Civil Engineering Reference
In-Depth Information
Thus, the variance of
()
(
)
(
)
x t
a
cos
t
a
cos
t
=⋅
ω
+⋅
ω
is then given by
1
1
2
2
⎧
⎪
2
(
)
aa
aa
2 if
1
+
ωω
=
1
2
2
1
⎪
⎪
⎪
2
2
2
1
2
if
is an integer
1
σ
=
+
ω
ω
≠
⎨
⎪
⎪
x
21
22
(
)
2
2
aaa
sin 2
πω
ω
a
21
1
1
2
2
if
is not an integer
⎪
+
⋅
+
ωω
21
2
/
/
2
πωωωω
−
⎪
⎩
21 12
Similar results would have been obtained if the cosines had been replaced by sinus functions.
Thus, if for instance
()
(
)
(
)
x t
a
cos
t
a
cos
t
1
=⋅
ω
+⋅
ω
and
ωω is an integer
≠
, then the
1
1
2
2
21
variance of
dx
()
(
)
(
)
x t
==− ⋅
a
ω
sin
ω
t
− ⋅
a
ω
sin
ω
t
11
1
2 2
2
dt
is given by
2
2
(
)
(
)
a
a
2
2
−
ω
−
ω
a
a
2
11
2 2
2
1
2
2
σ
=
+
=
ω
⋅
+
ω
⋅
x
1
2
2
2
2
2
n
dx
()
()
Likewise, the variance of the n
th
derivative of
x t
,
ft
=
, is given by
n
n
dt
2
2
a
a
2
2
n
2
n
1
2
σω
=⋅
+⋅
ω
f
1
2
2
2
Illustrating ensemble statistics, a situation where
N
different recordings of a stochastic
process within a time window between 0 and
T
are shown in Fig. 2.3. These may for
instance represent
N
simultaneous realisations of the along wind velocity in space, i.e.
they represent the wind velocity variation taken simultaneously and at a certain distance
(horizontal or vertical) between each of them. Extracting the recorded values at a given
time from each of these realisations will render a set of data
()
X
tk
,
1,....,
N
. On
this data set ensemble statistics may be performed. This is the type of statistics that
provides a stochastic description of the wind field distribution in space.
Another example of ensemble statistics is illustrated in Fig. 2.4.a, where the situation is
illustrated that
N
different observations of a stochastic process have been recorded, each
taken within a certain time window but in this case not necessarily at the same time. Each
of these time series is assumed to be stationary and Gaussian within the short term period
that has been considered. In wind engineering this may be an illustration of the situation
when a number of time series have been recorded of the wind velocity at a certain point in
space, each taken during different weather conditions. In that case one may only be
interested in performing statistics on the mean values and discard the rest of the recordings.
=
k
