A novel metamaterial multiple beam structure with internal local resonance

Abstract

This paper presents a novel locally resonant metamaterial structure and proposes a method to analyze its elastic wave dispersion properties. The structure is conceived as a multiple beam system with parallel beams transversely interconnected by periodic arrays of small resonators, each consisting of a spring-mass-spring subsystem in parallel with a couple of springs. Inserting the resonators between the beams, instead of attaching them as external appendages like in some alternative examples of locally resonant beams in the literature, makes the structure appealing for practical realization and use; furthermore, hosting several arrays of resonators gives the possibility to open multiple band gaps. The proposed method is a homogenization approach for flexural wave dispersion analysis, which removes the equations for the resonators from the set governing the dynamics of the system and reverts the original structure to an equivalent one featuring a tri-diagonal effective mass matrix with frequency dependent terms. The advantage of the homogenization approach is twofold: (1) it demonstrates that the band gaps arise in the frequency ranges where the effective mass matrix is negative definite, generalizing the well-established concept of band gaps attributable to negative mass effects in single locally resonant beams; (2) it provides dispersion curves and band gaps very efficiently, and the band gaps are identified upon calculating the eigenvalues of the tri-diagonal effective mass matrix. Analytical expressions of the band gap edges are obtained for the baseline case of a double beam system, to be readily used for design. Additionally, the exact transfer matrix method in conjunction with the Bloch theorem is formulated as alternative to the homogenization approach for wave dispersion analysis. Finally, the proposed concept of locally resonant structure and pertinent homogenization approach are validated by calculating the transmittance properties of the corresponding finite structure, via the standard finite element method.