Energy transmission in finite dissipative nonlinear periodic structures from excitation within a stop band

Abstract

We study nonlinear transmission of wave energy in a finite dissipative periodic structure, which is harmonically driven at one end at a forcing frequency lying within its stop band. We show that there is a threshold for the driving amplitude above which there is a sudden increase in the energy transmitted across the finite structure. This generic phenomenon for discrete nonlinear periodic systems is due to a loss of stability of the periodic solutions that are initially localized to the driven end of the structure. The transmission threshold is therefore predicted analytically based on the corresponding saddle-node bifurcation. The influence of damping, strength of coupling and the type of nonlinearity (hardening or softening) are assessed. In particular, we show that damping may eliminate the transmission phenomenon within a frequency range in the stop band. Increasing the strength of coupling between the units is found to increase the minimum force required for the onset of transmission, and the type of nonlinearity determines on which side of the pass band the enhanced transmission may occur.