## 5 Application of Parametric Forced TSBD for Suppression of Vertical Vibrations
From Equation (22), the minimum vertical vibration amplitude \(v_{0,min}\) and acceleration \(\bar{v}_{0,min}\), required to guarantee the occurrence of parametric resonance of the rolling body of the TSBD, can be defined by the following formulas,  
\[v_{0,min}\ =\ \ \zeta_{TSBD}\ \cdot\ \ \frac{g}{\omega_{TSBD}^{2}}\ \ \, \tag{23}\]  
\[\bar{v}_{0,min}\ =\ \ v_{0,min}\ \omega_{TSBD}^{2}=\ \ \zeta_{TSBD}\,g\ \ . \tag{24}\]  
Benchmark Study of a Scaled Harmonically Biaxial-Excited SDOF-Shear Frame Structure with TSBD Attached  
In this section, the application of the presented tuned solid ball damper (TSBD) to a scaled harmonically biaxial-excited SDOF-shear frame structure is demonstrated at first for classical vibration suppression, i.e., parametric resonance in the rolling body not used for vibration reduction. Figure 6 illustrates the mechanical model of the considered SDOF-main system with the TSBD attached; \(w(t)\) and \(\varphi(t)\) denote the horizontal displacement of the floor mass \(m_{H}\) and the angle of rotation of the rolling body in the runway. The shear frame structure is biaxial excited through the horizontal and vertical ground motions \(w_{g}(t)\) and \(v_{g}(t)\). This kind of biaxial dynamic excitation occurs in real structures such as skyscrapers, towers, masts, and wind turbines, for instance, in the event of an earthquake. The aim of this benchmark study is to investigate the effects of a parametric resonance of the rolling body in the roller track on the resulting vibration-damping effect and on the size of the achievable reduction of the horizontal vibration response \(w(t)\), respectively. To study the possible negative effects of parametric resonance of the rolling body, the excitation frequency of the vertical ground motion is set to the most critical value with the intention to induce parametric resonance, i.e., \(v_{g}(t)=v_{g,0}\cdot\cos(\Omega_{v}\ t)\), where \(v_{g,0}\) is the vertical excitation amplitude and \(\Omega_{v}=2\cdot\omega_{TSBD}\) is the circular excitation frequency at the most critical point on the \(\lambda\)-axis of the Ince-Strutt stability diagram (\(\lambda=0.25\), see Figure 4). According to Equation (20), the parametric resonance of the rolling body occurs at \(\lambda=0.25\) just in the case of a Lehr damping ratio \(\zeta_{TSBD}\) smaller than the value resulting from this equation. However, to indicate the difference between the achievable damping effect due t to the installation of the TSBD and the damping effect in the case of parametric resonance of the rolling body, in the first step, the vertical ground excitation is set to zero. In [30], an SDOF-shear frame structure with optimally tuned TSBD was already comprehensively investigated experimentally by the authors with regard just to the horizontal vibration damping effect, and the relevant system parameters of the main system are taken from there. The natural frequency and Lehr damping ratio of the main system were determined by free vibration measurements in [30] with \(f_{H}\) = 1.0 Hz and \(\zeta_{H}\) = 0.60% (in the quite high vibration amplitude range at approx. 2.50 m/s\({}^{2}\)). This was an experimental model setup with extremely low structural damping and hence, it was classified as very vibration prone. The kinetically equivalent mass of the SDOF-main system, including the dead weight of the roller track, is \(m_{H}\) = 4.0 kg. The semicircular runway of the TSBD was designed with a radius \(R\) = 0.2 m and the rolling body (cylindrical steel roller with density \(\rho\) = 7850 kg/m\({}^{3}\)) with radius \(r\) = 0.015 m. The mass of the cylindrical rolling body was chosen with \(m_{R}\) = 0.2 kg, resulting in a mass ratio of \(\mu\) = \(m_{R}/m_{H}\) = 0.05. The natural frequency and Lehr damping ratio of the rolling body in the runway were determined by free vibration tests and the evaluation of the time-decay function resulting in \(f_{TSBD}\) = 0.95 Hz and \(\zeta_{TSBD}\) = 4.0%.