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以侧向局域共振超构板为研究对象, 基于布洛赫定理及弯曲波传播理论, 建立了侧向局域共振超构板的振动响应、声辐射功率及辐射效率理论计算模型, 同时建立了侧向局域共振超构板的有限元模型以验证理论模型的有效性. 进一步开展了有/无周期附加侧向局域共振结构超构板的模态分析, 探索了侧向局域共振超构板辐射声功率及辐射效率随频率变化的关系. 结果表明, 侧向局域共振超构板在两条特定频段(带隙)内的板面均方振速及辐射声功率远低于均匀平板, 而辐射效率却高于均匀平板. 本研究可为推进侧向局域共振超构板在减振降噪领域的工程应用奠定理论基础.Based on the Bloch theory and the flexural wave propagation theorem, the model for calculating the vibration response, sound radiation power and radiation efficiency of an acoustic metamaterial plate is established. The modal analysis of a bare pate and the plate attached to periodical lateral local resonators are developed to investigate the vibration regulation. In addition, the patterns of the sound radiation power and radiation efficiency of the plate attached to periodical lateral local resonators versus frequency are studied. The results show that 1) in a specific frequency range, the mean square velocity and the sound radiation power are far lower than those of the bare plate, which is due to the resonance of the lateral local resonator; 2) unlike the vibration response and sound radiation power, the radiation efficiency of the plate attached to periodical lateral local resonator is higher than that of the bare plate. The investigation of the plate attached to lateral local resonator in this paper lays a solid foundation for the practical engineering in the field of vibration suppression and noise reduction.
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Keywords:
- acoustic metamaterial plate with periodic lateral local resonators /
- mean square velocity /
- sound radiation power /
- band gap
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图 1 浸润于空气中的侧向局域共振超构板 (a) 受外力激励的侧向局域共振超构板鸟瞰图; (b) 侧向局域共振超构板正视图
Fig. 1. Acoustic metamaterial plate with periodic local resonators submerged in air: (a) Perspective view of the acoustic metamaterial plate with external force applied; (b) front view of a single unit cell of the acoustic metamaterial plate.
图 2 侧向局域共振超构板振动加速度级的理论方法与有限元法结果对比
Fig. 2. Comparison of the theoretical and simulative results of the acceleration level for the metamaterial plate.
图 3 平板第1, 3, 4, 5阶模态
Fig. 3. Modal analysis of bare plate: the 1st, 3rd, 4th and 5th modes.
图 4 侧向局域共振超构板第1, 20, 134, 265阶模态
Fig. 4. Modal analysis of the metamaterial plate: the 1st, 20th, 134th and 265th modes.
图 5 平板与侧向局域共振超构板的(a)表面振速(MSV)与(b)辐射声功率(SRP)对比
Fig. 5. Comparison of (a) mean square velocity and (b) sound radiation power between the bare plate and the metamaterial plate.
图 6 平板与侧向局域共振超构板的辐射效率对比
Fig. 6. Comparison of radiation efficiency between the bare plate and the metamaterial plate.
表 1 侧向局域共振超构板模型计算参数
Table 1. Parameters for the metamaterial plate with periodic lateral local resonators.
参数名称 参数符号 参数取值 平板杨氏模量 E $2.1 \times {10^{11}}{\kern 1 pt} {\kern 1 pt} {\text{Pa}}$ 平板密度 $\rho $ $7850{\kern 1 pt} {\kern 1 pt} {\text{kg}}/{{\text{m}}^3}$ 平板泊松比 $\nu $ 0.3 平板厚度 h 0.003 m 垂向振子刚度 k1 $1 \times {10^4}\;{\text{N}}/{\text{m}}$ 垂向振子质量 m1 0.005 kg 侧向振子刚度 k2 $0.5 \times {10^4}\;{\text{N}}/{\text{m}}$ 侧向振子质量 m2 0.002 kg 四杆机构几何参数比 L/D 1 空气密度 ${\rho _0}$ $1.25\;{\text{kg}}/{{\text{m}}^3}$ 空气中声速 c $343\;{\text{m}}/{\text{s}}$ 
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表 2 平板模态频率
Table 2. Modal frequency of the bare plate.
模态阶数 1 2 3 4 5 6 7 频率/Hz 75.8 164.6 214.6 303.3 312.6 445.8 451.4
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表 3 侧向局域共振超构板模态频率
Table 3. Modal frequency of the metamaterial plate.
模态阶数 1 2 3 4 5 6 7 频率/Hz 70.8 147.8 179.9 200.95 201.7 206.3 206.4 模态阶数 7—11 12—130 131 132 133 134 135 频率/Hz 207.1—207.9 208.1—208.9 217.8 223.7 234.3 259.3 260.6 模态阶数 136 137 138 139 140—260 261 262 频率/Hz 267.9 268.0 269.1 269.9 270.0—271.1 278.6 282.0 模态阶数 263 264 265 266 267 频率/Hz 288.3 329.8 336.9 456.9 462.2
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[1] Lu Z, Yu X, Lau S K, Khoo B C, Cui F 2020 Appl. Acoust. 157 107003
Google Scholar
[2] 丁昌林, 董仪宝, 赵晓鹏 2018 物理学报 67 194301
Google Scholar
Ding C L, Dong Y B, Zhao X P 2018 Acta Phys. Sin. 67 194301
Google Scholar
[3] 刘娇, 侯志林, 傅秀军 2015 物理学报 64 154302
Google Scholar
Liu J, Hou Z L, Fu X J 2015 Acta Phys. Sin. 64 154302
Google Scholar
[4] Gao N, Tang L, Deng J, Lu K, Chen K 2021 Appl. Acoust. 175 107845
Google Scholar
[5] Shaat M, El Dhaba A R 2019 Compos. Part B-Eng. 172 506
Google Scholar
[6] Liu Z, Zhang X, Mao Y, Zhu Y Y, Yang Z, Chan C T, Sheng P 2000 Science 289 1734
Google Scholar
[7] Fang N, Xi D, Xu J, Ambati M, Srituravanich W, Sun C, Zhang X 2006 Nat. Mater. 5 452
Google Scholar
[8] Lee S H, Park C M, Seo Y M, Kim C K 2010 Phys. Rev. B 81 241102
Google Scholar
[9] 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
[10] Xiao Y, Wen J, Wen X 2012 J. Phys. D Appl. Phys. 45 195401
Google Scholar
[11] Xiao Y, Wen J, Wen X 2012 J. Sound Vib. 331 5408
Google Scholar
[12] Zhu R, Liu X N, Huang G L, Huang H H, Sun C T 2012 Phys. Rev. B 86 144307
Google Scholar
[13] Ma F, Meng H, Wu J H 2017 J. Appl. Phys. 121 015102
Google Scholar
[14] Peng H, Pai P F 2014 Int. J. Mech. Sci. 89 350
Google Scholar
[15] Peng H, Pai P F, Deng H 2015 Int. J. Mech. Sci. 103 104
Google Scholar
[16] 孙宏伟, 林国昌, 杜星文, Pai P F 2012 物理学报 61 154302
Google Scholar
Sun H W, Lin G C, Du X W, Pai P F 2012 Acta Phys. Sin. 61 154302
Google Scholar
[17] Wang T, Sheng M P, Guo Z W, Qin Q H 2016 Appl. Acoust. 114 118
Google Scholar
[18] Wang T, Sheng M, Ding X, Yan X 2018 J. Phys. D Appl. Phys. 51 115306
Google Scholar
[19] Guo Z, Sheng M, Pan J 2019 Appl. Sci. 9 3651
Google Scholar
[20] Song Y, Feng L, Liu Z, Wen J, Yu D 2019 Int. J. Mech. Sci. 150 744
Google Scholar
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