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In recent years, the vibration and noise reduction performance of military aircraft has become an important index to measure its performance. In order to solve the problem of low-frequency noise generated by military aircraft, a novel Helmholtz two-dimensional phononic crystal is constructed in this paper. The structure adopts maze-shaped air channel and adds rigid oscillators. On condition that the lattice constant is 62 mm, the lower limit of the first band gap is reduced to about 15 Hz. The structure has four complete band gaps in a range of 0–500 Hz, specifically, they being 15.223–17.464 Hz, 107.46–200.68 Hz, 231.18–310.68 Hz, and 341.14–404.49 Hz. In addition, the sound reduction index of the structure reaches 25 dB at 15 Hz, and two peaks higher than 150 dB appear at about 107 Hz and 231 Hz. which shows distinct sound insulation capability in the low-frequency range. It has engineering significance in controlling the low-frequency noise in the aircraft cabin. The cause of the band gap is explored by analyzing the vibration mode and sound pressure field. The “spring-oscillator” of the structure model is established by the method of “Mechanical-acoustic analogy”. The finite element method and transfer matrix method are used to calculate the upper limit and the lower limit of the first band gap. It is shown that for the first gap of the structure, the results obtained by the two methods are similar, which indicates the correctness of the model hypothesis. Secondly, the effects of structural parameters such as the lattice constant, the length of the air channel and the oscillator material on the first band gap are investigated by finite element method and equivalent model method. It is obtained that the increase of the length of air channel and lattice constant will reduce the lower limit of the first band gap, with other structural parameters remaining unchanged. Moreover, the increase of the density of the oscillator material can effectively reduce the upper limit and lower limit of the second band gap, which further reveals the essence of the formation of the band gap of the structure and verifies the accuracy of the equivalent model. This study provides theoretical support for low frequency noise control and broadens the design of low-frequency phononic crystals.
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Keywords:
- Helmholtz resonator /
- phononic crystal /
- low-frequency bandgap /
- noise insulation and reduction
[1] 吴永祥, 丁传东 1998 航天医学与医学工程 11 54
Wu Y X, Ding C D 1998 Space Med. Med. Eng. 11 54
[2] Kuo C Y, Hung C L, Chen H C, Shih C P, Lu R H, Chen C W, Hung L W, Lin Y C, Chen H K, Chu D M, Lin Y Y, Chen Y C, Wang C H 2021 Int. J. Environ. Res. Public. Health. 18 2982
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Zhang J L, Yao H, Du J, Zhao J B, DongY K, Qi P S 2016 J. Chin. Ceram. Soc. 44 1440
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Tian Y, Ge H, Lu M H, Chen Y F 2019 Acta Phys. Sin. 68 194301
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Liao T, Sun X W, Song T, Tian J H, Kang T F, Sun W B 2018 Acta Phys. Sin. 67 214208
Google Scholar
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[15] Gao N, Qu S, Li J, Wang J, Chen W 2021 Int. J. Mech. Sci. 208 106695
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[16] Wang Z G, Lee S H, Kim C K, Park C M, Nahm K, Nikitov S A 2008 J. Appl. Phys. 103 064907
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[17] Liu C R, Wu J H, Chen X, Ma F Y 2019 J. Phys. D:Appl. Phys. 52 105302
Google Scholar
[18] 陈鑫, 姚宏, 赵静波, 张帅, 贺子厚, 蒋娟娜 2019 物理学报 68 084302
Google Scholar
Chen X, Yao H, Zhao J B, Zhang S, He Z H, Jiang J N 2019 Acta Phys. Sin. 68 084302
Google Scholar
[19] Dai H, Liu T, Jiao J, Xia B, Yu D 2017 J. Appl. Phys. 121 135105
Google Scholar
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图 1 (a) 新型声子晶体元胞横截面; (b)新型声子晶体周期排列结构
Figure 1. (a) Cross section of the phononic crystal structure; (b) periodic arrangement structure of the phononic crystal.
图 2 正方形晶格不可约Brillouin区
Figure 2. The irreducible Brillouin region of the square lattice.
图 3 隔声量仿真示意图
Figure 3. Schematic diagram of sound insulation volume simulation.
图 4 (a) 结构带隙图; (b) 隔声曲线
Figure 4. (a) Band diagram; (b) transmission spectrum.
图 5 (a) A点处声场压力图; (b) B点处声场压力图; (c) A点处振型图; (d) B点处振型图
Figure 5. (a) The sound pressure distribution diagram of point A; (b) the sound pressure distribution diagram of point B; (c) the vibration mode diagram of point A; (d) the vibration mode diagram of point B.
图 6 (a) 平直带处声场压力图; (b) 平直带处声场振型图
Figure 6. (a) The sound pressure distribution diagram of the straight belt; (b) the vibration mode diagram of the straight belt.
图 7 (a)带隙下限系统示意图; (b)带隙上限系统示意图
Figure 7. (a) The system corresponding to the lower limit of band gap; (b) the system corresponding to upper limit of the band gap.
图 8 晶格常数a对第一低频带隙的影响
Figure 8. Influence of the lattice constant a on the first band gap.
图 9 空气通道长度l2对第一低频带隙的影响
Figure 9. Influence of the length of air channel l2 on the first band gap.
图 10 振子材料对第一、二低频带隙的影响
Figure 10. Influence of oscillator material on the first and second band gaps.
表 1 两种方法所得的带隙范围
Table 1. The low-frequency band gap range obtained by the two methods.
计算方法 第一带隙下限 第一带隙上限 有限元法 15.223 17.464 传递矩阵法 15.959 17.131 误差/% 4.83 1.91 
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表 2 材料参数
Table 2. Material parameters.
材料名称 铜 钢 铝 碳 环氧树脂 密度/(kg·m–3) 8950 7780 2730 1750 1180 杨氏模量/(1010 Pa) 16.46 21.06 7.76 23.01 0.45 泊松比 0.326 0.300 0.352 0.300 0.370
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[1] 吴永祥, 丁传东 1998 航天医学与医学工程 11 54
Wu Y X, Ding C D 1998 Space Med. Med. Eng. 11 54
[2] Kuo C Y, Hung C L, Chen H C, Shih C P, Lu R H, Chen C W, Hung L W, Lin Y C, Chen H K, Chu D M, Lin Y Y, Chen Y C, Wang C H 2021 Int. J. Environ. Res. Public. Health. 18 2982
Google Scholar
[3] Wolfe H F, Camden M P, Byrd L W, Paul D B, Simmons L W 2015 J. Aircr. 37 319
[4] Bagherzadeh S A, Salehi M 2021 Appl. Acoust. 178 108020
Google Scholar
[5] Atak O, Huybrechs D, Pluymers B, Desmet W 2014 J. Sound Vibr. 333 3367
Google Scholar
[6] 张佳龙, 姚宏, 杜军, 赵静波, 董亚科, 祁鹏山 2016 硅酸盐学报 44 1440
Zhang J L, Yao H, Du J, Zhao J B, DongY K, Qi P S 2016 J. Chin. Ceram. Soc. 44 1440
[7] Han D H, Zhao J B, Zhang G J, Yao H 2021 Symmetry 13 1379
Google Scholar
[8] Li X, Ning S, Liu Z, Yan Z, Luo C, Zhuang Z 2019 Comput. Methods Appl. Mech. Engrg. 361 112737
[9] Chen L, Guo Y, Yi H 2021 Phys. Lett. A 388 127054
Google Scholar
[10] Shao H, He H, He C, Chen G 2020 J. Mater. Res. 35 3021
Google Scholar
[11] 田源, 葛浩, 卢明辉, 陈延峰 2019 物理学报 68 194301
Google Scholar
Tian Y, Ge H, Lu M H, Chen Y F 2019 Acta Phys. Sin. 68 194301
Google Scholar
[12] 廖涛, 孙小伟, 宋婷, 田俊红, 康太凤, 孙伟彬 2018 物理学报 67 214208
Google Scholar
Liao T, Sun X W, Song T, Tian J H, Kang T F, Sun W B 2018 Acta Phys. Sin. 67 214208
Google Scholar
[13] Chen J S, Chen Y B, Cheng Y H, Chou L C 2020 Phys. Lett. A 384 126887
Google Scholar
[14] Kim M J 2019 Int. J. Mod. Phys. B 33 1950138
Google Scholar
[15] Gao N, Qu S, Li J, Wang J, Chen W 2021 Int. J. Mech. Sci. 208 106695
Google Scholar
[16] Wang Z G, Lee S H, Kim C K, Park C M, Nahm K, Nikitov S A 2008 J. Appl. Phys. 103 064907
Google Scholar
[17] Liu C R, Wu J H, Chen X, Ma F Y 2019 J. Phys. D:Appl. Phys. 52 105302
Google Scholar
[18] 陈鑫, 姚宏, 赵静波, 张帅, 贺子厚, 蒋娟娜 2019 物理学报 68 084302
Google Scholar
Chen X, Yao H, Zhao J B, Zhang S, He Z H, Jiang J N 2019 Acta Phys. Sin. 68 084302
Google Scholar
[19] Dai H, Liu T, Jiao J, Xia B, Yu D 2017 J. Appl. Phys. 121 135105
Google Scholar
[20] Li Y, Chen T, Wang X, Yu K, Song R 2015 Physica B (Amsterdam, Neth.) 456 261
Google Scholar
[21] Elford D P, Chalmers L, Kusmartsev F V, Swallowe G M 2011 J. Acoust. Soc. Am. 130 2746
Google Scholar
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