Wave Dispersion in One-Dimensional Nonlinear Local Resonance Phononic Crystals with Perturbation Method

Abstract

Nonlinear phononic crystals are receiving increasingly greater attention in the field of sound absorption and vibration reduction. In this paper, we use the perturbation method to investigate elastic wave propagation in one-dimensional discrete local resonance nonlinear phononic crystals. The nonlinear force on the inner resonator is expressed in the form of a linear part plus a cubic nonlinear fluctuation. By combining Bloch wave theory and the perturbation method, the nonlinear dispersion relation is obtained by a first-order approximate analytical solution. The results show that the band’s cut-off frequency is not only affected by the degree of nonlinearity but is closely related to the wave amplitude. In addition, the finite element method is used for comparison and verification. Finally, an application example of a wave filter is provided based on the nonlinear characteristics.

Phononic Band Structure by Calculating Effective Parameters of One-Dimensional Metamaterials

“…In the context of recent perspectives on the future progress in the field of study of the metamaterials, we can mention as examples the following three works: (1) Soo-Ho Jo, et al proposed the design of 2D phononic crystals introducing defects with piezoelectric patches; these structures allow the harvesting and localization of the energy of elastic waves in broadband frequencies [32]. (2) Yan Chen, et al applying a method based on the perturbative analysis, studied the dynamic behavior of a 1D nonlinear mass-in-mass phononic crystal to calculate the dispersion relation and demonstrate that by modifying the wave amplitude and the degree of nonlinearity, the band-gap frequency range can be adjusted (validating the results with finite element simulations) [33]. Lastly, (3) Luyun Chen, et al using a finite element model of a 2D hexachiral phononic crystal (whose main characteristic is that it does not have a complete mirror symmetry), investigated the acoustic transmission and tunneling in a Dirac-cone state, with applications oriented towards wave manipulation [34].…”