Civil Engineering Reference
In-Depth Information
1/2
() ( )
ˆ
ˆ
ª
º
G
S
,0
ω
=
¬
ω
(A.21)
11
j
xx
j
¼
1/2
m
1
ª
−
º
() ( )
()
ˆ
ˆ
ˆ
¦
2
G
S
,0
G
ω
=
ω
−
ω
(A.22)
«
»
mm
j
xx
j
mk
j
¬
¼
k
1
=
n
−
1
ˆ
( )
ˆ
() ()
()
ˆ
¦
S
,
s
G
G
ωΔ
−
ω
⋅
ω
xxj
kj kj
ˆ
()
k
1
G
=
ω
=
(A.23)
mn
j
ˆ
G
ω
nn
j
Example A.2
( )
A process
x
is statistically distributed in time and space. Its cross-spectrum
S
ω Δ
,
s
is
xx
()
defined by the product between the single point spectrum
S
ω shown in Fig. A.3 and its root-
x
( )
coherence function
Coh
ω Δ
,
s
shown in Fig. A.4. I.e.,
xx
( ) ( )
( )
S sS h s
,
,
ω
Δ=
ω
⋅
ω
Δ
xx
x
xx
()
exp
i
ϕ ω
The phase spectrum
ª
º
¼
is assumed equal to unity for all relevant values of ω and
¬
xx
Δ
s
. Let us set out to simulate the process at three points in space, each a distance 10 m apart.
Thus,
T
T
[
]
[
]
ǻ
s
s
s
s
010 20
=Δ Δ Δ =
1
2
3
Let us for simplicity settle with the three point frequency segmentation shown in Fig. A.3. I.e.
T
T
[
]
[
]
Ȧ
=
ωωω
=
0.3
0.7
1.1
and
Δ=
0.4
1
2
3
(It should be noted that this frequency segmentation is only justified by the wish of obtaining
mathematical expressions with reasonable length, such that a complete solution may be presented.
For any practical purposes such a coarse segmentation will most often render unduly inaccurate
results.) The single point spectrum at these frequency settings are then (see Fig. A.3)
T
T
() () ()
[
]
ª
S
S
S
º
4.0
7.6
3.0
S
=
ω
ω
ω
=
¬
¼
x
x
1
x
2
x
3
while the corresponding values of the root coherence function are given by (see Fig. A.4)
