Environmental Engineering Reference
In-Depth Information
the operating time range [ t, t + t ] to the product of the total number of
objects N and the operating time range ∆ t .
D
ntt t
(,
+D
)
ˆ ()
ft
=
.
[1.25]
Nt
D
Since
n ( t , t + t ) = n ( t + Δ t ) - n ( t ),
where n ( t + t ) is the number of objects that failed during the operating
time t + t , then the estimate of the p.d.f. is:
ˆ
D +D −
nt t nt
(
)
()
1
Qtt t
(,
+D
)
ˆ
ˆ
ˆ
ft
()
=
=
Qt t Qt
(
)
+D −
()
=
,
[1.26]
Nt
D
D
t
D
t
where ˆ (,
+D is the estimate of the FP in the operating time range, i.e.
the increment of FP in ∆ t .
The estimate of the p.d.f. is the frequency of failures, i.e. the number of
failures per operating time related to the initial number of objects.
Probabilistic definition of p.d.f. follows from [1.26] as the operating
time interval t t 0 and increase of the sample size N→∞
Qtt t
)
ˆ
[
]
d Pt
1
()
Q t t t n t dQ t
(,
+D −
)
()
()
dP t
()
ft
( )
=
lim
=
=
= −
.
[1.27]
D
t
dt
d t
()
dt
D→
t
0
The failure distribution density is essentially the distribution density
(probability density) of the random variable T of the operating time of the
object to failure.
Since Q ( t ) is a non-decreasing function of its argument, then f ( t ) > 0.
One of the possible types of graph f ( t ) is shown in Fig. 1.3.
As seen from Fig. 1.3, p.d.f. f ( t ) characterises the failure rate (or reduced
FP) with which the specific values of the operating time of all N objects
( t 1 , ..., t N ), forming the random value of operating time to failure T pf the
given object, are distributed. Let us say the tests show that the value of
operating time t i inherent to the greatest number of objects as indicated
by the maximum value of f ( t i ). On the other hand, longer operating time t j
was recorded only for a few objects and, therefore, the frequency f ( t j ) of
suc long operating tim being recorded on the general background is small.
Some operating time t and the infinitesimally interval of operating time
of width dt , adjacent to t , are plotted on the abscissa. Then the probability
that the random value of operating time T fits in the elementary section
of width dt is:
￿ ￿ ￿ ￿ ￿
{
}
{
}
P T t t dt
∈ + = < <+ ≈
(,
)
P t T t dt
f t dt
() ,
[1.28]
where f ( t ) dt is the element of the FP of the object in the interval [ t, t +
 
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