3 Structural dynamics of the new Komárom Bridge In this section the Komárom Bridge is discussed in detail.
The FEM model of the bridge (AxisVM software, X4) at
stage-10 can be seen in Fig. 4. The orthotropic steel deck
with the longitudinal trapezoidal ribs was simplified by
using beam elements. The top flanges of the two stiffening
girders were defined with respect to the effective width
and thickness of the deck plate. The crossbeams at every
3 m were merged into one beam at every 24 m; therefore,
the stiffness and mass properties were adjusted adequately.
The pylon was also modelled by using beams. The stay
cables were modelled by using truss elements. The sec-
ond order effect of the cable tension forces on the dynamic
properties was included. The vibration amplitudes to be
determined were expected to be moderate; therefore,
modal analysis was utilized, which is based on linearity.
In Fig. 4 the first bending mode shape is also illustrated
over the FEM model. The corresponding natural frequency
is f = 0.363 Hz. At the next stage, a new deck segment was
erected and welded, and a new cable pair was installed
on the riverbank side of the pylon (stage-11, f = 0.346 Hz).
The modelling concept is illustrated in Fig. 5. The vertical
vibration of the cantilever at the sensor position was deter-
mined by using modal analysis. By considering the first
relevant bending mode, scalar equation Eq. (1) was solved
by using the Newmark-beta time advancement scheme for
the unknown modal displacement y 1
y t y t y t f t 1
The modal force on the right-hand side includes the
vertical sinusoidal distributed force time function (p y )
representing the VIV forces that will be addressed later.
The linear spring and viscous forces acting from the k th
TMD to the proper point of the deck (Q TMD,k ) were calcu-
lated from the relative displacement and velocity.
f t z p t z dz z Q t y k TMD k 1
In Eq. (1) y 1
, ω 01
and γ are the modal displacement, the
circular frequency and the modal damping belonging to
the first mode. In Eq. (2) Ф
(z) is the normalized bend-
ing mode shape function, determined by the AxisVM soft-
ware. The structural model was validated by a series of
jumps performed by the designers at the end of the can-
tilever at stage-10. The TMDs were already active in this
stage, but they could be inactivated by fixation. By this
way, the damping of the pure steel structure as well as
the effect of the TMDs could be conveniently studied.
The measured and simulated vertical accelerations of the
cantilever without TMDs are shown in Fig. 6. The results
with TMDs can be seen in Fig. 7. The simulation time step
size Δt was selected to 0.01 s based on accuracy analysis.
The jumping load (ten series of jumps) was defined to
the end point of the cantilever as a periodic (piecewise-lin-
ear) discrete P(t) time function instead of the distributed
p y (t,z) in Eq. (1). The measured natural frequency with
fixed TMDs was in the range of f = 0.38–0.39 Hz, slightly
higher than the calculated (f = 0.363 Hz). The measured
logarithmic decrement of damping without TMDs was
close to δ = 0.02, which is proposed for steel structures in
Eurocode. Therefore, it was used in the further dynamic
analysis as intrinsic damping of the first mode of the
bridge. The measured damping with the activated TMDs
was δ = 0.08, which is very close to the value obtained from
the simulation. To conclude, the measured and calculated