Wavenumber-space band clipping in nonlinear periodic structures
WWavenumber-space band clipping in nonlinear periodic structures
Weijian Jiao ∗ and Stefano Gonella † Department of Civil, Environmental, and Geo- EngineeringUniversity of Minnesota, Minneapolis, MN 55455, USA
Abstract
In weakly nonlinear systems, the main effect of cubic nonlinearity on wave propagation is an amplitude-dependentcorrection of the dispersion relation. This phenomenon can manifest either as a frequency shift or as a wavenumbershift depending on whether the excitation is prescribed as a initial condition or as a boundary condition, respectively.Several models have been proposed to capture the frequency shifts observed when the system is subjected to harmonicinitial excitations. However, these models are not compatible with harmonic boundary excitations, which representthe conditions encountered in most practical applications. To overcome this limitation, we present a multiple scalesframework to analytically capture the wavenumber shift experienced by dispersion relation of nonlinear monatomicchains under harmonic boundary excitations. We demonstrate that the wavenumber shifts result in an unusual dispersioncorrection effect, which we term wavenumber-space band clipping. We then extend the framework to locally-resonantperiodic structures to explore the implications of this phenomenon on bandgap tunability. We show that the tuningcapability is available if the cubic nonlinearity is deployed in the internal springs supporting the resonators.
Recently, nonlinear periodic structures and metamaterials have drawn great attention due to their rich dynamic responseand uniquely tunable dispersive properties that endow them with a wider functionality space compared to their linearcounterparts. For example, metamaterials manufactured using soft materials can undergo large deformation that can beexploited to trigger strong nonlinear effects, including dramatic geometrical transformations (Bar-Sinai et al., 2020) andinstabilities (Wang et al., 2014). Similar nonlinear effects can also be achieved through Hertzian contact interactions ingranular crystals (Daraio et al., 2006; Boechler et al., 2010) and magnetic interactions in magneto-mechanical systems (Bilalet al., 2017). A special attribute of nonlinear systems is the tunability of their dynamic properties, which can overcomethe inherent passivity of linear phononic crystals and metamaterials (a review of these effects is given by Bertoldi et al.,2017). Nonlinearity can also be exploited to enable solitary wave propagation in soft metamaterials (Raney et al., 2016;Deng et al., 2017; Ziv and Shmuel, 2020).In weakly nonlinear systems, a well-known phenomenon associated with quadratic nonlinearity is the second harmonicgeneration (SHG). While SHG has been widely studied and observed in conventional solids and structures (Polyakova,1964; De Lima and Hamilton, 2003; Matlack et al., 2011; and an excellent review can be found in Hamilton et al., 1998),its implications for periodic structures has only come to prominence in recent years. Among the early contributions in thisdirection, we recall the work on SHG in monatomic chains (S´anchez-Morcillo et al., 2013; Mehrem et al., 2017). A numberof works have presented applications of this (or similar) nonlinear phenomenon germane to the metamaterial paradigm,including modal mixing (Ganesh and Gonella, 2017), acoustic diodes and switches (Liang et al., 2010; Boechler et al.,2011), and subwavelength energy harvesting (Jiao and Gonella, 2018a; Jiao and Gonella, 2018b).Other weakly nonlinear systems of interest are those featuring weak cubic nonlinearity. The main effect of cubicnonlinearity on wave propagation is an amplitude-dependent correction of the dispersion relation, which can manifesteither as a frequency shift or as a wavenumber shift depending on the variable that is controlled in the excitation (Jiaoand Gonella, 2019). For example, if the excitation is prescribed as an initial condition (i.e., an initial spatial profileprescribed over the domain), one can consider the wavenumber as a fixed externally-controlled parameter. In this case,the cubic nonlinear effect manifests as a frequency shift. In contrast, working with a boundary excitation implies thatthe input frequency can be treated as the fixed parameter, and therefore the effect has to manifest as a wavenumbershift. Various perturbation techniques have been employed to predict the dispersion shifts experienced by harmonic wavesin cubic nonlinear chains. Chakraborty and Mallik (2001) proposed a perturbation scheme to determine the amplitude-dependent characteristic of the frequency cutoffs of nonlinear monoatomic chains. Lazarov and Jensen (2007) employed the ∗ [email protected] † [email protected] a r X i v : . [ n li n . PS ] S e p ethod of harmonic balance to study the shift of bandgaps in chains with attached nonlinear resonators. Later, Narisettiet al. (2010) derived an explicit expression for frequency shifts using the Lindstedt-Poincar´e perturbation technique. Amore general analytical treatment based on multiples scales analysis can be found in textbooks on perturbation methods(Holmes, 2012), leading to a consistent result for frequency shifts if harmonic initial excitations are prescribed. In addition,these analytical tools have been applied to higher-dimensional and multi-degree of freedom periodic structures to capturethe spectro-spatial effects of the tuning of their dispersive properties (Narisetti et al., 2011; Manktelow et al., 2013;Bukhari and Barry, 2020).While the bulk of the literature on the subject is focused on frequency shifts because of the mathematical tractability ofthe problem, the theoretical framework cannot be automatically transported to the dual scenarios in which the excitationis prescribed as harmonic oscillations at the boundaries. This condition is of greater interest for practical engineeringapplications, in which the excitation is indeed prescribed using a point source through an actuation device, such asa shaker or transducer. Adapting an framework previously introduced to study doubly-nonlinear systems that featuresimultaneously quadratic and cubic nonlinearities (Jiao and Gonella, 2019), in this paper we use a multiple scales schemeto properly determine the dispersion correction of weakly nonlinear chains under boundary excitations (i.e., wavenumbershifts). In Section 2, we demonstrate that cubic nonlinearity, in combination with harmonic boundary excitations, givesrise to unusual dispersive properties that are fundamentally different from those associated with frequency shifts, andall these findings are supported by numerical simulations. In Section 3, the framework is extended to locally-resonantperiodic structures, in which the additional degree of freedom induced by the internal resonator provides opportunities toexplore the potential of wavenumber shifts for bandgap tunability under practical excitation constrains.
In this section, we develop a multiple scales framework to properly capture the dispersion relation of a nonlinear monatomicspring-mass chain under both initial and boundary excitations, and we investigate how different excitation conditions cansignificantly change the manifestation of cubic nonlinearity on the dispersive properties.
Consider an infinite monatomic chain in which adjacent masses are connected by springs featuring cubic nonlinearity.Under the assumption of weak nonlinearity, the restoring force in each spring can be expressed as f = kδ + (cid:15) Γ δ (1)where δ denotes the relative displacement, (cid:15) is a small parameters, and k and Γ are the linear and cubic spring constants,respectively. The equation of motion for the n th mass m can be derived as m ¨ u n + k (2 u n − u n − − u n +1 ) + (cid:15) Γ (cid:104) ( u n − u n − ) − ( u n +1 − u n ) (cid:105) = 0 (2)where u n represents the displacement of the n th mass and the superscripted dot denotes time differentiation.In the spirit of multiple scales analysis, we introduce a fast spatio-temporal variable θ n = ξn − ωt (where ξ and ω are the normalized wavenumber and frequency, respectively) to capture the fundamental wave response, and two slowvariables s = (cid:15)n (spatial), and τ = (cid:15)t (temporal) to capture the weakly nonlinear effects (i.e., dispersion corrections).Accordingly, the solution is assumed to have an expansion of the form u n = u n ( θ n , s, τ ) + (cid:15)u n ( θ n , s, τ ) + O ( (cid:15) ) (3)Substituting Eq. 3 into Eq. 2, we obtain the equations at each order of expansion: O (1) : ω m ∂ u n ∂θ n + k (2 u n − u n − − u n +1 ) = 0 (4) O ( (cid:15) ) : ω m ∂ u n ∂θ n + k (2 u n − u n − − u n +1 ) = f (5)where the forcing function f at O ( (cid:15) ) is given as f = 2 ωm ∂ u n ∂θ n ∂τ + k (cid:32) ∂u n +1 ∂s − ∂u n − ∂s (cid:33) − Γ (cid:104) ( u n − u n − ) − ( u n +1 − u n ) (cid:105) (6)The general solution at O (1) can be expressed as u n = A ( s, τ )2 e iθ n + A ∗ ( s, τ )2 e − iθ n (7)2here A is an arbitrary function of the slow variables ( s, τ ), and ( · ) ∗ denotes the complex conjugate. Imposing Blochconditions on the fast scale variable θ n between neighboring masses, we obtain u n ± = A ( s, τ )2 e iθ n e ± iξ + A ∗ ( s, τ )2 e − iθ n e ∓ iξ (8)Substituting Eq. 7 and Eq. 8 into Eq. 4, the linear dispersion relation is obtained ω = (cid:112) k (1 − cos( ξ )) /m (9) Linear dispersion curvePredicted by Eq.14Predicted by Eq.15
FIGURE 1: Dispersion relation corrections predicted by Eq. 14 and Eq. 15, compared with the linear dispersion relation of amonatomic chain.
We now shift our attention to the equation at O ( (cid:15) ), in which the forcing function f can be obtained by substitutingEq. 7 and Eq. 8 in Eq. 6. To prevent unbounded solution, the secular terms (i.e., the terms of the form e ± iθ n ) appearingin f must be eliminated, leading to the following condition ∂A∂τ + λ ∂A∂s + iµ | A | A = 0 (10)where λ = k sin ξ/ωm and µ = 6Γ sin ( ξ/ /ωm . The solution of A can be written in a polar form as A ( s, τ ) = α ( s, τ ) e − iβ ( s,τ ) (11)Substituting it into Eq. 10, yields a complex-variable algebraic equation, the solution of which requires that real andimaginary components vanish individually such that ∂α∂τ + λ ∂α∂s = 0 ∂β∂τ + λ ∂β∂s = µα (12)The general solutions of these equations are α = α ( s − λτ ) β = β ( s − λτ ) + β ∗ (13)The full expression for β encompasses a homogeneous solution β and a particular solution β ∗ , the latter of which can beexpressed either in terms of variable τ as β ∗ = µα τ , or in terms of variable s as β ∗ = µα s/λ , depending on whether initialconditions or boundary conditions are considered. For example, given the initial harmonic amplitude profile u n = A cos ξn at t = 0, it follows that α = A , β = 0 and β ∗ = µα τ . Thus, the fundamental solution at O (1) is u n = A e i (cid:104) ξn − ( ω + (cid:15)µA ) t (cid:105) + c.c. (14)3here c.c. denotes the complex conjugate of the preceding term. Eq. 14 shows that the nonlinear equation of motion(i.e., Eq. 2) produces a fundamental plane-wave solution, in which the frequency is modified by an amplitude-dependentcorrection term (cid:15)µA . This cubic nonlinear effect is well documented in mathematics textbooks (Holmes, 2012), as well asin literature of nonlinear periodic structures (Narisetti et al., 2010). In contrast, if a boundary condition u n = A cos ωt is imposed at one end of the chain (e.g., at n = 0), it follows that α = A , β = 0 and β ∗ = µα s/λ . With this, thefundamental solution becomes u n = A e i (cid:104) ( ξ − (cid:15)µA /λ ) n − ωt (cid:105) + c.c. (15)in which the correction term (cid:15)µA /λ takes place in the wavenumber domain.Clearly, either frequency shifts or wavenumber shifts can modify the dispersive properties of the nonlinear monatonicchain. To demonstrate their tuning effects on the dispersion relation, we plot the modified dispersion relations predictedby Eq. 14 and Eq. 15, in comparison with the linear one, using the following parameters: m = 1, k = Γ = 1, (cid:15) = 0 . A = 0 . ξ increases. While the twononlinear curves overlap for a large range of frequencies, the one predicted by Eq. 15 diverges when ω is close enough tothe cutoff frequency (2 rad/s). This divergence issue implies that the weakly nonlinear assumptions do not hold any more,and therefore the predictions of Eq. 15 are spurious. Linear dispersion curveIE-type NDRBE-type NDR
FIGURE 2: Corrected dispersion relations of a monatomic chain featuring hardening cubic nonlinearity obtained under differentexcitation conditions (a detailed comparison near the cutoff frequency is given in the inset).
To resolve this issue, we follow the approach proposed by Jiao and Gonella (2019), the basic notion of which is that theBloch condition (Eq. 8), which imposes a constrain on the wavenumber between neighboring masses, needs to be updatedonce a wavenumber correction is determined from the perturbation analysis. Specifically, the original wavenumber ξ should be replaced by the modified one ξ − (cid:15)β/s (this can be obtained by substituting Eq. 11 in Eq. 7 and collecting allthe wavenumber contributions). This treatment results in a new condition (replacing Eq. 10) for the elimination of thesecular terms at O ( (cid:15) ): ∂A∂τ + λ sin (cid:18) ξ − (cid:15) βs (cid:19) ∂A∂s + iµ | A | A = 0 (16)where λ = kωm . Substituting Eq. 11 in Eq. 16 yields the following equations for α and β∂α∂τ + λ sin (cid:18) ξ − (cid:15) βs (cid:19) ∂α∂s = 0 ∂β∂τ + λ sin (cid:18) ξ − (cid:15) βs (cid:19) ∂β∂s = µα (17)It can been shown that the above system of equations reduces to Eq. 12 to the first order approximation for cases wherethe wavenumber correction (cid:15)β/s is at a higher order when compared to ξ . However, it is possible that, for some specialcases, Eq. 17 gives profoundly different results from those solved by Eq. 12 as demonstrated below. Considering the sameboundary excitation condition, it is reasonable to assume a priori, if a plane wave solution is allowed, that α = A , β = 04nd β ∗ = Cs = (cid:15)Cn , where C is a real constant. Here, (cid:15)C is the “true” wavenumber shift that we intend to determine.Substituting these into Eq. 17, yields the following transcendental equation for Cλ C sin ˜ ξ = µA (18)where ˜ ξ = ξ − (cid:15)C . While Eq. 18 is not amenable for analytical treatments, it is not difficult to find possible solutionsnumerically. Once C is determined, the fundamental solution can be expressed as u n = A e i [ ( ξ − (cid:15)C ) n − ωt ] + c.c (19)For the same parameters used above, in Fig. 2 we plot the corrected dispersion relation according to Eq. 19, and we alsosuperimpose the linear dispersion curve and the nonlinear one predicted by Eq. 14, for comparison. We notice that thedivergence observed in Fig. 1 for frequencies close to 2 rad/s is successfully resolved. The two nonlinear dispersion curvesshown in Fig. 2 predict distinct dispersive characteristics near the cutoff frequency. Specifically, the cubic nonlinear effectmanifests as frequency shifts when initial excitations are imposed, leading to an extension of the dispersion relation inthe frequency domain (green dotted curve). In contrast, wavenumber shifts are induced when boundary excitations areimposed. As a consequence, the dispersion relation is clipped in wavenumber space near the π limit (black dash-dottedcurve). This wavenumber-space clipping effect is qualitatively consistent with observations recently reported by Bae andOh (2020) invoking different modeling arguments. However, as shown in the inset of Fig. 2, the frequency range of thepassing band is unaffected. For convenience, we will refer to the former nonlinear dispersion relation as IE-type NDR(IE standing for “initial excitation”) and to the latter one as BE-type NDR (BE standing for“boundary excitation”)throughout this article. Linear dispersion curveIE-type NDRBE-type NDR
FIGURE 3: Corrected dispersion relations of a monatomic chain featuring softening cubic nonlinearity obtained under differentexcitation conditions (a detailed comparison near the cutoff frequency is given in the inset).
In the previous example, we investigated the dispersion characteristics of a monatomic chain featuring hardening cubicnonlinearity with Γ = 0 .
1. We now proceed to examine the case with softening cubic nonlinearity (Γ = − . − .
1, which is plotted as a black dashed curvein Fig. 3 and compared against the IE-type NDE counterpart (green dotted curve). Again, we observe the band clippingeffect near the π limit. However, in contrast with the previous case, the cutoff frequency of the chain is modified usingboth nonlinear corrections. Compared to the IE-type NDR, the BE-type NDR further lowers the cutoff frequency for thesame excitation amplitude, thus enabling a lager degree of nonlinear tuning. To validate the above theoretical findings, we perform numerical simulations by integrating the equations of motion (Eq. 2)by means of the Verlet Algorithm (Swope et al., 1982). We study the wave response of a monatomic chain with hardeningcubic nonlinearity (with Γ = 0 . − .
1) at one excitation frequency (1.8 rad/s), and the corresponding result is given in Fig. 5.It can be seen that the simulation result reveals a right-shifted wavenumber, which is in excellent agreement with thetheoretical predication (red dashed bar). As an interesting side observation, we notice that the numerical simulation coulddiverge from the analytical model and multi-frequency quasi-periodic response would emerge if the harmonic excitationis set sufficiently close to the cutoff frequency and beyond certain amplitude thresholds (similar phenomena are reportedby Boechler et al. (2011)). A m p lit ud e (a) (b) (c) A m p lit ud e A m p lit ud e A m p lit ud e A m p lit ud e A m p lit ud e SimulationExcitation frequency SimulationExcitation frequency SimulationExcitation frequencySimulationLinearNonlinear SimulationLinearNonlinearSimulationLinearNonlinear
FIGURE 4: Spectra of the nonlinear response of a monatomic chain featuring hardening cubic nonlinearity (Γ = 0 .
1) under harmonicboundary excitations with three different frequencies: (a) 1.5 rad/s; (b) 1.6 rad/s; (c) 1.7 rad/s. First row: FFT of the outputsignal. Second row: FFT of the spatial profile after appropriately long simulation time.
The next task is to verify that the BE-type NDR preserves the cutoff frequency of the linear case. A characteristic thatmakes it distinct from the IE-type NDR for cases with hardening cubic nonlinearity. To this end, we set the excitationfrequency to 2.05 rad/s, which is above the linear cutoff frequency (2 rad/s) but below that of the IE-type NDR (2.075rad/s). The corresponding nonlinear response is shown in Fig. 6, in which we plot the input and output as functions oftime in Fig. 6(a), as well as the spatial profile in Fig. 6(b). From a visual inspection, we observe a low output-input ratioand a clear spatial attenuation, which indicate the establishment of bandgap conditions further confirming the validity ofthe BE-type NDR obtained from the analytical model.
Metamaterials featuring internal resonators are of great interest for their ability to open locally-resonant bandgap at lowfrequencies (Liu et al., 2000), as well as their implications for wave manipulation, including negative refraction (Zhu et al.,2014), subwavelength wave steering (Celli and Gonella, 2015), and seismic shielding (Colombi et al., 2016). In this section,we extend the multiple scales framework to nonlinear locally-resonant periodic structures to explore the availability ofbandgap tunability under the practical constrain of boundary excitations.
Consider a periodic structure featuring internal resonators, which can be conceptually modeled as a mass-in-mass chain(as depicted in Fig. 7). Cubic nonlinearity can be incorporated either in the springs connecting the neighboring massesof the main chain (configuration referred to as system A), or in the internal springs attached to the internal resonators6 A m p lit ud e SimulationExcitation frequency A m p lit ud e SimulationLinearNonlinear (a) (b)
FIGURE 5: Spectra of the nonlinear response of a monatomic chain featuring softening cubic nonlinearity (Γ = − .
1) underharmonic boundary excitations at 1.8 rad/s. (a) FFT of the output signal. (b) FFT of the spatial profile after appropriately longsimulation time. -0.500.5 InputOutput (a) (b) A m p lit ud e A m p lit ud e t n FIGURE 6: Nonlinear response of a monatomic chain under harmonic boundary excitation at frequency 2.05 rad/s, lying above thecutoff of the linear system but below that of the branch endowed with frequency correction. (a) Input signal vs. output signal. (b)Spatial profile after appropriately long simulation time. Both data sets suggest attenuation compatible with bandgap conditions,confirming that a frequency shift of the dispersion branch is not observed for boundary excitations. (system B). To obtain the BE-type NDR of the two systems, we extend the multiple scales analysis of the monatomicchain to the two-degree-of-freedom problem of the mass-in-mass chain.The equations of motion for the two nonlinear mass-in-mass systems can be written in the matrix form as (cid:20) m m (cid:21) (cid:26) ¨ u n ¨ v n (cid:27) + (cid:20) k + k − k − k k (cid:21) (cid:26) u n v n (cid:27) + (cid:20) − k
00 0 (cid:21) (cid:26) u n − v n − (cid:27) + (cid:20) − k
00 0 (cid:21) (cid:26) u n +1 v n +1 (cid:27) + f A(B) NL = (cid:26) (cid:27) (20)in which f ANL = (cid:15) Γ (cid:26) ( u n − u n − ) − ( u n +1 − u n ) (cid:27) for system A and f BNL = (cid:15) Γ (cid:26) ( u n − v n ) − ( u n − v n ) (cid:27) for system B, and u n and v n denote the displacements of m and m in the n th unit cell, respectively. The nodal displacements can be expressedup to the first order approximation as u n = (cid:26) u n ( t ) v n ( t ) (cid:27) = (cid:26) u n ( θ n , s, τ ) + (cid:15)u n ( θ n , s, τ ) v n ( θ n , s, τ ) + (cid:15)v n ( θ n , s, τ ) (cid:27) (21)7 System ASystem B k Γ k k k Γ FIGURE 7: Schematic of nonlinear mass-in-mass chains. System A: a mass-in-mass chain with nonlinear springs connecting themasses m . System B: a mass-in-mass chain with nonlinear springs attached to the local resonators m . By substituting Eq. 21 into Eq. 20, we obtain the following system of cascading equations O (1) : ω M ∂ u n ∂θ n + K u n + K u n − + K u n +1 = (22) O ( (cid:15) ) : ω M ∂ u n ∂θ n + K u n + K u n − + K u n +1 = F A(B) (23)where M = (cid:20) m m (cid:21) is the mass matrix; K = (cid:20) k + k − k − k k (cid:21) ; K = (cid:20) − k
00 0 (cid:21) and the forcing term at O ( (cid:15) ) is F A(B) = 2 ω M ∂ u n ∂θ n ∂τ − q A(B) + k (cid:18) ∂u n +1 ∂s − ∂u n − ∂s (cid:19) (24)in which q A = Γ (cid:26) ( u n − u n − ) − ( u n +1 − u n ) (cid:27) for system A and q B = Γ (cid:26) ( u n − v n ) − ( u n − v n ) (cid:27) for system B.A plane wave solution at O (1) can be assumed to have the form u n = A ( s, τ )2 φ e iθ n + A ∗ ( s, τ )2 φ ∗ e − iθ n (25)where φ = (cid:26) φ u φ v (cid:27) is a modal eigenvector. Accordingly, the Bloch condition (Eq. 8) becomes u n ± = A ( s, τ )2 φ e iθ n e ± iξ + A ∗ ( s, τ )2 φ ∗ e − iθ n e ∓ iξ (26)Substituting Eq. 25 and Eq. 26 into Eq. 22 yields the following eigenvalue problem (cid:16) − ω M + K ( ξ ) (cid:17) φ = (27)where K ( ξ ) = (cid:20) k (1 − cos ξ ) + k − k − k k (cid:21) . The linear dispersion relation of a mass-in-mass chain can be obtained bysolving the above eigenvalue problem, which can be analytically expressed as ω , = (cid:114) b ∓ ∆( ξ )2 a (28)where ∆( ξ ) = √ b − ac , with a = m m , b = ( m + m ) k + 2 m k (1 − cos ξ ), and c = 2 k k (1 − cos ξ ), and ω and ω denote the acoustic and optical branch, respectively. The corresponding normalized eigenvector is given by φ , = (cid:40) − ω , m + k k (cid:41) / (cid:113) ω , m − k m ω , + 2 k (29)8 Linear dispersion curveBE-type NDR Linear dispersion curveBE-type NDR (a) (b) B a ndg a p t un i ng FIGURE 8: Corrected dispersion relations of (a) system A with nonlinearity in the main chain and (b) system B with nonlinearityin the local resonators. The band clipping effect gives rise to a bandgap tuning functionality in system B, which is not available insystem A.
We now follow the same logic described in the section 1.1 to derive an equation for the elimination of the secularterms at O ( (cid:15) ). For multi-degree-of-freedom systems, we can write the solution u n as a superposition of the normal modes: u n = Φz n , where Φ is the modal matrix featuring the modal vectors φ as columns (Meirovitch, 2010; Manktelow et al.,2013). Premultiply Eq. 23 by φ H , where ( · ) H denotes conjugate transpose, and note that its linear kernel is the same asthe linear kernel of Eq. 23, leading to ω ¯ m ∂ z n ∂θ n + ¯ kz n = φ H F A(B) (30)where ¯ m = φ H M φ and ¯ k = φ H K φ are the modal mass and stiffness. The elimination of the secular terms in the RHSof Eq. 30 leads to the required condition using the updated wavenumber ξ − (cid:15) βs ∂A∂τ + λ sin (cid:18) ξ − (cid:15) βs (cid:19) ∂A∂s + iµ A(B) | A | A = 0 (31)where λ = k | φ u | ω ¯ m , µ A = 6Γ φ u sin ( ξ/ /ω ¯ m for system A, and µ B = 3Γ( φ u − φ v )( φ u − φ v + φ u φ v − φ u φ v ) / ω ¯ m for systemB. Repeating the procedure that leads to Eq. 18, we derive a similar transcendental equation, from which the wavenumbershift can be determined numerically. Then, we can easily construct the BE-type NDR of the two nonlinear systems. Fora set of parameters selected as m = 1, m = 0 . k = k = Γ = 1, (cid:15) = 0 .
1, and A = 0 .
5, we plot in Fig. 8 the BE-typeNDR of system A in comparison with that of system B, and we superimpose their linear dispersion relations for reference.As expected, in the BE-type NDR of system A, we observe band clipping similar to that observed in the monoatomicchain with a wavenumber shift that increases as ξ approaches π and no influence on the bandgap bounds. Interestingly,for system B, in addition to the π limit, clipping also appears at the origin of the optical branch. As a result, the brancheffectively starts at a finite wavenumber and the shift is accompanied by an upward shift of the cut-on frequency, resultingin a bandgap extension. This dispersive characteristic uniquely germane to system B endows it with a bandgap tuningfunctionality that is unachievable in system A where the nonlinearity is implemented in the main chain. We perform a suite of numerical simulations to demonstrate the bandgap tuning capability of system B and the lackthereof of system A. The excitation frequency is set at 1.76 rad/s, which is located in the range of the bandgap extensionshown in the inset of Fig. 8(b). First, in Fig. 9(a) and (b), we plot the temporal and spatial response of system Aand B, respectively, for the same high-amplitude excitation ( A = 0 . A = 0 . × − ). We observe that the attenuation effect is switched off, which furtherdemonstrates that the bandgap of system B can be tuned by controlling of the excitation amplitude.9
10 15 20 25 30 35 40 n -0.3-0.2-0.100.10.20.30 2 4 6 8 10 t -0.500.5 A m p lit ud e -0.500.5 5 10 15 20 25 30 35 40-0.3-0.2-0.100.10.20.3 InputOutput InputOutput (c)(a) (b) -505 10 -4 -4 InputOutput A m p lit ud e A m p lit ud e A m p lit ud e A m p lit ud e A m p lit ud e n nt t FIGURE 9: Temporal and spatial response of (a) system A and (b) system B for high amplitude of excitation A = 0 .
6, and (c)system B for low amplitude A = 0 . × − , under harmonic boundary excitation at frequency 1.76 rad/s. First row: Input signalvs output signal. Second row: Spatial profile after appropriately long simulation time, indicating that attenuation is activated onlyfor system B under high amplitude of excitation. In summary, we have presented a general framework based on multiple scales analysis to properly capture the dispersiveproperties of weakly nonlinear periodic structures. Through this framework, we have revisited the benchmark problemof a cubic nonlinear monatomic chain, demonstrating that the cubic nonlinearity manifests as wavenumber shifts underharmonic boundary excitations in contrast to the typical frequency shifts observed when harmonic initial excitationsare prescribed. Moreover, we have shown that wavenumber shifts have a strong influence on the dispersion relation,resulting in wavenumber-space band clipping. Then, we have extended the framework to nonlinear locally-resonantperiodic structures to explore the bandgap tuning potential resulting from the band clipping effect. We have determinedthat bandgap tunability is available in systems where the cubic nonlinearity is introduced in the internal springs supportingthe resonators, while no such effect arises in the presence of cubic nonlinearity in the main chain.
Acknowledgement
The authors acknowledge the support of the National Science Foundation (CAREER Award CMMI-1452488).
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