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

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 the 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 factorization for the spatial and temporal parts of the solution and a periodic distribution function as ansatz of a general solution of the temporal part of the nonlinear equations of motion. A comparison with numerical simulations shows the range of validity of this expression. This work provides a tool to design and study nonlinear dynamics for some classes of seismic metamaterials such as composite foundations, perturbative metamaterials, and metasurfaces.