Civil Engineering Reference
In-Depth Information
VB
ρ
(
)
ˆ
()
T
Qt
∫
φ Bv
dx
=
⋅
⋅
(6.33)
i
i
q
2
L
exp
T
()
x
⎡
⎤
where
φ
=
⎣
φφφ
. The Fourier transform of Eq. 6.33
⎦
i
y
z
θ
i
ρ
VB
(
)
ˆ
()
T
i
a
∫
φ Ba
dx
ω
=
⋅
⋅
(6.34)
q
v
Q
i
2
L
exp
where
T
[
]
(
)
x
,
a
a
(6.35)
contains the Fourier amplitudes of the
u
and
w
components. This will then render the
following modal load spectrum
a
ω
=
v
u
w
1
(
)
()
*
S
lim
a
a
ω
=
⋅
Q
Q
Q
π
T
i
i
i
T
→∞
T
⎧
⎫
⎡
⎤ ⎡
⎤
2
VB
1
⎛
ρ
⎞
⎪
(
)
(
)
⎪
ˆ
ˆ
⎢
T
*
⎥ ⎢
T
⎥
lim
∫
dx
∫
dx
=
φ Ba
⋅
⋅
⋅
φ Ba
⋅
⋅
⎨
⎬
⎜
⎟
i
q
v
i
q
v
2
T
⎢
⎥ ⎢
⎥
π
⎝
⎠
T
→∞
⎪
⎪
L
L
⎣
⎦ ⎣
⎦
exp
exp
⎩
⎭
2
VB
ρ
{
}
⎛
⎞
ˆ
ˆ
T
()
() (
)
T
()
()
∫∫
=
⋅
φ
x
⋅
B
x
⋅
S
Δω
x
,
⋅
B
x
⋅
φ
x
dx dx
⎜
⎟
i
1
q
1
v
q
2
i
2
1
2
2
⎝
⎠
L
exp
(6.36)
where
*
*
⎡
⎤
⎡
aa
aa
S
S
1
1
⎤
u
u
u
w
uu
uw
(
)
⎡
*
(
)
T
(
)
⎤
x
,
lim
x
,
x
,
lim
S
Δω
=
a
ω
⋅
a
ω
=
⎢
⎥
⎢
=
⎥
v
⎣
v
1
v
2
⎦
T
T
*
*
SS
π
π
T
→∞
T
→∞
aa
aa
⎢
⎥ ⎣
⎦
⎣
⎦
wu
ww
wu
ww
(6.37)
This is greatly simplified if the cross spectra between flow components are negligible,
i.e.
SS
=
≈
0
, see Eq. 6.17. Then
uw
wu
2
⎡
2
⎤
VB
ρ
()
()
S
ω
=
⋅
J
ω
(6.38)
⎢
⎥
i
Q
i
2
⎢
⎥
⎣
⎦
where:
{
}
ˆ
ˆ
ˆ
2
T
()
()
2
(
)
T
()
()
⎡
⎤
J
∫∫
φ
x
x
x
,
x
x
dx dx
=
⋅
B
⋅
I S
⋅
Δω
⋅
B
⋅
φ
(6.39)
i
i
1
q
1
⎣
v
v
⎦
q
2
i
2
1
2
L
exp
is the joint acceptance function, and where
