  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.
