Amplitude Dependent Dispersion in Essentially Nonlinear Periodic Chains

Abstract

Wave dispersion in one dimensional essentially nonlinear chains is investigated using generalized harmonic balance method. The general approach presented is applied to predict dispersion in nonlinear chains with Hertzian contact interaction [1]. A compressive load at the two ends of a chain determines the magnitude of the nonlinearity. The cut-off frequencies and band-gaps exhibited by mono-atomic and diatomic chains are shown to be amplitude dependent. Amplitude dependent dispersion predicted by harmonic balance method agrees with the trend predicted by perturbation analysis [2]. In contrast to other approximate methods which are limited to small amplitude waves, the main advantage of the harmonic balance method is the ability to examine the high amplitude effects on wave properties. This includes the generation of multi-harmonic frequencies and significant variation in band-structure which can be exploited to design tunable acoustic devices. Applying a numerical algorithm based on generalized harmonic balance framework, the present study depicts a significant variation in band-gap between low and high wave amplitudes in essentially nonlinear periodic chains.