Dispersion analysis of lattices with spatially modulated stiffness nonlinearity

In this work, we report amplitude-dependent band structures in mechanical lattices with spatially modulated weak cubic stiffness. In contrast to lattices characterized by spatially uniform nonlinearity, the lattices studied herein possess a remarkable ability to filter, guide, and redirect the propagation of nonlinear acoustic waves. We use a multiple scale perturbation analysis to derive amplitude-dependent dispersion relationships in a generalized spatially modulated lattice. Nonconventional results are presented: the existence of hardening and softening within the same branch, the merging of branches at specific amplitudes, and the broadband cancelation of dispersion shifting altogether. Direct numerical simulation of the lattice equations of motion provides validation for the analytical predictions, and a method of physically realizing the system is proposed. Findings may inspire nonlinear devices for enhanced vibration control and mitigation.