Civil Engineering Reference
In-Depth Information
2
1
Ddx
∫
φ
2
z
ˆ
⎛
σ
⎞
D D
1
ρ
λ
L
(
)
q
z
exp
2
2
1
g
VV
,
σ
=
⎜
⋅
⋅
⎟
⋅
⋅
⋅
⋅
η
⎜
⎟
R
72
74
2
2
1
m
b
1
2
π
St
ζζ
−
⎛
⎞
⎝
⎠
1
z
1
ae
1
2
1
∫
dx
⎜
φ
⎟
⎜
⎟
z
⎝
⎠
L
2
2
Ddx
∫
φ
2
z
ˆ
⎛
σ
⎞
D
D
1
ρ
λ
L
(
)
q
z
exp
2
2
2
g
VV
,
σ
=
⎜
⋅
⋅
⎟
⋅
⋅
⋅
⋅
η
⎜
⎟
R
72
74
2
2
2
m
b
2
2
π
St
ζζ
−
⎛
⎞
⎝
⎠
2
z
2
ae
2
2
2
∫
dx
⎜
φ
⎟
⎜
⎟
z
⎝
⎠
L
2
D
∫
dx
φ
2
3
θ
⎛
2
⎞
(
)
ˆ
BD
σ
1
ρ
1
λ
L
(
)
q
exp
2
⎜
⎟
2
3
θ
g
VV
,
σ
=
⋅
⋅
⋅
⋅
⋅
⋅
η
R
72
74
2
2
3
⎜
m
⎟
b
3
2
π
St
ζζ
−
⎛
⎞
3
3
ae
⎝
⎠
θ
3
2
3
∫
dx
⎜
φ
⎟
⎜
⎟
θ
⎝
⎠
L
where
32
2
⎫
⎡
⎤
⎛ ⎞
1
VV
/
⎛
−
⎞
V
1
(
)
R
⎪
⎢
⎥
1
gV V
,
=
⎜ ⎟
⋅
exp
−
⎜
⎟
1
R
⎜
⎟
⎪
⎜ ⎟
1
V
⎢
2
b
⎥
⎝
⎠
⎝ ⎠
R
z
1
⎣
⎦ ⎪
⎪
1
2
3
⎧
⎪
32
2
⎡
⎤
⎛ ⎞
1
−
VV
/
V
1
⎛
⎞ ⎪
D
ω
=⋅
(
)
R
⎢
⎥
i
gV V
,
exp
2
V
i
=
⎜ ⎟
⋅
−
where
=
⎜
⎟ ⎬
⎨
⎪
⎩
2
R
⎜
⎟
R
i
2
⎜ ⎟
V
⎢
2
b
⎥
⎪
2
St
⎝
⎠
⎝ ⎠
R
z
2
⎣
⎦
⎪
⎪
32
⎡
2
⎤
⎛ ⎞
1
VV
/
⎛
−
⎞
V
1
(
)
R
⎪
⎢
3
⎥
gV V
,
⎜ ⎟
exp
=
⋅
−
⎜
⎟
3
R
⎜
⎟
⎜ ⎟
3
V
⎢
2
b
θ
⎥
⎪
⎝
⎠
⎝ ⎠
R
3
⎣
⎦
⎭
What then remains are the aerodynamic damping contributions given in Eq. 6.68, from which the
following is obtained:
2
∫
dx
φ
z
1
⎧
2
⎫
()
2
2
⎡
σ
V
⎤
ρ
B
ρ
B
⎪
⎪
L
(
)
r
zz
()
*
1
exp
ζ
V
=
⋅
H
⋅
=
⋅
K
V
,
V
⋅
1
−
⎢
⎥
⎨
⎬
ae
a
R
2
1
4
m
∫
dx
4
m
z
1
a D
φ
⎢
⎥
⎪
⎪
⎣
⎦
1
1
z
z
1
⎩
⎭
L
2
∫
φ
dx
z
⎧
2
⎫
2
()
⎡
V
⎤
2
2
σ
ρ
B
ρ
B
⎪
⎪
L
(
)
r
zz
()
*
1
exp
ζ
V
=
⋅
H
⋅
=
⋅
K
V
,
V
⋅
1
−
⎢
⎥
⎨
⎬
ae
a
R
2
2
4
m
∫
dx
4
m
z
2
a D
φ
⎢
⎥
⎪
⎪
⎣
⎦
2
2
z
z
2
⎩
⎭
L
2
∫
dx
φ
θ
2
3
⎧
⎫
()
4
2
⎡
σ
V
⎤
ρ
B
ρ
B
(
)
⎪
⎪
L
rr
()
*
2
exp
V
A
K
V
,
V
1
θθ
ζ
=
⋅
⋅
=
⋅
⋅
−
⎢
⎥
⎨
⎬
ae
a
R
3
2
3
4
m
∫
dx
4
m
θ
a
φ
⎢
⎥
⎪
⎪
⎣
⎦
3
3
θ
θ
3
⎩
⎭
L
The relevant response diagrams are shown in Figs. 6.13 and 6.14 below.
