Nonlinear normal modes and band zones in granular chains with no pre-compression

Nonlinear DynDOI 10.1007/s11071-010-9809-0

O R I G I NA L PA P E R

Nonlinear normal modes and band zones in granular chainswith no pre-compression

K.R. Jayaprakash · Yuli Starosvetsky ·Alexander F. Vakakis · Maxime Peeters ·Gaetan Kerschen

Received: 4 March 2010 / Accepted: 11 August 2010© Springer Science+Business Media B.V. 2010

Abstract We study standing waves (nonlinear nor-mal modes—NNMs) and band zones in finite gran-ular chains composed of spherical granular beads inHertzian contact, with fixed boundary conditions. Al-though these are homogeneous dynamical systems inthe notation of Rosenberg (Adv. Appl. Mech. 9:155–242, 1966), we show that the discontinuous nature ofthe dynamics leads to interesting effects such as sep-aration between beads, NNMs that appear as travel-ing waves (these are characterized as pseudo-waves),and localization phenomena. In the limit of infinite ex-tent, we study band zones, i.e., pass and stop bands inthe frequency–energy plane of these dynamical sys-

K.R. Jayaprakash (�) · Y. Starosvetsky · A.F. VakakisDepartment of Mechanical Science and Engineering,University of Illinois at Urbana Champaign,1206 W. Green Street, Urbana, IL 61822, USAe-mail: [email protected]

Y. Starosvetskye-mail: [email protected]

A.F. Vakakise-mail: [email protected]

M. Peeters · G. KerschenDepartment of Aerospace and Mechanical Engineering,University of Liege, 1, Chemin des Chevreuils (B52/3),4000 Liege, Belgium

M. Peeterse-mail: [email protected]

G. Kerschene-mail: [email protected]

tems, and classify the essentially nonlinear responsesthat occur in these bands. Moreover, we show howthe topologies of these bands significantly affect theforced dynamics of these granular media subject tonarrowband excitations. This work provides a classi-fication of the coherent (regular) intrinsic dynamics ofone-dimensional homogeneous granular chains withno pre-compression, and provides a rigorous theoreti-cal foundation for further systematic study of the dy-namics of granular systems, e.g., the effects of dis-orders or clearances, discrete breathers, nonlinear lo-calized modes, and high-frequency scattering by lo-cal disorders. Moreover, it contributes toward the de-sign of granular media as shock protectors, and in thepassive mitigation of transmission of unwanted distur-bances.

Keywords Nonlinear normal modes · Granularmedia · Band gaps

1 Introduction

One defines nonlinear normal modes (NNMs) as syn-chronous periodic particular solutions of the nonlin-ear equations of motion of dynamical systems, keep-ing in mind the fact that the superposition principleis no longer valid in the nonlinear case. With such arestricted definition, a nonlinear generalization of theconcept of normal mode of linear vibration theory ispossible, and beginning with the works of Lyapunov

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K.R. Jayaprakash et al.

[1] several attempts were made in this direction. Lya-punov’s theorem proves the existence of n synchro-nous periodic solutions (NNMs) in neighborhoodsof stable equilibrium points of n degrees-of-freedom(DOF) Hamiltonian systems whose linearized eigen-frequencies are not integrably related. The formula-tion and development of the theory of NNMs can beattributed to Rosenberg and his co-workers who de-veloped general qualitative [2] and quantitative [3–6]techniques for analyzing NNMs in discrete conserva-tive oscillators. Rosenberg considered n DOF conser-vative oscillators and defined NNMs as vibrations-in-unison, i.e. synchronous time-periodic motions duringwhich all coordinates of the system vibrate equiperi-odically (in a synchronous fashion), reaching theirmaximum and minimum values at the same instant oftime. In linearizable systems with weak nonlinearitiesit is natural to suppose that NNMs are particular peri-odic solutions that, as the nonlinearities tend to zero,approach in limit the classical normal modes of thecorresponding linearized systems. On the contrary, es-sentially nonlinear systems do not show such a behav-ior, since the NNMs in these systems are not neces-sarily extensions of normal modes of linear systems.Thus the NNMs in these systems may exceed in num-ber the degrees-of-freedom of the system, and, in addi-tion, certain of the NNMs may not have counterparts inlinear theory. This is due to NNM bifurcations, whichbecome exceedingly more complicated as the numberof DOF of the systems increases and introduces newfeatures, such as nonlinear mode localization, in thedynamics.

The discussion of NNMs in the literature has fo-cused mainly on dynamical systems with smoothnonlinearities [1–8], although some works on sys-tems with non-smooth nonlinearities have appeared[27–29]. In this work we will consider NNMs in ho-mogeneous granular chains with no pre-compression,possessing essential (nonlinearizable) nonlinearitiesdue to Hertzian contact interactions between beads[9]. In such strongly nonlinear systems we can ex-tend the definition of NNM but we will need to in-troduce some modifications. For example, it will beshown that the condition of synchronicity is not validfor certain of the nonlinear modes of granular chains.We mention at this point that the dynamics of gran-ular chains has been studied extensively both analyt-ically and experimentally. Notably, in the works byNesterenko [9], Daraio et al. [10, 11], Sen et al. [12],

Nesterenko et al. [13] and Herbold et al. [14] differentaspects of disturbance transmission in homogeneousand disordered granular chains were studied.

Granular media is a highly complex and distinctclass of dynamical systems. The dynamics of thesemedia is highly tunable, and depending on the ap-plied pre-compression it can be either strongly orweakly nonlinear and smooth or non-smooth. Indeed,for strong pre-compression the dynamics of granu-lar media is weakly nonlinear, whereas when no pre-compression is applied the dynamics is strongly non-linear (in fact, not even linearizable) and separationbetween beads is possible (in the absence of com-pressive forces that keep neighboring beads in con-tact). Hence, the dynamics of granular chains withno (or sufficiently weak) pre-compression is eithersmooth—when neighboring beads are in contact—or non-smooth when separation between beads takesplace and collisions between beads take place. Thisforms an added source of strong nonlinearity in thegranular medium, i.e., in addition to the strongly non-linear Hertzian law interaction between beads in con-tact.

In this work we will consider one-dimensional ho-mogeneous granular media in their essentially non-linear limit of absence of pre-compression betweenbeads. In this case the force (F ) vs. displacement (d)law governing the Hertzian interaction between neigh-boring spherical beads is highly nonlinear:

F ={

kd3/2, d ≥ 0

0, d < 0

since it lacks a linear term in compression and is zeroin extension (when bead separation occurs); in theabove relation, k is the nonlinear coefficient. Such agranular system does not support oscillations of in-dividual beads about zero mean positions, but rathertheir oscillations are highly asymmetric and non-synchronous. Such systems have been characterizedby Nesterenko [9] as “acoustic vacua” or “sonic vac-uum,” meaning sound propagation is impossible inthese systems. More precisely, it implies the impos-sibility for linear waves described by standard waveequation to propagate in them. It follows that theanalysis of the dynamics requires the application ofstrongly nonlinear methods that are individually tai-lored for this class of systems since linearized orweakly nonlinear methods are not applicable.

Nonlinear normal modes and band zones in granular chains with no pre-compression

Our study of standing waves (NNMs) in one-dimensional granular media will be structured as fol-lows. In Sect. 2, we initiate our analysis by consid-ering a two-bead system with fixed boundary condi-tions, and study NNMs by numerical Poincaré mapsand analytical techniques in terms of hypergeomet-ric functions [15]. In Sect. 3 we will extend ouranalysis to three-bead granular systems, and com-pute the corresponding NNMs by numerical methods.In Sect. 4 we will focus on higher-dimensional (butfinite-dimensional) systems and consider exclusivelytwo specific classes of NNMs, namely the in-phaseand out-of-phase ones since, as we will show, theseform the boundaries of the frequency–energy bandwithin which all other NNMs are realized. In Sect. 5we will study the dynamics of infinite-dimensional ho-mogeneous granular chains, and define propagation(pass) and attenuation (stop) bands in the frequency–energy plane, within which disturbances either propa-gate or attenuate, respectively, as they are transmittedthough these media. The effect of these bands on theforced dynamics of high-dimensional systems will bestudied in Sect. 6, where numerical integrations willconfirm the theoretical predictions regarding the roleof the pass and stop bands on the dynamics. Some con-cluding remarks and a summary of the main findingswill be provided in Sect. 7.

2 Two-bead granular system

The two-bead granular system considered is depictedin Fig. 1. The system is homogeneous in the sense thatthe two spherical beads have identical geometric andmaterial properties. Moreover, the beads are placedbetween rigid boundaries (walls) with no gaps exist-ing between them or with the walls when they are intheir trivial equilibrium positions. The system is con-sidered without any pre-compression, and thus it ex-hibits strong (essential) stiffness nonlinearity and po-tentially non-smooth effects. Furthermore, no dissipa-tive effects are yet considered in this study, but weakdissipative forces in the bead interactions will be con-sidered later, in our computational study of the forceddynamics of high-dimensional granular media carriedout in Sect. 6. Assuming Hertzian contact interactionbetween beads and between the end beads and the rigidwalls, the kinetic and potential energies of the two-

Fig. 1 The two-beadsystem

bead system are defined as follows:

T = 1

2m

(u2

1 + u22

)

U = 2

5

E(2R)1/2

3(1 − μ2)

× [(−u1)

5/2+ + (u2)

5/2+ + (u1 − u2)

5/2+

](1)

where, ui is the displacement of the ith bead, R de-notes radius, E elastic modulus, μ Poisson’s ratio,ρ mass density, and m mass of each bead; the (+) sub-script in the expressions in (1) is used to emphasizethat the bracketed term is non-zero only if the terminside the bracket is positive and zero otherwise. Theequations of motion can be derived from Lagrange’sequation as follows:

u1 = A[(−u1)

3/2+ − (u1 − u2)

3/2+

]

u2 = A[−(u2)

3/2+ + (u1 − u2)

3/2+

] (2)

where A = E(2R)1/2/[3m(1 − μ2)].The displacements are non-dimensionalized by

means of the following normalizations:

X = u

R, τ = t

2R/C

(π(1 − μ2)√

2

)−1/2

C =√(

E

ρ

)

so the equations of motion are placed in the followingnormalized form:

X′′1 = (−X1)

3/2+ − (X1 − X2)

3/2+

X′′2 = −(X2)

3/2+ + (X1 − X2)

3/2+

(3)

where Xi = Xi(τ) in the terms of the normalizedtime τ , and prime denotes differentiation with respectto τ . System (3) will be employed in the followinganalysis. We re-emphasize at this point that the dy-namics of system (3) is not only essentially nonlin-


Fig. 2 Poincaré map of the global dynamics of the two-bead system for h = 0.0001

ear (as the stiffness terms do not possess linear com-ponents), but, in addition, they are non-smooth. Theloss of smoothness is due to the fact that the interac-tion force due to (compressive) Hertzian contact van-ishes when the center distance between the two beadsexceeds a length equal to twice the radius. From theabove, we expect the dynamics to be highly com-plex.

Since the system possesses two degrees-of-freedomand is conservative, it is possible to analyze its globaldynamics in terms of numerical Poincaré maps. In-deed, a nonlinear normal mode (NNM) of this systemis defined as a time-periodic oscillation where the beadoscillations possess identical frequencies but are notnecessarily synchronous. Moreover, whether the dy-namics is smooth or non-smooth, a NNM should bedepicted as a (modal) curve in the configuration plane.System (3) possesses a four-dimensional phase space,but owing to energy conservation (since no dissipative

effects considered),

E(X1,X

′1,X2,X

′2

) ≡ (1/2)(X′2

1 + X′22

)+ (2/5)

[(−X1)

5/2+ + (X2)

5/2+

+ (X1 − X2)5/2+

] = h

the dimensionality can be reduced by one, and thedynamics can be restricted to a three-dimensionalisoenergetic manifold. Intersecting the isoenergeticflow by a two-dimensional cut section defined as T :{X2 = 0}, we obtain a two-dimensional Poincaré mapP : Σ → Σ , where the Poincaré section Σ is definedas Σ = {X2 = 0, X′

2 > 0} ∩ {E(X1,X′1,X2,X

′2) = h}

and depicts the global nonlinear dynamics of the sys-tem on the cut section that is now parameterized by(X1,X

′1). In Fig. 2 we depict the numerical map at

the energy level h = 0.0001. Three types of periodicsolutions are detected, namely an in-phase NNM, an


Fig. 3 The in-phase NNM for the two-bead granular system and h = 0.0001: (a) time series, (b) modal curve in the configurationplane, (c) depiction in a projection of the phase plane

out-of-phase NNM and numerous subharmonic orbits.We note that the out-of-phase NNM is a synchronousoscillation and lies on the X1 axis on the cut section,whereas the in-phase NNM is offset from that axisand is an asynchronous oscillation. Moreover, the twomodes appear to be stable and, hence, physically real-izable.

In Fig. 3, we depict the in-phase NNM for the en-ergy level h = 0.0001 (region A of the Poincaré plotin Fig. 2). From the time series in Fig. 3a, we notethat the two beads pass through their equilibrium posi-

tions at different instants of time. Hence, such a modedoes not conform to the traditional definition of NNMas proposed by Rosenberg [3]; however, by extendingthe definition to account for non-synchronicity in thebead oscillation, we may classify this time-periodicmotion as in-phase NNM. The energy exchanges be-tween beads for that mode are of particular interest.Referring to Fig. 3a, at point 1 the entire energy ofthe system is elastic due to the interaction of the sec-ond bead with the wall, whereas the first bead is mo-tionless but offset from its equilibrium position. As


time progresses, at point 2 the beads start interactingwith each other, and at that time instant the first beadcollides with the second. Proceeding to point 3, thesecond bead loses contact with the wall as it passesthrough its equilibrium position. Once the first beadpasses through its equilibrium position at point 4, itstarts interacting with the rigid wall and at that pointthe second bead has transferred almost all of its en-

ergy to the first bead and is about to get stationary.At point 5, the second bead loses all its energy, be-comes stationary but offset from its equilibrium posi-tion, and the entire process repeats itself. It is inter-esting to note that from point 2 to 5, both beads re-main in contact with each other, and, as explained pre-viously, the second bead loses contact with the wallwhereas the first bead gains contact with it. Beyond

Fig. 4 Comparing the dynamics of the two-bead granular system to the dynamics of the corresponding vibro-impacting one: (a) timeseries, (b) modal curve in the configuration plane


point 5, the first bead interacts with the wall and sub-sequently loses contact with the second bead and thecomplete energy of the system is in form of elastic po-tential energy due to the interaction of the first beadand the wall (at point 6). Hence, complete exchangeof energy between the two beads occurs, and the in-phase NNM can be represented by the time-delayedrelation X1(τ ) = −X2(τ − T/2), where T is the pe-riod of the in-phase NNM. The previous discussionhighlights the complex and highly asymmetric oscil-lation of the granular system when it oscillates in thisin-phase periodic motion.

The non-smooth character of the in-phase NNMcan be inferred from the depiction of the dynamics inthe projection of the phase plane (cf. Fig. 3c), where adiscontinuity in slope can be readily noted. This hap-pens at the time instant when one of the beads losescontact with the wall and the other stands stationary. Itfollows that non-smooth effects in the oscillation areassociated with ‘silent’ periods of bead responses, i.e.,with phases of the motion where a bead is stationary atan offset position from the zero equilibrium. Moreover,the depiction of the modal curve in the configurationplane in Fig. 3b confirms our earlier observation re-garding the non-synchronicity of the bead oscillationssince they cross their equilibrium points at differentinstants of time.

Finally, a comparative study of the dynamics of thetwo-bead granular system to the dynamics of a two-DOF vibro-impacting system (corresponding to per-fectly rigid beads and purely elastic collisions betweenthem and the wall) is performed in Fig. 4. The compar-ison is purely qualitative and thus we have attributedany energy to neither of the Hertzian or vibro impactsystems. Due to the Hertzian contact law, deforma-tion of beads during collisions occurs, and as a resultthe time series is smooth. By contrast, in the vibro-impacting system bead deformation does not occur,and so there exist non-smooth transitions between dif-ferent phases of the dynamics. Still, the vibro-impactin-phase NNM could be considered as a generating so-lution to develop asymptotic approximations for thein-phase NNM of the granular medium in the limit ofsmall bead deformations.

Focusing now on the out-of-phase NNM (region Bin the Poincaré map in Fig. 2), in Fig. 5a we depict thecorresponding responses of the two beads. In this caseneither of the two beads reaches stationarity after sep-aration, so no ‘silent’ regions exist, and the dynamics

is smooth (in the sense that no non-smooth effects oc-cur in the phase plot in Fig. 5c). Indeed, the two beadsoscillate in synchronicity and in out-of-phase fashion,and at the precise time instant when they loose contactwith the rigid walls they get into contact and start inter-acting with each other. On the other hand, when theydetach from each other, they get into contact with thewalls. Hence, at any instant of time a bead would eitherbe interacting with the wall or with the other bead andthe dynamics is smooth. Moreover, both beads crosstheir equilibrium positions at the same instant of time,so this mode conforms to the Rosenberg’s definitionof NNM [6]. It is worth mentioning that for higher-dimensional systems the out-of-phase mode does notconform to the Rosenberg’s definition of NNM [6]. Incase of even number of beads, the pair of beads, whichare positioned symmetric about the center of mass ofthe chain, would pass through their equilibrium pointsat the same instant of time.

Due to the fact that the out-of-phase NNM corre-sponds to synchronous and symmetric oscillations ofthe two beads, it can be explicitly analyzed. To thisend, the oscillation is divided into two phases, dur-ing which the strongly nonlinear equations of mo-tion decouple and can be explicitly solved. In the firstphase of the oscillation, the total energy h of the gran-ular system is partitioned equally between the twobeads in the form of elastic potential energy due totheir interactions with the rigid walls. Hence, in thisphase the two beads remain in contact with the wallsand are detached from each other. Consequently, themutual interaction terms in the equations of motionvanish and the equations of motion decouple com-pletely:

X′′1 = (−X1)

3/2+

X′′2 = −(X2)

3/2+

(4)

subject to the initial conditions X1(0) = −(5h/4)2/5,X′

1(0) = 0, X2(0) = (5h/4)2/5, and X′2(0) = 0. De-

noting X1(τ ) = −X2(τ ) ≡ X(τ), the solution of (4) iscomputed in explicit form as

τ = h−1/10(5/4)2/52F1

(1

2,

2

5; 7

5;1

)

− X2√h

2F1

(1

2,

2

5; 7

5; 4X

5/22

5h

)(5)


Fig. 5 The out-of-phase NNM for the two-bead granular system and h = 0.0001: (a) time series, (b) modal curve in the configurationplane, (c) depiction in a projection of the phase plane

where 2F1(•,•; •; •) is a hypergeometric function de-fined as in [15],

2F1(a, b; c; z) = �(c)

�(a)�(b)

×∞∑

n=0

(�(a + n)�(b + n)

�(c + n)

)(zn

n!)

2F1(a, b; c;1) = �(c)�(c − a − b)

�(c − a)�(c − b)

and �(•) is the Gamma function [15]. This solutionis valid only until the two beads reach their respectiveequilibrium points and the first phase of the oscillationis completed.

At that time instant the second phase of the oscilla-tion starts. As the beads pass through their equilibriumpositions, they lose their contact with the rigid wallsand engage in mutual interaction. Again, the equa-tions of motion can be combined into one and solved


in closed form. To show this, we consider again theequations of motion

X′′1 = −(X1 − X2)

3/2+

X′′2 = (X1 − X2)

3/2+

(6)

and introduce the relative displacement variable δ =X1 − X2. Then, the equations of motion can be com-bined into the following single equation:

δ′′ = −2(δ)3/2+ (7)

which can be solved explicitly by quadratures. Wenote at this point that at the end of the first phase ofthe oscillation the total energy of the system, h, is inthe form of kinetic energy, equally distributed betweenthe two beads. Hence, at the beginning of the secondphase of the oscillation the beads have opposite veloc-ities, which provides us with the necessary initial con-ditions, δ(0) = 0, δ′(0) = 2

√h. Hence, the following

analytical solution for (7) is derived:

τ = δ

2√

h2F1

(1

2,

2

5; 7

5; 2δ5/2

5h

)(8)

The time period of the out-of-phase NNM can be sim-ilarly evaluated as

T = T (h)

= h−1/102F1

(1

2,

2

5; 7

5;1

)[2

(5

4

)2/5

+(

5

2

)2/5]

≈ 5.341h−0.1 (9)

which provides an explicit expression for the depen-dence of the frequency ω = 2π/T of the out-of-phaseNNM on the energy h. This analytical solution will befurther utilized in our later discussion of the dynamicsof the granular system in the frequency–energy plane.

The in-phase NNM corresponds to asynchronousand non-smooth oscillations of the two beads and itdoes not lend itself to a similar explicit solution. Infact, as many as five distinct phases of the dynamicsexist for the in-phase NNM, of which only two can beanalyzed explicitly (the other three phases correspondto concurrent interactions of the beads with each otherand the rigid walls). This prevents the decoupling ofthe nonlinear equations of motion and a closed-formsolution.

Focusing now on the subharmonic orbits of thetwo-bead system, in Figs. 6a–6c we present three

representative oscillations corresponding to 1:3, 2:3and 3:5 rational relations between the frequencies ofthe oscillations of the first and second beads, respec-tively. All these orbits are stable but the domains ofquasi-periodic (‘regular’) motions surrounding themare small (cf. Fig. 2). Moreover, due to the non-integrability of the dynamics of this system there existsa countable infinity of stable subharmonic orbits satis-fying general m : n rational frequency relationships,albeit with increasingly smaller domains of realizationas the mutually prime integers m and n increase.

The two NNMs and subharmonic orbits can bebetter represented in a frequency–energy plot (FEP).As discussed in [17], by depicting the dynamics ofa system in a FEP it is possible to study the influ-ence of these modes on the forced and damped dy-namics; moreover, it would be possible to better re-late the dynamics of the two-bead granular system tothe dynamics of higher-dimensional granular systemsthat will be considered later. To depict the two NNMsin the FEP it is necessary to compute their contin-uations for varying energy and derive the cor to re-sponding frequency–energy relationships. This can beperformed immediately for the out-of-phase NNM bymeans of relation (9). Since no similar explicit rela-tionship can be derived for the in-phase NNM, weformulated a numerical shooting method to study thisNNM for higher energy ranges. The shooting methodis applied by specifying initial displacements and zeroinitial velocities for the two beads, so that at τ = 0 theentire energy is stored in the second bead due to itselastic compression by the right rigid wall (cf. Fig. 7).It follows that at τ = 0 the first bead is neither in con-tact with the wall nor with the second bead. Hence,assuming that the total (conserved) energy of the sys-tem is equal to h, the initial conditions for the granularsystem are chosen as

X2(0) = (5h/2)2/5, X1(0) = α(2)1 (5h/2)2/5

X′2(0) = 0, X′

1(0) = 0

where 0 < α(2)1 < 1 is a constant that determines the

asymmetry in the initial conditions for the in-phasemode, and it is computed by the shooting method. Thisconstant is computed as α

(2)1 ≈ 0.5585, and it is inter-

esting to note that it remains constant for all energies.This is due to the homogeneous nature of the systemaccording to the definition of Rosenberg [6]. To per-


Fig. 6 Subharmonic orbits of the two-bead system and h = 0.0001: (a) 1:3, (b) 2:3, and (c) 3:5 ratios between bead frequencies


form numerical continuation of the subharmonic or-bits for higher energies we utilized the numerical algo-rithm developed by Peeters et al. as discussed in [17].

In Fig. 8 we present the different families of modes(NNMs and subharmonic orbits) of the two-bead sys-tem in the FEP. The frequency–energy curve of theout-of-phase NNM provides the upper limit of time-periodic orbits for this system, so no periodic oscilla-tions can be realized in the upper region of the FEP,which is labeled as ‘prohibited’ band. We note thatthe families of the two NNMs and subharmonic or-bits are defined over the entire energy range and arerepresented by smooth curves that bifurcate from theorigin of the FEP. This is in contrast to dynamical sys-tems with essential but smooth stiffness nonlinearities,where subharmonic orbits appear as ‘tongues’ over fi-nite energy intervals [17]. It will be of interest to studyhow the topological structure of the FEP changes aswe increase the number of beads, and in particular,how the ‘prohibited’ band changes as the number ofbeads tends to infinity and the granular chain becomes

Fig. 7 Initial conditions forapplying the shootingmethod for the in-phaseNNM (two-bead system)

of infinite extent. These questions are discussed in thefollowing sections.

3 Three-bead granular system

Considering the three-bead granular system and as-suming identical elastic and geometric properties forthe beads, the governing normalized equations of mo-tion are given by

X′′1 = (−X1)

3/2+ − (X1 − X2)

3/2+

X′′2 = −(X2 − X3)

3/2+ + (X1 − X2)

3/2+

X′′3 = −(X3)

3/2+ + (X2 − X3)

3/2+

(10)

where the previous normalizations hold. In this caseit is not feasible to compute Poincaré maps for study-ing the global dynamics since the dimensionality ofthe system is relatively high (it has a six-dimensionalphase space). Nevertheless, it is possible to numeri-cally compute the NNMs and subharmonic orbits ofthis system using the computational techniques dis-cussed in the previous section.

Since the in-phase NNM (labeled as ‘NNM 1’)plays an important role in the dynamics of the gran-ular system, we will examine this mode in detail. To

Fig. 8 Representation of the modes of the two-bead system in the frequency–energy plot (FEP)


compute this NNM we employ the shooting methodby assuming zero initial velocities and initial displace-ments in the form

X3(0) = (5h/2)2/5

X2(0) = α(3)2 (5h/2)2/5

X1(0) = α(3)1 (5h/2)2/5

where h is the (conserved) energy of the system, and,as previously, the coefficients α

(3)1,2 characterize the

asymmetry in the initial deformation of the in-phaseNNM. These coefficients are computed as 0 < α

(3)1 ≈

0.374 < α(3)2 ≈ 0.744 < 1 for all values of the en-

ergy h. In Figs. 9a–9c we depict the in-phase NNM; insimilarity to the two-bead system, we infer that thereare domains where beads become motionless and off-

Fig. 9 The in-phase NNM (pseudo-traveling wave) for the three-bead granular system and h = 0.0001: (a) time series, (b, c) depictionsin projections of the phase plane


Fig. 10 Velocity profiles for the in-phase NNM (pseudo-traveling wave) of the three-bead granular system and h = 0.0001

set from their zero equilibrium positions. In addition,non-smooth effects in the dynamics are clearly noted,and a high non-synchronicity between bead oscilla-tions is deduced. The most important (and unique)features of this mode are the patterns of separationand loss of contact between beads and between theend beads and the rigid walls (which lead to the non-smooth effects depicted in Figs. 9b, 9c). Indeed, theinitial conditions required for realization of the in-phase NNM are such that except for one of the beads,all other beads are detached from each other and withthe walls. It is this feature that prevents the study ofthis mode using continuum approximation approachesin higher-dimensional systems with increased numberof beads. Hence, the in-phase NNM is the mode mostaffected and influenced by the discrete nature of thegranular system.

An even more peculiar feature of the in-phase NNMis that it resembles a traveling wave propagating backand forth in the granular chain. This becomes moreapparent in higher-dimensional granular media (seethe next section), but a first hint is provided by study-ing the velocity profiles in Fig. 10 for the three-beadsystem. In that figure we present a superposition ofthe velocity profiles of the central and end beads andshow them on the same timescale. We note that the ve-locity profiles are in the form of single-hump ‘pulses’;

half of the velocity profiles of the end beads match ex-actly that of the central bead, whereas the other halfis strongly influenced by the interaction of each endbead with the wall. Moreover, there is a constant timedelay between the transmission of velocity ‘pulses’ inneighboring beads, so the in-phase NNM resemblesa traveling wave. This result will be generalized forhigher-dimensional systems where it will be shownthat the velocity profiles of all beads except for the twoend beads are identical but for a constant time shift.Hence, although the in-phase NNM is in actuality atime-periodic standing wave, the motion of each beadis followed by an extended period where it settles to anoffset stationary position until the bead executes a mo-tion in the reverse direction after a time interval equalto the half period of the NNM. Since each bead (ex-cept for the end ones) executes an identical motion butfor a constant time shift, the in-phase NNM appearsindeed as a traveling wave. Based on these observa-tions we will refer from here on to the in-phase NNMas a pseudo-wave.

The second NNM (labeled as ‘NNM 2’) of thethree-bead system corresponds to out-of-phase mo-tions between neighboring beads and is presented inFig. 11. Due to non-synchronicity of the oscillations ofthe three beads, this mode is again in non-conformanceto the definition of NNM given by Rosenberg [6],


Fig. 11 The out-of-phase NNM for the three-bead granular system and h = 0.0001

but in similarity to the two-bead case it representsthe highest frequency NNM of the three-bead sys-tem, and as such, it forms the upper bound of thedomain of periodic motions in the frequency–energyplane.

The third mode (labeled as ‘NNM 3’) of the three-bead system corresponds to stationarity of the centralbead (bead 2) for all times. As shown in Fig. 12, thedynamics of the system can be partitioned into twophases. In the first phase, the end beads share equallythe energy of the system, which at τ = 0 is purely elas-tic. At the end of the first phase, the energy is com-pletely transformed to kinetic and the two end beadshave opposite velocities. Since the central bead is sta-tionary, for this NNM it acts as a virtual wall. Hence,the dynamics of the system in the second phase isquite similar to the first one, and the equations of mo-tion decouple throughout enabling us to solve for thebead responses in closed form. Omitting the details ofthe analysis, we may express the period of this NNMas

T = T (h) = 4h−1/10(

5

4

)2/5

2F1

(1

2,

2

5; 7

5;1

)

≈ 6.436h−0.1 (11)

where h denotes the total energy.

The last type of NNM (labeled NNM 4) supportedby the three-bead system is localized, with one of theend beads interacting with the wall and oscillatingwith an amplitude that is much larger (about twice)than the corresponding amplitudes of the other twobeads (cf. Fig. 13). In addition, neighboring beads os-cillate in an out-of-phase fashion. It is clear that due tothe symmetry of the system this mode is degenerate asit may be realized in an alternative symmetric config-uration where the motion is localized to the other endbead. It is interesting to note that this type of nonlin-ear localization occurs in the homogeneous granularsystem, and in complete absence of pre-compression.To the authors’ knowledge, this is the first report ofthis type of (strongly) localized motion in a singlebead of a finite homogeneous granular system. Wemake the remark at this point that nonlinear localiza-tion in granular homogeneous media has been reportedin previous works (e.g., [18, 19]) but only under suf-ficiently strong pre-compression and/or in the pres-ence of disorder, so that no separation between beadswould be possible; moreover, spatially periodic stand-ing waves with recurring localization after a wave-length have been reported in infinite granular chainswith no pre-compression [16]. We were not able to de-tect an analogous localized NNM where the localiza-


Fig. 12 The NNM with stationarity of the central bead for the three-bead granular system and h = 0.0001

Fig. 13 The localized NNM for the three-bead granular system and h = 0.0001

tion takes place in the central bead of the three-beadsystem under consideration.

In addition to the four NNMs, the system supportsa countable infinity of subharmonic orbits, in similar-ity to the two-bead system. The FEPs of the NNMs

and subharmonic orbits of the three-bead system arepresented in Fig. 14. From the discussion above, thenumber of NNMs in this system exceeds its degreesof freedom, but this is possible in essentially nonlineardiscrete oscillators [7]. The upper boundary separat-


Fig. 14 Representation of different modes of the three-bead system in the frequency–energy plot (FEP)

ing the region of time-periodic orbits from the ‘pro-hibited’ band, again, is formed by the out-of-phaseNNM 2, whereas the lowest frequency mode is thein-phase NNM 1, which as discussed previously is apseudo-wave. Moreover, compared to the correspond-ing results of the two-bead system (cf. Fig. 8) we notethat with increasing number of beads the upper bound-ary (corresponding to the out-of-phase NNM) movestoward higher frequencies, whereas the curve corre-sponding to the in-phase (pseudo-wave) NNM movestoward lower frequencies. This trend will be con-firmed in the next section where higher-dimensionalgranular systems are considered. In the FEP of Fig. 14we depict also the frequency–energy curves of twosubharmonic orbits, which occur within the comple-ment of the ‘prohibited’ band. No time-periodic orbits(NNMs or subharmonic orbits) can occur in the ‘pro-hibited’ band, similarly to the case of the two-beadsystem.

4 Higher-dimensional granular systems

We now extend our analysis to homogeneous gran-ular chains of higher dimensionality. Our main aimis to study the changes in the structure of the FEPas the number of beads increases, and, in particu-lar, to identify regions in the frequency–energy plane

where spatial transmission of disturbances (i.e., en-ergy) is facilitated or prohibited by the intrinsic dy-namics of the granular medium. This information is ofpractical significance when such media are designedas passive mitigators of shocks or other types of un-wanted disturbances. Since the out-of-phase and in-phase NNMs represent the highest and lowest fre-quency NNMs of the granular medium, respectively,all other NNMs (localized or non-localized) are real-ized in the frequency–energy band defined by these‘bounding’ NNMs, irrespective of the dimensionalityof the granular medium. Hence, in what follows wewill only focus on these two NNMs and investigatehow the topology of the band of realization of NNMs(and its complementary high-frequency ‘prohibited’band) changes with increasing number of beads in thegranular chain.

In Fig. 15 we depict the in-phase and out-of-phaseNNMs in the FEP for systems composed of two toseven beads. We note that as the number of beads in-creases the out-of-phase NNM makes a transition to-ward higher frequencies and quickly converges (accu-mulates) to a definite upper bounding curve, whereasthe in-phase NNM makes a similar transition towardlower frequencies. No such quick convergence is notedfor the in-phase NNM, but rather, as the number ofbeads tends to infinity, the in-phase NNM tends toward


Fig. 15 Representation of the in-phase and out-of-phase NNMs in the frequency–energy plot (FEP) for granular systems from two toseven beads; in the limit of infinite number of beads, the FEP is partitioned into a propagation and an attenuation band

the zero-frequency axis. This raises an interestingquestion concerning the dynamics of the infinite gran-ular system as the in-phase NNM approaches the zero-frequency limit. Namely, as discussed in the previoussection, due to bead separation the in-phase NNM re-sembles a traveling wave corresponding to a single-hump velocity disturbance propagating back and forththrough the granular system (hence the previous char-acterization of this NNM as a pseudo-wave). More-over, the ‘silent’ period for each bead (correspondingto the time period where the bead remains motionlessat an offset position) progressively increases with in-creasing dimensionality of the system. As discussedpreviously, in actuality the in-phase NNM is a stablestanding wave, but for any specific bead the recur-rence of the disturbance happens after a ‘silent’ pe-riod equal to half the period of the in-phase NNM;in turn, this period increases with increasing num-ber of beads, as is the corresponding ‘silent’ periodof the in-phase NNM. Following this argument onestep further, one might deduce that in the limit of in-finite number of beads the period of the NNM tendsto infinity (and its frequency tends to zero as it ap-proaches the energy axis in the FEP), so the in-phasemode should degenerate to a true traveling wave, sim-ilar (or identical?) to the solitary wave studied byNesterenko [9].

In Fig. 16 we compare the velocity profile of thesolitary wave studied by Nesterenko to the single-hump velocity profiles of the in-phase NNMs of fixed–fixed three- to seven-bead granular systems. It is clearthat the velocity profiles of the in-phase NNMs do notconverge to the solitary wave studied by Nesterenko.Clearly, the in-phase NNM converges to a differentlimit as the dimensionality of the system tends to in-finity, which raises an obvious question concerning thetype of dynamics that this limit represents. The answercan be found by considering more closely the dynam-ics of the granular chain when it oscillates on the in-phase NNM. As discussed previously, for an n-beadgranular system the initial conditions necessary for ex-citing the in-phase NNM can be expressed in termsof non-zero initial normalized displacements and zeroinitial velocities as follows:

X1(0) = α(n)1 (5h/2)2/5 < X2(0)

= α(n)2 (5h/2)2/5 < · · · < Xn−1(0)

= α(n)n−1(5h/2)2/5 < Xn(0)

= (5h/2)2/5

where h is the (conserved) energy of the system. Thisindicates that in order to realize the in-phase NNM,


Fig. 16 Comparison of the velocity profiles of the in-phase NNMs of fixed–fixed n-bead granular systems to the velocity profile ofthe solitary wave studied by Nesterenko [9]

Fig. 17 Coefficients α(n)k , k = 1, . . . , n − 1, for systems with n = 2, . . . ,7 for the in-phase NNMs of the fixed–fixed granular system,

with α(n)n = 1

it is necessary to compress the nth bead at the rightend of the granular system, and then allow for gaps(clearances) between the rest of the beads, whichmonotonically decrease as the left end of the systemis reached. In Fig. 17 we depict the coefficients α

(n)k ,

k = 1, . . . , n − 1, characterizing the asymmetry in thein-phase NNM deformation (in terms of the corre-sponding gaps) for systems with n = 2, . . . ,7 beads,and note that as n increases the differences betweenthese coefficients decrease. Moreover, the numerical


Fig. 18 Velocity profiles of seven adjacent beads: (a) in-phase NNM, (b) the solitary wave [9]; time delays in terms of normalizedtime τ

results indicate that in the limit n → ∞ these gapsshould tend to zero, which indicates that the in-phaseNNM cannot be realized in the limit of infinite granu-lar chain. We conclude that the infinite limit representsa singularity for the family of in-phase NNMs, whichmakes physical sense since otherwise this family ofNNMs (which are non-synchronous standing wavesor pseudo-waves) would degenerate to an actual trav-

eling wave, which would be a clear inconsistency interms of the dynamics. We make the remark at thispoint that even though the realization of the in-phaseNNM is hindered as the dimensionality of the gran-ular chain increases, it can significantly influence thedynamics of lower-dimensional granular media.

To demonstrate more clearly the pseudo-wave char-acter of the in-phase NNM, in Fig. 18 we present the


Table 1 Normalized time delays for velocity pulse propagation

n 100-Bead chain(Solitary wave [9])

n-Bead system(In-phase NNM)

3 5.248 7.67

4 7.858 10.62

5 10.528 13.45

6 13.168 16.25

7 15.778 19.04

velocity profile over half the period of the mode foreach of the beads of the seven-bead granular systemwith fixed–fixed boundary conditions, and compare itto the corresponding velocity profiles for seven beadsfor solitary wave propagation [9]. Since theoreticallythe solitary wave can be realized only in the infinitegranular chain, the results in Fig. 18 were computedfor a chain with 100 beads and free-free boundary con-ditions; the magnitude of the impulse provided to thefirst bead was selected so that the velocity amplitudeof the resulting solitary wave matches the velocity am-plitude of the in-phase NNM of the seven-bead fixed–fixed granular system.

We note that, whereas the velocity profiles of allbeads match exactly for the case of the solitary wave(cf. Fig. 18b), the same holds only for the velocity pro-files of the five central beads for the case of the in-phase NNM (cf. Fig. 18a). For the end beads of thefixed–fixed granular system, only half of their veloc-ity profiles match those of the central beads, whereasthe other half are strongly influenced by the interac-tion of these beads with the rigid walls. Moreover,comparing the time delays of arrival of the pulsefrom the first bead to the last, we note that the soli-tary wave possesses a higher speed compared to thepseudo-wave in the fixed–fixed granular system. Wenote that in Fig. 18a the pseudo-wave is depicted foronly one half of the period of the in-phase NNM (forthe other half of the period the pseudo-wave is ‘re-flected’ by the right rigid wall and it ‘propagates’ inthe opposite direction—i.e., backwards—through themedium). A tabulation of these time delays for sys-tems with three to seven beads is presented in Table 1,from which we infer that the pseudo-wave (i.e., the in-phase NNM) becomes slower compared to the solitarywave studied by Nesterenko with increasing numberof beads.

5 Intrinsic dynamics of the infinite granularchain: propagation and attenuation bands

In the limit of infinite number of beads the frequency–energy plane is partitioned into two regions, namely apropagation band (PB) and an attenuation band (AB).The existence of this type of bands in linear [20–23] and nonlinear [19, 24] periodic media has beenwell documented in the literature. Typically, in PBstraveling waves exist, and these are the dynamicalmechanisms for spatially transferring energy throughthe medium to the far field. In ABs near-field mo-tions with exponentially decaying envelopes are real-ized that cannot propagate energy in the far field. Theboundaries separating the PBs and ABs are spatiallyextended time-periodic standing waves, which can beregarded as NNMs of the infinite periodic media.

Similar types of orbits are realized inside thePBs and ABs of the infinite homogeneous granularmedium with no pre-compression. The portioning ofthe FEP of the infinite granular chain in terms of PBsand ABs is realized by the limiting (accumulating)curves of the in-phase and out-of-phase NNMs as de-picted in Fig. 15. As the number of beads tends toinfinity, the in-phase NNM tends toward the energyaxis (i.e., zero frequency), and the spacing betweenthe frequency–energy curves of the NNMs and subhar-monic orbits tends to zero (or alternately, the NNMsand subharmonic orbits become densely ‘packed’ interms of frequency and energy inside the PB).

In the infinite limit, the dynamics of the granu-lar medium possesses continuous families of travel-ing waves inside the PB. These motions were studiedin [16], where it was shown that the infinite gran-ular medium supports a countable infinity of fami-lies of stable traveling waves in the form of prop-agating multi-hump velocity pulses with arbitrarywavelengths; these families are parameterized by the‘silent’ regions of zero velocity that separate suc-cessive maxima of the propagating velocity pulses.As these ‘silent’ regions increase, the correspondingwavelengths and periods of the traveling waves alsoincrease and the frequencies of the waves decrease.In the limit of infinite ‘silent’ region, the wave ceasesto be periodic and its frequency becomes zero; thisasymptotic limit of single-hump solitary waves is thesolitary wave studied by Nesterenko [9].

Different families of traveling waves were com-puted in [16] and their propagation properties were


studied. It was found that these periodic waves areslower than the solitary wave studied by Nesterenko,with their speeds strongly depending on their ampli-tudes (energies). We wish to depict the frequency–energy curves of these families of traveling peri-odic waves in the FEP of the infinite granular chain.Clearly, since each family of traveling waves corre-sponds to an infinite sequence of propagating veloc-ity pulses separated by finite ‘silent’ regions (whereno motion occurs), it carries an infinite amount ofenergy. In order to assign a finite energy measure toeach family of traveling periodic waves, we computedthe corresponding energy density, i.e., the energy car-ried by each wave family over a wavelength of themotion. Moreover, the corresponding frequency fol-lows directly from the time-periodic character of thewaves. As the ‘silent’ region of the traveling wave in-creases, so does its period. It follows that with increas-ing ‘silent’ region the frequency of the wave decreases,until in the limit of infinite ‘silent’ region (correspond-ing to the solitary wave studied by Nesterenko) the en-ergy axis is reached and the frequency of the wave iszero. In addition, it is interesting to note that for alltraveling waves with finite ‘silent’ regions there occurseparations between beads, whereas in the limit of in-finite ‘silent’ region there are no separations betweenbeads; in that limit a continuum approximation can beutilized to study the solitary wave (as performed byNesterenko in [9]).

In Fig. 19, we depict two of the families of travel-ing waves, which as discussed previously are distin-guished by the regions of zero velocity (the ‘silent’ re-gions) between successive maxima of velocity pulses.One family corresponds to traveling waves with three-bead periodicity, whereas the second to four-bead peri-odicity [16]. Indeed, we confirm that these families ofwaves are realized inside the PB of the infinite gran-ular medium. In the limit of infinite ‘silent’ region,we obtain the single-hump solitary wave studied byNesterenko, which lies on the energy axis of the FEP(as it corresponds to infinite period or zero frequency).

In addition to these families of traveling periodicwaves, additional motions can occur inside the PB, in-cluding subharmonic motions, standing waves with re-curring localization features (i.e., periodically spacedhumps) and chaotic orbits. A family of such standingwaves with three-bead periodicity [16] is depicted inthe FEP in Fig. 19, with one of the beads oscillating inout-of-phase fashion and with higher amplitude com-

pared to its neighboring beads. We conjecture that ad-ditional families of such standing waves occur insidethe PB of the infinite granular medium, distinguishedby the extent of bead-periodicity (spatial wavelength)of the standing wave, and type of localization charac-teristics. We end our discussion of the dynamics in-side the PB by noting that no similar standing waves(with or without localization characteristics) can occurin PBs of perfectly ordered linear periodic media, al-though standing waves with spatially localized slopescan occur in ordered periodic media with smooth stiff-ness nonlinearities [26]. The full study of standingwaves with localization features, however, is beyondthe scope of this work.

We now consider the dynamics of the infinite gran-ular system inside the attenuation band (AB) of theFEP in Fig. 19. No spatially-periodic standing or trav-eling waves can occur for frequency–energy ranges inthat band, so no spatial transfer of energy in the gran-ular medium is possible for motions inside the AB.Rather, near-field motions occur inside the AB, corre-sponding to spatially periodic oscillations of the beadsabout different positive offset positions with spatiallydecaying envelopes; overall, the motion of the gran-ular medium is a standing wave with decaying enve-lope, with each bead performing a time-periodic oscil-lation about its own (positive offset) equilibrium po-sition. Examples of this type of near-field standingwaves are provided in the next section where the in-fluence of the propagation and attenuation bands onthe forced response of the granular medium to narrow-band excitation is studied. We note that the responseof the granular medium inside the AB resembles theresponses of unforced linear periodic media [20–22],which possess similar decaying standing waves (andeven decaying ‘complex’ waves when more than onecoupling coordinates between individual substructuresexist [25]) in well defined attenuation bands.

In the next and final section, we demonstrate the in-fluence of the intrinsic dynamics of the unforced gran-ular system on the forced dynamics of the same systemforced by a narrowband excitation. We expect that thepartitioning of the FEP in terms of propagation andattenuation bands will affect in a significant way thecapacity of the granular medium to transmit or attenu-ate disturbances through it. This, in turn, has obviousimplications on the capacity of the granular chain toact as passive mitigator of unwanted disturbances.


Fig. 19 Traveling andstanding waves inside thepropagation band (PB 1) ofthe FEP of the infinitegranular chain [16], theattenuation band (AB 2) isalso indicated:(a) frequency–energycurves, (b) velocity profilesof traveling waves insidethe PB, with three- andfour-bead periodicity,(c) velocity profile of astanding wave inside thePB, with three-beadperiodicity

6 Influence of the intrinsic dynamics on the forcedresponses of the granular medium

In this final section, we show how the topology of thepropagation and attenuation zones in the frequency–

energy plot (FEP) affects the forced dynamics of the

multi-dimensional granular chain. The influence of the

intrinsic dynamics of the unforced chain on the forced

dynamics is inferred from our previous observation

that disturbances initiated inside the PB of the FEP can


spatially transfer energy within the granular medium(through excitation of traveling waves), whereas mo-tions initiated inside the AB are near-field solutionsthat cannot transfer energy through the medium. Tonumerically verify this theoretical prediction we ex-cited a 50-bead granular chain by imposing a har-monic excitation on its left boundary in the form y0 =A sinωτ , where ω is the cyclic frequency of the excita-tion in terms of the normalized time τ . Although this isa finite-dimensional medium, we expect that, with theexception of certain end effects, its intrinsic dynamicswill be close to the dynamics of the corresponding in-finite chain. In order to eliminate high-frequency com-ponents in the response of the chain that result due tothe excitation of low-amplitude chaotic motions (re-sulting from the strong non-integrability of the granu-lar chain—see the Poincaré map in Fig. 2), we addedweak viscous dissipative forces in the interactions be-tween neighboring beads with viscous damping co-efficients equal to d = 0.01. As shown below, theseweak dissipative terms successfully dampen out high-frequency chaotic contributions to the dynamics andhelp us reveal coherent features generated by the in-trinsic dynamics in the forced responses.

In Fig. 20 we depict the forced response of thegranular system for excitation parameters A = 1.5 andω = 1. This corresponds to dynamics that occur insidethe PB of the infinite chain, and this is confirmed bythe spatially extended dynamical response of the sys-tem. We note that the initial disturbance generated atthe left boundary of the 50-bead chain is transmittedthroughout the chain; in addition, the total energy inthe system gradually builds up until it forms an os-cillation about a high level. Moreover, from the snap-shots of the chain deformation at different time in-stants presented in Fig. 20b, we infer that the initialdisturbance travels from the left boundary through thechain and gets reflected from the right boundary. It fol-lows that the forced dynamics of the chain is mainlycaused by excitation of the spatially extended intrinsicmotions, which, as discussed in the previous section,are densely ‘packed’ inside the PB of the FEP. Hence,we conclude that in this case, the intrinsic dynamicsfavor the transfer of disturbances through the chain.

A qualitatively different picture for the dynamicsis inferred from the results in Fig. 21 that depict theforced response of the chain inside the AB of the FEP.In this case the excitation parameters are selected asfor A = 0.3 and ω = 1; so we keep the same fre-quency with the previous simulation by decrease the

amplitude of the excitation, or equivalently, the en-ergy input in the system. Judging from the FEP inFig. 19, we make the theoretical prediction that bydecreasing the energy input from a high to a suffi-ciently low level while keeping the frequency fixed,the dynamics should make a transition from the PB tothe AB. Hence, the dynamics should change qualita-tively, and from spatially extended become spatiallyconfined. This is confirmed by the numerical simula-tions presented in Fig. 21, where we note that in thiscase no energy transmission occurs through the chain,but rather the input energy is confined in a region closeto the left boundary where it is originally generatedby the oscillating left rigid wall. Moreover, from thetransient responses of system presented in Fig. 21a wenote that after some initial transients the dynamics set-tles into a pattern of a standing wave with a decayingenvelope; it follows that the response of the systembecomes increasingly smaller away from the point ofbase excitation. This is in agreement with our remarksregarding the near-field motions that typically occurinside the ABs. We conclude that, contrary to the pre-vious simulation, in this case the intrinsic dynamicsof the granular chain does not enable the transmissionof disturbances through the granular medium. In addi-tion, from the snapshot plots in Fig. 21b we concludethat with increasing time the motion inside the AB set-tles into a near-linear decaying configuration wherethe beads nearest to the excitation have the greatestoffsets from the trivial equilibrium, whereas the onesfurthest have negligible offsets (this is also confirmedby the waveforms in Fig. 21a). This indicates that thereis an effective pre-compression in the granular chainaway from the excitation point. Therefore, the dynam-ics in the AB can be studied by adopting a weakly non-linear approach. We may expand the response of eachbead in Taylor series about its mean offset position andretain only the leading-order nonlinear terms. Then weshould be able to derive analytical approximations tothe spatially decaying waves inside the AZ depicted inFig. 21a.

Comparing the temporal evolution of the total en-ergy in the system (i.e., Figs. 20c and 21c) we notethat for excitation frequency inside the AB the energydecays with time and reaches a zero steady-state value,whereas for excitation frequency inside the PB the en-ergy reaches a steady state where it fluctuates about anon-zero mean value (actually, the computation of thismean value can help us to represent approximately the


Fig. 20 Forced response of the 50-bead granular chain for harmonic wall excitation y0 = 1.5 sin τ : (a) transient responses of selectedbeads for 1500 < τ < 2000, (b) snapshots of chain deformation at selected time instants, (c) evolution of total energy in the chain


Fig. 21 Forced response of the 50-bead granular chain for harmonic wall excitation y0 = 0.3 sin τ : (a) transient responses of selectedbeads for 1000 < τ < 2000, (b) snapshots of chain deformation at selected time instants, (c) evolution of total energy in the chain


response depicted in Fig. 20 in the FEP of Fig. 19a).This is a clear demonstration of the capacity of the in-trinsic dynamics of the granular medium to attenuateenergy inside the AB of the FEP.

We conclude that the results presented in this sec-tion indicate that the topological structure of the PBand AB of the frequency–energy plot affects signifi-cantly the narrowband-forced dynamics of the granu-lar chain. Indeed, depending on the frequency–energycontent of the excitation, the intrinsic dynamics of themedium either facilitates or hinders the spatial prop-agation of disturbances within the granular medium,allowing the propagation or dissipating certain fre-quency components. To extend this study to broad-band (shock) excitation it will be necessary to adddissipative effects such as dry friction or plasticity tothe Hertzian law interaction considered herein. We ex-pect that such dissipative effects will reveal clearlythe effect of the intrinsic dynamics on the broad-band response of the granular medium. Moreover, itshould be possible to identify transitions in the dis-sipative dynamics in the FEP, in an exactly similarway performed for transient smooth nonlinear dynam-ics of dissipative-coupled oscillators [17]. Such stud-ies can be used in formulating predictive designs ofthis type of granular media as passive mitigators ofshocks or other types of unwanted transient distur-bances.

7 Concluding remarks

New types of NNMs in granular media have been stud-ied in this work. These media have the intriguing fea-ture of ‘sonic vacuum,’ that is, the corresponding ve-locity of sound in these media is zero due to the essen-tially nonlinear (nonlinearizable) Hertzian contact in-teractions between neighboring beads. Moreover, non-smooth effects are added to the dynamics due to beadseparation when in the absence of compressive forcesbetween them. Though the traditional terminology ofNNMs is restricted to smooth dynamical systems, theessentially nonlinear and non-smooth features of thedynamics of granular media force us to broaden the de-finition of NNMs to include time-periodic orbits thatare not necessarily synchronous. In the case of granu-lar systems, the non-synchronicity is caused by beadseparation, which can lead to ‘silent’ regions in thebead dynamics, whereby one or more beads become

motionless for a finite period of time at an offset posi-tion from the zero equilibrium.

Clearly, the most interesting of the NNMs of thehomogeneous granular systems studied in this work isthe in-phase NNM, which to the authors’ knowledgehas not been studied before in the scientific literature.This mode can be realized only when the initial stateof the beads of the granular medium have prescribedgaps between them, i.e., for zero initial velocities, theinitial bead displacements should be such that no beadis in touch with another bead or with the walls, exceptfor one of the end beads. The in-phase NNM resem-bles a traveling wave propagating backward and for-ward through the finite granular chain, and hence itwas labeled as pseudo-wave. In addition, it can onlyexist in the finite chain, since in the limit of infinitenumber of beads the gaps between beads (required forthe excitation of this mode) tend to zero. On the con-trary, the solitary wave studied by Nesterenko [9] doesnot involve bead separation, so it can be studied byan analysis based on continuum approximation of thestrongly nonlinear equations of motion.

The significance of the in-phase (pseudo-wave)NNM and the out-of-phase NNM of the granularchain is that they form the boundaries of the bandin the frequency–energy plane within which all otherNNMs are realized. It follows that as the number ofbeads tends to infinity this region forms the propaga-tion band (PB) in the frequency–energy plot (FEP),whereas the complementary region forms the attenu-ation band (AB). Motions inside the PB transfer en-ergy through the medium and are spatially extended,whereas the corresponding motions inside the AB arenear-field solutions. Hence, the topologies of thesebands in the frequency–energy plane affect signifi-cantly the forced dynamics of the granular medium.Indeed, frequency components inside the PB propa-gate unattenuated whereas components inside the ABare damped by the intrinsic dynamics of the medium.In the event of application of a high frequency im-pulsive/shock loading, the frequency components inthe PB are transmitted along the chain, whereas thefrequency components in the AB are attenuated. Asdiscussed previously, when the system is harmoni-cally excited at high frequency, the system settlesdown to a localized mode and the high frequencysignal is completely attenuated. This result, which isconfirmed by direct numerical simulations reportedin this work, has considerable practical significance


in designing this type of strongly nonlinear granu-lar media as passive mitigators of unwanted distur-bances.

Acknowledgements This work was funded by MURI grantUS ARO W911NF-09-1-0436, Dr. David Stepp is the grantmonitor. The authors would like to thank Professor Leonid I.Manevitch of the Institute of Chemical Physics, Russian Acad-emy of Science, and Professors John Lambros and PhilippeGeubelle of the University of Illinois at Urbana-Champaign forvaluable discussions and suggestions.

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Nonlinear normal modes and band zones in granular chains with no pre-compression

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