Civil Engineering Reference
In-Depth Information
θ
ωω=
1
and
θ
ωω=
3
. To simplify the relevant expressions, let us introduce the
z
z
following notation:
2
4
B
B
m
ω
ω
ρ
ρ
ˆ
r
β
=
=
0.05
β
=
=
0.33
γ
=
and
ω
=
z
θ
r
ω
ω
z
θ
z
θ
Due to the flat plate type of aerodynamic properties it is in this particular case only static
divergence and flutter that may occur. The flat plate aerodynamic derivatives are given in Eq. 5.27
(and shown in Fig. 5.3), i.e.:
π
⎡
⎤
ˆ
ˆ
2
FV
FV
−
π
−
⎢
⎥
2
⎡
*
*
⎤
⎢
⎢
⎥
HA
1
1
π
(
)
π
(
)
⎥
⎢
⎥
ˆˆ
ˆˆ
1
++
F
4
GV V
−
1
−−
F
4
GV V
*
*
HA
HA
⎢
⎥
⎢
⎥
2
8
2
2
=
⎢
⎥
⎢
⎥
⎢
*
*
π
(
)
(
)
⎥
⎢
⎥
ˆ
ˆ
ˆ
ˆ
2
FV
G
4
V
FV
G
4
V
3
3
π
−
−
⎢
⎥
⎢
⎥
⎢
2
*
*
HA
⎥
⎣
⎦
⎢
4
4
π
(
)
π
ˆ
ˆ
⎥
14
+
GV
GV
⎢
⎥
⎣
2
2
⎦
ˆ
()
where
VVB V
=
⎡
ω
⎤
⎦
is the reduced velocity, and where
⎣
i
(
)
(
)
J
JY YYJ
ˆ
2
⋅
++⋅
−
ˆ
2
JJ YY
⎛⎞
=
ω
⎛⎞
=−
ω
⋅
+⋅
i
1
1
0
1
1
0
i
10 10
2
F
G
and
⎜
⎝⎠
⎜
⎝⎠
2
2
2
(
)
(
)
(
)
(
)
JY
YJ
JY
YJ
+
+
−
+
+
−
1
0
1
0
1
0
1
0
(
)
J
ˆ
2
are the real and imaginary parts of the so-called Theodorsen's circulatory function.
ω
and
ni
(
)
Y
ˆ
2
n
=
0 or 1
, are first and second kind of Bessel functions with order
n
.
ˆ
ω
ω
,
is the non-
ni
()
ˆ
dimensional resonance frequency, i.e.
ωω
=
B
VV
/
. For an ideal flat plate type of cross
i
i
section
′ = (see quasi static solution given in Eq. 5.29). Thus, the stability limit with
respect to static divergence is identified by
C
π
2
M
12
V
⎛
2
m
1
⎞
V
B
2
cr
cr
B
θ
1.96
=
⋅
⇒
=
≈
⎜
⎟
B
4
C
ω
′
ω
ρ
βπ
⎝
⎠
M
θ
θ
With respect to the flutter stability limit an approximate solution can be obtained from Eq. 8.40
(the Selberg formula), rendering
1/2
⎧
2
⎫
⎡
⎤
1/2
(
)
mm
V
B
⎛⎞
⋅
⎪
ω
⎪
cr
⎢
z
⎥
z
B
θ
−
2
0.6
1
1.67
1
=⋅
−
⋅
≈ ⋅
−
γ
⎨
⎜⎟
⎬
3
ω
⎢
ω
⎥
ρ
⎝⎠
⎪
⎪
θ
θ
⎩
⎣
⎦
⎭
