## 5 Application of Parametric Forced TSBD for Suppression of Vertical Vibrations
In contrast, to achieve parametric resonance of the rolling body of the TSBD in a wide range of the force excitation frequency and, hence, to get the highest-possible vibration damping effect under vertical time harmonic force excitation of a vertical vibrating main system, the circular natural frequency of the parametric-forced TSBD \(\omega_{TSBD}\) must be chosen equal to the most critical parametric resonance frequency with the major areas of instability at \(\lambda=0.25\) (cf. Equation (19) and Figure 4),  
\[\omega_{TSBD} = \frac{1}{2}\cdot\frac{\omega_{H}}{1+\mu}\quad. \tag{21}\]  
In Equation (21) a forced resonance excitation of the main system is assumed, i.e., \(\Omega_{H}=\omega_{H}/(1+\mu)\), where the frequency shift of the main system natural frequency due to the self weight of the rolling body \(m_{R}\) is taken into account by the factor \(1/(1+\mu)\). It is recommended to choose the mass ratio \(\mu=m_{R}/m_{H}\) in analogy to the classical Den Hartog tuned TSBD with 0.5 to 5%. Figure 5: Illustration of the mechanical principle of action of the parameter excited TSBD attached to a vibration prone main system during two consecutive vibration cycles of the main system, (**a**) rolling body location left upwards with velocity \(\dot{\varphi}=0\) and main system displacement \(v(t)=0\), (**b**) rolling body location in the runway center with velocity \(\dot{\varphi}=\dot{\varphi}_{max}\) and main system displacement \(v(t)=v_{max}\) (**c**) rolling body location right upwards with velocity \(\dot{\varphi}=0\) and main system displacement \(v(t)=0\), (**d**) rolling body location in the runway center with velocity \(\dot{\varphi}=\dot{\varphi}_{max}\) and main system displacement \(v(t)=v_{max}\). The achievable damping effect is, of course, also dependent on the size of the Lehr damping ratio \(\zeta_{TSBD}\) of the TSBD. On the one hand, the damping ratio of the TSBD should be chosen as small as possible to achieve a fast onset of parametric resonance of the rolling body of the TSBD. On the other hand, a certain level of energy dissipation is necessary to limit the maximum vibration amplitudes of the rolling body in the runway and to dissipate the absorbed vibration energy of the main system into heat. It is recommended to limit the amplitude of the rolling body with \(\varphi_{limit}=60^{\circ}\) to prevent grip losses in the runway and to fulfill the assumed condition of pure rolling (see assumed rolling condition in Equation (1)). To guarantee the occurrence of parametric resonance of the rolling body of the TSBD and to achieve a significant reduction of the forced vertical vibrations of the main system via the application of the parametric-excited TSBD, the Lehr damping ratio must be in any case smaller than the cut-off value defined in Equation (20), i.e.,  
\[\zeta_{TSBD}\ <\ \ \omega_{TSBD}^{2}\,\frac{v_{0}}{g}\ \ . \tag{22}\]  
Note that Equation (22) differs from Equation (20) with regard to the inequality sign in order to excite parametric resonance vibrations of the rolling body in the runway. Equation (22) shows that the cut-off value depends on the value of the vertical vibration amplitude \(v_{0}\) of the main system. Hence, it is necessary to calculate the expected maximum vertical vibration amplitude \(v_{0}=v_{0,max}\) within the predesign of the TSBD. Once \(v_{0,max}\) is known, the value of the Lehr damping ratio should be chosen as a fraction of the cut-off value defined in Equation (22) to achieve a fast onset of parametric resonance of the rolling body of the TSBD and reduction of the vertical structural vibration amplitude, respectively.