1D / Quasi-one-dimensional Variable Cross-sectional Study Of Phononic Crystal And The Application Of Ultrasound In Power

Phononic crystals (PCs) are the periodic structures or functional materials made of different elastic properties materials, which exhibit acoustic or elastic wave band gaps (BGs). When the elastic/acoustic wave propagates in the PCs, the wave will be affected by the periodic structure, and the phononic BGs produce, i.e. sound and vibration in the band gaps are forbidden. Due to the unique and novel BGs, localization, sound focusing, negative refraction characteristics and so on, the PCs have a broad potential application in vibration attenuation and isolation, noise control and sound functional devices.In some applications such as vibration attenuation and isolation, the realization of small size and low frequency device (tens of Hz, and even a few Hz or even lower) is a difficult issue. In the front part of this paper, the characteristics of the band gap are studyed. We found that, compared with the constant cross-section phononic crystals (CCPCs), the variable cross-section phononic crystals (VCPCs) have the wider bandwidth, the lower BGs frequency, and the better performance in the BGs. In the latter part of the article, the relationship between the band structure of VCPCs and the resonant frequency of the structure in power ultrasound is investigated, and a new method for the calculation of the complex structure’s resonance frequency is provided.The main research contents and conclusions of this paper include:(1) Several commonly used methods are introduced for the BG calculation of1D PCs. Compared with the transfer matrix method (TM), the convergence of the lump-mass method (LM) which is mainly used in this paper is discussed.(2) Based on the idea of LM, the1D hourglass PCs with conical section are simplified to the infinite periodic mass-spring structures; and the formula of the BG calculation for longitudinal wave propagated in the hourglass PCs is derived.(3) Combined with the LM and finite-difference time-domain method (FDTD), the frequency response function (FRF) for the longitudinal wave propagating in the quasi-ID PCs with exponential section is derived.(4) The influence of the radius on the BGs. For the hourglass PCs with conical section, the greater difference of the radii, the greater influence of the BGs, i.e. the initial frequency will be lower and the cut-off frequency will be higher; it is helpful for the increasement of the bandwidth and the BGs’movement to the low-frequency. For the quasi1D VCPCs, the initial frequency decreases and the cut-off frequency increases as the increasement of the radius of the output terminal; it is also helpful for the increasement of the bandwidth and the BGs’ movement to the low-frequency; on the contrary, the bandwidth is reduced and the band gap moves to the high-frequency. While the ratios of the radii are the same, the hourglass PCs have wider bandwidth and lower band BG frequency than those of the quasi1D VCPCs.(5) The influence of the radius on the attenuation performance within the BGs. The analysis results calculated via finite element (FE) method and FDTD method show that, for the two structures investigated in this paper, the FRF values within the BG will be reduced if the radii are changed. Compared the two structures with each other, the FRF values of the hourglass PCs are less than those of quasi1D VCPCs, so, the attenuation performance is better than that of quasi1D VCPCs. For the CCPCs, the attenuation performance within the BGs is improved by increasing the number of the cell. However, in the paper, another method is presented, i.e. the attenuation performance is improved by changing the ratio of the radii. It is helpful for the PCs in the practical applications.(6) The influence of the lattice constant on the BGs. For the VCPCs and CCPCs, the BGs will be moved to the low-frequency and the bandwidth will be reduced, as the lattice constant increases. But for the VCPCs, the initial frequency is less and the cut-off frequency is higher than those of CCPCs.(7) The influence of the filling fraction of materials on the BGs. As the filling fraction changed, the quasi1D VCPCs exhibit BGs as the CCPCs. For the1D hourglass PCs, the relationships between the filling fraction and the BGs are more complex. For the CCPCs, when the proportion of one material closed to0or1, the bandwidth will be closed to zero; however, the bandwidth of the VCPCs will be closed to a non-zero value.(8) The influence of the material parameters on the BGs. The parameters of one material remain unchanged, when the density (or Young’s modulus) of the other material is much larger (or far less) than the density (or Young’s modulus) of this material, the effects of the material parameters and BGs are discussed. The results show that the VCPCs exhibit the same relationships as the CCPCs.(9) The rod with variable cross-section can be seen as a special VCPC which contains only one material. For the hourglass variable cross-section rod, compared the band structure via LM method, and response frequency curve via FE method with the resonance frequency via resonance frequency function, we found that, the initial frequency of the odd band gap is equal to the odd-order resonance frequency of cell with free boundary, and the cut-off frequency of the even band gap is equal to the even-order resonance frequency of cell with free boundary.The difference between the VCPCs and the CCPCs can be explained as:first, the equivalent lumped parameters (equivalent lumped mass and equivalent stiffness coefficient) are changed as the radius changed, and the BGs frequencies are changed by the change of the equivalent lumped parameters. Secondly, the VCPCs can be seen as a special multi-component CCPC, and the material parameters are changed with the position, the change of the material parameters will cause the changement of the frequency of the band gap. Finally, when the acoustic wave propagated in the medium, the volume velocity is continuous, i.e. the particle vibration velocity will be decreased as the radius increase, and the vibration displacement will be decreased, too. So, for the quasi1D finite period PCs, the frequency response curves will move down, and cause the change of the BGs characteristics.We hope, the investigations in this paper are helpful for the PCs in vibration attenuation and isolation, noise control and the calculation of the ultrasonic horn (working tool) with complex structure in power ultrasound.