Civil Engineering Reference
In-Depth Information
2
∫
∫
φ
dx
φφ
dx
z
z
θ
2
3
B
B
ρ
ω
ρ
ω
L
L
*
1
exp
*
2
exp
r
r
H
H
ζ
=
ζ
=
ae
zz
ae
z
2
2
4
m
θ
4
m
ω
∫
dx
ω
∫
dx
φ
φ
zz
zz
z
z
L
L
2
∫
dx
∫
dx
φ
φφ
z
θ
θ
4
3
ρ
B
ω
ρ
B
ω
L
L
*
2
exp
*
1
exp
r
r
A
A
ζ
=
ζ
=
ae
ae
z
2
2
θθ
4
m
θ
4
m
ω
∫
dx
ω
∫
dx
φ
φ
θθ
θθ
θ
θ
L
L
The solution procedure demands iterations, because the aerodynamic derivatives can
only be read off if the outcome,
and
c
V
, are known. The theory of flutter was first
presented by Theodorsen [28]. In cases where
ω
r
/
is larger than about 1.5, then
Selberg's formula [22] may be used to provide a first estimate of the mean wind velocity
that defines the flutter stability limit
θ
ωω
z
1/2
2
⎧
⎡
⎤
1/2
⎫
(
)
mm
⋅
⎛
⎞
ω
⎪
⎪
z
z
B
θ
V
0.6
B
⎢
1
⎥
=
ω
⋅
−
⋅
(8.40)
⎨
⎜ ⎟
⎬
cr
θ
3
⎢
ω
⎥
ρ
⎪
⎝ ⎠
⎪
θ
⎣
⎦
⎩
⎭
Example 8.1
Let us consider a slender horizontal beam type of bridge with a cross section whose aerodynamic
properties are close to those of an ideal flat plate, and set out to calculate the possible stability
limits associated with the two mode shapes
0
T
T
[
]
[
]
0
00
φ
=
φ
φ
=
φ
1
2
with corresponding eigen-frequencies
ω
and
ω
, and with modally equivalent and evenly
z
distributed masses
m
and
m
θ
. It is for simplicity assumed that
φφ
≈
and that
L
=
L
. Let
z
θ
exp
us allot the following values to the necessary structural quantities
ρ
(kg/m
3
)
B (m)
m
(kg/m)
(kgm
2
/m)
m
θ
ω
(rad/s)
ζζ
=
z
z
θ
1.25
20
0.8
0.005
4
5
610
10
⋅
We wish to investigate the properties of the instability limits at various values of the frequency
ratio
θ
ωω, and thus it is assumed that
ω
=
0.8
rad/s while
ω
is arbitrary between
z
z
