## 5 Application of Parametric Forced TSBD for Suppression of Vertical Vibrations  
Typically, the classical tuned solid ball damper (TSBD) considered in this study is applied to engineering structures such as high-rise buildings or towers that are prone to horizontal vibrations only. However, the idea is that, in the case of a vertical vibrating structure (e.g., pedestrian or railway bridges), the parametric forced rolling body of the TSBD absorbs the kinetic energy from the vertical vibrating main system analogous to the classical horizontal orientated vibration dampers in case of a horizontal vibration structure. Hence, the goal of this section is to study the application of a parametric forced TSBD for the reduction of vertical vibrations of engineering structures by optimally tuning the frequency and Lehr damping ratio of the rolling body in the runway to the parametric forcing frequency of the main system. Figure 5 illustrates the mechanical principle of action to damp the vertical vibrations of structures by the attached parameter excited TSBD. Assuming a vertically forced resonance excitation \(F_{v}(t)=F_{v,0}\cos(\Omega_{v}\ t)\) of a 2-DOF-main system where the vertical forcing frequency is given as twice the natural circular frequency of the TSBD, i.e., \(\Omega_{v}=2\cdot\omega_{TSBD}\) and hence \(\lambda=0.25\); according to Equation (19), the main system performs two cycles of vibration while the rolling body in the runway performs just one vibration cycle. Figure 5a depicts the instant of time at which the rolling body in the runway is at the position of its maximum rotational amplitude \(\varphi=\varphi_{max}\) with the corresponding value of angular velocity \(\dot{\varphi}=0\). At this point in time, the main system is located at the point of zero crossing, \(v(t)=0\), of the forced vibration and is not affected by the movement of the rolling body. However, at the location of maximum vertical displacement \(v(t)=v_{max}\) (Figure 5b), the direction of vibration of the main system changes from downward to upward and at this instant of time the rolling body in the runway exhibits its maximum angular velocity \(\dot{\varphi}=\dot{\varphi}_{max}\) and, hence, its maximum centrifugal force \(F_{y}\). Since the generated centrifugal force \(F_{y}\) acts against the direction of the movement of the main system, the forced vibration amplitude \(v(t)\) is significantly reduced by the movement of the rolling body in the runway. The same amplitude reduction effect occurs within the second vibration cycle of the main system when the rolling body reaches its maximum rotational amplitude on the right side of the  
runway and moves back to the position of maximum angular velocity and generates again the maximum centrifugal force that acts against the direction of movement of the main system (Figure 5c,d). To achieve the highest-possible reduction of the vertical vibration amplitude of the main system the attached parametric forced TSBD must be optimally tuned to the parametric resonance frequency. It is noted that the tuning process of the parametric-excited TSBD attached to a vertical vibrating main system differs significantly from the conventional Den Hartog tuning. The method of Den Hartog tuning was already presented in Section 3, and it was shown that in the case of a time harmonic force excitation of the horizontal vibrating main system, the optimal design parameters of the TSBD (natural circular frequency \(\omega_{TSBD}\) and Lehr damping ratio \(\zeta_{TSBD}\)) are just a function of the chosen mass ratio \(\mu=m_{R}/m_{H}\) (see Equation (14)). It was also mentioned that the mass ratio should be chosen in the range of 0.5 to 5% to get a sufficiently high damping effect of the vibration-prone main system.