Civil Engineering Reference
In-Depth Information
whose properties are such that
()
*
T
S
ω =⋅
G G
(A.15)
xx
xx
xx
Assuming a frequency segmentation of
N
equidistant points, the simulated
simultaneous time series at
are then given by
m
=
1,2,....,
M
mN
()
(
)
()
¦¦
xt G
2c s
t
(A.16)
=
ω
⋅
Δω
⋅
ω
⋅ +
ψ
m
mn
j
j
nj
nj
11
==
where
j
is the frequency segment number and
is an arbitrary phase angle between
zero and
2
π . In most cases of a homogeneous wind field (see Eq. 2.87)
ψ
nj
ˆ
( ) ( ) ( )
S sS S s
ωΔ
,
=
ω
⋅
ωΔ
,
(A.17)
xx
x
xx
S
is the single-point spectral density of the process,
where
ss s
is the
Δ =
−
x
mn
spatial separation between points
x
and
x
, and where
ˆ
( )
( )
()
S
,
s
Coh
,
s
exp
i
ωΔ
=
ωΔ
⋅
ª
ϕ ω
º
¼
(A.18)
¬
xx
xx
xx
ˆ
ˆ
ˆ
T
()
*
Thus, defining a Cholesky decomposition
G G
, then the time series at
S
ω =⋅
xx
xx
xx
m
1,2,....,
M
are given by
=
mN
() () (
)
ˆ
()
¦¦
xt G S
=
ω
⋅
2
ω Δω
⋅ ⋅
cos
ω
⋅ +
t
ψ
(A.19)
m
mn
j
x
j
j
nj
nj
11
==
where
ˆ
m
G
is the content of
ˆ
x
G
(i.e. the reduced versions of
m
G
in Eq. A.14)
ˆ
ª
º
G
GG
000
""
0
0
11
«
»
ˆ
ˆ
«
00
0
0
»
""
21
22
«
»
#
#
#
#
#
ˆ
«
»
()
G
ω
=
«
(A.20)
xx
»
ˆ
ˆ
ˆ
ˆ
GG G G
""
00
mm n m
1
2
«
»
«
#
#
#
#
#
»
«
»
ˆ
ˆ
ˆ
ˆ
ˆ
GG G G G
"""
«
»
¬
¼
MM n m M
1
2
and where a Cholesky decomposition will render
