Civil Engineering Reference
In-Depth Information
I
V
0.08
1) the turbulence intensity
=
σ
=
(see Eq. 3.14)
ww
0.3
x
f
z
⎛⎞
L
x
x
f
f
f
u
L
100
162
m
L
2) the integral length scales:
=⋅
=
,
=
(see Eq. 3.36),
⎜
⎝⎠
u
w
10
12
x
f
()
S
ω
1.5
LV
⋅
w
w
x
f
3) the auto spectral density:
=
(see Eq. 3.25)
2
(
)
5 3
σ
12.25
LV
w
+
⋅
ω
⋅
w
x
⎛
ω
⋅Δ
⎞
ˆ
(
)
Co
,
x
exp
C
4) the normalised co-spectrum:
ω
Δ=
−
⋅
⎠
(see Eq. 3.41)
⎜
⎟
ww
wx
V
⎝
()
CC
6.5 / 2
1.0
where
=
=
π
≈
.
wx
wy
f
Let us allot the following values to the remaining constants that are necessary for a numerical
calculation of the relevant dynamic response quantities at
x
=
L
2
:
r
(kg/m
3
)
ρ
B
(m)
D
(m)
m
(kg/m)
m
(kgm
2
/m)
(rad/s)
ω
(rad/s)
ω
ζ
ζ
1
2
1
2
1.25
20
4
0.8
2.0
0.005
0.005
4
5
610
10
⋅
Since
m
and
m
are constant along the span, then the modally equivalent and evenly distributed
masses
mm
=
and
mm
=
. It should be noted that
1
1
2
2
T
2
2
T
2
2
φφ
⋅
= =
φ
sin
π
x L
………..
and
………..
φφ
⋅
= =
φ
sin
π
x L
11
1
z
22
2
L
m
L
⎫
and that
∫
φφ
⋅
dx
=
for any combination of
=
z
or
θ
.
⎬
⎭
mn
1
2
2
n
0
Finally, let us for simplicity adopt quasi-static values to the aerodynamic derivatives, except for
*
2
*
2
(
)
A
which is responsible for aerodynamic damping in torsion. Adopting
A
C
′
V B
=−
β
ω
MM
i
0.2
and
β = provides a good approximation to the flat plate properties. Thus, the aerodynamic
derivatives associated with motion in the across wind vertical direction and torsion are given by
(see Eq. 5.26):
ˆ
*
1
*
2
*
*
1
*
⎡
⎤
*
*
⎡⎤
⎡⎤
H
ˆ
0
ˆ
⎡⎤
A
AC
V
⎡
HA
HA
HA
⎤
−
V
−
⎢⎥
⎢⎥
=⋅
4
4
⎢⎥
⎢
⎥
⎢
⎥
ˆ
2
*
*
HC
H
=⋅ −⋅
′
⎢
β
V
⎥
=
0
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎢⎥
⎣⎦
⎢⎥
⎢
⎥
L
2
*
MM
5
5
⎢⎥
⎢
⎥
⎢
⎥
ˆ
2
2
*
*
V
A
V
⎢
⎥
⎢
⎣⎦
⎣
⎢
⎥
⎣⎦
3
3
⎣
6
6
⎦
⎦
ˆ
(
)
where:
. The aerodynamic coefficients associated with changes in stiffness and damping
are then given by (see Eq. 6.51 and 6.52, or the fully expanded versions in Eqs. 6.53 and 6.54):
VVB
ω
=
i
2
B
(
)
ρ
(
)
*
2
*
2
2
∫
BH
B A
dx
∫
dx
κ
=
⋅
φ φ
+
φ φ
φ
+
φ
ae
z
3
3
z
θ
θ
θ
θ
ij
2
m
i
j
i
j
i
i
i
L
L
exp
2
B
(
)
ρ
(
)
*
*
2
*
2
2
∫
H
BA
B A
dx
∫
dx
ζ
=
⋅
φ φ
+
φ φ
+
φ φ
φ
+
φ
ae
z
z
1
z
1
2
z
θ
θ
θ
θ
ij
4
m
i
j
i
j
i
j
i
i
i
L
L
exp
