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Fundamentals of Reliability Engineering and Applications

E. A. Elsayedelsayed@rci.rutgers.edu

Rutgers University

Quality Control & Reliability Engineering (QCRE)IIE

February 21, 2012

OutlinePart 1. Reliability Definitions

Reliability Definition…Time dependent characteristics

Failure Rate Availability MTTF and MTBF Time to First Failure Mean Residual Life Conclusions

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OutlinePart 2. Reliability Calculations

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

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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?

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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

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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

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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

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Calculations

Time Interval

Failure Density( )f t x 410

Hazard rate ( )h t x 410

0-1000

1001-2000

2001-3000

……

6001-7000

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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

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Failure Density vs. Time

1 2 3 4 5 6 7 x 103

Time in hours9

×10-

4

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Hazard Rate vs. Time

1 2 3 4 5 6 7 × 103

Time in Hours

10

×10-

4

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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

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Reliability vs. Time

1 2 3 4 5 6 7 x 103

Time in hours

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Exponential Distribution

Definition

( ) exp( )f t t

( ) exp( ) 1 ( )R t t F t

( ) 0, 0h t t

(t)

Time

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Exponential Model Cont’d

1

MTTF

2

1Variance

12Median life ︵ln ︶

Statistical Properties

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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

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Exponential Model Cont’d

1

MTTF

Statistical Properties

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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.

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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

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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

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Empirical Estimate of F(t) and R(t)

Assume that we use the mean rank estimator

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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

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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

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Empirical Estimate of F(t) and R(t)

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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

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Calculations

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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

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Reliability Function

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0

0.2

0.4

0.6

0.8

1

1.2

0 200 400 600 800 1000 1200

Reliability

Time

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Probability Density Function

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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

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Constant Failure Rate

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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

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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.

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Input Data

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Plot of the Data

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Exponential Fit

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Exponential Analysis

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.

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