Amplitude-activated mechanical wave manipulation devices using nonlinear metamaterials

Transcription

1 Advanced Composites and Hybrid Materials (2018) 1: ORIGINAL RESEARCH Amplitude-activated mechanical wave manipulation devices using nonlinear metamaterials James M. Manimala 1 & Prateek P. Kulkarni 1 & Karthik Madhamshetty 1 Received: 4 September 2018 /Revised: 11 October 2018 /Accepted: 5 November 2018 /Published online: 15 November 2018 # Springer Nature Switzerland AG 2018 Abstract We report device implications for acoustic metamaterials with various nonlinear oscillator microstructures as passive amplitudeactivated mechanical wave filters and waveguides using simulations on their representative one-dimensional discrete element models. Linear and various nonlinear hardening and softening stiffness cases and combinations thereof are considered for the local oscillators. The propagation and attenuation characteristics of harmonic waves in a tunable frequency range are found to correspond to the excitation amplitude and nonlinearity-dependent shifts in the local resonance bandgap for such nonlinear acoustic metamaterials. Three passive acoustic devices (i) amplitude-activated selective filter, (ii) amplitude band pass or band rejection filter, and (iii) direction-biased waveguide are demonstrated numerically. Constituent frequency components in bifrequency excitations are shown to be retained or diminished to varying degrees within acoustic metamaterials with either hardening or softening local oscillators depending on their individual amplitudes. Using trilinear hardening or softening oscillators instead switches the response between attenuation and propagation respectively, outside of a tunable bandwidth of amplitude for the same excitation frequency. Whereas, amplitude-dependent, passive direction-bias in propagation characteristics for a given excitation frequency is demonstrated in a metamaterial waveguide composed of units with tuned combinations of linear and nonlinear hardening oscillators deployed in sequence. These predictions indicate the possibility of realizing acoustic metamaterials having passive adaptive, amplitude-activated dynamics using tailored combinations of nonlinear oscillators to elicit prescribed wave transformations across them. Keywords Acoustic metamaterial. Nonlinearity. Wave propagation 1 Introduction As engineering missions become more multifarious and constrained, there is a greater demand for the next generation of structural materials to integrate several functionalities and adapt to dynamic variations in their in-service environment. With additive and hybrid manufacturing technologies attaining critical commercial maturity in recent years, the potential to realize advanced composite designs, which were beyond the scope of conventional fabrication techniques, provides an opportune time to explore emergent approaches to enrich the performance of structural materials. * James M. Manimala james.manimala@okstate.edu 1 School of Mechanical & Aerospace Engineering, Oklahoma State University, 218, Engineering North, Stillwater, OK 74078, USA One promising route to attain multifunctionality and adaptivity in structural materials is via acoustic metamaterials (AM). AM which are artificial materials akin to manmade composites trace their conceptual origins to their electromagnetic counterparts. They exhibit unique dynamic properties not seen in natural materials, not just due to their material constituent but more so due to their engineered local configurations. Depending on the scale of implementation, these configurations may be deployed as microscopic inclusions in meta-composites or even as macroscale endo-structures within load-bearing exo-structures. This latitude in their design approach has made AM attractive to be explored as a potential solution for hitherto unaddressed challenges related to structural materials. In as much as these designs can be realized in practice, tailored acoustic performance could be extracted using AM without some of the penalties related to weight and volume that are encountered through the use of traditional acoustic materials.

2 798 Adv Compos Hybrid Mater (2018) 1: Fig. 1 Amplitude-activated dispersion curve shifts from approximate analytical solutions for the nonlinear acoustic metamaterial with cubically nonlinear oscillators [15] Fig. 3 Force versus displacement curves for the linear and the various nonlinear cases Due to their unique mechanical wave manipulation capabilities, AM have seen much research [1 5] and application [6 10] interest in past years. Several types of local engineering features [11 19] that lend enriched dynamic behavior have been explored for AM. Further, AM concepts have been shown to be scalable resulting in variants ranging from micro-scale devices [20] and performance-tailored materials [21] to large-scale structural elements [16, 22]. Of particular, focus recently have been AM with smart [23, 24] oradaptive [25] behavior. Both active [26, 27] and passive [23] means have been investigated to enable these. In certain contexts, a solely passive adaptive manipulation of mechanical waves is desirable, especially when not just frequency but also amplitude-activated modification in propagation characteristics need to be built-in within the AM. AM utilizing nonlinear local oscillator inclusions within a host material or structure have been shown to enable amplitudedependent shifts in the dispersion branches leading to unusual propagation characteristics [3, 14, 15, 28 30]. These excitation amplitude-dependent shifts due to the presence of local oscillators having cubically nonlinear hardening or softening behavior can be achieved within a tunable bandgap frequency range (Fig. 1). Significantly, an upshift or downshift in the lower bound of the bandgap is shown to be possible for the hardening or softening case respectively. The propagation and attenuation characteristics of harmonic waves were found to correspond to the amplitude and nonlinearity-dependent shifts in the local resonance bandgap for such nonlinear acoustic metamaterials (NLAM). Utilizing this phenomenon, the possibility of realizing passive, amplitude-activated acoustic filters and waveguides using nonlinear acoustic metamaterials is demonstrated through numerical simulations on their representative discrete element models. The simulation models, lattice parameters, and device schemes are described in Section 2 followed by device implications from the discussion of simulation results in Section 3. Conclusions are presented in Section 4. 2Numericalmodels Discrete mass-spring lattice models that are representative of the one-dimensional NLAM being considered are employed Fig. 2 Discrete mass-spring lattice model representation of the nonlinear acoustic metamaterial (NLAM)

3 Adv Compos Hybrid Mater (2018) 1: Fig. 4 Simulation model for the direction-biased waveguide for simulations. The simulation model for the NLAM used to demonstrate amplitude-dependent filters (AASF and ABFW) is shown in Fig. 2. It consists of a mass-in-mass lattice chain of 1000 unit cells. Each unit cell consists of an external mass (m 1 ) and linear spring (k 1 ) representing the discretized host material along with an internal mass (m 2 ) and nonlinear spring (k 2n ) representing the nonlinear resonator inclusion. The lattice parameters used in these simulation were [m 1, m 2, k 1, k 2, k n, L, N] = [0.1, 0.9, 1E5, 1E3, 10, 40, 1000], where L is the lattice length and N is the total number of units. Fig. 5 Normalized input (n =1) and output (n =100) displacement histories (u 1 (n) /u 1 (1) ) from simulations and their frequency spectra for an excitation frequency, Ω =1.005 (just within the LRAM bandgap) at excitation amplitudes of a A = 0.1 (quasi-linear) and b A =10

4 800 Adv Compos Hybrid Mater (2018) 1: Fig. 6 Normalized input (n =1) and output (n =100) displacement histories (u 1 (n) /u 1 (1) ) from simulations and their frequency spectra for an excitation frequency, Ω = 0.9(just below the linear LRAM bandgap) at excitation amplitudes of a A = 0.1 (quasi-linear) and b A =10 Four nonlinear stiffness cases as shown in the force versus displacement plots for compatible units in Fig. 3 were considered. Stiffness and force descriptions for the cubically nonlinear hardening (NLH) case are respectively k 2n ¼ k 2 þjk n jx 2 NLH : Fx ðþ¼k 2 x þjk n jx 3 ð1þ ð2þ An approximate bilinear softening (BLS) case defined as 8 sffiffiffiffiffiffiffiffiffiffi k 2 x jk n jx 3 k 2 >< if jxj 3jk n j BLS : Fx ðþ¼ sffiffiffiffiffiffiffiffiffiffi ð3þ 2k 3=2 2 sgnðþ x p 3 ffiffi k 2 >: if jxj > 3j kn j 1 2 3jk n j is considered. A purely nonlinear softening case is not considered so as to avoid instabilities arising from a monotonically decreasing negative stiffness regime past the zero stiffness point. Two trilinear cases are also considered. The trilinear hardening (TLH) case is defined as TLH : Fx ðþ¼ 8 < k 2 x if jxj 1 k 2 ð2x sgnðþ x Þ if 1 < jxj 5 : k 2 ðx þ 4sgnðÞ x Þ if jxj > 5 while the trilinear softening (TLS) is defined as 8 < k 2 x if jxj 1 TLS : Fx ðþ¼ k 2 sgnðþ x if 1 < jxj 5 : k 2 ðx 4sgnðÞ x Þ if jxj > 5 ð4þ ð5þ

5 Adv Compos Hybrid Mater (2018) 1: Fig. 7 Normalized input (n =1) and output (n =100) displacement histories, u 1 (n) / u 1 (1) (top) and their frequency spectra (bottom) for the NLAM with nonlinear hardening (NLH) oscillators using bifrequency excitations of different frequency and amplitude combinations: a [Ω 1, A 1 ]=[1.005,0.1]+[Ω 2, A 2 ]=[3,0.1]andb [Ω 1, A 1 ]=[0.9,10]+[Ω 2, A 2 ] = [1, 3] The simulation model for the direction-biased waveguide consists of four sections of lattice chains, each with a different number of unit cells in series as shown in Fig. 4. These sections are, in order, from left to right a400-unit monatomic massspring lattice chain representing the elastic host medium without inclusions; a 200-unit mass-in-mass lattice chain with linear resonators representing an LRAM; a 200-unit mass-in-mass lattice chain with nonlinear hardening (NLH) oscillators representing an NLAM; and finally, yet another 400-unit monatomic chain for the host medium. The parametric settings for the directionbiased waveguide simulation model are detailed in Section 3.3. For each case, depending on the device being interrogated, single or bifrequency harmonic excitation at various amplitudes and frequencies are applied to one end of the model used. 3 Device implications 3.1 Amplitude-activated selective filter In order to verify the shifts in dispersion curves due to the presence of cubically nonlinear local oscillators predicted by the approximate analytical solution, firstly, single frequency input excitations were applied to the first unit cell of the NLAM model at low and high amplitude levels. For the nonlinear hardening case, the input and output time histories and frequency spectra for an excitation frequency of Ω =1.005at low (A = 0.1) and high (A = 10) amplitudes are shown in Fig. 5. As this excitation frequency is just within the linear bandgap, at low amplitude, a quasi-linear behavior is expected

6 802 Adv Compos Hybrid Mater (2018) 1: Fig. 8 Normalized input (n =1) and output (n =100) displacement histories u 1 (n) / u 1 (1) (top) and their frequency spectra (bottom) for the NLAM with nonlinear hardening (NLH) oscillators using bifrequency excitations of different frequency and amplitude combinations: a [Ω 1, A 1 ] = [0.45, 0.1] + [Ω 2, A 2 ] = [1.005, 0.1] and b [Ω 1, A 1 ]=[0.45,10]+[Ω 2, A 2 ] = [1.005, 10] and observed. The output at low amplitude is drastically diminished. At high amplitude, a propagating output is clearly seen consistent with the upshift in the bandgap frequency range due to the presence of hardening local attachments. While the output spectrum at high amplitude is predominantly at the excitation frequency, a significant downshift is also noticed. This mechanism of frequency downshift could be utilized to realize unique wave manipulation devices. For the bilinear softening case, the input and output are shown in Fig. 6. The excitation frequency was Ω = 0.9, which lies just under the lower bound of the bandgap. At low amplitude (A = 0.1), a propagating output is obtained as predicted for the linear case. When the amplitude is high (A = 10), the output behavior switches to attenuation indicating the downshift in the bandgap frequency range due to the presence of the softening local attachments. Next, the nonlinearity-induced dispersion curve shifts in NLAM are employed to demonstrate amplitude-activated selective filtering. For a composite signal consisting of two distinct frequency components, amplitude-dependent selective filtering can be realized using NLAM. Some examples are shown in Figs. 7 and 8. In each case, the output is selectively altered with a switch from low to high amplitude. Considering the nonlinear hardening case shown in Fig. 7, when a bifrequency input signal defined by [Ω 1, A 1 ] = [1.005, 0.1] + [Ω 2, A 2 ] = [3, 0.1] is provided, both components are diminished in the output, since they are both within the linear bandgap and at low amplitude. In contrast, a bifrequency input of [Ω 1, A 1 ] = [0.9,

7 Adv Compos Hybrid Mater (2018) 1: Fig. 9 Input (n = 1) and output (n = 100) displacement histories for the trilinear hardening (TLH) case at an excitation frequency of Ω = and at excitation amplitudes of a 0.05, b 2, c 5, and d 50 10] + [Ω2, A2] = [1, 3] results in the high-frequency component being removed while retaining the low-frequency component. As noted before, the propagated spectrum displays a downshifted component in addition to the input component that is retained due to the nonlinearity. For the bilinear softening case shown in Fig. 8, a bifrequency input of [Ω 1, A1] = [0.45, 0.1] + [Ω2, A2] = [1.005, 0.1] results in the lowfrequency component alone being propagated. This is consistent with the linear dispersion behavior. For the same frequencies, a high-amplitude signal results in even the low-frequency component being considerably diminished as the softening response shifts the bandgap frequency range towards the low frequencies. By tuning the bandgap frequency range and the nature and degree of the nonlinear response, amplitudeactivated selective filtering can be accomplished for specific combinations of frequencies. 3.2 Amplitude band pass and band rejection filters The possibility of realizing amplitude band pass or band rejection filters is explored using NLAMs with trilinear hardening or softening resonators. The simulation model remains the same as that shown in Fig. 2; however, the trilinear forcedisplacement responses (Fig. 3) are prescribed for the local

8 804 Adv Compos Hybrid Mater (2018) 1: Fig. 10 Input (n = 1) and output (n = 100) displacement histories for the trilinear softening (TLH) case at an excitation frequency of Ω = 0.9 and at excitation amplitudes of a 0.1, b 1, c 2, and d 0.2 Fig. 11 Transmissibility at the 100th unit (D*100) versus excitation amplitude at fixed excitation frequencies for the trilinear hardening (TLH) and softening (TLS) cases attachments. The trilinear hardening and softening responses are defined as per Eqs. (4) and (5). For the trilinear hardening case, input excitations at a frequency of Ω = 1.005, which is just within the linear bandgap are applied at various amplitudes spanning a range that results in the resonator s motion to transition through the trilinear regime. Comparison of input and output displacement histories for selected excitation amplitudes for the trilinear hardening case are shown in Fig. 9. At low amplitude (A = 0.05), the output is attenuated as expected, since the excitation frequency is within the bandgap and the behavior of the resonator is quasi-linear. At higher amplitudes within a

9 Adv Compos Hybrid Mater (2018) 1: Fig. 12 a Input (n = 1000) and output (n = 1) displacement histories for right to left propagation; b input (n =1)and output (n = 1000) displacement histories for left to right propagation; c input and output frequency spectra for left to right and right to left propagation for direction-biased waveguide at low (A = 0.1) excitation amplitude for Ω =2 bandwidth from about A =0.2 to A = 5, the output displays a propagating behavior. When the amplitude is increased further, transition to attenuation behavior is observed. For the trilinear softening case, input excitations at a frequency of Ω = 0.9, which is just below the onset of the linear bandgap are applied at amplitudes in the same range as those for the trilinear hardening case. Selected trilinear softening comparison cases are shown in Fig. 10. At low amplitude (A = 0.1), propagation is seen, whereas with increase in amplitude, the output transitions to attenuation within the amplitude range of A =0.2toA = 20. Above this amplitude range, propagation behavior is recovered. Transmissibility for trilinear hardening and softening cases is showninfig.11. An amplitude band pass behavior is observed for the trilinear hardening case within the linear bandgap, while an amplitude band rejection behavior is observed for the trilinear softening case below the linear bandgap. Depending on the resonator s motion, transition to a higher or lower effective stiffness response within a tunable input excitation amplitude bandwidth can be engineered by tailoring the trilinear forcedisplacement response of the local attachments. This phenomenon could be used to create acoustic devices that act as amplitude-activated passive adaptive transducers or filters. 3.3 Direction-biased waveguide Consider a device composed of a linear LRAM with 100 unit cells arranged in series with an NLAM with NLH oscillators with the same number of unit cells connecting between two elastic media having the same static mass and stiffness as the LRAM and NLAM. The simulation model for this case is showninfig.4. For the purposes of this discussion, the LRAM is assumed to be located to the left of the NLAM and the elastic media on either side are distinguished using the designations of left and right. The discrete spring-mass simulation model has a total of 1000 unit cells (400 each for the elastic media and 100 each for the LRAM and NLAM). The NLAM has a parametric setting of [m 1, m 2, k 1, k 2, k- n] = [0.1, 0.9, 1E5, 1E3, 10], the positive value of k n lending a hardening response, whereas the LRAM has a parametric

10 806 Adv Compos Hybrid Mater (2018) 1: Fig. 13 a Input (n = 1000) and output (n = 1) displacement histories for right to left propagation; b input (n = 1) and output (n = 1000) displacement histories for left to right propagation; c input and output frequency spectra for left to right and right to left propagation in the direction-biased waveguide at high (A = 10) excitation amplitude for Ω = 2 setting of [m1, m2, k1, k2] = [0.1, 0.9, 1E5, 3610]. Therefore, the local resonance frequency (LRF) for the LRAM is set to 1.9ω0 where ω0 is the linear LRF for the NLAM when kn is zero (or when the excitation amplitude is very low). The bandgap frequency range for the LRAM extends from 1.9ω0 to about 6.01ω0 while the NLAM s bandgap for low amplitudes (quasi-linear behavior) ranges from about ω0 to 3.16ω0. The relative bandgap frequency ranges for the LRAM and NLAM are chosen to take advantage of the downshift in the propagated spectrum for the NLAM at high amplitude. As the excitation amplitude is increased, the NLAM permits propagation at excitation frequencies more and more within its linear bandgap range (ω0 to 3.16ω0) due to the shifting of the acoustic mode towards higher frequencies. In order to demonstrate the direction-biased behavior for this waveguide, a single frequency excitation input at Ω = ω/ω0 = 2 with low (A = 0.1) and high (A = 10) excitation amplitudes applied at either the left end or the right end of the device for each case is considered in the simulations. The output response at the other end of the device is recorded for each case. The results are discussed in the following paragraphs. As can be seen from Fig. 12, the output response for a low amplitude input at Ω = 2 for both left to right and right to left displays the same level of attenuation. At low excitation amplitudes, the NLAM behaves like an LRAM with an LRF, ω0. Since the excitation frequency of Ω = 2 falls within the bandgap frequency range for both the LRAM (1.9 < Ω < 6.01) and the NLAM (1 < Ω < 3.16), the wave is attenuated irrespective of the direction of propagation (left to right or right to left) at low excitation amplitude. The frequency spectra of the output displacement histories (Fig. 12) reveal that the excitation frequency components are almost completely diminished for output in both directions. At high amplitude (A = 10) as shown in Fig. 13, the output displacement responses for left to right and right to left directions are drastically different. For the left to right case at the same excitation frequency of Ω = 2, the input is first attenuated as it passes through the LRAM since it is just within its bandgap frequency range (1.9 < Ω < 6.01). This attenuated (low amplitude) wave does not activate the nonlinear response when it next passes through the NLAM and therefore the NLAM behaves as a quasi-linear AM for which the frequency (Ω = 2) still falls within its bandgap (1 < Ω < 3.16). This results in a nearly complete reduction of the excitation frequency component in the output response for left to right propagation at high amplitude (Fig. 13). In contrast, when the same input is given from right to left at high amplitude, as it passes first through the NLAM, it permits propagation due to the

11 Adv Compos Hybrid Mater (2018) 1: shifting of the acoustic mode to higher frequencies as predicted by the approximate solutions (Fig. 13). However, this propagation is accompanied by a shifting of the wave s frequency content to lower frequencies as observed for the single frequency excitations using the NLAM with NLH. Thus, after traversing the NLAM, the propagated wave has a predominant frequency content significantly below the original excitation frequency of Ω = 2 which falls within the propagation regime (acoustic mode) for the subsequent LRAM that it encounters. The resultant output at the left end of the device, therefore, retains the high input amplitude while its frequency content is shifted significantly to lower frequencies as is evident from its spectrum in Fig. 13. Therefore, at high amplitudes, this device acts as a direction-biased waveguide for mechanical waves. Simulations for the direction-biased waveguide demonstrate the feasibility of utilizing the phenomenon of downshift in the frequency content of a propagated wave as it passes through an NLAM in conjunction with the tunable bandgap for an LRAM to realize amplitudeactivated direction-bias for mechanical waves. 4 Conclusion The potential to realize amplitude-activated, passive adaptive mechanical wave manipulation devices using nonlinear acoustic metamaterials is demonstrated through numerical simulations on their representative discrete element models. Different types of nonlinear behavior such as cubically nonlinear hardening, bilinear softening, and trilinear hardening and softening are considered for the force-displacement response of local oscillator attachments. The nonlinearityinduced amplitude-activated shifts in the dispersion curves were verified using single frequency excitations. Depending on their relative amplitudes and frequencies, selective filtering of a constituent component of bifrequency signals is demonstrated, both for the nonlinear hardening as well as for the bilinear softening case. Utilizing trilinear hardening or softening local oscillators, amplitude band pass or band rejection filters could be realized. The amplitude bandwidth for this filtering effect can be tuned by tailoring the trilinear response. Further, using the phenomenon of downshifts in the propagated spectrum observed for the nonlinear hardening case, a tuned combination of a linear acoustic metamaterial and a nonlinear hardening metamaterial is employed to obtain amplitude-activated, passive direction-bias in propagation behavior. These results indicate the possibility of creating acoustic devices that can self-adapt to frequency as well as amplitude variations for transduction and filtering applications. With current additive and hybrid manufacturing processes reaching critical commercial maturity, meta-composites that incorporate engineered local features that enable such unusual dynamic characteristics could conceivably be realized for enriched performance in a variety of structural applications. Funding information J. M. and K. M. received support through Defense Advanced Research Projects Agency (DARPA) Grant No. D16AP00032 during the completion of this work. Compliance with ethical standards Conflict of interest interest. References The authors declare that they have no conflict of 1. Martinez-Sala R, Sancho J, Sanchez JV, Gomez V, Llinares J, Meseguer F (1995) Sound attenuation by sculpture. Nature 378: Sánchez-Pérez JV, Caballero D, Mártinez-Sala R, Rubio C, Sánchez- Dehesa J, Meseguer F et al (1998) Sound attenuation by a twodimensional array of rigid cylinders. Phys Rev Lett 80: Vincent JH (1898) LX. On the construction of a mechanical model to illustrate Helmholtz s theory of dispersion. Philos Maga 46: Liu Z, Zhang X, Mao Y, Zhu YY, Yang Z, Chan CT et al (2000) Locally resonant sonic materials. Science 289: Huang HH, Sun CT, Huang GL (2009) On the negative effective mass density in acoustic metamaterials. Int J Eng Sci 47: Popa BI, Zigoneanu L, Cummer SA (2011) Experimental acoustic ground cloak in air. Phys Rev Lett 106: Ambati M, Fang N, Sun C, Zhang X (2007) Surface resonant states superlensing in acoustic metamaterials. Phys Rev B 75: Zhang S, Yin L, Fang N (2009) Focusing ultrasound with an acoustic metamaterial network. Phys Rev Lett 102: Yang Z, Mei J, Yang M, Chan NH, Sheng P (2008) Membrane-type acoustic metamaterial with negative dynamic mass. Phys Rev Lett 101: Chen Y, Liu H, Reilly M, Bae H, Yu M (2014) Enhanced acoustic sensing through wave compression pressure amplification in anisotropic metamaterials. Nat Commun 5: Kulkarni PP, Manimala JM (2016) Longitudinal elastic wave propagation characteristics of inertant acoustic metamaterials. Jpn J Appl Phys 119: Wang G, Wen X, Wen J, Shao L, Liu Y (2004) Two-dimensional locally resonant phononic crystals with binary structures. Phys Rev Lett 93: Lazarov BS, Jensen JS (2007) Low-frequency b gaps in chains with attached non-linear oscillators. Int J Nonlin Mech 42: Manimala JM, Sun CT (2016) Numerical investigation of amplitude-dependent dynamic response in acoustic metamaterials with nonlinear oscillators. J Acoust Soc Am 139: Kulkarni PP, Manimala JM (2018) Nonlinear Inertant Acoustic Metamaterials their Device Implications, in proceedings of the SEM Annual Conference on Experimental Mechanics, Indiana, USA 16. Xiao Y, Wen J, Wang G, Wen X (2013) Theoretical experimental study of locally resonant Bragg band gaps in flexural beams carrying periodic arrays of beam-like resonators. J Vib Acoust 135: Lee SH, Park CM, Seo YM, Wang ZG, Kim CK (2009) Acoustic metamaterial with negative density. Phys Lett A 373: Lee SH, Park CM, Seo YM, Wang ZG, Kim CK (2010) Composite acoustic medium with simultaneously negative density modulus. Phys Rev Lett 104:054301

12 808 Adv Compos Hybrid Mater (2018) 1: Boechler N, Theocharis G, Daraio C (2011) Bifurcation-based acoustic switching rectification. Nat Mater 10: Zhang H, Xiao Y, Wen J, Yu D, Wen X (2016) Ultra-thin smart acoustic metasurface for low-frequency sound insulation. Appl Phys Lett 108: Manimala JM, Huang HH, Sun CT, Snyder R, Bl S (2014) Dynamic load mitigation using negative effective mass structures. Eng Struct 80: Smith TL, Rao K, Dyer I (1986) Attenuation of plate flexural waves by a layer of dynamic absorbers. The National Academies of Sciences, Engineering, Medicine 23. Nadkarni N, Arrieta AF, Chong C, Kochmann DM, Daraio C (2016) Unidirectional transition waves in bistable lattices. Phys Rev Lett 116: Zhai S, Chen H, Ding C, Shen F, Luo C, Zhao X (2015) Manipulation of transmitted wave front using ultrathin planar acoustic metasurfaces. Appl Phys A - Mater 120: Zhu R, Chen Y, Barnhart M, Hu G, Sun C, Huang G (2016) Experimental study of an adaptive elastic metamaterial controlled by electric circuits. Appl Phys Lett 108: Popa BI, Cummer SA (2014) Non-reciprocal highly nonlinear active acoustic metamaterials. Nat Commun 5: Xiao S, Ma G, Li Y, Yang Z, Sheng P (2015) Active control of membrane-type acoustic metamaterial by electric field. Appl Phys Lett 106: Maznev AA, Every AG, Wright OB (2013) Reciprocity in reflection transmission: what is a phonon diode?wave Motion50: Narisetti RK, Leamy MJ, Ruzzene M (2010) A perturbation approach for predicting wavepropagation in one-dimensional nonlinear periodic structures. J Vib Acoust 132: Casalotti A, El-Borgi S, Lacarbonara W (2018) Metamaterial beam with embedded nonlinear vibration absorbers. Int J Nonlin Mech 98:32 42