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where in this case
i
and
j
are equal to 1 or 2. Introducing the choice of aerodynamic derivatives
given above, then:
*
3
∫
φφ
BH dx
z
θ
12
2
2
3
3
ρ
B
ρ
B
ρ
B
⎛ ⎞
V
L
exp
*
3
,
κ
=
0
κ
=
⋅
=
⋅
H
=
⋅
C
′
⋅
⎜
⎝ ⎠
ae
ae
L
11
2
12
2
m
∫
dx
2
m
2
m
B
ω
φ
1
1
1
1
z
1
L
22*
3
∫
φ
BAdx
θ
2
2
2
4
4
ρ
B
ρ
B
ρ
B
⎛ ⎞
V
L
exp
*
3
,
κ
=
0
κ
=
⋅
=
⋅
A
=
⋅
C
′
⋅
⎜
⎝ ⎠
ae
ae
M
21
2
22
2
m
∫
dx
2
m
2
m
B
ω
φ
2
2
2
2
θ
2
L
2*
1
∫
Hdx
φ
z
1
2
2
2
ρ
B
ρ
B
ρ
B
V
L
exp
*
1
H
C
,
0
ζ
=
⋅
=
⋅
= −
⋅
′
⋅
ζ
=
ae
L
ae
11
2
12
4
m
∫
dx
4
m
4
m
B
ω
φ
1
z
1
1
1
1
L
*
1
∫
BA dx
φφ
θ
z
21
2
3
3
ρ
B
ρ
B
ρ
B
V
L
exp
*
1
A
C
ζ
=
⋅
=
⋅
= −
⋅
′
⋅
ae
M
21
2
4
m
∫
dx
4
m
4
m
B
ω
φ
2
2
2
2
θ
2
L
22*
2
∫
BAdx
φ
θ
2
2
2
4
4
⎛ ⎞
ρ
B
ρ
B
ρ
B
V
L
exp
*
2
A
C
′
ζ
=
⋅
=
⋅
= −
⋅
β
⋅
⎜
⎝ ⎠
ae
M
M
22
2
4
m
∫
dx
4
m
4
m
B
ω
φ
2
2
2
2
θ
2
L
The non-dimensional frequency response function is then given by (see Eq. 6.49)
−
1
2
⎧
⎫
⎛
⎞
⎡⎤
1
⎡⎤
1
⎪
⎪
ˆ
()
(
)
H
ω
=−
I
κ
− ⋅
ω
diag
+
2
i
ω
⋅
diag
⋅
ζζ
−
=
⎜
⎟
⎨
⎢⎥
⎢⎥
⎬
η
ae
⎜
⎟
⎣⎦
ae
ω
ω
⎪
⎣⎦
⎪
⎝
⎠
i
i
⎩
⎭
1
−
⎧
0
0
⎫
⎡
κ
⎤
−
2
−
1
⎛
⎡
ζ
⎤
⎞
10
⎡
0
⎤
⎡
0
⎤
⎡
0
ω
ω
ζ
⎤
⎪
⎡ ⎤
⎪
ae
ae
12
1
1
1
11
2
2
i
⎜
⎟
−
⎢
⎥
−
ω
⎢
⎥
+
ω
⎢
⎥
−
⎢
⎥
⎨
⎬
⎢ ⎥
⎢
⎥
01
0
2
1
⎜
0
ζ
⎟
κ
−
−
ζ
ζ
⎢
⎥
⎢
0
ω
⎥
⎢
0
ω
⎥
⎢
⎥
⎣ ⎦
⎣
⎦
⎪
⎪
⎣
ae
⎦
⎣
⎦
⎣
⎦
2
⎣
ae
ae
⎦
⎩
⎝
⎠
⎭
22
2
2
21
22
42
42
4
−
−
−
where:
κ
=
97.66 10
⋅
⋅
V
,
κ
=
1.563 10
⋅
⋅
V
,
ζ
=−
39.06 10
⋅
⋅
V
,
ae
ae
ae
12
22
11
4
42
−
−
ζ
=−
1.563 10
⋅
⋅
V
,
ζ
=−
0.1563 10
⋅
⋅
V
, and where all other quantities are given
ae
ae
21
22
above. The aerodynamic stiffness and damping coefficients
κ
,
κ
,
ζ
,
ζ
,
ζ
ae
ae
ae
ae
ae
12
21
11
21
22
are shown in Fig. 6.8. The absolute value of the determinant of the non-dimensional frequency
response function (at
V
= ) is shown in Fig. 6. 9 together with the single point spectral density
and normalised co-spectrum of the wind turbulence
w
component.
0
