## 7 Benchmark Study of a Harmonically Biaxial Force Excited 2-DOF-Bridge Structure with TSBD Attached
runway to occur during vertical resonance vibrations of the bridge structure with installed TSBD, the optimal natural frequency of the rolling bodies is set to the critical value with the largest portion of the unstable region in the Ince-Strutt stability map (see Figure 4) with \(f_{\text{TSBD}}=0.5f_{\text{v},0}\cdot 1/(1+\mu)=0.67\) Hz (see Section 4, Equation (19) and Figure 4, stability parameter \(\lambda=\lambda_{critical}=0.25\)). The associated geometry parameters of the single TSBD are obtained by transforming Equation (9) to \((R-r)=0.38\) m and with \(r=D/2=0.225\) m the radius for the semicircular runway results in \(R=0.618\) m. With the aim of raising the parameter resonance of the rolling body as quickly as possible, the Lehr damping ratio of the TSBD is chosen to be very small with \(\zeta_{\text{TSBD}}=0.01=1\%\). To determine the vibration responses of the bridge deck resulting from biaxial dynamic pedestrian excitation without and with activated optimally tuned TSBD, the coupled nonlinear differential equations of the 3-DOF system given in Equation (12) are solved numerically in the time domain. For this purpose, the MATLAB/Simulink software is used and the continuous solution method ode45 is selected. This is the Runge-Kutta (4, 5) equation solver, which performs a fifth-order integration method with a fourth-order error. The biaxially acting time harmonic excitation forces are assumed in the equations of motion of the equivalent discrete dynamic system to be \(F_{h}(t)=F_{h,0}\cdot\cos(\Omega_{h}~{}t)\) and \(F_{v}(t)=F_{v,0}\cdot\cos(\Omega_{v}~{}t)\), with excitation force amplitudes chosen to be \(F_{h,0}=10\) kN and \(F_{v,0}=1\) kN. Figure 12 shows the calculated time histories of the vertical dynamic deformation \(v(t)\) of the bridge deck due to biaxial pedestrian excitation at the resonance frequency with and without activated parametric-excited TSBD. It can be seen that the vertical vibration amplitudes reach a maximum value of \(v_{dyn,max}=0.14\) m without activated TSBD. After activation of the TSBD, strongly reduced dynamic deformation amplitudes result in the steady state (from \(t=50\) s) with \(v_{dyn,max}=0.009\) m. The installation of the parametric-excited TSBD hence results in a reduction of the maximum vertical dynamic deformations by approx. 93% in the steady state. It can also be seen that vibrational beat effects are present in the transient condition. To investigate the vibration damping effect of the installed vertically parametric forced TSBD in the entire frequency range of interest, the vertical excitation frequency \(\Omega_{v}\) is varied in the next step in the range around the resonance point of the third vertical natural frequency of the bridge deck with TSBD and the amplitude-frequency response function is calculated numerically. The result of this calculation is shown in Figure 13. In these calculations, the horizontal excitation frequency was assumed to be \(\Omega_{h}=2\pi f_{1,h}\). Figure 13  
Figure 12: Numerically calculated time history of the vertical dynamic displacement of the bridge deck without and with parametric forced activated TSBD. indicates that the vibration damping effect of the installed vertically parametric forced TSBD is given over the entire excitation frequency range of interest. The amplification of the parametric resonance vibration in the rolling body of the TSBD leads to the absorption of the vibration energy, which is building up in the bridge structure due to the external dynamic excitations. The absorbed energy is dissipated into heat by solid friction between the rolling body and the runway. The stabilization of the initially unstable parametric resonance vibration of the rolling body results from the reduction of the vertical vibration amplitudes of the bridge structure through the restoring effect of the installed TSBDs. The vibration-control efficiency of the classical optimally tuned TMD is additionally displayed in Figure 13 for comparison.