Nonlinear dynamics of a model of acoustic metamaterials with local resonators

Nonlinear dynamics of a model of acoustic metamaterials with local resonators are investigated numerically and theoretically. We focus on dynamics of band edge modes (BEMs) and zone boundary modes (ZBMs) which are on the upper bounds of acoustic bands and optical bands of the phonon dispersion band. It is found that, in a region of weak anharmonicity, higher harmonics of a fundamental mode and static displacement are excited in both BEM and ZBM if the geometrical relation between the main lattice and the local resonators has evenorder nonlinearity. Numerical solutions of nonlinear periodic orbits which are continued from vibrations in the small amplitude limit by the shooting method indicate that structure of the periodic orbits of the local resonators depends on the form of nonlinear terms of the geometrical relation. Moreover, the nonlinear periodic orbits become unstable when the amplitude of the periodic orbit becomes larger. Direct numerical simulations show that unstable dynamics occur due to modulational instability. After destabilization of the nonlinear periodic orbits, spatial energy localization is also observed.