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7/26/2019 Mon Practical Reliability Theory - Dodson
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Managed by UT-Battellefor the Department of Energy
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$$%mpt#on h#&h #mpl#e$ that the
fa#l%re rate #$ &on$tant #f #t #$ %$ed a$ the
$ole mea$%re of e*%#pment+$ rel#ab#l#ty,
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Modeling
There are essentially 5 types of models Static
is constant Easy' if only life $ere this simple
Dynamic has a comple( functional form
To build a model6
7reate a logical structure of components Specify the reliability of each component
Drill do$n the structure as deep as you need to and8or ha"e data
( )t
( )t
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for the Department of Energy
SNS Static Model / is constant0Uses Maro" 7hains
( )t
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Dynam#& Model
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U$e$ of the Model
De$#gn ha$e
Model #$ a $#mple hat #f tool for eal%at#ngperforman&e to &ompare the pro5e&ted $y$tem rel#ab#l#ty#th the &%$tomer6$ expe&tat#on$,
7perat#onal ha$e
8al#date model parameter$ #th mea$%red performan&e,9re yo% gett#ng hat yo% expe&ted:
;f not *%e$t#on$ to a$< #n&l%de a$ the $y$tem= De$#gned rong
B%#lt rong ;n$talled rong 7perated rong Ma#nta#ned rong ;n a $#&
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> 4ognormal Distribution
> ,eibull Distribution
Time Distributions /Models0 of the.ailure !ate .unction
E(ponential Distribution
Normal Distribution
-( ) tf t e =
2
2
( - )-
2!
( )2
t
f t e
=
2
2
(ln - )-
2!
( )2
t
f t et
=
-! -
( )
tt
f t e
=
8ery &ommonly %$ed een #n &a$e$ to
h#&h #t doe$ not apply ($#mple)?9ppl#&at#on$= Ele&tron#&$ me&han#&al
&omponent$ et&,
8ery $tra#ghtforard and #dely %$ed?9ppl#&at#on$= Ele&tron#&$ me&han#&al
&omponent$ et&,
8ery poerf%l and &an be appl#ed to de$&r#be ar#o%$ fa#l%re pro&e$$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
7onstant .ailure !ate
!egion
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E(ponential Distribution /Model0
7onstant .ailure !ate
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
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