## 7 Benchmark Study of a Harmonically Biaxial Force Excited 2-DOF-Bridge Structure with TSBD Attached
Due to the expected very low Lehr damping ratio in the existing pedestrian bridge and the consideration of the vibration critical case of synchronization of groups of people as well as the purposeful resonance excitation, the decision was made to install classical passive vibration dampers (mass-spring dampers) to reduce the vertical person-induced vibrations of the relevant third natural frequency \(f_{3v}\). The planning and installation of these vibration dampers, as well as the acceptance measurements, carried out after the opening of the bridge for passenger traffic to check the vibration damper effectiveness, are documented in [43]. The actual natural frequencies and Lehr damping ratio of the finished pedestrian bridge were also determined by in situ measurements, using the forced-vibration testing method. The results matched very well with the values determined in advance by the numerical calculation (e.g.,\(f_{3v,measured}=1.42\) Hz and \(\zeta_{H,3,measured}=0.43\%\), cf. with Figure 10). The results of the in situ measurements are also documented in [43]. In this article, the parametric-excited TSBD, which can be used as an alternative to the classical vibration dampers (mass-spring dampers), is investigated for reducing the vertical pedestrian-induced vibrations of the biaxial-excited bridge. Here, the parametric resonance of the rolling body in the TSBD is deliberately induced and used to achieve the corresponding vibration damping of the considered bridge structure. Again, the decisive third natural frequency with the corresponding vertical vibration mode \(f_{3v}=1.37\) Hz of the structure is considered within the tuning process of the TSBD and an optimal tuning of the TSBD with steel balls as rolling bodies is carried out. Figure 10: Calculated natural frequencies and corresponding mode shapes of the considered pedestrian bridge: (**a**)\(f_{1v}=0.58\) Hz; (**b**)\(f_{2v}=0.78\) Hz; (**c**)\(f_{3v}=1.37\) Hz; (**d**)\(f_{4v}=2.14\) Hz; and (**e**)\(f_{1v}=2.53\) Hz; (**f**)\(f_{5v}=3.08\) Hz. The mass ratio of the rolling body to the kinetically equivalent mass of the bridge structure is chosen as \(\mu=m_{R}/m_{H}=0.02\). The continuous dynamic system of the bridge structure is reduced to an equivalent discrete 2-DOF system with displacements \(w(t)\) and \(v(t)\) for the investigation of a biaxial vibration for the third vertical and for the first horizontal eigenmode (see Figure 10). For the third vertical and first horizontal eigenmode, the kinetically equivalent mass with \(m_{H}=112.240\) kg was taken from the results of the numerical calculation of the eigenfrequencies and the corresponding vibration modes. The total mass of the rolling body of the TSBD results therefore to \(m_{R}=\mu\cdot m_{H}=2250\) kg (rounded up). If the TSBD is designed with a single steel ball as the rolling element, the required diameter and the corresponding volume would be \(D_{R}=0.82\) m and \(V_{R}=0.29\) m\({}^{3}\). However, the bridge deck, which is designed with a low overall height, does not provide sufficient space for those requirements and, therefore, the single TSBD is divided into a total of six TSBDs acting in parallel and arranged in the vibration maxima of the third vertical eigenmode. Figure 11a,b show the placement of the total of six TSBD at the discrete locations along the bridge and an installation example in the cross-section of the bridge deck. The mass of the rolling body for each individual TSBD is \(m_{R,i}=375\) kg and the associated diameter of the steel sphere results in \(D_{R,i}=0.45\) m.  
The optimal tuning of the parametric-excited TSBD is done by the suitable selection of their natural frequency. In order for the parametric resonance of the rolling body in the  
Figure 11: Installation of parametric excited TSBD to the vibration prone pedestrian bridge, (**a**) placement of the total of six TSBD at the discrete locations along the bridge in the vibration maxima of the third mode shape, (**b**) cross section of the bridge deck with exemplary illustrated installed TSBD.