Mon Practical Reliability Theory - Dodson

7/26/2019 Mon Practical Reliability Theory - Dodson

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George Dodson

Spallation Neutron Source

Practical pplicationsof

!eliability Theory


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Topics

!eliability Terms and Definitions

!eliability Modeling as a tool for e"aluating system performance #n the design phase $hat are the tradeoffs of cost "s% reliability performance&

#n the operational phase' does the performance meet e(pectations&

nalysis of the failure rate of systems or components )o$ do systems fail&

#s the failure rate *reasonable+ &

nalytical calculation for the number of Spares ,hat inds of spares are there&

,hat is a *reasonable+ number of spares&


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!eliability Terms

Mean Time To .ailure /MTT.0 for non-repairable

systems

Mean Time Bet$een .ailures for repairable systems/MTB.0

!eliability Probability /sur"i"al0 !/t0

.ailure Probability /cumulati"e density function 0 ./t012-!/t0

.ailure Probability Density f/t0

.ailure !ate /ha3ard rate0 /t0

Mean residual life /M!40


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#mportant !elationships

00

0

( ) ( ) exp - ( ) ( ) / ( ) ( )

( ) !- ( ) exp - ( ) ( ) ( ) / ( )

tt

t

f t t u du dF t dt F t f u du

R t F t u du t f t R t

= = =

= = =

( ) ( ) !R t F t+ =

"here ( )t #$ the fa#l%re rate f%n&t#on


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MTB.

The MTB #$ #dely %$ed a$ themea$%rement of e*%#pment+$ rel#ab#l#ty and

performan&e, Th#$ al%e #$ often &al&%lated

by d##d#ng the total operat#ng t#me of the%n#t$ by the total n%mber of fa#l%re$

en&o%ntered, Th#$ metr#& #$ al#d onlyhen

the data #$ exponent#ally d#$tr#b%ted, Th#$ #$

a poor a$$%mpt#on h#&h #mpl#e$ that the

fa#l%re rate #$ &on$tant #f #t #$ %$ed a$ the

$ole mea$%re of e*%#pment+$ rel#ab#l#ty,


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Modeling

There are essentially 5 types of models Static

is constant Easy' if only life $ere this simple

Dynamic has a comple( functional form

To build a model6

7reate a logical structure of components Specify the reliability of each component

Drill do$n the structure as deep as you need to and8or ha"e data

( )t

( )t


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SNS Static Model / is constant0Uses Maro" 7hains

( )t


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Dynam#& Model


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U$e$ of the Model

De$#gn ha$e

Model #$ a $#mple hat #f tool for eal%at#ngperforman&e to &ompare the pro5e&ted $y$tem rel#ab#l#ty#th the &%$tomer6$ expe&tat#on$,

7perat#onal ha$e

8al#date model parameter$ #th mea$%red performan&e,9re yo% gett#ng hat yo% expe&ted:

;f not *%e$t#on$ to a$< #n&l%de a$ the $y$tem= De$#gned rong

B%#lt rong ;n$talled rong 7perated rong Ma#nta#ned rong ;n a $#&


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> 4ognormal Distribution

> ,eibull Distribution

Time Distributions /Models0 of the.ailure !ate .unction

E(ponential Distribution

Normal Distribution

-( ) tf t e =

2

2

( - )-

2!

( )2

t

f t e

=

2

2

(ln - )-

2!

( )2

t

f t et

=

-! -

( )

tt

f t e

=

8ery &ommonly %$ed een #n &a$e$ to

h#&h #t doe$ not apply ($#mple)?9ppl#&at#on$= Ele&tron#&$ me&han#&al

&omponent$ et&,

8ery $tra#ghtforard and #dely %$ed?9ppl#&at#on$= Ele&tron#&$ me&han#&al

&omponent$ et&,

8ery poerf%l and &an be appl#ed to de$&r#be ar#o%$ fa#l%re pro&e$$e$?9ppl#&at#on$= Ele&tron#&$ mater#al

$tr%&t%re et&,

8ery poerf%l and &an be appl#ed to

de$&r#be ar#o%$ fa#l%re pro&e$$e$?

9ppl#&at#on$= Ele&tron#&$ me&han#&al&omponent$ mater#al $tr%&t%re et&,


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E(ponential Model

Definition6 7onstant .ailure !ate

( )e( @ ) ( @ ) ( )

t xx

r tR x t P T t x T t e R x

e

+

= > + > = = =

( ) exp( ) 0 0f t t t = >

( ) exp( ) ! ( )R t t F t= =

( ) ( ) / ( )t f t R t = =

l/t0

t


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E(ponential Model 7ont%

! 0,3.1

( ) MTTFR MTTF e

e

=

=

=

!MTTF

=

2

!( )Var T

=

!Med#an l#fe (ln 2) 0,.13!4 MTTF

= =

>Atat#$t#&al ropert#e$


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!

( ) exp 0 0 0t tf t t

= > >

,eibull Model Definition

is the Shape Parameter and

is the 7haracteristic 4ifetime /28e0 sur"i"al

!

( ) ( ) / ( ) t

t f t R t

= =

( ) exp ! ( )t

R t F t

= =


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,eibull Model 7ontinued6

!/

0

!(! )tMTTF t e dt

= = +

2

2 2 !(! ) (! )Var

= + +

!/Med#an l#fe ((ln 2) )=

>Atat#$t#&al ropert#e$


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9ersatility of ,eibull Model

!

( ) ( ) / ( ) t

t f t R t

= =

.ailure !ate6

Time t

!=

7onstant .ailure !ate!egion

.ailure!ate

:

Early 4ife

!egion

0 !< +


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Maintenance6

Time t

F

,ear-;ut

!egion

)a3ard!ate

:

9n #mportant a$$%mpt#on foreffe&t#e ma#ntenan&e #$ that

&omponent$ #ll eent%ally hae an

;n&rea$#ng a#l%re Cate,

Ma#ntenan&e &an ret%rn the

&omponent to the on$tant a#l%reCeg#on,

5

7onstant .ailure !ate

!egion


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Terminal Mortality /,ear-;ut0

Time t

F

,ear-;ut

!egion

)a3ard!ate

:

omponent$ #ll eent%ally enter

the "ear-7%t Ceg#on here the

a#l%re Cate #n&rea$e$ een #th an

effe&t#e Ma#ntenan&e rogram,

Io% need to be able to dete&t theon$et of Term#nal Mortal#ty

5


!egion


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E(ponential Distribution /Model0


Single8Multiple .ailure Modes


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E(ample

The higher the failure rate is' the faster thereliability drops $ith time

l#n&rea$e$


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>,aloddi ,eibull' a Aed#$h #nentor and eng#neer #nented

the"e#b%ll d#$tr#b%t#on #n !13, The U,A, 9#r or&e

re&ogn#Sed the mer#t of "e#b%ll6$ method$ and f%nded h#$

re$ear&h to !1',

>4eonard Hohnsonat eneral Motor$ #mproed "e#b%ll6$

method$, e $%gge$ted the %$e of med#an ran< al%e$ for

plott#ng,

>The eng#neer$ at ratt V "h#tney fo%nd that the "e#b%ll

method or


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.ailure Probability Density is related to the.ailure Probability by6

!eliability .unction is related to the .ailureProbability Density by6

0

( ) ( )

x

f x f s ds= ( ( ))( ) d F xf x dx=

( ) ! ( ) ( )t

R t F t f u du

= =

!

2 2 #$ better than !:


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.ailure !ate .unction

#ncreasing failure rate /#.!0 "%s% decreasingfailure rate /D.!0

E(amples( ) or ( ) re$pe&t#elyt t Z ]

( ) here & #$ a &on$tant

( ) here 0

!( ) for t 0

!

t c

t at a

tt

=

= >

= >+

Z

]


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.ormal Statistical Test Procedures

2

>Test for assumption in a more statistical

$ay

> Goodness-of-.it test

>Bartlett=s test for E(ponential

>Mann=s test for ,eibull

>Iomogoro"-Smirno" /IS0 test


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Graphical Model 9alidation

,eibull Plot

( ) ! ( ) ! exp

! ln ln ln ln

! ( )

tF t R t

tF t

= =

=

B ( )iF t

#$ l#near f%n&t#on of ln(t#me),

> Estimate at tiusing Bernard=s .ormula

0,3B ( )i

F t

=

or nob$ered fa#l%re t#me data ! 2( ,,, ,,, )i nt t t t

Mon Practical Reliability Theory - Dodson

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