Predicting wave propagation in periodic lattices with internal resonators

Abstract

Periodic lattice metamaterials like that of the square honeycomb, hexagonal honeycomb, and kagomè lattices have proven themselves useful in structural applications with their ability to strongly suppress vibrations over a broad range of frequencies at subwavelength scales. There have been many attempts in recent years to supplement their dynamic features with the use of internal resonators, creating additional band gaps for heightened overall performance. However, the broad definition of an internal resonator in this context means that it can be difficult to model all cases extensively. In an attempt to help bridge the analytical gap, we consider 2-D lattice designs with incorporated cantilever beam-type internal resonators. Doing so allows for a FEM technique to be used that treats each lattice, and its embedded resonators, as a collection of Timoshenko beams. With it, we can create global mass and stiffness matrices that serve as the driving force behind the characterization of band diagrams and their modal structure. A semi-analytical modeling approach is introduced which highlights the explicit dependence of band diagram features on resonator parameters. The general methodology is discussed, and numerical results are compared with simulated results generated through COMSOL Multiphysics.