Chapter 6 Time dependent reliability of components and

system

6.2 Failure rate time curve

6.3 Reliability and Hazard functions

Reliability:

(6.1)

(6.2)

(6.3)

Failure rate (instantaneous rate of failure, hazard function or hazard rate):

(6.4,6.5)

dt

tdRN

dt

tdNdt

tdN

Ndt

tdR

N

tN

N

tNN

N

tNtR

f

f

ffs

)()(

)(1)(

)(1

)()()(

dt

tdR

tRdt

tdR

N

N

dt

tdN

tNth

s

f

s

)(

)(

1)()(

)(

1)(

reliability at time t:

The distribution function:

The reliability of the component:

t

TT dxxftF0

])()(

tt

tRtR

tdRdxxh

00)(ln

)(

)()(

t

dxxhtR0

])(exp[)(

t

T

dxxh

tFtTPtTPtR

0])(exp[

)(1)(1)()(

Differentiation, yield

6.4 Modeling of failure Rates

t

t dxxhtF0

)()](1ln[

)(1

)(

)(

)()(..

)()(1

)(

tF

tf

tR

tfthei

thtF

tf

T

TT

T

T

timeunitperfailurexh )(

and

Assuming linear variation:

t tedhth

tFthtf

0])(exp[)(

)](1)[()(

tt

edhtR ])(exp[)(0

21)( ctcth

6.5 Estimation of failure rate from emperical data

The estimate reliability function at time t:

(6.23)

The failure rate can be computed as

(6.24)

example:6.1

N

tNtR s )()(

ttN

ttNtNth

s

ss

)(

)()()(

6.6 Mean time before failure (MTBF)

Since all system fail after a finite time, we have

(6.26)

0000

)()()(

)( dttRttRdtdt

tdRtdtttfMTBF t

0)(

00)(

0)(

dttRMTBF

tatttRand

tastR

6.7 Series system

n

iins RRRRRR

1321 ......

ini

s tt

1min

n

ii

n

iisss tFttPttPttPtF

11

)(11)(1)(1)()(

Failure time of the series system is

The failure time distribution :

The probability function of the failure time:

n

jiii

n

jj

n

j

j

j

sss tFtf

t

F

F

F

dt

tdFtf

,111

)](1[)()(

)(

ti

ietR )(

ttn

iis

s

n

ii

eetRtR

1

)(

1

)()(

n

jis

1

6.7.1 Failure rate of the system

Where the failure rate of the system:

The reliability can be expressed as:

])(exp[)(0t

ii dhtR

n

iiis

n

iis

dxxhdxxh

n

i

dxxhn

iis

consishifhthor

thth

ee

etRtR

t n

ii

n

i

t

i

t

i

1

1

))(()(

1

)(

1

)(

)()(

)()(

011

0

0

6.7.2 MTBF of the system

n

ii

t

n

ii

t

n

ii

n

ii

n

ii

e

dteMTBF

dttRdttRMTBF

10

)(

1

0

)(

01

0

11

))(()(

1

1

6.8 Parallel SystemReliability of parallel system, first seven events

6.8.1 Failure Rate of the system

The system failure rate is given by:

n

ii

n

j

n

jiiij

p

tF

tFtf

tF

tfth

1

1 ,1

)(1

)()(

)(1

)()(

6-8-2 MTBF of the system

0

01

01

0

1....111

11

))(11)(

21 dteee

dte

dttRdttRMTBF

ttt

n

i

t

n

ii

n

i

For special case for n=2

inin

kik tFtF

i

ntR

)()(1)(

6-9 (k,n) systems

2121

0

111

111 21

dteeMTBF tt

11

00

tt edte

Where relation is used:

Probability distribution function of the system

inik

ikk tFtF

i

ntRtF

)()(1)(1)(

1

0

6.9.1 MTBF of the system

001)( dtee

i

ndttRMTBF

n

ki

intitk

Failure time of the system

)()(1)()!1()!(

!)()( 1 tftFtF

kkn

n

dt

tdFtf kknk

k

6.10 Mixed series and parallel system

The system failure rate is given by:

The reliability R0 shown in fig. 6-8

6.11 complex systems

A

B

C

D

E

6.11.1 Enumeration method

EBDB RRRR

No component fail 1. ABCDE 0

6.11.2 Conditional Probability method

conditionfailedin

Ccomponent

theofliability

conditionfailedin

Ccomponentwithsystem

theofliability

conditionoperatingin

Ccomponent

theofliability

conditionoperatingin

Ccomponentwithsystem

theofliability

systemthe

ofliability

cricri

cricri

ReRe

ReReRe

AC

D

AB

A

B

6.11.3 Cut set method

1. Identify the minimal cut sets of the system

2. Model the components of each minimal cut set to be in parallel .

3. Assume that the various cut sets are in series

4. Find the reliabilityof the system using the parallel-series model.

6.12 Reliability Enhancement

6.12.1 Series system

nidRRR

RRdRRRRRdRR

j

n

j

n

ijjjj

niiii

.....2,1;

.......)(.......

1 ,1

112100

Constant constrain:

n

jkkkj

n

ikkki RdRRdRdR

,1,10

That is

j

iji

jj

ii

R

RdRdR

R

RdR

R

RdR

00

The cost involved in achieving the new system

jjnj

ii

jj

iijjii

jj

ii

RcRc

Rc

RcdRcdRc

R

RdR

R

RdR

min,.....,2,1

00

6.12.2 parallel system

(6.73)

ni

R

dRdR

n

ikkk

i ......,2,1;

)1(,1

0

For minimum cost:

)1(min)1(.....,3,2,1

jjnj

ii RcRc

6.13 reliability allocation- agree method

i

i

i

imt

m

t

m

te ii

111 /

tconscT

tw

i

ii tan

6.87