Reliability Final Exam Solutions

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EMP5103 Final Exam 2001

1. A robot system can either fail completely or it undergoes preventative

maintenance. Prove, using the Markov method that its steady state availability is

given by:

state failed from rate repair system robot the is e maintenanc ve preventati respect to with rate repair the is rate failure system robot the is rate e maintenanc ve preventati system robot the is ty availabili steady system robot the is where f p f p SS p f f p p f p f SS μ AV AV              

Solution:

t t P t t Pp t t P t t P t t P t t P t t P t t P t t t P t t P p o p p f o f f f p p f f p f o o                                  ) ( ) 1 )( ( ) ( ) ( ) 1 )( ( ) ( ) ( ) ( ) 1 )( 1 )( ( ) ( p p f f o p f o p p f f o p f o o t p p f f o p f o o

t

P

t

P

t

P

dt

t

dP

t

P

t

P

t

P

t

P

t

P

t

P

t

P

t

P

t

P

t

P













)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

lim

)

(

)

(

)

(

)

(

)

(

)

(

0









































  0 ) 0 ( ) 0 ( , 1 ) 0 ( , 0 _ _ ) ( ) ( ) ( ) ( ) ( ) (         f p o f o f f f p o p p p P P P t time At t P t P dt t dP t P t P dt t dP    

Final value Theorem: _tlim_₀ f(t)_slim_₀sf(s)

Robot Operating Robot Failed Robot down for preventive maintenance

λ

_p

λ

_f

μ

_p

μ

_f

f

p

o

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







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 p o p p o p p p f f p p o p f o f f p p o p f o o ) ( ) ( ) ( ) ( ) ( ) ( ) )( ( ) ( : get to above (s) P into (s) P and (s) P Plug ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 1 ) ( ) ( ) ( ) ( ) ( ) ( ) ( 2 2 o p f                                                                                  

2. (a) What are the four classifications of reliability cost? Discuss each category in

detail.

(b) List at least ten major responsibilities of a reliability engineering department.

Solutions:

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

Prevention Cost:

- Redundancy - Parts

- Hourly cost and overhead rates for design engineers, reliability engineers, etc… Appraisal Cost: p f f p p f p f o s o ss P sP s A











      ( ) lim 0

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- Hourly cost and overhead rates for evaluation, reliability qualification, reliability demonstration, life-testing, etc…

- Vendor assurance cost for new component qualification, inspection, etc… - Etc…

Internal Failure Cost:

- Hourly cost and overhead rates for troubleshooting and repair, retesting, failure analysis, etc…

- Replaced part’s cost. - Spare parts inventory. - Etc…

External Failure Cost:

- Cost to failure or repair. - Replaced parts cost. - Cost of failure analysis.

- Warranty administration and reporting cost. - Liability insurance.

- Etc…

(b)

 Establishing reliability policy, plan, and procedures.

 Reliability allocation.

 Reliability prediction (MIL-HDBK-217).

 Specification and design reviews with respect to reliability.

 Reliability growth monitoring.

 Providing reliability related inputs to design specification and proposals.

 Reliability demonstration (MIL-STD-471).

 Training reliability manpower and performing reliability-related research and development work.

 Monitoring subcontractors’, if any, reliability activities.

 Auditing the reliability activities.

 Failure data collection and reporting.

 Failure data analysis.

 Consulting.

 Etc…

3. (a) List and discuss at least 10 tasks of a Reliability Engineer.

(b) Describe the following:

(i) Bathtub hazard rate curve

(ii) AND gate

(iii)

OR gate

(iv)Cumulative distribution function

(v) Exponential distribution

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λ(t)

t

Burn-in period Useful life period Wear out period

(a)

- Performing analysis of a proposed design. - Analyzing customer complaints with reliability. - Investigating field failures.

- Running tests on the system, sub-system and parts.

- Developing tests on the system, subsystem and components.

- Budgeting the tolerable system failure down to the component level. - Developing a reliability program plan.

- Determining reliability of alternative designs.

- Providing information to designers or management concerning reliability. - Monitoring sub-contractor’s reliability performance.

- Participating in evaluating requests for proposals. - Developing reliability models and techniques. - Participating in design reviews.

- Etc…

(b) Bathtub Hazard Rate Curve:

Has three time periods: burn-in period, useful life period, and wear out period.

b t e b bt k c ct k t    ( ) 1(1 ) 1

For b,c,β,λ > 0

0 ≤ k≤ 1

t ≥ 0

And c = 0.5 and b = 1 to get the shape above

b,c = shape parameters

β,λ = scale parameters

t = time

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The AND gate denotes that an output event occurs if and only if all the input events

occur.

OR Gate:

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

occur.

Cumulative Distribution Function:



 t f x dx t F 0 ) ( ) (

where

f(x)

is the probability density function

Exponential Distribution:





















                        



) ( ) ( ) ( exp ) ( 1 ) ( exp 1 exp exp ) ( exp ) (

|

0 0 t R t f t t t F t R t x dx x t F t t f t t

4. Prove that the mean time to failure of a parallel system is given by:

output

inputs

output

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



n j

j

MTTF

1

1 

system in the units of number total the is times failure d distribute lly exponentia unit with a of failure to mean time the is n 

State any assumptions associated with your derivations.

Solution:

For a given unit, the reliability is denoted as:

exp



t



For a given unit, the MTTF is denoted as:







    0 1 exp   t dt

In a parallel system with n identical components, the reliability is:

































_

_

1 , 0 0 1 exp exp exp 1 exp 1 1 ) ( exp 1 1 1 1 0 0                             



  u t u t u du t du dt dt t du t u dt t dt t R MTTF t R R R n p p n p n p        







_

_ 

_





n i n i i n i i n p

i

u

du

u

du

u

MTTF

1 1 1 0 1 0 1 1 1 0

1

1 _

_

_|

_



EMP5103 Final Exam 2000

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2. Same as question 3 from 2001.

3. Prove, using the Markov method that a system’s steady state unavailability,

UVSS

is given by:

UVSS  ___

where

Solution:



























                  ) ( ) ( ) ( ) ( ) 0 ( ) ( ) ( 1 ) ( ) ( 1 ) ( ) ( ) ( ) 0 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 0 0 and 1 ) 0 ( 0, t At time ) ( 1 ) ( ) ( 1 ) ( 0 1 0 1 1 1 1 0 1 0 1 0 0 0 0 1 1 1 0 0 1 0 0 1 1 1 0 0 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                                        

System operating

0 System failed

1 



system the of rate repair constant the is system the of rate failure constant the is  

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



















 



                                                                     sP s _s 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 ₀ 1 ₀ 2 2 1 2 0 0 0 0 0 1 lim ) ( lim ) ( ) ( 1 1 ) ( ) ( 1 ) ( ) ( ) (

4. Obtain hazard rate expressions for the following failure probability density,

f(t), and reliability, R(t), functions:

(i)

_f₍_t₎___et

(ii)

_R

_t

_e

t 1

)

(





where t is time

parameter shape the is parameter scale the is rate failure constant the is   

Solution:

















1 1 1 exp 1 exp 1 ) ( ) ( 1 ) ( ) ( exp exp ) ( ) ( ) ( exp ) ( 1 ) ( exp 1 ) ( ) (   _       _                                                t t t t dt t dR t R t ii t t t R t f t t t F t R t t F i    ss UAV

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