Wave propagation in one-dimensional nonlinear acoustic metamaterials

Abstract

The propagation of waves in nonlinear acoustic metamaterial (NAM) is fundamentally different from that in conventional linear ones. In this article we consider two one-dimensional (1D) NAM systems featuring respectively a diatomic and a tetratomic meta unit-cell. We investigate the attenuation of waves, band structures, and bifurcations to demonstrate novel nonlinear effects, which can significantly expand the bandwidth for elastic wave suppression and cause nonlinear wave phenomena. The harmonic averaging approach, continuation algorithm, and Lyapunov exponents (LEs) are combined to study the frequency responses, nonlinear modes, bifurcations of periodic solutions, and chaos. The nonlinear resonances are studied, and the influence of damping on hyperchaotic attractors is evaluated. Moreover, a 'quantum' behavior is found between the low-energy and high-energy orbits. This work provides a theoretical base for furthering understandings and applications of NAMs.in NPSs would be relevant to those properties. Acoustic devices such as diodes [36] and lenses [37] can be built upon them. The strong NPS has been proved to increase the velocity of sound and therefore the acoustic impedance [38]. Moreover, the bandgap properties in NPSs attract much attention [39][40][41][42][43][44]. The perturbation approach and the harmonic balance method (HBM) are adopted [40][41][42] to study the amplitude-dependent dispersions, stop band properties, and wave beaming in granular crystals; and experimental works highlighted the role played by the critical amplitude in energy transmission [43] and bifurcation-induced bandgap reconfiguration [44] in NPSs. Actually, the granular crystals are suitable for ultrasonic applications; it is hard to consider them at the low-frequency regime because of the high contact stiffness they inherently feature.Because of their promising applications, AMs with both low-frequency and broadband properties attract much attention. However, the mechanisms for both properties to occur simultaneously are difficult to realize. LAMs consist of linear 'meta-atoms', but when this meta-atom becomes nonlinear in NAMs, wave propagation properties show different patterns. In our recent works [45,46], the wave propagation in diatomic and tetratomic NAMs are analyzed using the homotopy analysis method, and we found that the chaotic bands resulting from bifurcations can significantly enlarge the width of the wave-suppressing bands. This finding demonstrates that chaos is a novel and promising mechanism to simultaneously achieve low-frequency and broadband in both mono-bandgap NAMs and multi-bandgap NAMs; this finding also reiterates still that a strong nonlinearity is beneficial to expand the bandwidth by several times.However, there are many phenomena arising in NAMs that have not yet been fully explained nor demonstrated. For example, why are the responses in the first passband similar to those observed with LAMs? Under which conditions can the elastic energy propagate in the bandgap? When will the wave be amplified by c...

Ultra-low and ultra-broad-band nonlinear acoustic metamaterials

“…Therefore, the passbands of the strongly nonlinear AMs can significantly attenuate the elastic waves and enables subwavelength properties with a = λ f /9 (see Methods). In the first passband, the influences of the nonlinearity on the three resonances are: significant for 31.56 Hz, moderate for 16 Hz ( H ( ω ) decreases by 8 dB) and weak for 6 Hz ( H ( ω ) unchanged), suggesting that the lower the resonance frequency below LR1, the weaker the effect of the nonlinearity 47, 49 .…”

“…Recently, a mechanism was theoretically predicted in discrete NAMs 47–49 : the chaotic bands. Chaos is an aperiodic long-term behavior in a deterministic/nonlinear system exhibiting a strong dependence on the initial conditions 50 .…”

“…Chaos is an aperiodic long-term behavior in a deterministic/nonlinear system exhibiting a strong dependence on the initial conditions 50 . In NAMs 49 , chaotic bands are those passbands in which an incident low-frequency periodic wave becomes a chaotic emerging wave, reducing wave transmission. The chaotic wave features a high-frequency continuous spectrum evidencing the dispersion of energy 47, 49 .…”

“…In NAMs 49 , chaotic bands are those passbands in which an incident low-frequency periodic wave becomes a chaotic emerging wave, reducing wave transmission. The chaotic wave features a high-frequency continuous spectrum evidencing the dispersion of energy 47, 49 . These waves have lower amplitudes than the corresponding linear resonances; thus, NAMs can suppress wave propagation in the passbands.…”

“…These waves have lower amplitudes than the corresponding linear resonances; thus, NAMs can suppress wave propagation in the passbands. The wave suppression effect of the strong chaos is broadband and it depends on the frequency but not the width of the nonlinear LR bandgap 49 . Therefore, we can design a NAM with nonlinear meta-cells that generate ultra-low frequency but narrow linearized LR bandgaps.…”