Nonlinear harmonic generation in two-dimensional lattices of repulsive magnets
NNonlinear harmonic generation in two-dimensional lattices of repulsive magnets
Weijian Jiao and Stefano Gonella
Department of Civil, Environmental, and Geo- EngineeringUniversity of Minnesota, Minneapolis, MN 55455, USA
In this Letter, we provide experimental evidence of nonlinear wave propagation in a triangularlattice of repulsive magnets supported by an elastic foundation of thin pillars and we interpret allthe individual features of the nonlinear wavefield through the lens of a phonon band calculation thatprecisely accounts for the inter-particle repulsive forces. We confirm the co-existence of two spec-trally distinct components (homogeneous and forced) in the wave response that is induced via secondharmonic generation (SHG), a well-known effect of quadratic nonlinearity (here embedded in themagnetic interaction). We show that the modal and spatial characteristics of the second harmoniccomponents are complementary to those exhibited by the fundamental harmonic. This endows thelattice with a functionality enrichment capability, whereby additional modes and directivity patternscan be triggered and tuned by merely increasing the amplitude of excitation.
In recent years, nonlinear periodic structures andacoustic metamaterials have been extensively studied be-cause of their rich dynamical behavior and for their tun-ability and adaptivity characteristics. A number of stud-ies have focused on the propagation of solitary wavesand discrete breathers in a variety of material systems,such as granular crystals [1–4], magnetic systems [5–7], and mechanical metamaterials [8–10]. Other no-table works have explored metastructures equipped withbistable or bucklable elements and exhibiting tuning andenergy harvesting functionalities [11–14]. In general,achieving these nonlinear effects requires activating thestrongly nonlinear response associated with large defor-mation that is achievable, for instance, working with softmaterials or thin structures.Tuning effects can also be triggered and harnessed inweakly nonlinear systems, and their implications havebeen investigated for wave control of phononic media. Inthe case of weak cubic nonlinearity, the main manifes-tation of nonlinearity is an amplitude-dependent correc-tion of the dispersion relation, which, in principle, en-ables shifting the onset and width of bandgaps via asimple control of the excitation amplitude [15–18]. Re-cently, this tuning effect has been employed to controledge states in topological phononic lattices [19]. Anotherweakly nonlinear effect of great relevance is the secondharmonic generation (SHG), which is the main signatureof quadratic nonlinearity - a dominant contribution inmany nonlinear physical systems [17, 20–28]. Harness-ing SHG in nonlinear acoustic metamaterials has openednew doors for a broad range of applications, includingacoustic diodes and switches [29, 30], subwavelength en-ergy trapping [25], and adaptive spatial directivity [31].The opportunity spectrum gets even wider if we considersystems that feature simultaneously cubic and quadraticnonlinearities, where the correction of the band diagraminduced by cubic nonlinearity affects indirectly the man-ifestation of SHG, providing a secondary tuning capabil-ity, as recently shown in [32]. While SHG has been widely studied in one-dimensionalnonlinear metamaterials and waveguides, the investiga-tion of its effects on the spatial characteristics of nonlin-ear wavefields in 2D metamaterials has been more spo-radic [26, 31] and still lacks a definitive experimental ob-servation of nonlinear response at amplitude levels thatare suitable for practical applications. In this Letter, weattempt to bridge this gap by experimentally investigat-ing SHG in a discrete system consisting of a periodicnetwork of repulsively interacting magnets supported byan elastic foundation of thin pillars. The system can beinterpreted as a practical realization of a triangular lat-tice of particles with on-site potentials. Specifically, wefirst verify the existence of SHG in the spectrum of theresponse. To this end, we take advantage of the strengthof the nonlinear effects granted by the particle nature ofthe system, compared to the case of structural lattices,even without the establishment of phase matching condi-tions. Moreover, we experimentally confirm that the sec-ond harmonic encompasses two contributions, tradition-ally referred to as the forced and homogeneous compo-nents [24, 26]. Finally, we show that the second harmonicfeatures distinctive and complementary modal and spa-tial characteristics when we compare the response withthat of the fundamental harmonic.The specimen used for these tasks, shown in Fig. 1(a),consists of an array of pillars arranged to form a tri-angular lattice occupying a half-hexagon domain. Eachpillar consists of a magnetic ring (Grade N42, with 1/4inch outer diameter, 1/16 inch inter diameter and 1/8inch thickness) inserted at the tip of a slender Aluminumbeam whose other end is plugged into an Acrylic basethrough a drilled hole. The pillars in the interior of thelattice feature slender beams (1/16 inch in diameter).For the exterior pillars, the magnets are simply glued tothe tips of thick beams (1/4 inch in diameter) featur-ing large bending stiffness to effectively establish fixedboundary conditions along the contour of the hexagonaldomain. The magnets are arranged as to experience side- a r X i v : . [ n li n . PS ] J u l FIG. 1. (a) Lattice specimen consisting of magnets supportedby thin beams in the interior (shown in the inset) and thickbeams along the boundary. The interior magnets are coveredby reflective tape to enhance laser measurements. (b) Equiv-alent spring-mass model of the lattice, showing the functionof the pillars acting as an elastic foundation. (c) Linear dis-persion relation obtained from the analytical model in [33],accounting for the effect of the static inter-particle repulsiveforces. by-side repulsive interaction in their own plane, and eachmagnet is initially in equilibrium under the action of theself-balancing static forces exerted by its neighbors. Theconfiguration guarantees that, for the amplitudes of in-terest for this study, the motion of the magnets remainsconfined within the plane of the lattice. This setup is im-ported and adapted from a previous experimental effort,in which we used this platform to characterize the linearresponse of lattices of magnetically interacting particlesystems [33]. Through those experiments, we were ableto demonstrated a series of non-intuitive correction ef-fects induced on the lattice band structure by the staticinter-particle repulsive forces. In this work, we leveragethese key results as a precious guideline for the nonlinearinvestigation.As shown in Fig. 1(b), the system is modeled as atriangular spring-mass lattice in which each node is con-nected to ground through a flexural spring that capturesthe elastic foundation effect of the supporting beam. Thein-plane repulsive interaction between neighboring mag-nets is modeled as a nonlinear spring featuring an in-verse power law f ( r ) = βr − α , with α = 4 . β = 1 . × − . Here, the parameters have beenobtained by fitting the force-displacement relation be-tween two magnets acquired experimentally through amicrometer equipped with a highly sensitive load cell (see details in the SI of [33]). The spring constant of the flex-ural springs in the foundation is taken as the equivalentflexural stiffness of a cantilever beam with the cross sec-tional and material properties of the pillar, and foundto be k f = 19 . m = 7 . × − kg is the mass of each magnet and L = 0 .
01 m is the initial spacing between two nodes inthe lattice. In our previous study [33], we have shownthat the correct dispersive properties of repulsive latticescannot be resolved using conventional harmonic spring-mass models. Specifically, we have demonstrated thatthe static repulsive forces due to the magnets, whichconstraint the lattice at equilibrium, can induce largesoftening corrections of dispersion branches - an effectespecially felt by the shear mode. Adapting the result in[33], the band diagram of a repulsive lattice with an elas-tic foundation can be obtained by solving the followingeigenvalue problem (cid:104) − ω M + D ( k ) (cid:105) φ = (1)where ω = 2 πν , k is the wavevector, M = (cid:20) m m (cid:21) is themass matrix and D ( k ) = 2 (cid:88) l =1 (cid:110) f r ( L ) e l ⊗ e l (cid:2) cos( k · R l ) − (cid:3)(cid:111) + 2 (cid:88) l =1 (cid:26) f ( L ) L ( I − e l ⊗ e l ) (cid:2) cos( k · R l ) − (cid:3)(cid:27) + K f (2)is a wavevector-dependent dynamical matrix that al-ready incorporates Bloch conditions, and R = L e = L [1 0] T , R = L e = L [1 / √ / T , and R = L e = L [ − / √ / T are the lattice vectors. In Eq. 2,the first term is the conventional stiffness matrix for har-monic particle systems, where f r ( L ) is the first deriva-tive of the repulsive force f ( r ) evaluated at the initialnodal spacing L , the second term is an additional stiff-ness contribution capturing the aforementioned softeningeffect and K f = (cid:20) k f k f (cid:21) represents the stiffness of theelastic foundation. The solution of Eq. 1 yields the banddiagram plotted in Fig. 1(c). As expected, the band di-agram is fully gapped at low frequencies as a result ofthe elastic foundation, and we observe the two canonicalacoustic branches, the first is dominated by shear mech-anisms and the second by longitudinal mechanisms.We now proceed to investigate the nonlinear response.To experimentally capture the in-plane response of thelattice, we employ a 3D scanning laser Doppler vibrom-eter (SLDV, Polytec PSV-400-3D) by which we measurethe displacement of the individual magnets. The spec-imen is excited in the vertical direction by a force ap-plied to the center magnet of the bottom layer (as in-dicated by the red dot in Fig. 2(a)) through a Bruel & (a) (b) ( ξ , ν )(2 ξ , 2 ν ) FIG. 2. Spectra of the experimental responses for tone-burst excitations at 50 Hz. (a) Response to low-amplitude excitation,showing the fundamental harmonic activating both shear and longitudinal modes (red solid curves). (b) Response to high-amplitude excitation, showing unequivocally two distinct signatures at the second harmonic (i.e., 100 Hz), confirming theexistence of both homogeneous and forced components of the nonlinearly generated harmonic. The white dashed curvesrepresent all the possible spectral points that can be activated by the forced component. ν Total wavefieldFiltered wavefield at ν (b)(a)(c) FIG. 3. Snapshots of the wavefields experimentally acquired by laser scans at six successive time instants, showing distinctspatial patterns at the different harmonics. (a) Total wavefield. (b) Wavefield filtered at ν , highlighting the fundamentalharmonic. (c) Wavefield filtered at 2 ν , highlighting the second harmonic. The two harmonics feature complementary modaland directional characteristics, thus exposing the functionality enrichment that is achieved by triggering a nonlinear response. Kjaer Type 4809 shaker, powered by a Bruel & KjaerType 2718 amplifier. The excitation is prescribed as afive-cycle tone burst with carrier frequency ν = 50 Hz.First, we use a low-amplitude excitation to elicit a linearresponse of the specimen. In Fig. 2(a) we plot the colormap of the normalized spectral amplitude obtained via2D discrete Fourier transform (2D-DFT) in space andtime of the experimental spatio-temporal response sam-pled along one lattice vector (i.e., along the green lineshown in Fig. 1(a)). Then, we progressively raise theamplitude of excitation until the deformations are suffi- ciently large to activate non-negligible nonlinearity in theresponse, and the corresponding spectral amplitude mapis given in Fig. 2(b). In both figures, we superimpose theΓ-M portion of the linear dispersion branches (red curves)obtained via Bloch analysis informed using our modifiedlattice model and periodically extended into the secondBrillouin zone for convenience. In addition, the whitedashed curves denote the parametric locus of the 2 ξ -2 ν , ( ξ ) pairs, i.e., the spectral points that feature simul-taneously twice the frequency and twice the wavenum-ber of the acoustic phonons at the fundamental harmonic( ξ, ν , ( ξ )). The 2 ξ - 2 ν , ( ξ ) points represent the pairsof frequency and wavenumber that can be displayed bythe forced component of a nonlinearly generated secondharmonic. Since these pairs do not live on any disper-sion branch (and therefore do not conform to any modeof the linear system), their activation is conditional uponthe generation of harmonics and is therefore the mostrobust detector of nonlinearity in the wave response. Inother words, phonons that live on these curves must begenerated through nonlinear mechanisms intrinsic to thelattice and cannot be merely induced by an external ex-citation prescribed at 2 ν . This consideration provides apowerful tool to distinguish with absolute certainty themanifestation of nonlinearly generated harmonics fromother spurious signatures of high-frequency componentsthat could be already embedded in the excitation signal(for example due to nonlinearities in the signal gener-ation and amplification). As expected, in Fig. 2(a) weobserve that the main spectral contribution is located atthe prescribed frequency ( ν = 50 Hz), and both shearand longitudinal modes are activated, with no apprecia-ble signature at the second harmonic (2 ν = 100 Hz). Incontrast, in the spectrum of the nonlinear response (i.e.,Fig. 2(b)) we clearly recognize two additional spectralsignatures at the second harmonic. The one overlappingthe longitudinal mode corresponds to the homogeneouscomponent, while the other which lies precisely on thedashed curve, is unequivocally identified as the forcedcomponent. To the authors’ best knowledge, this is thefirst experimental study that explicitly reveals the co-existence of the two distinct nonlinear contributions in atwo-dimensional periodic structure.From inspection of the band diagram, we have alreadyobserved that the fundamental and second harmonics fea-ture distinct and complementary modal characteristics.The response at the fundamental harmonic, while blend-ing shear and longitudinal effects, is dominated by theshear, which represents a softer mechanism. In contrast,the response at the second harmonic (more precisely, thehomogeneous part, for which a clear modal structure canbe invoked) is dominated by longitudinal mechanisms.By this observation, we can characterize this result as aninstance of modal enhancement, whereby the nonlinearactivation of the second harmonic introduces in the re-sponse modal characteristics that are complementary tothose exhibited by the linear response. The vibrometerscan also allows exploring the manifestation of SHG onthe spatial pattern of the wave response. In Fig. 3 we plotsix snapshots of the propagating wavefield. In Fig. 3(a),we show the total wavefield, encompassing fundamentaland second harmonics, while in Fig. 3(b) and (c) we iso-late the filtered components at ν and 2 ν , respectively.The wavefields in Fig. 3 reveal spatial complementaritybetween the predominantly vertical directivity of the sec-ond harmonic and the quasi-isotropic pattern of the fun-damental harmonic. This result indicates that, in addi- tion to modal enrichment, nonlinearity is also responsiblefor a directivity enrichment. (a)(c) (d) (b) (e) FIG. 4. k -plane amplitude spectra of (a) the filtered secondharmonic of a nonlinear response and (b) a linear wavefieldexcited directly at 2 ν . (c) and (d) Filtered wavefields of thespectral components inscribed in amber and magenta boxes,respectively. To highlight separately and filter the spatial contribu-tions of the homogeneous and forced components thatcoexist at the second harmonic, we subject the last snap-shot of the wavefield filtered at 2 ν (i.e., the last snap-shot in Fig. 3(c)) to 2D-DFT in space. The resulting k -space spectrum is plotted in Fig. 4(a). For compari-son, we repeat the exercise for a linear wavefield obtainedwith a low-amplitude excitation prescribed directly at2 ν (Fig. 4(e)), whose spectrum is shown in Fig. 4(b).In Fig. 4(a), we identify two spectral signatures at thesecond harmonic (appearing with their mirror counter-parts due to the symmetry folding operations involvedin the 2D-DFT). The dominant component, inscribed bymagenta boxes, is consistent with the spectrum of the lin-ear wavefield excited directly at 2 ν (i.e., Fig. 4(b)), andis therefore interpreted as the homogeneous component.On the other hand, the secondary contribution, inscribedby amber boxes, which is germane to the nonlinear re-sponse, must be interpreted as the forced response. Byzeroing out the two signatures, one at a time, and car-rying out an inverse 2D-DFT of the remainder, we canfilter out the separate wavefields of the two components,as shown in Fig. 4(c) and (d), respectively. Clearly, thepattern in the magenta box of Fig. 4(d), which exhibitslongitudinal behavior, is reminiscent of the wavefield inFig. 4(e), further supporting, from a spatial perspective,the notion that the homogeneous component conformsto the linear response that would be observed in the lat-tice if the excitation were prescribed directly at 2 ν . Theother, shown in Fig. 4(c) and corresponding to the forcedcomponent, displays a more dispersive spatial pattern. Itis worth emphasizing again that, while it is possible thatthe homogeneous component may be in part triggeredby nonlinearities extrinsic to the mechanical system anddue, for instance, to harmonics buried in the excitationsignal, the forced component is germane to the SHG es-tablished within the structure. Therefore, the existenceof the force second harmonic is the most powerful detec-tor of nonlinearities in the lattice.In summary, we have experimentally characterized thenonlinear wave response of a lattice of repulsive mag-nets on an elastic foundation. First, we have demon-strated the existence of SHG in the specimen, sepa-rately pinpointing its two second harmonic contributions.Then, we have shown that the nonlinear response fea-tures modal characteristics that are complementary tothe those of the fundamental wave. Finally, we havereconstructed the spatial characteristics of the two non-linear components, revealing additional complementaritybetween them in terms of spatial characteristic and di-rectivity. The magnetic lattice prototype reveals to be anideal platform for the experimental observation of non-linear wave propagation. On one hand, its compliance isconducive to displacements that are at least one order ofmagnitude larger than what is achievable in hard solidspecimens and approaching the advantages of soft mate-rials without their pitfalls in terms of damping. On theother hand, its discrete nature results in low modal com-plexity, compared to a structural lattice, thus facilitatinga clear identification of all the spectral components. ACKNOWLEDGEMENT
This work is supported by the National Science Foun-dation (CAREER Award CMMI-1452488). The authorsare indebted to Lijuan Yu for her precious help with thespecimen assembly and to Joseph Labuz, Xiaoran Wangand Chen Hu for sharing their invaluable expertise withthe force-displacement testing apparatus. [1] C. Daraio, V. F. Nesterenko, E. B. Herbold, and S. Jin,Phys. Rev. E , 016603 (2005).[2] C. Daraio, V. Nesterenko, E. Herbold, and S. Jin, Phys.Rev. E , 026610 (2006).[3] A. Leonard, F. Fraternali, and C. Daraio, ExperimentalMechanics , 327 (2013).[4] N. Boechler, G. Theocharis, S. Job, P. G. Kevrekidis,M. A. Porter, and C. Daraio, Phys. Rev. Lett. ,244302 (2010).[5] F. M. Russell, Y. Zolotaryuk, J. C. Eilbeck, andT. Dauxois, Phys. Rev. B , 6304 (1997).[6] M. Molern, A. Leonard, and C. Daraio, J. Appl. Phys. , 184901 (2014). [7] M. Moler´on, C. Chong, A. J. Mart´ınez, M. A. Porter,P. G. Kevrekidis, and C. Daraio, New J. Phys. ,063032 (2019).[8] B. G. Chen, N. Upadhyaya, and V. Vitelli, Proc. Natl.Acad. Sci. , 13004 (2014).[9] B. Deng, J. R. Raney, V. Tournat, and K. Bertoldi,Phys. Rev. Lett. , 204102 (2017).[10] B. Deng, P. Wang, Q. He, V. Tournat, and K. Bertoldi,Nat. Commun. , 3410 (2018).[11] T. Mullin, S. Deschanel, K. Bertoldi, and M. C. Boyce,Phys. Rev. Lett. , 084301 (2007).[12] P. Wang, F. Casadei, S. Shan, J. C. Weaver, andK. Bertoldi, Phys. Rev. Lett. , 014301 (2014).[13] N. Nadkarni, A. F. Arrieta, C. Chong, D. M. Kochmann,and C. Daraio, Phys. Rev. Lett. , 244501 (2016).[14] S. Shan, S. H. Kang, J. R. Raney, P. Wang, L. Fang,F. Candido, J. A. Lewis, and K. Bertoldi, AdvancedMaterials , 4296 (2015).[15] R. Narisetti, M. Leamy, and M. Ruzzene, ASME. J. Vib.Acoust. , 031001 (2010).[16] R. Narisetti, M. Ruzzene, and M. Leamy, ASME. J. Vib.Acoust. , 061020 (2011).[17] J. Cabaret, V. Tournat, and P. B´equin, Phys. Rev. E , 041305 (2012).[18] L. Bonanomi, G. Theocharis, and C. Daraio, Phys. Rev.E , 033208 (2015).[19] R. K. Pal, J. Vila, M. Leamy, and M. Ruzzene, Phys.Rev. E , 032209 (2018).[20] V. Tournat, V. E. Gusev, V. Y. Zaitsev, andB. Castagnede, Europhys. Lett. , 798 (2004).[21] K. H. Matlack, J. Y. Kim, L. J. Jacobs, and J. Qu, J.Appl. Phys. , 014905 (2011).[22] V. S´anchez-Morcillo, I. P´erez-Arjona, V. Romero-Garc´ıa,V. Tournat, and V. Gusev, Phys. Rev. E , 043203(2013).[23] A. Mehrem, N. Jim´enez, L. Salmer´on-Contreras,X. Garc´ıa-Andr´es, L. Garc´ıa-Raffi, R. Pic´o, andV. S´anchez-Morcillo, Phys. Rev. E , 012208 (2017).[24] W. Jiao and S. Gonella, J. Mech. Phys. Solids , 1(2018).[25] W. Jiao and S. Gonella, Phys. Rev. Appl. , 024006(2018).[26] R. Ganesh and S. Gonella, J. Mech. Phys. Solids , 272(2017).[27] S. Wallen and N. Boechler, Wave Motion , 22 (2017).[28] I. Grinberg and K. Matlack, Wave Motion , 102466(2020).[29] B. Liang, X. Guo, J. Tu, D. Zhang, and J. C. Cheng,Nat. Mater. , 989 (2010).[30] N. Boechler, G. Theocharis, and C. Daraio, Nat. Mater. , 665 (2011).[31] R. Ganesh and S. Gonella, Appl. Phys. Lett. , 084101(2017).[32] W. Jiao and S. Gonella, Phys. Rev. E99