presentation weibull analysis_english

8/12/2019 Presentation Weibull Analysis_English

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Version 3 November 2005 VE School

Weibull analysis

Nicolas Forissier

january the26 th 2007


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What is a Weibull analysis?

A Weibull analysis on warranty data consists in:

1 - Studying reliability as a function of mileage.

2 - Making the assumption that the Reliability lawfollows a Weibull distribution

3 - Finding the Weibull distribution which is mostappropriate to the data observed.


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Weibull distribution is the most appropriate tool for reliability analyses can be adjusted to most observed failure modes

is capable of describing each phase of the life cycle of a device bymeans of a point-in-time failure rate given in its simplified form by:

The failure probability is:

The reliability at a given mileage is written as:

( )1

=

tt

What is a Weibull distribution?

=

ttR exp)(

=

ttF exp1)(


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= scale parameter:

corresponds to the mileage at which there are 63.2%

of defective devices

is hard to interpret as it depends on the shapeparameter

= shape parameter:

characterises the phase of life

WEIBULL DISTRIBUTION: its parameters


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wear, fatiguebreakdown in youthdebugging

= 1 (or close to 1)

maturity

randombreakdownuseful life

< 1

youth

> 1

ageing

Characterising a product's phases of lifeusing the parameter

often process

faults*Inappropriate process

*Non robust DT/process variability

*Product characteristic too dispersed

in relation to the functional requirement

often product design

flawsIf it appears prematurely:

*Design flaw



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Weibull distribution: approximation of F(t) in 0

For a Weibull distribution: F(t) = 1- exp(-(t/))

development limited in 0 to the order 1 of F(t) is

F(t) (t/)ie F(t) constant* t

therefore, if we multiply t by 2, F(t) is multiplied by 2

examples:

0,1%

0,1%

0,1%0,1%

0,1%

F(10,000km)

8 *0,1%=0,8%3

4*0,1% = 0,4 %2

2* *0,1% 0,283 %1,52*0,1% = 0,2%1

*0,1% 0,141 %0,5

F(20,000 km)Beta

2

2


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WEIBULL DISTRIBUTION: graphical

representationUsing the data observed and the calculationof F, we change the variables:

Graphical representation:

( ) ( )X t t= ln ( )( )

Y tF t

=

ln ln

1

1and

F Y

X

t

The possibility of representing thereliability law by a 2 parameterWeibull model is judged according tothe "alignment of points":

NB: The Weibull straight line characterisesjust one failuremode

( ) ( ) ( )

LnXodt

tRtF

== =Y',exp11

( ) LnX =Y

The parameter therefore corresponds to the gradient of the straight line


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wear, fatiguebreakdown in youthdebugging

= 1 (or close to 1)

maturity

randombreakdownuseful life

< 1

youth

> 1

ageing

often process

faults*Inappropriate process


*Product characteristic too dispersed

in relation to the functional requirement

often product design

flawsIf it appears prematurely:

*Design flaw


F F F

Characterising a product's phases of lifeusing the parameter


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Description of Weibull models

The Weibull distribution is representedgraphically according to different models

3 types of models encountered3 types of models encountered

11 Mode22 Parameters

11 Mode33 Parameters

22 Modes44 Parameters


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Distribution at 1 mode 2 parametersDistribution at 1 mode 2 parameters

Representation of the 2 parameter2 parameter Weibull

distribution describing one failure modeone failure mode

F tt

( ) exp=

1



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Example of a line, for 1 mode 2 parameters with a Beta = 0.57(therefore < 1).The Weibull curve below shows the failures of the Turbo GARRET GT 15-44 mounted on engines X,

delivered during the 11/99-10/00 period. These failures are manifested in the engine's lack of power.

Example of a line

Cumulatedpercentageofdefectivedevices

Multiplier coefficient : 100 km

Weibull curve

Description of the analisis :

Vehicle : all vehiclesEngine : engine X

Delivery period : No 99 Oct 00NITG : M430 M431 turbo compressorRC : shortage of powerCountry : France


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Example of a line

Analysis of this case showed that the

problem is due to the process.

Poor placement of the silicone seal

between the crankshaft and the casing.

we find silicone particles in the oil.

the turbo lubrication drilled holes are

blocked

the turbo is blocked

the customer notices a shortage of

power

Cylinder casingCrankshaft

bearing 1

SILICONE DEPOSIT

Silicone seam

When the bearing 1 is tightened to the crankshaft, thesiliconeoverflows into the cylinder casing


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DistributionDistribution atat 11 ModeMode 33 parametersparameters Representation of the 33 parameterparameter Weibull distribution describing

oneone failurefailure modemode

=

ttF exp1)(

= positioning parameter,

corresponds to the mileage from

which it is considered possible thatthere be a failure.

This is usually set at 0.

Taille duparc : 60226 Mode de calcul : 1 mode/ 3 paramtres

Nombre de points : 31 Paramtre de forme (beta) : 0,98

Kilomtrage max. des incidents : 80834 Dure de vie caractristique (eta ) : 47198873

Paramtre de position(gamma) : 6448

Coefficient d'ajustement : 0,985

B(0,5) en km : 223725

B(0,03) en km : 18879

Nombre de points hors cadre : 0

1000 5000 10000 50000 100000

0.0001

0.0002

0.0005

0.001

0.002

0.005

0.01

0.02

0.03

0.05

0.1

0.2

0.5

0.7

1

Km

%cumuldedfaillants


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Distribution at 2 modes 4 parametersDistribution at 2 modes 4 parameters Representation of the 22 parameterparameter Weibull distribution

describing 2 failure modes2 failure modes

Mode 1distribution

Mode 2

distribution

Note: the model used in West is different

West Model:

F( t ) = 1 R1(t) * R2 (t)

with R1(t) = 1 F1(t)with R2(t) = 1 F2(t)


Cm for mode change

ttif

texp1F(t)

ttif

texp1F(t)

Cm

Cm

2

2

1

1

>

=

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