引 en
×

分享给微信好友或者朋友圈

使用微信“扫一扫”功能。

圆柱壳弹性波超材料分级排列的带隙拓宽方法研究*

  • 姚凌云1,2

    机 构：

    1. 西南大学 工程技术学院,重庆 400715

    2. 丘陵山区农业装备重庆市重点实验室, 重庆 400715

    ×
  • ，姚敦辉1,2

    机 构：

    1. 西南大学 工程技术学院,重庆 400715

    2. 丘陵山区农业装备重庆市重点实验室, 重庆 400715

    ×

1. 西南大学 工程技术学院,重庆 400715；
2. 丘陵山区农业装备重庆市重点实验室, 重庆 400715；

Research on the Widening Method of Bandgap with Graded Arrangement for Elastic Wave Metamaterial in Cylindrical Shells*

  • Yao Lingyun1,2

    Affiliation：

    1. College of Engineering and Technology, Southwest University, Chongqing 400715 , China

    2. Chongqing Key Laboratory of Agricultural Equipment in Hilly and Mountainous Areas, Chongqing 400715 , China

    ×
  • ，Yao Dunhui1,2

    Affiliation：

    1. College of Engineering and Technology, Southwest University, Chongqing 400715 , China

    2. Chongqing Key Laboratory of Agricultural Equipment in Hilly and Mountainous Areas, Chongqing 400715 , China

    ×

1. College of Engineering and Technology, Southwest University, Chongqing 400715 , China；
2. Chongqing Key Laboratory of Agricultural Equipment in Hilly and Mountainous Areas, Chongqing 400715 , China；

通讯作者:

姚凌云,E-mail:lingyunyao@swu.edu.cn

中图分类号:O321;TH113;O799

文献标识码:A

文章编号:1672-6553-2023-21(7)-038-005

DOI:10.6052/1672-6553-2023-065

参考文献 1
肖勇.局域共振型结构的带隙调控与减振降噪特性研究[D].长沙：国防科学技术大学,2012.XIAO Y.Research on bandgap regulation and vibration and noise reduction characteristics of local resonant structures [D].Changsha:National University of Defense Science and Technology,2012.(in Chinese)
查找原文
参考文献 2
温激鸿.声子晶体振动带隙及减振特性研究 [D].长沙：国防科技大学,2005.WEN J H.Vibration attenuation and band gap characteristics of phononic crystals [D].Changsha:National University of Defence Technology,2005.(in Chinese)
查找原文
参考文献 3
KUSHWAHA M S,HALEVI P,MARTINEZ G,et al.Theory of acoustic band structure of periodic elastic composites [J].Physical Review B,1994,49(4):2313-2322.
查找原文
参考文献 4
WANG G,WEN X,WEN J,et al.Two-dimensional locally resonant phononic crystals with binary structures [J].Physical Review Letters,2004,93(15):154302.
查找原文
参考文献 5
BILAL,OSAMA R,HUSSEIN M.Trampoline metamaterial:local resonance enhancement by springboards [J].Applied Physics Letters,2013,103(11):111901.
查找原文
参考文献 6
XIAO Y,WEN J H,WEN X S.Broadband locally resonant beams containing multiple periodic arrays of attached resonators [J].Physics Letters A,2012,376:1384-1390.
查找原文
参考文献 7
YUA X,LU Z,CUI F,et al.Tunable acoustic metamaterial with an array of resonators actuated by dielectric elastomer [J].Extreme Mechanics Letters,2017,12:37-40.
查找原文
参考文献 8
YAO D,XIONG M,LUO J,et al.Flexural wave mitigation in metamaterial cylindrical curved shells with periodic graded arrays of multi-resonator [J].Mechanical Systems and Signal Processing,2022,168:108721.
查找原文
参考文献 9
LI Q,YANG D,MAO X.Pressure-resistant cylindrical shell structures comprising graded hybrid zero Poisson’s ratio metamaterials with designated band gap characteristics [J].Marine Structures,2022,84:103221.
查找原文
参考文献 10
NATEGHI A,BELLE L V,CLAEYS C,et al.Wave propagation in locally resonant cylindrically curved metamaterial panels [J].International Journal of Mechanical Sciences,2017,127:73-90.
查找原文
参考文献 11
罗金雨,姚凌云,江国期,等.一种圆柱壳类声子晶体振动带隙及振动特性研究 [J].振动与冲击,2019,(8):133-138.LUO J Y,YAO L Y,JIANG G Q,et al.Study on the vibration band gap and vibration characteristics of a cylindrical shell like phononic crystal [J].Journal of Vibration and Shock,2019,08,133-138.(in Chinese)
查找原文
参考文献 12
SONG Y,WEN J,YU D,et al.Reduction of vibration and noise radiation of an underwater vehicle due to propeller forces using periodically layered isolators [J].Journal of Sound and Vibration,2014,333(14):3031-3043.
查找原文
参考文献 13
LEE G Y,CHONG C,KEVREKIDIS P G,et al.Wave mixing in coupled phononic crystals via a variable stiffness mechanism [J].Journal of the Mechanics and Physics of Solids,2016,95:501-516.
查找原文
目录contents

    摘要

    圆柱壳弹性波超材料的弯曲波带隙拓宽问题限制其满足实际工程中的宽频隔振要求,针对该问题,本文首先研究了基于局域共振机理的圆柱壳弹性波超材料弯曲波带隙特点,研究局域谐振器质量和弹簧劲度系数的关系,然后将周期分级排列的组合方式应用于圆柱壳类弹性波超材料的带隙拓宽中,并利用有限元法进行能带结构和振动传输特性计算.研究结果显示,该方法可实现弯曲波带隙的拓宽；利用组合法构建的轴向周期分级排列圆柱壳弹性波超材料可实现705Hz~1226Hz频率范围内弯曲波的高效衰减,带隙拓宽至分别为单一谐振器的2.55倍,这表明该方法在宽频减振方面具有明显优势,应用前景广阔.

    Abstract

    The problem of bending wave band gap widening of cylindrical shell elastic wave Metamaterial restricts it to meet the requirements of wide frequency vibration isolation in practical engineering. Aiming at this problem, this paper first studies the characteristics of bending wave band gap of cylindrical shell elastic wave Metamaterial based on local resonance mechanism, studies the relationship between local resonator mass and spring stiffness coefficient, and then applies the combination of periodic hierarchical arrangement to the band gap widening of cylindrical shell elastic wave Metamaterial, the finite element method is used to calculate the Electronic band structure and vibration transmission characteristics. The research results show that this method can achieve widening of the bending wave bandgap; The elastic wave Metamaterial of axially periodic graded cylindrical shells constructed by the combination method can achieve efficient attenuation of bending waves in the frequency range of 705Hz~1226Hz, and the band gap is widened to 2.55 times of that of a single resonator, which shows that the method has obvious advantages in broadband vibration reduction and has broad application prospects.

  • 引言

  • 圆柱壳是轮船、飞机、火箭、航空航天以及精密仪器的基本结构，其振动噪声问题是机械工业中重要的噪声源[1-4]，而圆柱壳结构的减振降噪研究显得十分迫切且非常有必要性.目前针对圆柱壳减振降噪的研究存在受限于频率范围的影响，对宽频的隔振降噪问题适用性不强.弹性波超材料作为目前一种新颖的减振降噪结构，正被人们应用到圆柱壳减振降噪领域，为圆柱壳的减振降噪设计提供新思路.

  • 弹性波超材料是一种由人工设计，按照周期性或非周期性排布后表现出自然材料所不具备的超常物理性质的结构或材料[5-7]，其独特的弹性波带隙特性使得其在减振，降噪等方面得到了广泛的运用.在其目前的研究中，对于梁、板类声学超材料的研究已经广为报道，然而，对于圆柱壳弹性波超材料的研究却很少，研究圆柱壳弹性波超材料带隙特性并利用其进行减振降噪已成为人们关注的热点.Yao等人[8]最近提出了一种局部谐振圆柱壳弯曲波带隙的理论模型，并研究了一种利用超材料圆柱壳进行宽带波抑制的技术，Li等人[9]在模型中实现周期边界条件和布洛赫定理，利用有限元法计算并进行优化设计具有指定承载性能和带隙特性的零泊松比（zero Poisson's ratio，ZPR）圆柱壳超材料，Nateghi等人[10]研究了圆柱弯曲超材料板在不同方向上的禁带行为，并讨论了圆柱壳半径对带隙的影响.

  • 弹性波超材料的带隙特性为圆柱壳结构的振动控制提供了可能，而为了使带隙范围覆盖结构振动的主要频段，这就要求带隙宽度尽量大且连续.现有的圆柱壳带隙研究大都通过调整原胞的相关参数来实现带隙拓宽的目的，设计的超材料结构较为复杂，且实际工程中某些理想参数难以实现[11-13].因此，研究一种简单可行的带隙拓宽方法是圆柱壳弹性波超材料理论研究方面十分重要的问题.

  • 本文首先研究局域谐振器质量和弹簧劲度系数与弯曲波带隙的关系，获得带隙的调控规律，然后利用周期分级排列技术进行圆柱壳类弹性波超材料弯曲波带隙拓宽，为圆柱壳的宽频减振降噪提供新的方法，具有重要的理论研究意义和工程应用价值.

  • 1 圆柱壳超材料的弯曲波带隙计算模型

  • 基于局域共振理论的圆柱壳弹性波超材料模型如图1所示.其中，假设第（i，j）个局域谐振器的弹簧劲度系数和质量分别用和表示. 本文中对弹簧振子结构进行设计，具体胞元结构型式及尺寸参数如表1和图2所示.

  • 图1 基于局域共振机理的圆柱壳弹性波超材料模型

  • Fig.1 Cylindrical shell elastic wave metamaterial based on local resonance mechanism

  • 表1 圆柱壳类弹性波超材料的几何和聚酰胺纤维材料参数

  • Table1 Geometric and material parameters of cylindrical shell-like elastic wave metamaterials

  • 利用有限元法对圆柱壳弹性波超材料胞元进行特征值计算：

  • K-ω2MU=0
    (1)

  • 式中K、M和U分别代表胞元的刚度矩阵、质量和位移向量.胞元的周期条件设置Floquet周期边界条件（periodic boundary conditions，PBC）.考虑到圆柱壳圆周方向才有圆柱坐标，其周期条件需要进行坐标变换，须将周向波矢ks分解到x和y方向，分别为ks·cos（a/R）·π/a、ks·sin（a/R）·π/a；z方向的Floquet周期矢量为ky·π/a.其中，R为圆柱壳中面半径，a为原胞晶格常数.图4所示为胞元的能带结构图，从图中可以发现，圆柱壳类弹性波超材料弯曲波带隙范围为1060Hz~1220Hz.

  • 图2 局域共振型的圆柱壳超材料胞元及其几何参数

  • Fig.2 Cylindrical shell metamaterial based on local resonance mechanism and its geometrical parameters

  • 图3 周期边界条件施加示意图

  • Fig.3 Schematic diagram of the cyclic boundary condition imposed

  • 图4 能带结构

  • Fig.4 Energy band structure

  • 2 谐振参数与带隙特性关系研究

  • 谐振质量及其弹簧劲度系数对圆柱壳弹性波超材料的带隙特性影响很大，为此，将计算四种不同谐振器的胞元能带结构，分析带隙特性与谐振参数的关系.四种胞元差异在谐振器的尺寸上，具体信息如表2所示.

  • 表2 不同胞元的几何参数

  • Table2 Geometric parameters information in different cell

  • 运用有限元法计算四种胞元的能带结构如图5所示.从能带图中可以看出A、B、C、D四种具有不同谐振器质量的圆柱类弹性波超材料弯曲波带隙分别为1060Hz~1220Hz、950Hz~1150Hz、800Hz~1000Hz和680Hz~887Hz.

  • 图5 有限元法计算A、B、C、D四种胞元能带图

  • Fig.5 Energy band of A, B, C and D cells obtained by finite element method

  • 3 周期分级排列带隙拓宽方法

  • 在对四种具有不同谐振器的圆柱壳弹性波超材料胞元进行能带分析时获得了带隙范围与位置.在此基础上为扩大弯曲波的带隙范围，本节将具有不同谐振器的原胞进行组合，形成一种周期分级排列的带隙拓宽方法，获得更宽的弯曲波带隙范围.该方法是将n个具有相同参数的原胞在某一方向上拼接成一个整体模块，然后将不同参数的模块像周期结构一样周期性地组合在一起.由于壳体弯曲波形成了以轴向波矢ky和环向波矢ks为参数的轴向传播螺旋波，同时为了研究所设计的圆柱壳类弹性波超材料对非完整圆柱壳结构的弯曲波衰减效果，本文分别在圆柱壳的轴向和周向上利用组合法进行弯曲波带隙拓宽研究.具体方法是将8个原胞A沿轴向或周向布置形成原胞I，然后以相同的方式对原胞B、C和D进行布置并分别被命名为原胞II、III和IV；最后，沿轴向或周向将原胞I、II、III和IV拼接在一起，获得周期分级排列的圆柱壳类弹性波超材料.沿轴向和周向周期分级排列的圆柱壳类弹性波超材料分别如图6（a）、（b）所示.

  • 图6 组合法圆柱壳类弹性波超材料

  • Fig.6 Cylindrical shell elastic wave metamaterials with combined method

  • 利用有限元法计算弯曲波在结构中的传播情况以验证有效性，研究对轴向周期分级排列的圆柱壳类弹性波超材料进行振动传输特性分析.计算模型如图7所示.施加垂向力T来模拟沿入射弯曲波，完美匹配层（PML）应用于超材料两端.

  • 图8（b）给出弹性波在结构中的传递损失，从中可以发现：所研究的轴向周期分级排列圆柱壳类弹性波超材料在705Hz~1226Hz的宽频范围内对弯曲波具有良好的衰减效果，最大衰减量为68dB，平均衰减量在34dB以上，带隙宽度为原胞A带隙宽度的2.55倍，极大的拓宽了弯曲波的带隙范围.图8（a）是通过将原胞A、B、C、D四个弯曲波带隙拼接成一个图而获得的能带结构图，可以看出通过周期分级排列的方式可以很好的将四个原胞的弯曲波带隙进行有效拼接，周期分级排列圆柱壳类弹性波超材料的弯曲波带隙为695Hz~1226Hz.振动传输特性曲线的起始频率比能带结构的起始频率高30Hz，导致带隙宽度缩小了30Hz，由于此频率范围内该结构对弯曲波的衰减量很小，而在有限周期结构中无法体现出来，符合弹性波超材料有限周期结构的特点，有限周期结构的弯曲波带隙范围与无限周期结果的能带结构带隙范围吻合良好.

  • 图7 轴向周期分级排列圆柱壳类弹性波超材料

  • Fig.7 Axial periodic graded arrangement of cylindrical shell-like elastic wave metamaterials

  • 图8 能带与振动传输特性：（a）能带；（b）传输特性曲线

  • Fig.8 Energy band and transmission characteristics： (a) Energy band; (b) Transmission characteristic

  • 4 结论

  • 本文在进行了弯曲波带隙与谐振器参数的关系研究、获得带隙调控规律后，将组合分级排列方法应用于圆柱壳类弹性波超材料弯曲波带隙拓宽，进一步提高该超材料的实用性.主要有以下结论：

  • （1）谐振器质量增加时，带隙的起始频率和截止频率均下降，但起始频率下降的速度比截止频率下降的速度更快，导致带隙宽度快速增加.

  • （2）利用组合法进行弯曲波带隙拓宽时并不会影响各谐振带隙的位置和宽度，各胞元带隙是相互独立的.

  • （3）该方法可以显著拓宽弯曲波带隙范围且易于实现，为圆柱壳结构中弯曲波的宽频调控和振动控制提供了可行的解决方案.

  • 参考文献

    • [1] 肖勇.局域共振型结构的带隙调控与减振降噪特性研究[D].长沙：国防科学技术大学,2012.XIAO Y.Research on bandgap regulation and vibration and noise reduction characteristics of local resonant structures [D].Changsha:National University of Defense Science and Technology,2012.(in Chinese)

    • [2] 温激鸿.声子晶体振动带隙及减振特性研究 [D].长沙：国防科技大学,2005.WEN J H.Vibration attenuation and band gap characteristics of phononic crystals [D].Changsha:National University of Defence Technology,2005.(in Chinese)

    • [3] KUSHWAHA M S,HALEVI P,MARTINEZ G,et al.Theory of acoustic band structure of periodic elastic composites [J].Physical Review B,1994,49(4):2313-2322.

    • [4] WANG G,WEN X,WEN J,et al.Two-dimensional locally resonant phononic crystals with binary structures [J].Physical Review Letters,2004,93(15):154302.

    • [5] BILAL,OSAMA R,HUSSEIN M.Trampoline metamaterial:local resonance enhancement by springboards [J].Applied Physics Letters,2013,103(11):111901.

    • [6] XIAO Y,WEN J H,WEN X S.Broadband locally resonant beams containing multiple periodic arrays of attached resonators [J].Physics Letters A,2012,376:1384-1390.

    • [7] YUA X,LU Z,CUI F,et al.Tunable acoustic metamaterial with an array of resonators actuated by dielectric elastomer [J].Extreme Mechanics Letters,2017,12:37-40.

    • [8] YAO D,XIONG M,LUO J,et al.Flexural wave mitigation in metamaterial cylindrical curved shells with periodic graded arrays of multi-resonator [J].Mechanical Systems and Signal Processing,2022,168:108721.

    • [9] LI Q,YANG D,MAO X.Pressure-resistant cylindrical shell structures comprising graded hybrid zero Poisson’s ratio metamaterials with designated band gap characteristics [J].Marine Structures,2022,84:103221.

    • [10] NATEGHI A,BELLE L V,CLAEYS C,et al.Wave propagation in locally resonant cylindrically curved metamaterial panels [J].International Journal of Mechanical Sciences,2017,127:73-90.

    • [11] 罗金雨,姚凌云,江国期,等.一种圆柱壳类声子晶体振动带隙及振动特性研究 [J].振动与冲击,2019,(8):133-138.LUO J Y,YAO L Y,JIANG G Q,et al.Study on the vibration band gap and vibration characteristics of a cylindrical shell like phononic crystal [J].Journal of Vibration and Shock,2019,08,133-138.(in Chinese)

    • [12] SONG Y,WEN J,YU D,et al.Reduction of vibration and noise radiation of an underwater vehicle due to propeller forces using periodically layered isolators [J].Journal of Sound and Vibration,2014,333(14):3031-3043.

    • [13] LEE G Y,CHONG C,KEVREKIDIS P G,et al.Wave mixing in coupled phononic crystals via a variable stiffness mechanism [J].Journal of the Mechanics and Physics of Solids,2016,95:501-516.

  • 基本信息

    中图分类号: O321;TH113;O799
    文献标识码: A
    DOI: 10.6052/1672-6553-2023-065
    文章编号: 1672-6553-2023-21(7)-038-005

    基金信息

    国家自然科学基金资助项目(52175121)；
    National Natural Science Foundation of China (52175121)；

    引用信息

    稿件历史

    收稿日期: 2023-06-21

  • 参考文献

    • [1] 肖勇.局域共振型结构的带隙调控与减振降噪特性研究[D].长沙：国防科学技术大学,2012.XIAO Y.Research on bandgap regulation and vibration and noise reduction characteristics of local resonant structures [D].Changsha:National University of Defense Science and Technology,2012.(in Chinese)

    • [2] 温激鸿.声子晶体振动带隙及减振特性研究 [D].长沙：国防科技大学,2005.WEN J H.Vibration attenuation and band gap characteristics of phononic crystals [D].Changsha:National University of Defence Technology,2005.(in Chinese)

    • [3] KUSHWAHA M S,HALEVI P,MARTINEZ G,et al.Theory of acoustic band structure of periodic elastic composites [J].Physical Review B,1994,49(4):2313-2322.

    • [4] WANG G,WEN X,WEN J,et al.Two-dimensional locally resonant phononic crystals with binary structures [J].Physical Review Letters,2004,93(15):154302.

    • [5] BILAL,OSAMA R,HUSSEIN M.Trampoline metamaterial:local resonance enhancement by springboards [J].Applied Physics Letters,2013,103(11):111901.

    • [6] XIAO Y,WEN J H,WEN X S.Broadband locally resonant beams containing multiple periodic arrays of attached resonators [J].Physics Letters A,2012,376:1384-1390.

    • [7] YUA X,LU Z,CUI F,et al.Tunable acoustic metamaterial with an array of resonators actuated by dielectric elastomer [J].Extreme Mechanics Letters,2017,12:37-40.

    • [8] YAO D,XIONG M,LUO J,et al.Flexural wave mitigation in metamaterial cylindrical curved shells with periodic graded arrays of multi-resonator [J].Mechanical Systems and Signal Processing,2022,168:108721.

    • [9] LI Q,YANG D,MAO X.Pressure-resistant cylindrical shell structures comprising graded hybrid zero Poisson’s ratio metamaterials with designated band gap characteristics [J].Marine Structures,2022,84:103221.

    • [10] NATEGHI A,BELLE L V,CLAEYS C,et al.Wave propagation in locally resonant cylindrically curved metamaterial panels [J].International Journal of Mechanical Sciences,2017,127:73-90.

    • [11] 罗金雨,姚凌云,江国期,等.一种圆柱壳类声子晶体振动带隙及振动特性研究 [J].振动与冲击,2019,(8):133-138.LUO J Y,YAO L Y,JIANG G Q,et al.Study on the vibration band gap and vibration characteristics of a cylindrical shell like phononic crystal [J].Journal of Vibration and Shock,2019,08,133-138.(in Chinese)

    • [12] SONG Y,WEN J,YU D,et al.Reduction of vibration and noise radiation of an underwater vehicle due to propeller forces using periodically layered isolators [J].Journal of Sound and Vibration,2014,333(14):3031-3043.

    • [13] LEE G Y,CHONG C,KEVREKIDIS P G,et al.Wave mixing in coupled phononic crystals via a variable stiffness mechanism [J].Journal of the Mechanics and Physics of Solids,2016,95:501-516.

    • 微信公众号二维码

    手机版网站二维码

    动力学与控制学报 ® 2024 版权所有
    技术支持：北京勤云科技发展有限公司