Reliability engg

8/11/2019 Reliability engg

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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


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Reliability Engineering

Outline

Reliability definition

Reliability estimation

System reliability calculations

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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)

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Reliability Importance

Business data

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Warranty costs measured in million dollars for several large

American manufacturers in 2006 and 2005.

(www.warrantyweek.com)


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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).

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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

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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.

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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).

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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.

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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?

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Mean Time Between Failure: MTBF

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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

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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:

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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

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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

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

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Failure Density vs. Time

1 2 3 4 5 6 7 x 103

Time in hours

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10-4


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Hazard Rate vs. Time

1 2 3 4 5 6 7 103

Time in Hours

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10-4


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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

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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, 0t t

(t)

Time

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Exponential Model Contd

1MTTF

2

1Variance

12Median life (ln )

Statistical Properties

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6Failures/hr5 10

MTTF=200,000 hrs or 20 years

Median life =138,626 hrs or 14

years


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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 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


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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


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Assume that we use the mean rank estimator

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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

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( ) ( ) ( ).( 1)

1 ( ).( 1)

i ii i i i

i

i

i

F t F tf t t t tt n

f tt n


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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


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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

DensityFunction

Time


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Failure Rate

Constant

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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)asdefined 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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E ti l A l i


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Exponential Analysis


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Go Beyond Constant Failure Rate

- Weibull Distribution (Model) and

Others

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The General Failure Curve

Time t

1

Early Life

Region

2

Constant Failure Rate

Region

3

Wear Out

Region

Failu

reRate

0

ABC

Module

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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.

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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


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Motivation - Use of Burn-in

Improve reliability using cull eliminator

1

2

MTTF=5000 hours

Company

Company

After burn-inBefore burn-in

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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?

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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

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Weibull Model Cont.

1/

0

1(1 )tMTTF t e dt

2

2 2 1(1 ) (1 )Var

1/Median life ((ln2) )

Statistical properties

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Weibull Model

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Weibull Analysis: Shape Parameter

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Normal Distribution

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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


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Input Data


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Plots of the Data


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Test for Weibull Fit


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Test for Weibull Fit

Parameters for Weibull


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Parameters for Weibull


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E l 2 I t D t


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Example 2: Input Data


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Example 2: Plots of the Data


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Example 2:Test for Weibull Fit


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Example 2:Test for Weibull Fit

Example 2: Parameters for Weibull


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Example 2: Parameters for Weibull

Weibull Analysis


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Weibull Analysis


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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

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( ) 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

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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

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Reliability engg