Granular gases of rod-shaped grains in microgravity

Anomalous diffusion in polydisperse granular gases

We investigate both ensemble and time-averaged mean-squared displacements of particles in a polydisperse granular system in a homogeneous cooling state and derive rigorous analytical expressions valid both at short and long time scales. The discrepancies in ensemble and time-averaged mean-squared displacements indicate ergodicity breaking in granular systems consisting of an arbitrary number of species of different sizes and masses. We compare the results of our study with Monte Carlo simulations in terms of a powerful low-rank algorithm and find a nice agreement.

Rotational and translational motions in a homogeneously cooling granular gas

A granular gas composed of monodisperse spherical particles was studied in microgravity experiments in a drop tower. Translations and rotations of the particles were extracted from optical video data. Equipartition is violated, the rotational degrees of freedom were excited only to roughly 2/3 of the translational ones. After stopping the mechanical excitation, we observed granular cooling of the ensemble for a period of three times the Haff time, where the kinetic energy dropped to about 5% of its initial value. The cooling rates of all observable degrees of freedom were comparable, and the ratio of rotational and translational kinetic energies fluctuated around a constant value. The distributions of translational and rotational velocity components showed slight but systematic deviations from Gaussians at the start of cooling.

Cooling of a granular gas mixture in microgravity

Granular gases are fascinating non-equilibrium systems with interesting features such as spontaneous clustering and non-Gaussian velocity distributions. Mixtures of different components represent a much more natural composition than monodisperse ensembles but attracted comparably little attention so far. We present the observation and characterization of a mixture of rod-like particles with different sizes and masses in a drop tower experiment. Kinetic energy decay rates during granular cooling and collision rates were determined and Haff's law for homogeneous granular cooling was confirmed. Thereby, energy equipartition between the mixture components and between individual degrees of freedom is violated. Heavier particles keep a slightly higher average kinetic energy than lighter ones. Experimental results are supported by numerical simulations.

Platonic solids bouncing on a vibrating plate

The energy transfer between bouncing particles and rigid boundaries during impacts is crucially influenced not only by restitution coefficients of the material but also by particle shapes. This is particularly important when such particles are mechanically agitated with vibrating plates. Inertial measurement units are able to measure all acceleration and rotational velocity components of an object and store these data for subsequent analysis. We employ them to measure the dynamics of cubes and icosahedra on vibrating plates to study the efficiency of energy transfer into the individual degrees of freedom (DOFs) of the excited object. The rotational DOFs turn out to be much less excited than the vertical translational motion. Most remarkably, there is only little difference between the two Platonic solids in both the absolute energies and the energy partition ratios.

Statistics of a two-dimensional immersed granular gas magnetically forced in volume

We present an experimental study of the dynamics of a set of magnets within a fluid in which a remote torque applied by a vertical oscillating magnetic field transfers angular momentum to individual magnets. This system differs from previous experimental studies of granular gas where the energy is injected by vibrating the boundaries. Here, we do not observe any cluster formation, orientational correlation and equipartition of the energy. The magnets' linear velocity distributions are stretched exponentials, similar to three-dimensional boundary-forced dry granular gas systems, but the exponent does not depend on the number of magnets. The value of the exponent of the stretched exponential distributions is close to the value of 3/2 previously derived theoretically. Our results also show that the conversion rate of angular momentum into linear momentum during the collisions controls the dynamics of this homogenously forced granular gas. We report the differences among this homogeneously forced granular gas, ideal gas, and nonequilibrium boundary-forced dissipative granular gas.

Non-Gaussian, transiently anomalous, and ergodic self-diffusion of flexible dumbbells in crowded two-dimensional environments: Coupled translational and rotational motions

We employ Langevin-dynamics simulations to unveil non-Brownian and non-Gaussian center-of-mass self-diffusion of massive flexible dumbbell-shaped particles in crowded two-dimensional solutions. We study the intradumbbell dynamics of the relative motion of the two constituent elastically coupled disks. Our main focus is on effects of the crowding fraction ϕ and of the particle structure on the diffusion characteristics. We evaluate the time-averaged mean-squared displacement (TAMSD), the displacement probability-density function (PDF), and the displacement autocorrelation function (ACF) of the dimers. For the TAMSD at highly crowded conditions of dumbbells, e.g., we observe a transition from the short-time ballistic behavior, via an intermediate subdiffusive regime, to long-time Brownian-like spreading dynamics. The crowded system of dimers exhibits two distinct diffusion regimes distinguished by the scaling exponent of the TAMSD, the dependence of the diffusivity on ϕ, and the features of the displacement-ACF. We attribute these regimes to a crowding-induced transition from viscous to viscoelastic diffusion upon growing ϕ. We also analyze the relative motion in the dimers, finding that larger ϕ suppress their vibrations and yield strongly non-Gaussian PDFs of rotational displacements. For the diffusion coefficients D(ϕ) of translational and rotational motion of the dumbbells an exponential decay with ϕ for weak and a power-law variation D(ϕ)∝(ϕ-ϕ^{★})^{2.4} for strong crowding is found. A comparison of simulation results with theoretical predictions for D(ϕ) is discussed and some relevant experimental systems are overviewed.

Direct observation of crystal nucleation and growth in a quasi-two-dimensional nonvibrating granular system

We study a quasi-two-dimensional macroscopic system of magnetic spherical particles settled on a shallow concave dish under a temporally oscillating magnetic field. The system reaches a stationary state where the energy losses from collisions and friction with the concave dish surface are compensated by the continuous energy input coming from the oscillating magnetic field. Random particle motions show some similarities with the motions of atoms and molecules in a glass or a crystal-forming fluid. Because of the curvature of the surface, particles experience an additional force toward the center of the concave dish. When decreasing the magnetic field, the effective temperature is decreased and diffusive particle motion slows. For slow cooling rates we observe crystallization, where the particles organize into a hexagonal lattice. We study the birth of the crystalline nucleus and the subsequent growth of the crystal. Our observations support nonclassical theories of crystal formation. Initially a dense amorphous aggregate of particles forms, and then in a second stage this aggregate rearranges internally to form the crystalline nucleus. As the aggregate grows, the crystal grows in its interior. After a certain size, all the aggregated particles are part of the crystal and after that crystal growth follows the classical theory for crystal growth.

Dynamics of a vibration-driven single disk

Granular particles exhibit rich collective behaviors on vibration beds, but the motion of an isolated particle is not well understood even for uniform particles with a simple shape such as disks or spheres. Here we measured the motion of a single disk confined to a quasi-two-dimensional horizontal box on a vertically vibrating stage. The translational displacements obey compressed exponential distributions whose exponent [Formula: see text] increases with the frequency, while the rotational displacements exhibit unimodal distributions at low frequencies and bimodal distributions at high frequencies. During short time intervals, the translational displacements are subdiffusive and negatively correlated, while the rotational displacements are superdiffusive and positively correlated. After prolonged periods, the rotational displacements become diffusive and their correlations decay to zero. Both the rotational and the translational displacements exhibit white noise at low frequencies, and blue noise for translational motions and Brownian noise for rotational motions at high frequencies. The translational kinetic energy obeys Boltzmann distribution while the rotational kinetic energy deviates from it. Most energy is distributed in translational motions at low frequencies and in rotational motions at high frequencies, which violates the equipartition theorem. Translational and rotational motions are not correlated. These experimental results show that the random diffusion of such driven particles is distinct from thermal motion in both the translational and rotational degrees of freedom, which poses new challenges to theory. The results cast new light on the motion of individual particles and the collective motion of driven granular particles.

Visual analysis of density and velocity profiles in dense 3D granular gases

Granular multiparticle ensembles are of interest from fundamental statistical viewpoints as well as for the understanding of collective processes in industry and in nature. Extraction of physical data from optical observations of three-dimensional (3D) granular ensembles poses considerable problems. Particle-based tracking is possible only at low volume fractions, not in clusters. We apply shadow-based and feature-tracking methods to analyze the dynamics of granular gases in a container with vibrating side walls under microgravity. In order to validate the reliability of these optical analysis methods, we perform numerical simulations of ensembles similar to the experiment. The simulation output is graphically rendered to mimic the experimentally obtained images. We validate the output of the optical analysis methods on the basis of this ground truth information. This approach provides insight in two interconnected problems: the confirmation of the accuracy of the simulations and the test of the applicability of the visual analysis. The proposed approach can be used for further investigations of dynamical properties of such media, including the granular Leidenfrost effect, granular cooling, and gas-clustering transitions.

Particle Dynamics at the Onset of the Granular Gas-Liquid Transition

We study experimentally the dynamical behavior of few large tracer particles placed in a quasi-2D granular "gas" made of many small beads in a low-gravity environment. Multiple inelastic collisions transfer momentum from the uniaxially driven gas to the tracers whose velocity distributions are studied through particle tracking. Analyzing these distributions for an increasing system density reveals that translational energy equipartition is reached at the onset of the gas-liquid granular transition corresponding to the emergence of local clusters. The dynamics of a few tracer particles thus appears as a simple and accurate tool to detect this transition. A model is proposed for describing accurately the formation of local heterogeneities.

Cluster dynamics in dense granular gases of rod-like particles

Granular gases are interesting multiparticle systems which, irrespective of the apparent simplicity of particle interactions, exhibit a rich scenario of so far only little understood features. We have numerically investigated a dense granular gas composed of frictional spherocylinders which are excited mechanically by lateral vibrating container walls. This study was stimulated by experiments in microgravity on parabolic flights. The formation of spatial inhomogeneities (clusters) was observed in a region near the corners of the container, about halfway from the excitation plates. The particles in the clusters show a tendency to align parallel to the container walls, seemingly increasing the stabilizing effect of friction. The simulation results provide hints that the phase difference of the vibrations of the two excitation walls might affect the cluster dynamics.

Continuously heated granular gas of elongated particles

Some years ago, Harth et al. experimentally explored the steady state dynamics of a heated granular gas of rod-like particles in microgravity [K. Harth et al. Phys. Rev. Lett. 110, 144102 (2013)]. Here, we report numerical results that quantitatively reproduce their experimental findings and provide additional insight into the process. A system of sphero-cylinders is heated by the vibration of three flat side walls, resulting in one symmetrically heated direction, one non-symmetrically heated direction, and one non-heated direction. In the non-heated direction, the speed distribution follows a stretched exponential distribution $$p(\upsilon )\, \propto \,{\rm{exp}}\left( { - {{\left( {{{\left| \upsilon \right|} \mathord{\left/ {\vphantom {{\left| \upsilon \right|} C}} \right. \kern-\nulldelimiterspace} C}} \right)}^{1.5}}} \right)$$. In the symmetrically heated direction, the velocity statistics at low speeds is similar but it develops pronounced exponential tails at high speeds. In the non-symmetrically heated direction (not accessed experimentally), the distribution also follows $$p(\upsilon )\, \propto \,{\rm{exp}}\left( { - {{\left( {{{\left| \upsilon \right|} \mathord{\left/ {\vphantom {{\left| \upsilon \right|} C}} \right. \kern-\nulldelimiterspace} C}} \right)}^{1.5}}} \right)$$ , but the velocity statistics of rods moving toward the vibrating wall resembles the indirectly excited direction, whereas the velocity statistics of those moving away from the wall resembles the direct excited direction.

Crowding-Enhanced Diffusion: An Exact Theory for Highly Entangled Self-Propelled Stiff Filaments

We study a strongly interacting crowded system of self-propelled stiff filaments by event-driven Brownian dynamics simulations and an analytical theory to elucidate the intricate interplay of crowding and self-propulsion. We find a remarkable increase of the effective diffusivity upon increasing the filament number density by more than one order of magnitude. This counterintuitive "crowded is faster" behavior can be rationalized by extending the concept of a confining tube pioneered by Doi and Edwards for highly entangled, crowded, passive to active systems. We predict a scaling theory for the effective diffusivity as a function of the Péclet number and the filament number density. Subsequently, we show that an exact expression derived for a single self-propelled filament with motility parameters as input can predict the nontrivial spatiotemporal dynamics over the entire range of length and timescales. In particular, our theory captures short-time diffusion, directed swimming motion at intermediate times, and the transition to complete orientational relaxation at long times.

Velocity Distribution of a Homogeneously Cooling Granular Gas

In contrast to molecular gases, granular gases are characterized by inelastic collisions and require therefore permanent driving to maintain a constant kinetic energy. The kinetic theory of granular gases describes how the average velocity of the particles decreases after the driving is shut off. Moreover, it predicts that the rescaled particle velocity distribution will approach a stationary state with overpopulated high-velocity tails as compared to the Maxwell-Boltzmann distribution. While this fundamental theoretical result was reproduced by numerical simulations, an experimental confirmation is still missing. Using a microgravity experiment that allows the spatially homogeneous excitation of spheres via magnetic fields, we confirm the theoretically predicted exponential decay of the tails of the velocity distribution.

Crystallization via shaking in a granular gas with van der Waals interactions

We investigate the effect of van der Waals forces on a collection of granular particles by means of molecular dynamics simulations of a vibrated system in three dimensions. The van der Waals interactions introduce two phase coexistences: one between a random close packing and a gas and a second between a polycrystalline dense state and a gas, where the dense, disordered component crystallizes when the driving amplitude exceeds a threshold value. The region of stability of the ordered state in the nonequilibrium phase diagram grows in size as the Hamaker constant increases or the degree of dissipation increases.

Dynamics and thermodynamics of air-driven active spinners

We report on the collective behavior of active particles in which energy is continuously supplied to rotational degrees of freedom. The active spinners are 3D-printed disks, 1 cm in diameter, that have an embedded fan-like structure, such that a sub-levitating up-flow of air forces them to spin. Single spinners exhibit Brownian motion with a narrow Gaussian velocity distribution function, P(v), for translational motion. We study the evolution of P(v) as the packing fraction and the average single particle spin speeds are varied. The interparticle hydrodynamic interaction is negligible, and the dynamics is dominated by hyperelastic collisions and dissipative forces. As expected for nonequilibrium systems, P(v) for a collection of many spinners deviates from Gaussian behavior. However, unlike translationally active systems, phase separation is not observed, and the system remains spatially homogeneous. We then search for a near-equilibrium counterpart for our active spinners by measuring the equation of state. Interestingly, it agrees well with a hard-sphere model, despite the dissipative nature of the single particle dynamics.

An instrument for studying granular media in low-gravity environment

A new experimental facility has been designed and constructed to study driven granular media in a low-gravity environment. This versatile instrument, fully automatized, with a modular design based on several interchangeable experimental cells, allows us to investigate research topics ranging from dilute to dense regimes of granular media such as granular gas, segregation, convection, sound propagation, jamming, and rheology-all without the disturbance by gravitational stresses active on Earth. Here, we present the main parameters, protocols, and performance characteristics of the instrument. The current scientific objectives are then briefly described and, as a proof of concept, some first selected results obtained in low gravity during parabolic flight campaigns are presented.

Free Cooling of a Granular Gas of Rodlike Particles in Microgravity

Granular gases as dilute ensembles of particles in random motion are at the basis of elementary structure-forming processes in the Universe, involved in many industrial and natural phenomena, and also excellent models to study fundamental statistical dynamics. The essential difference to molecular gases is the energy dissipation in particle collisions. Its most striking manifestation is the so-called granular cooling, the gradual loss of mechanical energy E(t) in the absence of external excitation. We report an experimental study of homogeneous cooling of three-dimensional granular gases in microgravity. The asymptotic scaling E(t)∝t^{-2} obtained by Haff's minimal model [J. Fluid Mech. 134, 401 (1983)JFLSA70022-112010.1017/S0022112083003419] proves to be robust, despite the violation of several of its central assumptions. The shape anisotropy of the grains influences the characteristic time of energy loss quantitatively but not qualitatively. We compare kinetic energies in the individual degrees of freedom and find a slight predominance of translational motions. In addition, we observe a preferred rod alignment in the flight direction, as known from active matter or animal flocks.

Cooperative standing-horizontal-standing reentrant transition for numerous solid particles under external vibration

We report the collective behavior of numerous plastic bolt-like particles exhibiting one of two distinct states, either standing stationary or horizontal accompanied by tumbling motion, when placed on a horizontal plate undergoing sinusoidal vertical vibration. Experimentally, we prepared an initial state in which all of the particles were standing except for a single particle that was placed at the center of the plate. Under continuous vertical vibration, the initially horizontal particle triggers neighboring particles to fall over into a horizontal state through tumbling-induced collision, and this effect gradually spreads to all of the particles, i.e., the number of horizontal particles is increased. Interestingly, within a certain range of vibration intensity, almost all of the horizontal particles revert back to standing in association with the formation of apparent 2D hexagonal dense-packing. Thus, phase segregation between high and low densities, or crystalline and disperse domains, of standing particles is generated as a result of the reentrant transition. The essential features of such cooperative dynamics through the reentrant transition are elucidated with a simple kinetic model. We also demonstrate that an excitable wave with the reentrant transition is observed when particles are situated in a quasi-one-dimensional confinement on a vibrating plate.

Anisotropic particles strengthen granular pillars under compression

We probe the effects of particle shape on the global and local behavior of a two-dimensional granular pillar, acting as a proxy for a disordered solid, under uniaxial compression. This geometry allows for direct measurement of global material response, as well as tracking of all individual particle trajectories. In general, drawing connections between local structure and local dynamics can be challenging in amorphous materials due to lower precision of atomic positions, so this study aims to elucidate such connections. We vary local interactions by using three different particle shapes: discrete circular grains (monomers), pairs of grains bonded together (dimers), and groups of three bonded in a triangle (trimers). We find that dimers substantially strengthen the pillar and the degree of this effect is determined by orientational order in the initial condition. In addition, while the three particle shapes form void regions at distinct rates, we find that anisotropies in the local amorphous structure remain robust through the definition of a metric that quantifies packing anisotropy. Finally, we highlight connections between local deformation rates and local structure.

Mechanical excitation of rodlike particles by a vibrating plate

The experimental realization and investigation of granular gases usually require an initial or permanent excitation of ensembles of particles, either mechanically or electromagnetically. One typical method is the energy supply by a vibrating plate or container wall. We study the efficiency of such an excitation of cylindrical particles by a sinusoidally oscillating wall and characterize the distribution of kinetic energies of excited particles over their degrees of freedom. The influences of excitation frequency and amplitude are analyzed.

Segregation and pattern formation in dilute granular media under microgravity conditions

Space exploration and exploitation face a major challenge: the handling of granular materials in low-gravity environments. Indeed, grains behave quite differently in space than on Earth, and the dissipative nature of the collisions between solid particles leads to clustering. Within poly-disperse materials, the question of segregation is highly relevant but has not been addressed so far in microgravity. From parabolic flight experiments on dilute binary granular media, we show that clustering can trigger a segregation mechanism, and we observe, for the first time, the formation of layered structures in the bulk.

On the coarsening dynamics of a granular lattice gas

We investigated experimentally and theoretically the dynamics of a driven granular gas on a square lattice and discovered two characteristic regimes: Initially, given the dissipative nature of the collisions, particles move erratically through the system and start to gather on selected sites called traps. Later on, the formation of those traps leads to a strong decrease of the grain mobility and slows down dramatically the dynamics of the entire system. We realize detailed measurements linking a trap's stability to the global evolution of the system and propose a model reproducing the entire dynamics of the system. Our work emphasizes the complexity of coarsening dynamics of dilute granular systems.

Ordering in granular-rod monolayers driven far from thermodynamic equilibrium

The orientational order in vertically agitated granular-rod monolayers is investigated experimentally and compared quantitatively with equilibrium Monte Carlo simulations and density functional theory. At sufficiently high number density, short rods form a tetratic state and long rods form a uniaxial nematic state. The length-to-width ratio at which the order changes from tetratic to uniaxial is around 7.3 in both experiments and simulations. This agreement illustrates the universal aspects of the ordering of rod-shaped particles across equilibrium and nonequilibrium systems. Moreover, the assembly of granular rods into ordered states is found to be independent of the agitation frequency and strength, suggesting that the detailed nature of energy injection into such a nonequilibrium system does not play a crucial role.

Translational and rotational temperatures of a 2D vibrated granular gas in microgravity

We present an experimental study performed on a vibrated granular gas enclosed into a 2D rectangular cell. Experiments are performed in microgravity conditions achieved during parabolic flights. High speed video recording and optical tracking allow to obtain the full kinematics (translation and rotation) of the particles. The inelastic parameters are retrieved from the experimental trajectories as well as the translational and rotational velocity distributions. We report that the experimental ratio of translational versus rotational temperature decreases with the density of the medium but increases with the driving velocity of the cell. These experimental results are compared with existing theories and we point out the differences observed. We also present a model which fairly predicts the equilibrium experimental temperatures along the direction of vibration.

Driven inelastic Maxwell gases

We consider the inelastic Maxwell model, which consists of a collection of particles that are characterized by only their velocities and evolving through binary collisions and external driving. At any instant, a particle is equally likely to collide with any of the remaining particles. The system evolves in continuous time with mutual collisions and driving taken to be point processes with rates τ(c)(-1) and τ(w)(-1), respectively. The mutual collisions conserve momentum and are inelastic, with a coefficient of restitution r. The velocity change of a particle with velocity v, due to driving, is taken to be Δv=-(1+r(w))v+η, where r(w)∈[-1,1] and η is Gaussian white noise. For r(w)∈(0,1], this driving mechanism mimics the collision with a randomly moving wall, where r(w) is the coefficient of restitution. Another special limit of this driving is the so-called Ornstein-Uhlenbeck process given by dv/dt=-Γv+η. We show that while the equations for the n-particle velocity distribution functions (n=1,2,...) do not close, the joint evolution equations of the variance and the two-particle velocity correlation functions close. With the exact formula for the variance we find that, for r(w)≠-1, the system goes to a steady state. Also we obtain the exact tail of the velocity distribution in the steady state. On the other hand, for r(w)=-1, the system does not have a steady state. Similarly, the system goes to a steady state for the Ornstein-Uhlenbeck driving with Γ≠0, whereas for the purely diffusive driving (Γ=0), the system does not have a steady state.

Clustering and segregation in driven granular fluids

In microgravity, the successive inelastic collisions in a granular gas can lead to a dynamical clustering of the particles. This transition depends on the filling fraction of the system, the restitution of the used materials and on the size of the particles. We report simulations of driven bi-disperse gas made of small and large spheres. The size as well as the mass difference imply a strong modification in the kinematic chain of collisions and therefore alter significantly the formation of a cluster. Moreover, the different dynamical behaviors can also lead to a demixing of the system, adding a few small particles in a gas of large ones can lead to a partial clustering of the taller type. We realized a detailed phase diagram recovering the encountered regimes and developed a theoretical model predicting the possibility of dynamical clustering in binary systems.

Symmetry of interactions rules in incompletely connected random replicator ecosystems

The evolution of an incompletely connected system of species with speciation and extinction is investigated in terms of random replicators. It is found that evolving random replicator systems with speciation do become large and complex, depending on speciation parameters. Antisymmetric interactions result in large systems, whereas systems with symmetric interactions remain small. A co-dominating feature is within-species interaction pressure: large within-species interaction increases species diversity. Average fitness evolves in all systems, however symmetry and connectivity evolve in small systems only. Newcomers get extinct almost immediately in symmetric systems. The distribution in species lifetimes is determined for antisymmetric systems. The replicator systems investigated do not show any sign of self-organized criticality. The generalized Lotka-Volterra system is shown to be a tedious way of implementing the replicator system.

Collision dynamics of particle clusters in a two-dimensional granular gas

In a granular gas, inelastic collisions produce an instability in which the constituent particles cluster heterogeneously. These clusters then interact with each other, further decreasing their kinetic energy. We report experiments of the free collisions of dense clusters of particles in a two-dimensional geometry. The particles are composed of solid CO(2), which float nearly frictionlessly on a hot surface due to sublimated vapor. After two dense clusters of ≈100 particles collide, there are two distinct stages of evolution. First, the translational kinetic energy rapidly decreases by over 90% as a "jamming front" sweeps across each cluster. Subsequently, the kinetic energy decreases more slowly as the particles approach the container boundaries. In this regime, the measured velocity distributions are non-Gaussian with long tails. Finally, we compare our experiments to computer simulations of colliding, two-dimensional, granular clusters composed of circular, viscoelastic particles with friction.

Energy dissipation in driven granular matter in the absence of gravity

We experimentally investigate the energy dissipation rate in sinusoidally driven boxes which are partly filled by granular material under conditions of weightlessness. We identify two different modes of granular dynamics, depending on the amplitude of driving, A. For intense forcing, A>A(0), the material is found in the collect-and-collide regime where the center of mass of the granulate moves synchronously with the driven container, while for weak forcing, A<A(0), the granular material exhibits gaslike behavior. Both regimes correspond to different dissipation mechanisms, leading to different scaling with amplitude and frequency of the excitation and with the mass of the granulate. For the collect-and-collide regime, we explain the dependence on frequency and amplitude of the excitation by means of an effective one-particle model. For both regimes, the results may be collapsed to a single curve characterizing the physics of granular dampers.