Nonlinear Dynamics of Spatially Modulated Topological Metamaterials with Nonlinear Local Resonators

Low-frequency wave control is essential in vibration isolation, structural acoustics, and energy harvesting, where compact devices must operate away from high resonant bands. While non-linear topological metamaterials can provide amplitude-dependent waveguiding and filtering, they commonly suffer from having high operational frequencies, with weak effects at lower frequencies. To address this, the current work leverages nonlinear local resonators in a spatially modulated, locally resonant topological metamaterial to shift the strongest nonlinear effects toward lower frequencies. In the current work, the proposed nonlinear metamaterial is modeled as a spring-mass chain with spatial modulation in the main cell stiffnesses and with each mass in the chain coupled to a local resonator. Different combinations of nonlinearities in main cell stiffness and the resonator's stiffness are analyzed. The method of multiple scales is used to derive an approximate analytical solution for the amplitude-dependent dispersion relations of an infinite chain, while the harmonic balance method is used to determine the frequency band structure and localized mode shapes. Furthermore, all results are validated numerically through direct numerical integration. From the resulting band structures and mode shapes, we observe that the presence of nonlinearity in the resonators causes the strongest nonlinear effects at frequencies nearest to the resonator frequency. In particular, resonator-localized nonlinearity shifts the locally resonant gap and can yield amplitude-tunable edge modes and discrete breathers at lower frequencies in comparison with chain-only nonlinearity. The observations suggest that the proposed metamaterial is capable of tuning its nonlinear effects and producing a larger operational frequency.