Civil Engineering Reference
In-Depth Information
The envelope process is wider band. If X(t) is
Gaussian
, E(t) obeys a
Rayleigh
law.
8.4.5.3.
Statistical study of the absolute maximum reached by a zero-average
process in a given time period
This section will be directly useful for the seismic analyses described above. We
will present below a model that can be applied to wide-band Gaussian processes
before trying to extend it to narrow-band processes.
8.4.5.3.1.
Average number of threshold crossings per time unit
We consider a positive threshold x
m
. We will deal with the crossings of that
threshold owing to the process:
Xt
x or Xt
!
x
m
m
The crossing average frequency is given by:
f
³
f x
v px , v dv
[8.27]
m
m
f
where p (x,v) are the joint probability density of the process and its derivative.
For a Gaussian process we have:
^
`
1
2
2
2
px, v
1/2ʌıı exp
x
/
V
v
/ı
[8.28]
der
2
2
der
Once the calculations are over, by using [8.12] and introducing the a-
dimensional threshold r = x
m
/V we obtain:
1/ 2
2
fr
2m/m
exp
r/2
[8.29]
2
0
1/ 2
REMARK.-
f0/2
m/m
represents the average frequency of the process
2
0
according to the
Rice
formula.
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