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Earthquake Engineering Research - HKU Libraries - The University ...

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451<br />

due to the Den Hartog criteria. <strong>The</strong> effectiveness of the TMD could be observed as the maximum<br />

displacements are reduced to 1mm in comparison to the undamped motion with maximum value of<br />

about 7mm.<br />

When performing a numerical modal analysis of a structure, natural frequencies higher than the real<br />

ones are obtained in general, as the masses and material stiffness are assumed on the save side (usually<br />

the mass is overestimated and the material stiffness is underestimated). <strong>The</strong>se assumptions finally<br />

result in calculated frequencies to be about 10-20% higher than the real occurring frequencies.<br />

Experimental testing on the footbridge always is recommended to optimize the TMD. Nevertheless, in<br />

order to show the influence of higher real system stiffness than the assumed one (resulting in a nonoptimal<br />

configuration of the TMD), the dynamic response of the bridge is shown in Fig. 4.3 for a<br />

running pedestrian.<br />

-no TWO<br />

~TMD with optimal tuned passive damper<br />

0004<br />

0005<br />

""TMD with subopttmal tuned passve damper<br />

-TMD with suboptimal tuned semi-active damper<br />

Fig. 4.3 Bridge response with non-optimal TMD properties (stiffness<br />

variation)<br />

It can be observed that in case of a non-optimal tuned TMD, the amplitudes are app. 35% higher than<br />

in case of optimal tuning. In case of a non-optimal tuned semi-active TMD, the maximum<br />

displacements (and accelerations) do not exceed the values of the optimal tuned passive TMD. This<br />

shows the excellent performance of a semi-active controlled TMD.<br />

Due to presence of live loads on the bridge deck, the mass of the structure may significantly vary<br />

which again results in a non-optimal configuration of the TMD. In case the total mass of the bridge is<br />

very high, only little influence on the natural frequency has to be expected, as it varies proportional to<br />

the square root of the mass. However, in case of the footbridge in Forchheim, the live loads may<br />

increase the total bridge mass by a factor of 2. Thus, the passive TMD cannot be tuned optimal for<br />

different load cases.<br />

In order to show this effect, the bridge mass of the simplified model was increased by a factor 1.6<br />

(taking into account that the corresponding eigenform only represents a part of the total mass). <strong>The</strong><br />

resulting response of the bridge is shown in Fig. 4.4 for the undamped case as well as for the cases<br />

with a passive TMD and with a semi-active TMD.

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