Study of wave propagation in strongly nonlinear periodic lattices using a harmonic balance approach

This paper presents a general harmonic balance method for studying plane wave propagation in strongly nonlinear periodic media. The proposed approach starts by assuming a multi-wavenumber and frequency solution for the unit cell degrees of freedom. A Galerkin projection then transforms the nonlinear differential equations of motion into a set of nonlinear algebraic equations, which are subsequently solved numerically through a Newton-like iteration scheme. These solutions reveal amplitude-dependent dispersion behavior and group velocities. Specific example systems studied include one-dimensional chains and two-dimensional lattices, both formed by a periodic arrangement of spheres interacting under a Hertzian contact law. Amplitude-dependent dispersion is noted in monatomic and diatomic chains, and in hexagonally close-packed two-dimensional lattices. The validity of the presented technique is assessed through direct numerical simulation of the equations governing finite-extent lattices. Strong agreement is documented for results calculated using the harmonic balance approach and the direct numerical simulations. (c) 2011 Elsevier B.V. All rights reserved.