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flow system. This implies that the resonance frequency at
V
0
≠
is different from the
V
0
initial value that was determined at
(or in vacuum). In general the consequence of
this effect is that any response calculation involving the aerodynamic derivatives
contained in
a
K
will demand iterations. However, under normal circumstances the
effects of
a
K
will only be of significant importance in the velocity region at or
immediately below an instability limit. At a characteristic mean wind velocity well
below such an instability limit it is usually the aerodynamic derivatives contained in
=
a
C
ω with increasing
V
to the
determination of the aerodynamic derivatives are most often only of minor importance,
especially as compared to other uncertainties in the theory (see further discussion in
chapters 6.3 and 8). On the other hand, at or in the vicinity of an instability limit the flow
induced changes to the resonance frequency will in most cases be of great importance,
and thus, for the determination of an instability limit this effect can usually not be
ignored (see chapter 8).
Aerodynamic derivatives for an ideal flat plate type of cross section were first
developed by Theodorsen [28]. They are given by:
that play the leading role, and the effects of the changes of
i
⎡
π
⎤
ˆ
ˆ
2
FV
FV
−
π
−
⎢
⎥
i
i
2
*
*
⎢
⎥
⎡
⎤
HA
1
1
π
(
)
π
(
)
⎢
⎥
⎢
⎥
ˆˆ
ˆˆ
1
F
4
GV
V
1
F
4
GV
V
++
−
−−
*
*
HA
HA
⎢
i
i
i
i
⎥
⎢
⎥
2
8
2
2
=
(5.27)
⎢
⎥
⎢
⎥
*
*
(
)
π
(
)
⎢
ˆ
ˆ
ˆ
ˆ
⎥
⎢
⎥
2
π
FV
−
G
4
V
FV
−
G
4
V
3
3
i
i
i
i
⎢
⎥
⎢
2
⎥
*
*
HA
⎣
⎦ ⎢
⎥
4
4
π
(
)
π
ˆ
ˆ
⎢
⎥
14
+
GV
GV
i
i
2
2
⎣
⎦
ˆ
i
()
VVBV
where
=
⎡
ω
⎤
⎦
is the reduced velocity, and
⎣
i
(
)
(
)
JJY YYJ
ˆ
2
⋅
++⋅
−
⎫
⎛⎞
=
ω
i
1
1
0
1
1
0
F
⎪
⎜
⎝⎠
2
2
(
)
(
)
JY YJ
JJ YY
+
+
−
⎪
⎬
1
0
1
0
(5.28)
ˆ
2
ω
⋅
+⋅
⎛⎞
⎪
i
10 10
2
G
=−
⎜⎟
⎪
2
(
)
(
)
⎝⎠
JY
YJ
+
+
−
⎭
1
0
1
0
are the real and imaginary parts of the so-called Theodorsen's circulatory function. Their
content
(
)
(
)
J
ω
ˆ
2
and
Y
ω
ˆ
2
,
n
=
0 or 1
, are first and second kinds of Bessel
ni
ni
functions with order
n
, and
ˆ
ω
is the non-dimensional resonance frequency, i.e.
ˆ
−
= =
. The flat plate aerodynamic derivatives given in Eq. 5.27 are
plotted in Fig. 5.3. (The division of ω
()
1
ˆ
B
VV V
/
ωω
i
i
with 2 in Eq. 5.28 stems from Theodorsen's
B
/
2
rather than
B
which is chosen herein.).
choice of frequency normalization with
