## Abstract

Granular dampers are systems used to attenuate undesired vibrations produced by mechanical devices. They consist of cavities filled by granular particles. In this work, we consider a granular damper filled with a binary mixture of frictionless spherical particles of the same material but different size using numerical discrete element method simulations. We show that the damping efficiency is largely influenced by the composition of the binary mixture.

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## 1 Introduction

Granular dampers are devices that exploit the dissipative nature of granular interactions to passively attenuate vibrations. In their most simple design, they consist of an enclosure partially filled by granular particles. When subjected to mechanical vibrations, the particles collide with each other and with the walls of the enclosure, thus, inelastic interactions (particle-particle and particle-wall) dissipate mechanical energy into heat. Granular dampers reveal distinctive characteristics beneficial for their application in technical devices: The design of granular dampers is simple and requires minimal maintenance, which makes them interesting for aerospace technologies [1] and applications in weightlessness [2,3,4]. They operate across a wide range of temperature [5], making them particularly useful in harsh environments. Due to these features, granular dampers have numerous technical applications, for instance, in medicine [6], construction [7], and vibration control during earthquakes [8,9,10].

Granular dampers have been subject to research for a long time, also using numerical discrete element method (DEM) simulation techniques [11,12,13]. In numerical studies and in experiments, it has been shown that the performance of granular dampers is sensitive to a variety of parameters such as the mass ratio between the granulate and the container [14], number and material of the particles [15, 16], particle size [1, 17, 18] and shape [19], size and shape of the enclosure [20,21,22], the free volume inside the container (gap clearance) [23, 24], and others [25,26,27]. Yet, the effect of particle size dispersion on the efficiency of granular dampers was much less investigated. Only recently, the addition of micrometer-sized particles to a system of millimeter-sized monodisperse spheres was reported to have a significant effect on damping performance [21]. An explanation of this effect, as well as the influence of the dispersion on damping efficiency in general, remain open questions.

In this paper, we consider granular dampers using bidisperse granular mixtures by means of DEM simulation, which is a reliable numerical tool to study granular systems [28, 29].

## 2 System setup and numerical model

Our damper of cubical shape (\(8\times 8\times 8\text { cm}^3\)) is filled partially by *N* frictionless spheres. For both container and particles, we assume the same material characteristics corresponding to steel (density \(\rho = 7850\, \text {kg/m}^3\), Young's modulus \(E = 2\times 10^{9}\,\text {Pa}\), Poisson's ratio \(\nu =0.3\)).

The damper is driven by oscillations, \(z(t)=A_\text {damp} \cos (2 \pi f_\text {damp} t)\) with \(f_\text {damp} = 70\,\text {Hz}\) where gravity, \(g = 9.8\,\text {m/s}^2\), acts in negative *z* direction. Figure 1 shows a sketch of the setup.

The interaction force between viscoelastic frictionless spheres is given by the elastic Hertz law [30] and an appropriate dissipative force. The Hertz law results from the assumption that the elastic stress is a linear function of the strain, therefore, for the dissipative force it is reasonable to assume that the dissipative stress is a linear function of the strain rate [31]. Combining these forces, we obtain the absolute value of the force between particles *i* and *j* of radii \(R_i\) and \(R_j\) at positions \(\vec {r}_i\) and \(\vec {r}_j\) at velocities \(\vec {v}_i\) and \(\vec {v}_j\). When in contact, that is, if \(\xi _{ij}\equiv R_i+R_j - \left| \vec {r}_i-\vec {r}_j\right| \ge 0\), the force is

with \(1/R_{ij}^\text {eff}\equiv 1/R_i + 1/R_j\). The dissipative constant, *A*, is a function of the viscosities of the particles’ materials [31], which are difficult to determine. A much easier way to determine *A* is through the coefficient of restitution at a certain impact rate [32]. For our simulations, to determine *A*, we assume the coefficient of restitution \(\varepsilon =0.75\) for steel particles colliding at impact rate 1 m/s. For bidisperse mixtures, we obtain *A* for the collision of particles of each species and calculate its arithmetic mean. In some cases, it can happen that the dissipative force overcompensates the restoring elastic force, resulting in a total attractive force (see [28] for a detailed explanation). The maximum rule in Eq. 2) ensures that the interaction force is always repulsive. In our force model, we neglect frictional forces between contacting particles, justified by previous evidence [19, 33, 34], which shows that the main dissipation mechanism in granular dampers is due to normal interaction forces. For our DEM simulations, we use Yade [35]. The time step used in the simulations is \(1 \cdot 10^{-7}\) s.

## 3 Energy dissipation per cycle of oscillation

We quantify energy dissipation efficiency by

where \(E_\text {pp}\) is given by

and \(E_\text {pw}\) by

with \(T=1/f\). Equation (3) relates the energy dissipated by particle-particle (\(E_\text {pp}\)) and particle-wall (\(E_\text {pw}\)) contacts during one period *T* of oscillation to the maximum energy, \(E_\text {max}\), that can be dissipated in one oscillation cycle. The quantity \(E_\text {max}\) was defined in [2]. It is obtained by assuming that all particles collide twice per cycle with the bottom and top walls of the container at maximal relative velocity and dissipate the energy of the relative motion in each of these collisions.

Note that \(E_\text {d}\) is similar but not identical to the quantity *b* defined in Eq. (4) of [13], which relates the dissipated energy to the total kinetic energy.

The dissipated energy depends on the dynamics of the granulate and is, thus, a fluctuating quantity. Therefore, we compute an average, \(\left<E_\text {d}\right>\), from a linear fit to \(E_\text {d}(t)\).

## 4 Modes of operation of granular dampers

When operating in weightlessness, depending on the amplitude, \(A_\text {damp}\), of the oscillation, granular dampers reveal different modes [2]. For small \(A_\text {damp}\), the granular material exhibits gas-like behavior. In this state, only a small fraction of the kinetic energy is dissipated. More relevant for granular damping is the *collect-and-collide* regime [34] observed for large \(A_\text {damp}\), where the center of mass of the granulate moves synchronously with the container. In this state, during the inward stroke, the granulate accumulates at the wall of the container forming a packed layer. When the acceleration of the box decreases, this layer leaves the wall collectively and eventually collides with the opposing wall where a large fraction of the kinetic energy is dissipated. The modes of operation are separated by a sharp transition at a critical value of \(A_\text {damp}\); here the dissipated energy per oscillation achieves its maximum value [2, 4]. Motivated by this argument, in the results section, we consider the interval \(A_\text {damp}\in [2,4]\,\text {cm}\) containing the critical value.

Although initially found and described for conditions of weightlessness, the modes of operations of granular dampers and the sharp transition between them were also found to happen for vertical and horizontal vibrations under gravity [21, 36]. A more detailed classification of modes of behavior of vibrated boxes filled by granular material was presented in [37]; however, not in the context of granular damping.

## 5 Results

### 5.1 Classes of equal binary mixtures

We consider three classes of binary mixtures in comparison with a monodisperse reference system: (a) the total mass of the granulate (\(M_\text {tot}\)) in the bidisperse system equals the mass in the monodisperse reference system, but not the number of particles; (b) the total number of particles (\(N_\text {tot}\)) in the bidisperse system equals the number of particles in the reference system, but not the total mass of the granulate; (c) both the number of particles and the mass of the granulate in the bidisperse system equal the corresponding quantities in the monodisperse reference system, thus, isolating the effect of size dispersion from particle number and mass. We summarize the properties of the classes (a)–(c) in Table 1.

### 5.2 Class (a): equal total mass

Figure 2
shows the average energy dissipation efficiency per cycle, \(\left<E_\text {d}\right>\), normalized by the total mass, *m*, of the granulate, as a function of the vibration amplitude for different size ratios, \(\sigma\). For all values of \(\sigma\), the system reveals a similar behavior. An optimal value of the driving amplitude, \(A_\text {opt}\), divides the dynamics into two regimes: For \(A_\text {damp}\lesssim 2.8\,\text {cm}\), the granulate is in a gaseous state where only a small fraction of the particles collides with the walls during an oscillation period. In contrast, for \(A_\text {damp}\gtrsim 2.8\,\text {cm}\), the center of mass of the granulate moves synchronously with the external driving, in the collect-and-collide regime. Both modes of dynamic behavior, gaseous and collect-and-collide, and their dissipative properties are discussed in detail in [2,3,4, 34] (Fig. 3).

Figure 4 shows snapshots of the simulation at the times of minimal and maximal velocity for the reference system and for a bidisperse system with \(\sigma =0.1\) in the collect-and-collide regime. We notice that for bidisperse systems where the size ratio is high, the particles spread over a larger volume in the container. Moreover, we observe size segregation (Fig. 3). A reverse Brazil nut-like effect ([38, 39]) is observed, where small particles are found at the top of the big particles. Segregation is observed to originate after the granular material collides with the top and bottom walls of the container. When the granulate collides with a wall (top or bottom), big particles need more time to overcome inertia than small particles. Small particles leave the wall first, as shown by Fig. 3 and the phase plot in Fig. 7, and percolate through the interstices between big particles, leading to the observed segregation.

For \(A_\text {damp} > A_\text {opt}\) (Fig. 2), the bidisperse system with \(\sigma = 0.5\) achieves higher values of \(\left<E_\text {d}\right>/m\) than the reference system, i.e., it dissipates energy more efficiently than the monodisperse reference system, in the collect-and-collide mode.

To understand this behaviour, we consider the particles’ collisions with the top and bottom walls, which dominate the dissipation [33]. In each time step of the integration, we record the number of particles contacting one of these walls (Fig. 5).

We note that for bidisperse systems, the collision frequency is higher than for the reference system (\(\sigma =1\)). Detailed analysis shows that this difference can be attributed to collisions of small particles with the walls.

The number of particle contacts alone is, however, not sufficient to explain the difference in energy dissipation. Therefore, we record the force in *z* direction (\(F_z\)) exerted by the granulate on the top and bottom walls (Fig. 6). The bidisperse damper with \(\sigma =0.5\) shows a higher value of \(F_z\), compared to the reference system. Higher absolute values of \(F_z\) lead to higher momentum transfer and, in turn, to larger loss of kinetic energy in dissipative collisions, which explains the difference in the energy dissipation efficiency shown in Fig. 2 (Fig. 6).

### 5.3 Class (b): equal number of particles

Figure 8
shows the average energy dissipation efficiency per cycle, \(\left<E_\text {d}\right>\), normalized by the total mass, *m*, of the granulate, as a function of the vibration amplitude for different size ratios, \(\sigma\), corresponding to Fig. 2 for class (a) binary mixtures. For all values of \(\sigma\), the systems exhibit a similar behavior. Again, we see two regimes for small and large amplitude (gaseous and collect-and-collide) and again the dissipation efficiency for bidisperse mixtures exceeds that of the reference system (\(\sigma =1\)).

From Fig. 9 we observe that the number of contacts with the top and bottom walls of the reference system (\(\sigma =1\)) exceeds that of mixtures for all values \(\sigma <1\). Similarly, the force exerted on the top and bottom walls by the particles, \(F_z\), is higher for (\(\sigma =1\)) compared with mixtures for all values of \(\sigma\) (Fig. 10). Consequently, the difference in dissipation shown in Fig. 8 cannot be attributed to the number of contacts nor \(F_z\).

From the phase plot of the center-of-mass position in the direction of the oscillation, *z*, (Fig. 11),
we see that both small and large particles of the bidisperse system (with \(\sigma =0.5\)) assume larger velocities than the particles of the reference system (\(\sigma =1\)). Higher absolute velocities give rise to higher impact velocities and, thus, larger loss of kinetic energy in dissipative collisions, which explains the difference in the damping efficiency shown in Fig. 8.

### 5.4 Class (c): equal total mass and number of particles

Similar as for the previous classes, also for class (c), \(\left<E_\text {d}\right>/m\), as a function of \(A_\text {damp}\) (Fig. 12),
shows two well separated regimes for large and small \(A_\text {damp}\). In contrast to classes (a) and (b), here the transition value of \(A_\text {damp}\) depends on \(\sigma\). Also in contrast to classes (a) and (b), here the values of \(\left<E_\text {d}\right>/m\) of monodisperse reference system for \(A_\text {damp} > 2.8\,\text {cm}\) exceed the values of dampers with \(\sigma <1\). Here, the number of contacts (Fig. 13) of the reference system exceeds the value for the granular mixtures. Furthermore, the force in *z* direction, \(F_z\), (Fig. 14), exerted by the particles of the reference system is considerably larger than for bidisperse dampers.

From the phase plot for \(\sigma =0.5\), Fig. 15, we see that large particles follow approximately the same path as the particles of the reference system while the small particles achieve larger velocities. Large and small particles in the bidisperse system do not move collectively. In contrast to dampers of class (b), there is no energy dissipation enhancement due to a higher center of mass velocity of both large and small particles.

## 6 Conclusion

The efficiency of granular dampers operating with a bidisperse granular mixture depends sensitively on the composition of the mixture. Thus, the choice of the composition can enhance energy dissipation in granular dampers significantly as compared to granular dampers operating with a monodisperse granulate of the same material. For the comparison with a monodisperse reference system, we considered three classes of mixtures: (a) both monodisperse and bidisperse systems are of the same total mass of the granulate (but different number of particles); (b) both systems have the same total number of particles (but different mass); (c) the sizes of the particles in the mixture are scaled such that both systems contain the same number of particles of the same total mass.

For dampers of class (a), the efficiency increases due to a larger frequency of collisions and higher exerted force during impact with the top and bottom walls in comparison with the reference system.

For dampers of class (b), the efficiency increases due to the larger relative velocity between the granular particles and the wall of the container, shown by the phase plot of the center of mass velocity of the binary system in comparison with the monodisperse reference system.

For dampers of class (c), the energy dissipation efficiency is smaller than for the monodisperse reference system due to a smaller frequency of collisions and lower exerted force during impact with the top and bottom walls. Furthermore, there is no energy dissipation improvement due to a higher center of mass velocity of both large and small particles.

In conclusion, we find that the dissipative properties of mixtures of differently sized particles differ from the dissipative properties of monodisperse systems. Therefore, the particle size distribution of the granulate is an important feature for the design of granular dampers.

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## Acknowledgements

The work was supported by the Interdisciplinary Center for Nanostructured Films (IZNF), the Competence Unit for Scientific Computing (CSC), and the Interdisciplinary Center for Functional Particle Systems (FPS) at Friedrich-Alexander Universität Erlangen-Nürnberg. N. R. V.-R. and M. E. acknowledge funding by the Deutsche Forschungsgemeinschaft (DFG) through project number 406658237. A. S. and T. P. acknowledge funding by DFG under project number 411517575 and through the Research Training Group GRK 2423 “Fracture across Scales frascal”.

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Varela-Rosales, N.R., Santarossa, A., Engel, M. *et al.* Granular binary mixtures improve energy dissipation efficiency of granular dampers.
*Granular Matter* **25**, 49 (2023). https://doi.org/10.1007/s10035-023-01337-8

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DOI: https://doi.org/10.1007/s10035-023-01337-8