Reliability Final Exam Solutions

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: AV SS 

f p  f  p  p f  f  p

where AV SS is the robot system steady availability p is the robot system preventative maintenance rate

f

is the robot system failure rate

μp

is the repair rate with respect to preventative maintenance

f

is the robot system repair rate from failed state

Solution: μp Robot down for preventive

Robot Operating

maintenance

p

λf

λp

o

Robot Failed

μf

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

Po (t  t )  Po (t )  ( f   p ) Po (t )t  P f (t )  f t  Pp (t )  p t P (t  t )  Po (t ) lim o  ( f   p ) Po (t )  P f (t )  f  Pp (t )  p t t  0 dPo (t )  ( f   p ) Po (t )  P f (t )  f  Pp (t )  p dt dPp (t ) dt dP f (t )

  p Pp (t )  Po (t ) p

  f P f (t )  Po (t ) f dt At _ time _ t  0, Po (0)  1, Pp (0)  P f (0)  0 f (t )  lim sf ( s ) Final value Theorem: tlim 0 s 0

f

sPo ( s )  Po ( s )  ( f   p ) Po ( s )  Pp ( s )  p  Pf ( s )  f sPo ( s )  1  ( f   p ) Po ( s )  Pp ( s )  p  Pf ( s )  f sPp ( s )   p Pp ( s )  Po ( s ) p Pp ( s ) 

Po ( s ) p s  p

sPf ( s )   f Pf ( s )  Po ( s ) f Pf ( s ) 

Po ( s ) f s f

Plug Pf (s) and Pp (s) into Po (s) above to get : Po ( s ) 





( s   f )( s   p )

s s  s(  f   p   p   f )   f  p   p  f   f  p 2

A  s 2  s( f   p   p   f )   f  p   p  f   f  p Pp ( s )  Pf ( s ) 





 f (s   p ) sA  p (s   f ) sA

Ass  Po  lim sPo ( s )  s 0

f p  f  p  p f   f  p

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:

-

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 Solution:

(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:

λ(t)

Burn-in period

Useful life period

Wear out period

t Has three time periods: burn-in period, useful life period, and wear out period.  (t )  kct c1  (1  k )bt b1 e  t

b

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 AND Gate:

output

inputs The AND gate denotes that an output event occurs if and only if all the input events occur. OR Gate:

output

inputs The OR gate denotes that an output event occurs if any one or more of the input events occur. Cumulative Distribution Function: t

F (t ) 

 f ( x)dx 0

where f (x ) is the probability density function Exponential Distribution: f (t )   exp  t  t

F (t )    exp  x  dx   exp  x  |  1  exp  t  0

t

0

R (t )  1  F (t )  exp  t  f (t )  (t )   R(t )

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

n

1 j 1 j

MTTF   

 is the mean time to failure of a unit with exponentially distributed failure times n is the total number of units in the system

State any assumptions associated with your derivations. Solution: For a given unit, the reliability is denoted as: exp  t  

1

For a given unit, the MTTF is denoted as:  exp  t  dt     0 In a parallel system with n identical components, the reliability is: R p  1  1  R 

n

R p  1  1  exp  t  

MTTF p 

n

 1  1  exp  t  





0

0

 R p (t )dt 

n

dt

u  1  exp  t  du   exp  t  dt  dt 

du du     exp  t 1 u

t  0  u  0, t    u  1 1

1

n n n 1 un ui 1 1 MTTFp    du     u i 1du    |    0 1 u i 1 i i 1 i 0 0 i 1

EMP5103 Final Exam 2000 1. Same as question 2 from 2001.

2. Same as question 3 from 2001. 3. Prove, using the Markov method that a system’s steady state unavailability, UV SS is given by: UVSS 

 

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

Solution:

System operating



0



P0  t  t   P0 (t )1  t   P1 (t ) t P1  t  t   P1 (t )1  t   P0 (t )t

At time t  0, P0 (0)  1 and P1( 0 )  0 dP0 (t )   P0 (t )  P1 (t )  dt dP1 (t )   P1 (t )   P0 (t ) dt sP0 ( s )  P0 (0)   P0 ( s )  P1 ( s ) 

 s    P0 ( s)  1  P1 ( s)  P0 ( s ) 

1   P1 ( s ) s s

sP1 ( s )  P1 (0)   P1 ( s )   P0 ( s )

 s    P1 ( s)  P0 ( s)

System failed 1

P1 ( s ) 

 P0 ( s ) s

P0 ( s ) 

1    P0 ( s ) s s s

    1 P0 ( s ) 1    s   s   s     s s P0 ( s )   2  s    s      s       s P1 ( s ) 

 s   s   s2     s s2     s

UAV ss  lim sP1 ( s )  lim s 0

s 0

UAV ss 

   s       

 

4. Obtain hazard rate expressions for the following failure probability density, f(t), and reliability, R(t), functions: (i) f (t )  e  t 1

(ii) R (t )  e  t



where t is time

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

Solution: (i ) F (t )  1  exp  t  R (t )  1  F (t )  exp  t 

 (t ) 

f (t )  exp  t    R (t ) exp  t 

(ii ) (t )  

1 dR (t )  R (t ) dt

  1 1  exp  t    t    

1 1  exp  t     



  1    t     

***All other exams are a repetition of these questions***