Closed-form formulas for bandgap estimation and design of metastructures undergoing longitudinal or torsional vibration

Abstract

Metastructures can be employed to control waves/vibrations owing to their unique bandgap behaviour. There are two problems that are of great concern in the design of metastructures: (1) How to predict the bandgap frequency range quickly and accurately? (2) How to design a desired bandgap with broad width and large attenuation performance within the weight and size constraints? This work aims to explore these problems by addressing a kind of metastructures composed of a periodic array of local resonators attached to a host structure undergoing longitudinal/torsional vibration. Such structures have both fundamental and practical significance. For the first problem, closed-form formulas of the bandgap edge and centre frequencies are derived. These formulas enable a direct and accurate estimation of the lowest two bandgaps of the considered metastructures within the entire design space, being more general and applicable than well-known existing closed-form formulas, which are only applicable to the lowest bandgap created in the deep-subwavelength regime. For the second problem, a guideline is proposed for the design of a sub-wavelength region bandgap with simultaneously significant and robust attenuation performance at its centre frequency, and an improved bandgap width than the conventional deep-subwavelength design. Furthermore, closed-from design formulas for optimal selection of system parameters are provided. The applicability of the proposed closed-form formulas and guideline is validated by numerical examinations of simple and advanced realizable metastructures.