chapter 6 time dependent reliability of components and system
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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
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