Integrated vibration absorption and isolation design method for metamaterial beams based on bandgap wave-insulating vibration isolatior

Wei Wei; Guan Feng; Fang Xin

doi:10.7498/aps.73.20241135

Abstract

Advanced vibration control technology is widely needed in the fields of aerospace and shipbuilding. Currently, separate vibration absorption and isolation design of most systems are separated, and existing isolation designs cannot effectively enhance the isolation of low-frequency line spectra. There is an urgent need to develop integrated vibration absorption and isolation designs and strengthen low-frequency line spectrum control. In response to this need, this paper focuses on a typical Euler beam and the investigation of the propagation characteristics of vibrations in transverse direction and longitudinal direction, the principles of integrated vibration absorption and isolation design, and the synergistic regulation of bandgaps, based on acoustic metamaterial bandgap wave-insulating vibration control configurations and analytical methods. Ultimately, without adding additional structures, the wave-insulating vibration control device is used to generate multiple modes of vibration absorption and isolation simultaneously, achieving an integrated low-frequency, broadband, and high-efficiency vibration absorption and isolation design. This method achieves broadband vibration isolation in the transverse vibration isolation path while also introducing local resonance bandgaps that significantly improve low-frequency vibration isolation. In the longitudinal (forward propagation) path, in addition to near-zero and Bragg bandgaps, multilayer isolators generate multimodal local resonant bandgaps, achieving low-frequency broadband vibration absorption and effective control in the entire frequency range. This paper elucidates the synergistic modulation of longitudinal and transverse bandgaps, showing that by superimposing these bandgaps, an impressive bandgap ratio of 87.3% below 100 Hz across the entire frequency range can be achieved. Furthermore, an entity structure is designed, and the accuracy of the analytical results is verified by using the finite element method. The findings provide feasible design ideas for realizing the integrated vibration absorption and isolation of complex structures such as beams, plates, pipelines, and frames.

Keywords:

Authors and contacts

1.
College of Intelligent Science, National University of Defense Technology, Changsha 410073, China
2.
Naval Research Academy, Beijing 100161, China

Corresponding author: Fang Xin, xinfangdr@sina.com

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 52241103, 52322505) and the Natural Science Foundation of Hunan Province, China (Grant No. 2023JJ10055).

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References

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[3]	Jiang S D, Bi K M, Ma R S, Xu K 2024 J. Sound Vib. 586 118510 Google Scholar
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[5]	Nigdeli S M, Bekdas G 2017 KSCE J. Civ. Eng. 912 21 Google Scholar
[6]	Lu Z, Wang Z X, Zhou Y, Lu X L 2018 J. Sound Vib. 423 18 Google Scholar
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[8]	Rong K J, Yang M, Lu Z, Zhang J W, Tian L, Wu S Y 2024 J. Build. Eng. 89 109253 Google Scholar
[9]	Wei C 2024 J. Braz. Soc. Mech. Sci. Eng. 46 120 Google Scholar
[10]	Zhou J H, Zhou J X, Pan H B, Wang K, Cai C Q, Wen G L 2024 Appl. Math. Mech. 45 1189 Google Scholar
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Cited By

图 1 传统周期支撑梁的振动传递特性分析　(a) 振动传播路径示意图; (b)不同f₀下的近端支撑的横向振动传递率; (c) 不同f₀下的远端支撑的横向振动传递率; (d)右端截面的纵向振动传递率

Figure 1. Analysis of vibration propagation in a traditional periodic support structure of a beam: (a) Schematic diagram of the vibration propagation path; (b) transverse vibration transmission ratio of the proximal support at different f₀; (c) transverse vibration transmission ratio of the distal support at different f₀; (d) longitudinal vibration transmission ratio of the right-end section.

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图 2 吸隔振一体超材料梁的结构设计　(a)理论模型; (b)元胞模型

Figure 2. Structural design of a vibration isolation and absorption integrated metamaterial beam: (a) Theoretical model; (b) metacell model.

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图 3 传统结构和吸隔振一体结构在近端的支撑结构的振动传动率　(a)传递率曲线; (b)传统隔振结构; (c)多层隔振结构; (d)吸隔振一体结构

Figure 3. Vibration transmission ratio of the traditional structure and the integrated vibration isolation and absorption structure at proximal support structure: (a) Transmission ratio curve; (b) traditional vibration isolation structure; (c) multi-layer vibration isolation structure; (d) integrated vibration isolation and absorption structure.

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图 4 横向波矢虚部和近端支撑结构的传递率曲线　(a)波矢虚部; (b)近端支撑结构的传递率曲线

Figure 4. Imaginary part of the transverse wave vector and transmission ratio curve of the proximal support structure: (a) Imaginary part of the wave vector; (b) transmission ratio curve of the proximal support structure.

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图 5 不同形式支撑梁的元胞模型和对应纵向衰减系数　(a)多层隔振模型纵向衰减系数; (b)吸隔振一体模型纵向衰减系数; (c) 多层隔振模型示意图; (d) 吸隔振一体模型示意图

Figure 5. Metacell models and corresponding longitudinal attenuation coefficients of support beams in different configurations: (a) Longitudinal attenuation coefficient of the multi-layer vibration isolation model; (b) longitudinal attenuation coefficient of the integrated vibration isolation and absorption model; (c) schematic diagram of multi-layer vibration isolation model; (d) schematic diagram of integrated vibration isolation and absorption model.

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图 6 吸隔振一体结构的振动传递特性　(a)纵向和横向上的弹性波衰减系数; (b)远端支撑结构的传递率曲线

Figure 6. Vibration transmission characteristics of the integrated vibration isolation and absorption structure: (a) Longitudinal and transverse attenuation coefficient; (b) transmission ratio curve of the distal support structure.

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图 7 频率比G的调制规律　(a) G对于纵向带隙的调制规律; (b) G对于横向带隙的调制规律; (c) G对于叠加带隙的影响; (d) G对于传递率的影响

Figure 7. Modulation rules of frequency ratio G: (a) Modulation rules of G for longitudinal bandgaps; (b) modulation rules of G for transverse bandgaps; (c) the effect of G on superimposed band gaps; (d) the effect of G on transmission ratio.

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图 8 质量比r和附加质量比$\beta $的调制规律　(a)横向LR带隙起始频率的调制; (b) $\beta $对于横向LR带隙的宽度影响; (c) r对于横向LR带隙的宽度影响; (d)附加质量比$\beta $的归一化传递率

Figure 8. Modulation law of the mass ratio r and the additional mass ratio $\beta $: (a) Modulation of the starting frequency of the transverse LR bandgap; (b) the effect of the additional mass ratio $\beta $ on the width of the transverse LR bandgap; (c) the effect of r on the width of the transverse LR bandgap; (d) the normalized transfer rate of the additional mass ratio $\beta $.

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图 9 Bragg带隙与LR带隙的协同调制　(a) 3阶LR带隙与Bragg的协同调制; (b) 3种近似耦合带隙的衰减系数曲线

Figure 9. Synergistic modulation of Bragg band gaps and LR bandgaps: (a) Collaborative modulation of third-order LR bandgap and Bragg; (b) attenuation coefficient curves of three approximate coupled bandgaps.

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图 10 Bragg带隙与1st LR带隙的协同调制　(a)纵向和横向的衰减系数; (b)近端和远端的传递率曲线

Figure 10. Synergistic modulation of Bragg bandgaps and 1st LR bandgap: (a) Longitudinal and transverse band structures; (b) transmission ratio curves of proximal and distal supports.

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图 11 损耗因子${\eta _1}$和${\eta _2}$对于衰减系数和传递率曲线的影响　(a) 不同损耗因子${\eta _1}$对应的衰减系数; (b) 不同损耗因子${\eta _1}$对应的传递率曲线; (c) 不同损耗因子${\eta _2}$对应的衰减系数; (d) 不同损耗因子${\eta _2}$对应的传递率曲线

Figure 11. Effects of attenuation factor ${\eta _1}$ and ${\eta _2}$ on attenuation coefficient and transmission ratio curves: (a) Attenuation coefficient corresponding to different ${\eta _1}$; (b) transmission ratio curves corresponding to different ${\eta _1}$; (c) attenuation coefficient corresponding to different ${\eta _2}$; (d) transmission ratio curves corresponding to different ${\eta _2}$.

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图 12 吸隔振一体超材料梁的示意图　(a) 超材料梁模型; (b)超材料梁的元胞; (c)元胞中波阻隔振器

Figure 12. Schematic diagram of the vibration isolation and absorption integrated metamaterial beam: (a) Metamaterial beam model; (b) metacell of the metamaterial beam; (c) metacell of the wave-insulating vibration isolator.

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图 13 5周期的吸隔振一体超材料梁的振动特性　(a)谱元法和有限元法计算得到的振动传递率对比; (b)特定频率下的吸隔振一体超材料的振动响应

Figure 13. Vibration characteristics of the 5-period vibration isolation and absorption integrated metamaterial beam: (a) Comparison of vibration transmission ratios obtained by the spectral element method and the finite element method; (b) vibration response of the integrated vibration isolation and absorption metamaterial at specific frequencies.

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图 14 (a)不同损耗因子下的传递率曲线; (b)与传统单自由度隔振器的传递率对比

Figure 14. (a) Transmission ratio curves under different attenuation factors; (b) comparison with the transmission ratio of a traditional single-degree-of-freedom isolator.

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表 1 三种耦合方式的带隙宽度和总宽度

Table 1. Bandgap width and total width of the three coupling modes.

晶格常数		a/mm
晶格常数		3000	3850	3360	3220
1st	起始频率/Hz	21.4—26.0	18.2—29.4	19.9—25.2	20.3—25.2
1st	宽度/Hz	4.4	11.2	5.3	4.9
2nd	起始频率/Hz	31.7—33.8	32.6—34.5	30.2—34.7	30.6—32.8
2nd	宽度/Hz	2.1	1.9	4.5	2.2
3rd	起始频率/Hz	35.5—36.1	35.3—36.2	35.4—36.3	34.7—36.3
3rd	宽度/Hz	0.6	0.9	0.9	1.6
4th	起始频率/Hz	36.8—37.0	36.3—40	36.4—43.7	36.4—45.5
4th	宽度/Hz	0.2	4.7	7.3	9.1
总宽度/Hz		7.3	18.7	18.0	17.8

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[1]	苏常伟, 梁冉, 王雪仁, 周涛, 李海超 2023 舰船科学技术 45 1 Google Scholar Su C W, Liang R, Wang X R, Zhou T, Li H C 2023 Ship Sci. Technol. 45 1 Google Scholar
[2]	杜德锋, 何江贤, 孟凡凯 2023 声学技术 42 552 Google Scholar Du D F, He J X, Meng F K 2023 Tech. Acoust. 42 552 Google Scholar
[3]	Jiang S D, Bi K M, Ma R S, Xu K 2024 J. Sound Vib. 586 118510 Google Scholar
[4]	Djedoui N, Ounis A 2022 Pract. Period. Struct. Des. Constr. 27 3 Google Scholar
[5]	Nigdeli S M, Bekdas G 2017 KSCE J. Civ. Eng. 912 21 Google Scholar
[6]	Lu Z, Wang Z X, Zhou Y, Lu X L 2018 J. Sound Vib. 423 18 Google Scholar
[7]	熊怀, 孔宪仁, 刘源 2015 振动工程学报 28 785 Google Scholar Xiong H, Kong X R, Liu Y 2015 J. Vib. Eng. 28 785 Google Scholar
[8]	Rong K J, Yang M, Lu Z, Zhang J W, Tian L, Wu S Y 2024 J. Build. Eng. 89 109253 Google Scholar
[9]	Wei C 2024 J. Braz. Soc. Mech. Sci. Eng. 46 120 Google Scholar
[10]	Zhou J H, Zhou J X, Pan H B, Wang K, Cai C Q, Wen G L 2024 Appl. Math. Mech. 45 1189 Google Scholar
[11]	Liu T, Li A Q, Zhang H Y 2024 Mech. Syst. Signal Proc. 206 110852 Google Scholar
[12]	Zhao F, Ji J C, Cao S Q, Ye K, Luo Q T 2024 Nonlinear Dyn. 112 1815 Google Scholar
[13]	Yan G, Lu J J, Qi W H, Liu F R, Yan H, Zhao L C, Wu Z Y, Zhang W M 2024 Nonlinear Dyn. 112 5955 Google Scholar
[14]	Xing X D, Chen Z B, Feng Z Q 2023 J. Vib. Eng. Technol. 11 1595 Google Scholar
[15]	Ma H Y, Wang K, Zhao H F, Zhao C, Xue J, Liang C, Yan B 2023 J. Sound Vib. 565 117905 Google Scholar
[16]	Wang Q, Chen Z X, Wang Y H, Gong N, Yang J, Li W H, Sun S S 2024 Mech. Syst. Signal Proc. 208 111029 Google Scholar
[17]	Kushwaha M S, Halevi P, Dobrzynski L, Djafari-Rouhani B 1993 Phys. Rev. Lett. 71 2022 Google Scholar
[18]	Montero De Espinosa F R, Jiménez E, Torres M 1998 Phys. Rev. Lett. 80 1208 Google Scholar
[19]	Hu G B, Austin A C M, Sorokin V, Tang L H 2021 Mech. Syst. Signal Proc. 146 106982 Google Scholar
[20]	Liu Z Y, Zhang X X, Mao Y M, Zhu Y Y, Yang Z Y, Chan C T, Sheng P 2000 Science 289 1734 Google Scholar
[21]	尹剑飞, 蔡力, 方鑫, 肖勇, 杨海滨, 张弘佳, 钟杰, 赵宏刚, 郁殿龙, 温激鸿 2022 力学进展 52 508 Google Scholar Yin J F, Cai L, Fang X, Xiao Y, Yang H B, Zhang H J, Zhong J, Zhao H G, Yu D L, Wen J H 2022 Adv. Mech. 52 508 Google Scholar
[22]	Redondo J, Godinho L, Staliunas K, Sánchez-Pérez J V 2023 Appl. Acoust. 211 109555 Google Scholar
[23]	Wei W, Guan F, Fang X 2024 Appl. Math. Mech. 45 1171 Google Scholar
[24]	Sheng P, Fang X, Wen J H, Yu D L 2021 J. Sound Vib. 492 115739 Google Scholar
[25]	杜春阳, 郁殿龙, 刘江伟, 温激鸿 2017 物理学报 66 140701 Google Scholar Du C Y, Yu D L, Liu J W, Wen J H 2017 Acta Phys. Sin. 66 140701 Google Scholar
[26]	Cai C Q, Zhou J X, Wang K, Pan H B, Tan D G, Xu D L, Wen G L 2022 Mech. Syst. Signal Proc. 174 109119 Google Scholar
[27]	Wang G, Wan S K, Hong J, Liu S, Li X H 2023 Mech. Syst. Signal Proc. 188 110036 Google Scholar
[28]	Fan X L, Li J Q, Zhang X Y, Li F M 2022 Int. J. Mech. Sci. 236 107742 Google Scholar
[29]	Yao D H, Xiong M K, Luo J Y, Yao L Y 2022 Mech. Syst. Signal Proc. 168 108721 Google Scholar
[30]	Liu Y, Cheng L, Du J T 2022 J. Acoust. Soc. Am. 152 3471 Google Scholar
[31]	Wang S, Wang M Q, Guo Z W 2021 Phys. Lett. A 417 127671 Google Scholar
[32]	Zhou W, Li Y, Yan G, Zhou J 2024 Eur. J. Mech. Solid. 103 105143 Google Scholar
[33]	Zhuang D, Dimitrios C, Jian Y 2021 Appl. Acoust. 178 108013 Google Scholar
[34]	吴健, 白晓春, 肖勇, 耿明昕, 郁殿龙, 温激鸿 2016 物理学报 65 064602 Google Scholar Wu J, Bai X C, Xiao Y, Geng M X, Yu D L, Wen J H 2016 Acta Phys. Sin. 65 064602 Google Scholar
[35]	张鑫浩, 赵才友, 郑钧元, 牛亚文, 未娜超 2023 机械科学与技术 42 338 Google Scholar Zhang X H, Zhao C Y, Zheng J Y, Niu Y W, Wei N C 2023 Mechanical Science and Technology for Aerospace Engineering 42 338 Google Scholar
[36]	肖勇 2012 博士学位论文 (长沙: 国防科学技术大学 Xiao Y 2012 Ph. D. Dissertation (Changsha: National University of Defense Technology
[37]	Gao Y Q, Wang L F, Sun W, Wu K, Hu H Y 2022 Acta Mech. 233 477 Google Scholar
[38]	温激鸿, 蔡力, 郁殿龙, 肖勇, 赵宏刚, 尹剑飞, 杨海滨 2018 声学超材料基础理论与应用 (北京: 科学出版社) 第313页 Wen J H, Cai L, Yu D L, Xiao Y, Zhao H G, Yin J F, Yang H B 2018 Basic Theory and Application of Acoustic Metamaterials (Beijing: Science Press) p313
[39]	Xiao L, Bursi O S, Li H, Wang M, Du X L 2023 Int. J. Mech. Sci. 237 107770 Google Scholar
[40]	Mu D, Wang K, Shu H S, Lu J H 2022 Int. J. Mech. Sci. 236 107761 Google Scholar
[41]	郁殿龙 2006 博士学位论文 (长沙: 国防科学技术大学) Yu D L 2006 Ph. D. Dissertation (Changsha: National University of Defense Technology
[42]	Xiao Y, Wang S, Li Y, Wen J H 2021 Mech. Syst. Signal Proc. 159 107777 Google Scholar

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Vol. 73, Issue 22 - 2024-11-20

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