Civil Engineering Reference
In-Depth Information
2
∫
φ
dx
θ
2
(
)
4
V
B
⎛
ω
⎞
ρ
L
exp
θ
cr
*
3
where:
A
κ
=
⎜
⎟
ae
⎜
⎟
2
θθ
2
m
ω
∫
φ
dx
⎝
⎠
θ
θ
θ
L
ˆ
(
)
E
0,
V
0
1
It is seen that
θ
κ = . Thus, a static divergence type of
instability limit may be identified under the condition that
ω
=
=
when
η
r
cr
ae
2
∫
dx
φ
θ
2
(
)
4
⎛
ω
V
⎞
B
ρ
L
cr
exp
θ
*
3
A
1
=
(8.18)
⎜
⎟
⎜
⎟
2
m
2
ω
∫
dx
φ
⎝
⎠
θ
θ
θ
L
Since this is a purely static type of unstable behaviour the quasi-static version of
*
A
from Eq. 5.26 applies, and thus, the following critical mean wind velocity for static
divergence is obtained
1/2
⎛
⎞
2
∫
dx
φ
⎜
⎟
θ
2
m
⎜
L
⎟
VB
θ
=⋅
ω
⋅
⋅
(8.19)
cr
θ
⎜
4
2
⎟
BC
′
∫
dx
ρ
φ
M
θ
⎜
⎟
L
⎝
⎠
exp
8.3 Galloping
Let
φ
be the mode shape with the lowest eigen-frequency
ωω
=
whose main
1
z
component is
φ
, i.e.
z
0
T
[
]
φ
≈
0
φ
(8.20)
1
z
Since the resonance frequency associated with this mode is
()
z
V
, then
ω
(
)
V
(8.21)
ωω
=
r
z
cr
and the impedance in Eq. 8.11 is reduced to
(
)
ˆ
2
(
)
(
)
EV
,
1
/
2
i
/
ω
=−
κ
−
ω
ω
+
ζζ
−
ωω
(8.22)
η
r r
e
r z
z
e
r z
zz
zz
