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

## TL;DR

This paper derives an analytical nonlinear dispersion relation for anharmonic periodic mass-spring systems, aiding the design of nonlinear seismic metamaterials and advancing understanding of wave propagation in nonlinear regimes.

## Contribution

It introduces a novel analytical dispersion relation for anharmonic chains, validated for up to 50% anharmonic potential energy, facilitating nonlinear metamaterial design.

## Key findings

- Analytical dispersion relation valid for significant anharmonicity
- Numerical validation confirms accuracy up to 50% anharmonic energy
- Provides a practical tool for designing nonlinear seismic metamaterials

## 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.

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Source: https://tomesphere.com/paper/1812.07953