Nonlinear dynamics of spatially modulated topological metamaterials with nonlinear local resonators

Abstract Low-frequency wave control is essential in vibration isolation, structural acoustics, and energy harvesting, where compact devices must function below dominant resonant bands. Although nonlinear topological metamaterials enable amplitude-dependent waveguiding and filtering, their practical response is often concentrated at relatively high frequencies, with weak influence in the low-frequency regime. Therefore, in this study, nonlinear local resonators are incorporated into a spatially modulated, locally resonant topological lattice to shift the dominant nonlinear effects to lower frequencies. In this analysis, we model the metamaterial as a spring–mass chain with spatially modulated primary-cell stiffnesses and a local resonator attached to each mass. Three combinations of stiffness nonlinearity are examined: primary-cell only, resonator only, and simultaneous primary-cell–resonator nonlinearity. The method of multiple scales is used to obtain an approximate amplitude-dependent dispersion relation for an infinite chain, while harmonic balance is employed to compute the band structure and associated localized mode profiles. These analytical results are verified by direct time-domain numerical integration. From the analysis, we observe that nonlinear effects are strongest in the vicinity of the resonator frequency. Furthermore, when nonlinearity is localized in the resonators, the topologically trivial locally resonant bandgap shifts with amplitude and low-frequency amplitude-tunable edge modes and discrete breathers emerge at substantially lower frequencies than in the chain-only nonlinear case, where the strongest amplitude-dependent effects remain concentrated in the higher-frequency bands. Overall, the proposed metamaterial architecture enables controllable tuning of nonlinear wave phenomena and broadens the effective operating range.