Nonlinear dispersion relation in anharmonic periodic mass-spring and mass-in-mass systems

Abstract

The study of wave propagation in chains of anharmonic periodic systems is of fundamental importance to understand the response of dynamical absorbers of vibrations and acoustic metamaterials working in nonlinear regime. Here, we derive an analytical nonlinear dispersion relation for periodic chains of anharmonic mass-spring and mass-in-mass systems resulting from considering the hypothesis of weak anharmonic energy and a periodic distribution function as ansatz of a general solution of the nonlinear equations of motion. Numerical simulations show that this expression is valid for anharmonic potential energy up to 50% of the harmonic one. This work provides a simple tool to design and study nonlinear dynamics for a class of seismic metamaterials.

Low thermal conductivity in a modular inorganic material with bonding anisotropy and mismatch

“…3C). The anharmonicity of the TA phonons is demonstrated by the non-linearity in the ω(k) dispersion at low k (35)(36)(37). The second derivative of the dispersion of BiOCl (Fig.…”

Stochastic analysis of locally resonant linear and hysteretic metamaterials for seismic isolation of process equipment

“…Moreover, they also found that optical wave modes in nonlinear metamaterial are sensitive to parameters of the nonlinear cubic constitutive relation. Zivieri et alii [24] started from the factorization of the spatial and temporal parts of the solution and a periodic distribution function as ansatz of a general solution of the temporal part of the nonlinear equations of motion (EoM). Hence, they derived an analytical nonlinear dispersion relation for nonlinear periodic mass-spring and mass-in-mass systems.…”

“…More precisely, for a SDoF system with N = 1, the nonlinear differential equation Eq. ( 7) becomes, (24) where ceq and keq are linearization coefficients that are "equivalent" in a statistical sense [27,28,37,38]. At this stage, it is useful to introduce a state-space formulation of Eq.…”

Localized Energy Absorbers in Hertzian Chains

“…Mass-in-mass (MIM) systems [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] and nonlinear energy sinks (NESs) [19][20][21][22] are promising strategies for designing tailored metamaterials with unusual acoustical and mechanical properties. MIM lattices have been investigated for more than a decade [1][2][3], leading to stimulating concepts and applications.…”