Introduction of Local Resonators to a Nonlinear Metamaterial With Topological Features

Abstract Metamaterials have been shown to benefit from the addition of local resonators, nonlinear elements, or topological properties, gaining features such as additional bandgaps and localized vibration modes. However, there is currently no work in the literature that examines a metamaterial system including all three elements. In this work, we model a 1-dimensional metamaterial lattice as a spring-mass chain with coupled local resonators. Quasiperiodic modulation in the nonlinear connecting springs is utilized to achieve topological features. For comparison, a similar system without local resonators is also modeled. Both analytical and numerical methods are used to study this system. The infinite chain response of the proposed system is solved through the perturbation method of multiple scales. This analytical solution is compared to the finite chain response, estimated using the method of harmonic balance and solved numerically. The resulting band structures and mode shapes are used to study the effects of quasiperiodic parameters and excitation amplitude on the system behavior both with and without the presence of local resonators. Specifically, the impact of local resonators on topological features such as edge modes is established, demonstrating the appearance of a trivial bandgap and multiple localized edge states for both main cells and local resonators.