## 8 Conclusions  
In the first part of the paper, the nonlinear equations of motions of a two-degree-of-freedom (2-DOF) main system with a single TSBD attached were derived taking a combined horizontal and vertical force excitation into account. It was demonstrated that, under certain conditions, the vertical forcing frequency and vibration amplitude of the main system parametric resonance of the rolling body in the runway occurs which can negatively influence or even completely cancel out the vibration-damping effect of tuned vibration absorbers. The effects of a vertical parametric excitation on the vibration damping effect of the TSBD were demonstrated on an SDOF-shear frame structure with TSBD attached. A cut-off value for avoiding parametric resonance of the rolling body of the TSBD was defined in the form of a minimum-required value of the Lehr damping ratio.  
In the second part of the paper, it was investigated whether the deliberate induction of parametric resonance of the rolling body of the TSBD can be applied for the reduction of vertical vibrations of bridge structures that are biaxially excited by passenger traffic. It was shown that with a suitable choice of natural frequency and Lehr damping ratio of the TSBD, a stable vibration of the rolling body is formed despite the occurrence of the parametric resonance, and the forced vertical vibrations of the bridge deck can be reduced up to 93% over the total frequency range of interest.  
The special feature of the investigated parametric-excited solid-body damper is its use for the reduction of vertical vibrations of engineering structures. Hereby, the optimal choice of the damper parameters is no longer made according to [PERSON]'s optimization criteria, but by suitable choice of the parameters of the Ince-Strutt stability map and the size of the Lehr damping ratio of the TSBD and its rolling body, respectively. To achieve the  
Figure 13: Numerically calculated frequency response function of the vertical dynamic displacement of the bridge deck without and with parametric forced TSBD activated and additionally with classical optimally tuned TMD activated.  
highest-possible vibration damping effect and a fast response behavior of the TSBD, the stability parameter \(\lambda\) has to be chosen with \(\lambda=(\omega_{TSBD}/\Omega_{H})^{2}=0.25\) and the Lehr damping ratio \(\zeta_{TSBD}\) should be chosen as small as possible.  
In an ongoing research project, further investigations are being carried out to optimize the presented parametric-excited solid-body damper and to extend the passive to an adaptive (frequency-adaptable) vibration damper system. In the course of this work, the application of the parametric-excited solid-body vibration dampers for the reduction of vertical vibrations of railway bridges during train passage is also being investigated. Additionally, experimental studies regarding the evaluation of the vibration-control effectiveness of TSBD and a comparison to the results gained from the numerical simulation are also planned.  
Conceptualization, M.R.; introduction, M.R. and J.M.; mechanical model, M.R.; optimization, M.R. and J.M.; numerical studies, M.R.; writing--original draft preparation, M.R.; writing--review and editing, J.M.; visualization, M.R.; supervision, M.R.; project administration, M.R. All authors have read and agreed to the published version of the manuscript.  
Open Access Funding by TU Wien.  
Not applicable.  
The data presented in this article is available on request from the corresponding author.  
The authors declare no conflict of interest.  
The following abbreviations are used in this manuscript:  
FRFFrequency Response Function  
MDOFMultiple Degrees Of Freedom  
MWMegawatt  
SDOFSingle Degree Of Freedom  
TMDTuned Mass Damper  
TLCDTuned Liquid Column Damper  
TLDTuned Liquid Damper  
TSDTuned Sloshing Damper