Civil Engineering Reference
In-Depth Information
symbols shown in chapter 4.4, the frequency domain versions of
a
C
and
a
K
are given
by
P PP
HHH
A
P PP
HHH
A
⎡
⎤
⎡
⎤
1
5
2
4
6
3
⎢
⎥
⎢
⎥
C
=
⎢
and
K
=
⎢
(5.23)
ae
5
1
2
⎥
ae
6
4
3
⎥
⎢
AA
⎥
⎢
AA
⎥
⎣
⎦
⎣
⎦
5
1
2
6
4
3
The coefficients contained in
a
C
and
a
K
are then functions of the frequency of
motion, the mean wind velocity and the type of cross section (and to some extent the
initial or mean angle of incidence and the turbulence properties in the oncoming flow).
Usually, they have been experimentally determined in wind tunnel aeroelastic section
model tests, limited to vertical and torsion displacements. Since their main use lies in the
detection of unstable motion at high wind velocities, the primary modal mass and
stiffness properties of the section model will intentionally only contain the eigen-
frequencies associated with the most onerous modes with respect to unstable structural
oscillations. For a plate-like bridge cross section this is usually the lowest mode in
torsion together with the shape-wise similar and lowest vertical mode. (Shape-wise
similarity is required because the effect of aerodynamic coupling between the two modes
is often important.) Since the along wind motion is absent in the section model, all terms
associated with this direction must either be disregarded or taken from the quasi static
buffeting theory (see Eqs. 5.25 and 5.26). The tests may be performed in three
alternative ways. The original procedure was to extract the motion induced forces from
the changes in resonance frequency and damping properties in transient (i.e. decay)
recordings at various wind velocities under the conditions of pure vertical motion, pure
torsion and finally combined vertical and torsion (see appendix C). Another procedure is
to perform ambient vibration tests, again at various wind velocities, and use the theory of
system identification to extract the sought flow-structure interaction properties. The third
procedure is to use a section model that undergoes forced oscillations at various
frequencies, amplitudes and wind velocities. From such a steady-state situation cross
sectional forces are measured by pressure tap recordings on the surface of the model
hull. Subtraction of the forces at zero motion will then render net motion induced effects.
The method of forced oscillations is demanding and generally not in use. Thus, the
frequency at which the aerodynamic derivatives are determined will most often be
associated with the mass and stiffness properties of the relevant section model, as well as
the motion induced forces themselves at various mean wind velocities. I.e., the
aerodynamic derivatives will be associated with the eigen-frequencies of the chosen set
of section model mode shapes, and thus, they will be functions of the reduced velocity
(
ˆ
)
VVB
ω
. For the purpose of full scale calculations the similarity requirements
between model scale and full scale conditions must be fulfilled, and thus, the
aerodynamic derivatives will have to be extracted as functions of
=
i
ˆ
(
)
VVB
ω
=
.
i
