Civil Engineering Reference
In-Depth Information
In this paper, the authors present the results from an experimental campaign, aimed at determining the changes in mass
and damping of a vertically vibrating footbridge due to pedestrians. The results are presented in a generalized manner, i.e. as
the added/subtracted mass or concentrated damping per single pedestrian, at varying amplitudes of vibration and for varying
probabilities of occurrence.
4.2 Methodology
The methodology used for the determination of the changes in mass and damping of a footbridge due to pedestrians is
conceptually simple. First, a detailed experimental modal analysis of the laboratory footbridge without pedestrians
is undertaken. From this, eigenmodes, frequencies, modal mass and damping of the empty footbridge are determined.
Next, a hydraulic actuator is attached to the center of the footbridge; the purpose of this being the displacement- and
frequency-controlled excitation of the footbridge at mid-span. The dynamic properties of the footbridge are once
again determined through vertical dynamic actuation of the footbridge at mid-span. An excitation frequency sweep
around the first natural frequency of the empty footbridge reveals the frequency at which minimum effort is needed
for the actuator to excite the footbridge vertically. Once it is established that the footbridge's dynamic properties have
not changed due to the introduction of the actuator, streaming pedestrians with varying flow rates are introduced to the
footbridge. Longer measurements of the pedestrian-induced loads ensure that the excitation is stationary in nature. For
the experiments reported herewith, vertical excitation was undertaken at predetermined bridge mid-span amplitudes of
1, 5 and 10 mm. The observed shift in frequency can be attributed to two separate effects, a change in structural mass
and/or a change in damping.
If the vibration of the footbridge is restricted to motion in the first eigenmode, then the damped circular frequency of the
empty footbridge will be:
s
k
2
ð
1
ξ
Þ
ω
D
¼
(4.1)
m
where
k
is the modal stiffness,
m
is the modal mass and
is the modal damping ratio of the footbridge's first eigenmode.
When pedestrians are traversing over the footbridge, a change in the footbridge's dynamic properties leads to a new damped
circular frequency:
ξ
s
k
0
ð
ξ
0
2
1
Þ
ω
0
D
¼
(4.2)
m
0
where
k
0
is the modified modal stiffness,
m
0
is the modified modal mass and
ξ
0
is the modified modal damping ratio of the
footbridge's first eigenmode. As the footbridge's modal stiffness is expected to remain constant, it is assumed that
k
0
¼
k
.
By solving (
4.2
) for
k
0
and substituting into (
4.1
), the footbridge's new modal mass as a result of the traversing pedestrians
can be expressed as:
ξ
0
2
m
ω
D
ð
Þ
1
m
0
¼
(4.3)
ω
0
D
2
2
ð
1
ξ
Þ
The footbridge's modified damped circular frequency is readily determined by performing an excitation frequency
sweep, at predetermined displacement levels, close to the original circular frequency of the unloaded footbridge. The
frequency at which the work required by the actuator is minimized is the modified natural frequency for the excited
eigenmode.
The modified damping can be determined indirectly by computing the energy dissipated per vibration cycle by the
footbridge. The energy dissipated per cycle by the footbridge is a function of the load imparted by the actuator and the
velocity of the footbridge at the point of load application, so that [
5
]:
Z
T
E
D
¼
F
ð
t
Þνð
t
Þ
dt
(4.4)
0
