Elastic Wave Propagation in Strongly Nonlinear Lattices and Its Active Control

Abstract

This work studies elastic wave propagation in strongly nonlinear periodic systems and its active control with specific attention to an infinite mass-in-mass lattice. Piezoelectric materials are applied to it to provide active control loads to manipulate band structures of the lattice. Governing equations of the active mass-in-mass lattice with cubic nonlinearities are established. The control loads are modeled by using linear piezoelectric springs. Due to phase differences among vibrations of different cells during wave propagation, a series of delay functions with different delays are used to represent the steady-state of a traveling wave. The incremental harmonic balance method for delay dynamic systems is employed in this case to calculate periodic solutions of the lattice. The fast Fourier transform is employed to construct the Jacobian matrix of the Newton–Raphson iteration to avoid a large number of Galerkin integrations, and thus, the efficiency is significantly improved. Amplitude-dependent dispersion curves are calculated using results of the linearized system as an initial guess for the iteration. The results are compared with existing results in the literature, which demonstrates that the present method is efficient for wave propagation analysis of strongly nonlinear structures. Effects of nonlinearities, the mass ratio, and different control actions on band structures of the mass-in-mass lattice are investigated through a comprehensive parametric study. Numerical results show that the band structures can be influenced by nonlinearities of the lattice. New stopbands and critical wave numbers can be created by the control actions.