## 5 Application of Parametric Forced TSBD for Suppression of Vertical Vibrations
It is noted that this measured value of the Lehr damping ratio of the TSBD is below the optimal value that would result from the application of the optimization equations according to [PERSON] [4] (evaluating Equation (14) results \(\zeta_{TSBD,opt}\) = 12.7%). However, for the presented investigations of the effect of the parameter resonance on the vibration damping behavior, carried out in this study, the measured value of \(\zeta_{TSBD}\) = 4.0% is retained. Figure 6: Mechanical model of the considered biaxial-excited SDOF-shear frame structure with TSBD attached. The investigation of the possible negative effect of parametric resonance on the considered biaxial-excited dynamic system with optimally tuned TSBD attached, see Figure 6, is carried out by numerically by solving the coupled nonlinear differential equations of motion with MATLAB/Simulink software. Therefore, Equation (12) is rewritten for the biaxial dynamic action of a combined horizontal and vertical displacement excitation \(w_{t}(t)=w_{g}(t)+w(t)\) and \(v_{t}(t)=v_{g}(t)\) to the considered 2-DOF-main system as follows,  
\[ \begin{split}&(1+\mu)\ddot{w}+\mu(R-r)\ \ddot{\varphi}-\mu(R-r)\ \dot{\varphi}^{2}\varphi+2\zeta_{H,h}\ \omega_{H,h}\ \dot{w}+\omega_{H,h}^{2}\ w=-(1+\mu)\ddot{w}_{g}\\ &\ddot{\varphi}\ +\ \omega_{TSBD}^{2}\frac{\ddot{w}}{g}\ +\ 2\ \zeta_{TSBD}\ \omega_{TSBD}\ \dot{\varphi}\ +\ \omega_{TSBD}^{2}\left[1+\frac{v_{g}}{g}\right]\ \varphi=-\omega_{TSBD}^{2}\frac{\ddot{w}_{g}}{g}\ \.\end{split}  \tag{25}\]  
The horizontal and vertical ground excitations were assumed to be time harmonic \(w_{g}(t)=w_{g,0}\cdot\cos(\Omega_{w}\ t)\) and \(v_{g}(t)=v_{g,0}\cdot\cos(\Omega_{w}\ t)\), whereby the ground motion amplitudes \(w_{g,0}=0.001\) m and \(v_{g,0}=0.021\) m were selected. The horizontal excitation frequency was assumed to be a resonance excitation for the considered SDOF-shear frame structure, i.e., \(\Omega_{w}=2f_{H}\). The vertical excitation frequency was chosen to achieve a parametric resonance of the rolling body of the TSBD at the most critical parameter excitation frequency point \(\lambda=0.25\) (see Figure 4) with \(\Omega_{\varphi}=2\cdot\omega_{TSBD}=4\pi f_{TSBD}\) (see Section 4). To study the influence of the parametric resonance of the rolling body to the damping effect of the SDOF-main system, the calculations are in comparison also performed by setting the vertical ground excitation \(v_{g,0}\) to zero. Figure 7a shows the time history of the horizontal dynamic deformation of the floor mass \(w(t)\) under solely horizontal resonance excitation \(w_{g}(t)=w_{g,0}\cdot\cos(\Omega_{w}t)\) calculated numerically with MATLAB/Simulink without and with activated TSBD in case of absence of the vertical base excitation, i.e., \(v_{g,0}=0\). In the case of the solely horizontal excitation of the SDOF-shear frame structure, a quite-high reduction of the maximum horizontal deformation amplitudes \(w(t)\) results, which corresponds to the damping effect of the optimally tuned TSBD. Without the activated damper, the maximum horizontal vibration amplitude grows up to \(w_{max}=0.08\) m. After activation of the optimally tuned TSBD, the maximum vibration amplitudes are just in the range of 0.005 m, i.e., the reduction of the maximum structural response induced by the optimally tuned TSBD is around 95%. Contrarily, in the case of combined horizontal and parametric resonance most critical vertical excitation of the SDOF-shear frame structure, parametric resonance occurs in the rolling body of the TSBD due to its above-mentioned special selected parameters, the excitation frequency as well as the value of the excitation amplitude. It is seen that the parametric resonance of the rolling body leads to a continuous increase of the horizontal deformations of the shear frame floor mass over time (Figure 7b). Under the given conditions, the vertical excitation hence leads to a significant deterioration up to the complete loss of the vibration damping effect induced by the TSBD.