PHYSICAL REVIEW E 88, 042205 (2013)

Small-amplitude acoustics in bulk granular media

David L. Henann,1,* John J. Valenza, II,2 David L. Johnson,2 and Ken Kamrin1

1Department of Mechanical Engineering, MIT, Cambridge, Massachusetts 02139, USA2Sensor Physics, Schlumberger-Doll Research, Cambridge, Massachusetts 02139, USA

(Received 3 August 2013; revised manuscript received 8 October 2013; published 31 October 2013)

We propose and validate a three-dimensional continuum modeling approach that predicts small-amplitudeacoustic behavior of dense-packed granular media. The model is obtained through a joint experimental and finite-element study focused on the benchmark example of a vibrated container of grains. Using a three-parameter linearviscoelastic constitutive relation, our continuum model is shown to quantitatively predict the effective mass spectrain this geometry, even as geometric parameters for the environment are varied. Further, the model’s predictionsfor the surface displacement field are validated mode-by-mode against experiment. A primary observation is theimportance of the boundary condition between grains and the quasirigid walls.

DOI: 10.1103/PhysRevE.88.042205 PACS number(s): 45.70.−n, 83.10.Ff, 43.20.Gp, 43.20.Jr

I. INTRODUCTION

The propagation of small-amplitude elastic waves throughgranular media is central to a variety of phenomena. Forexample, carefully designed granular crystals can exhibitfrequency band gaps or be designed as waveguides [1–3].Alternatively, many industries, such as automotive, aerospace,and oil and gas, are interested in utilizing bulk granularmedia—a large amorphous collection of densely packedgrains—to attenuate, or damp, elastic waves [4–8]. Modelingtechniques are crucial for the design and optimization ofdevices involving granular acoustics. Due to the periodicarrangement of grains, discrete methods are appropriate formodeling wave propagation in granular crystals; however,continuum methods are desired to model bulk granular mediadue to the large number of grains and amorphous structure.Continuum modeling of wave propagation in bulk granularmaterials has remained an open modeling issue. In particular,the importance of material nonlinearity due to the Hertziannature of the intergranular contacts is not clear (see [9,10]),nor is it clear to what extent complex grain-wall interactionsmay be simplified within a model.

As a straightforward example, we consider the benchmarkproblem of a shaker cup filled with grains under one-dimensional excitation, pictured schematically in Fig. 1. Thecup is subjected to a broad range of frequencies, and at eachfrequency, f , the complex-valued acceleration of the cup,a(f ), and the force exerted upon the cup, F (f ), may bemeasured. The effective mass of the granular media is thendefined as

M(f ) = F (f )

a(f )− MC, (1)

where MC is the (real-valued) static mass of the cup. On amacroscopic scale, the dynamic effective mass is useful forunderstanding the frequency-dependent acoustic propertiesand dissipative capacity of the granular system. A notableobservation from experiments is that resonant peaks do notoccur at ratios of 1 : 3 : 5 : 7 : . . . , as would be expected for

*Permanent address: School of Engineering, Brown University,Providence, RI 02906, USA; david_henann@brown.edu

the one-dimensional vibration of a linear elastic solid, andthe relative importance of material nonlinearity and boundaryeffects in determining the ratio of resonant peaks is not clear.Therefore, in this paper, we propose a general method formodeling small-amplitude bulk granular acoustics and validatethat it produces quantitative predictions of experimentalmeasurements in this benchmark case.

II. EXPERIMENTAL METHODS AND RESULTS

We consider fine-grained (∼10−1 mm) tungsten (W) par-ticles with irregular grain shape, described in [11]. Effectivemass measurements were conducted using the cylindrical cupapparatus shown schematically in Fig. 1, with a diameter of2R = 25.4 mm. The filling depth, L, of the granular mediawas varied to test the effect of the aspect ratio, R/L. Prior tothe measurement of M(f ), the tungsten grains were coatedwith a thin (� grain size) layer of polydimethylsiloxane(PDMS), using the process reported in [12]. The result ofthis process is a thin, solid viscoelastic film on each grain,which increases the damping capacity of the grains, as isdesired in industrial applications. The dry, coated grains werethen poured into the cup and subjected to a stress-controlledmechanical compaction protocol, described extensively in[11], to achieve some degree of consolidation.

A. Effective mass measurement

The complex-valued effective mass of the granular mediaM(f ) was then experimentally measured for different aspectratios, R/L, in the following manner. After employing theconsolidation protocol, the cup was affixed to the top ofa force gauge, which in turn was affixed to a minishaker.Two accelerometers were glued to the bottom of the cup, atopposite ends of a single diameter. The shaker was driven bythe amplified sinusoidal output of a signal generator. Beforeperforming the frequency sweep, L was inferred from thedifference between the depth of the cup and the depth tothe surface of the compacted granular medium. With thisinformation the sample density is given by ρ = M0/(πR2L),where M0 is the static mass of the granular media. Theeffective mass M(f ) was determined from the simultaneousmeasurement of the force, F (f ), and the acceleration, a(f ),

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HENANN, VALENZA II, JOHNSON, AND KAMRIN PHYSICAL REVIEW E 88, 042205 (2013)

FIG. 1. (Color online) Schematic cross section of the cylindricalcup apparatus.

at the bottom of the cylindrical cup [see Eq. (1)], where a(f )was determined by averaging that measured at the two pointson opposite ends of the cup diameter. The measurements wereperformed at roughly 1200 evenly spaced discrete frequenciesover a relatively broad band, f = 0.1–8 kHz. The magnitudesof F (f ) and a(f ) were measured with a lock-in amplifier,where the frequency of the shaker driving signal is thereference. Both of these quantities exhibited a phase shiftwith respect to the shaker drive, confirming that they arecomplex-valued and, consequently, that the effective mass isalso a complex-valued quantity. We represent the real partof the effective mass as M1(f ) and the imaginary part asM2(f ). The real and imaginary effective mass for tungstengrains subjected to the mechanical compaction protocol areshown as solid lines in Figs. 2(a) and 2(b), respectively, foraspect ratios R/L = 0.71, 1.05, and 2.01, demonstrating thatthe ratios of resonant peaks for a given aspect ratio are not1 : 3 : 5 : 7 : . . . . The utility of these quantities for predicting theeffect of a granular load on the broadband acoustic response ofa resonant structure has been previously demonstrated [11,13].

B. Mode-shape measurement

To gather information on the mode shapes at resonance [14],we employed a laser vibrometer to experimentally measurethe complex-valued displacement, u(i,j ), at discrete points(i,j ) on the free surface of the granular medium at fixedfrequencies. The laser was mounted on a right-angle bracket,attached to two translational stages. The position of each stagewas controlled by a step motor, capable of translating eachstage in perpendicular directions. At each position (i,j ) onthe top surface of the granular medium, the velocity, v(i,j ) =u(i,j )(2πf ), was determined from the doppler shift of thelight reflected off of the granular medium. The magnitude ofthe velocity was measured with a lock-in amplifier, with thefrequency of the shaker driving signal as the reference. Afterlocating the center of the cup, the laser was systematicallytranslated along the surface of the granular medium. Theseexperiments were performed with a 25.4-mm-diameter cup,employing a step size of 0.5 mm. The spot size of the laserwas on the order of ∼1 mm, so we characterize the modeshape at several locations as specified through a squaretwo-dimensional array that has 48 elements per side. Theindices i,j = 1,2,3, . . . ,48 represent the position of each

0 1000 2000 3000 4000 5000 6000 7000 80000

100

200

300

400

500

600

700

800

f (Hz)

M2

(g)

Experiment – R/L = 0.71Simulation – R/L = 0.71Experiment – R/L = 1.05Simulation – R/L = 1.05Experiment – R/L = 2.01Simulation – R/L = 2.01

0 1000 2000 3000 4000 5000 6000 7000 8000−300

−200

−100

0

100

200

300

400

500

f (Hz(a)

(b)

)

M1

(g)

Experiment – R/L = 0.71Simulation – R/L = 0.71Experiment – R/L = 1.05Simulation – R/L = 1.05Experiment – R/L = 2.01Simulation – R/L = 2.01

FIG. 2. (Color online) The (a) real M1 and (b) imaginaryM2 effective mass spectra for several aspect ratios, R/L = 0.71(black lines), 1.05 [red (gray)], and 2.01 [green (light gray)]. Solidlines represent experimental measurements while dashed lines aresimulation results.

measurement in both dimensions. We did not take data atthose points in the square array that correspond to locationsoutside of the cylindrical cup. At the same time, we monitoredthe acceleration at the bottom of the cup, aC(i,j ). The magni-tude of this value was also determined with a lock-in amplifierwith the shaker driving signal as the reference. Assuming thecup is rigid, the displacement amplitude of the cup is givenby uC(i,j ) = aC(i,j )/(2πf )2. The purpose of measuring thecup displacement is twofold. First, we want to ensure thatthe motion of the cup is uniform so that the displacement ofthe cup is a constant value, uC, at each position (i,j ). Second,we utilize uC to normalize the displacement magnitude at eachpoint on the free surface of the granular medium to determinethe relative displacement between the granular medium andthe cup, defined as

Urel(i,j ) = |u(i,j )||uC| − 1. (2)

Figures 3(a) and 3(b) show intensity plots for the normalizeddisplacement field, Urel(i,j ), of the two lowest frequencyresonant modes (f0 = 2740 Hz and f1 = 4785 Hz) formechanically compacted tungsten grains and an aspect ratioof R/L = 0.71 (solid black lines in Fig. 2). Figures 3(a) and3(b) demonstrate approximately axisymmetric mode shapes.Therefore, we compute the average value of Urel at each

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0 2 4 6 8 10 120

5

10

15

r (mm

(a) (b)

(c))

Ure

l

Experiment - First ModeSimulation - First ModeExperiment - Second ModeSimulation - Second Mode

FIG. 3. (Color online) (a,b) Intensity plots from the laser vi-brometer measurements showing the normalized displacement field,Urel(i,j ), of the two lowest frequency modes (f0 = 2740 Hz and f1 =4785 Hz) corresponding to the M(f ) shown in Fig. 2 (R/L = 0.71).The black circular outlines indicate the perimeter of the cup. (c) Theradial average of the relative displacement fields, along with thatpredicted by the finite-element model.

radial point r (measured from the central axis of the cup)and denote this quantity by 〈Urel〉. This average relativedisplacement is plotted in Fig. 3(c) for the first two resonantpeaks [black and red (gray) points, respectively]. Figure 3(c)allows for two important observations: (i) the mode shape issmooth, indicating continuum behavior, and (ii) the relativedisplacement is zero at the wall, indicating a no-slip wallcondition.

In light of (ii), at a physical level we recognize that thegrain-wall interaction is more precisely dictated by Coulombicfriction, a nonlinear boundary condition. Our explanation forwhy this reduces to an effectively no-slip condition in actualrealizations of granular packings hinges on the smallness ofthe vibration amplitudes. Noting that inertial forces on thegrains decrease with the vibration amplitude, as amplitudesbecome small enough, the material tractions at the wallcan be brought well beneath the wall’s frictional resistance,precluding wall slip. We have verified this explanation withadditional simulations. Using the commercial finite-elementpackage ABAQUS/EXPLICIT [15], we performed simulationsutilizing frictional boundary conditions in the configurationdescribed above and observed the related assertion thatfor fixed, small-amplitude driving the no-slip condition isobserved along the walls’ entirety for typical wall frictioncoefficients. Recognizing our experiments are in this regime

allows us to make the no-slip idealization, opening the doorfor linear modeling.

III. SIMULATIONS

As is demonstrated in the following, accounting for theno-slip wall condition is crucial to rationalize the observedexperimental results. In fact, we find that excellent agreementmay be achieved by using a three-dimensional version ofthe Kelvin-Voigt viscoelastic constitutive model, in which thestress tensor σ is given as the sum of linear elastic and linearviscous parts: σ = σ e + σ v. The elastic contribution is givenby the familiar linear elastic relation

σ e = E

1 + νε + Eν

(1 + ν)(1 − 2ν)(tr ε)1, (3)

where ε is the infinitesimal strain tensor, E is Young’smodulus, and ν is Poisson’s ratio. For the viscous part, weassume Rayleigh proportional stiffness damping, in which theviscous stress is simply given by

σ v = β

[E

1 + νε + Eν

(1 + ν)(1 − 2ν)(tr ε)1

], (4)

where ε is the strain-rate tensor and β is the Rayleigh stiffnessdamping parameter, having units of time. This model reliesupon three material parameters, E, ν, and β, to fully describethe dynamic response of the material. We understand theintrinsic limitation of a linear model with constant materialparameters—as vibrational amplitudes get larger the nonlinearHertzian interactions between grains becomes important [16].However, we show that a linear constitutive model becomes anacceptable simplification for the purpose of general modelingof small-amplitude acoustics in bulk, dense-packed grains.Moreover, despite the well-known Hertzian contact mechanicsof particles [17,18], we have verified in separate numerical cal-culations that nonlinear constitutive models based on Hertziancontacts (i.e., Jiang-Liu elasticity [9,10]) yield negligibleimprovement over the strictly linear constitutive model weuse here.

A. Effective mass

In order to predict the frequency spectrum of the effectivemass, we perform finite-element calculations, utilizing thedirect-solution steady-state dynamics analysis procedure inABAQUS/STANDARD [15]. This linear solution technique calcu-lates the complex-valued, steady-state response of a discretizedsystem under harmonic excitation at a given frequency, f . Theinitial mesh configuration for an aspect ratio of R/L = 0.71 isshown in Fig. 4, and a similar mesh resolution is used for otheraspect ratios. We make an axisymmetric simplification, withthe axis of symmetry denoted in the figure. The nodes on thebottom and right-hand side of the mesh represent the granularmaterial in contact with the cup floor and wall. In line with theobservation of no slip at the wall, we apply the base excitation,i.e., the motion of the cup, directly to these nodes, usingdisplacement control. The complex-valued reaction force isthen determined from the simulation results at each frequency,allowing for the calculation of the effective mass spectra viaEq. (1). Since we have chosen a linear elastic constitutive

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HENANN, VALENZA II, JOHNSON, AND KAMRIN PHYSICAL REVIEW E 88, 042205 (2013)

FIG. 4. Initial finite-mesh configuration for axisymmetric effec-tive mass simulation (R/L = 0.71).

model, gravity has no effect on the simulated results and istherefore omitted.

Figure 2 shows an example of model calculations (dashedlines) for three of the various aspect ratios employedexperimentally. For these calculations the Young’s modulusof the mechanically compacted tungsten grains is taken tobe E = 162 MPa. This value provides the best fit for thefrequency of the first resonant peak f0 for the three aspectratios shown. The value of Poisson’s ratio has little effect onthe first resonant peak but does affect the location of the secondpeak f1. A value of ν = 0.1 was chosen to best fit this secondpeak in all cases. Finally, a single value of β = 3.7 × 10−6 swas taken to best fit the heights of the resonant peaks f0 andf1 in Fig. 2(b). The simple linear material model having onlythree material parameters does a good job of capturing thesalient features of the effective mass spectra for the broadrange of aspect ratio employed experimentally.

B. Mode shapes

The model results also well describe the mode shapes ofthe first two resonant peaks. Figure 5 shows the calculatedcomplex-valued displacement fields for the first two resonantpeaks (f0 = 2711 Hz and f1 = 5115 Hz in the simulations)for R/L = 0.71. Half of the cylinder is displayed for clarity.Since the displacement field is complex valued, it may beparametrized by a phase angle, ranging from 0 to 2π rad.The displacement fields at π/2 and 3π/2 rad are shown inFigs. 5(a) and 5(b), respectively, for the first resonant peak,while the same results are shown in Figs. 5(c) and 5(d) for thesecond resonant peak. (Movies of the full cycles are includedin the Supplemental Material [19].) At these two points duringthe steady-state cycle, the displacement magnitudes in thebulk are at their maximum, and the displacements for phaseangles of π/2 and 3π/2 rad are negatives of each other. Asexpected for driving near resonance, the displacements in thebulk are out of phase with the cup motion, which attainsmaximums in magnitude at phase angles of 0 and π rad.From these simulation results, we calculate the radial profile

FIG. 5. (a) The simulated displacement fields for the first resonantpeak (f0 = 2711 Hz) at phase angles of π/2 and 3π/2 rad for R/L =0.71. (b) The same simulation result for the second resonant peak(f1 = 5115 Hz) at phase angles of π/2 and 3π/2 rad for R/L = 0.71.

of the average relative displacement on the top surface forR/L = 0.71 via Eq. (2) and plot the profiles for the first tworesonant peaks using dashed lines in Fig. 3(c). The comparisonto the experiments is excellent.

C. Generalization to other granular systems

To this point, we have shown that our simple model ispredictive for a single granular system and several aspectratios. To demonstrate that our approach is general, weconsider different preparation protocols, grain materials, anda wider range of aspect ratios. As the most important featureof the effective mass spectrum, we consider the frequency ofthe first resonant peak f0 and its dependence upon the modelgeometry (R and L) and material and preparation protocol(ρ, E, and ν). (For the lightly damped case, i.e., small β, thenumerical value of β has a negligible effect on f0 [20].) Usingdimensional analysis, the functional dependence should be ofthe form

f0L√E/ρ

= g

(R

L,ν

). (5)

It is straightforward to probe this functional dependence usingour simulation capability. We find that the function g isinsensitive to Poisson’s ratio in a range from 0.0 to 0.2 andplot the normalized frequency f0L/

√E/ρ as a function of the

aspect ratio R/L as a solid line in Fig. 6 for ν = 0.1.We consider several consolidation protocols for the tung-

sten grains, achieving different degrees of compaction and thus

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0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

R/L

f0L

/E

/ρ

SimulationW Experiment – Tapped – E = 23 MPaW Experiment – 5 m/s2 – E = 46 MPaW Experiment – 30 m/s2 – E = 120 MPaW Experiment – Compacted – E = 162 MPaSS Experiment – Tapped – E = 17 MPa

FIG. 6. Normalized frequency corresponding to f0 (data points)from effective mass measurements with two metallic granular systemsafter a variety of handling protocols as a function of aspect ratio. Theone notable free parameter, E, used to normalize the independentdata sets is noted in the legend. Also shown is the prediction oflinear finite-element simulations (solid line). Theoretical high and lowaspect ratio limits are plotted as dotted and dashed lines, respectively.

acoustic properties. The lowest consolidation was achieved bysimply tapping the side of the cylindrical cup, while slightlygreater compaction was achieved by vibrating the granularmedium at f = 1 kHz and a = 5 m/s2. The latter protocolwas also employed with a higher acceleration amplitude,a = 30 m/s2, to achieve greater consolidation, and the greatestdegree of consolidation was achieved with the aforementionedcompaction protocol. While the mechanically compactedgrains were coated with PDMS, all other samples weretested without coating. We also consider a stainless steel (SS)granular material with fairly spherical, monodisperse particles(∼10−1 mm), subjected to tapping. While the aspect ratios forall tungsten samples were varied by changing the filling heightwith a fixed 25.4-mm-diameter cup, the cup radius was alsovaried for the SS samples, using cups with one of the followingdiameters (in mm): 6.35, 12.7, 25.4, or 37.5. The symbols inFig. 6 show a compilation of the data for the normalized lowestfrequency mode as a function of aspect ratio for each of thefive granular systems described. To perform the normalizationfor each system, a value of the Young’s modulus E must bechosen. The value of E used in the normalization processis noted in the legend for each granular system. The valuesof E determined in this way are reasonable and follow thetrend of increasing stiffness with greater compaction [11].Once this free parameter is determined, the simulation welldescribes the effect of aspect ratio for all granular systems,providing further confidence that the simple linear model isphysically representative of the behavior of the bulk granularmedia.

The high and low aspect ratio limits of the simulationmay be understood using simple toy models. In the limit ofR/L → ∞, i.e., an infinitely wide cup, in which the wallshave no effect, we recover the one-dimensional plane-waveresponse. In this case, due to the closed bottom and open topof the cup, the first resonant peak corresponds to a quarterwavelength longitudinal wave, expressed for a linear elasticmaterial as f0L/vl = 0.25, where vl is the longitudinal wavespeed. Expressed in terms of the elastic material parameters,

we have

f0L√E/ρ

= 0.25

√(1 − ν)

(1 + ν)(1 − 2ν). (6)

In the opposite limit of R/L → 0, i.e., an infinitely tall cup,the displacements may be idealized as varying only with radialposition and not vertical position, effectively ignoring endeffects. Vibrational modes in this case correspond to that ofa circular drum head with displacement increments along thedirection of the cylinder axis, the lowest frequency mode beinggiven by 2πf0R/vt = 2.4, where vt is the tangential wavespeed and 2.4 arises as the first root of the zeroth-order Besselfunction of the first kind. In terms of E and ν, the expressionbecomes

f0L√E/ρ

= 2.4

2π (R/L)√

2(1 + ν). (7)

For both cases, the dependence on the Poisson’s ratio is weakin the range of 0.0 to 0.2. We plot the R/L → ∞ limit [Eq. (6)]as a dotted line and the R/L → 0 limit [Eq. (7)] as a dashedline in Fig. 6 for ν = 0.1. In both cases, the limits are welldescribed by the simple toy models, giving a sense of theunderlying physics.

IV. CONCLUSION

In closing, we recall that the results reported here are limitedto (i) small-amplitude acoustics in (ii) densely packed matter.Regarding the first condition, small-amplitude excitations arecrucial to achieving the effective no-slip wall interactionand linear material behavior, opening the door for the lin-ear modeling approach pursued here. As larger amplitudeexcitations are considered, more complex, nonlinear frictionalwall interactions as well as a nonlinear elastic material model,such as the Jiang-Liu model [9], may need to be included.Regarding the second condition, as we leave the dense-packedregime toward looser packings, we expect the predictivecapability across the entire effective mass spectrum to lessen.However, the dominating behavior, i.e., the details of the lowestfrequency mode, appear to be well captured over an array ofconsolidation levels (see Fig. 6). Under these two conditions,we have shown that a simple linear constitutive model isrepresentative of the dynamic response of a dense-packedgranular media, providing an excellent quantitative descriptionof the effect of geometry on the effective mass spectra aswell as the resonant mode shapes. The simulation capabilityworks for various materials and preparations, and we expectthe capability to be amenable to changes in geometry or otherconditions, such as the application of a confining pressure tothe top surface [20].

ACKNOWLEDGMENTS

D.L.H. and K.K. acknowledge support from Schlumberger-Doll Research. Larry McGowan is gratefully acknowledgedfor assistance with experiments.

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[1] C. Daraio, V. F. Nesterenko, E. B. Herbold, and S. Jin, Phys.Rev. E 72, 016603 (2005).

[2] C. Daraio and V. F. Nesterenko, Phys. Rev. E 73, 026612 (2006).[3] C. Daraio, V. F. Nesterenko, E. B. Herbold, and S. Jin, Phys.

Rev. E 73, 026610 (2006).[4] W. Kuhl and H. Kaiser, Acustica 2, 179 (1962).[5] L. Cremer and M. Heckl, Structure Borne Sound (Springer,

Berlin, 1973).[6] J. M. Bourinet and D. Le Houedec, in Powders & Grains, edited

by R. Behringer and J. Jenkins (Balkema, Rotterdam, 1997),Vol. 97.

[7] J. C. Sun, H. B. Sun, L. C. Chow, and E. J. Richards, J. SoundVib. 104, 243 (1986).

[8] S. A. Nayfeh, J. M. Verdirame, and K. K. Varanasi, Proc. SPIE4697, 158 (2002).

[9] Y. Jiang and M. Liu, Phys. Rev. Lett. 91, 144301 (2003).[10] L. Bonneau, B. Andreotti, and E. Clement, Phys. Rev. E 75,

016602 (2007).[11] J. Valenza, C.-J. Hsu, R. Ingale, N. Gland, H. A. Makse, and

D. L. Johnson, Phys. Rev. E 80, 051304 (2009).

[12] J. J. Valenza and D. L. Johnson, Phys. Rev. E 85, 041302 (2012).[13] J. J. Valenza, C.-J. Hsu, and D. L. Johnson, J. Acoust. Soc. Am.

128, 2768 (2010).[14] Regarding our “resonance” terminology, we drive the system

at a frequency equal to the real part of the complex-valuednormal mode frequency, which has the effect of maximizingthe contribution of the mode in question to the observeddisplacement field.

[15] ABAQUS, reference manuals, 2010.[16] L. R. Gomez, A. M. Turner, M. van Hecke, and V. Vitelli, Phys.

Rev. Lett. 108, 058001 (2012).[17] K. L. Johnson, Contact Mechanics (Cambridge University Press,

Cambridge, UK, 1985).[18] H. A. Makse, N. Gland, D. L. Johnson, and L. Schwartz, Phys.

Rev. E 70, 061302 (2004).[19] See Supplemental Material at http://link.aps.org/supplemental/

10.1103/PhysRevE.88.042205 for full-cycle movies of the firsttwo resonant modes for R/L = 0.71.

[20] Y. Hu, D. L. Johnson, J. J. Valenza, and H. A. Makse(unpublished).

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