d data t 2. Density functions 3 Reliability function 3. 4. Hazard and failure rates
Failure Density Hazard rate f (t ) x 104 h(t ) x 104
100 5 .0 200 103 100 5 .0 200 103
0-1000
1001 2000 1001-2000 2001-3000 2001 3000 …… 6001-7000
Exponential Model Cont’d Cont d
Statistical Properties
r h / s e r u l i a F 6 0 1
1
5
F T T M
MTTF=200,000 hrs or 20 years
1
r h / s e r u l i a F 6 0 1 5
1
Part 1.
Outline Reliability Definitions
Reliability Definition…Time Definition Time dependent characteristics Failure Rate Availability MTTF and MTBF Time to First Failure Mean Residual Life C Conclusions l i
in
2 Mean rank estimator 2.
1
in in
3 Median 3. M di rank k estimator ti t (B (Bernard) d) 4. Median rank estimator (Blom)
0 3 0 4 . .
/ /
3 8 1 4
17
Empirical Estimate of F(t) and R(t)
Time Interval 0 1000 0-1000 1001-2000 2001-3000 …… 6001-7000
Reliability R (t ) 5/5=1 0 5/5=1.0 / 2.0/4.0=0.5 1/3.33=0.33 …… 0.35/10=.035
11
Reliability y vs. Time
5
Basic Calculations
Suppose n0 identical units are subjected to a test. During the interval (t, t +∆t ), we observed nf (t ) failed components components. Let ns (t ) be the surviving components at time t , then we define: Failure density function Failure rate function Reliability function
4
Part 2.
Outline Reliability Calculations
1 Use 1. U of f failure f il d data t
a) b)
Interval data (no censoring) Exact failure times are known
2. Density functions 3. Reliability y function 4. Hazard and failure rates
t
1
t
t
t
t
t
t
2
1
1
1
. . .
. . .
i
i
i
n
n
16
Empirical p Estimate of F(t) ( ) and R(t) ()
F (ti )
is obtained by y several methods
in
1 Uniform “naive” 1. naive estimator
n f (t ) ˆ f (t ) n 0 t
nf (t ) ˆ h (t ) , ns (t )t
n s (t ) ˆ R (t ) Pr (T t ) n0
6
Basic Definitions Cont’d
The unreliability F(t) is
︵︶ ︵︶ Example: 200 light bulbs were tested and the failures in 1000-hour intervals are
Assume that we use the mean rank estimator i ˆ F (ti ) n 1 n 1 i ˆ R (ti ) ti t ti 1 i 0,1, 2,..., n n 1 Since f (t ) is the derivative of F(t ), then
40 2 .0 0 200 103
40 4 .0 0 100 103 20 3 .3 3 60 103
20 1 .0 200 103
……..
7 0 .3 5 200 103
……
7 10 7 103
8
Failure Density vs. Time
×10-4
1
2
3
4
5
6
7
x 103
Time in hours
12
Exponential Distribution
Definition
h (t )
(t)
0, t 0
Time
f (t ) exp( t ) R (t ) exp( p( t ) 1 F (t )
13
ˆ (t ) F ˆ (t ) F i 1 i ˆ (t ) f i ti .( (n 1) ˆ (t ) f i 1 ti .(n 1) ti ti 1 ti
18
Empirical Estimate of F(t) and R(t)
ˆ (t ) 1 i ti .( (n 1 i ) ˆ (t ) ln ( R ˆ (t ) H i i
Statistical Properties
r h / s e r u l i a F 6 0 1 5
1
F T T M
MTTF=200,000 hrs or 20 years
It is important to note that the MTTF= MTTF 1/failure is only applicable for the constant failure rate case ( (failure time follow exponential p distribution.