Environmental Engineering Reference
In-Depth Information
Units of running time
-1
Operating time
t, s
1. 4
Possible changes of FP
l
(
t
).
t
t
dP t
()
t
∫
∫
Pt
=
ln ()|
Pt
=
ln ()
Pt
=−l
() ,
tdt
()
0
0
0
from the basic equation of the relationship of the main reliability indicators
is as follows:
t
∫
P t
()
=
exp
()
−l
t dt
.
[1.35]
0
The value of λ(
t
)
dt
is the likelihood that the element worked flawlessly
in the operating time range [0,
t
] and fails in the interval [
t, t + dt
].
The equation [1.35] shows that all the reliability indicators
P
(
t
),
Q
(
t
),
f
(
t
) and λ(
t
) are equal in the sense that knowing one of them, we can define
others.
Numerical characteristics of the reliability of non-renewable items.
Mean operating time to failure
The previously discussed functional reliability indicators
P
(
t
),
Q
(
t
),
f
(
t
) and λ(
t
) completely describe the random value of operating time
T
=
{
t
}. At the same time, to solve some practical problems of reliability it is
enough to know some numerical characteristics of this random variable
and, above all,
the mean operating time to failure
.
The statistical estimate
of the mean operating time to failure
1
N
ˆ
=
∑
T
t
,
[1.36]
0
i
1
where
t
i
is the operating time to failure of
i
-th object.
The
probabilistic definition
of the mean operating time to failure is the
expected value
(EV) of the random value
T
defined by:
∞
{ }
∫
T M T
=
tf t dt
() .
=
[1.37]
0
0
Using the expression for the density distribution of failures
f l(t).
()
= −
dP t dt
()/
,
and integration by parts, equation [1.37] can be transformed to the form
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