An improved method for bandgap calculation of a locally resonant plate with multi-periodic of multiple degree-of-freedom resonators

Abstract

An improved plane wave expansion (PWE) method is developed based on the governing equations of a lattice unit cell, providing support for calculating the band structures of a locally resonant (LR) plate with multiple arrays of multiple degree-of-freedom resonators. The extended plane wave expansion (EPWE) method is correspondingly presented to obtain the complex band structures of the LR plate. Then the bandgap properties of LR plate systems with single/multiple arrays of single/multiple degree-of-freedom resonators are analyzed, which present a good agreement with the results calculated by the classical theory or finite element method. Compared with the classical theory, the improved PWE and EPWE methods have excellent efficiency and broad applicability for LR plate systems with complex attachments. Furthermore, when the damping of resonators is taken into account, the merging of the multiple bandgaps can be observed using the EPWE method adopted in this paper, which gives a useful tool to analyze the band structures of an LR plate with merged bandgaps for broadband vibration suppression.

A multi-bandgap metamaterial with multi-frequency resonators

“…According to the PWE method, 41,44,52 the solutions of equations (4) and (5) are first sought as harmonic solutions; that is, w∼(boldx∼,t∼)=w¯(truex∼)e−iω∼t∼ and truez∼ij(truet∼)=truez¯ije−iω∼t∼. Moreover, due to the space periodicity of the resonators lattice, according to the Floquet-Bloch theorem, the plate and resonators displacements are quasi-periodic functions (in space) expressed in a Fourier series of the reciprocal-lattice vector trueG∼mn=a(mboldb1+nboldb2) in which the basis vectors of the reciprocal lattice are given by boldb1=2π/a(bolde1−1/3bolde2) and boldb2…”

“…42 The one-dof resonator system is characterized by one complete LF bandgap only, whereas, by introducing multi-dof resonators, the number of bandgaps of the LR structures can be increased. 43 Li and Sheng 44 studied a LR isotropic plate, hosting multiple arrays of multi-dof spring-mass resonators, addressing the location, bandwidth, and attenuation strength of the bandgaps.…”