Theoretical calculations of vibration reduction band gaps characteristics of novel phononic-like crystal Euler beam structure

The elastic wave band gap (BG) property of locally resonant phononic crystal (LRPC) enables it to control and attenuate the propagation of elastic waves in its internal structure in the BG frequency range. In particular, it has unique advantages in obtaining low-frequency BG and can be used for the control and attenuation of low-frequency vibration. Therefore, the LRPC is of great significance in the field of low-frequency vibration reduction and has broad application value and prospect. However, the LRPC has the defects of narrow BG width and only one BG, which makes it limited in engineering application. In response to the problems of narrow BG width and only one BG in traditional LRPC Euler beams, this paper establishes a novel phononic-like crystal (PLC) Euler beam model that can open up multi-frequency and low-frequency vibration reduction BGs. The PLC Euler beam model considers the non rigidity of the contact interface between the wrapping layer and the scatterer and matrix, the foundation constraint effect, and the damping characteristics of the wrapping layer material. The vibration reduction BG characteristics of the PLC Euler beam are studied through theoretical calculations and analysis. Firstly, the improved transfer matrix method (ITMM) is derived for calculating the band structure of the PLC Euler beam, and the results are compared and verified with those obtained from the finite element method (FEM). Secondly, the spectral element method (SEM) is derived for calculating the frequency response function (FRF) of the PLC Euler beam, and to evaluate its attenuation effect on vibration elastic waves. The theoretical calculation results show that the PLC Euler beam opens up 6 low-frequency vibration reduction BGs, exhibiting good attenuation effects on vibrational elastic waves within the BG frequency range, with the maximum attenuation value at the BG starting frequency. The novel PLC Euler beam has a broad prospect in controlling and attenuating low-frequency and multi-frequency vibrations, and its design idea has a unique novelty, which provides theoretical support and design idea for broadening the BG width and number of LRPC Euler beam structures, and provides theoretical methods and ideas for designing and studying LRPC with low-frequency and multi-frequency BGs.