###### Abstract
On the other hand, the combination of a ball-shaped damper mass with a spherical container can be described as a tuned solid ball damper (TSBD) which is used for multidirectional vibration damping. Both can also be described as tuned rolling (rotary) mass dampers (TRMD) [25]. The investigations of TSBD focus mainly on high and comparatively slim buildings such as chimneys and wind turbines, as they are exposed to multidirectional wind and ocean wave loads [26] and furthermore do only provide little space for additional installations. Regarding the limited available space, TSBDs or TSRDs have two major advantages: they provide high-density damper masses such as steel or other metals and demand less space than conventional pendulum and spring-mass dampers. In [25] the research went even further and investigated the application of a rolling mass damper in voided concrete slabs with promising results. Although the motion of a rolling damper mass in a cylindrical or spherical container is more difficult to describe than the movement of a tuned mass damper (TMD), TSBDs and TSRDs can be designed using the same optimization equations according to [PERSON] [4]. The most important characteristic of TRMDs is their way of dissipating energy through rolling friction between the rolling mass and the container. Furthermore, even with multiple rolling masses, those types of dampers behave like a system with two degrees of freedom if combined with a single degree of freedom main system [27]. Therefore, either a friction model e.g., Coulomb friction, or an equivalent viscous damping term, needs to be considered [27]. In addition, combinations of tuned rolling mass dampers with well-investigated damper types, such as tuned liquid column dampers (TLCDs), have recently been examined more closely [28; 29]. In this article, a parametric forced tuned solid ball damper (TSBD) for use in vibration control of engineering structures is presented and investigated in two relevant application examples. The damping effectiveness of the TSBD was already investigated experimentally by the authors in [30] on an SDOF shear-frame structure under solely horizontal forced excitation. The investigated TSBD consists of a rolling body that performs a frictionally driven rolling motion in a semicircular runway. The damper derives the restoring effect from gravity alone and the energy dissipation takes place via rolling friction that is properly tuned by varying the friction level of the runway. It is shown that due to the vertical excitation of the TSBD with specific values of excitation frequencies, the vibration response of the rolling body becomes unstable due to parametric resonance and these vibrations lead to a significant loss of the damping effectiveness in case of dominant horizontal vibrating main systems. In the first part of the article, the parametric forcing is studied, and a cut-off value of the Lehr damping ratio is defined for the TSBD to prevent any unwanted parametric resonance vibrations in the rolling body of the damper. In the second part, the conscious use of the parametric resonance of the rolling body for the reduction of vertical forced vibrations of engineering structures, e.g., pedestrian bridges, is studied and evaluated. Therefore, the parametric resonance of the rolling body is forced consciously by properly tuning the frequency and damping of the TSBD. The special feature is the natural frequency tuning of the TSBD. To achieve the highest-possible vibration energy absorption from the vertical forced main system to the vibration damper, the natural frequency of the TSBD is tuned to \(f_{TSBD}=0.5f_{H}\cdot 1/(1+\mu)\). In this case, the natural frequency of the rolling body in the runway is equal to the critical value of parametric resonance with the largest portion of the unstable region in the Ince-Strutt stability map (stability parameter \(\lambda=\lambda_{critical}=0.25\)).