Application of Parametric Forced Tuned Solid Ball Dampers for Vibration Control of Engineering Structures

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.

1. Introduction

The increasing slenderness of recently built engineering structures makes them highly susceptible to vibrations, especially in the case of low natural structural damping. Tower-like structures such as wind turbines, skyscrapers, transmission towers, and chimneys are dynamically stressed by earthquakes, machinery, wind, and ocean waves and are excited by forced vibrations. In the case of a coincidence of an excitation frequency with a natural frequency of the structure, resonance vibrations occur which can lead to structural damage or even failure. In the case of frequently repeated dynamic stresses with resonance excitation acting over a long period of time, the problem of material fatigue comes to the fore, and the service life of the structure is significantly reduced. By installing a vibration absorber that is optimally tuned to the relevant resonance frequency, the vibration amplitudes that occur during resonance can be reduced to a residual minimum and the initial low existing structural damping is significantly increased. Vibration absorbers are usually secondary structures consisting of a mass, a spring, and a damper element, which are preferably coupled to the vibration-prone main system at the location of the largest dynamic deflections. The vibration absorber extracts a dominant part of the vibration energy from the dynamically stressed main system and dissipates it into heat via viscous damper elements or via friction effects [1]. If the vibration absorber is optimally tuned to a selected natural frequency of the main system, reductions of the maximum vibration 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]. Frahm 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 Ormondroyd and Den Hartog [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 Den Hartog [4] in 1936. Den Hartog 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 Den Hartog’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]. The extension of passive solid and liquid dampers to adaptive and active vibration dampers has also been extensively investigated scientifically and is partly applied in practice [21,22,23].

In the case of liquid dampers, the apparent low density of the moving damper mass (e.g., water ρ = 1000 kg/m³) compared to conventional solid dampers (e.g., steel ρ = 7850 kg/m³) results in a considerable disadvantage in terms of the space required at the installation site of the damper. For example, a 5MW offshore wind turbine with a 112 m steel tube length has a diameter of 5.5 m at the tower head [24]. The total vibrating mass of such a wind turbine is approx. 560 tons and it consists of the head mass (nacelle, hub, and rotor blades) with approx. 435 tons and the kinetically equivalent tower mass with approx. 125 tons. If the ratio of the damper mass to the vibrating mass of the main system is 5%, which is common in the design of vibration absorber systems, the required damper mass results in 28 tonnes, which corresponds in the case of a fluid damper to approx. 28 m³ of horizontally moving (active) fluid mass. To achieve the desired bidirectional vibration damper effect, at least two liquid dampers must be installed in the two possible vibration directions, resulting in a total space requirement of 56 m³ at the tower head of a wind turbine. Assuming an available tower diameter of 5.5 m, the total floor space is just 23.76 m² and the horizontal section of the liquid dampers would have to be at least 2.36 m high. Thus, it is evident that even if liquid dampers offer several advantages compared to solid dampers, their application to real structures often fails because of the limited available installation space in the case of vibration-prone slender high-rise structures such as, e.g., wind turbines. Compared to liquid dampers, the installation space required for the moving mass of pendulum dampers is, in principle, significantly smaller due to the raw density of steel. However, the very low fundamental natural frequency in the range of approx. 0.20 Hz usually found in on- and offshore wind turbines requires a very large pendulum length of approx. 6.21 m. Hence, the space gained by the small steel pendulum mass compared to the fluid mass of liquid dampers is compensated by the very long pendulum lengths.

The optimal tuning of vibration absorbers is, in the first step, performed to a single selected resonant frequency of the main system by applying the optimization equations according to Den Hartog [4]. The two optimization parameters (natural frequency and damping) of the vibration damper result solely from the selected mass ratio μ = m_A/m_H (active damper mass m_A to the kinetically equivalent mass of the main system in the considered vibration mode). As an example: With a resonance frequency of the forced-excited main system of f_H = 1 Hz and μ = 0.05, the optimum natural frequency of the damper results in f_A,opt = 0.95 Hz, and the optimum Lehr damping ratio to ζ_A,opt = 13%. Once the damper is optimally tuned, vibration reductions of up to 95% can be achieved over the entire frequency range of interest. However, in the case of coupled vibration modes (not well separated) and tuning of multiple dampers to a single selected vibration mode, a state space optimization method is preferably applied to determine the optimal absorber parameters [14,15].

Lately, newer types of solid-body vibration absorbers, with ball or cylinder-shaped damper masses, have been investigated in quite a bit of detail. They consist of a cylindrical- or spherical-shaped container which allows a rolling movement of the damper mass. The combination of the cylinder-shaped mass and container can be described as a tuned solid roll damper (TSRD) which is used for unidirectional vibration damping. 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 Den Hartog [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 = 0.5·f_H·1/(1 + μ). 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 λ = λ_critical = 0.25). The damping ratio of the TSBD is chosen to be very low, to ensure the occurrence of parametric resonance in case of vertical forced vibrations of the main system. It is shown that despite the occurrence of parametric resonance vibrations, a stable motion of the rolling body is established in the TSBD and a very high vibration damping with a reduction of up to 93% of the main system is achieved compared to the uncontrolled case without any vibration absorber. The initially unstable motion of the rolling body becomes stable due to the energy absorption from the main system to the damper and the associated reduction of the maximum vertical vibration amplitudes. The exploitation of parametric excitation for vibration damping proves to be extremely effective and it is shown that the vibration reductions of the main system can be achieved over the entire excitation frequency range of interest.

2. Mechanical Model

2.1. Description of Considered TSBD

In the case of uniaxial forced vibrations of the main system, the passive tuned solid ball damper (TSBD) consists of a cylindrical roller as the active damper mass, which performs a rolling motion involving friction in a semicircular runway (see Figure 1a). When the TSBD is extended to general plane vibration-control applications of arbitrary forced excited main systems, a sphere is designed as the active damper mass and a calotte as the roller track (see Figure 1b). The advantage of TSBD against conventional vibration absorbers such as the tuned mass damper (TMD) or the tuned pendulum damper (TPD) is the fact that there are no mechanical components such as springs and/or damping elements that require maintenance during the service life and therefore the TSBD is extremely cost efficient. The TSBD derives its restoring effect from gravity alone and energy dissipation is achieved via rolling friction. By changing the surface roughness of the semicircular runway, the self-damping of the rolling body can be increased to the desired optimum value [30].

2.2. Free-Body Diagram of the TSBD

The substructure method is used to derive the coupled nonlinear equations of motion of a biaxial-excited two-degree-of-freedom (2-DOF) main system with a single TSBD attached. For this purpose, in the first step, the free-body diagram of the TSBD is considered in a general plane deflected position (Figure 2).

Figure 3 illustrates the free-body diagram of the biaxial-excited 2-DOF main system with applied reaction Forces F_x and F_y resulting from the movement of the rolling body in the runway of the TSBD.

In Figure 2, the chosen reference point A is shifted by the combined horizontal and vertical dynamic displacement of the 2-DOF main system w(t) and v(t) with respect to the origin O. The coordinates of the origin are denoted by (x, y) and the moving coordinates are denoted by (x′, y′). m_R is the mass of the rolling body and r is the corresponding radius. The runway is assumed to be semicircular with a radius of R. The distance of the chosen reference point A to the center of gravity of the rolling body is (R − r). The degrees of freedom of the rolling body are defined by φ(t) and α(t) and are linked via the rolling condition (pure rolling assumed) as follows,

$$\dot{\alpha }=\dot{\phi } \frac{\left(R-r\right)}{r} .$$

(1)

The nonlinear equation of motion of the rolling body in the runway is determined by applying Lagrange’s equation [31],

$$\frac{d}{dt}\left(\frac{\mathsf{\partial }T}{\mathsf{\partial }\dot{\phi }}\right)-\frac{\mathsf{\partial }T}{\mathsf{\partial }\phi }+\frac{\mathsf{\partial }V}{\mathsf{\partial }\phi }=Q_{\phi } .$$

(2)

In Equation (2), T defines the kinetic energy, V the potential energy, and Q_φ the generalized forces. In the generalized forces Q_φ, the energy dissipation occurring during the rolling process of the rolling body due to friction effects (Coulomb friction) is considered and introduced as equivalent viscous damping in the equation of motion [1]. The kinetic energy T of the horizontally and vertically excited TSBD is composed of the rotational and translational components as follows,

$$T=\frac{1}{2}{I}_S {\dot{\phi }}^2+\frac{1}{2}{m}_R v_S^2+\frac{1}{2}{I}_S {\dot{\alpha }}^2 .$$

(3)

In Equation (2), I_S defines the mass moment of inertia in relation to the center of gravity of the rolling body and v_S its corresponding velocity which results from the time derivative of the displacement variables in x- and y-directions,

$$v_S^2={\dot{w}}^2+2\dot{w} \dot{\phi }\left(R-r\right)\cos \phi +{\dot{v}}^2+2\dot{v} \dot{\phi }\left(R-r\right)\sin \phi +{\left(R-r\right)}^2{\dot{\phi }}^2 .$$

(4)

The potential energy V of the TSBD is given by,

$$V=m_R g \left[v-\left(R-r\right)\cos \phi \right] .$$

(5)

Substituting the center of gravity velocity (Equation (4)) into the kinetic-energy expression (Equation (3)) and performing the derivatives of T and V according to Lagrange’s equation (Equation (2)) yields the nonlinear parametric-excited equation of motion of the TSBD given below,

$$\ddot{\phi } + 2 {\zeta }_{TSBD} {\omega }_{TSBD} \dot{\phi } + {\omega }_{TSBD}^2 \left[1+\frac{\ddot{v}}{g}\right] \sin \phi =-{\omega }_{TSBD}^2\frac{\ddot{w}}{g} \cos \phi .$$

(6)

For small vibration amplitudes, i.e., small rotation angles φ(t) and, hence, sin φ ≅ φ and cos φ ≅ 1; Equation (6) becomes a linear equation of motion with constant natural angular frequency ω_TSBD, see Equation (12). The friction damping of the rolling body in the runway is introduced in Equation (6) by the equivalent viscous damping term ζ_TSBD. It is seen that the vertical forced vibration of the main system (denoted by the acceleration $\ddot{v}$), leads to a time-variant stiffness parameter in the equation of motion of the TSBD and this represents a parametric excitation for the rolling body in the runway. This feature is discussed in detail in Section 4. The linear natural angular frequency of the TSBD is given by,

$${\omega }_{TSBD}^{} = \sqrt{ \frac{1}{1+\frac{I_S}{m_R r^2}+\frac{I_S}{m_R{\left(R-r\right)}^2}} \cdot \frac{g}{\left(R-r\right)} } .$$

(7)

Neglecting the expression [I_S/m_R (R − r)²] in the denominator of Equation (7), the linear natural frequency f_TSBD results in dependence of the geometric shape of the rolling body and the runway to,

$$f_{TSBD} = \frac{1}{2 \pi } \cdot \sqrt{ \frac{g}{ \frac{3}{2} \left(R-r\right) } } ,$$

(8)

for a cylindrical roller running in a semicircular runway and to

$$f_{TSBD} = \frac{1}{2 \pi } \cdot \sqrt{ \frac{g}{ \frac{7}{5} \left(R-r\right) } } ,$$

(9)

for a sphere running in a calotte.

It is seen from Equations (8) and (9) that the natural frequencies of the TSBD are lower than those of a conventional pendulum damper [1] by a factor of (2/3) and (5/7), respectively. An interesting observation arises when the geometric shape of the runway is designed as a cycloid [32]. In this case, the natural frequency of the TSBD results independent of the magnitude of the vibration amplitude and this is a great advantage over conventional pendulum dampers.

The nonlinear reaction forces F_x and F_y resulting from the movement of the rolling body in the runway are determined by applying the conservation of momentum [27] to the free-body diagram in Figure 1 (with respect to the reference point A, dead weight of the rolling body not included),

$$\begin{gathered} F_x=m_R \left[\ddot{w}+\left(R-r\right) \ddot{\phi } \cos \phi -\left(R-r\right) {\dot{\phi }}^2\sin \phi \right] , \\ {F}_y=m_R \left[\ddot{v}+\left(R-r\right) \ddot{\phi } \sin \phi +\left(R-r\right) {\dot{\phi }}^2\cos \phi \right] . \end{gathered}$$

(10)

Both of the reaction forces F_x and F_y contain the terms of absolute rolling-body acceleration (superposition of the translational and rotational acceleration components) as well as the term of centrifugal acceleration. It is seen that the maximum of F_y occurs for φ = 0. In Section 6 this part of the parametric-excited rolling body interaction force is used to reduce the vertical forced vibrations of the considered vibration-prone main system.

2.3. Substructure Synthesis of the Biaxial-Excited 2-DOF Main System with TSBD Attached

For the determination of the coupled nonlinear equations of motion of the considered 2-DOF main system—degrees of freedom denoted as horizontal and vertical dynamic displacement amplitudes w(t) and v(t)—with a single TSBD attached; the derived reaction forces F_x and F_y (Equation (10)) resulting from the motion of the rolling body in the runway are applied in the selected reference point A of the deflected free-body diagram of the main system (Figure 3).

In the considered general plane case the main system is excited horizontally and vertically to forced vibrations. The horizontal and vertical dynamic excitation forces F_h and F_v act at the center of gravity of the 2-DOF main system. It is assumed that the center of gravity coincides with the center of stiffness of the main system. Therefore, no coupled bending and torsional vibrations occur due to the external dynamic forces and inertia forces. The two displacement coordinates of the main system are denoted by w(t) and v(t). Application of d’Alembert’s principle [31] to the free-body diagram of the main system (Figure 2) in the x- and y-directions yields the following two coupled equations of motion,

$$\begin{gathered} \ddot{w}+2{\zeta }_{H,h} {\omega }_{H,h} \dot{w}+{\omega }_{H,h}^2 w=\frac{1}{m_H}\left[F_h-F_x\right] , \\ \ddot{v}+2{\zeta }_{H,v} {\omega }_{H,v} \dot{v}+{\omega }_{H,v}^2 v=\frac{1}{m_H}\left[F_v-F_y\right] . \end{gathered}$$

(11)

The energy dissipation of the main system was introduced into the equations of motion in the horizontal and vertical direction of vibration as linear viscous damping ζ_H,h and ζ_H,v. ω_H,h and ω_H,v define the natural angular frequencies of the main system in the horizontal and vertical vibration directions. The kinetically equivalent moving mass of the main system m_H is assumed to be identical for both vibration directions. Inserting the derived reaction forces of the rolling body F_x and F_y into Equation (11) and including the equation of motion of the TSBD (Equation (6)) yields the following three coupled nonlinear differential equations of motion for describing the motions of the biaxially excited main system with a single attached TSBD,

$$\begin{gathered} \left(1+\mu \right)\ddot{w}+\mu \left(R-r\right) \ddot{\phi }-\mu \left(R-r\right) {\dot{\phi }}^2\phi +2{\zeta }_{H,h} {\omega }_{H,h} \dot{w}+{\omega }_{H,h}^2 w=\frac{F_h}{m_H} , \\ \left(1+\mu \right)\ddot{v}+\mu \left(R-r\right) \ddot{\phi } \phi +\mu \left(R-r\right) {\dot{\phi }}^2+2{\zeta }_{H,v} {\omega }_{H,v} \dot{v}+{\omega }_{H,v}^2 v=\frac{F_v}{m_H} , \\ \ddot{\phi } + {\omega }_{TSBD}^2\frac{\ddot{w}}{g} + 2 {\zeta }_{TSBD} {\omega }_{TSBD} \dot{\phi } + {\omega }_{TSBD}^2 \left[1+\frac{\ddot{v}}{g}\right] \phi =0 . \end{gathered}$$

(12)

In Equation (12) the nonlinear functions sin φ ≅ φ and cos φ ≅ 1 were linearized for simplification of the equations of motion. However, the time-variant stiffness parameter and the quadratic terms of the angular velocity of the rolling body are still left in the equations for further analysis of the damping effectiveness using the parametric resonance effect of the TSBD (see Section 6) in the case of vertical forcing of the main system and TSBD, respectively. μ defines the ratio of the damper mass m_R (i.e., rolling body mass) to the kinetically equivalent mass of the main system m_H,

$$\mu =\frac{m_R}{m_H}$$

(13)

3. Den Hartog Tuning of a Single TSBD Attached to an SDOF-Main System

To achieve the optimum damping effect of the single attached TSBD attached to an SDOF-main system, the two design parameters δ = ω_TSBD/ω_H = f_TSBD/f_H and ζ_TSBD must be selected appropriately. δ and ζ_TSBD define the ratio of the natural frequency of the TSBD to the natural frequency of the SDOF-main system and the linearized viscous damping of the TSBD. The analytical formulas for determining the optimum design parameters under time-harmonic force excitation F(t) = F₀·cos(Ωt) of an SDOF-main system, where F₀ and Ω are the excitation force amplitude and the circular excitation frequency, and under the conservative assumption of an undamped main system, were first derived by Den Hartog [4],

$${\delta }_{opt}= \frac{{\omega }_{TSBD}}{{\omega }_H} = \frac{f_{TSBD}}{f_H} = \frac{1}{ 1+\mu }, {\zeta }_{TSBD,opt}= \sqrt{ \frac{3\mu }{ 8 {\left(1+\mu \right)}^3 } }$$

(14)

Equation (14) shows that the optimum design parameters of the harmonic force excited the SDOF-main system with a single TSBD attached depend solely on the mass ratio μ = m_R/m_H. In practical applications of the TSBD, the mass ratio is chosen in the range of 0.5 to 5%, and, therefore, the optimum frequency ratio results in 0.995–0.952. For instance, in the case of a natural frequency of the SDOF-main system of f_H = 1.0 Hz and μ = 5%, the optimum natural frequency of the vibration damper results in f_TSBD = 0.95 Hz. The corresponding optimal value of the Lehr damping ratio, in this case, would be ζ_TSBD = 0.127 = 12.7%. A deviation from the optimal design parameters, which, for example, may result from a change in the natural frequency of the main system over its lifetime, leads to a decrease in the effectiveness of the vibration damper. Petersen [1] shows that deviations from the optimal natural frequency of the vibration absorber leads to significantly higher losses of effectiveness compared to deviations from the optimal Lehr damping ratio. For instance, a detuning of the vibration damper’s eigenfrequency by 15% leads to a tripling of the maximum vibration response of the main system, for μ = 4%. In contrast, an increase of the Lehr damping ratio by the factor of 1.5 leads to an insignificant loss of the optimum vibration absorber effect, for μ = 4%. The losses in the effectiveness of not optimally tuned vibration absorbers become smaller at higher mass ratios, i.e., the damper is less prone to detuning that occurs naturally during its service life. To counteract the detuning effect of vibration dampers, the application of semi-active dampers proves to be advantageous. Hence, an innovative approach for the extension of the passive TSBD to a semi-active damping device is presented in Section 6.

It is noted that the optimal design parameters given in Equation (14) are applicable in the case of a harmonic force-excited SDOF-main system and under the assumption of an undamped main system ζ_H = 0. For other possible excitations and response parameters of an SDOF-main system, the optimum design parameters are given in [1,33]. An extension of Den Hartog’s tuning formulas to Multi-Degree-Of-Freedom (MDOF) main systems are presented in [34] and a state space optimization approach for MDOF-main systems and multiple attached vibration absorbers is given in [14,15]. In addition, Pocanschi and Phocas [35] give the following analytical equations for the correction of the optimal design parameters considering the structural damping of the main system ζ_H > 0 when optimizing the vibration damper,

$$\begin{gathered} {\tilde{\delta }}_{opt} = {\delta }_{opt} - \left(0.241+1.7 \mu -2.6 {\mu }^2\right) {\zeta }_H - \left(1-1.9 \mu +{\mu }^2\right) {\zeta }_H^2 \\ {\tilde{\zeta }}_{A,opt} = {\zeta }_{A,opt}+ \left(0.13+0.12 \mu +0.4 {\mu }^2\right) {\zeta }_H-\left(0.01+0.9 \mu +3 {\mu }^2\right) {\zeta }_H^2 \end{gathered}$$

(15)

4. Parametric Excitation of the TSBD Due to Vertical Forcing

The nonlinear equation of motion of the combined horizontal and vertical excited TSBD was derived in Section 2.2 (Equation (6)) and can be written in linearized form (sin φ ≅ φ and cos φ ≅ 1) as follows,

$$\ddot{\phi } + 2 {\zeta }_{TSBD} {\omega }_{TSBD} \dot{\phi } + {\omega }_{TSBD}^2 \left[1+\frac{\ddot{v}}{g}\right] \phi =-{\omega }_{TSBD}^2\frac{\ddot{w}}{g} .$$

(16)

Neglecting the viscous damping term of the rolling body (ζ_TSBD = 0) and assuming the absence of any horizontal excitation ($\ddot{w}\left(t\right)=0$), Equation (16) becomes the so-called Hill differential equation and can be written in the following form [32],

$$\ddot{\phi } + P\left(t\right) \phi =0 \cdots \mathrm{with}\cdots P\left(t\right) ={\omega }_{TSBD}^2 \left[1+\frac{\ddot{v}}{g}\right] .$$

(17)

Hill’s differential equation is a linear differential equation of second order with a time-variant stiffness parameter P(t) and it is characterized by the fact that the parameter P(t) is a periodic function of time. In the case of an engineering structure, this could be, for example, the vertical periodic human-induced vibration of footbridges. In such vibrating systems, so-called parametric-excited vibrations occur, which are characterized by an unstable behavior of the vibration process in the case of parametric resonance, i.e., the vibration amplitude increases towards infinity despite the presence of structural damping of the dynamic system [32,36]. This fact is the fundamental difference to the conventional resonance phenomenon of linear dynamic systems, in which the vibration amplitude in the steady state reaches a stable constant value in the presence of structural damping. Another fundamental difference is the need for an initial disturbance of the dynamic system to raise parametric-excited vibrations and the infinite number of excitation frequencies at which parametric resonance can occur. A well-known and often-cited example of parametric-excited vibrations is the classical pendulum with a vertically moving suspension point [31,32].

For the special case of a time-harmonic vertical excitation function v(t) = v₀·cos(Ω_v t), where v₀ and Ω_v are the excitation amplitude and the circular excitation frequency, Equation (17) becomes the so-called Mathieu differential equation which is not solvable in closed form. Its stability behavior, i.e., the statement about the limitedness or unlimitedness of the solution functions, has been studied, e.g., by Merkin [36], Wittenburg [37], and Klotter [38]. For a closer examination of the stability behavior, we transform Equation (17) into the mathematically more common normal form by introducing the dimensionless time τ = Ω_v t [38],

$${\phi }^{″} + \left[\lambda +\gamma \cos \tau \right] \phi =0 \mathrm{with} u^{\prime }=\frac{du}{d\tau } .$$

(18)

In Equation (18) the dimensionless parameters λ and γ are defined as follows,

$$\lambda =\frac{{\omega }_{TSBD}^2}{{\Omega }_v^2} \mathrm{and} \gamma =- {\omega }_{TSBD}^2\frac{v_0}{g} .$$

(19)

Depending on the choice of the two stability parameters λ and γ, one obtains either stable or unstable solutions for Mathieu’s equation, whereby an initial disturbance of the dynamic system is assumed. The calculation of the stability areas and their boundary lines can be approximated with the application of the perturbation calculation [36] or the method of slowly varying amplitude and phase according to Krylov and Bogoljubov [39]. The results of these calculations are presented in a λ, γ diagram. Such a stability diagram, calculated for the first time by Ince and Strutt for an undamped dynamic system, is shown in Figure 4a. The areas of the λ, γ diagram, which lead to stable solutions of the Mathieu equation, are shown shaded. One can see that the regions of instability decrease with increasing values of λ.

Figure 4a indicates that the stability diagram is symmetrical to the λ-axis and parametric resonance of undamped dynamic systems may occur with increasing λ for an infinite number of excitation frequencies Ω_v. For small values of γ, i.e., small vertical excitation amplitudes v₀, it is seen that instability occurs at the values λ = n²/4, where n = 1, 2, …, ∞. The fact that parametric resonance occurs at an infinite number of sub- and superharmonic frequencies is a significant difference from the conventional resonance phenomenon, which occurs just for λ = 1. Figure 4b illustrates a detailed view of the stability diagram from λ = −0.5 ÷ 1.7 for a damped dynamic system. It is indicated that the presence of structural damping decreases the regions of instability significantly. The effects of the damping influence on the change of the stability regions and boundary curves have been comprehensively described by, e.g., Klotter [38] or Kotowski [40].

From Figure 4, one can see, that the major areas of instability occur at λ = 0.25 in the Ince–Strutt stability diagram. Thus, the instability phenomenon of parametric resonance occurs with the highest probability in the case when the vertical excitation frequency is twice the natural frequency of the TSBD, i.e., Ω_v = 2·ω_TSBD (see Equation (19)). It can be shown that for a stable forced vibration of a damped dynamic system, with parametric resonance at the critical point λ = 0.25, the following cut-off value of the Lehr damping ratio is required [41,42],

$${\zeta }_{TSBD} > {\omega }_{TSBD}^2 \frac{v_0}{g} .$$

(20)

In Equation (20), the parameters of the TSBD have already been used for the definition of the value of the required Lehr damping ratio. Assuming one wants to deliberately induce parametric resonance in the rolling body of the TSBD, the magnitude of the actual existing Lehr damping ratio must be correspondingly smaller than the value given in Equation (20).

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(Ω_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., Ω_v = 2·ω_TSBD and hence λ = 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 φ = φ_max with the corresponding value of angular velocity $\dot{\phi }=0$. 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 $\dot{\phi }={\dot{\phi }}_{max}$ 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 ω_TSBD and Lehr damping ratio ζ_TSBD) are just a function of the chosen mass ratio μ = 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.

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 ω_TSBD must be chosen equal to the most critical parametric resonance frequency with the major areas of instability at λ = 0.25 (cf. Equation (19) and Figure 4),

$${\omega }_{TSBD} = \frac{1}{2}\cdot \frac{{\omega }_H}{ 1+\mu } .$$

(21)

In Equation (21) a forced resonance excitation of the main system is assumed, i.e., Ω_H = ω_H/(1 + μ), 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 + μ). It is recommended to choose the mass ratio μ = m_R/m_H in analogy to the classical Den Hartog tuned TSBD with 0.5 to 5%.

The achievable damping effect is, of course, also dependent on the size of the Lehr damping ratio ζ_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 φ_limit = 60° 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.,

$${\zeta }_{TSBD} < {\omega }_{TSBD}^2 \frac{v_0}{g} .$$

(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 v₀ of the main system. Hence, it is necessary to calculate the expected maximum vertical vibration amplitude v₀ = 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.

From Equation (22), the minimum vertical vibration amplitude v_0,min and acceleration ${\ddot{v}}_{0,min}$, required to guarantee the occurrence of parametric resonance of the rolling body of the TSBD, can be defined by the following formulas,

$$v_{0,min} = {\zeta }_{TSBD} \cdot \frac{g}{{\omega }_{TSBD}^2} ,$$

(23)

$${\ddot{v}}_{0,min} = v_{0,min} {\omega }_{TSBD}^2= {\zeta }_{TSBD} g .$$

(24)

6. Benchmark Study of a Scaled Harmonically Biaxial-Excited SDOF-Shear Frame Structure with TSBD Attached

In this section, the application of the presented tuned solid ball damper (TSBD) to a scaled harmonically biaxial-excited SDOF-shear frame structure is demonstrated at first for classical vibration suppression, i.e., parametric resonance in the rolling body not used for vibration reduction. Figure 6 illustrates the mechanical model of the considered SDOF-main system with the TSBD attached; w(t) and φ(t) denote the horizontal displacement of the floor mass m_H and the angle of rotation of the rolling body in the runway. The shear frame structure is biaxal excited through the horizontal and vertical ground motions w_g(t) and v_g(t). This kind of biaxial dynamic excitation occurs in real structures such as skyscrapers, towers, masts, and wind turbines, for instance, in the event of an earthquake. The aim of this benchmark study is to investigate the effects of a parametric resonance of the rolling body in the roller track on the resulting vibration-damping effect and on the size of the achievable reduction of the horizontal vibration response w(t), respectively.

To study the possible negative effects of parametric resonance of the rolling body, the excitation frequency of the vertical ground motion is set to the most critical value with the intention to induce parametric resonance, i.e., v_g(t) = v_g,0·cos(Ω_v t), where v_g,0 is the vertical excitation amplitude and Ω_v = 2·ω_TSBD is the circular excitation frequency at the most critical point on the λ-axis of the Ince–Strutt stability diagram (λ = 0.25, see Figure 4). According to Equation (20), the parametric resonance of the rolling body occurs at λ = 0.25 just in the case of a Lehr damping ratio ζ_TSBD smaller than the value resulting from this equation. However, to indicate the difference between the achievable damping effect due to the installation of the TSBD and the damping effect in the case of parametric resonance of the rolling body, in the first step, the vertical ground excitation is set to zero.

In [30], an SDOF-shear frame structure with optimally tuned TSBD was already comprehensively investigated experimentally by the authors with regard just to the horizontal vibration damping effect, and the relevant system parameters of the main system are taken from there. The natural frequency and Lehr damping ratio of the main system were determined by free vibration measurements in [30] with f_H = 1.0 Hz and ζ_H = 0.60% (in the quite high vibration amplitude range at approx. 2.50 m/s²). This was an experimental model setup with extremely low structural damping and hence, it was classified as very vibration prone. The kinetically equivalent mass of the SDOF-main system, including the dead weight of the roller track, is m_H = 4.0 kg. The semicircular runway of the TSBD was designed with a radius R = 0.2 m and the rolling body (cylindrical steel roller with density ρ = 7850 kg/m³) with radius r = 0.015 m. The mass of the cylindrical rolling body was chosen with m_R = 0.2 kg, resulting in a mass ratio of μ = m_R/m_H = 0.05. The natural frequency and Lehr damping ratio of the rolling body in the runway were determined by free vibration tests and the evaluation of the time-decay function resulting in f_TSBD = 0.95 Hz and ζ_TSBD = 4.0%. It is noted that this measured value of the Lehr damping ratio of the TSBD is below the optimal value that would result from the application of the optimization equations according to Den Hartog [4] (evaluating Equation (14) results ζ_TSBD,opt = 12.7%). However, for the presented investigations of the effect of the parameter resonance on the vibration damping behavior, carried out in this study, the measured value of ζ_TSBD = 4.0% is retained.

The investigation of the possible negative effect of parametric resonance on the considered biaxial-excited dynamic system with optimally tuned TSBD attached, see Figure 6, is carried out by numerically by solving the coupled nonlinear differential equations of motion with MATLAB/Simulink software. Therefore, Equation (12) is rewritten for the biaxial dynamic action of a combined horizontal and vertical displacement excitation w_t(t) = w_g(t) + w(t) and v_t(t) = v_g(t) to the considered 2-DOF-main system as follows,

$$\begin{gathered} \left(1+\mu \right)\ddot{w}+\mu \left(R-r\right) \ddot{\phi }-\mu \left(R-r\right) {\dot{\phi }}^2\phi +2{\zeta }_{H,h} {\omega }_{H,h} \dot{w}+{\omega }_{H,h}^2 w=-\left(1+\mu \right){\ddot{w}}_g \\ \ddot{\phi } + {\omega }_{TSBD}^2\frac{\ddot{w}}{g} + 2 {\zeta }_{TSBD} {\omega }_{TSBD} \dot{\phi } + {\omega }_{TSBD}^2 \left[1+\frac{{\ddot{v}}_g}{g}\right] \phi =-{\omega }_{TSBD}^2\frac{{\ddot{w}}_g}{g} . \end{gathered}$$

(25)

The horizontal and vertical ground excitations were assumed to be time harmonic w_g(t) = w_g,0·cos(Ω_w t) and v_g(t) = v_g,0·cos(Ω_v t), whereby the ground motion amplitudes w_g,0 = 0.001 m and v_g,0 = 0.021 m were selected. The horizontal excitation frequency was assumed to be a resonance excitation for the considered SDOF-shear frame structure, i.e., Ω_w = 2·f_H. The vertical excitation frequency was chosen to achieve a parametric resonance of the rolling body of the TSBD at the most critical parameter excitation frequency point λ = 0.25 (see Figure 4) with Ω_v = 2·ω_TSBD = 4π f_TSBD (see Section 4). To study the influence of the parametric resonance of the rolling body to the damping effect of the SDOF-main system, the calculations are in comparison also performed by setting the vertical ground excitation v_g,0 to zero.

Figure 7a shows the time history of the horizontal dynamic deformation of the floor mass w(t) under solely horizontal resonance excitation w_g(t) = w_g,0·cos(Ω_wt) calculated numerically with MATLAB/Simulink without and with activated TSBD in case of absence of the vertical base excitation, i.e., v_g,0 = 0. In the case of the solely horizontal excitation of the SDOF-shear frame structure, a quite-high reduction of the maximum horizontal deformation amplitudes w(t) results, which corresponds to the damping effect of the optimally tuned TSBD. Without the activated damper, the maximum horizontal vibration amplitude grows up to w_max = 0.08 m. After activation of the optimally tuned TSBD, the maximum vibration amplitudes are just in the range of 0.005 m, i.e., the reduction of the maximum structural response induced by the optimally tuned TSBD is around 95%.

Contrarily, in the case of combined horizontal and parametric resonance most critical vertical excitation of the SDOF-shear frame structure, parametric resonance occurs in the rolling body of the TSBD due to its above-mentioned special selected parameters, the excitation frequency as well as the value of the excitation amplitude. It is seen that the parametric resonance of the rolling body leads to a continuous increase of the horizontal deformations of the shear frame floor mass over time (Figure 7b). Under the given conditions, the vertical excitation hence leads to a significant deterioration up to the complete loss of the vibration damping effect induced by the TSBD. To avoid the occurrence of the parametric resonance, the Lehr damping ratio of the TSBD would have to be selected with ζ_TSBD,opt > 7.7% in the examined case, according to Equation (20). This would again ensure the optimum vibration damping effect induced by the TSBD, even with the vertical displacement excitation given here.

7. Benchmark Study of a Harmonically Biaxial Force Excited 2-DOF-Bridge Structure with TSBD Attached

In this second benchmark study, a harmonically biaxial force-excited 2-DOF-bridge structure with a single TSBD attached is considered (see Figure 8). The horizontal and vertical dynamic displacement of the structure is denoted with w(t) and v(t). F_h(t) and F_v(t) define the horizontal and vertical excitation forces which are assumed to be time harmonic, i.e., F_h(t) = F_h,0·cos(Ω_h t) and F_v(t) = F_v,0·cos(Ω_v t).

The special feature of this section is the evaluation of the potential application of parametric forcing of the rolling body in the runway of the TSBD, for reducing vertical vibrations of a vibration-prone main system. The optimal tuning of the TSBD to the vertical parametric resonance frequency of the main system is performed according to Section 5 (Equations (21) and (22)).

All parameters of the considered 2-DOF-bridge structure such as the geometry, bending stiffness, and mass, as well as the dynamic parameters that were determined by numerical calculations and dynamic in situ measurements, were taken from a real existing pedestrian bridge [43] located in Austria. The considered bridge is a curved suspension bridge (radius = 85 m) with a span of 104.50 m (measured in the curvature of the bridge deck), see Figure 9. The bridge pylon, which is made of steel S355 J2 + N, has a diameter of 900 to 1000 mm and a total height of 40.5 m. The suspension cable has a cross section of 101 cm² and the total of 31 hangers connected to the bridge deck on one side have a diameter of 20.1 to 24.4 mm. The bridge deck is a slender triangular-shaped steel box girder made of steel S460 NL and reinforced with longitudinal stiffeners. On top of the steel box girder, a concrete layer with a thickness of 9 to 11 cm was constructed and a 3 cm thick asphalt pavement is placed on top of it. During the design phase of the bridge, an analysis of the pedestrian-induced vibrations was carried out according to the HIVOSS guideline [44].

The first six natural frequencies and corresponding vibration modes calculated in a spatial numerical model of the bridge structure are shown in Figure 10. It is seen that the determined vertical eigenmodes are analogous to a single-span beam with articulated bearing on both sides (Bernoulli–Euler beam). For the pedestrian-induced vibrations, the third natural frequency f_3v = 1.37 Hz with the corresponding vertical vibration mode was classified as resonance critical. At this natural frequency, resonance vibrations with high vibration amplitudes can be triggered by groups of people walking “slowly” across the bridge and synchronizing themselves. Due to the extremely slender design of the structure and the generally extremely low energy-dissipation capacity of steel as a building material, the Lehr damping ratio was selected as ζ_H,i = 0.40% within the dynamic calculations for each vibration mode.

Due to the expected very low Lehr damping ratio in the existing pedestrian bridge and the consideration of the vibration critical case of synchronization of groups of people as well as the purposeful resonance excitation, the decision was made to install classical passive vibration dampers (mass-spring dampers) to reduce the vertical person-induced vibrations of the relevant third natural frequency f_3v. The planning and installation of these vibration dampers, as well as the acceptance measurements, carried out after the opening of the bridge for passenger traffic to check the vibration damper effectiveness, are documented in [43]. The actual natural frequencies and Lehr damping ratio of the finished pedestrian bridge were also determined by in situ measurements, using the forced-vibration testing method. The results matched very well with the values determined in advance by the numerical calculation (e.g., f_3v,measured = 1.42 Hz and ζ_H,3,measured = 0.43%, cf. with Figure 10). The results of the in situ measurements are also documented in [43].

In this article, the parametric-excited TSBD, which can be used as an alternative to the classical vibration dampers (mass-spring dampers), is investigated for reducing the vertical pedestrian-induced vibrations of the biaxial-excited bridge. Here, the parametric resonance of the rolling body in the TSBD is deliberately induced and used to achieve the corresponding vibration damping of the considered bridge structure. Again, the decisive third natural frequency with the corresponding vertical vibration mode f_3v = 1.37 Hz of the structure is considered within the tuning process of the TSBD and an optimal tuning of the TSBD with steel balls as rolling bodies is carried out.

The mass ratio of the rolling body to the kinetically equivalent mass of the bridge structure is chosen as μ = m_R/m_H = 0.02. The continuous dynamic system of the bridge structure is reduced to an equivalent discrete 2-DOF system with displacements w(t) and v(t) for the investigation of a biaxial vibration for the third vertical and for the first horizontal eigenmode (see Figure 10). For the third vertical and first horizontal eigenmode, the kinetically equivalent mass with m_H = 112.240 kg was taken from the results of the numerical calculation of the eigenfrequencies and the corresponding vibration modes. The total mass of the rolling body of the TSBD results therefore to m_R = μ·m_H = 2250 kg (rounded up). If the TSBD is designed with a single steel ball as the rolling element, the required diameter and the corresponding volume would be D_R = 0.82 m and V_R = 0.29 m³. However, the bridge deck, which is designed with a low overall height, does not provide sufficient space for those requirements and, therefore, the single TSBD is divided into a total of six TSBDs acting in parallel and arranged in the vibration maxima of the third vertical eigenmode. Figure 11a,b show the placement of the total of six TSBD at the discrete locations along the bridge and an installation example in the cross-section of the bridge deck. The mass of the rolling body for each individual TSBD is m_R,i = 375 kg and the associated diameter of the steel sphere results in D_R,i = 0.45 m.

The optimal tuning of the parametric-excited TSBD is done by the suitable selection of their natural frequency. In order for the parametric resonance of the rolling body in the runway to occur during vertical resonance vibrations of the bridge structure with installed TSBD, the optimal natural frequency of the rolling bodies is set to the critical value with the largest portion of the unstable region in the Ince–Strutt stability map (see Figure 4) with f_TSBD = 0.5·f_v,3·1/(1 + μ) = 0.67 Hz (see Section 4, Equation (19) and Figure 4, stability parameter λ = λ_critical = 0.25). The associated geometry parameters of the single TSBD are obtained by transforming Equation (9) to (R − r) = 0.38 m and with r = D/2 = 0.225 m the radius for the semicircular runway results in R = 0.618 m. With the aim of raising the parameter resonance of the rolling body as quickly as possible, the Lehr damping ratio of the TSBD is chosen to be very small with ζ_TSBD = 0.01 = 1%.

To determine the vibration responses of the bridge deck resulting from biaxial dynamic pedestrian excitation without and with activated optimally tuned TSBD, the coupled nonlinear differential equations of the 3-DOF system given in Equation (12) are solved numerically in the time domain. For this purpose, the MATLAB/Simulink software is used and the continuous solution method ode45 is selected. This is the Runge–Kutta (4, 5) equation solver, which performs a fifth-order integration method with a fourth-order error.

The biaxially acting time harmonic excitation forces are assumed in the equations of motion of the equivalent discrete dynamic system to be F_h(t) = F_h,0·cos(Ω_h t) and F_v(t) = F_v,0·cos(Ω_v t), with excitation force amplitudes chosen to be F_h,0 = 10 kN and F_v,0 = 1 kN. Figure 12 shows the calculated time histories of the vertical dynamic deformation v(t) of the bridge deck due to biaxial pedestrian excitation at the resonance frequency with and without activated parametric-excited TSBD. It can be seen that the vertical vibration amplitudes reach a maximum value of v_dyn,max = 0.14 m without activated TSBD. After activation of the TSBD, strongly reduced dynamic deformation amplitudes result in the steady state (from t = 50 s) with v_dyn,max = 0.009 m. The installation of the parametric-excited TSBD hence results in a reduction of the maximum vertical dynamic deformations by approx. 93% in the steady state. It can also be seen that vibrational beat effects are present in the transient condition.

To investigate the vibration damping effect of the installed vertically parametric forced TSBD in the entire frequency range of interest, the vertical excitation frequency Ω_v is varied in the next step in the range around the resonance point of the third vertical natural frequency of the bridge deck with TSBD and the amplitude-frequency response function is calculated numerically. The result of this calculation is shown in Figure 13. In these calculations, the horizontal excitation frequency was assumed to be Ω_h = 2 π f_1,h. Figure 13 indicates that the vibration damping effect of the installed vertically parametric forced TSBD is given over the entire excitation frequency range of interest. The amplification of the parametric resonance vibration in the rolling body of the TSBD leads to the absorption of the vibration energy, which is building up in the bridge structure due to the external dynamic excitations. The absorbed energy is dissipated into heat by solid friction between the rolling body and the runway. The stabilization of the initially unstable parametric resonance vibration of the rolling body results from the reduction of the vertical vibration amplitudes of the bridge structure through the restoring effect of the installed TSBDs. The vibration-control efficiency of the classical optimally tuned TMD is additionally displayed in Figure 13 for comparison.

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 Den Hartog’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 highest-possible vibration damping effect and a fast response behavior of the TSBD, the stability parameter λ has to be chosen with λ = (ω_TSBD/Ω_H)² = 0.25 and the Lehr damping ratio ζ_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.