reliability final exam solutions

8/19/2019 Reliability Final Exam Solutions

1/9

EMP5103 Final Exam 2001

1. A robot system can either fail completely or it underoes pre!entati!e

maintenance. Pro!e" usin the Mar#o! method that its steady state a!ailability is

i!en by$

statefailedfromraterepairsystemrobottheis

emaintenancve preventatirespect towithraterepairtheis

ratefailuresystemrobottheis

rateemaintenancve preventatisystemrobottheis

tyavailabilisteadysystemrobottheis

where

f

p

f

p

SS

p f f p p f

p f

SS

μ

AV

AV

µ

λ

λ

µ λ µ λ µ µ

µ µ

++=

%olution$

t t P t t Ppt t P

t t P t t P t t P

t t P t t P t t t P t t P

po p p

f o f f f

p p f f p f oo

∆+∆−=∆+

∆+∆−=∆+∆+∆+∆−∆−=∆+

λ µ

λ µ

µ µ λ λ

)()1)(()(

)()1)(()(

)()()1)(1)(()(

p p f f o p f o

p p f f o p f oo

t

p p f f o p f oo

t P t P t P dt

t dP

t P t P t P t

t P t t P

t t P t t P t t P t P t t P

µ µ λ λ

µ µ λ λ

µ µ λ λ

)()()()()(

)()()()()()(

lim

)()()()()()(

0

+++−=

+++−=∆

−∆+

∆+∆+∆+−=∆+

→∆

0)0()0(,1)0(,0 _ _

)()()(

)()()(

====

=+

=+

f po

f o f f f

po p p p

P P P t time At

t P t P dt

t dP

t P t P

dt

t dP

λ µ

λ µ

Final value Theorem: )(lim)(lim00

s sf t f st →→

=

Robot

Operating

Robot FailedRobot downfor preventive maintenance

p

f

! p

!f

f p o


2/9

[ ]

[ ]

sA

s s P

sA

s s P

s s A

s s s

s s s P

s

s P s P

s P s P s sP

s

s P s P

s P s P s sP

s P s P s P s sP

s P s P s P s P s sP

f p

f

p f

p

p f f p p f f p p f

p f f p p f f p p f

p f

o

f

f o

f

f o f f f

p

po p

po p p p

f f p po p f o

f f p po p f oo

)()(

)()(

)(

)(

))(()(

"gettoabove(s)#into(s)#and(s)##lug

)()(

)()()(

)()(

)()()(

)()()()(1)(

)()()()()()(

$

$

o pf

µ λ

µ λ

µ λ µ λ µ µ λ λ µ µ

µ λ µ λ µ µ λ λ µ µ

µ µ

µ

λ

λ µ

µ

λ

λ µ

µ µ λ λ

µ µ λ λ

+

=

+

=

+++++++=

+++++++

++

=

+

=

=+

+

=

=+

+=++−

+=++−

2. &a' (hat are the four classifications of reliability cost) *iscuss each cateory in

detail.

&b' +ist at least ten ma,or responsibilities of a reliability enineerin department.

%olutions$

(a) Reliability cost = PC+ AC + IFC + EFC

Prevention Cost:

% Redundancy

% Parts

% ourly cost and overhead rates !or desi"n en"ineers# reliability en"ineers#

etc$

Appraisal Cost:

p f f p p f

p f

o s

o ss s sP P A µ λ µ λ µ µ

µ µ

++===

→)(lim

0


3/9

% ourly cost and overhead rates !or evaluation# reliability %uali!ication#

reliability demonstration# li!e&testin"# etc$

% 'endor assurance cost !or ne comonent %uali!ication# insection# etc$

% Etc$

Internal Failure Cost:

% ourly cost and overhead rates !or troubleshootin" and reair# retestin"#!ailure analysis# etc$

% Relaced art*s cost

% ,are arts inventory

% Etc$

External Failure Cost:

% Cost to !ailure or reair

% Relaced arts cost

% Cost o! !ailure analysis

% -arranty administration and reortin" cost

% .iability insurance

% Etc$

(b)• Establishin" reliability olicy# lan# and rocedures

• Reliability allocation

• Reliability rediction (/I.&012&345)

• ,eci!ication and desi"n revies ith resect to reliability

• Reliability "roth monitorin"

• Providin" reliability related inuts to desi"n seci!ication and roosals

• Reliability demonstration (/I.&,T0&654)

• Trainin" reliability manoer and er!ormin" reliability&related research and

develoment or7

•

/onitorin" subcontractors*# i! any# reliability activities• Auditin" the reliability activities

• Failure data collection and reortin"

• Failure data analysis

• Consultin"

• Etc$

3. &a' +ist and discuss at least 10 tas#s of a -eliability Enineer.

&b' *escribe the folloin$

&i' /athtub haard rate cur!e

&ii' A* ate

&iii' - ate

&i!'umulati!e distribution function

&!' Exponential distribution

%olution$


4/9

λ(t)

t

Burn-in period Useful life period Wear out period

(a)% Per!ormin" analysis o! a roosed desi"n

% Analy8in" customer comlaints ith reliability

% Investi"atin" !ield !ailures

% Runnin" tests on the system# sub&system and arts

% 0eveloin" tests on the system# subsystem and comonents% 1ud"etin" the tolerable system !ailure don to the comonent level

% 0eveloin" a reliability ro"ram lan

% 0eterminin" reliability o! alternative desi"ns

% Providin" in!ormation to desi"ners or mana"ement concernin" reliability

% /onitorin" sub&contractor*s reliability er!ormance

% Particiatin" in evaluatin" re%uests !or roosals

% 0eveloin" reliability models and techni%ues

% Particiatin" in desi"n revies

% Etc$

(b) &athtub 'aard Rate urve"

'as three time periods" burn%in period, useful life period, and wear out period*

bt e

bbt k

cct k t

β β λ λ 1)1(1)( −−+−=

For bc!λ " # # $ k$ % t & #

And c ' # and b ' % to *et t+e s+ape abo,e

bc ' s+ape parameters

!λ ' scale parameters

t ' time

+- .ate"


5/9

/he +- gate denotes that an output event occurs if and only if all the input events

occur*

OR .ate"

/he OR gate denotes that an output event occurs if any one or more of the input events

occur*

umulative -istribution Function"

∫ =t

d- - f t F 0

)()(

where )( - f is the probability density function

ponential -istribution"

( )

( ) ( ) ( )

( )

λ λ

λ

λ λ λ λ

λ λ

==

−=−=

−−=−−=−=

−=

∫

)(

)()(

ep)(1)(

ep1epep)(

ep)(

20

0

t .

t f t

t t F t .

t -d- -t F

t t f

t t

4. Pro!e that the mean time to failure of a parallel system is i!en by$

output

inputs

output

inputs


6/9

∑=

=n

/ / 011F

1

1θ

systemin theunitsof numbertotaltheis

timesfailureddistributellyeponentiaunit withaof failuretomean timetheis

n

θ

%tate any assumptions associated ith your deri!ations.

%olution$

For a given unit, the reliability is denoted as" ( )t λ −ep

For a given unit, the 3//F is denoted as" ( )∫ ∞

==−0

1ep θ

λ λ dt t

4n a parallel system with n identical components, the reliability is"

( )

( )( )

( )( )[ ]

( )

( ) ( )

1,00

1epep

ep1

ep11)(

ep11

11

00

=⇒∞==⇒=

−=−=→−=

−−=

−−−==

−−−=

−−=

∫ ∫ ∞∞

ut ut

u

du

t

du

dt dt t du

t u

dt t dt t . 011F

t .

. .

n

p p

n

p

n

p

θ λ λ λ λ

λ

λ

λ

∑∑∫ ∑∫ ===

− ===−−=

n

i

n

i

in

i

in

pii

uduudu

u

u 011F

11

1

0

1

0 1

11

0

1

1

12 θ θ θ θ

EMP5103 Final Exam 2000

1. %ame as uestion 2 from 2001.


7/9

2. %ame as uestion 3 from 2001.

3. Pro!e" usin the Mar#o! method that a system6s steady state una!ailability"

SS UV is i!en by$

µ λ λ +=SS

UV

here

%olution$

( ) ( )

( ) ( )

( )

( ) λ µ

λ µ

λ

µ

λ

µ λ

µ λ

λ µ

µ λ

λ µ

µ λ

)()(

)()()0()(

)(

1

)(

)(1)(

)()()0()(

)()()(

)()()(

00and1)0(0,t+t time

)(1)(

)(1)(

01

0111

10

10

1000

01

1

10

0

10

011

100

s P s P s

s P s P P s sP

s P s s s P

s P s P s

s P s P P s sP

t P t P dt

t dP

t P t P dt

t dP

)( P P

t t P t t P t t P

t t P t t P t t P

=+

+−=−

+++=

+=+

+−=−

+−=

+−=

===

∆+∆−=∆+

∆+∆−=∆+

5ystem operating

0

5ystem failed

1

λ

µ

systemtheof raterepairconstanttheis

systemtheof ratefailureconstanttheis

µ

λ


8/9

( ) ( ) ( )

( ) ( )

( ) λ µ λ

λ µ

λ

λ µ

λ

λ µ

µ

µ

λ

λ µ

µ

µλ µ λ

µ

λ µ λ

λ µ

µ

λ

λ

µ

λ

µ

λ

+=

++==

++=

+++

+=

+++=

−+++=

+=

++−

+++

+=

+=

→→ s s sP UAV

s s s s

s

s s P

s s

s

s s

s s P

s s s s P

s P s s s

s P

s P s

s P

s s ss

01

0

$$1

$0

0

00

01

lim)(lim

)(

)(

11)(

)(1

)(

)()(

4. btain haard rate expressions for the folloin failure probability density"

f&t'" and reliability" -&t'" functions$

&i't

et f λ λ −=)(

&ii'β

θ t

et .1

)(−

=

here t is time

parameter shapetheis

parameter scaletheis

ratefailureconstanttheis

β

θ

λ

%olution$

( )

( )

( )( )

11 1ep1

ep

1)(

)(

1)()(

epep

)()()(

ep)(1)(

ep1)()(

−− =

−−

−

−=−=

=−−==

−=−=−−=

β β β

β θ

β

θ θ

β

θ

λ

λ λ λ λ λ

λ

λ

t t t

t dt

t d.

t .t ii

t t

t .t f t

t t F t .

t t F i

µ λ

λ

+= ssUAV


9/9

777All other exams are a repetition of these uestions777

reliability final exam solutions

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