Higher-Order Dispersion, Stability, and Waveform Invariance in Nonlinear Monoatomic and Diatomic Systems

Abstract

Recent studies have presented first-order multiple time scale approaches for exploring amplitudedependent plane-wave dispersion in weakly nonlinear chains and lattices characterized by cubic stiffness. These analyses have yet to assess solution stability, which requires an analysis incorporating damping. Furthermore, due to their first-order dependence, they make an implicit assumption that cubic stiffness influences dispersion shifts to a greater degree than quadratic stiffness, and they thus ignore quadratic shifts. This paper addresses these limitations by carryingout higher-order, multiple scales perturbation analyses of linearly-damped nonlinear monoatomic and diatomic chains. The study derives higher-order dispersion corrections informed by both quadratic and cubic stiffness, and quantifies plane wave stability using evolution equations resulting from the multiple scales analysis and numerical experiments. Additionally, by reconstructing plane waves using both homogeneous and particular solutions at multiple orders, the study introduces a new interpretation of multiple scales results in which predicted waveforms are seen to exist over all space and time, constituting an invariant, multi-harmonic wave of infinite A c c e p t e d M a n u s c r i p t N o t C o p y e d i t e dDownloaded From: http://vibrationacoustics.asmedigitalcollection.asme.org/pdfaccess.ashx?url=/data/journals/jvacek/0/ on 04/24/2017 Terms of Use: http://www.asme.org/aboutVIB-16-1265, Leamy 2 extent analogous to cnoidal waves in continuous systems. Using example chains characterized by dimensionless parameters, numerical studies confirm that the spectral content of the predicted waveforms exhibits less growth/decay over time as higher-order approximations are used in defining the simulations' initial conditions. Thus the study results suggest that higher-order multiple scales perturbation analysis captures long-term, non-localized invariant plane waves, which have the potential for propagating coherent information over long distances.

Emergent subharmonic band gaps in nonlinear locally resonant metamaterials induced by autoparametric resonance

“…Thus, at the first-order of perturbation, we assume that the localized excitation (boundary condition) is given by a time-harmonic displacement applied at j = 0 of the form: u e (τ ) = εû e (τ ) = ε(Û e e iω p τ +φ ep + εÛ e e iω s τ +φ es ). (11) where i is the imaginary unit, the primary frequency corresponds to the angular frequency of excitation, i.e.,ω p ≡¯ , ω s is the secondary frequency,Û e is the magnitude of the excitation, and φ e p and φ e s are, respectively, the phases of the primary and secondary harmonics. Note that the amplitude of the secondary harmonic is taken much smaller than the primary one.…”

“…In general, nonlinear interactions involve either cubic and/or quartic potentials, such as the famous Fermi-Pasta-Ulam-Tsingou (FPUT) problem [5,6], or more complex models such as the Hertz interaction law in granular materials [7][8][9][10]. Recent studies on FPUT-type phononic lattices focused mainly on weakly nonlinear interactions and the effect of nonlinearity on the dispersion relations [11][12][13][14][15][16]. In these papers, it was shown that nonlinearity induces amplitude-dependent dispersion and group velocity, i.e., demonstrating the tunability potential of nonlinear phononic structures.…”

Experimental proof of emergent subharmonic attenuation zones in a nonlinear locally resonant metamaterial

“…Despite of great interest, the number of investigations carried out on the nonlinear dynamic behavior of locally resonant metamaterials exhibiting attenuation frequency ranges is quite limited in comparison with linear ones, also because of the conceptual and modelling difficulties 16 – 20 . Only few works investigated the energy transfer mechanisms induced by the nonlinear coupling between the resonator and the host medium.…”

Transient analysis of nonlinear locally resonant metamaterials via computational homogenization

“…Recently, the role of nonlinearity on the metamaterial response has raised attention as it would enable the design of tunable structures [1]. Nonlinearities may induce amplitude-dependent dispersion and group velocities, and also the propagation of emergent waveforms, such as solitons [27][28][29]. However, only a few papers in the literature have addressed the effect of nonlinearity on locally resonant metamaterials [30][31][32].…”