Biomedical Engineering Reference
In-Depth Information
Another important statistical function for the reliability engineering and failure analysis
is failure rate. Failure rate is the frequency with which an engineered system or compo-
nent fails. It is often denoted by the Greek letter
λ
and is important in reliability theory. By
calculating the failure rate for smaller and smaller intervals of Δ
x
, the interval becomes
infinitely small. This results in the hazard function, which is the instantaneous failure rate
at any point for the continuous sense, as shown in Figure 6.31. The mathematical formulas
are shown in Equations 6.31 and 6.32.
( )
−
(
)
R x
R x
+
∆
x
( )
=
λ
x
lim
∆
(6.31)
⋅
( )
∆
x R x
x
→
0
(
)
f x
R x
*
*
(
)
=
−
(
)
λ
x
*
)
= −
ln
1
F x
*
(6.32)
(
Unlike the above-mentioned probability density function
f
(
x
), the statistical significance
of the hazard function means the failure probability of a specimen, which is failed at the
infinitely small interval of Δ
x
.
The Weibull Distribution Function
Statisticians always prefer large samples of data, but engineers and clinicians are forced to
do statistical analysis with very small samples, even as few as three to five failures. When
the result of a failure involves safety or extreme costs, it is inappropriate to request more
failures. The Student's
t
test and the analysis of variance (ANOVA) analysis [100,120,218] are
the most used statistical analysis methods for biological application. These two analysis
methods can simply decide the statistical significance between several groups with differ-
ent experimental variables and parameters. In addition, to characterize the strength data
fluctuation, reliability, failure probability, and failure mechanism of materials, Equation
6.33 represents a powerful statistical distribution function. This function is the Weibull
distribution function, invented by Waloddi Weibull in 1937 and delivered in his hallmark
American paper in 1951 [240]. He claimed that this model can be applied to a wide range
of problems, and the Weibull models have been used in many different applications for
solving a variety of problems from many different disciplines [240-247].
m
x
−
x
F x
(
)
= −
1
exp
−
i
o
(6.33)
i
η
A powerful theoretical argument explains why the Weibull distribution function works
so well for many failure mechanisms. The Weibull model can be derived theoretically as
a form of extreme value distribution, governing the time to occurrence of the “weakest
link” of many competing failure processes. This argument provides the theoretical basis
for choosing a Weibull life distribution model for a particular mechanism. The Weibull
distribution function has been successfully modeling failures for several diverse appli-
diverse appli-
cations such as capacitor, gate oxide, ball bearing, relay, and material strength failure
has been successfully modeling failures for several diverse appli-
several diverse appli-
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