Civil Engineering Reference
In-Depth Information
where
2
2
∫
dx
∫
dx
φ
φ
z
z
2
2
2
ρ
B
⎛⎞
ω
ρ
B
ω
L
L
exp
exp
r
*
4
r
*
1
H
H
κ
=
and
ζ
=
⎜⎟
⎝⎠
ae
zz
ae
zz
2
m
2
4
m
2
ω
∫
dx
ω
∫
dx
φ
φ
z
z
zz
z
z
L
L
Setting the real and imaginary parts of Eq. 8.22 equal to zero, a dynamic stability limit
may then be identified at an in-wind resonance frequency
−
1/2
2
⎛
∫
⎞
φ
dx
z
⎜
⎟
2
ρ
B
L
⎜
*
4
exp
⎟
1
H
ωω
=
+
(8.23)
r
z
2
⎜
2
m
⎟
∫
dx
φ
z
z
⎜
⎟
L
⎝
⎠
when the damping properties are such that
2
∫
φ
dx
z
2
B
ρ
ω
L
*
1
exp
r
H
ζζ
=
=
(8.24)
z
ae
zz
2
4
m
ω
∫
dx
φ
zz
z
L
This type of stability problem is called galloping. It is seen that a galloping instability
can only occur if
*
*
H
is consistently
negative, see Fig. 5.3, but this is a property that vanishes for cross sections with
increasing bluffness.)
Adopting the quasi-static versions of the aerodynamic derivatives given in Eq. 5.26,
then the stability limit is defined by the following mean wind velocity
H
attains positive values. (For a flat plate
2
∫
dx
φ
z
4
m
ζ
z
z
L
VB
=
ω
⋅
⋅
⋅
(8.25)
cr
z
(
)
2
2
−+⋅
CCDB
/
B
∫
dx
ρ
φ
z
LD
L
exp
An analytical solution to the problem of galloping
w
as first presented by den Hartog
[29], showing that galloping can only occur if
CCDB
′
<−
/
.
⋅
L
D
