Review Rel Meas

8/10/2019 Review Rel Meas

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2. Reliability measures Objectives: Learn how to quantify reliability of a system Understand and learn how to compute the following measures

Reliability function Expected life Failure rate and hazard function

Learn some common probability density functions of time to failureand learn when to apply them Exponential

Normal Weibull

Learn how to estimate hazard functions from data Learn how to select a reliability function for a given problem


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Reliability function Assumption: New equipment T =Failure time, random variable because we do

not know when a system will fail

Probability density function of failure time, f T (t). Units: # of failures per unit time Reliability function , R(t)= probability that system

will work properly at time t

Failure distribution function, F T (t)= probabilitythat a system will fail by time t


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Expected life of a component or system,E(T)

t

f T(t)

E(T)

Expected life


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Hazard function

Hazard function: h(t)=probability that, given that asystem has survived until time t, it will fail

between times t and t+ t, divided by t. Units ofh(t): 1/unit time h(t)= f T(t)/R(t) Example start with N=1000 light bulbs, at T=1000

hrs, 300 light bulbs are still working. After 10 hrs5 more bulbs fail. The hazard function isapproximately: h(1005)=5/(300*10)


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Shape of hazard function of mostreal-life systems: bathtub

function

t

h(t)

Debugging,or infantmortality

Aging


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Relation between reliability measures

)(t f T )(t F T )(t R )(t h

)(t f T - dt t dF T )(

dt t dR )( t

dt t h

et h R 0

')'(

)()0(

)(t F T t

T dt t f 0

')'( - )(1 t R t

dt t he R 0

')'()0(1

)(t Rt

T dt t f 0

')'(1 )(1 t F T - t

dt t h

e R 0

')'(

)0(

)(t h t T

T

dt t f

t f

0')'(1

)(

)(1

)(

t F dt

t dF

T

T

)(

)(

t Rdt

t dR -


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Reliability and hazard functions

for well known distributions Exponential

Good choice for systems or components whose

strength does not change with time and whichare subjected to extreme disturbances occurringcompletely at random and independently.

f T(t)=1/ *exp(-t/ )

R(t)= exp(-t/ ) h(t)=1/


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Shape of exponential distribution

t

f T(t)

1/

E(T)=


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Normal distribution

t

f T(t)

Standard deviation,

Two parameter distribution


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No closed form analytical expression forcumulative distribution

Cumulative distribution of standard normal, (z),has been tabulated. We also have excellent

polynomial approximations. Standard normal haszero mean and unit standard deviation. Very easy to do reliability computations with

normal distributions

Finding F T(t) if T is normal. Transform T intostandard normal.

FT(t)=P(T t)=P[(T- )/ (t- )/ ]= (z)


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Cumulative distribution of standardnormal variable

z (z)

0 0.5

-1 0.16

-2 0.02

-3 0.001

-4 3 10 -5


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If we model the time to failure using a normal distribution thenthere is small probability of the time to failure being negative.This does not make sense. Always check that the probabilityof the time being negative is small compared to the

probabilities we are calculating in the problem at hand. Forexample, if the we are working with systems whose failure

probabilities are about 10 -3, then the probability of the time to

failure being negative should be about 10 -5 or less.

t

f T(t)

The area under thecurve to the left of zerois the probabilityof t being negative

0


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Weibull distribution Good choice for systems or components whose strengthdeteriorates with time and which are subjected to extreme

disturbances occurring completely at random andindependently.

Consider a building in Greece that is expected to be sustaina very strong earthquake (say above 6.5 in the Richterscale) once every ten years. Like any real life system, thestrength of the building deteriorates with time. A Weibulldistribution is a good candidate for modeling the time tofailure (or length of the life) of the building.

Very popular for describing strength and life length Generalizes the exponential distribution


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Reliability function

for t greater than

Three parameter distribution: use shape parameter, , to control shape is the scale parameter, affects dispersion use location parameter, , to shift the mean value shape parameter=1, Weibull reduces to the exponential

distribution

)(

)( t

et R


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Shape of Weibull probability density functionif shape parameter less than 3.6, density is skewed to the right

if shape parameter is greater than 4, density is skewed to the left .

0 2 4 60

2

4

6

8

theta=0.5

theta=1

theta=4

bet a=0.5

t

f ( t )


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Shape of Weibull probability density function.

0 2 4 60

1

2

theta=0.5

theta=1

theta=4

bet a=1

t

f ( t )


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.

0 2 4 60

1

2

3

4

theta=0.5theta=1theta=4

beta=4

t

f ( t )

Shape of Weibull probability density function


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Shape of Weibull probability density function

0 2 4 60

2

4

6

8

theta=0.5theta=1theta=4

bet a=1 0

t

f ( t )

.


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Effect of shape parameter

Consider building exposed toearthquakes:The larger the value of the shape

parameter, the larger the rate ofdeterioration in strengthIf the shape parameter is one then thereis no deterioration in the strength


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Statistics

3665.0

222

)(

])}1

1({)2

1([)(

)1

1()()(

eT median

T E

Median: 50% probability lower than median,50% higher than median


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Other common distributions

Lognormal; If x is normal then exp(x) islognormal

Gamma: quite similar to Weibull


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Estimating hazard function, failuredensity function and reliability function

from dataCase 1: Large sample of data about failures (N

greater than 30)

Start with N systems.

t N t t N t N

t f

t t N t t N t N t h

N (t) N

R(t)

t N

)()()(

)()()()(

tat time,lysuccessfuloperatethatsystemsof number),(


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Case 2: Small samples

Study homework 3


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Selecting a probability distribution on

the basis of knowledge of the particular physical situation causing failures

In most real life problems, we do not have enoughdata to estimate probability distributions.Therefore, we rely on experience or onanalytically obtained associations of physical

situations causing failure and probabilitydistributions to select type of probabilitydistribution to failure.


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Weibull and exponential models Extreme disturbances occurring completely at

random and independently. Example: time ofoccurrence and intensity of a strong earthquake

does not affect the time of occurrence andintensity of the next. Probability of occurrence of one earthquake

during [t, t+dt] is dt. Average rate of occurrenceof extreme disturbances is disturbances/unit time

Probability of a system failing because of adisturbance, p(t)


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Earthquake intensity versus time

t

IntensitySevere earthquakes

severe earthquakes per yearreturn period, 1/


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Reliability

pt T

pt

t P T

d pt P

pet f

et R

et pt f

eet R

t

)(

)(

:constantis p(t)If

)()(

)()(

)()( 0


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Suggested reading

Fox, E., The Role of Statistical Testing in NDA, Engineering Design Reliability

Handbook, CRC press, 2004, p. 26-1.

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