Analysis of stability, waveform invariance, and non-reciprocity in nonlinear periodic structures

Abstract

Wave propagation in periodic structures and metamaterials has attracted the interest of the engineering community due to the presence of bandgaps, or forbidden frequency ranges of propagation, which have inspired devices to include filters, diodes, switches, and waveguides. Geometric and material nonlinearities present in periodic structures result in propagation features (e.g. direction, cut-off/cut-on frequencies, group velocity) dependent on wave amplitude, thereby providing an additional means to manipulate wave propagation. This work explores waveform invariance, stability, and non-reciprocity as enabled by nonlinear periodic structures. One-dimensional and two-dimensional discrete lattices are considered. Waveform invariance, in which a multi-harmonic solution persists without dispersing for all space and time, and plane wave stability, in which high amplitude waves undergo significant distortion of their spectral content, are informed by perturbation analysis of the lattices’ equations of motion. Extensions of the invariant plane wave solutions include special fundamental frequencies that avoid higher-harmonic generation and a slow-scale energy modulation between internally-resonant wave propagation modes. Non-reciprocity exploits a preferred energy transfer that occurs from large to small scales, coupled by strongly nonlinear restoring forces, to create a periodic lattice with highly asymmetrical wave propagation. Theoretical findings are validated with numerical simulations and experimental testing and may apply to damage detection, data encryption, and shock and vibration mitigation.