Information Technology Reference
In-Depth Information
- the lifetime of rails based on a Weibull survival model for the remaining
time before removal of rail sections;
- the velocity of decay of grindable rail defects;
- the impact of grinding on the survival function estimated before;
A Weibull survival model is used to model the rail removal function.
The proba-bility for removing one meter section of rail is calculated. The
cumulated distribution function of the Weibull distribution can be written as
follows:
F
β
. The variable
t
describes
the age of the rail. The parameters
β
and
η
are estimated separately for every
UIC group or HSL. The removals are considered as events; the rail sections
without any removal are used as reference population and are right censored,
i.e. there is no observed defect before the renewal of the line or before the
end of the observation time. Thus, the date of their renewal or the date of
the study is used as lower limit for the removal date. The parameters and are
estimated by applying the least square method on the double logarithm of the
empiric distribution function. The empiric distribution function is obtained
by the Kaplan-Meier method, adapted for censored data. This method is
used to estimate three distribution functions: the distribution of all defects,
of grindable defects and of non-grindable defects. An example is given in
Figure 1.
For each defect listed in the defect database, it is possible to find all
control meas-urements that were carried out on this defect. Let
P
i
be the
depth of a defect at the visit number
i
and
t
i
the corresponding date; the
variable
i
then counts the number of visits carried out on the same defect.
We calculate the increase of the defect between two visits,
ΔP
i
=
(
t
)=1
−
exp
(
−
(
t/η
)
)
where
β
,
η
and
t>
0
P
i−
1
.
Several possible methods to estimate the propagation velocity were tested.
The most intuitive estimator:
P
i
−
=
n
ΔP
i
Δt
i
υ
ˆ
(1)
observations
n
is the number of all defect growths entered into the database and
Δt
i
=
t
i
−
t
i−
1
, is not used, as it depends highly on the number of visits per defect.
The results were obtained with the estimator that is based on a linear
regression for every defect. In general, the results obtained on the defect
propagation concerns defects deeper than 5mm. The defects cannot be ob-
served by ultrasonic inspections. The grinding interventions concern defects
smaller than 1mm. We use an approximation as shown in Figure 2.
The observations in the defect database reveal that the most frequent
defect grows under an angle of about
38
◦
to the vertical. We estimate the
velocity
v
in the graph. Assuming the speed in the propagation direction
remains constant, we have
sin
(
α
)
υ
ˆ
=
υ
·
)
≈
υ
·
0
,
2204
(2)
cos
(
β
