PSI - Issue 78

Anna Brunetti et al. / Procedia Structural Integrity 78 (2026) 1729–1736

1735

N is the degrees of freedom of the problem. The uncoupled equation involving forces transferred by the TMD is ̃ Z̈ + ̃ Ż + ̃ Z = +Φ , ( + ̇ ) (4) where Φ , is the modal displacement of the i -th mode at the node t where the TMD is attached. The displacement ( ) of node t can be expressed as a combination of the modes; however, when the external force has the circular frequency ω of a vibration mode, this mode dominates over the others, so that the following approximation applies: ≈ , (5) By suitably deriving and substituting equation (5) into (4), the problem condensation on the degree of freedom of the node where the TMD is attached can be obtained: ̃ Φ 2 , ̈ + ̃ Φ 2 , ̇ + ̃ Φ 2 , = Φ , +( + ̇ ) (6) The following dynamic equation of the TMD adds to equation (6): ̈ + ̇ + =− ̈ (7) in order to formulate a problem characterised by 2 DOFs, reconciling the problem with the typical one addressed in the literature. Equations (6) and (7) govern a dynamic problem constituted by 2 DOFs in the unknowns and . ̃ , ̈ + ̃ , ̇ + ̃ , = , +( + ̇ ) (8) Exploiting the above formulation, four devices were designed for the case study, one for each frequency, to mitigate vertical accelerations. Each device is positioned at the point of maximum modal displacement for the specific frequency. The properties of all devices are shown in Table 3.

(ton) (kN/m) (%) (kNs/m) 0.3 27.15 0.10 0.57

Table 3. Mechanical Properties of the TMD Devices. TMD f (Hz)

1 2 3 4

1.52 1.82 2.22 3.77

0.3 0.3 0.3

39.05 58.31

0.10 0.10 0.10

0.68 0.84 1.42

167.94

In order to assess the effectiveness of the devices and to verify their impact on the structural response, a new vibration analysis of the footbridge is carried out, following the methodology outlined in the Sétra guidelines. The analysis was again performed using SAP2000, where the devices were modelled as Two-Joint Link elements, with stiffness and damping properties assigned as specified in Table 3. A mass of 0.3 ton is applied at the free end of each link to represent the TMD mass. Fig. 6 illustrates the results by comparing the configuration before and after the TMD modelling. The installation of the TMDs generates two new frequencies close to the original one. In the graphs of Fig. 6, they are highlighted using the same colour. These frequencies correspond to two distinct modal shapes: one characterised by a phase shift of the TMD with respect to the bridge and the other by out-of-phase movement between the two elements. At these frequencies, two acceleration peaks with significantly reduced amplitude are observed compared to the configuration without devices, which highlights the effectiveness of the damping system. The final adopted configuration significantly improves user comfort by limiting the maximum vertical accelerations to values lower than 0.5 m/s². This value is four times lower than the maximum acceleration recorded in the configuration without devices.

Fig. 6. Comparison of vertical acceleration results for the structural model before and after the implementation of TMDs.