###### Abstract  
In this paper, parametric forced tuned solid ball dampers (TSBD) are considered for vibration control of engineering structures in an untypical way. The special feature of the presented investigation is to evaluate the potential application of parametric forcing of the rolling cylindrical or spherical body in the runway for reducing the vertical vibrations of a vibration-prone main system. Typically, tuned solid ball dampers are applied to structures that are prone to horizontal vibrations only. The coupled nonlinear differential equations of motion are derived and the phenomenon of parametric resonance of the rolling body in the runway is analyzed. A criterion for avoiding parametric resonance is given to achieve the optimal damping effect of the TSBD. In the second part of the article, a method for the targeted use of parametric resonance to reduce the vertical vibrations of engineering structures is presented and verified, considering a biaxially harmonic excited pedestrian bridge. It is shown that, with a suitable choice of damper parameters, a stable vibration of the rolling body in the runway is formed over the course of the vibration despite the occurrence of parametric resonance and that the maximum vertical vibration amplitudes of the main system can be reduced up to 93%. Hence, the here presented untypical application of parametric forced TSBD for reducing the vertical forced vibrations of vibration-prone main systems could be successfully demonstrated. t +
Footnote †: journal: Applied Physics Letters  
amplitudes of up to 95% can be achieved over the entire relevant excitation frequency range [1]. The first known application of vibration absorbers dates to 1909 [2]. [PERSON] patented a so-called slinger tank for damping the rolling motion of ships which was installed, for example, in the passenger liners \"Bremen\" and \"Europa\", whose rolling motion could be reduced by approx. 2/3 by means of the installed slinger tanks. The theoretical principles of tuned mass dampers (TMDs) were first published by [PERSON] and [PERSON] [3] in 1928. The procedure for optimal tuning of the TMD to the natural frequency of an undamped single-degree-of-freedom system under harmonic force excitation was treated extensively and in detail by [PERSON] [4] in 1936. [PERSON] stated that the optimum effect of the vibration damper after the choice of the mass ratio succeeds solely through the suitable determination of its natural frequency and its Lehr damping ratio. The extension of [PERSON]'s theory to damped single- and multiple-degree-of-freedom systems and to different types of force and displacement excitation was presented in [5; 6; 7; 8; 9]. Vibration absorbers can be divided into solid and liquid dampers in terms of the mechanical properties of the moving damper mass [1]. In the case of liquid dampers of the tuned liquid column damper (TLCD) type, usually, a fluid medium (e.g., water) is enclosed in a U-shaped piping system. When the main system vibrates due to external dynamic forces, a phase-shifted movement of the entire liquid column starts. The vibration reduction results from the transfer of structural vibration energy to the liquid column movement inside of the piping system and the occurrence of energy dissipation via friction and turbulence effects. The magnitude of energy dissipation is properly increased to the desired optimal level by orifices built into the fluid flow [10; 11; 12; 13; 14; 15]. A second type of liquid damper is the so-called tuned liquid sloshing damper (TLSD) in which the phase-shifted sloshing motion of the liquid surface is used for vibration damping of the main system [16]. Solid dampers of the mass-spring damper type (tuned mass damper--TMD) and pendulum type (tuned pendulum damper--TPD) are widely used in the application for vibration damping of engineering structures and have been extensively studied scientifically [1]. Recent studies on vibration control using passive dampers are given in [17; 18; 19; 20].