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

970 Chapter 29 Statistics and probability distributions
29.15 RELIABILITYENGINEERING
Reliability engineering is an important area of study. Unreliable products lead to human
frustration, financial loss and in the case of life-critical systems can lead to death.
As the complexity of engineering systems has increased, mathematical methods of assessing
reliability have grown in importance. Probability theory forms a central part of
thedesignofhighlyreliablesystems.Formostitemsthefailureratechangeswithtime.
A common pattern is exhibited by the appropriately named ‘bath tub’ curve, illustrated
inFigure 29.22.
Carpartfailuresarequitewellmodelledbythisdistribution.Forexample,acrankshaft
may fail quite quickly as a result of a manufacturing defect. If it does not, then there is
usuallyalongperiodduringwhichthelikelihoodoffailureislow.Aftermanyyearsthe
probability offailure increases.
ConsidertheprobabilityofanitemfailingoveratotaltimeperiodT.Thisperiod,T,
isthetimeduringwhichtheitemisfunctioning.Thetimetakentorepairtheitemisnot
consideredinthiscalculationbutisconsideredalittlelateron.Supposetheprobabilityof
failureisevenlydistributedoverthisperiod;thatis,theprobabilityoffailureismodelled
bytheuniformdistribution(seeSection29.12).Itisimportanttonotethatthisisafairly
simplistic assumption. IfN = number of failures of the item over a time periodT, it is
possible todefine a mean failure rate,k, by
k = N T
For example, if an item fails 10 times in a period of 5 years we define the mean failure
ratetobek = 10/5 = 2,thatistwofailuresperyear.Becauseoftheuniformdistribution
ofthefailuresacrossthetimeperiod,thequantitykisconstant.Toillustratethisconsider
the previous example with a period of 10 years. During this time the item will fail 20
times and so
k = 20 = 2 failures per year
10
as found earlier.
Another usefultermisthemean timebetween failures(MTBF), which isgiven by
MTBF = 1 k
The termMTBF isonly used foritems thatarerepairable.
Let the interval T be divided into n small sub-intervals each of length δT, that is
nδT = T. Suppose each sub-interval is so small that only one failure can occur during
it. Note that since repair time has been neglected it is always possible to have a failure
in a sub-interval. So theN failures which occur during timeT occur inN distinct sub-
Failure rate
Time
Figure29.22
The‘bath tub’curve.