Civil Engineering Reference
In-Depth Information
2
∫
∫
φ
dx
φφ
dx
z
z
θ
()
()
2
V
3
V
B
ω
B
ω
ρ
ρ
L
L
exp
exp
z
*
1
z
*
2
H
H
ζ
=
ζ
=
ae
zz
ae
z
2
2
4
m
ω
θ
4
m
ω
∫
φ
dx
∫
φ
dx
z
z
z
z
z
z
L
L
(8.14)
2
∫
dx
∫
dx
φ
φφ
θ
θ
z
()
()
4
3
ω
V
ω
V
B
B
ρ
ρ
L
L
exp
exp
θ
*
2
θ
*
1
A
A
ζ
=
ζ
=
ae
ae
z
θθ
4
m
2
θ
4
m
2
ω
∫
dx
ω
∫
dx
φ
φ
θ
θ
θ
θ
θ
θ
L
L
(8.15)
where
()
()
and
are the mean wind velocity dependent resonance frequencies
ω
z
V
ω
V
T
0
T
()
[
]
()
[
]
x
00
associated with
φ
. A purely single mode
unstable behaviour contains motion either in the vertical direction (i.e. galloping) or in
torsion. Such an instability limit may then be identified from the first or the second row
of the matrices in Eq. 8.11 alone, in which case
x
0
and
≈
φ
φ
≈
φ
1
z
2
2
1
(
)
(
)
.
Otherwise, the unstable behaviour contains a combined motion in the vertical direction
and torsion (i.e. flutter), in which case the instability limit may be identified from Eq.
8.11, and
V
or
V
ωω
=
ωω
=
r
z
cr
r
cr
θ
(
)
(
)
. Motion induced coupling effects between
r
and
r
θ
(i.e. flutter) will only occur if the off-diagonal terms in Eq. 8.11 are unequal to zero, i.e.
if
ωω
=
V
=
ω
V
r
z
cr
θ
cr
∫
(see Eqs. 8.12 - 8.15).
φφ
dx
≠
0
z
θ
L
exp
8.2 Static divergence
Let
φ
be the mode shape in predominantly torsion that has the lowest eigen-frequency.
Let us for simplicity assume that
T
[
]
00
φ
≈
φ
(8.16)
2
At
ω = , the instability effect is static and not dynamic. It is simply a problem of
loosing torsion stiffness due to interaction effects with the air flow. Thus, the impedance
in Eq. 8.11 is reduced to
0
ˆ
(
)
E
ω
=
0,
V
=
1
−
κ
(8.17)
r
cr
ae
η
θθ
