非线性周期结构动力学与波动调控研究进展

高鹏林; 龚凌云; 王国旭; 罗宇; 朱俊喆; 高昊; 马海斌; 瞿叶高

doi:10.6052/1000-0992-24-047

非线性周期结构动力学与波动调控研究进展

doi: 10.6052/1000-0992-24-047 cstr: 32046.14.1000-0992-24-047

高鹏林^{1, 2},
龚凌云^{1, 2},
王国旭^{1, 2},
罗宇^{1, 2},
朱俊喆^{1, 2},
高昊^{1, 2},
马海斌^{1, 2},
瞿叶高^{1, 2, ,}

1.
上海交通大学机械系统与振动全国重点实验室, 上海 200240
2.
上海交通大学振动、冲击、噪声研究所, 上海 200240

基金项目: 国家自然科学基金(12202267, 12425202, 12393781, U2141244, 12121002, 12202260)、上海市市级科技重大专项资助的课题和上海交通大学机械系统与振动全国重点实验室自主课题(MSVZD202404)的支持

详细信息

作者简介:
高鹏林, 上海交通大学机械与动力工程学院副教授, 博士生导师. 主要研究方向为人工周期结构动力学与振动噪声控制, 在《Physical Reviews Letters》《Nature Communications》《Journal of Sound and Vibration》等学术期刊发表论文30余篇, 入选上海市浦江人才计划、上海市海外高层次青年人才, 获上海市力学学会优秀青年学者一等奖

瞿叶高, 上海交通大学科研院副院长、前沿创新研究院院长, 机械系统与振动全国重点实验室副主任. 主要从事非线性动力学设计及振动与声波操控研究. 主持国家杰出青年科学基金以及基金委重大项目课题、叶企孙基金重点项目等. 获教育部自然科学奖一等奖、上海市科技进步一等奖、中国力学学会青年科技奖、中国振动工程学会青年科技奖等奖励. 担任《Applied Acoustics》副主编以及多个国内外期刊编委. 兼任中国力学学会理性力学和力学中的数学方法专业委员会委员, 中国振动工程学会副秘书长、非线性振动专业委员会副主任等

通讯作者:
quyegao@sjtu.edu.cn

中图分类号: O322
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出版历程
- 收稿日期: 2024-12-26
- 录用日期: 2025-04-14
- 网络出版日期: 2025-05-06

Review on the dynamics and wave control in nonlinear periodic structures

GAO Penglin^{1, 2},
GONG Lingyun^{1, 2},
WANG Guoxu^{1, 2},
LUO Yu^{1, 2},
ZHU Junzhe^{1, 2},
GAO Hao^{1, 2},
MA Haibin^{1, 2},
QU Yegao^{1, 2
, ,}

1.
State Key Laboratory of Mechanical System and Vibration, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 20024, China
2.
Institute of Vibration, Shock and Noise, Shanghai Jiao Tong University, Shanghai 200240, China

More Information

Corresponding author: quyegao@sjtu.edu.cn

摘要

摘要: 周期结构是构建自然界和人类工程结构的一种重要形式, 平移对称性引起的Bloch调制作用赋予其独特的能带结构和丰富的时/频域动力学特性, 为弹性波/声波传播调控、新型声功能器件设计、振动与噪声控制等提供了新思路. 非线性效应可以突破线性理论框架的束缚, 能够增强甚至拓展人工周期结构的功能. 但非线性周期系统在单胞结构设计和建模分析方面存在诸多困难, 也面临时空平移对称性破缺、非线性响应特性及机理复杂等深层次科学难题, 给非线性周期结构动力学设计与实际应用带来挑战. 针对上述问题, 学者们融合力学、声学、材料学和能带物理等多学科研究方法开展了一些卓有成效的研究, 本文旨在及时总结非线性周期结构动力学和波动调控领域的重要研究进展, 梳理研究中存在的不足之处和关键难题, 凝聚力量, 促进该领域的深入发展. 首先, 归纳了力学周期结构非线性效应来源、单胞结构设计方法、非线性周期结构动力学建模与分析方法; 之后, 综述了非线性周期结构在通带、带隙和能量局域束缚等方面的主要特性, 着重介绍了非线性导致的能带频移、模式耦合、低频宽带拓展和带隙内波动模式局域束缚等丰富动力学现象, 梳理了非线性周期结构在波调控装置和减振降噪设计方面的一些应用探索. 最后, 针对现有研究存在的一些不足和关键难题, 展望了未来理论研究和应用探索亟需关注的若干发展方向.
- 非线性周期结构 /
- 动力学设计 /
- 非线性振动 /
- 振动控制 /
- 波动调控
Abstract: Periodic structures are an important form of constructing structures in natural and human engineering. The Bloch modulation caused by translational symmetry gives them unique band structures and rich time/frequency domain dynamic characteristics, providing new avenues for elastic/acoustic wave propagation regulation, novel wave-based functional device design, vibration and noise control, etc. Nonlinear effects can break through the constraints of linear theoretical frameworks and can enhance or even expand the functions of artificial periodic structures. However, nonlinear periodic systems have many difficulties in unit cell design and modeling analysis. They also face key scientific problems such as the broken of space-time invariance, the complexity of nonlinear response characteristics and mechanisms, which brings challenge to the dynamic design and practical application of nonlinear periodic structures. In response to the above problems, scholars have carried out some fruitful research by integrating multidisciplinary research methods in fileds such as mechanics, acoustics, materials science and band physics. This article aims to timely summarize the important research progress concerning about wave dynamics and control in nonlinear periodic structures, sort out the shortcomings and key problems, gather strength and promote the in-depth development of this field. First, the sources of nonlinear effects of periodic structures, unit cell structure design methods, and nonlinear dynamics modeling and analysis methods are summarized. Then, the main characteristics of nonlinear periodic structures in terms of passband, bandgap, and local energy confinement are reviewed, and the rich dynamic phenomena such as amplitude-induced band shift, wave mode coupling, low-frequency broadband bandgap, and spatial confinement of wave modes in the bandgap caused by nonlinearity are introduced. Some application explorations of nonlinear periodic structures in wave control devices, vibration and noise reduction are sorted out. Finally, in view of some shortcomings and key problems in existing research, several development directions that need special attention in future theoretical research and application exploration are prospected.
- nonlinear periodic structure /
- dynamic design /
- nonlinear vibration /
- vibration control /
- wave control

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图 1 自然界和人类工程中存在的典型周期结构. (a) 蓝闪蝶翅鳞结构(Plattner 2004), (b) 微纤毛驱动流体运动示意图(Wang et al. 2024b), (c) 蜘蛛网, (d) 由周期桁架结构构成的星载天线(Zheng et al. 2022), (e) 运载火箭中由周期格栅构成的复合材料壳体结构(Vasiliev et al. 2012)

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图 2 线性周期结构和非线性周期结构发展历程简图. 来自文献(Liu et al. 2000, 陈毅等 2016, Nash et al. 2015, Gao et al. 2022b, Merkel et al. 2011, Meaud 2020, Liang et al. 2010, Fang et al. 2017a, Alfahmi et al. 2022)

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图 3 WOS数据库收录的关于周期结构的论文数目变化过程(数据截止于2024年12月)

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图 4 周期结构非线性效应来源. (a) 几种代表性非线性效应 (b) 非线性作用力示意图

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图 5 通过几何构型设计实现非线性回复力. (a) 线性弹簧组合结构, 实现三次非线性回复力(Bae & Oh 2022), (b) 线性弹簧组合结构实现双稳态(Nadkarni et al. 2014), (c) 折纸双稳态结构(Gillman et al. 2018), (d) 连杆−软铰链组合结构(Zhang & Rudykh 2024)

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图 6 利用几何大变形和材料非线性实现非线性回复力. (a) 屈曲梁周期结构(Bidhendi 2022), (b) 大变形超弹性梁−弹簧双稳态结构(Raney et al. 2016), (c) 曲梁−直梁−弹簧多稳态结构(Xu & Xiang 2024), (d) 大变形屈曲壳可重构多稳态结构(Khajehtourian & Kochmann 2021), (e) 多稳态网络结构(Jin et al. 2020)

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图 7 利用接触或碰撞实现非线性回复力. (a) 带间隙的弹簧−质量振子结构(Yu et al. 2024), (b) 颗粒晶体(Allein et al. 2020), (c) 木桩结构(Kim et al. 2015)

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图 8 通过外界物理场引入非线性效应. (a) 具有非线性力−磁耦合作用的梁结构(Xue et al. 2023), (b) 含永磁体的柔性铰链结构(Jiao et al. 2024), (c) 由介电弹性材料构成一维周期结构(Chen et al. 2020)

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图 9 非线性惯性力和非线性阻尼力. (a) 惯性放大旋转结构, 实现非线性阻尼力(Zhao et al. 2024), (b) 弹簧−连杆惯性放大机构, 同时实现非线性惯性力和非线性阻尼力(Settimi et al. 2021), (c) 弹簧摆结构, 同时实现非线性惯性力和非线性阻尼力(Zhou et al. 2024a), (d)弹簧−连杆组合结构, 同时实现非线性惯性力、非线性阻尼力和非线性回复力(Zhou et al. 2019)

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图 10 一维非线性原子链能带结构时/频域分析. (a) 弱非线性原子链(Zhou et al. 2018), (b) 强非线性原子链, 实线为谐波平衡法计算结果, 背景阴影色为时域计算获得的2D FFT结果(Gong et al. 2023)

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图 11 基于不同频域计算方法获得的非线性周期结构能带曲线. (a) Lindstedt-Poincaré摄动法与传递矩阵法对比(Manktelow et al. 2013a), (b) Lindstedt-Poincaré摄动法与多尺度法对比, $ {\omega _0} $为线性频散曲线, $ \omega $为单谐波近似下两种频域方法的计算结果(Manktelow et al. 2011)

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图 12 可编程压电周期结构及其带隙性质. (a) 压电梁周期结构, (b) 基于可编程数字电路的力−电耦合单胞, (c) 不同非线性系数$ {\alpha _{{\text{nl}}}} $下能带曲线变化, 青色圆圈和背景颜色分别为采用谐波平衡法和2D FFT方法获得的能带曲线

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图 13 压电周期结构的频散曲线与2D FFT谱. 青色圆圈和背景颜色分别为采用谐波平衡法和2D FFT计算的能带曲线, 时域仿真中激励脉冲中心频率为82 Hz. (a) 线性电路$ {\alpha _{{\text{nl}}}} = 0 $, (b) 非线性电路$ {\alpha _{{\text{nl}}}} = {10^{12}} $

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图 14 非线性压电周期结构与缺陷束缚态. (a) 由负电容分流电路构造的压电梁示意图, 通过移除中间区域的2对压电片构造局部缺陷, (b) 不同非线性系数下的传递率频响曲线, 青色背景表示初始线性系统的Bragg带隙范围, (c) 不同非线性系数下压电梁传递率频响, 横坐标为频率, 纵坐标为梁长度方向的归一化坐标

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图 15 非线性原子链能带频移特性. (a) 不同波幅下单原子颗粒晶体的能带曲线, (b) 不同波幅下二维原子链的群速度方向图, (c) 小振幅激励下弹性波的传播, (d) 大振幅激励下弹性波的传播. 来自文献(Narisetti et al. 2012, 2011)

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图 16 非线性引起的不同频率间能量转换. (a) 由钢芯谐振结构组成的一维周期结构(Jiao & Gonella 2018a); (b) 基波与超谐波模式耦合效应示意图(Jiao & Gonella 2018a); (c) 由粗糙接触界面构成的非线性周期结构(Patil & Matlack 2021); (d) 某时刻的弹性波位移信号, 红色虚线表示周期结构所在区域(Patil & Matlack 2021); (e) 空间卷绕型声学周期结构与非线性混频现象(Jeon & Oh 2021)

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图 17 一维非线性原子链能带曲线与带隙特征. (a) Bragg带隙, (b) 局域共振带隙. 图中黑色实线为线性系统的结果, 蓝色曲线为考虑非线性效应的结果. 来自文献(Bae & Oh 2022)

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图 18 弹性矢量孤波及幅度带隙. (a) 柔性铰链组成的非线性周期结构, 其可承载弹性矢量孤波, (b) 孤波幅度带隙范围与结构角$ {\varphi _0} $的变化规律, (c) 幅度传递率$ {{{A_{40}}} /{{A_2}}} $, (d) $ {\theta _{40}} $和$ {\theta _2} $相关性关系. (c)和(d)为$ {\varphi _0} = 5^\circ $时的仿真和实验结果. 来自文献(Deng et al. 2018)

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图 19 非线性周期结构带隙频移特性. (a) 非线性双原子链模型与实物结构, (b) 曲梁结构载荷−变形曲线, (c) 不同激励强度下透射谱实验结果, 橙色虚线表示理论预测的带隙边界, 深蓝区域为带隙范围. 来自文献(Zhang et al. 2023)

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图 20 非线性带隙时空相关特性. (a) 非线性单原子链模型, (b) 不同激励幅度下单原子链透射频谱变化过程, (c) 三原子单胞及原子链模型, 支链采用两自由度非线性振子, (d) 200 Hz激励下, 透射系数随点阵位置$ n = {x / a} $变化过程, 图例中L和N分别表示线性和非线性原子链, $ {d_0} $为阻尼系数. 来自文献(Bae & Oh 2020, Fang et al. 2020)

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图 21 通带和带隙之间频率成分转换与能量转换. (a) 左图为单频激励$ {f_0} $作用下, 颗粒晶体的透射功率谱密度示意图, 右图施加了频率为$ {f_{\mathrm{c}}} $的高幅控制信号, (b) 施加控制信号$ {f_{\mathrm{c}}} $后, 实验测量的透射功率谱密度, (c) 含非线性振子的一维原子链模型, (d) 混沌对通带和带隙透射率影响示意图, (e) 单频激励下($ {\Omega _{\text{e}}} = 0.678 $), 非线性原子链功率谱密度随激励波幅变化图. 来自文献(Li et al. 2014, Fang et al. 2017b)

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图 22 惯性放大机构. (a) 弹簧−质量位移放大机构及其组成的二维周期结构(Yilmaz et al. 2007), (b) 由磁流变弹性体构成的主动可调惯性放大机构, 右下角插图为传递率, 随着杠杆比$ \mu = {{{l_{\mathrm{b}}}} / {{l_{\mathrm{a}}}}} $增加带隙位置向低频拓展(Xue et al. 2024)

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图 23 准零刚度周期结构. (a) 曲梁单元及其构成的准零刚度周期结构(Zhang et al. 2021b), (b) 附加准零刚度谐振子的周期梁(Zhou et al. 2019), (c) 基于弹簧正、负刚度抵消效应设计的准零刚度谐振子(Zhou et al. 2019, 2017)

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图 24 传统的带隙拓宽方法. (a) 单相多重谐振结构及其能带结构(Gao et al. 2019a), (b) 超材料梁透射率随阻尼$ \zeta $和梯度参数$ \delta $的变化图, 参数$ \left( {\zeta ,\delta } \right) $从上到下分别为(0.6, 0.1)、(0, 0)、(0, 0.05)和(0, 0.1)(Hu et al. 2021), (c) 融合负刚度和惯性放大效应的单胞结构(Zhou et al. 2024b), (d) 带隙低频宽带拓展示意图(Zhou et al. 2024b)

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图 25 基于非线性效应的带隙拓宽方法. (a) 非线性弹簧−质量三原子模型, (b) 三原子系统等效质量, (c) 包含20个单胞的非线性原子链频响曲线, 黑色代表线性原子链, 彩色对应不同的非线性系数, $ {{{k_n}} / {{k_1}}} = 100 $(洋红色)、$ {{{k_n}} / {{k_1}}} = 500 $(红色)、$ {{{k_n}} / {{k_1}}} = 1000 $(青色), (d) 附加双稳态振子的周期梁结构, (e) 双稳态振子相图, 周期梁的基底加速度为$ 0.3g $, 激励频率为12.5 Hz, (f) 多重局域共振非线性原子链模型, (g) 压电梁与非线性分流电路. 来自文献(Fang et al. 2018, Xia et al. 2020, Zhao et al. 2023c, Chen et al. 2024a)

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图 26 非线性周期结构能量局域束缚效应. (a) 经典呼吸子波形与频散曲线, (b) 波数带隙呼吸子波形与频散曲线, (c) 颗粒晶体缺陷束缚态频率$ {f_b} $随激励幅度$ {{\max \left( {{u_{40}}} \right)} / {{\delta _1}}} $变化曲线, 插图为颗粒晶体模型和束缚态波形, (d) 大振幅激励下颗粒晶体中弹性波传播和束缚态时空演化过程, 该算例中缺陷束缚态频率为22.47 kHz, (e) 非线性Aubry–André原子链特征频率$ \Omega $随波幅$ A $变化过程, IPR表征波模式的局域状态, 数值越大表示局域束缚程度越高, 插图为非线性Aubry–André模型示意图, (f) 特征模式I和II空间束缚状态随波幅$ A $变化过程. 来自文献(Chong et al. 2024, Theocharis et al. 2009, Rosa et al. 2023)

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图 27 非线性效应调控拓扑相变. (a) 上图: 基于SSH模型构造的含软化和硬化非线性的原子链模型, 下图: 入射波振幅调控拓扑相变示意图, 大振幅激励可以产生频率为$ {\Omega _{\mathrm{m}}} $的拓扑束缚态, (b) 归一化模态幅度随激励幅度$ A $变化过程, 颜色越深表示能量局域束缚程度越高, (c) 非线性效应破缺四重简并点(青色虚线在$ \Gamma $点构成双Dirac锥结构), 形成拓扑非平庸带隙, (d) F = 1 N激励力作用下六方点阵的位移场分布, 呈现出非局域体波传播现象, (e) F = 100 N 激励力作用下六方点阵的位移场分布, 呈现出界面传播现象. 来自文献(Chaunsali & Theocharis 2019, Darabi & Leamy 2019)

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图 28 强非线性效应对拓扑束缚态的影响. (a) 基于SSH模型构造的非线性原子链, (b) 带隙内束缚态特征频率$ \omega $随激励能量的变化趋势, 图中阴影区域对应线性原子链的体能带, (c) 不同激励能量下界面态局域状态变化, $ {a_1} $ ~ $ {j_1} $对应图(b)中标识点, (d) 在线性SSH模型中引入手性对称缺陷后, 束缚态频率随缺陷强度$ \left| \delta \right| $的变化, (e) 手性对称缺陷对非线性界面态鲁棒性的影响, 模式浓度表示波模式的局域束缚程度, 数值越大, 能量局域程度越高. 来自文献(Tempelman et al. 2021)

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图 29 多稳态周期结构与转变波. (a) 由预压缩复合材料曲梁构成的双稳态周期结构, 利用永磁体在结构单元间产生非线性斥力(Nadkarni et al. 2016), (b) 充气式主动可调双稳态壳体结构(Vasios et al. 2021), (c) 由双稳态壳体构成的一维周期结构(Vasios et al. 2021), (d) 三稳态周期结构与转变波碰撞现象(Yasuda et al. 2020), (e) 利用缺陷和边界引导拉胀周期结构中转变波的传播(Jin et al. 2020)

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图 30 弹性孤波与非互易波. (a) 上图: 张拉整体结构, 下图: 不同冲击速度$ {v_0} $下压缩脉冲转变为稀疏型孤波的演化过程(Fraternali et al. 2014), (b) 上图: 由柔性铰链构成的一维周期结构, 下图: 矢量孤波传播与碰撞过程(Deng et al. 2019b), (c) 由线性周期结构和非线性介质构成的非互易系统, 实现弹性波/声波二极管功能(Li et al. 2019)

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图 31 梁/板/壳等代表性结构形式与周期结构设计. (a) 受磁力控制的非线性梁结构(Xue et al. 2023), (b) 周期性排布动力吸振器的转子结构(Xu et al. 2023), (c) 几种轻质夹芯结构的相对密度−带隙宽度对比图(Gao et al. 2024), (d) 含惯性放大共振结构的三明治夹芯板(Yang et al. 2024), (e) 附加准零刚度单元的圆柱壳周期结构(Cai et al. 2024)

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图 32 基于非线性分流电路的压电周期结构与低频振动控制. (a) 压电周期结构实验装置, (b) 压电梁在微幅激励下的振动频响, 黄色阴影区域表示非线性带隙, (c) 压电梁非线性动力学频响特性, (d) 压电梁传递率频响曲线, 调节纯非线性系数实现宽带可调振动控制, (e) 主动胞元结构, 压电片用于实现时滞反馈控制, (f) 优化时滞参数获得的超宽带隙(灰色阴影区域). 来自文献(Zhang et al. 2021c, Gong et al. 2025, Gao & Wang 2023)

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图 33 非线性周期结构在航空航天领域振动控制中的应用. (a) 超音速来流下功能梯度壁板颤振控制示意图, (b)壁板结构局部位移极值分岔图, 壁板上周期排布非线性动力吸振器, (c) 附加非线性振子的薄板周期结构, 局部放大图展示了由永磁体构成的强非线性共振结构, (d) 线性和非线性振子作用下, 薄板结构振动和辐射噪声频谱, (e) 准零刚度周期结构及其在航天器控制力矩陀螺隔振系统中的应用. 来自文献(Tian et al. 2022a, Fang et al. 2022, Zhao et al. 2023b)

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图 34 水下结构声振耦合控制. (a) 左图: 准零刚度周期结构, 右图: 水下滑翔机超低频隔振应用(Liu et al. 2024), (b) 压电周期结构用于水下壳体多模态声振耦合控制(Zheng et al. 2024)

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图 35 高声压级下微孔结构非线性声学特性. (a) 孔口粒子振速与激励声压之间呈现非线性关系, (b) 90 dB下Helmholtz共鸣器孔口粒子振速分布, (c) 140 dB下孔口粒子振速分布. 来自文献(Ingard & Ising 1967, Zhu et al. 2022)

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图 36 吸隔声阵列结构. (a) Helmholtz共鸣器平面阵列结构(Huang et al. 2020), (b) 轻质蜂窝夹芯板吸声结构(Tang et al. 2020), (c) 周期性多层嵌套吸声结构(插图)与高声强混响室吸声性能测试装置(Zhu et al. 2023a), (d) 整流罩模型布置吸声器前后, 内声腔噪声频谱(Zhu et al. 2023a), (e) 含Helmholtz共鸣器或微孔阵列的一维声管道(Fan et al. 2013, Lan et al. 2017)

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图 37 基于异质界面和有源主动系统构造的非互易整流器件. (a) 钢珠和聚四氟乙烯珠(PTFE)构成的颗粒晶体, 顶端通过磁体施加预紧力(Nesterenko et al. 2005), (b) 弹性波整流器件与工作原理示意图, 整流器件由颗粒晶体和层状周期结构构成(Devaux et al. 2015), (c) 有源声学整流器件(Popa & Cummer 2014), (d) Floquet拓扑绝缘体与单向边界态, 通过对单胞声容进行时变调制引入非互易效应(Fleury et al. 2016)

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图 38 基于弹性波传播特性设计的逻辑门器件. (a) 颗粒晶体开关功能示意图及OR/AND逻辑运算信号功率谱密度(Li et al. 2014), (b) 全加法器原理图与对应的周期结构实现装置, 右上角插图为弹性波开关原理示意图, 随着单胞构型变化传输信号会在开/关两种状态间切换(Bilal et al. 2017)

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图 39 非线性波束操纵器件. (a) 由颗粒晶体构成的非线性聚焦透镜, 调节静态预紧力空间分布可改变焦点位置(Spadoni & Daraio 2010), (b) 二维非线性声透镜, 彩色表示水中声学子弹恒压面(Donahue et al. 2014), (c) 柔性铰链构成的非线性周期结构(Deng et al. 2019a), (d) 不同冲击角下的孤波脉冲速度场分布(Deng et al. 2019a)

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图 40 基于周期结构实现无损检测或健康监测功能. (a) 利用颗粒晶体中孤波反射特性识别复合材料弱粘接区域(Singhal et al. 2017), (b) 颗粒晶体用于复合材料分层损伤实时监测(Yoon et al. 2023), (c) 由周期结构滤波区域(MM1)、周期结构聚焦区域(MM2)和混沌腔构成的缺陷检测装置(Gliozzi et al. 2015)

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图 41 基于波传播的致动器件. (a) 毛毛虫爬行运动的“二极管”输运特性, 以此可设计能够单向运动的周期结构致动器件(Zhou & Wang 2024), (b) 由刚性杆和双稳态关节组成的机械连杆机构, 利用转变波现象实现空间展开部署(Zareei et al. 2020), (c) 双稳态旋转方块结构单元与其构成的二维周期结构致动器件(Deng et al. 2022a)

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表 1 常见的非线性周期结构动力学建模方法

示意图	动力学模型	代表性文献
	原子链模型	(Chakraborty & Mallik 2001) (Lazarov & Jensen 2007)
	连续极限模型	(Deng et al. 2019a) (Deng et al. 2020b)
	有限元模型	(Manktelow et al. 2013b) (Mosquera-Sánchez et al. 2024)
	混合模型	(Casalotti et al. 2018) (Fang et al. 2022)

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表 2 压电单胞几何及材料参数

参数	基底层 (铝)	压电层 (PZT-5H)
长度 (mm)	50.00	50.00
宽度 (mm)	25.00	25.00
厚度 (mm)	0.50	0.25
密度 (kg/m³)	2700	7450
杨氏模量 (GPa)	69.0	76.9
压电应力常数$ {\bar e_{31}} $ (C/m²)	—	−14.3
恒应力介电常数$ {\overline{\epsilon }}_{33}^{\text{S}} $ (nF/m)	—	37.2

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