Civil Engineering Reference
In-Depth Information
It is seen that an instability in pure torsion can only occur if
*
A
attains positive values.
*
(For a flat plate
A
is consistently negative, see Fig. 5.3.) Since the quasi-static value of
*
A
is zero, it is futile to define a stability limit based on the quasi-static theory.
8.5 Flutter
As mentioned above, flutter is a dynamic stability problem where
r
couples with
r
θ
.
Such coupling occurs via the off-diagonal terms
κ
and
κ
in Eq. 8.11 above, and
ae
z
θ
ae
z
θ
therefore, it is most prone to occur between modes
φ
and
φ
that are shape-wise
similar and whose main components are
. Experimental observations show
that it is usually the aerodynamic forces associated with the motion in torsion that are the
driving forces in the coupling process.
Let
φ
and
φ
z
φ
be the mode shape with the lowest eigen-frequency
ωω
=
whose main
2
θ
component is
φ
, i.e.
T
[
]
00
φ
≈
φ
(8.31)
2
Let
φ
be another mode that shape-wise is similar to
φ
and whose main component is
φ
, i.e.
z
0
T
[
]
0
φ
≈
φ
(8.32)
1
z
and whose eigen-frequency is
ωω
=
. A flutter stability limit is then identified by
1
z
(
)
ˆ
ˆ
(
)
(
)
is given in Eq. 8.11. Since
r
couples with
r
det
E
,
V
0
where
E
,
r r
V
ω
=
ω
r r
θ
η
η
into a joint resonant motion, then
(
)
(
)
V
V
(8.33)
ωω
=
=
ω
r
z
cr
θ
cr
From a computational point of view it is convenient to split
ˆ
E
into four parts, i.e.
η
(
)
ˆ
ˆ
ˆ
ˆ
ˆ
2
i
EEE EE
(8.34)
η
=++
+
1
2
3
4
