On the treatment of connecting conditions and the choice of admissible functions in energy method to investigate band gap characteristics of periodic beams

Abstract

Periodic structures are known to have specific frequency ranges called band gaps, where the transmission of elastic waves is suppressed. Research on band gap formation in mechanical structures has recently gained considerable attention, and there is a need for analytically versatile and cost-effective methods to address the band gap effect in various structural elements. In this paper, we investigate band gap characteristics and frequency response analysis of periodic beams using an energy method that allows for easy extension to plates and shells. In using the energy method for vibration problems with non-uniform systems like periodic structures, the selection of admissible functions becomes crucial, and the treatment of boundary and connecting conditions is also essential. This study focuses on those subjects by employing elastic beams with a periodic structure of unit cells; a unit cell consists of two parts with different structural characteristics. Since band gaps can be determined by dispersion curves, the analytical process for obtaining dispersion curves by the energy method is described in detail in this paper. Three treatments to express a unit cell and three admissible functions are tested. Calculated dispersion curves are compared with those obtained by the differential quadrature method (DQM), which is widely used in band gap research. The frequency response analysis is also carried out using the energy method and compared with the results by finite element analysis using ANSYS and experimental results, and good agreement has been demonstrated among them. No resonant peaks exist in the frequency range corresponding to the band gap in the dispersion curves for both experimental and numerical results, and this shows that the formation of the band gap in the actual periodic beam and the validity of the numerical approach using the energy method for dispersion analysis and FRF calculation of periodic beams.

Keywords

• band gap
• periodic structure
• vibration analysis
• energy principle
• dispersion curve
• frequency response analysis