Civil Engineering Reference
In-Depth Information
Usually, vortex shedding induced dynamic response is largely resonant and narrow-
banded. It will then suffice to only consider the resonant part of the frequency domain
integration in Eq. 6.82, and discard the background part. Thus,
∞
∞
2
ˆ
2
2
()
()
(
)
∫
Sd
x
∫
H
d S
σ
=
ω
≈
φ
⋅
ωω
⋅
ω
r
r
n
r
ˆ
n
η
n
n
n
Q
n
0
0
(
)
S
πω
ω
z
ˆ
⎧
n
n
Q
n
2
2
()
x
n
⇒
σφ
=
⋅
=
(6.89)
⎨
⎩
(
)
r
n
r
n
θ
4
ζζ
−
nae
n
As mentioned above, it is also a reasonable assumption that the integral length-scale
D
q
and
q
for
are identical. Adopting the convenient notation (see Eq. 6.41)
λ
θ
z
⎧
2
M
m x
∫
n
=
φ
=
(6.90)
⎨
⎩
n
n
n
θ
L
S
S
and introducing
and
from Eqs. 6.87 and 5.33, then the following is obtained
Q
n
q
n
12
⎛
⎞
⎜
2
⎟
Ddx
∫
φ
1/2
z
⎜
⎟
⎡
⎤
()
ˆ
σ
φ
x
σ
ρ
BD
λ
L
(
)
⎝
⎠
r
zr
q
exp
⎢
⎥
z
z
g
VV
,
=
⋅
⋅
⋅
⋅
⋅
(
)
z
72
7/4
2
2
D
m
⎢
⎥
z
2
St
b
∫
dx
π
⋅
ζζ
−
φ
z
z
z
ae
z
z
⎣
⎦
L
(6.91)
12
⎛
⎞
⎜
⎟
2
Ddx
∫
φ
1/2
θ
⎜
⎟
()
(
2
⎡
⎤
)
x
σ
ˆ
φ
BD
ρ
λ
L
(
)
q
⎝
exp
⎠
θ
r
⎢
⎥
θ
g
VV
,
σ
=
⋅
⋅
⋅
⋅
⋅
(
)
r
R
θ
72
7/4
m
2
2
θ
2
St
⎢
⎥
∫
dx
θ
π
b
φ
⋅
ζζ
−
θ
⎣
θ
θ
ae
⎦
θ
θ
L
(6.92)
where
3/2
⎡
2
⎤
⎛ ⎞
1
−
VV
/
⎛
⎞
z
V
1
⎧
(
)
R
n
⎢
⎥
gV V
,
exp
n
=
⎜ ⎟
⋅
−
=
(6.93)
⎜
⎟
⎨
⎩
n
n
⎜
⎟
⎜ ⎟
V
⎢
2
b
⎥
θ
⎝
⎠
⎝ ⎠
R
n
n
⎣
⎦
(
)
and where
n
VD
/2
t
.
=
ωπ
⋅
R
n
