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Article(id=1227620261831307868, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1227620260010979924, articleNumber=null, orderNo=null, doi=10.16385/j.cnki.issn.1004-4523.2024.05.013, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1663603200000, receivedDateStr=2022-09-20, revisedDate=1669478400000, revisedDateStr=2022-11-27, acceptedDate=null, acceptedDateStr=null, onlineDate=1770617078501, onlineDateStr=2026-02-09, pubDate=1716825600000, pubDateStr=2024-05-28, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1770617078501, onlineIssueDateStr=2026-02-09, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1770617078501, creator=13701087609, updateTime=1770617078501, updator=13701087609, issue=Issue{id=1227620260010979924, tenantId=1146029695717560320, journalId=1225147924628267009, year='2024', volume='37', issue='5', pageStart='729', pageEnd='902', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1770617078068, creator=13701087609, updateTime=1770795280844, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1228367696677499202, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1227620260010979924, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1228367696677499203, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1227620260010979924, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=847, endPage=855, ext={EN=ArticleExt(id=1227620262011662943, articleId=1227620261831307868, tenantId=1146029695717560320, journalId=1225147924628267009, language=EN, title=Semi analytical coupling modeling and vibration characteristics of ABH plate with constrained damping layer, columnId=null, journalTitle=Journal of Vibration Engineering, columnName=null, runingTitle=null, highlight=null, articleAbstract=

In recent years，acoustic black hole （ABH） has shown an extremely broad application prospect in the fields of structural vibration and noise suppression，acoustic wave control，energy recovery，etc，due to its excellent performance. However，the truncation of ABH edge will lead to the existence of non-zero reflection coefficient，thus weakening the acoustic black hole effect. In this paper，the constrained layer damping is introduced into ABH plates. Under the framework of Rayleigh Ritz method，Gaussian function is selected as the basis function，and the distribution of basis function is determined according to the shape of ABH plate to avoid the singularity of mass matrix and stiffness matrix. A semi analytical model of ABH plate with constrained layer damping is established. By comparing with the results of finite element analysis，the correctness of the semi analytical modeling method is verified. The influence of structural parameters of constrained layer damping on the bending vibration characteristics of ABH plate is studied，and the damping mechanism and energy dissipation of constrained layer damping are revealed. The experiment further verifies the damping effect of ABH plate with constrained layer damping. The research provides a design reference for the application of constrained layer damping in ABH structures.

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声学黑洞（ABH）以其优异的性能在结构减振降噪、声波调控、能量回收等领域展示了极其广阔的应用前景。但声学黑洞边缘截断会导致非零反射系数的存在，从而弱化声学黑洞效应。为此，本文在声学黑洞结构中引入约束阻尼材料，在Rayleigh-Ritz法框架下，选择高斯函数作为基函数，根据声学黑洞板的形状确定高斯基函数的分布，避免质量矩阵和刚度矩阵的奇异化，建立了声学黑洞约束阻尼板的半解析分析模型。通过与有限元分析结果对比，验证了半解析建模方法的正确性。研究了约束阻尼结构参数对声学黑洞板弯曲振动特性的影响规律，揭示了约束阻尼的减振机理和能量耗散作用。实验进一步验证了声学黑洞约束阻尼板的减振效果。

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郑玲（1963―），女，博士，教授。E-mail： 。
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李军军（1997—），男，硕士。E-mail： 。

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李军军（1997—），男，硕士。E-mail： 。

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李军军（1997—），男，硕士。E-mail： 。

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Mechanical Systems and Signal Processing， 2019， 118： 461-476., articleTitle=Passive constrained viscoelastic layers to improve the efficiency of truncated acoustic black holes in beams, refAbstract=null)], funds=[Fund(id=1227675686794625502, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227620261831307868, awardId=51875061, language=CN, fundingSource=国家自然科学基金资助项目(51875061), fundOrder=null, country=null)], companyList=[AuthorCompany(id=1227675678619926690, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227620261831307868, xref=1, ext=[AuthorCompanyExt(id=1227675678632509602, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227620261831307868, companyId=1227675678619926690, language=EN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=1College of Mechanical and Vehicle Engineering, Chongqing University, Chongqing 400044, China), 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Geometric parameters

, figureFileSmall=null, figureFileBig=null, tableContent=
参数数值
矩形板长度a/m0.195
矩形板宽度b/m0.145
矩形板厚度huni/m0.005
声学黑洞半径rabh/m0.1015
声学黑洞中心截断厚度hcenter/m0.001
声学黑洞剖面变化幂指数m2.5
声学黑洞剖面的斜率ξ1.22
阻尼层半径rv/m0.1015
阻尼层厚度hv/m0.0021
约束层半径rc/m0.1015
约束层厚度hc/m0.001
A（xA,yA,zA）/m（0.195,0.0075,0.005）
B（xB,yB,zB）/m（0.195,0.015,0.005）
C（xC,yC,zC）/m（0.195,0.015,0.005）
), ArticleFig(id=1227675686391972301, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227620261831307868, language=CN, label=表1, caption=

几何参数

, figureFileSmall=null, figureFileBig=null, tableContent=
参数数值
矩形板长度a/m0.195
矩形板宽度b/m0.145
矩形板厚度huni/m0.005
声学黑洞半径rabh/m0.1015
声学黑洞中心截断厚度hcenter/m0.001
声学黑洞剖面变化幂指数m2.5
声学黑洞剖面的斜率ξ1.22
阻尼层半径rv/m0.1015
阻尼层厚度hv/m0.0021
约束层半径rc/m0.1015
约束层厚度hc/m0.001
A（xA,yA,zA）/m（0.195,0.0075,0.005）
B（xB,yB,zB）/m（0.195,0.015,0.005）
C（xC,yC,zC）/m（0.195,0.015,0.005）
), ArticleFig(id=1227675686467469779, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227620261831307868, language=EN, label=Tab.2, caption=

Material parameters

, figureFileSmall=null, figureFileBig=null, tableContent=
组件密度ρ/(kg·m‒3）弹性模量E/GPa泊松比ν损失因子η
声学黑洞板78902060.260.013
阻尼层10000.060.30.3
约束层78902060.260.013
), ArticleFig(id=1227675686559744469, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1227620261831307868, language=CN, label=表2, caption=

材料参数

, figureFileSmall=null, figureFileBig=null, tableContent=
组件密度ρ/(kg·m‒3）弹性模量E/GPa泊松比ν损失因子η
声学黑洞板78902060.260.013
阻尼层10000.060.30.3
约束层78902060.260.013
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声学黑洞约束阻尼板半解析耦合建模与振动特性研究
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李军军 1 , 李剑辉 1 , 郑玲 1 , 邓杰 2
振动工程学报 | 2024,37(5): 847-855
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振动工程学报 | 2024, 37(5): 847-855
声学黑洞约束阻尼板半解析耦合建模与振动特性研究
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李军军1 , 李剑辉1, 郑玲1 , 邓杰2
作者信息
  • 1重庆大学机械与运载工程学院，重庆　400044
  • 2西北工业大学航海学院，陕西西安　710072

    • 李军军（1997—），男，硕士。E-mail： 。

通讯作者:

郑玲（1963―），女，博士，教授。E-mail： 。
Semi analytical coupling modeling and vibration characteristics of ABH plate with constrained damping layer
Jun-jun LI1 , Jian-hui LI1, Ling ZHENG1 , Jie DENG2
Affiliations
  • 1College of Mechanical and Vehicle Engineering, Chongqing University, Chongqing 400044, China
  • 2School of Marine Science and Technology, Northwestern Polytechnical University, Xi’an 710072, China
出版时间: 2024-05-28 doi: 10.16385/j.cnki.issn.1004-4523.2024.05.013
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摘要
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声学黑洞（ABH）以其优异的性能在结构减振降噪、声波调控、能量回收等领域展示了极其广阔的应用前景。但声学黑洞边缘截断会导致非零反射系数的存在，从而弱化声学黑洞效应。为此，本文在声学黑洞结构中引入约束阻尼材料，在Rayleigh-Ritz法框架下，选择高斯函数作为基函数，根据声学黑洞板的形状确定高斯基函数的分布，避免质量矩阵和刚度矩阵的奇异化，建立了声学黑洞约束阻尼板的半解析分析模型。通过与有限元分析结果对比，验证了半解析建模方法的正确性。研究了约束阻尼结构参数对声学黑洞板弯曲振动特性的影响规律，揭示了约束阻尼的减振机理和能量耗散作用。实验进一步验证了声学黑洞约束阻尼板的减振效果。

振动控制  /  声学黑洞  /  约束阻尼  /  高斯展开法  /  半解析模型

In recent years，acoustic black hole （ABH） has shown an extremely broad application prospect in the fields of structural vibration and noise suppression，acoustic wave control，energy recovery，etc，due to its excellent performance. However，the truncation of ABH edge will lead to the existence of non-zero reflection coefficient，thus weakening the acoustic black hole effect. In this paper，the constrained layer damping is introduced into ABH plates. Under the framework of Rayleigh Ritz method，Gaussian function is selected as the basis function，and the distribution of basis function is determined according to the shape of ABH plate to avoid the singularity of mass matrix and stiffness matrix. A semi analytical model of ABH plate with constrained layer damping is established. By comparing with the results of finite element analysis，the correctness of the semi analytical modeling method is verified. The influence of structural parameters of constrained layer damping on the bending vibration characteristics of ABH plate is studied，and the damping mechanism and energy dissipation of constrained layer damping are revealed. The experiment further verifies the damping effect of ABH plate with constrained layer damping. The research provides a design reference for the application of constrained layer damping in ABH structures.

vibration control  /  acoustical black hole （ABH）  /  constrained damping layer  /  Gaussian expansion method  /  semi-analytical model
李军军, 李剑辉, 郑玲, 邓杰. 声学黑洞约束阻尼板半解析耦合建模与振动特性研究. 振动工程学报, 2024 , 37 (5) : 847 -855 . DOI: 10.16385/j.cnki.issn.1004-4523.2024.05.013
Jun-jun LI, Jian-hui LI, Ling ZHENG, Jie DENG. Semi analytical coupling modeling and vibration characteristics of ABH plate with constrained damping layer[J]. Journal of Vibration Engineering, 2024 , 37 (5) : 847 -855 . DOI: 10.16385/j.cnki.issn.1004-4523.2024.05.013
正文
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引言
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声学黑洞是利用薄壁结构几何参数或材料参数的梯度变化，使波在结构中的传播速度逐渐减小，理想情况下波速减小至零从而不发生反射的现象。由于结构简单、易于加工以及具有独特的能量聚集效应，声学黑洞在振动噪声控制［1-3］、能量回收［4-6］、声辐射控制［7-9］、信号处理［10］等方面均展示了广阔的应用前景。
针对声学黑洞存在的非均匀波长分布和波数快速变化问题，国内外学者提出了不同的建模方法，包括几何声学法［11-12］、传递矩阵法［13-14］、有限元法［15-16］和半解析法［17-18］等。其中半解析法吸收了解析法的准确性和数值方法处理边界问题的灵活性，具有较高的计算精度和效率。Deng等［19-20］采用基于克罗内克积表示的二维高斯基函数对周期声学黑洞板的位移场进行了拟合，但采用克罗内克积表示的二维基函数容易导致质量矩阵和刚度矩阵的奇异化。
为了克服边缘截断带来的非零反射系数和声学黑洞效应弱化问题，阻尼材料一直被广泛研究。Krylov［21］的研究表明声学黑洞和自由阻尼的耦合效应可以使反射系数明显减小。Zeng等［22］研究了圆锥形声学黑洞梁的能量聚集效应以及阻尼层作用。Ji等［23］通过在声学黑洞梁的能量聚集区域引入阻尼层，进一步减小了声学黑洞梁的振动。Liu等［24］将超构表面与声学黑洞相结合，设计了圆形声学黑洞超构表面，少量阻尼材料强化了超构表面的能量吸收。Deng等［25］研究了约束阻尼在补偿声学黑洞梁边缘截断方面的明显作用。
本文基于高斯展开法，建立声学黑洞与约束阻尼耦合系统的半解析分析模型。分析约束阻尼结构参数对声学黑洞板弯曲振动特性的影响规律，探索约束阻尼在声学黑洞结构中的减振机理和能量耗散作用，为约束阻尼在声学黑洞减振结构中的应用提供设计参考。
1　声学黑洞约束阻尼板半解析模型
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1.1　问题描述
将一个圆形的二维声学黑洞嵌入到一个矩形薄板中，在声学黑洞中心位置处粘贴约束阻尼材料，构建声学黑洞约束阻尼板，如图1所示。声学黑洞板的厚度为huni，长度和宽度分别为和。A点为谐波激励作用点，其坐标为（xA，yA，zA）。声学黑洞位于矩形板的中心，其厚度变化服从幂指数规律h（r）=ξrm+hcenter，其中r=［（x-a）2+（x-b）2］1/2为坐标点到声学黑洞中心的距离，ξ=（huni-hcenter）/r为声学黑洞剖面变化的斜率，hcenter为声学黑洞中心截断厚度，rabh为声学黑洞半径，m为厚度变化幂指数。约束阻尼处于声学黑洞的中心位置，其中阻尼层半径rv和约束层半径rc保持一致，阻尼层的厚度为hv，约束层的厚度为hc。边界为自由边界。
1.2　高斯基函数的选取
为了建立声学黑洞与约束阻尼耦合系统运动方程，遵循参考文献［25］中的约束假设。基板及各层的位移关系如图2所示。其中，w为横向挠度；up，uv和uc分别为ABH板、阻尼层和约束层在x方向的位移；vp，vv和vc分别为ABH板、阻尼层和约束层在y方向的位移。
基于约束假设条件，声学黑洞板与约束阻尼各层的位移关系可表示为：
式中　 hp，hv和hc分别为ABH板、阻尼层和约束层的厚度；，分别为阻尼层在yOx和zOx平面内的夹角变化。
通过式（1）可以得到阻尼层中面的位移和应变：
式中　 d = （hp+hc）/2+hv表示约束层和基板的中面距离。
在Rayleigh-Ritz法框架下，将w，up，uc，vp，vc进行分解，表示为基函数和权重系数的组合形式：
式中　 ，ξu，ξv，φu，φv为基函数的集合；，，，为权重系数集合。定义系统权重系数向量q为：
由于基函数集合是二维的，可以进一步用克罗内克积进行分解：
式中　 pψxi和pψyi分别为x方向和y方向的尺度因子；qψxi和qψyi分别为x方向和y方向的平移因子。
尺度因子pψxi和pψyi的下限为：
式中　ceil（x）表示大于或等于x的最小整数。
平移因子qψxi和qψyi的取值范围为：
同理，ξu，ξv，φu和φv也可以用克罗内克积表示。
直接采用克罗内克积表示φu和φv，其基函数为矩形分布，但文中约束层为圆形，这将导致质量矩阵和刚度矩阵奇异化。为此，首先对约束层基函数进行筛选，筛选后的形函数和各自的中心坐标（qφuxi，qφuyi），（qφvxi，qφvyi）应满足：
式中　 ，和分别为约束层基础数和在x方向和y方向的尺度因子；，为形函数中心到约束层边界的距离。由此可记录，中的元素在φu和φv中的位置，用于剔除质量矩阵、刚度矩阵和外力的幅值向量中多余元素对应的行和列。
1.3　运动方程
声学黑洞与约束阻尼耦合系统的动能K由声学黑洞板的动能Kp、阻尼层的动能Kv、约束层的动能Kc组成：
其中：
式中　 ρp，ρv，ρc分别为声学黑洞板、阻尼层和约束层的密度。
耦合系统的势能U为：
式中　 Up为声学黑洞板的势能，Uv为阻尼层的势能，Uc为约束层的势能。
式中　 σi和εi（i=p，v，c）为每层的应力和应变；γxz和γyz分别为在xOz平面和yOz平面的切应变；E=Ep（1+iηp）和E=Ec（1+iηc）分别为声学黑洞板和约束层的复弹性模量，其中Ep和Ec分别为声学黑洞板和约束层的杨氏模量，ηp和ηc分别为声学黑洞板和约束层的阻尼损耗因子；为阻尼层的复剪切模量。
激励点A处的外力功可表示为：
式中　 为作用在A点的简谐外力。
由此，耦合系统的拉格朗日函数为：
式中　 M和K为系统的质量矩阵和刚度矩阵；f为外力向量。
在计算得到M，K和f后，根据和中的元素在φu和φv中的位置剔除多余元素对应的行和列，便可得到系统的质量矩阵，刚度矩阵和外力向量。
根据欧拉-拉格朗日方程：
声学黑洞与约束阻尼耦合系统的运动方程为：
将和代入式（23），可以获得耦合系统运动方程的频域表达式：
2　模型验证
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基于Comsol多场耦合有限元分析软件，建立声学黑洞与约束阻尼耦合系统的有限元模型，验证半解析建模方法的正确性。表1和表2分别是其几何和材料参数。
图3为声学黑洞约束阻尼板的模态分析结果。从图3（a）可以看出，半解析分析模型计算结果与有限元模型计算结果吻合。图3（b）为半解析与有限元模态频率计算结果的相对误差，可知，其模态频率有很好的收敛性，除了极少数模态频率的误差为2%~3%，其余误差都在1.5%以内。
图4为第1，8，12和18阶模态振型对比。第一行为半解析模型结果，第二行为有限元模型结果。可以看出两种方法得到的模态振型完全一致。
3　约束阻尼参数对弯曲振动特性的影响
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图5为声学黑洞板和均匀板的平均平方速度，其表达式为：
式中　 S为板的面积，均匀板粘贴了与声学黑洞板相同的约束阻尼材料。可以看到，声学黑洞板在声学黑洞起始频率1190 Hz以上时，其减振效果更加明显。对300~1190 Hz范围内的振动也有所抑制，这主要得益于约束阻尼层的作用。
3.1　阻尼层损耗因子的影响
图6为阻尼层损耗因子对黑洞中心减振特性的影响。可以看出，随着损耗因子的增加，声学黑洞板的减振作用更加明显，这是因为约束阻尼层的阻尼效应是由约束层和基板位移差导致的黏弹性层剪切形变引起的，这意味着系统的能量耗散与振动衰减速度由阻尼层损耗因子主导。但减振作用与损耗因子之间并不呈现线性关系，没有必要过分增加阻尼损耗因子。
3.2　阻尼层厚度的影响
图7为阻尼层厚度对声学黑洞板减振特性的影响。可以看出，随着阻尼层厚度的增加，声学黑洞板的减振抑制作用会增强。这是因为阻尼层厚度增大，发生相同弯曲变形时，阻尼层上下表面的位移差会增大，从而导致应变能和能量耗散增大。但剪切应变将会减小，阻尼层厚度过大，其减振效果会变差。
3.3　约束层刚度和厚度的影响
约束层是影响黏弹性层发生剪切变形的主要因素之一，因此约束层的刚度和厚度也会影响约束阻尼对系统的减振效果。图8为约束层杨氏模量对减振特性的影响。随着约束层刚度的增加，在1700 Hz以下的振动减小，1700 Hz以上的振动增大。一般而言，随着约束层刚度增大，阻尼层的剪切变形会增大，减振效果会增强，这显然与上述现象不符。为此，对比粘贴相同约束阻尼材料的声学黑洞板和均匀板的平均平方速度，如图9所示。随着约束层刚度增大，均匀板的振动在高频和低频均会减小，说明约束层刚度的增大的确有助于提升约束阻尼的减振效果。但声学黑洞板（见图9（a））的减振规律与图8相同，这可能是因为约束阻尼材料粘贴在声学黑洞中心，增加约束层刚度改变了弯曲波的波长与波速，削弱了声学黑洞的能量集中效应。声学黑洞现象在低频的影响相对较小，随着频率升高其能量集中效应将越来越明显，因此形成了随约束层刚度增大，低频减振效果增强，高频减振效果减弱的现象。
图10为约束层厚度对声学黑洞板和均匀板减振特性的影响。与图9对比可以看出，改变约束层厚度对声学黑洞板减振效果的影响规律与改变刚度的规律一致，这是因为增大约束层刚度和厚度都会削弱声学黑洞的能量集中效应。而在3.2节中，随着阻尼层厚度的增加，不会出现低频与高频在减振效果上的差异，这是因为阻尼层杨氏模量（60 MPa）远远小于ABH板的杨氏模量（206 GPa），所以对声学黑洞效应的影响很小。
3.4　阻尼层与约束层比例的影响
在工程应用中，阻尼材料的厚度和质量是受到限制的，因此探究阻尼层与约束层的比例（rvc=hv/hc）有着十分重要的意义。图11（a）为约束阻尼层总厚度为3 mm时不同rvc的对比结果，随着rvc的减小，声学黑洞板在低频的减振效果增强，而在高频的减振效果变差，这与单独改变约束层厚度的规律一致。这是因为约束层的密度和刚度都远大于阻尼层的密度和刚度，故约束层厚度改变产生的影响要大于阻尼层厚度。
此外，由于约束层的密度比较大，减小rvc会导致约束阻尼层总质量的增加。图11（b）为约束阻尼层总质量为431.6 g不同rvc的对比结果。可以看出，随着rvc的减小，ABH板的减振效果会变差，这是因为阻尼层厚度改变产生的影响要大于约束层厚度，其规律也与单独改变阻尼层厚度一致。
4　实验验证
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声学黑洞板的实验平台如图12所示。平台由约束阻尼声学黑洞板实验样件、信号发生器（SINOC-ERA PIEZOTRONIC JZK-5）、激振器、信号功率放大器（SINOCERA PIEZOTRONICS，YE5871-A）、加速度传感器与 LMS 数据采集器组成。实验中，利用弹性绳将实验样件悬挂起来，以模拟自由边界条件。通过激振器对样件上A点施加激励，在声学黑洞板B点和C点处获取响应信号，计算B点到C点的速度平方传递率（VST）。
实验结果如图13所示。图13（a）为声学黑洞板的速度平方传递率，可以看出，实验结果和理论计算结果的变化趋势具有良好的一致性。但实验结果比理论计算结果的速度平方传递率稍小一些，这是因为实验样件通过弹性绳悬挂于铝合金架子上，激振器产生的振动能量通过弹性绳传递到架子上会产生能量损失。图13（b）为声学黑洞约束阻尼板的速度平方传递率，实验结果比理论结果的速度平方传递率也要小一些，在200 Hz和500 Hz处峰值偏差较大，主要原因可能有两个：（1）声学黑洞板的声学黑洞加工存在误差；（2）实验中存在结构阻尼的作用，而理论计算并没有考虑结构阻尼的影响。总体上，理论结果和实验结果的主要趋势符合较好，表明所建立的半解析模型能较准确地预测声学黑洞约束阻尼板的振动响应。
另外，从图14可以看出，在粘贴约束阻尼材料后，声学黑洞板的传递率明显减小，说明约束阻尼能明显减弱声学黑洞板的振动。
5　结论
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本文在Rayleigh-Ritz法框架下，选择高斯函数作为基函数，考虑高斯基函数的分布和质量矩阵、刚度矩阵的奇异化问题，建立了声学黑洞约束阻尼板的半解析模型。分析了约束阻尼结构参数对声学黑洞板弯曲振动特性的影响规律。本文的主要结论如下：
（1）基于高斯展开法的半解析理论分析模型能准确预测声学黑洞约束阻尼板的振动特性，避免了质量矩阵和刚度矩阵的奇异化，为研究约束阻尼参数对声学黑洞板弯曲振动特性的影响规律奠定了基础。
（2）约束层阻尼结构参数的研究表明，阻尼层的阻尼损耗因子对声学黑洞板的减振效果影响显著，损耗因子越大，减振效果越好；此外，增加阻尼层的厚度，由于强化了剪切变形和能量耗散，会进一步提升声学黑洞板的减振作用。
（3）由于约束层刚度和厚度的增大会削弱声学黑洞的能量聚集效应，从而导致低频减振作用增强和高频减振作用减弱。阻尼层与约束层的厚度比决定了其各自的贡献和占比，也反映出阻尼层和约束层类似的规律。
在后续的研究中，将进一步探索约束阻尼结构参数之间的最佳匹配关系，并研究以目标频率为导向的约束阻尼拓扑优化方法，实现约束阻尼结构的轻量化设计。
基金
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  • 国家自然科学基金资助项目(51875061)
文献
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2024年第37卷第5期
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doi: 10.16385/j.cnki.issn.1004-4523.2024.05.013
  • 接收时间：2022-09-20
  • 首发时间：2026-02-09
  • 出版时间：2024-05-28
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  • 收稿日期：2022-09-20
  • 修回日期：2022-11-27
基金
国家自然科学基金资助项目(51875061)
作者信息
    1重庆大学机械与运载工程学院，重庆　400044
    2西北工业大学航海学院，陕西西安　710072

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郑玲（1963―），女，博士，教授。E-mail： 。
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2种不同金属材料的力学参数
科
Family		 属数
Number of
genus		 种数
Number of
species		 占总种数比例
Percentage of
total species (%)		 属
Genus		 种数
Number of
species		 占总种数比例
Percentage of total
species (%)
鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
小菇科 Mycenaceae 2 12 5.74 丝盖伞属 Inocybe 5 2.39
多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
红菇科 Russulaceae 3 23 11.00 小皮伞属 Marasmius 6 2.87
小菇属 Mycena 11 5.26
光柄菇属 Pluteus 5 2.39
红菇属 Russula 17 8.13
栓菌属 Trametes 5 2.39
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