Research Article Time-Periodic Solutions of Driven-Damped

Research ArticleTime-Periodic Solutions of Driven-Damped TrimerGranular Crystals

E G Charalampidis12 F Li3 C Chong24 J Yang3 and P G Kevrekidis25

1School of Civil Engineering Faculty of Engineering Aristotle University of Thessaloniki 54124 Thessaloniki Greece2Department of Mathematics and Statistics University of Massachusetts Amherst MA 01003-4515 USA3Aeronautics amp Astronautics University of Washington Seattle WA 98195-2400 USA4Department of Mechanical and Process Engineering (D-MAVT) Swiss Federal Institute of Technology (ETH)8092 Zurich Switzerland5Center for Nonlinear Studies andTheoretical Division Los Alamos National Laboratory Los Alamos NM 87544 USA

Correspondence should be addressed to F Li lifciompaccn

Received 19 November 2014 Revised 29 April 2015 Accepted 5 May 2015

Academic Editor Oded Gottlieb

Copyright copy 2015 E G Charalampidis et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We consider time-periodic structures of granular crystals consisting of alternate chrome steel (S) and tungsten carbide (W) sphericalparticles where each unit cell follows the pattern of a 2 1 trimer S-W-S The configuration at the left boundary is driven by aharmonic in-time actuation with given amplitude and frequency while the right one is a fixed wall Similar to the case of a dimerchain the combination of dissipation driving of the boundary and intrinsic nonlinearity leads to complex dynamics For fixeddriving frequencies in each of the spectral gaps we find that the nonlinear surface modes and the states dictated by the lineardrive collide in a saddle-node bifurcation as the driving amplitude is increased beyond which the dynamics of the system becomeschaotic While the bifurcation structure is similar for solutions within the first and second gap those in the first gap appear to beless robust We also conduct a continuation in driving frequency where it is apparent that the nonlinearity of the system results ina complex bifurcation diagram involving an intricate set of loops of branches especially within the spectral gap The theoreticalfindings are qualitatively corroborated by the experimental full-field visualization of the time-periodic structures

1 Introduction

Granular chains which consist of closely packed arrays ofparticles that interact elastically have proven over the lastseveral decades to be an ideal testbed to theoretically andexperimentally study novel principles of nonlinear dynamics[1ndash3] Examples include but are not limited to solitary waves[1 2 4 5] and dispersive shocks [6ndash8] as well as brightand dark discrete breathers [9ndash15] Beyond such fundamentalaspects their extreme tunability makes granular crystalsrelevant for numerous applications such as shock and energyabsorbing layers [16ndash19] actuating devices [20] acousticlenses [21] acoustic diodes [12] and switches [22] as well assound scramblers [23 24]

Our emphasis in the present work will be on coher-ent nonlinear waveforms that are time-periodic A specialinstance of this is when the spatial profile is localizedin which case the structure is termed a discrete breatherThe study of discrete breathers has been a topic of intensetheoretical and experimental interest during the 25 yearssince their theoretical inception as has been summarized forexample in [25] The broad and diverse span of fields wheresuch structures have been of interest includes among othersoptical waveguide arrays or photorefractive crystals [26]micromechanical cantilever arrays [27] Josephson-junctionladders [28 29] layered antiferromagnetic crystals [30 31]halide-bridged transition metal complexes [32] dynamical

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 830978 15 pageshttpdxdoiorg1011552015830978

2 Mathematical Problems in Engineering

models of the DNA double strand [33] and Bose-Einsteincondensates in optical lattices [34]

In Fermi-Pasta-Ulam type settings (which are intimatelyconnected to the case of precompressed granular crystals inthe weakly nonlinear regime see eg (1)) it was proven in[35 36] (see also the discussion in [25]) that small amplitudediscrete breathers are absent in spatially homogeneous (iemonoatomic) chains Instead dark breathers (those on top ofa non-vanishing background) have been found therein [15]It is for that reason that the first theoretical and experimentalinvestigations of breathers with a vanishing background (iebright breathers) have taken place in granular chains withsome degree of spatial heterogeneity which plays a criticalrole in the emergence of such patterns Examples includechains with defects [9 10 12] (see also [37] for recentexperiments) and a spatial periodicity of two (ie dimerlattices) [11 13] Motivated by these works a theoreticalstudy of breathers in granular crystals with higher orderspatial periodicity (such as trimers and quadrimers) wasrecently conducted in [14] Therein it was demonstrated thatbreathers with a frequency in the highest finite gap appearto be more robust than their counterparts with frequencyin the lower gaps Another relevant study is [38] whereexperimental and numerical analysis of band gap effectsrelated to transient behavior of driven dissipative granularchains were studied

The goal of the present work is the systematic studyof time-periodic solutions (including breathers) of trimergranular crystals with frequency in the first or second gapas well as in the acoustic and optical bands In particularwe investigate the robustness of the breathers experimen-tally using a full-field visualization technique based onlaser Doppler vibrometry This is a significant improvementover the aforementioned experimental observation of brightbreathers [11ndash13] where force sensors are placed at isolatedparticles within the granular chain which does not allow afull-field representation of the breather We complement thisstudy with a detailed theoretical probing of the more realisticdamped-driven variant of the pertinentmodel Our extensiveanalysis of such modes consists of the study of their familyunder continuations in both the amplitude and the frequencyparameters of the external drive and a detailed comparisonof the findings between numerically exact (up to a prescribedtolerance) periodic orbit solutions and experimentally tracedcounterparts thereof We also note that there are limitedstudies reporting the full-field experimental visualization oftime-periodic solutions in granular crystals Indeed the onlywork to this effect that the authors are aware of is [15] whichdeals with the much simpler setting of a monomer granularcrystal where surface breathers are not addressed Althoughthe theme of heterogeneous granular crystals has been receiv-ing considerable theoretical and even experimental attentionrecently most of it has been focused on the realm of travelingwaves [39ndash42] while the emphasis herein will be on the time-periodic exponentially localized discrete breather states

The paper is structured as follows In Sections 2 and 3we describe the experimental and theoretical setups respec-tively The main results are presented in Section 4 wheretime-periodic solutions with frequency in the firstsecond

Table 1 Parameter values of the trimer granular chain (S-W-S)

Material Parameter119864 [Nm2] 119903 [m] ] 120588 [kgm3]

Chrome steel (S) 200 times 109 9525 times 10minus3 03 7780Tungsten carbide (W) 628 times 109 9525 times 10minus3 028 14980

gaps and in the spectral bands are studied in both of thesesetups and compared accordingly Finally Section 5 providesour conclusions and discusses a number of future challenges

2 Experimental Setup

Figure 1 shows a schematic of the experimental setup consist-ing of a granular chain and a laser Doppler vibrometer Inthis study we consider a granular chain composed of119873 = 21spherical beads The beads are made out of chrome steel (Sgray particles in Figure 1) and tungsten carbide (W blackparticles) materials See Table 1 for nominal values of thematerial parameters used hereafter The granular chain has aspatial periodicity of three particles and each unit cell followsthe pattern of a 2 1 trimer S-W-SThe spheres are supportedby four polytetrafluoroethylene (PTFE) rods which allowaxial vibrations of particles with minimal friction whilerestricting their lateral movements The granular chain iscompressed by a moving wall at one end of the chain thatapplies static force in a controllable manner via a linear stage(see Figure 1) We measure the preapplied static force (1198650 =10N in this study) by using a static force sensor mountedon the moving wall We assume that this moving wall isstationary throughout our analysis since it exhibits orders-of-magnitude larger inertia compared to the particlersquosmasses

The granular chain is driven by a piezoelectric actuatorpositioned on the other side of the chain We imposeactuation signals of chosen amplitude and frequency throughan external function generator and a power amplifier Thedynamics of individual particles are scoped by a laserDoppler vibrometer (LDV Polytec OFV-534) which is capa-ble of measuring particlesrsquo velocities with a resolution of002 120583msHz12 The LDV scans the granular chain throughthe automatic sliding rail and measures the vibrationalmotions of each particle three times for statistical pur-poses We obtain the full-field map of the granular chainrsquosdynamics by synchronizing and reconstructing the acquireddata

3 Theoretical Setup

The equation we use to model the experimental setup is aFermi-Pasta-Ulam-type lattice with a Hertzian potential [1]leading to

119899

=119860119899minus1119872119899

[1205750119899minus1 +119906119899minus1 minus119906119899]32+

minus119860119899

119872119899

[1205750119899 +119906119899 minus119906119899+1]32+

minus119899

120591

(1)

Mathematical Problems in Engineering 3

Poweramplifier

Chrome steel beadTungsten carbide bead

Functiongenerator

Moving wall

Staticforce

sensorGuiding rod

Laser Dopplervibrometer

Automaticsliding rail

ActuatorF0

n = 21

n = 1

Figure 1 Experimental setup of a 21-particle granular chain composed of chrome steel and tungsten carbide beads in a 2 1 ratio Actuationand sensing systems based on a piezoelectric force sensor and a laser Doppler vibrometer are also illustrated

where 119899 = 1 119873 119906119899

= 119906119899

(119905) isin R is the displacement ofthe 119899th bead from its equilibrium position at time 119905 isin R119872

119899

is the mass of the 119899th bead 1205750119899 is a precompression factorinduced by the static force 1198650 = 119860

119899

120575320119899 and the bracket

is defined by [119909]+

= max(0 119909) The 32 power accountingfor the nonlinearity of the model is a result of the sphere-to-sphere contact that is the so-calledHertzian contact [43]Wemodel the dissipation as a dash-pot which can be interpretedas the friction between individual grains and PTFE rodsThis form of dissipation has been utilized in the context ofgranular crystals in several previous works [12 15] but it isworth noting that works such as [6 7] considered the internalfriction caused by contact interaction between grains Thestrength of the dissipation is captured by the parameter 120591which serves as the sole parameter used to fit experimentaldata (120591 = 21ms in this study) The elastic coefficient 119860

119899

depends on the interaction of bead 119899 with bead 119899 + 1 and forspherical point contacts has the form [44]

119860119899

=

4119864119899

119864119899+1radic119903119899119903119899+1 (119903119899 + 119903119899+1)

3119864119899+1 (1 minus ]2

119899

) + 3119864119899

(1 minus ]2119899+1)

(2)

where 119864119899

]119899

and 119903119899

are the Youngrsquos modulus Poissonrsquos ratioand the radius respectively of the 119899th beadThe left boundaryis an actuator and the right one is kept fixed that is

1199060 (119905) = 119886 cos (2120587119891119887119905)

119906119873+1 (119905) = 0

(3)

where 119886 and 119891119887

represent the driving amplitude and fre-quency respectively Note that at the boundaries (ie 119899 = 0

and 119899 = 119873+1) we consider flat surface-sphere contacts whilethe same material properties are assumed thus for examplethe elastic coefficient at the left boundary takes the form

1198600 =21198641radic1199031

3 (1 minus ]21) (4)

31 Linear Regime and Dispersion Relation Assuming thedynamic strains that are small relative to the static precom-pression that is

1003816100381610038161003816119906119899 minus 119906119899+11003816100381610038161003816

1205750119899≪ 1 (5)

equation (1) can then be well approximated by its linearizedform

119899

=119870119899minus1119872119899

(119906119899minus1 minus119906119899) minus

119870119899

119872119899

(119906119899

minus119906119899+1) minus

119899

120591 (6)

with 119870119899

= 32119860119899

120575120119899 corresponding to the linearized stiff-

nesses Subsequently (6) can be converted into a system of2119873 first-order equations and written conveniently in matrixform as

Y = AY (7)

with Y = (1199061 119906119873 V1(= 1) V119873(= 119873))119879 and

A = (

O IC (minus1120591) I

) (8)

whereO and I represent the119873times119873 zero and identitymatricesrespectively and the119873times119873 (tridiagonal) matrixC is given by


5 10 20

4

2

6

Eigenfrequency index0

015

8f

(kH

z)

(a)

0 1 2 30

2

4

6

8

f(k

Hz)

fc6

fc4

fc5

fc2

fc3

fc1

Wavenumber (k120572)

(b)

Figure 2 Eigenfrequency spectrum of the trimer granular chain for parameter values given in Table 1 and 120591 = 21ms (a) Spectrumcorresponding to (10) for 119873 = 21 beads Note that the horizontal dashed black lines correspond to the cut-off frequencies listed in Table 2(b) Numerically obtained frequencies of the infinite lattice using (14) as a function of the wavenumber (119896120572)

C =

(((((((((

(

minus1198700 + 11987011198721

11987011198721

0 sdot sdot sdot 0

11987011198722

minus1198701 + 11987021198722

11987021198722

0 d d d 0

119870119873minus2

119872119873minus1

minus119870119873minus2 + 119870119873minus1119872119873minus1

119870119873minus1

119872119873minus1

0 sdot sdot sdot 0119870119873minus1119872119873

minus119870119873minus1 + 119870119873119872119873

)))))))))

)

(9)

Note that in the above linearization we applied fixed bound-ary conditions at both ends of the chain that is1199060 = 119906119873+1 = 0(we account for the actuator in the following subsection)Thisequation is solved by 119884

119899

= 119910119899

119890119894120596119905 where 120596 corresponds to the

angular frequency (with 120596 = 2120587119891) and where

Ay = 120582y (10)

with (120582 y) corresponding to the eigenvalue-eigenvector pairwhile 120582 = 119894120596 and y = (1199101 1199102119873)

119879 The eigenfrequencyspectrum (119891 = minus119894(1205822120587)) using the values of Table 1 and 120591 =21ms is shown in Figure 2(a)

On the other hand the dispersion relation for the trimerchain configuration within an infinite lattice can be obtainedusing a Bloch wave ansatz [45] We first rewrite (6) in thefollowing convenient form

3119895minus2 =11987031198721

(1199063119895minus3 minus1199063119895minus2) minus11987011198721

(1199063119895minus2 minus1199063119895minus1)

minus

3119895minus2

120591

(11a)

3119895minus1 =11987011198722


(1199063119895minus1 minus1199063119895)

minus

3119895minus1

120591

(11b)

3119895 =11987011198721

(1199063119895minus1 minus1199063119895) minus11987031198721

(1199063119895 minus1199063119895+1) minus3119895

120591 (11c)

with 119895 isin Z+ where we made use of the spatial periodicityof the trimer lattice that is1198723119895 = 1198723119895minus2 equiv 1198721(= 1198723) and1198703119895minus2 = 1198703119895minus1 equiv 1198701(= 1198702) and1198723119895minus1 equiv 1198722 and 1198703119895 equiv 1198703The Bloch wave ansatz has the form

1199063119895minus2 = 1205781119890119894(119896120572119895+120596119905)

(12a)

1199063119895minus1 = 1205782119890119894(119896120572119895+120596119905)

(12b)

1199063119895 = 1205783119890119894(119896120572119895+120596119905)

(12c)

where 1205781 1205782 and 1205783 are the wave amplitudes 119896 is thewavenumber and 120572 is the size (or the equilibrium length)of one unit cell of the lattice Substitution of (12a) (12b) and(12c) into (11a) (11b) and (11c) yields


((

(

1205962minus1198701 + 11987031198721

minus 119894120596

120591

11987011198721

11987031198721

119890minus119894119896120572

11987011198722

1205962minus211987011198722

minus 119894120596

120591

11987011198722

11987031198721

119890119894119896120572

11987011198721

1205962minus1198701 + 11987031198721

minus 119894120596

120591

))

)

(

1205781

1205782

1205783

) = (

000) (13)

The nonzero solution condition of the matrix-system (13) ordispersion relation (ie the vanishing of the determinant ofthe above homogeneous linear system) has the form

119872211198722120596

6minus 119894

3119872211198722120591

1205965

minus1198721 [2 [11987031198722 +1198701 (1198721 +1198722)] +3119872111987221205912

]1205964

+ 1198941198721120591[4 [11987031198722 +1198701 (1198721 +1198722)] +

119872111987221205912

]1205963

minus 4119870211198703sin

2(119896120572

2)

+ [1198701 (1198701 + 21198703) (21198721 +1198722)

+211987211205912

[11987031198722 +1198701 (1198721 +1198722)]] 1205962minus 1198941198701120591(1198701

+ 21198703) (21198721 +1198722) 120596 = 0

(14)

which has (nontrivial) solutions at 119896 = 0 and 119896 = 120587120572 (iefirst Brillouin zone) given by

1198911198881 (119896 = 0) = 119894

12120587120591

(lower acoustic)

1198911198882 (119896 = 0) = plusmn

12120587radic1198701 + 211987031198721

minus141205912

+ 1198941

4120587120591

(second lower optic)

1198911198883 (119896 = 0) = plusmn

12120587radic1198701 (21198721 +1198722)

11987211198722minus

141205912

+ 1198941

4120587120591

(first upper optic)

1198911198884 (119896=

120587

120572) = plusmn

12120587radic11987011198721

minus141205912

+ 1198941

4120587120591

(first lower optic)

1198911198885 (119896=

120587

120572)

= plusmn12120587radic1198701 (21198721 +1198722) + 211987031198722 minus 119891

211987211198722minus

141205912

+ 1198941

4120587120591(upper acoustic)

1198911198886 (119896=

120587

120572)

= plusmn12120587radic1198701 (21198721 +1198722) + 211987031198722 + 119891

211987211198722minus

141205912

+ 1198941

4120587120591(second upper optic)

(15)

where 119891 = radic[1198701(21198721 +1198722) + 211987031198722]2 minus 161198701119870311987211198722

The solutions (15) correspond to the cut-off frequencies of thespectral bands The above values are (in principle) complexsince we consider dissipative dynamics in the system (1)(embodied by the

119899

120591 term) in order to model the exper-imental setup Therefore the reported frequencies hereafterwill correspond to the real part of these values that is thepart contributing to the dispersion properties of the solutionsrather than their decay In Table 2 the predicted values of thecut-off frequencies are presented (up to two decimal places)using the values of the material parameters of Table 1 and120591 = 21ms

Finally upon solving numerically the dispersion relation(cf (14)) the frequency as a function of the dimensionlesswavenumber (120581120572) is presented in Figure 2(b) together withthe cut-off frequencies of Table 2 Note that these cut-offfrequencies are also plotted as horizontal dashed black linesin Figure 2(a) for comparison It can be discerned from bothpanels of Figure 2 that there exist two finite gaps in thefrequency spectrum (in general their number is one less thanthe period of the granular crystal) namely [119891

1198885 1198911198884] and[1198911198883 1198911198882] together with one semi-infinite gap [119891

1198886infin) Wealso note the presence of two additional localized modesdepicted in Figure 2(a) between the first and second opticalpass bands due to the fixed boundary conditions Thesemodes denoted by black dots in Figure 2(a) correspond


Table 2 Calculated cut-off frequencies (in kHz) using (15)

1198911198885 119891

1198884 1198911198883 119891

1198882 1198911198886

304 423 604 671 737

to surface modes and their existence has been previouslydiscussed for example in [11]

32 Steady-State Analysis in the Linear Regime In order toaccount for the actuation in the linear problem we add anexternal forcing term to (7) in the form

Y = AY+ F (119905) (16)

where the sole nonzero entry of F is at the (119873 + 1)th nodeand has the form 119865

119873+1 = 119886(11987001198721)cos(120596119887119905) and the matrixA is given by (8) An inhomogeneous solution of (16) can beobtained by introducing the ansatz

Y =

(((((((

(

1198861 cos (120596119887119905) + 1198871 sin (120596119887119905)

119886119873

cos (120596119887

119905) + 119887119873

sin (120596119887

119905)

minus1198861120596119887 sin (120596119887119905) + 1198871120596119887 cos (120596119887119905)

minus119886119873

120596119887

sin (120596119887

119905) + 119887119873

120596119887

cos (120596119887

119905)

)))))))

)

(17)

with unknown (real) parameters 119886119895

and 119887119895

(119895 = 1 119873)Inserting (17) into (16) yields a system of linear equationswhich can be easily solved

AX = F (18)

where

A = (

C (120596

120591) I

minus(120596

120591) I C

) (19)

C = minus(C + 1205962119887

I) with C given by (9) while X = (1198861 1198861198731198871 119887119873) is the vector containing the unknown coefficientsand F = (119886(11987001198721) 0 0) We refer to (17) as theasymptotic equilibrium of the linear problem (as dictatedby the actuator) since all solutions of the linear problemapproach it for 119905 rarr infin Clearly this state will be spatiallylocalized if the forcing frequency 119891

119887

lies within a spectralgap otherwise it will be spatially extended (with the formercorresponding to a surface breather which we discuss inthe following section) For small driving amplitude 119886 wefind that the long-term dynamics of the system approachesan asymptotic state that is reasonably well approximated by(17) However for larger driving amplitudes the effect of thenonlinearity becomes significant and we can no longer relyon the linear analysis

In order to understand the dynamics in the high-amplitude regime we perform a bifurcation analysis of time-periodic solutions of the nonlinear problem (1) using aNewton-Raphson method that yields exact time-periodicsolutions (to a prescribed tolerance) and their linear sta-bility through the computation of Floquet multipliers (seeappendix for details) We use the asymptotic linear state (17)as an initial guess for the Newton iterations

4 Main Results

Besides the linear limit considered above another relevantsituation is that of the Hamiltonian system that is with1120591 rarr 0 in (1) and 119886 rarr 0 in (3) It is well known that time-periodic solutions that are localized in space (ie breathers)exist in a host of discrete Hamiltonian systems [25] includinggranular crystals with spatial periodicity [11] In particularHamiltonian trimer granular chains were recently studied in[14] There it was observed that breathers with frequencyin the second spectral gap are more robust than those withfrequency in the first spectral gap Features that appear toenhance the stability of the higher gap breathers are (i)tails avoiding resonances with the spectral bands and (ii)lighter beads oscillating out-of-phase In that sense thebreathers found in the second spectral gap of trimers aresomewhat reminiscent of breathers found in the gap ofdimer lattices [11] Furthermore breathers of damped-drivendimer granular crystals (ie (1) with a spatial periodicityof two) were studied recently in [13] In that setting thebreathers become surface breathers since they are localizedat the surface rather than the center of the chain Yet ifone translates the surface breather to the center of the chainit bears a strong resemblance to a ldquobulkrdquo breather Thusnonlinearity periodicity and discreteness enabled two classesof relevant states The nonlinear surface breathers and onestuned to the external actuator (ie those proximal to theasymptotic linear state dictated by the actuator) These twowaveforms were observed to collide and disappear in a limitcycle saddle-node bifurcation as the actuation amplitude wasincreased [13] Beyond this critical point no stable periodicsolutions were found to exist in the dimer case of [13] andthe dynamics were found to ldquojumprdquo to a chaotic branch Weaim to identify similar features in the case of the trimer witha particular emphasis on the differences arising due to thehigher order periodicity of the system To that end we firstpresent results on surface breathers with frequency in thesecond gap and perform parameter continuation in drivingamplitude to draw comparisons to the bifurcation structurein dimer granular lattices Following that we investigatebreathers in the first spectral gap and compare them totheir second gap breather counterparts Finally we identifyboth localized and spatially extended states as the drivingfrequency is varied through the entire range of spectral valuescovering both gaps and the three pass bands between whichthey arise

41 Driving in the Second Spectral Gap We first considera fixed driving frequency of 119891

119887

= 63087 kHz which


5 10 15 20

minus001

0

001

n(m

s)

n

(a)

0 004 008 0120

0004

0008

0012

(A)

(D)

(B)

(C)

4(m

s)

a (120583m)

(b)

0 02 04 06 080

002

004

006

008

01

4(m

s)

a (120583m)

(c)

Figure 3 Comparison between experimental and theoretical results for a driving frequency119891119887

= 63087 kHz which lies in the second spectralgap (a) Profile of solution for a driving amplitude of 119886 = 3216 nm Experimental values (green shaded areas) are shown as a superpositionof the velocities measured over the 50ms time window of recorded data The numerically predicted extrema (over one period) of the exactsolution obtained via Newtonrsquosmethod are shownwith blue (open) circles (b)The quantity V4 as a function of the amplitude 119886 of the drive Forthe numerically exact solutions V4 is themaximumvelocity of the 4th bead over one period ofmotionNote that the blue segments correspondto stable parametric regions while the red and green ones correspond to real and oscillatory unstable parametric regions respectively Blacksquares with error bars represent the experimentally measured mean values of the quantity V4 over three experimental runs where V4 wascalculated as the maximum measured over the 50ms time window of recorded data Finally the dashed gray line is the maximum velocity(over one period) of bead 4 as predicted by the linear solution of (17) (c) Zoomed out version of panel (b) The gray circles correspond tonumerical results that were obtained by dynamically running the equations of motion and monitoring the maximum velocity V4 over theexperimental time window The experimental data in this panel are shown again by black squares

lies within the second spectral gap (see Table 2) For lowdriving amplitude there is a single time-periodic state (iethe one proximal to the driven linear state) which theexperimentally observed dynamics follows (see panels (a)and (b) in Figure 3) A continuation in driving amplitudeof numerically exact time-periodic solutions (see the bluered and green lines of Figure 3(b)) reveals the existence ofthree branches of solutions for a range of driving amplitudes

The branch indicated by label (A) is proximal to the lineardriven state given by (17) (shown as a gray dashed line inFigure 3(b)) Each solutionmaking up this branch is in-phasewith the boundary actuator (see Figure 4(a)) and has lightermasses that are out-of-phase with respect to each other (seeeg Figure 7(a)) These solutions are asymptotically stablewhich is also evident in the experiments (see Figure 3(a) andthe blackmarkers with error bars in Figure 3(b)) At a driving


0 10 20minus5

0

5

10

minus1 0 1minus1

minus05

0

05

n

un

(120583m

)

Im(120582

)

Re(120582)

times10minus4

1

(a)

0 10 20minus01

0

01

02

03

minus1 0 1minus1

minus05

0

05

1

n

un

(120583m

)

Im(120582

)

Re(120582)

(b)

0 10 20

minus03

minus015

0

015

03

minus1 0 1minus1

minus05

0

05

1

n

SteelTungsten

un

(120583m

)

Im(120582

)

Re(120582)

(c)

0 10 20

minus04

minus02

0

02

04

minus1 0 1minus1

minus05

0

05

1

n

SteelTungsten

un

(120583m

)

Im(120582

)

Re(120582)

(d)

Figure 4 Displacement profiles and corresponding Floquet multipliers of time-periodic solutions obtained at a driving frequency of 119891119887

=

63087 kHz for a driving amplitude of (a) 119886 = 02 nm (b) 119886 = 30 nm (c) 119886 = 40 nm and (d) 119886 = 01 120583m These solutions correspond tothe (A)ndash(D) labels of Figure 3(b) The lighter masses (ie the Steel beads) oscillate (nearly) out-of-phase and are shown as blue diamondswhereas the heavier masses (the Tungsten Carbide ones) are shown as orange squares

amplitude of 119886 asymp 2275 nm an unstable and stable branchof nonlinear surface breathers arise through a saddle-nodebifurcation (see labels (B) and (C) of Figure 3(b) resp) Atthe bifurcation point 119886 asymp 2275 nm the profile stronglyresembles that of the corresponding Hamiltonian breather(when shifted to the center of the chain) hence the namesurface breather It is important to note that the presence ofdissipation does not allow for this branch to bifurcate near119886 rarr 0 (where this bifurcation would occur in the absenceof dissipation) Instead the need of the drive to overcome thedissipation ensures that the bifurcation will emerge at a finitevalue of 119886 As 119886 is increased the solutions constituting theunstable ldquoseparatrixrdquo branch (b) resemble progressively morethe ones of branch (a) see Figure 4(b) For example branch(b) progressively becomes in-phase with the actuator as 119886 isincreased Indeed these two branches collide and annihilateat 119886 asymp 3226 nm On the other hand the (stable at least fora parametric interval in 119886) nonlinear surface breather (c) isout-of-phasewith the actuator (see Figure 4(c))This solutionloses its stability through a Neimark-Sacker bifurcationwhich is the result of a Floquet multiplier (lying off the realline) acquiring modulus greater than unity (see label (D) of

Figures 3(b) and 4(d)) Such a Floquet multiplier indicatesconcurrent growth and oscillatory dynamics of perturbationsand thus the instability is deemed as an oscillatory oneSolutions with an oscillatory instability are marked in greenin Figure 3(b) whereas red dashed lines correspond to purelyreal instabilities (see also Figure 4(b) as a case example)and solid blue lines denote asymptotically stable regionsThe reason for making the distinction between real andoscillatory instabilities is that quasi periodicity and chaosoften lurk in regimes in parameter space where solutionspossess such instabilities [13] Indeed past the above men-tioned saddle-node bifurcation as the amplitude is furtherincreased for an additional narrow parametric regime thecomputational dynamics (gray circles in Figure 3(c)) followsthe upper nonlinear surfacemode of branch (c) yet once thisbranch becomes unstable the dynamics appears to reach achaotic state In the experimental dynamics (black squares inFigure 3(c)) a very similar pattern is observed qualitativelyalthough the quantitative details appear to differ some-what Admittedly as more nonlinear regimes of the systemrsquosdynamics are accessed (as 119886 increases) such disparities areprogressively more likely due to the opening of gaps between


0 5 10 15 20

minus4

minus2

0

2

4

n

times10minus3 n

(ms

)

(a)

0 05 1 150

0005

001

0015

(A)

(B)

(D)

(C)

a (120583m)

4(m

s)

(b)

0 05 1 150

003

006

009

a (120583m)

4(m

s)

(c)

Figure 5 Same as Figure 3 but for a driving frequency of 119891119887

= 37718 kHz which lies in the first spectral gap (a) Profile of solution fora driving amplitude of 119886 = 024816 120583m (b) The bifurcation diagram for 119891

119887

= 37718 kHz (shown here) is similar to the one presented inFigure 3(b) which was for a driving frequency in the second gap (119891

119887

= 63087 kHz) However the bifurcation points occur for much largervalues of the driving amplitude and the structure of the solutions themselves varies considerably in comparison to the second gap (see text)(c) Zoomed out version of panel (b) In this case the jump of the experimental data is likely due to driving out of a controllable range (seetext) rather than to chaos as in the case of the second gap breathers

the spheres and the limited applicability (in such regimes)of the simple Hertzian contact law See also the discussionin Section 43 for more details on possible explanations fordisagreement in high amplitude regimes

42 Driving in the First Spectral Gap We now consider afixed driving frequency of119891

119887

= 37718 kHz which lies withinthe first spectral gap Qualitatively the breather solutionsand their bifurcation structure are similar to those in thesecond gap (compare Figures 3 and 4 with Figures 5 and 6)However there are several differences which we highlighthere For example the lighter masses now oscillate (nearly)in-phase rather that out-of-phase (for comparison see panels(a) and (b) in Figure 7) The emergence of the nonlinear

surface modes occurs for a much larger value of the drivingamplitude (119886 asymp 01778 120583m) as well as the bifurcationof the state (a) dictated by the actuator with the relevantbranch of the nonlinear surface modes (119886 asymp 13 120583m)Indeed the latter turns out to be out-of-range for exper-imentally controllable driving amplitudes given the strokeof the piezoelectric actuator and the power by the electricamplifier in our experimental setup Thus applications suchas bifurcation based rectification [12] are not suitable forbreather frequencies in the first gap in the present settingAlthough the range of amplitudes yielding stable solutionsis much larger these solutions are still effectively linear (seethe dashed gray line in Figure 5(b)) Another peculiar featureparticular to this case is that the branch (a) loses its stabilitythrough a Neimark-Sacker bifurcation (119886 asymp 09 120583m) rather


0 10 20

minus003

minus0015

0

0015

003

minus1 0 1minus1

minus05

0

05

1

n

un

(120583m

)

Im(120582

)Re(120582)

(a)

0 10 20minus04

minus02

0

02

04

06

minus1 0 1minus1

minus05

0

05

1

n

un

(120583m

)

Im(120582

)

Re(120582)

(b)

SteelTungsten

0 10 200

05

1

15

minus1 0 1minus1

minus05

0

05

1

n

un

(120583m

)

Im(120582

)

Re(120582)

(c)

SteelTungsten

0 10 200

1

2

3

minus18 minus09 0 09 18minus18

minus09

0

09

18

n

un

(120583m

)

Im(120582

)

Re(120582)

(d)


=

37718 kHz for a driving amplitude of (a) 119886 = 50 nm (b) 119886 = 094 120583m (c) 119886 = 12 120583m and (d) 119886 = 094 120583mThese solutions correspond to the(A)ndash(D) labels of Figure 5(b)The lighter masses (ie the Steel beads) oscillate in-phase and are shown as blue diamonds whereas the heaviermasses (the Tungsten Carbide ones) are shown as orange squares

than through a saddle-node bifurcation with the nonlinearsurface mode although it regains stability shortly thereafter(119886 asymp 09876 120583m) Interestingly however the correspondingexperimental branch deviates from the theoretical (near-linear) branch close to this destabilization point apparentlyleading to large amplitude chaotic behavior thereafter

As an additional feature worth mentioning there arebreathers within the first gap that resonate with the linearmodes For example any breather with a frequency 119891

119887

isin

[33550 36850] will have a second harmonic that lies in thesecond optical band Indeed the breather solution depictedin Figure 7(c) has a frequency 119891

119887

= 34429 but the tailoscillates with twice that frequency (ie it is resonating withthe linear mode at a frequency of 2119891

119887

) Finally we note thatthe magnitude of the instabilities tends to be much larger inthe first spectral gap due possibly to their spatial structure(see eg Figure 6(d)) in agreement with what was found in[14]

While a detailed probing of the spatial profiles and thecorresponding bifurcation structure in the first and secondgap reveal several differences a common theme is that theexperimental data points generally follow rather accuratelythe theoretically predicted curve In particular the agreementbetween experiment and theory is rather satisfactory as longas the data points are within the stable parametric regions

This is clearly depicted in panel (b) of Figures 3 and 5 Notethat the linear asymptotic equilibrium (shown by the dashedgray line and corresponding to |119906

119899

minus119906119899+1|1205750 rarr 0) becomes

closer to the theoretically predicted curve for small relativedisplacements (see eg (5)) a feature that we also use as aconsistency check for the numerics In both cases (first orsecond gap) once the amplitude 119886 of the actuator reachesa critical value (119886 asymp 032016 120583m for the second gap and119886 asymp 082416 120583m for the first gap) the experimental dataexperience jumps seemingly leading to chaotic behaviorImportantly these jumps arise close to the destabilizationpoints of the respective branches Nevertheless it should alsobe pointed out that in some of these regimes (especiallyso in the case of Figure 5) these driving amplitudes maybe near or beyond the regime of (accurately) controllableexperimentally accessible amplitudes See the discussion inSection 43 for more details on possible explanations fordisagreement in high amplitude regimes

43 A Full Driving Frequency Continuation Our previousconsiderations suggest that the driving frequency plays animportant role in the observed dynamics of the time-periodicsolutions Figure 8 complements those results by showing afrequency continuation for a fixed driving amplitude 119886 =

024816 120583m For this driving amplitude the response is highly


0 004 008 012minus02

0

02

04

t (ms)

Disp

lace

men

ts (120583

m) u3

u4

(a)

0 01 02minus004

minus002

0

002

004

t (ms)

Disp

lace

men

ts (120583

m)

u3

u4

(b)

49 492 494 496 498 50

minus001

minus0005

0

0005

001

Velo

citie

s (m

s)

t (ms)

15

2Tb

Tb

2

(c)

Figure 7 (a) Displacement versus time over one period of motion of two adjacent steel masses 1199063(119905) and 1199064(119905) for a solution in the secondspectral gap (with 119891

119887

= 63087 kHz and 119886 = 40 nm) In this case the motion is (nearly) out-of-phase (b) Same as (a) but for a solution in thefirst spectral gap (with 119891

119887

= 37718 kHz and 119886 = 50 nm) in which case the motion is in-phase (c) Velocities versus time for the motion ofsteel masses V2(119905) and V15(119905) for a solution with frequency 119891

119887

= 34429 kHz and amplitude 119886 = 50 nm Solid lines correspond to numericalresults while the dash-dotted lines correspond to experimental data (mean of the three runs) Notice that the tail oscillates with twice thefrequency of the primary nodes near the left boundary where 2119891

119887

lies in the second optical band

nonlinear where the ratio of dynamic to static compressionranges as high as for example |119906

119899

minus 119906119899+1|1205750119899 = 15 In

the low amplitude (ie near linear) range the peaks in theforce response coincide with the eigenfrequency spectrumpresented in Figure 2(a) with all solutions being stable Forthe chosen value of driving amplitude the features of thediagram at a low driving frequency are similar to their linearcounterparts (see eg the smooth resonant peaks in theacoustic band of Figure 8(a)) Similar to Figures 3 and 5the experimentally obtained values match the numericallyobtained dynamically stable response quite well with a fewnotable exceptions (i) in narrow parametric regions around

119891119887

asymp 235 kHz and 119891119887

asymp 27 kHz where the experimentalpoints experience jumps and (ii) the experimentally observedvelocities for the same frequency are upshifted when com-pared with the theoretical curve (see eg the region 119891

119887

asymp

1minus52 kHz in Figure 8(a)where the peaks donot align exactly)Possible explanations for such discrepancies are given belowbut we also note that discrepancies in experimental andtheoretical dispersion curves of granular crystals have alsobeen previously discussed see for a recent example [45]

In general as the driving amplitude is increased the peaksin the bifurcation diagram bend (in some cases they bendinto the gap) in a way reminiscent of the familiar fold-over


0 1 2 3 4 5 6 7 8 90

0005001

0015002

0025003

0035004

0045 1st gap 2nd gap

fb (kHz)

4(m

s)

(a)

54 56 58 6 62 64 66 68 70

0005

001

0015

002

0025

003

0035

004

fb (kHz)

4(m

s)

(b)

Figure 8 The bifurcation diagram corresponding to the maximum velocity of bead 4 as a function of the actuation frequency 119891119887

and fora value of the actuation amplitude of 119886 = 024816 120583m is presented with smooth curves The color indexing is the same as in Figures 3 and5 The black squares with error bars represent the experimentally measured mean values of the velocities which were obtained using threeexperimental runs Furthermore the light gray areas enclosed by vertical dash-dotted black lines correspond to the frequency gaps accordingto Table 2 (see also Figure 2) Note that panel (b) is a zoom-in of panel (a)

event of driven oscillators see for example [46] for a recentstudy extending this to chains This in turn causes a seriesof intricate bifurcations and a loss of stability (see eg thesecond gap in Figure 8(a)) In particular large regions ofoscillatory instabilities are born and hence we expect regionsof quasi periodicity and of potentially chaotic response(see eg Section 41) Not surprisingly the experimentallyobserved values depart from the theoretical prediction oncestability is lost However the general rising and falling patternare captured qualitatively see Figure 8(b) Nevertheless itis worthwhile to note that the latter figure contains a veryelaborate set of loops and a highmultiplicity of corresponding(unstable) periodic orbit families for which no clear physicalintuition is apparent at present More troubling however isthe fact that there are also some stable regions where theexperimental points are off the theoretical curve A systematicstudy using numerical simulations revealed that within thesenarrow stable regions at hand (see eg Figure 8(b)) a largenumber of periods are required in order to converge to theexact periodic solution In particular it happens that theFloquet multipliers are inside the unit circle but also veryclose to it suggesting that the experimental time window of50ms used is not sufficiently long to observe experimentallythe periodic structures predicted theoretically by our modelThus the reported experimental observations in this regimeappear to be capturing a transient stage and not the eventualasymptotic formof the dynamical profile Afinal comment onthe possible discrepancy between themodel and experimentsis due to the fact that in this highly nonlinear regime (oflarge driving amplitudes) the beads come out of contact (thedynamic force exceeds the static force) in which case otherdynamic effects such as particlesrsquo rotations could have asignificant role Such features are not described in our simplemodel (1) In fact the contact points of the particles cannot beperfectly aligned in experiments due to the clearance betweenthe beads and the support rods This is likely to result in

dynamic buckling of the chain under the strong excitationswhich are considered in some of the cases above Here alsoit is plausible that additional forms of dissipation affect theparticle motions leading the experimental findings (in suchsettings) to underpredict the velocities observed in directnumerical simulations Nonetheless the overall experimentaltrends of bifurcation in Figure 8(a) are in fair qualitativeagreement with the theoretical predictions

5 Conclusions and Future Challenges

We have presented a natural extension of earlier consid-erations of (i) Hamiltonian breathers in trimer chains and(ii) surface breathers in damped-driven dimers by study-ing a damped-driven trimer granular crystal through acombination of experimental and theoreticalcomputationalapproaches We found that the breathers with frequencyin the second gap are in analogy with those in the solegap of the damped-driven dimer in which the interplayof damped-driven dynamics with nonlinearity and spatialdiscreteness gives rise to saddle-node bifurcations of time-periodic solutions and turning points beyond which thereis no stable ordered dynamics as well as surface modes andgenerally rather complex and tortuous bifurcation diagramsWhile similar structures are found in the first gap they appearto be less robust (given the magnitude of their instabilities)and can resonate with the higher-order linear bands afeature interesting in its own right A continuation in drivingfrequency revealed that the nonlinearity causes the resonantpeaks to bend possibility into a spectral gap However thisnonlinear bending also causes the solutions to lose stabilityin various ways such as Neimark-Sacker and saddle-nodebifurcations More importantly all of these features werevalidated experimentally through laser Doppler vibrometryallowing for the first time to our knowledge the full-fieldvisualization of surface breathers in granular crystals


Several interesting questions remain unexplored includ-ing for example the mechanism leading to the emergenceof the observed chaotic dynamics and the more accuratequantitative description of that regimeWe also observed thatthe bifurcation structure is generally more complex in thecase of breathers that are resonant with the linear modeswhich bears further probing Other possible avenues forfuture research include the investigation of dark breathersin such chains Recall that as in [15] dark breathers maybifurcate from the top edge of the acoustic first or secondoptical band under suitable drive Lastly it would be quiteinteresting to generalize considerations of breathers andsurface modes in two-dimensional hexagonal lattices withboth homogeneous and heterogeneous compositions Suchstudies are currently under consideration andwill be reportedin future publications

Appendix

Numerical Methods and Linear Stability

In this Appendix we shortly discuss the numerical methodsemployed for finding exact time-periodic solutions of (1)together with the parametric continuation techniques foridentifying families of such solutions as either the value ofthe actuation frequency 119891

119887

(or equivalently the period 119879119887

=

1119891119887

) or amplitude 119886 changes The interested reader can findmore information along these directions for example in[47 48] (among others) and references therein

In order to find time-periodic solutions of (1) we convertthe latter into a system of first order ODEswritten in compactform

x = F (119905 u k) (A1)

with x = (u k)119879 while u = (1199061 119906119873)119879 and k = u

represent the 119873-dimensional position and velocity vectorsrespectively Next we define the Poincare map P(x(0)) =x(0) minus x(119879

119887

) where x(0) is the initial condition and x(119879119887

) isthe result of integrating (A1) forward in time until 119905 = 119879

119887

(using eg the DOP853 [49] or ODE [50] time integrators)Then a periodic solution with period 119879

119887

(ie satisfying theproperty x(0) = x(119879

119887

)) will be a root of the map P To thisend Newtonrsquosmethod is applied to themapP and thus yieldsthe following numerical iteration scheme

x(0119896+1) = x(0119896) minus [J]minus1x(0119896)P (x(0119896)) (A2)

together with the Jacobian of the map P namely J = I minus119881(119879119887

)where I is the 2119873times2119873 identitymatrix119881 is the solutionto the variational problem = (119863xF)119881 with initial data119881(0) = I while119863xF is the Jacobian of the equations ofmotion(A1) evaluated at the point x(0119896)

Subsequently the iteration scheme (A2) is fed by a suit-able initial guess (for fixed values of the actuation amplitude119886 and frequency 119891

119887

) and applied until a user-prescribedtolerance criterion is satisfiedThus a time-periodic solutionis obtained upon successful convergence (andwith high accu-racy) together with the eigenvalues or Floquet multipliers

(denoted by 120582) of themonodromymatrix119881(119879119887

)which conveyimportant information about the stability of the solutionin question In particular a periodic solution is deemedasymptotically stable (or a stable limit cycle) if there are noFloquet multipliers outside the unit circle whereas a periodicsolution with Floquet multipliers lying outside the unit circleis deemed to be linearly unstable Note that the iterationscheme (A2) can be initialized using the linear asymptoticequilibrium given by (17) Alternatively one may start withzero initial data and directly integrate the equations ofmotion(A1) forward in time Then (A2) can be initialized with theoutput waveform of the time integrator at a particular time

Having obtained an exact periodic solution to (1) for givenvalues of 119886 and119891

119887

we perform parametric continuations overthese parameters using the computer software AUTO [51]which employs a pseudo-arclength continuation and thusenables the computation of branches past turning pointsThisway we are able to trace entire families of periodic solutionsand study their corresponding stability characteristics interms of the Floquet multipliers

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

E G Charalampidis gratefully acknowledges financial sup-port from the FP7-People IRSES-606096 ldquoTopological Soli-tons from FieldTheory to Cosmosrdquo P G Kevrekidis acknowl-edges support from the National Science Foundation (NSF)under Grants CMMI-1000337 and DMS-1312856 from theERC and FP7-People under Grant IRSES-606096 andfrom the US-AFOSR under Grant FA9550-12-1-0332 P GKevrekidisrsquos work at Los Alamos is supported in part by theUS Department of Energy C Chongwas partially supportedby the ETH Zurich Foundation through the Seed ProjectESC-A 06-14 F Li and J Yang thank the support of the NSF(CMMI-1414748) and the US-ONR (N000141410388) E GCharalampidis and C Chong thank M O Williams (PACMPrinceton University) for insight regarding the AUTO codesused for the bifurcation analysis performed in this work

References

[1] V F Nesterenko Dynamics of Heterogeneous MaterialsSpringer New York NY USA 2001

[2] S Sen J Hong J Bang E Avalos and R Doney ldquoSolitary wavesin the granular chainrdquo Physics Reports vol 462 no 2 pp 21ndash662008

[3] GTheocharis N Boechler and C Daraio ldquoNonlinear periodicphononic structures and granular crystalsrdquo in Acoustic Meta-materials and Phononic Crystals vol 173 of Springer Series inSolid-State Sciences chapter 6 pp 217ndash251 Springer New YorkNY USA 2013

[4] C Coste E Falcon and S Fauve ldquoSolitary waves in a chainof beads under Hertz contactrdquo Physical Review E Statistical


Physics Plasmas Fluids and Related Interdisciplinary Topicsvol 56 no 5 pp 6104ndash6117 1997

[5] K Ahnert and A Pikovsky ldquoCompactons and chaos in stronglynonlinear latticesrdquo Physical Review E vol 79 no 2 Article ID026209 2009

[6] E B Herbold and V F Nesterenko ldquoShock wave structure ina strongly nonlinear lattice with viscous dissipationrdquo PhysicalReview E vol 75 no 2 Article ID 021304 8 pages 2007

[7] A Rosas A H Romero V F Nesterenko and K LindenbergldquoObservation of two-wave structure in strongly nonlineardissipative granular chainsrdquo Physical Review Letters vol 98 no16 Article ID 164301 4 pages 2007

[8] AMolinari andCDaraio ldquoStationary shocks in periodic highlynonlinear granular chainsrdquoPhysical ReviewE vol 80 Article ID056602 2009

[9] GTheocharisM Kavousanakis P G Kevrekidis C DaraioMA Porter and I G Kevrekidis ldquoLocalized breathing modes ingranular crystals with defectsrdquo Physical Review E vol 80 no 6Article ID 066601 2009

[10] S Job F Santibanez F Tapia and F Melo ldquoWave localization instrongly nonlinear Hertzian chains with mass defectrdquo PhysicalReview E vol 80 Article ID 025602 2009

[11] N Boechler G Theocharis S Job P G Kevrekidis M APorter and C Daraio ldquoDiscrete breathers in one-dimensionaldiatomic granular crystalsrdquo Physical Review Letters vol 104 no24 Article ID 244302 2010

[12] N Boechler G Theocharis and C Daraio ldquoBifurcation-basedacoustic switching and rectificationrdquo Nature Materials vol 10no 9 pp 665ndash668 2011

[13] CHoogeboomYManN Boechler et al ldquoHysteresis loops andmulti-stability from periodic orbits to chaotic dynamics (andback) in diatomic granular crystalsrdquo EPL vol 101 no 4 ArticleID 44003 2013

[14] C Hoogeboom and P G Kevrekidis ldquoBreathers in periodicgranular chainswithmultiple band gapsrdquoPhysical ReviewE vol86 Article ID 061305 2012

[15] C Chong F Li J Yang et al ldquoDamped-driven granularchains an ideal playground for dark breathers and multi-breathersrdquo Physical Review EmdashStatistical Nonlinear and SoftMatter Physics vol 89 no 3 Article ID 032924 2014

[16] C Daraio V F Nesterenko E B Herbold and S Jin ldquoEnergytrapping and shock disintegration in a composite granularmediumrdquo Physical Review Letters vol 96 no 5 Article ID058002 2006

[17] J Hong ldquoUniversal power-law decay of the impulse energy ingranular protectorsrdquo Physical Review Letters vol 94 Article ID108001 2005

[18] F Fraternali M A Porter and C Daraio ldquoOptimal design ofcomposite granular protectorsrdquoMechanics of Advanced Materi-als and Structures vol 17 no 1 p 1 2010

[19] R Doney and S Sen ldquoDecorated tapered and highly nonlineargranular chainrdquo Physical Review Letters vol 97 Article ID155502 2006

[20] D Khatri C Daraio and P Rizzo ldquoHighly nonlinear wavesrsquosensor technology for highway infrastructuresrdquo in Nonde-structive Characterization for Composite Materials AerospaceEngineering Civil Infrastructure and Homeland Security 2008vol 6934 of Proceedings of SPIE p 69340U San Diego CalifUSA April 2008

[21] A Spadoni and C Daraio ldquoGeneration and control of soundbullets with a nonlinear acoustic lensrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 107 pp 7230ndash7234 2010

[22] F Li P Anzel J Yang P G Kevrekidis andCDaraio ldquoGranularacoustic switches and logic elementsrdquo Nature Communicationsvol 5 article 5311 2014

[23] C Daraio V F Nesterenko E B Herbold and S Jin ldquoStronglynonlinear waves in a chain of Teflon beadsrdquo Physical Review Evol 72 no 1 Article ID 016603 2005

[24] V F Nesterenko C Daraio E B Herbold and S Jin ldquoAnoma-lous wave reflection at the interface of two strongly nonlineargranular mediardquo Physical Review Letters vol 95 no 15 ArticleID 158702 2005

[25] S Flach and A V Gorbach ldquoDiscrete breathersmdashadvances intheory and applicationsrdquo Physics Reports vol 467 no 1ndash3 pp1ndash116 2008

[26] F Lederer G I Stegeman D N Christodoulides G AssantoM Segev and Y Silberberg ldquoDiscrete solitons in opticsrdquo PhysicsReports vol 463 no 1ndash3 pp 1ndash126 2008

[27] M Sato B E Hubbard and A J Sievers ldquoColloquium nonlin-ear energy localization and its manipulation in micromechan-ical oscillator arraysrdquo Reviews of Modern Physics vol 78 no 1pp 137ndash157 2006

[28] P Binder D Abraimov A V Ustinov S Flach and YZolotaryuk ldquoObservation of breathers in Josephson laddersrdquoPhysical Review Letters vol 84 no 4 pp 745ndash748 2000

[29] E Trıas J J Mazo and T P Orlando ldquoDiscrete breathers innonlinear lattices experimental detection in a Josephson arrayrdquoPhysical Review Letters vol 84 no 4 pp 741ndash744 2000

[30] L Q English M Sato and A J Sievers ldquoModulational insta-bility of nonlinear spin waves in easy-axis antiferromagneticchains II Influence of sample shape on intrinsic localizedmodes and dynamic spin defectsrdquo Physical Review B vol 67Article ID 024403 2003

[31] U T Schwarz L Q English and A J Sievers ldquoExperimentalgeneration and observation of intrinsic localized spin wavemodes in an antiferromagnetrdquo Physical Review Letters vol 83article 223 1999

[32] B I Swanson J A Brozik S P Love et al ldquoObservationof intrinsically localized modes in a discrete low-dimensionalmaterialrdquo Physical Review Letters vol 82 no 16 pp 3288ndash32911999

[33] M Peyrard ldquoNonlinear dynamics and statistical physics ofDNArdquo Nonlinearity vol 17 no 2 pp R1ndashR40 2004

[34] O Morsch and M Oberthaler ldquoDynamics of Bose-Einsteincondensates in optical latticesrdquo Reviews of Modern Physics vol78 no 1 pp 179ndash215 2006

[35] G James ldquoExistence of breathers on FPU latticesrdquo ComptesRendus de lrsquoAcademie des Sciences Series I Mathematics vol332 no 6 pp 581ndash586 2001

[36] G James ldquoCentre manifold reduction for quasilinear discretesystemsrdquo Journal of Nonlinear Science vol 13 no 1 pp 27ndash632003

[37] Y Man N Boechler G Theocharis P G Kevrekidis andC Daraio ldquoDefect modes in one-dimensional granular crys-talsrdquo Physical Review EmdashStatistical Nonlinear and Soft MatterPhysics vol 85 no 3 Article ID 037601 2012


[38] E B Herbold J Kim V F Nesterenko S Y Wang and CDaraio ldquoPulse propagation in a linear and nonlinear diatomicperiodic chain effects of acoustic frequency band-gaprdquo ActaMechanica vol 205 no 1ndash4 pp 85ndash103 2009

[39] K R Jayaprakash A F Vakakis and Y Starosvetsky ldquoNonlinearresonances in a general class of granular dimers with no pre-compressionrdquo Granular Matter vol 15 no 3 pp 327ndash347 2013

[40] K R Jayaprakash Y Starosvetsky A F Vakakis and O VGendelman ldquoNonlinear resonances leading to strong pulseattenuation in granular dimer chainsrdquo Journal of NonlinearScience vol 23 no 3 pp 363ndash392 2013

[41] K R Jayaprakash A F Vakakis and Y Starosvetsky ldquoSolitarywaves in a general class of granular dimer chainsrdquo Journal ofApplied Physics vol 112 no 3 Article ID 034908 2012

[42] R Potekin K R Jayaprakash D M McFarland K Remick LA Bergman and A F Vakakis ldquoExperimental study of stronglynonlinear resonances and anti-resonances in granular dimerchainsrdquo Experimental Mechanics vol 53 no 5 pp 861ndash8702013

[43] H Hertz ldquoUber die Beruhrung fester elastischer KorperrdquoJournal fur die Reine und Angewandte Mathematik vol 92 pp156ndash171 1881

[44] K L Johnson Contact Mechanics Cambridge University PressCambridge UK 1985

[45] N Boechler J Yang G Theocharis P G Kevrekidis and CDaraio ldquoTunable vibrational band gaps in one-dimensionaldiatomic granular crystals with three-particle unit cellsrdquo Journalof Applied Physics vol 109 no 7 Article ID 074906 2011

[46] AHNayfeh andDTMookNonlinearOscillations JohnWileyamp Sons Weinheim Germany 2004

[47] A H Nayfeh and B Balachandran Applied Nonlinear Dynam-ics Analytical Computational and Experimental MethodsWiley Series in Nonlinear Science John Wiley amp Sons 1995

[48] E Doedel H B Keller and J-P Kernevez ldquoNumerical analysisand control of bifurcation problems (I) bifurcation in finitedimensionsrdquo International Journal of Bifurcation andChaos vol1 no 3 pp 493ndash520 1991

[49] E Hairer S P Noslashrsett and G Wanner Solving Ordinary Dif-ferential Equations I vol 8 of Springer Series in ComputationalMathematics Springer Berlin Germany 1993

[50] L F Shampine and M K Gordon Computer Solution ofOrdinary Differential Equations The Initial Value Problem WH Freeman New York NY USA 1975

[51] E Doedel AUTO httpindycsconcordiacaauto

Submit your manuscripts athttpwwwhindawicom

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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


models of the DNA double strand [33] and Bose-Einsteincondensates in optical lattices [34]

In Fermi-Pasta-Ulam type settings (which are intimatelyconnected to the case of precompressed granular crystals inthe weakly nonlinear regime see eg (1)) it was proven in[35 36] (see also the discussion in [25]) that small amplitudediscrete breathers are absent in spatially homogeneous (iemonoatomic) chains Instead dark breathers (those on top ofa non-vanishing background) have been found therein [15]It is for that reason that the first theoretical and experimentalinvestigations of breathers with a vanishing background (iebright breathers) have taken place in granular chains withsome degree of spatial heterogeneity which plays a criticalrole in the emergence of such patterns Examples includechains with defects [9 10 12] (see also [37] for recentexperiments) and a spatial periodicity of two (ie dimerlattices) [11 13] Motivated by these works a theoreticalstudy of breathers in granular crystals with higher orderspatial periodicity (such as trimers and quadrimers) wasrecently conducted in [14] Therein it was demonstrated thatbreathers with a frequency in the highest finite gap appearto be more robust than their counterparts with frequencyin the lower gaps Another relevant study is [38] whereexperimental and numerical analysis of band gap effectsrelated to transient behavior of driven dissipative granularchains were studied

The goal of the present work is the systematic studyof time-periodic solutions (including breathers) of trimergranular crystals with frequency in the first or second gapas well as in the acoustic and optical bands In particularwe investigate the robustness of the breathers experimen-tally using a full-field visualization technique based onlaser Doppler vibrometry This is a significant improvementover the aforementioned experimental observation of brightbreathers [11ndash13] where force sensors are placed at isolatedparticles within the granular chain which does not allow afull-field representation of the breather We complement thisstudy with a detailed theoretical probing of the more realisticdamped-driven variant of the pertinentmodel Our extensiveanalysis of such modes consists of the study of their familyunder continuations in both the amplitude and the frequencyparameters of the external drive and a detailed comparisonof the findings between numerically exact (up to a prescribedtolerance) periodic orbit solutions and experimentally tracedcounterparts thereof We also note that there are limitedstudies reporting the full-field experimental visualization oftime-periodic solutions in granular crystals Indeed the onlywork to this effect that the authors are aware of is [15] whichdeals with the much simpler setting of a monomer granularcrystal where surface breathers are not addressed Althoughthe theme of heterogeneous granular crystals has been receiv-ing considerable theoretical and even experimental attentionrecently most of it has been focused on the realm of travelingwaves [39ndash42] while the emphasis herein will be on the time-periodic exponentially localized discrete breather states

The paper is structured as follows In Sections 2 and 3we describe the experimental and theoretical setups respec-tively The main results are presented in Section 4 wheretime-periodic solutions with frequency in the firstsecond

Table 1 Parameter values of the trimer granular chain (S-W-S)

Material Parameter119864 [Nm2] 119903 [m] ] 120588 [kgm3]

Chrome steel (S) 200 times 109 9525 times 10minus3 03 7780Tungsten carbide (W) 628 times 109 9525 times 10minus3 028 14980

gaps and in the spectral bands are studied in both of thesesetups and compared accordingly Finally Section 5 providesour conclusions and discusses a number of future challenges

2 Experimental Setup

Figure 1 shows a schematic of the experimental setup consist-ing of a granular chain and a laser Doppler vibrometer Inthis study we consider a granular chain composed of119873 = 21spherical beads The beads are made out of chrome steel (Sgray particles in Figure 1) and tungsten carbide (W blackparticles) materials See Table 1 for nominal values of thematerial parameters used hereafter The granular chain has aspatial periodicity of three particles and each unit cell followsthe pattern of a 2 1 trimer S-W-SThe spheres are supportedby four polytetrafluoroethylene (PTFE) rods which allowaxial vibrations of particles with minimal friction whilerestricting their lateral movements The granular chain iscompressed by a moving wall at one end of the chain thatapplies static force in a controllable manner via a linear stage(see Figure 1) We measure the preapplied static force (1198650 =10N in this study) by using a static force sensor mountedon the moving wall We assume that this moving wall isstationary throughout our analysis since it exhibits orders-of-magnitude larger inertia compared to the particlersquosmasses

The granular chain is driven by a piezoelectric actuatorpositioned on the other side of the chain We imposeactuation signals of chosen amplitude and frequency throughan external function generator and a power amplifier Thedynamics of individual particles are scoped by a laserDoppler vibrometer (LDV Polytec OFV-534) which is capa-ble of measuring particlesrsquo velocities with a resolution of002 120583msHz12 The LDV scans the granular chain throughthe automatic sliding rail and measures the vibrationalmotions of each particle three times for statistical pur-poses We obtain the full-field map of the granular chainrsquosdynamics by synchronizing and reconstructing the acquireddata

3 Theoretical Setup

The equation we use to model the experimental setup is aFermi-Pasta-Ulam-type lattice with a Hertzian potential [1]leading to

119899

=119860119899minus1119872119899

[1205750119899minus1 +119906119899minus1 minus119906119899]32+

minus119860119899

119872119899

[1205750119899 +119906119899 minus119906119899+1]32+

minus119899

120591

(1)


Poweramplifier


Functiongenerator

Moving wall

Staticforce

sensorGuiding rod



ActuatorF0

n = 21

n = 1


where 119899 = 1 119873 119906119899

= 119906119899


119899


119899




119899


119860119899

=

4119864119899

119864119899+1radic119903119899119903119899+1 (119903119899 + 119903119899+1)

3119864119899+1 (1 minus ]2

119899

) + 3119864119899

(1 minus ]2119899+1)

(2)

where 119864119899

]119899

and 119903119899


1199060 (119905) = 119886 cos (2120587119891119887119905)

119906119873+1 (119905) = 0

(3)

where 119886 and 119891119887



1198600 =21198641radic1199031

3 (1 minus ]21) (4)


1003816100381610038161003816119906119899 minus 119906119899+11003816100381610038161003816

1205750119899≪ 1 (5)


119899

=119870119899minus1119872119899


119870119899

119872119899

(119906119899

minus119906119899+1) minus

119899

120591 (6)

with 119870119899

= 32119860119899



Y = AY (7)

with Y = (1199061 119906119873 V1(= 1) V119873(= 119873))119879 and

A = (


) (8)



5 10 20

4

2

6


015

8f

(kH

z)

(a)

0 1 2 30

2

4

6

8

f(k

Hz)

fc6

fc4

fc5

fc2

fc3

fc1


(b)


C =

(((((((((

(

minus1198700 + 11987011198721

11987011198721

0 sdot sdot sdot 0

11987011198722

minus1198701 + 11987021198722

11987021198722

0 d d d 0

119870119873minus2

119872119873minus1


119870119873minus1

119872119873minus1


minus119870119873minus1 + 119870119873119872119873

)))))))))

)

(9)


119899

= 119910119899



Ay = 120582y (10)




3119895minus2 =11987031198721



minus

3119895minus2

120591

(11a)

3119895minus1 =11987011198722


(1199063119895minus1 minus1199063119895)

minus

3119895minus1

120591

(11b)

3119895 =11987011198721


(1199063119895 minus1199063119895+1) minus3119895

120591 (11c)


1199063119895minus2 = 1205781119890119894(119896120572119895+120596119905)

(12a)

1199063119895minus1 = 1205782119890119894(119896120572119895+120596119905)

(12b)

1199063119895 = 1205783119890119894(119896120572119895+120596119905)

(12c)



((

(

1205962minus1198701 + 11987031198721

minus 119894120596

120591

11987011198721

11987031198721

119890minus119894119896120572

11987011198722

1205962minus211987011198722

minus 119894120596

120591

11987011198722

11987031198721

119890119894119896120572

11987011198721

1205962minus1198701 + 11987031198721

minus 119894120596

120591

))

)

(

1205781

1205782

1205783

) = (

000) (13)


119872211198722120596

6minus 119894

3119872211198722120591

1205965

minus1198721 [2 [11987031198722 +1198701 (1198721 +1198722)] +3119872111987221205912

]1205964

+ 1198941198721120591[4 [11987031198722 +1198701 (1198721 +1198722)] +

119872111987221205912

]1205963

minus 4119870211198703sin

2(119896120572

2)

+ [1198701 (1198701 + 21198703) (21198721 +1198722)

+211987211205912

[11987031198722 +1198701 (1198721 +1198722)]] 1205962minus 1198941198701120591(1198701

+ 21198703) (21198721 +1198722) 120596 = 0

(14)


1198911198881 (119896 = 0) = 119894

12120587120591

(lower acoustic)

1198911198882 (119896 = 0) = plusmn

12120587radic1198701 + 211987031198721

minus141205912

+ 1198941

4120587120591


1198911198883 (119896 = 0) = plusmn

12120587radic1198701 (21198721 +1198722)

11987211198722minus

141205912

+ 1198941

4120587120591

(first upper optic)

1198911198884 (119896=

120587

120572) = plusmn

12120587radic11987011198721

minus141205912

+ 1198941

4120587120591

(first lower optic)

1198911198885 (119896=

120587

120572)


211987211198722minus

141205912

+ 1198941


1198911198886 (119896=

120587

120572)

= plusmn12120587radic1198701 (21198721 +1198722) + 211987031198722 + 119891

211987211198722minus

141205912

+ 1198941


(15)



119899







1198911198885 119891

1198884 1198911198883 119891

1198882 1198911198886

304 423 604 671 737



Y = AY+ F (119905) (16)



Y =

(((((((

(

1198861 cos (120596119887119905) + 1198871 sin (120596119887119905)

119886119873

cos (120596119887

119905) + 119887119873

sin (120596119887

119905)

minus1198861120596119887 sin (120596119887119905) + 1198871120596119887 cos (120596119887119905)

minus119886119873

120596119887

sin (120596119887

119905) + 119887119873

120596119887

cos (120596119887

119905)

)))))))

)

(17)


and 119887119895


AX = F (18)

where

A = (

C (120596

120591) I

minus(120596

120591) I C

) (19)

C = minus(C + 1205962119887


119887



4 Main Results



119887

= 63087 kHz which


5 10 15 20

minus001

0

001

n(m

s)

n

(a)

0 004 008 0120

0004

0008

0012

(A)

(D)

(B)

(C)

4(m

s)

a (120583m)

(b)

0 02 04 06 080

002

004

006

008

01

4(m

s)

a (120583m)

(c)






0 10 20minus5

0

5

10

minus1 0 1minus1

minus05

0

05

n

un

(120583m

)

Im(120582

)

Re(120582)

times10minus4

1

(a)

0 10 20minus01

0

01

02

03

minus1 0 1minus1

minus05

0

05

1

n

un

(120583m

)

Im(120582

)

Re(120582)

(b)

0 10 20

minus03

minus015

0

015

03

minus1 0 1minus1

minus05

0

05

1

n

SteelTungsten

un

(120583m

)

Im(120582

)

Re(120582)

(c)

0 10 20

minus04

minus02

0

02

04

minus1 0 1minus1

minus05

0

05

1

n

SteelTungsten

un

(120583m

)

Im(120582

)

Re(120582)

(d)


=





0 5 10 15 20

minus4

minus2

0

2

4

n

times10minus3 n

(ms

)

(a)

0 05 1 150

0005

001

0015

(A)

(B)

(D)

(C)

a (120583m)

4(m

s)

(b)

0 05 1 150

003

006

009

a (120583m)

4(m

s)

(c)



119887


119887




119887




0 10 20

minus003

minus0015

0

0015

003

minus1 0 1minus1

minus05

0

05

1

n

un

(120583m

)

Im(120582

)Re(120582)

(a)

0 10 20minus04

minus02

0

02

04

06

minus1 0 1minus1

minus05

0

05

1

n

un

(120583m

)

Im(120582

)

Re(120582)

(b)

SteelTungsten

0 10 200

05

1

15

minus1 0 1minus1

minus05

0

05

1

n

un

(120583m

)

Im(120582

)

Re(120582)

(c)

SteelTungsten

0 10 200

1

2

3


minus09

0

09

18

n

un

(120583m

)

Im(120582

)

Re(120582)

(d)


=




119887

isin


119887


119887




119899






0 004 008 012minus02

0

02

04

t (ms)

Disp

lace

men

ts (120583

m) u3

u4

(a)

0 01 02minus004

minus002

0

002

004

t (ms)

Disp

lace

men

ts (120583

m)

u3

u4

(b)

49 492 494 496 498 50

minus001

minus0005

0

0005

001

Velo

citie

s (m

s)

t (ms)

15

2Tb

Tb

2

(c)


119887


119887


119887


119887



119899

minus 119906119899+1|1205750119899 = 15 In


119891119887

asymp 235 kHz and 119891119887


119887

asymp




0 1 2 3 4 5 6 7 8 90

0005001

0015002

0025003

0035004


fb (kHz)

4(m

s)

(a)

54 56 58 6 62 64 66 68 70

0005

001

0015

002

0025

003

0035

004

fb (kHz)

4(m

s)

(b)









Appendix



119887


=

1119891119887



x = F (119905 u k) (A1)



119887



119887


119887


119887






119887





119887




Acknowledgments


References






























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Poweramplifier


Functiongenerator

Moving wall

Staticforce

sensorGuiding rod



ActuatorF0

n = 21

n = 1


where 119899 = 1 119873 119906119899

= 119906119899


119899


119899




119899


119860119899

=

4119864119899

119864119899+1radic119903119899119903119899+1 (119903119899 + 119903119899+1)

3119864119899+1 (1 minus ]2

119899

) + 3119864119899

(1 minus ]2119899+1)

(2)

where 119864119899

]119899

and 119903119899


1199060 (119905) = 119886 cos (2120587119891119887119905)

119906119873+1 (119905) = 0

(3)

where 119886 and 119891119887



1198600 =21198641radic1199031

3 (1 minus ]21) (4)


1003816100381610038161003816119906119899 minus 119906119899+11003816100381610038161003816

1205750119899≪ 1 (5)


119899

=119870119899minus1119872119899


119870119899

119872119899

(119906119899

minus119906119899+1) minus

119899

120591 (6)

with 119870119899

= 32119860119899



Y = AY (7)

with Y = (1199061 119906119873 V1(= 1) V119873(= 119873))119879 and

A = (


) (8)



5 10 20

4

2

6


015

8f

(kH

z)

(a)

0 1 2 30

2

4

6

8

f(k

Hz)

fc6

fc4

fc5

fc2

fc3

fc1


(b)


C =

(((((((((

(

minus1198700 + 11987011198721

11987011198721

0 sdot sdot sdot 0

11987011198722

minus1198701 + 11987021198722

11987021198722

0 d d d 0

119870119873minus2

119872119873minus1


119870119873minus1

119872119873minus1


minus119870119873minus1 + 119870119873119872119873

)))))))))

)

(9)


119899

= 119910119899



Ay = 120582y (10)




3119895minus2 =11987031198721



minus

3119895minus2

120591

(11a)

3119895minus1 =11987011198722


(1199063119895minus1 minus1199063119895)

minus

3119895minus1

120591

(11b)

3119895 =11987011198721


(1199063119895 minus1199063119895+1) minus3119895

120591 (11c)


1199063119895minus2 = 1205781119890119894(119896120572119895+120596119905)

(12a)

1199063119895minus1 = 1205782119890119894(119896120572119895+120596119905)

(12b)

1199063119895 = 1205783119890119894(119896120572119895+120596119905)

(12c)



((

(

1205962minus1198701 + 11987031198721

minus 119894120596

120591

11987011198721

11987031198721

119890minus119894119896120572

11987011198722

1205962minus211987011198722

minus 119894120596

120591

11987011198722

11987031198721

119890119894119896120572

11987011198721

1205962minus1198701 + 11987031198721

minus 119894120596

120591

))

)

(

1205781

1205782

1205783

) = (

000) (13)


119872211198722120596

6minus 119894

3119872211198722120591

1205965

minus1198721 [2 [11987031198722 +1198701 (1198721 +1198722)] +3119872111987221205912

]1205964

+ 1198941198721120591[4 [11987031198722 +1198701 (1198721 +1198722)] +

119872111987221205912

]1205963

minus 4119870211198703sin

2(119896120572

2)

+ [1198701 (1198701 + 21198703) (21198721 +1198722)

+211987211205912

[11987031198722 +1198701 (1198721 +1198722)]] 1205962minus 1198941198701120591(1198701

+ 21198703) (21198721 +1198722) 120596 = 0

(14)


1198911198881 (119896 = 0) = 119894

12120587120591

(lower acoustic)

1198911198882 (119896 = 0) = plusmn

12120587radic1198701 + 211987031198721

minus141205912

+ 1198941

4120587120591


1198911198883 (119896 = 0) = plusmn

12120587radic1198701 (21198721 +1198722)

11987211198722minus

141205912

+ 1198941

4120587120591

(first upper optic)

1198911198884 (119896=

120587

120572) = plusmn

12120587radic11987011198721

minus141205912

+ 1198941

4120587120591

(first lower optic)

1198911198885 (119896=

120587

120572)


211987211198722minus

141205912

+ 1198941


1198911198886 (119896=

120587

120572)

= plusmn12120587radic1198701 (21198721 +1198722) + 211987031198722 + 119891

211987211198722minus

141205912

+ 1198941


(15)



119899







1198911198885 119891

1198884 1198911198883 119891

1198882 1198911198886

304 423 604 671 737



Y = AY+ F (119905) (16)



Y =

(((((((

(

1198861 cos (120596119887119905) + 1198871 sin (120596119887119905)

119886119873

cos (120596119887

119905) + 119887119873

sin (120596119887

119905)

minus1198861120596119887 sin (120596119887119905) + 1198871120596119887 cos (120596119887119905)

minus119886119873

120596119887

sin (120596119887

119905) + 119887119873

120596119887

cos (120596119887

119905)

)))))))

)

(17)


and 119887119895


AX = F (18)

where

A = (

C (120596

120591) I

minus(120596

120591) I C

) (19)

C = minus(C + 1205962119887


119887



4 Main Results



119887

= 63087 kHz which


5 10 15 20

minus001

0

001

n(m

s)

n

(a)

0 004 008 0120

0004

0008

0012

(A)

(D)

(B)

(C)

4(m

s)

a (120583m)

(b)

0 02 04 06 080

002

004

006

008

01

4(m

s)

a (120583m)

(c)






0 10 20minus5

0

5

10

minus1 0 1minus1

minus05

0

05

n

un

(120583m

)

Im(120582

)

Re(120582)

times10minus4

1

(a)

0 10 20minus01

0

01

02

03

minus1 0 1minus1

minus05

0

05

1

n

un

(120583m

)

Im(120582

)

Re(120582)

(b)

0 10 20

minus03

minus015

0

015

03

minus1 0 1minus1

minus05

0

05

1

n

SteelTungsten

un

(120583m

)

Im(120582

)

Re(120582)

(c)

0 10 20

minus04

minus02

0

02

04

minus1 0 1minus1

minus05

0

05

1

n

SteelTungsten

un

(120583m

)

Im(120582

)

Re(120582)

(d)


=





0 5 10 15 20

minus4

minus2

0

2

4

n

times10minus3 n

(ms

)

(a)

0 05 1 150

0005

001

0015

(A)

(B)

(D)

(C)

a (120583m)

4(m

s)

(b)

0 05 1 150

003

006

009

a (120583m)

4(m

s)

(c)



119887


119887




119887




0 10 20

minus003

minus0015

0

0015

003

minus1 0 1minus1

minus05

0

05

1

n

un

(120583m

)

Im(120582

)Re(120582)

(a)

0 10 20minus04

minus02

0

02

04

06

minus1 0 1minus1

minus05

0

05

1

n

un

(120583m

)

Im(120582

)

Re(120582)

(b)

SteelTungsten

0 10 200

05

1

15

minus1 0 1minus1

minus05

0

05

1

n

un

(120583m

)

Im(120582

)

Re(120582)

(c)

SteelTungsten

0 10 200

1

2

3


minus09

0

09

18

n

un

(120583m

)

Im(120582

)

Re(120582)

(d)


=




119887

isin


119887


119887




119899






0 004 008 012minus02

0

02

04

t (ms)

Disp

lace

men

ts (120583

m) u3

u4

(a)

0 01 02minus004

minus002

0

002

004

t (ms)

Disp

lace

men

ts (120583

m)

u3

u4

(b)

49 492 494 496 498 50

minus001

minus0005

0

0005

001

Velo

citie

s (m

s)

t (ms)

15

2Tb

Tb

2

(c)


119887


119887


119887


119887



119899

minus 119906119899+1|1205750119899 = 15 In


119891119887

asymp 235 kHz and 119891119887


119887

asymp




0 1 2 3 4 5 6 7 8 90

0005001

0015002

0025003

0035004


fb (kHz)

4(m

s)

(a)

54 56 58 6 62 64 66 68 70

0005

001

0015

002

0025

003

0035

004

fb (kHz)

4(m

s)

(b)









Appendix



119887


=

1119891119887



x = F (119905 u k) (A1)



119887



119887


119887


119887






119887





119887




Acknowledgments


References






























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










5 10 20

4

2

6


015

8f

(kH

z)

(a)

0 1 2 30

2

4

6

8

f(k

Hz)

fc6

fc4

fc5

fc2

fc3

fc1


(b)


C =

(((((((((

(

minus1198700 + 11987011198721

11987011198721

0 sdot sdot sdot 0

11987011198722

minus1198701 + 11987021198722

11987021198722

0 d d d 0

119870119873minus2

119872119873minus1


119870119873minus1

119872119873minus1


minus119870119873minus1 + 119870119873119872119873

)))))))))

)

(9)


119899

= 119910119899



Ay = 120582y (10)




3119895minus2 =11987031198721



minus

3119895minus2

120591

(11a)

3119895minus1 =11987011198722


(1199063119895minus1 minus1199063119895)

minus

3119895minus1

120591

(11b)

3119895 =11987011198721


(1199063119895 minus1199063119895+1) minus3119895

120591 (11c)


1199063119895minus2 = 1205781119890119894(119896120572119895+120596119905)

(12a)

1199063119895minus1 = 1205782119890119894(119896120572119895+120596119905)

(12b)

1199063119895 = 1205783119890119894(119896120572119895+120596119905)

(12c)



((

(

1205962minus1198701 + 11987031198721

minus 119894120596

120591

11987011198721

11987031198721

119890minus119894119896120572

11987011198722

1205962minus211987011198722

minus 119894120596

120591

11987011198722

11987031198721

119890119894119896120572

11987011198721

1205962minus1198701 + 11987031198721

minus 119894120596

120591

))

)

(

1205781

1205782

1205783

) = (

000) (13)


119872211198722120596

6minus 119894

3119872211198722120591

1205965

minus1198721 [2 [11987031198722 +1198701 (1198721 +1198722)] +3119872111987221205912

]1205964

+ 1198941198721120591[4 [11987031198722 +1198701 (1198721 +1198722)] +

119872111987221205912

]1205963

minus 4119870211198703sin

2(119896120572

2)

+ [1198701 (1198701 + 21198703) (21198721 +1198722)

+211987211205912

[11987031198722 +1198701 (1198721 +1198722)]] 1205962minus 1198941198701120591(1198701

+ 21198703) (21198721 +1198722) 120596 = 0

(14)


1198911198881 (119896 = 0) = 119894

12120587120591

(lower acoustic)

1198911198882 (119896 = 0) = plusmn

12120587radic1198701 + 211987031198721

minus141205912

+ 1198941

4120587120591


1198911198883 (119896 = 0) = plusmn

12120587radic1198701 (21198721 +1198722)

11987211198722minus

141205912

+ 1198941

4120587120591

(first upper optic)

1198911198884 (119896=

120587

120572) = plusmn

12120587radic11987011198721

minus141205912

+ 1198941

4120587120591

(first lower optic)

1198911198885 (119896=

120587

120572)


211987211198722minus

141205912

+ 1198941


1198911198886 (119896=

120587

120572)

= plusmn12120587radic1198701 (21198721 +1198722) + 211987031198722 + 119891

211987211198722minus

141205912

+ 1198941


(15)



119899







1198911198885 119891

1198884 1198911198883 119891

1198882 1198911198886

304 423 604 671 737



Y = AY+ F (119905) (16)



Y =

(((((((

(

1198861 cos (120596119887119905) + 1198871 sin (120596119887119905)

119886119873

cos (120596119887

119905) + 119887119873

sin (120596119887

119905)

minus1198861120596119887 sin (120596119887119905) + 1198871120596119887 cos (120596119887119905)

minus119886119873

120596119887

sin (120596119887

119905) + 119887119873

120596119887

cos (120596119887

119905)

)))))))

)

(17)


and 119887119895


AX = F (18)

where

A = (

C (120596

120591) I

minus(120596

120591) I C

) (19)

C = minus(C + 1205962119887


119887



4 Main Results



119887

= 63087 kHz which


5 10 15 20

minus001

0

001

n(m

s)

n

(a)

0 004 008 0120

0004

0008

0012

(A)

(D)

(B)

(C)

4(m

s)

a (120583m)

(b)

0 02 04 06 080

002

004

006

008

01

4(m

s)

a (120583m)

(c)






0 10 20minus5

0

5

10

minus1 0 1minus1

minus05

0

05

n

un

(120583m

)

Im(120582

)

Re(120582)

times10minus4

1

(a)

0 10 20minus01

0

01

02

03

minus1 0 1minus1

minus05

0

05

1

n

un

(120583m

)

Im(120582

)

Re(120582)

(b)

0 10 20

minus03

minus015

0

015

03

minus1 0 1minus1

minus05

0

05

1

n

SteelTungsten

un

(120583m

)

Im(120582

)

Re(120582)

(c)

0 10 20

minus04

minus02

0

02

04

minus1 0 1minus1

minus05

0

05

1

n

SteelTungsten

un

(120583m

)

Im(120582

)

Re(120582)

(d)


=





0 5 10 15 20

minus4

minus2

0

2

4

n

times10minus3 n

(ms

)

(a)

0 05 1 150

0005

001

0015

(A)

(B)

(D)

(C)

a (120583m)

4(m

s)

(b)

0 05 1 150

003

006

009

a (120583m)

4(m

s)

(c)



119887


119887




119887




0 10 20

minus003

minus0015

0

0015

003

minus1 0 1minus1

minus05

0

05

1

n

un

(120583m

)

Im(120582

)Re(120582)

(a)

0 10 20minus04

minus02

0

02

04

06

minus1 0 1minus1

minus05

0

05

1

n

un

(120583m

)

Im(120582

)

Re(120582)

(b)

SteelTungsten

0 10 200

05

1

15

minus1 0 1minus1

minus05

0

05

1

n

un

(120583m

)

Im(120582

)

Re(120582)

(c)

SteelTungsten

0 10 200

1

2

3


minus09

0

09

18

n

un

(120583m

)

Im(120582

)

Re(120582)

(d)


=




119887

isin


119887


119887




119899






0 004 008 012minus02

0

02

04

t (ms)

Disp

lace

men

ts (120583

m) u3

u4

(a)

0 01 02minus004

minus002

0

002

004

t (ms)

Disp

lace

men

ts (120583

m)

u3

u4

(b)

49 492 494 496 498 50

minus001

minus0005

0

0005

001

Velo

citie

s (m

s)

t (ms)

15

2Tb

Tb

2

(c)


119887


119887


119887


119887



119899

minus 119906119899+1|1205750119899 = 15 In


119891119887

asymp 235 kHz and 119891119887


119887

asymp




0 1 2 3 4 5 6 7 8 90

0005001

0015002

0025003

0035004


fb (kHz)

4(m

s)

(a)

54 56 58 6 62 64 66 68 70

0005

001

0015

002

0025

003

0035

004

fb (kHz)

4(m

s)

(b)









Appendix



119887


=

1119891119887



x = F (119905 u k) (A1)



119887



119887


119887


119887






119887





119887




Acknowledgments


References






























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










((

(

1205962minus1198701 + 11987031198721

minus 119894120596

120591

11987011198721

11987031198721

119890minus119894119896120572

11987011198722

1205962minus211987011198722

minus 119894120596

120591

11987011198722

11987031198721

119890119894119896120572

11987011198721

1205962minus1198701 + 11987031198721

minus 119894120596

120591

))

)

(

1205781

1205782

1205783

) = (

000) (13)


119872211198722120596

6minus 119894

3119872211198722120591

1205965

minus1198721 [2 [11987031198722 +1198701 (1198721 +1198722)] +3119872111987221205912

]1205964

+ 1198941198721120591[4 [11987031198722 +1198701 (1198721 +1198722)] +

119872111987221205912

]1205963

minus 4119870211198703sin

2(119896120572

2)

+ [1198701 (1198701 + 21198703) (21198721 +1198722)

+211987211205912

[11987031198722 +1198701 (1198721 +1198722)]] 1205962minus 1198941198701120591(1198701

+ 21198703) (21198721 +1198722) 120596 = 0

(14)


1198911198881 (119896 = 0) = 119894

12120587120591

(lower acoustic)

1198911198882 (119896 = 0) = plusmn

12120587radic1198701 + 211987031198721

minus141205912

+ 1198941

4120587120591


1198911198883 (119896 = 0) = plusmn

12120587radic1198701 (21198721 +1198722)

11987211198722minus

141205912

+ 1198941

4120587120591

(first upper optic)

1198911198884 (119896=

120587

120572) = plusmn

12120587radic11987011198721

minus141205912

+ 1198941

4120587120591

(first lower optic)

1198911198885 (119896=

120587

120572)


211987211198722minus

141205912

+ 1198941


1198911198886 (119896=

120587

120572)

= plusmn12120587radic1198701 (21198721 +1198722) + 211987031198722 + 119891

211987211198722minus

141205912

+ 1198941


(15)



119899







1198911198885 119891

1198884 1198911198883 119891

1198882 1198911198886

304 423 604 671 737



Y = AY+ F (119905) (16)



Y =

(((((((

(

1198861 cos (120596119887119905) + 1198871 sin (120596119887119905)

119886119873

cos (120596119887

119905) + 119887119873

sin (120596119887

119905)

minus1198861120596119887 sin (120596119887119905) + 1198871120596119887 cos (120596119887119905)

minus119886119873

120596119887

sin (120596119887

119905) + 119887119873

120596119887

cos (120596119887

119905)

)))))))

)

(17)


and 119887119895


AX = F (18)

where

A = (

C (120596

120591) I

minus(120596

120591) I C

) (19)

C = minus(C + 1205962119887


119887



4 Main Results



119887

= 63087 kHz which


5 10 15 20

minus001

0

001

n(m

s)

n

(a)

0 004 008 0120

0004

0008

0012

(A)

(D)

(B)

(C)

4(m

s)

a (120583m)

(b)

0 02 04 06 080

002

004

006

008

01

4(m

s)

a (120583m)

(c)






0 10 20minus5

0

5

10

minus1 0 1minus1

minus05

0

05

n

un

(120583m

)

Im(120582

)

Re(120582)

times10minus4

1

(a)

0 10 20minus01

0

01

02

03

minus1 0 1minus1

minus05

0

05

1

n

un

(120583m

)

Im(120582

)

Re(120582)

(b)

0 10 20

minus03

minus015

0

015

03

minus1 0 1minus1

minus05

0

05

1

n

SteelTungsten

un

(120583m

)

Im(120582

)

Re(120582)

(c)

0 10 20

minus04

minus02

0

02

04

minus1 0 1minus1

minus05

0

05

1

n

SteelTungsten

un

(120583m

)

Im(120582

)

Re(120582)

(d)


=





0 5 10 15 20

minus4

minus2

0

2

4

n

times10minus3 n

(ms

)

(a)

0 05 1 150

0005

001

0015

(A)

(B)

(D)

(C)

a (120583m)

4(m

s)

(b)

0 05 1 150

003

006

009

a (120583m)

4(m

s)

(c)



119887


119887




119887




0 10 20

minus003

minus0015

0

0015

003

minus1 0 1minus1

minus05

0

05

1

n

un

(120583m

)

Im(120582

)Re(120582)

(a)

0 10 20minus04

minus02

0

02

04

06

minus1 0 1minus1

minus05

0

05

1

n

un

(120583m

)

Im(120582

)

Re(120582)

(b)

SteelTungsten

0 10 200

05

1

15

minus1 0 1minus1

minus05

0

05

1

n

un

(120583m

)

Im(120582

)

Re(120582)

(c)

SteelTungsten

0 10 200

1

2

3


minus09

0

09

18

n

un

(120583m

)

Im(120582

)

Re(120582)

(d)


=




119887

isin


119887


119887




119899






0 004 008 012minus02

0

02

04

t (ms)

Disp

lace

men

ts (120583

m) u3

u4

(a)

0 01 02minus004

minus002

0

002

004

t (ms)

Disp

lace

men

ts (120583

m)

u3

u4

(b)

49 492 494 496 498 50

minus001

minus0005

0

0005

001

Velo

citie

s (m

s)

t (ms)

15

2Tb

Tb

2

(c)


119887


119887


119887


119887



119899

minus 119906119899+1|1205750119899 = 15 In


119891119887

asymp 235 kHz and 119891119887


119887

asymp




0 1 2 3 4 5 6 7 8 90

0005001

0015002

0025003

0035004


fb (kHz)

4(m

s)

(a)

54 56 58 6 62 64 66 68 70

0005

001

0015

002

0025

003

0035

004

fb (kHz)

4(m

s)

(b)









Appendix



119887


=

1119891119887



x = F (119905 u k) (A1)



119887



119887


119887


119887






119887





119887




Acknowledgments


References






























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198911198885 119891

1198884 1198911198883 119891

1198882 1198911198886

304 423 604 671 737



Y = AY+ F (119905) (16)



Y =

(((((((

(

1198861 cos (120596119887119905) + 1198871 sin (120596119887119905)

119886119873

cos (120596119887

119905) + 119887119873

sin (120596119887

119905)

minus1198861120596119887 sin (120596119887119905) + 1198871120596119887 cos (120596119887119905)

minus119886119873

120596119887

sin (120596119887

119905) + 119887119873

120596119887

cos (120596119887

119905)

)))))))

)

(17)


and 119887119895


AX = F (18)

where

A = (

C (120596

120591) I

minus(120596

120591) I C

) (19)

C = minus(C + 1205962119887


119887



4 Main Results



119887

= 63087 kHz which


5 10 15 20

minus001

0

001

n(m

s)

n

(a)

0 004 008 0120

0004

0008

0012

(A)

(D)

(B)

(C)

4(m

s)

a (120583m)

(b)

0 02 04 06 080

002

004

006

008

01

4(m

s)

a (120583m)

(c)






0 10 20minus5

0

5

10

minus1 0 1minus1

minus05

0

05

n

un

(120583m

)

Im(120582

)

Re(120582)

times10minus4

1

(a)

0 10 20minus01

0

01

02

03

minus1 0 1minus1

minus05

0

05

1

n

un

(120583m

)

Im(120582

)

Re(120582)

(b)

0 10 20

minus03

minus015

0

015

03

minus1 0 1minus1

minus05

0

05

1

n

SteelTungsten

un

(120583m

)

Im(120582

)

Re(120582)

(c)

0 10 20

minus04

minus02

0

02

04

minus1 0 1minus1

minus05

0

05

1

n

SteelTungsten

un

(120583m

)

Im(120582

)

Re(120582)

(d)


=





0 5 10 15 20

minus4

minus2

0

2

4

n

times10minus3 n

(ms

)

(a)

0 05 1 150

0005

001

0015

(A)

(B)

(D)

(C)

a (120583m)

4(m

s)

(b)

0 05 1 150

003

006

009

a (120583m)

4(m

s)

(c)



119887


119887




119887




0 10 20

minus003

minus0015

0

0015

003

minus1 0 1minus1

minus05

0

05

1

n

un

(120583m

)

Im(120582

)Re(120582)

(a)

0 10 20minus04

minus02

0

02

04

06

minus1 0 1minus1

minus05

0

05

1

n

un

(120583m

)

Im(120582

)

Re(120582)

(b)

SteelTungsten

0 10 200

05

1

15

minus1 0 1minus1

minus05

0

05

1

n

un

(120583m

)

Im(120582

)

Re(120582)

(c)

SteelTungsten

0 10 200

1

2

3


minus09

0

09

18

n

un

(120583m

)

Im(120582

)

Re(120582)

(d)


=




119887

isin


119887


119887




119899






0 004 008 012minus02

0

02

04

t (ms)

Disp

lace

men

ts (120583

m) u3

u4

(a)

0 01 02minus004

minus002

0

002

004

t (ms)

Disp

lace

men

ts (120583

m)

u3

u4

(b)

49 492 494 496 498 50

minus001

minus0005

0

0005

001

Velo

citie

s (m

s)

t (ms)

15

2Tb

Tb

2

(c)


119887


119887


119887


119887



119899

minus 119906119899+1|1205750119899 = 15 In


119891119887

asymp 235 kHz and 119891119887


119887

asymp




0 1 2 3 4 5 6 7 8 90

0005001

0015002

0025003

0035004


fb (kHz)

4(m

s)

(a)

54 56 58 6 62 64 66 68 70

0005

001

0015

002

0025

003

0035

004

fb (kHz)

4(m

s)

(b)









Appendix



119887


=

1119891119887



x = F (119905 u k) (A1)



119887



119887


119887


119887






119887





119887




Acknowledgments


References






























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










5 10 15 20

minus001

0

001

n(m

s)

n

(a)

0 004 008 0120

0004

0008

0012

(A)

(D)

(B)

(C)

4(m

s)

a (120583m)

(b)

0 02 04 06 080

002

004

006

008

01

4(m

s)

a (120583m)

(c)






0 10 20minus5

0

5

10

minus1 0 1minus1

minus05

0

05

n

un

(120583m

)

Im(120582

)

Re(120582)

times10minus4

1

(a)

0 10 20minus01

0

01

02

03

minus1 0 1minus1

minus05

0

05

1

n

un

(120583m

)

Im(120582

)

Re(120582)

(b)

0 10 20

minus03

minus015

0

015

03

minus1 0 1minus1

minus05

0

05

1

n

SteelTungsten

un

(120583m

)

Im(120582

)

Re(120582)

(c)

0 10 20

minus04

minus02

0

02

04

minus1 0 1minus1

minus05

0

05

1

n

SteelTungsten

un

(120583m

)

Im(120582

)

Re(120582)

(d)


=





0 5 10 15 20

minus4

minus2

0

2

4

n

times10minus3 n

(ms

)

(a)

0 05 1 150

0005

001

0015

(A)

(B)

(D)

(C)

a (120583m)

4(m

s)

(b)

0 05 1 150

003

006

009

a (120583m)

4(m

s)

(c)



119887


119887




119887




0 10 20

minus003

minus0015

0

0015

003

minus1 0 1minus1

minus05

0

05

1

n

un

(120583m

)

Im(120582

)Re(120582)

(a)

0 10 20minus04

minus02

0

02

04

06

minus1 0 1minus1

minus05

0

05

1

n

un

(120583m

)

Im(120582

)

Re(120582)

(b)

SteelTungsten

0 10 200

05

1

15

minus1 0 1minus1

minus05

0

05

1

n

un

(120583m

)

Im(120582

)

Re(120582)

(c)

SteelTungsten

0 10 200

1

2

3


minus09

0

09

18

n

un

(120583m

)

Im(120582

)

Re(120582)

(d)


=




119887

isin


119887


119887




119899






0 004 008 012minus02

0

02

04

t (ms)

Disp

lace

men

ts (120583

m) u3

u4

(a)

0 01 02minus004

minus002

0

002

004

t (ms)

Disp

lace

men

ts (120583

m)

u3

u4

(b)

49 492 494 496 498 50

minus001

minus0005

0

0005

001

Velo

citie

s (m

s)

t (ms)

15

2Tb

Tb

2

(c)


119887


119887


119887


119887



119899

minus 119906119899+1|1205750119899 = 15 In


119891119887

asymp 235 kHz and 119891119887


119887

asymp




0 1 2 3 4 5 6 7 8 90

0005001

0015002

0025003

0035004


fb (kHz)

4(m

s)

(a)

54 56 58 6 62 64 66 68 70

0005

001

0015

002

0025

003

0035

004

fb (kHz)

4(m

s)

(b)









Appendix



119887


=

1119891119887



x = F (119905 u k) (A1)



119887



119887


119887


119887






119887





119887




Acknowledgments


References






























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20minus5

0

5

10

minus1 0 1minus1

minus05

0

05

n

un

(120583m

)

Im(120582

)

Re(120582)

times10minus4

1

(a)

0 10 20minus01

0

01

02

03

minus1 0 1minus1

minus05

0

05

1

n

un

(120583m

)

Im(120582

)

Re(120582)

(b)

0 10 20

minus03

minus015

0

015

03

minus1 0 1minus1

minus05

0

05

1

n

SteelTungsten

un

(120583m

)

Im(120582

)

Re(120582)

(c)

0 10 20

minus04

minus02

0

02

04

minus1 0 1minus1

minus05

0

05

1

n

SteelTungsten

un

(120583m

)

Im(120582

)

Re(120582)

(d)


=





0 5 10 15 20

minus4

minus2

0

2

4

n

times10minus3 n

(ms

)

(a)

0 05 1 150

0005

001

0015

(A)

(B)

(D)

(C)

a (120583m)

4(m

s)

(b)

0 05 1 150

003

006

009

a (120583m)

4(m

s)

(c)



119887


119887




119887




0 10 20

minus003

minus0015

0

0015

003

minus1 0 1minus1

minus05

0

05

1

n

un

(120583m

)

Im(120582

)Re(120582)

(a)

0 10 20minus04

minus02

0

02

04

06

minus1 0 1minus1

minus05

0

05

1

n

un

(120583m

)

Im(120582

)

Re(120582)

(b)

SteelTungsten

0 10 200

05

1

15

minus1 0 1minus1

minus05

0

05

1

n

un

(120583m

)

Im(120582

)

Re(120582)

(c)

SteelTungsten

0 10 200

1

2

3


minus09

0

09

18

n

un

(120583m

)

Im(120582

)

Re(120582)

(d)


=




119887

isin


119887


119887




119899






0 004 008 012minus02

0

02

04

t (ms)

Disp

lace

men

ts (120583

m) u3

u4

(a)

0 01 02minus004

minus002

0

002

004

t (ms)

Disp

lace

men

ts (120583

m)

u3

u4

(b)

49 492 494 496 498 50

minus001

minus0005

0

0005

001

Velo

citie

s (m

s)

t (ms)

15

2Tb

Tb

2

(c)


119887


119887


119887


119887



119899

minus 119906119899+1|1205750119899 = 15 In


119891119887

asymp 235 kHz and 119891119887


119887

asymp




0 1 2 3 4 5 6 7 8 90

0005001

0015002

0025003

0035004


fb (kHz)

4(m

s)

(a)

54 56 58 6 62 64 66 68 70

0005

001

0015

002

0025

003

0035

004

fb (kHz)

4(m

s)

(b)









Appendix



119887


=

1119891119887



x = F (119905 u k) (A1)



119887



119887


119887


119887






119887





119887




Acknowledgments


References






























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 5 10 15 20

minus4

minus2

0

2

4

n

times10minus3 n

(ms

)

(a)

0 05 1 150

0005

001

0015

(A)

(B)

(D)

(C)

a (120583m)

4(m

s)

(b)

0 05 1 150

003

006

009

a (120583m)

4(m

s)

(c)



119887


119887




119887




0 10 20

minus003

minus0015

0

0015

003

minus1 0 1minus1

minus05

0

05

1

n

un

(120583m

)

Im(120582

)Re(120582)

(a)

0 10 20minus04

minus02

0

02

04

06

minus1 0 1minus1

minus05

0

05

1

n

un

(120583m

)

Im(120582

)

Re(120582)

(b)

SteelTungsten

0 10 200

05

1

15

minus1 0 1minus1

minus05

0

05

1

n

un

(120583m

)

Im(120582

)

Re(120582)

(c)

SteelTungsten

0 10 200

1

2

3


minus09

0

09

18

n

un

(120583m

)

Im(120582

)

Re(120582)

(d)


=




119887

isin


119887


119887




119899






0 004 008 012minus02

0

02

04

t (ms)

Disp

lace

men

ts (120583

m) u3

u4

(a)

0 01 02minus004

minus002

0

002

004

t (ms)

Disp

lace

men

ts (120583

m)

u3

u4

(b)

49 492 494 496 498 50

minus001

minus0005

0

0005

001

Velo

citie

s (m

s)

t (ms)

15

2Tb

Tb

2

(c)


119887


119887


119887


119887



119899

minus 119906119899+1|1205750119899 = 15 In


119891119887

asymp 235 kHz and 119891119887


119887

asymp




0 1 2 3 4 5 6 7 8 90

0005001

0015002

0025003

0035004


fb (kHz)

4(m

s)

(a)

54 56 58 6 62 64 66 68 70

0005

001

0015

002

0025

003

0035

004

fb (kHz)

4(m

s)

(b)









Appendix



119887


=

1119891119887



x = F (119905 u k) (A1)



119887



119887


119887


119887






119887





119887




Acknowledgments


References






























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20

minus003

minus0015

0

0015

003

minus1 0 1minus1

minus05

0

05

1

n

un

(120583m

)

Im(120582

)Re(120582)

(a)

0 10 20minus04

minus02

0

02

04

06

minus1 0 1minus1

minus05

0

05

1

n

un

(120583m

)

Im(120582

)

Re(120582)

(b)

SteelTungsten

0 10 200

05

1

15

minus1 0 1minus1

minus05

0

05

1

n

un

(120583m

)

Im(120582

)

Re(120582)

(c)

SteelTungsten

0 10 200

1

2

3


minus09

0

09

18

n

un

(120583m

)

Im(120582

)

Re(120582)

(d)


=




119887

isin


119887


119887




119899






0 004 008 012minus02

0

02

04

t (ms)

Disp

lace

men

ts (120583

m) u3

u4

(a)

0 01 02minus004

minus002

0

002

004

t (ms)

Disp

lace

men

ts (120583

m)

u3

u4

(b)

49 492 494 496 498 50

minus001

minus0005

0

0005

001

Velo

citie

s (m

s)

t (ms)

15

2Tb

Tb

2

(c)


119887


119887


119887


119887



119899

minus 119906119899+1|1205750119899 = 15 In


119891119887

asymp 235 kHz and 119891119887


119887

asymp




0 1 2 3 4 5 6 7 8 90

0005001

0015002

0025003

0035004


fb (kHz)

4(m

s)

(a)

54 56 58 6 62 64 66 68 70

0005

001

0015

002

0025

003

0035

004

fb (kHz)

4(m

s)

(b)









Appendix



119887


=

1119891119887



x = F (119905 u k) (A1)



119887



119887


119887


119887






119887





119887




Acknowledgments


References






























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 004 008 012minus02

0

02

04

t (ms)

Disp

lace

men

ts (120583

m) u3

u4

(a)

0 01 02minus004

minus002

0

002

004

t (ms)

Disp

lace

men

ts (120583

m)

u3

u4

(b)

49 492 494 496 498 50

minus001

minus0005

0

0005

001

Velo

citie

s (m

s)

t (ms)

15

2Tb

Tb

2

(c)


119887


119887


119887


119887



119899

minus 119906119899+1|1205750119899 = 15 In


119891119887

asymp 235 kHz and 119891119887


119887

asymp




0 1 2 3 4 5 6 7 8 90

0005001

0015002

0025003

0035004


fb (kHz)

4(m

s)

(a)

54 56 58 6 62 64 66 68 70

0005

001

0015

002

0025

003

0035

004

fb (kHz)

4(m

s)

(b)









Appendix



119887


=

1119891119887



x = F (119905 u k) (A1)



119887



119887


119887


119887






119887





119887




Acknowledgments


References






























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2 3 4 5 6 7 8 90

0005001

0015002

0025003

0035004


fb (kHz)

4(m

s)

(a)

54 56 58 6 62 64 66 68 70

0005

001

0015

002

0025

003

0035

004

fb (kHz)

4(m

s)

(b)









Appendix



119887


=

1119891119887



x = F (119905 u k) (A1)



119887



119887


119887


119887






119887





119887




Acknowledgments


References






























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Appendix



119887


=

1119891119887



x = F (119905 u k) (A1)



119887



119887


119887


119887






119887





119887




Acknowledgments


References






























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Time-Periodic Solutions of Driven-Damped