Civil Engineering Reference
In-Depth Information
Using that
(
)
(
)
sin
cos
cos
sin
sin
sin
2
sin
α
⋅
β
−
α
⋅
β
=
α
−
β
and
−+
απ
=−
α
this simplifies
into
1
⎡
1
1
⎡
⎤
ˆ
2
(
)
(
)
(
)
∫
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
∫
ˆ
ˆ
ˆ
ˆ
J
=
4
1
−Δ
x
cos
π
Δ +
x
sin
π
Δ
x
exp
−Δ
ω
x d x
Δ =
4
cos
π
Δ⋅
x
exp
−Δ
ω
x d x
Δ
⎢
⎢
⎥
y
π
⎣
⎦
⎢
⎣
0
0
1
1
⎤
1
(
)
(
)
∫
x
ˆ
cos
x
ˆ
exp
ˆ
x d x
ˆ
ˆ
∫
sin
x
ˆ
exp
ˆ
x d x
ˆ
ˆ
−
Δ⋅
π
Δ⋅
−
ω
ΔΔ+
π
Δ⋅
−
ω
ΔΔ
⎥
⎥
⎦
π
0
0
1
⎧
⎡
(
)
(
)
1
−Δ
x
ˆ
exp
− Δ
ω
ˆ
x
ˆ
⎤
⎪
(
)
4
ˆ
cos
x
ˆ
sin
x
ˆ
=
⋅ −
ωπ π π
Δ
+
Δ
⎨⎢
⎥
2
2
ˆ
ωπ
+
⎪
⎩
⎣
⎦
0
1
⎡
⎤
(
)
exp
ˆˆ
x
−
ω
Δ
(
)
⎢
(
)
⎥
2
2
ˆ
ˆ
ˆ
+
ωπ
−
cos
π
Δ
x
−
2
π π
sin
Δ
x
⎢
⎥
(
)
2
2
2
ˆ
ωπ
+
⎢
⎥
⎣
⎦
0
1
⎫
(
)
exp
ˆˆ
x
1
⎡
−Δ
ω
⎤
⎪
(
)
ˆ
sin
x
ˆ
cos
x
ˆ
−
⋅
ωπ π π
Δ
+
Δ
⎢
⎥
⎬
2
2
π
ωπ
ˆ
+
⎣
⎦ ⎪
⎭
0
Thus, the following is obtained:
⎡
⎤
(
)
1exp
ˆ
ˆ
+
−
ω
ω
⎢
⎥
ˆ
()
2
2
()
()
ˆ
ˆ
ˆ
ψω
=
+
2
π
⇒
J
ω
=⋅
4
ψ
ω
where
⎢
⎥
y
2
2
(
)
2
ˆ
ωπ
+
2
2
ˆ
ωπ
+
⎢
⎥
⎣
⎦
The standard deviation of the dynamic response at
x
=
L
2
is then given by
r
12
()
⎡
∞
⎤
S
ω
2
ˆ
ˆ
(
)
42
(
)
u
2
()
−
∫
ˆ
σ
L
2 . 8 0
=
⋅
⋅
V
H
ω
⋅
⋅
J
ω
d
ω
⎢
⎥
r
y
y
2
y
σ
⎢
⎥
⎣
⎦
0
u
ˆ
()
2
2
()
ˆ
and ω are defined above, and where
where
S
ωσ,
J
ω
u
u
y
−
1
(
)
ˆ
⎡
2
⎤
()
(
)
4
H
1
0.4
2
i
0.005
4.375 10
−
V
0.4
=− + + ⋅ ⋅ ⋅
⎢
⎣ ⎦
The chosen single point spectral density and corresponding normalised co-spectrum of the
turbulent
u
component are shown on the top left and right hand side diagrams in Fig. 6.5. The
non-dimensional frequency response function and the squared normalised joint acceptance
functions are shown on the lower left and right hand side diagrams in Fig. 6.5. The response
spectrum of the along wind
r
component at
ω
ω
ω
y
x
L
2
Vms
40 /
is shown in Fig. 6.6.
As can be seen, it contains a broad banded background part and a narrow banded resonant part at
0.4
=
and
=
r
r
x L
= is plotted versus
the mean wind velocity in Fig. 6.7. [It should be noted that the effect of aerodynamic damping is
considerable (see Example 6.3), and that the validity of the quasi-static theory may be limited.]
rad s
/
2
ω =
. The standard deviation of the dynamic response at
