Abstract
The central wavelength of the first Bragg scattering bandgap is approximately twice that of the lattice. Therefore, a low-frequency Bragg scattering bandgap with a small structural dimension for phononic crystals is difficult to obtain. In this study, a folded S-type periodic structure is developed to reduce the dimension in the direction of vibration suppression by folding unit cells. According to the foregoing, an improved folded S-type periodic structure with different unit cell arrangements is designed to widen the bandgap frequency range. Energy band diagrams and frequency responses are calculated based on the Bloch theory and using the finite element method. Furthermore, a prototype of the improved folded S-type periodic structure is fabricated using a three-dimensional printing technique, and a vibration experiment is conducted. To verify the vibration reduction performance of the structure, numerical simulation and experimental results are compared. This type of folded periodic structure can effectively reduce dimensions to satisfy the dimension requirements pertaining to the direction of vibration suppression. Hence, the foregoing can aid in promoting the use of elastic bandgap structures in engineering.
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References
Wang YF, Wang YZ, Wu B, Chen WQ, Wang YS. Tunable and active phononic crystals and metamaterials. Appl Mech Rev. 2020;72:040801.
Liu ZQ, Zhang X, Mao Y, Zhu YY, Yang Z, Chan CT, Sheng P. Locally resonant sonic materials. Science. 2000;289:1734–6.
He JX, Li HL, Tian YH, Zhang QZ, Lu ZX, Lan JX. Numerical analysis of viscous dissipation in microchannel sensor based on phononic crystal. Micromachines. 2021;12(8):994.
Li LJ, Gang XY, Sun ZY, Zhang XX, Zhang F. Design of phononic crystals plate and application in vehicle sound insulation. Adv Eng Softw. 2018;125:19–26.
Gu Y, Lan P, Cui Y, Li K, Yu Z. Dynamic interaction between the transmission wire and cross-frame. Mech Mach Theory. 2021;155:104068.
Sigalas MM, Economou EN. Elastic and acoustic wave band structure. J Sound Vib. 1992;158:377–82.
Martinez-Salar R, Sancho J, Sanchez JV, Gomez V, Linares J, Meseguer F. Sound attenuation by sculpture. Nature. 1995;378:241.
Ferrando V, Castro-Palacio JC, Marí B, Monsoriu JA. Study on band gap structure of Fibonacci quantum superlattices by using the transfer matrix method. Mod Phys Lett B. 2014;28:1450053.
Dal Poggetto VF, Serpa AL. Flexural wave band gaps in a ternary periodic metamaterial plate using the plane wave expansion method. J Sound Vib. 2021;495:115909.
Xie LX, Xia BZ, Liu J, Huang GL, Lei JR. An improved fast plane wave expansion method for topology optimization of phononic crystals. Int J Mech Sci. 2017;120:171–81.
Sainidou R, Stefanou N, Psarobas IE, Modinos A. A layer-multiple-scattering method for phononic crystals and hetero structures of such. Comput Phys Commun. 2005;166(3):197–240.
Sun JH, Wu TT. Propagation of acoustic waves in phononic-crystal plates and waveguides using a finite-difference time-domain method. Phys Rev B. 2007;76(10):104304.
Shi LL, Liu N, Zhou JY, Zhou YG, Wang JM, Liu QH. Spectral element method for band-structure calculations of 3D phononic crystals. J Phys D-Appl Phys. 2016;49(45):455102.
Wu ZJ, Li FM, Zhang C. Vibration band-gap properties of three-dimensional Kagome lattices using the spectral element method. J Sound Vib. 2015;341:162–73.
E LZY, Wu Z, Zou G, Li F, Zhang C, Sun A, Du Q. Band-gap characteristics of elastic metamaterial plate with axial rod core by the finite element and spectral element hybrid method. Mech Adv Mater Struct. 2020. https://doi.org/10.1080/15376494.2020.1863531.
Chin EB, Mokhtari AA, Srivastava A, Sukumar N. Spectral extended finite element method for band structure calculations in phononic crystals. J Comput Phys. 2021;427:35.
Krushynska AO, Kouznetsova VG, Geers MG. Towards optimal design of locally resonant acoustic metamaterials. J Mech Phys Solids. 2014;71:179–96.
Lu QF, Liu CC, Qin ZH, Ma WS, Li FM. Vibration control and band gap tuning of finite periodic structure composed by active functionally graded metamaterial bars. Mech Adv Mater Struct. 2022;10:1–14.
Sorokin VS. Effects of corrugation shape on frequency band-gaps for longitudinal wave motion in a periodic elastic layer. J Acoust Soc Am. 2016;139:1898–908.
Croenne C, Lee EJS, Page JH. Multimode propagation in phononic crystals with overlapping Bragg and hybridization effects. Appl Phys Lett. 2022;120:033104.
Zhang S, Shi Y, Gao Y. Tunability of band structure in a two-dimensional magnetostrictive phononic crystal plate with stress and magnetic loadings. Phys Lett A. 2017;381(12):1055–66.
Chen L, Guo Y, Yi H. Optimization study of band gaps properties for two-dimensional chiral phononic crystals base on lightweight design. Phys Lett A. 2021;388(127054):1–7.
Meng H, Bailey N, Chen Y, Wang L, Elmadih W. 3D rainbow phononic crystals for extended vibration attenuation bands. Sci Rep. 2020;10(1):2045–322.
Ren T, Li FM, Chen YO, Liu CC, Zhang CZ. Improvement of the band-gap characteristics of active composite laminate metamaterial plates. Compos Struct. 2020;254:112831.
Jiang P. Low-frequency band gap and defect state characteristics in a multi-stub phononic crystal plate with slit structure. J Appl Phys. 2017;121:015106.
Jin YB, Pennec Y, Pan YD, Djafari-Rouhani B. Phononic crystal plate with hollow pillars connected by thin bars. J Phys D Appl Phys. 2017;50:035301.
Jin YB, Pennec Y, Bonello B, Honarvar H, Dobrzynski L, Djafari-Rouhani B, Hussein MI. Physics of surface vibrational resonances: pillared phononic crystals, metamaterials, and metasurfaces. Rep Prog Phys. 2021;84:086502.
Djafari-Rouhani B, Pennec Y, Larabi H. Band structure and phonon transport in a phononic crystal made of a periodic array of dots on a membrane. IUTAM Symp Recent Adv Acoust Waves Solids. 2010;26:127–38.
Zhao C, Sai Y, Chen J. Tunable Lamb wave band gaps in two-dimensional magneto elastic phononic crystal slabs by an applied external magneto static field. Ultrasonics. 2016;71:69–74.
Zhang YF, Yu DL, Wen JH. Study on the band gaps of phononic crystal pipes with alternating materials in the radial and axial directions. Extreme Mech Lett. 2017;12:2–6.
Wu ZJ, Li FM, Zhang CZ. Band-gap analysis of a novel lattice with a hierarchical periodicity using the spectral element method. J Sound Vib. 2018;421:246–60.
Wen S, Xiong Y, Hao S, Li F, Zhang C. Enhanced band-gap properties of an acoustic metamaterial beam with periodically variable cross-sections. Int J Mech Sci. 2019;166(13):105229.
Hao S, Wu Z, Li F, Zhang C. Numerical and experimental investigations on the band-gap characteristics of metamaterial multi-span beams. Phys Lett A. 2019;383:126029.
Funding
This research is supported by the National Natural Science Foundation of China (Nos. 12072086, 12211540384 and 11761131006) and the Fundamental Research Funds for the Central Universities (No. 3072022CF0203).
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TH and ZW conceived the idea and contributed to methodology, software, formal analysis, experiment, data compilation, writing original draft and validation. ZW and FL were involved in funding acquisition. FL contributed to validation and supervision.
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Huo, T., Wu, Z. & Li, F. Bandgap Properties for the Folded S-Type Periodic Structure: Numerical Simulation and Experiment. Acta Mech. Solida Sin. 36, 624–632 (2023). https://doi.org/10.1007/s10338-023-00389-w
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DOI: https://doi.org/10.1007/s10338-023-00389-w