AMSSA RELIABILITY GROWTH

8/7/2019 AMSSA RELIABILITY GROWTH

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Bernardo Almada-Lobo (2007)

GESTO DA MANUTENO

RELIABILITY GROWTH


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Failure Rate

1. Linear trend or random: IID (independent identically distributed failures)

T

N(T)

( )( )

T

TNT =


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Failure Rate

2. Logarithmic trend

( ) 1 = TT

N(T)

T

(Crow Model)


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The first prototypes produced during the development of a new complex systemwill contain:

design, manufacturing and/or engineering deficiencies.

the initial reliability of the prototypes may be below the system's reliabilitygoal or requirement.

The prototypes are often subjected to a rigorous testing program.

Problem areas are identified and appropriate corrective actions (or redesign)

are taken.

Reliability growth is the improvement in the reliability of a product over a periodof time due to changes in the product's design / manufacturing process.

Reliability Growth


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Reliability Growth

Reliability growth occurs from corrective and/or preventive actions based onexperience gained from failures and from analysis of the equipment, design,production and operation processes.

The reliability growth test, analysis and fix concept in design is applied by

uncovering weaknesses during the testing stages and performing appropriatecorrective actions before full-scale production. A corrective action takes place atthe problem and root cause level. Therefore, a failure modeis a problem androot cause. Reliability growth addresses failure modes.

Rework, repair and temporary fixes do not constitute reliability growth.


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Reliability Growth

The initial MTBF is the value actually achieved by the basic reliability tasks.

The growth potential is the MTBF, with the current management strategy, thatcan be attained if the test is conducted long enough.

The effectiveness of the corrective actions is part of the overall managementstrategy


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Reliability Growth

Reliability growth analysis can be conducted using different data types:

Time-to-failure (continuous) data

the most commonly observed type It involves recording the times-to-failure for the unit(s) under test. can be applied to a single unit or system or to multiple units or systems

Success/Failure Data

also referred to as discrete or attribute data It involves recording data from a test for a unit when there are only two

possible outcomes: success or failure.


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Non-Homogeneous Poisson Process (NHPP)

HPP NHPP

ExpectednumberoffailuresN(T)

T

ExpectednumberoffailuresN(T)

T


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Duane Model

The Duane model is a two parametermodel. Therefore, to use this model as

a basis for predicting the reliabilitygrowth that could be expected in anequipment development program,procedures must be defined forestimating these parameters as a

function of equipment characteristics.Note that, while these parameters canbe estimated for a given data set usingcurve-fitting methods, there exists nounderlying theory for the Duane

model that could provide a basis for apriori estimation.

1.00

10000.00

10.00

100.00

1000.00

100.00 1000.00

ReliaSoft's RGA 6 - RGA.ReliaSoft.com

Cumulative Number of Failures vs Time

Time

Cum.

NumberofFailures

9/12/2006 11:01Stoneridge TEDKim Pries

DuaneData 1DevelopmentalLS

Alpha=-1.9467, b=18364.7224


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Duane Model

Slope of tangent i

Expectednumberoffailures

N(T)

Slope of chord = c

Cumulative test timeT

( ) = 11t

btN

( )( )( ) ( ) ( ) ci TbdT

TNEd === 111

( ) ( ) ( ) Tb

ci ln1

ln1lnln1lnln

+=+=

ln(s)

ln(T)

( )iln

( )cln( )1ln

c

Cumulative test time, T

( ) == TbT

TNc

1


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Duane Model

( )

( )

( )

( ) ( )

1,1

1

lnlnlnln

ln

ln

=

=

+==

=

=

=

+=

=

ci

c

c

c

c

MTBFMTBF

bTMTBF

TbMTBFbc

m

Tx

MTBFy

cmxy

TN

TMTBF

1/b = the cumulative failure rate at T= 1, or at the beginning of the test, orthe earliest time at which the first c is predicted, or the c for theequipment at the start of the design and development process

= 1 implies infinite MTBF growth.


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Duane Model

< 0.2 Reliability has low priority (minimum effort on the improvement ofthe products reliability)

= 0.2-0.3 Corrective action taken for important failure modes only

= 0.3-0.4 Well managed programme with reliability as a high priority

= 0.4-0.6 Programme dedicated to the removal of design weakness and toreliability

Both the failure rate or MTBF at time Tcan be obtained through graphical

extrapolation.

inaccurate when there is a poordata adjustment


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Crow Model

Larry H. Crow noted that the Duane model could be stochasticallyrepresented as a Weibull process, allowing for statistical procedures to

be used in the application of this model in reliability growth.

The reliability growth pattern for the Crow model is exactly the same patternas for the Duane postulate: the cumulative number of failures is linear whenplotted on ln-ln scale.

Unlike the Duane postulate the Crow model is statistically based.

Duane postulate: the failure rate is linear on ln-ln scale. Crow model: the failure intensity of the underlying NHPP is linear whenplotted on ln-ln scale.


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Crow Model

N(t): cumulative number of failures observed in cumulative test time t(t)=i(t): failure intensity for the Crow model

Under the NHPP model, is approximately the probablyof a failure occurring over the interval for small

The expected number of failures experienced over the test interval [0, T]

The Crow model assumes that may be approximated by the Weibull failure rate function

( ) ( ) 1==

ttt i

( )T

( )[ ] ( )=T

dttTNE

0

( ) tt t[ ]ttt +,


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Crow Model

If the intensity function, or the instantaneous failure intensity, , is defined as:

In the special case of exponential failure times there is no growth (=1) and the failureintensity, , is equal to .

In this case, the expected number of failures is given by:

and

In the general case: and

CROWDUANE

CROW

DUANEb

=

=

1

1

1= ( )T ( )ti

( ) ( ) 0and0,0with,1 >>>== TTTNdT

dTi

( )T

( )[ ] ( ) TdttTNE

T

== 0

( )[ ] ( )

T

dttTNE

T

=

=

0

( )[ ]( )

,...2,1,0;!

Pr ===

nn

eTnTN

Tn

( )[ ]( )

,...2,1,0;!

Pr ===

nn

eTnTN

Tn


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Crow Model


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Crow Model graphic method

( ) ( ) ( ) 001

00

1TTNTTTtN

dt

d

===

( ) TtN =

ln N(t)

ln(T)

( )( ) lnlnln += ttN

declive

( )( ) lnlnln += ttN


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Crow Model

Maximum Likelihood Estimators

The probability density function (pdf) of the ith event given that the (i - 1)th event occurred at Ti-1

is:

The likelihood function is

where T* is the termination time and is given by:

Taking the natural log on both sides:

And differentiating with respect to yields:

Set equal to zero and solve for :

0*

^

T

N

T

N==

( )( )

11

11

=ii TT

iii eTTTf

==

=

n

iit

i

n

i

n

etL1

11

1


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Crow Bounds on Instantaneous MTBF

Time Terminated Data

( ) 1 = TLMTBF

( ) 2 = TUMTBF

limite inferior:

limite superior:

(Tabela 2)

( )1

1

===

TNTTMTBF

Failure Terminated Data (less usual)

( ) 1 pMTBF TL =

( ) 2 pMTBF TU =

limite inferior:

limite superior:

( ) == NTTMTBF n

(Tabela 1)

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AMSSA RELIABILITY GROWTH