082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

29.15 Reliability engineering 971
intervals. The probability of failure occurring in a particular sub-interval, P f
, is then
given by
P f
=
number of sub-intervals inwhich failure occurs
total number ofsub-intervals
= N n
Ineach sub-interval, δT, the probability ofan item notfailing,P nf
, is
P nf
=1−P f
=1− N n = n −N
n
Astheitemprogressesthrougheachofthesuccessivesub-intervals δT itcanbethought
ofasundergoingaseriesoftrials,theresultofwhichisfailureornon-failure.Therefore,
the probability of the item not failing as it passes through all of thensub-intervals can
be obtained by multiplying each of the sub-interval probabilities of notfailing, thatis
( ) n −N n
probability of notfailingintimeT =
n
As the sub-interval δT becomes small, that is δT → 0, the number of sub-intervals,n,
becomes large, thatisn → ∞. Hence weneed toconsider the quantity
( ) n −N n
lim
n→∞
n
This limitcan be shown tobe e −N (see Appendix IV). Hence
probability of notfailingintimeT = e −N
Finally,theprobabilityofanitemfailingoneormoretimes--itemscanberepairedand
fail again -- inthe time intervalT isgiven by
P(T)=1−e −N =1−e −kT
P(T ) is the probability of at least one failure in the time period T. As a consequence
of the constancy of k, this formula can be used to calculate the probability of an item
failingone or more times inan arbitrarytimeperiod,t, inwhich case
P(t) =1−e −kt (29.1)
Itisimportanttostressthatthisformulaonlyappliesiftheprobabilityoffailureisevenly
distributed.
Engineeringapplication29.8
Breakdownsonafactoryprocessline
A factory process line makes use of 12 controllers to maintain process variables at
their correct values; all 12 controllers need to be working in order for the process
line to be operational. Records show that each controller fails, on average, once
every six months. Calculate the probability of the process line being stopped as a
resultofacontrollerfailure within a time periodofone month.
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