Civil Engineering Reference
In-Depth Information
8.4 Dynamic stability limit in torsion
A stability problem in torsion is related to galloping in the sense that it involves a single
mode type of motion. Let
φ
be the mode shape with the lowest eigen-frequency
ωω
=
whose main component is
φ
, i.e.
2
θ
T
[
]
φ
≈
00
φ
(8.26)
2
Since the resonance frequency associated with this mode is
()
V
, then
ω
(
)
V
(8.27)
ωω
=
r
cr
θ
and the impedance in Eq. 8.11 is reduced to
(
)
ˆ
2
(
)
(
)
EV
,
1
/
2
i
/
ω
=−
κ
−
ω
ω
+
ζζ
−
ωω
(8.28)
η
r r
e
r
θ
θ
e
r
θ
θθ
θθ
where
2
2
∫
dx
∫
dx
φ
φ
θ
θ
2
4
4
B
⎛ ⎞
ω
B
ω
ρ
ρ
L
L
exp
exp
r
*
3
r
*
2
A
A
κ
=
and
ζ
=
⎜ ⎟
⎝ ⎠
ae
ae
2
m
2
4
m
2
θθ
ω
∫
dx
θθ
ω
∫
dx
φ
φ
θ
θ
θθ
θ
θ
L
L
Setting the real and imaginary parts of Eq. 8.28 equal to zero, a dynamic stability limit
may then be identified at an in-wind resonance frequency
−
1/2
2
⎛
⎞
∫
dx
φ
θ
⎜
⎟
4
B
ρ
L
exp
⎜
*
3
⎟
1
A
ωω
=
+
⋅
⋅
(8.29)
r
θ
⎜
2
m
2
⎟
∫
dx
φ
θ
θ
⎜
⎟
L
⎝
⎠
when the damping properties are such that
2
∫
dx
φ
θ
4
B
ω
ρ
L
exp
r
*
2
A
ζζ
=
=
(8.30)
θ
ae
θθ
4
m
2
ω
∫
dx
φ
θθ
θ
L
