## 7 Benchmark Study of a Harmonically Biaxial Force Excited 2-DOF-Bridge Structure with TSBD Attached  
In this second benchmark study, a harmonically biaxial force-excited 2-DOF-bridge structure with a single TSBD attached is considered (see Figure 8). The horizontal and vertical dynamic displacement of the structure is denoted with \(w(t)\) and \(v(t)\). \(F_{h}(t)\) and \(F_{v}(t)\) define the horizontal and vertical excitation forces which are assumed to be time harmonic, i.e., \(F_{h}(t)=F_{h,0}\cdot\cos(\Omega_{h}\ t)\) and \(F_{v}(t)=F_{v,0}\cdot\cos(\Omega_{v}\ t)\). The special feature of this section is the evaluation of the potential application of parametric forcing of the rolling body in the runway of the TSBD, for reducing vertical vibrations of a vibration-prone main system. The optimal tuning of the TSBD to the vertical parametric resonance frequency of the main system is performed according to Section 5 (Equations (21) and (22)). All parameters of the considered 2-DOF-bridge structure such as the geometry, bending stiffness, and mass, as well as the dynamic parameters that were determined by numerical calculations and dynamic in situ measurements, were taken from a real existing pedestrian bridge [43] located in Austria. The considered bridge is a curved suspension bridge (radius = 85 m) with a span of 104.50 m (measured in the curvature of the bridge deck), see Figure 9. The bridge pylon, which is made of steel S355 J2 + N, has a diameter of 900 to 1000 mm and a total height of 40.5 m. The suspension cable has a cross section  
Figure 7: Time history of the horizontal dynamic deformation of the shear frame floor mass \(w(t)\) with and without activated TSBD: (**a**) solely horizontal resonance excitation \(w_{g}(t)\), i.e., \(v_{g}(t)=0\); (**b**) combined horizontal and vertical excitation \(w_{g}(t)\) and \(v_{g}(t)\) of the SDOF-shear frame structure. of 101 cm\({}^{2}\) and the total of 31 hangers connected to the bridge deck on one side have a diameter of 20.1 to 24.4 mm. The bridge deck is a slender triangular-shaped steel box girder made of steel S460 NL and reinforced with longitudinal stiffeners. On top of the steel box girder, a concrete layer with a thickness of 9 to 11 cm was constructed and a 3 cm thick asphalt pavement is placed on top of it. During the design phase of the bridge, an analysis of the pedestrian-induced vibrations was carried out according to the HIVOSS guideline [44]. Figure 8: Biaxial force excited 2-DOF-bridge structure with a single TSBD attached (the aim here is to reduce the vertical vibrations of the main system with the parametric forced TSBD). Figure 9: Considered pedestrian bridge in the benchmark study of the parametric forced TSBD for reduction of vertical vibrations of the main system (the bridge is executed as a curved suspension bridge in the ground view). The first six natural frequencies and corresponding vibration modes calculated in a spatial numerical model of the bridge structure are shown in Figure 10. It is seen that the determined vertical eigenmodes are analogous to a single-span beam with articulated bearing on both sides (Bernoulli-Euler beam). For the pedestrian-induced vibrations, the third natural frequency \(f_{3v}=1.37\) Hz with the corresponding vertical vibration mode was classified as resonance critical. At this natural frequency, resonance vibrations with high vibration amplitudes can be triggered by groups of people walking \"slowly\" across the bridge and synchronizing themselves. Due to the extremely slender design of the structure and the generally extremely low energy-dissipation capacity of steel as a building material, the Lehr damping ratio was selected as \(\zeta_{H,i}=0.40\%\) within the dynamic calculations for each vibration mode.