Spectro-spatial analyses of a nonlinear metamaterial with multiple nonlinear local resonators

Recent focus has been given to spectro-spatial analysis of nonlinear metamaterials since they can predict interesting nonlinear phenomena not accessible by spectral analysis (i.e., dispersion relations). However, current studies are limited to a nonlinear chain with single linear resonator or linear chain with nonlinear resonator. There is no work that examines the combination of nonlinear chain with nonlinear resonators. This paper investigates the spectro-spatial properties of wave propagation through a nonlinear metamaterials consisting of nonlinear chain with multiple nonlinear local resonators. Different combinations of softening and hardening nonlinearities are examined to reveal their impact on the traveling wave features and the band structure. The method of multiple scales is used to obtain closed-form expressions for the dispersion relations. Our analytical solution is validated via the numerical simulation and results from the literature. The numerical simulation is based on spectro-spatial analysis using signal processing techniques such as spatial spectrogram, wave filtering, and contour plots of 2D Fourier transform. The spectro-spatial analysis provides a detailed information about wave distortion due to nonlinearity and classify the distortion into different features. The observations suggest that nonlinear chain with multiple nonlinear resonators can affect the waveform at all wavelength limits. Such nonlinear metamaterials are suitable for broadband vibration control and energy harvesting, as well as other applications such as acoustic switches, diodes, and rectifiers, allowing wave propagation only in a pre-defined direction.