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Article(id=1243220383927743297, tenantId=1146029695717560320, journalId=1242844143416819734, issueId=1243220377556595432, articleNumber=null, orderNo=null, doi=10.15959/j.cnki.0254-0053.2025.03.010, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1741622400000, receivedDateStr=2025-03-11, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1774336437445, onlineDateStr=2026-03-24, pubDate=1758729600000, pubDateStr=2025-09-25, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1774336437445, onlineIssueDateStr=2026-03-24, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1774336437445, creator=13701087609, updateTime=1774336437445, updator=13701087609, issue=Issue{id=1243220377556595432, tenantId=1146029695717560320, journalId=1242844143416819734, year='2025', volume='46', issue='3', pageStart='541', pageEnd='810', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1774336435926, creator=13701087609, updateTime=1774336648182, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1243221267898282005, tenantId=1146029695717560320, journalId=1242844143416819734, issueId=1243220377556595432, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1243221267898282006, tenantId=1146029695717560320, journalId=1242844143416819734, issueId=1243220377556595432, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=684, endPage=694, ext={EN=ArticleExt(id=1243220384493974344, articleId=1243220383927743297, tenantId=1146029695717560320, journalId=1242844143416819734, language=EN, title=A New Triangular Element and Its Application in Vibration Analysis of Membranes, columnId=null, journalTitle=Chinese Quarterly of Mechanics, columnName=null, runingTitle=null, highlight=null, articleAbstract=

The free vibration analysis of membranes is of significant importance in engineering structures, especially in the design and optimization of membrane structures. This paper presents a new type of triangular element, aiming to improve the computational accuracy in free vibration analysis of membranes. Traditional 3-node triangular elements in membrane vibration analysis typically rely on polynomial shape functions, but this method often suffers from insufficient accuracy in complex vibration modes and high-order frequencies. To address this issue, this paper constructs a 10-node triangular element with the shape function incorporating trigonometric functions. The proposed 10-node triangular element consists of three corner nodes, two points of trisection for every edge, and the centroid node with its shape functions derived using the area coordinate method. The stiffness matrix and mass matrix are derived, and the frequencies and modes for free vibration of the membrane are computed, thereby the dynamic characteristics can be studied. To evaluate the effectiveness of this element, several typical examples are chosen, including the free vibration analysis of rectangular membrane and triangular membrane. By comparing with theoretical solutions and the 3-node element calculations in Ansys, the obtained results show that the 10-node triangular element can approximate the theoretical solutions with few computational elements. And the precision of the presented 10-node element is similar with that of the standard 10-node triangular element. The high precision of the proposed element is demonstrated in analysis of free vibration of membrane, which has the potential of further research and promotion.

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薄膜的自由振动分析在工程结构中具有重要意义，尤其在薄膜结构的设计与优化过程中．本文提出一种新型三角形单元，旨在提高薄膜自由振动分析中的计算精度．传统的三节点三角形单元在处理薄膜振动时，通常依赖于多项式形函数，但这种方法在复杂的振动模式和高阶频率计算中存在精度不够理想的问题．为了解决这一问题，本文构建了一种形函数包含三角函数的10节点三角形单元，所提三角形单元由三个角点、三条边的两个三等分点及重心点组成，其形函数通过面积坐标表示．推导所提单元的刚度矩阵和质量矩阵，计算薄膜自由振动的频率和模态，从而研究其动力学特性．为了评估该单元的效果，本文选取矩形薄膜和三角形薄膜自由振动作为典型算例．算例中给出本文结果、Ansys三节点单元的计算结果及理论解．结果之间的对比表明，10节点三角形单元在结构划分为较少计算单元的情况下，能够逼近理论解，且本文10节点单元与标准10节点单元的精度相当．从而说明所提有限单元在分析膜自由振动问题时具有高精度，具有进一步研究和推广的潜力．

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鲍四元．E-mail：
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鲍四元，教授．研究方向：结构新型有限元及应用研究．

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鲍四元，教授．研究方向：结构新型有限元及应用研究．

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鲍四元，教授．研究方向：结构新型有限元及应用研究．

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authorNames=骆江南, journalName=null, refType=null, unstructuredReference=骆江南. 基于无网格法的薄膜振动特性研究[D]. 西安: 西安理工大学, 2014., articleTitle=基于无网格法的薄膜振动特性研究, refAbstract=null), Reference(id=1243220409517191790, tenantId=1146029695717560320, journalId=1242844143416819734, articleId=1243220383927743297, doi=null, pmid=null, pmcid=null, year=2014, volume=null, issue=null, pageStart=null, pageEnd=null, url=null, language=null, rfNumber=[1], rfOrder=1, authorNames=Luo J N, journalName=null, refType=null, unstructuredReference=Luo J N. Study on membrane vibration characteristics based on meshless method[D]. Xi'an: Xi'an University of Technology, 2014 (in Chinese)., articleTitle=Study on membrane vibration characteristics based on meshless method, refAbstract=null), Reference(id=1243220409756267122, tenantId=1146029695717560320, journalId=1242844143416819734, articleId=1243220383927743297, doi=null, pmid=null, pmcid=null, year=2010, volume=29, issue=6, pageStart=768, pageEnd=771, url=null, language=null, rfNumber=[2], rfOrder=2, authorNames=赵凤群, 王忠民, journalName=机械科学与技术, refType=null, unstructuredReference=赵凤群, 王忠民. 运动矩形薄膜的非线性振动分析[J]. 机械科学与技术, 2010, 29(6): 768-771., articleTitle=运动矩形薄膜的非线性振动分析, refAbstract=null), Reference(id=1243220409890484858, tenantId=1146029695717560320, journalId=1242844143416819734, articleId=1243220383927743297, doi=null, pmid=null, pmcid=null, year=2010, volume=29, issue=6, pageStart=768, pageEnd=771, url=null, language=null, rfNumber=[2], rfOrder=3, authorNames=Zhao F Q, Wang Z M, journalName=Mechanical Science and Technology for Aerospace Engineering, refType=null, unstructuredReference=Zhao F Q, Wang Z M. Nonlinear vibration analysis of a moving rectangular membrane[J]. Mechanical Science and Technology for Aerospace Engineering, 2010, 29(6): 768-771 (in Chinese)., articleTitle=Nonlinear vibration analysis of a moving rectangular membrane, refAbstract=null), Reference(id=1243220410079228543, tenantId=1146029695717560320, journalId=1242844143416819734, articleId=1243220383927743297, doi=null, pmid=null, pmcid=null, year=2022, volume=41, issue=3, pageStart=56, pageEnd=60, url=null, language=null, rfNumber=[3], rfOrder=4, authorNames=周纪晨, 郭琴, journalName=大学物理, refType=null, unstructuredReference=周纪晨, 郭琴. 矩形薄膜受迫横振动问题的求解及可视化[J]. 大学物理, 2022, 41(3): 56-60., articleTitle=矩形薄膜受迫横振动问题的求解及可视化, refAbstract=null), Reference(id=1243220410288943751, tenantId=1146029695717560320, journalId=1242844143416819734, articleId=1243220383927743297, doi=null, pmid=null, pmcid=null, year=2022, volume=41, issue=3, pageStart=56, pageEnd=60, url=null, language=null, rfNumber=[3], rfOrder=5, authorNames=Zhou J C, Guo Q, journalName=College Physics, refType=null, unstructuredReference=Zhou J C, Guo Q. Solution and visualization of forced transverse vibration of rectangular thin films[J]. College Physics, 2022, 41(3): 56-60 (in Chinese)., articleTitle=Solution and visualization of forced transverse vibration of rectangular thin films, refAbstract=null), Reference(id=1243220410397995658, tenantId=1146029695717560320, journalId=1242844143416819734, articleId=1243220383927743297, doi=null, pmid=null, pmcid=null, year=2019, volume=null, issue=null, pageStart=null, pageEnd=null, url=null, language=null, rfNumber=[4], rfOrder=6, authorNames=郑志刚, journalName=null, refType=null, unstructuredReference=郑志刚. 空间薄膜结构自由振动非线性效应研究[D]. 哈尔滨: 哈尔滨工业大学, 2019., articleTitle=空间薄膜结构自由振动非线性效应研究, refAbstract=null), Reference(id=1243220410574156434, tenantId=1146029695717560320, journalId=1242844143416819734, articleId=1243220383927743297, doi=null, pmid=null, pmcid=null, year=2019, volume=null, issue=null, pageStart=null, pageEnd=null, url=null, language=null, rfNumber=[4], rfOrder=7, authorNames=Zheng Z G, journalName=null, refType=null, unstructuredReference=Zheng Z G. Research on nonlinear effect of free vibration of membrane structures in space[D]. Harbin: Harbin Institute of Technology, 2019 (in Chinese)., articleTitle=Research on nonlinear effect of free vibration of membrane structures in space, refAbstract=null), Reference(id=1243220410708374169, tenantId=1146029695717560320, journalId=1242844143416819734, articleId=1243220383927743297, doi=null, pmid=null, pmcid=null, year=2004, volume=272, issue=1-2, pageStart=450, pageEnd=460, url=null, language=null, rfNumber=[5], rfOrder=8, authorNames=Kang S W, Lee J M, journalName=Journal of Sound and Vibration, refType=null, unstructuredReference=Kang S W, Lee J M. Free vibration analysis of an unsymmetric trapezoidal membrane[J]. Journal of Sound and Vibration, 2004, 272(1-2): 450-460., articleTitle=Free vibration analysis of an unsymmetric trapezoidal membrane, refAbstract=null), Reference(id=1243220410821620381, tenantId=1146029695717560320, journalId=1242844143416819734, articleId=1243220383927743297, doi=null, pmid=null, pmcid=null, year=2004, volume=272, issue=1-2, pageStart=39, pageEnd=53, url=null, language=null, rfNumber=[6], rfOrder=9, authorNames=Kang S W, journalName=Journal of Sound and Vibration, refType=null, unstructuredReference=Kang S W. Free vibration analysis of composite rectangular membranes with a bent interface[J]. Journal of Sound and Vibration, 2004, 272(1-2): 39-53., articleTitle=Free vibration analysis of composite rectangular membranes with a bent interface, refAbstract=null), Reference(id=1243220410922283683, tenantId=1146029695717560320, journalId=1242844143416819734, articleId=1243220383927743297, doi=null, pmid=null, pmcid=null, year=2017, volume=34, issue=3, pageStart=1, pageEnd=14, url=null, language=null, rfNumber=[7], rfOrder=10, authorNames=岑松, 尚闫, 周培蕾, journalName=工程力学, refType=null, unstructuredReference=岑松, 尚闫, 周培蕾, 等. 形状自由的高性能有限元方法研究的一些进展[J]. 工程力学, 2017, 34(3): 1-14., articleTitle=形状自由的高性能有限元方法研究的一些进展, refAbstract=null), Reference(id=1243220411014558374, tenantId=1146029695717560320, journalId=1242844143416819734, articleId=1243220383927743297, doi=null, pmid=null, pmcid=null, year=2017, volume=34, issue=3, pageStart=1, pageEnd=14, url=null, language=null, rfNumber=[7], rfOrder=11, authorNames=Cen S, Shang Y, Zhou P L, journalName=Engineering Mechanics, refType=null, unstructuredReference=Cen S, Shang Y, Zhou P L. Advances in shape-free finite element methods: A review[J]. Engineering Mechanics, 2017, 34(3): 1-14 (in Chinese)., articleTitle=Advances in shape-free finite element methods: A review, refAbstract=null), Reference(id=1243220411131998890, tenantId=1146029695717560320, journalId=1242844143416819734, articleId=1243220383927743297, doi=null, pmid=null, pmcid=null, year=2023, volume=40, issue=2, pageStart=8, pageEnd=16, url=null, language=null, rfNumber=[8], rfOrder=12, authorNames=傅向荣, 陈璞, 孙树立, journalName=工程力学, refType=null, unstructuredReference=傅向荣, 陈璞, 孙树立, 等. 弹性力学有限元分析中的平衡与协调理论[J]. 工程力学, 2023, 40(2): 8-16., articleTitle=弹性力学有限元分析中的平衡与协调理论, refAbstract=null), Reference(id=1243220411211690670, tenantId=1146029695717560320, journalId=1242844143416819734, articleId=1243220383927743297, doi=null, pmid=null, pmcid=null, year=2023, volume=40, issue=2, pageStart=8, pageEnd=16, url=null, language=null, rfNumber=[8], rfOrder=13, authorNames=Fu X R, Chen P, Sun S L, journalName=Engineering Mechanics, refType=null, unstructuredReference=Fu X R, Chen P, Sun S L, et al. Research on equilibrium and conforming theory of the finite element method in elasticity[J]. Engineering Mechanics, 2023, 40(2): 8-16 (in Chinese)., articleTitle=Research on equilibrium and conforming theory of the finite element method in elasticity, refAbstract=null), Reference(id=1243220411274605234, tenantId=1146029695717560320, journalId=1242844143416819734, articleId=1243220383927743297, doi=null, pmid=null, pmcid=null, year=2020, volume=37, issue=1, pageStart=17, pageEnd=25, url=null, language=null, rfNumber=[9], rfOrder=14, authorNames=袁驷, 孙浩涵, journalName=工程力学, refType=null, unstructuredReference=袁驷, 孙浩涵. 二维自由振动问题的自适应有限元分析初探[J]. 工程力学, 2020, 37(1): 17-25., articleTitle=二维自由振动问题的自适应有限元分析初探, refAbstract=null), Reference(id=1243220411576595126, tenantId=1146029695717560320, journalId=1242844143416819734, articleId=1243220383927743297, doi=null, pmid=null, pmcid=null, year=2020, volume=37, issue=1, pageStart=17, pageEnd=25, url=null, language=null, rfNumber=[9], rfOrder=15, authorNames=Yuan S, Sun H H, journalName=Engineering Mechanics, refType=null, unstructuredReference=Yuan S, Sun H H. A preliminary study on adaptive finite element analysis of two-dimensional free vibration problems[J]. Engineering Mechanics, 2020, 37(1): 17-25 (in Chinese)., articleTitle=A preliminary study on adaptive finite element analysis of two-dimensional free vibration problems, refAbstract=null), Reference(id=1243220411715007162, tenantId=1146029695717560320, journalId=1242844143416819734, articleId=1243220383927743297, doi=null, pmid=null, pmcid=null, year=2024, volume=58, issue=4, pageStart=428, pageEnd=437, url=null, language=null, rfNumber=[10], rfOrder=16, authorNames=黄涛, 何泽青, 宋林, journalName=上海交通大学学报, refType=null, unstructuredReference=黄涛, 何泽青, 宋林. 圆柱薄膜振动特性[J]. 上海交通大学学报, 2024, 58(4): 428-437., articleTitle=圆柱薄膜振动特性, refAbstract=null), Reference(id=1243220411836641984, tenantId=1146029695717560320, journalId=1242844143416819734, articleId=1243220383927743297, doi=null, pmid=null, pmcid=null, year=2024, volume=58, issue=4, pageStart=428, pageEnd=437, url=null, language=null, rfNumber=[10], rfOrder=17, authorNames=Huang T, He Z Q, Song L, journalName=Journal of Shanghai Jiaotong University, refType=null, unstructuredReference=Huang T, He Z Q, Song L. Vibration characteristics of cylindrical membrane[J]. Journal of Shanghai Jiaotong University, 2024, 58(4): 428-437 (in Chinese)., articleTitle=Vibration characteristics of cylindrical membrane, refAbstract=null), Reference(id=1243220411954082500, tenantId=1146029695717560320, journalId=1242844143416819734, articleId=1243220383927743297, doi=null, pmid=null, pmcid=null, year=2009, volume=28, issue=5, pageStart=84, pageEnd=86, url=null, language=null, rfNumber=[11], rfOrder=18, authorNames=林文静, 陈树辉, 李森, journalName=振动与冲击, refType=null, unstructuredReference=林文静, 陈树辉, 李森. 圆形薄膜自由振动的理论解[J]. 振动与冲击, 2009, 28(5): 84-86., articleTitle=圆形薄膜自由振动的理论解, refAbstract=null), Reference(id=1243220412063134409, tenantId=1146029695717560320, journalId=1242844143416819734, articleId=1243220383927743297, doi=null, pmid=null, pmcid=null, year=2009, volume=28, issue=5, pageStart=84, pageEnd=86, url=null, language=null, rfNumber=[11], rfOrder=19, authorNames=Lin W J, Chen S H, Li S, journalName=Journal of Vibration and Shock, refType=null, unstructuredReference=Lin W J, Chen S H, Li S. Analytical solution of the free vibration of circular membrane[J]. Journal of Vibration and Shock, 2009, 28(5): 84-86 (in Chinese)., articleTitle=Analytical solution of the free vibration of circular membrane, refAbstract=null), Reference(id=1243220412239295180, tenantId=1146029695717560320, journalId=1242844143416819734, articleId=1243220383927743297, doi=null, pmid=null, pmcid=null, year=2005, volume=282, issue=1-2, pageStart=401, pageEnd=410, url=null, language=null, rfNumber=[12], rfOrder=20, authorNames=Houmat A, journalName=Journal of Sound and Vibration, refType=null, unstructuredReference=Houmat A. Free vibration analysis of membranes using the h-p version of the finite element method[J]. Journal of Sound and Vibration, 2005, 282(1-2): 401-410., articleTitle=Free vibration analysis of membranes using the h-p version of the finite element method, refAbstract=null), Reference(id=1243220412323181267, tenantId=1146029695717560320, journalId=1242844143416819734, articleId=1243220383927743297, doi=null, pmid=null, pmcid=null, year=2006, volume=44, issue=9, pageStart=943, pageEnd=951, url=null, language=null, rfNumber=[13], rfOrder=21, authorNames=Houmat A, journalName=Thin-Walled Structures, refType=null, unstructuredReference=Houmat A. Free vibration analysis of arbitrarily shaped membranes using the trigonometric p-version of the finite element method[J]. Thin-Walled Structures, 2006, 44(9): 943-951., articleTitle=Free vibration analysis of arbitrarily shaped membranes using the trigonometric p-version of the finite element method, refAbstract=null), Reference(id=1243220412461593300, tenantId=1146029695717560320, journalId=1242844143416819734, articleId=1243220383927743297, doi=null, pmid=null, pmcid=null, year=2005, volume=280, issue=1-2, pageStart=407, pageEnd=414, url=null, language=null, rfNumber=[14], rfOrder=22, authorNames=Buchanan G R, journalName=Journal of Sound and Vibration, refType=null, unstructuredReference=Buchanan G R. Vibration of circular membranes with linearly varying density along a diameter[J]. Journal of Sound and Vibration, 2005, 280(1-2): 407-414., articleTitle=Vibration of circular membranes with linearly varying density along a diameter, refAbstract=null), Reference(id=1243220412570645211, tenantId=1146029695717560320, journalId=1242844143416819734, articleId=1243220383927743297, doi=null, pmid=null, pmcid=null, year=2005, volume=43, issue=9, pageStart=1444, pageEnd=1454, url=null, language=null, rfNumber=[15], rfOrder=23, authorNames=Buchanan G R, Peddieson J J, journalName=Thin-Walled Structures, refType=null, unstructuredReference=Buchanan G R, Peddieson J J. A finite element in elliptic coordinates with application to membrane vibration[J]. Thin-Walled Structures, 2005, 43(9): 1444-1454., articleTitle=A finite element in elliptic coordinates with application to membrane vibration, refAbstract=null), Reference(id=1243220412688085724, tenantId=1146029695717560320, journalId=1242844143416819734, articleId=1243220383927743297, doi=null, pmid=null, pmcid=null, year=2014, volume=79, issue=null, pageStart=216, pageEnd=251, url=null, language=null, rfNumber=[16], rfOrder=24, authorNames=Fantuzzi N, Tornabene F, Viola E, journalName=International Journal of Mechanical Sciences, refType=null, unstructuredReference=Fantuzzi N, Tornabene F, Viola E. Generalized differential quadrature finite element method for vibration analysis of arbitrarily shaped membranes[J]. International Journal of Mechanical Sciences, 2014, 79: 216-251., articleTitle=Generalized differential quadrature finite element method for vibration analysis of arbitrarily shaped membranes, refAbstract=null), Reference(id=1243220412847469282, tenantId=1146029695717560320, journalId=1242844143416819734, articleId=1243220383927743297, doi=null, pmid=null, pmcid=null, year=2016, volume=32, issue=2, pageStart=163, pageEnd=173, url=null, language=null, rfNumber=[17], rfOrder=25, authorNames=Liu D S, Lin I H, journalName=International Review of the Red Cross, refType=null, unstructuredReference=Liu D S, Lin I H. Vibration analysis of the multiple-hole membrane by using the coupled Diem-Fe scheme[J]. International Review of the Red Cross, 2016, 32(2): 163-173., articleTitle=Vibration analysis of the multiple-hole membrane by using the coupled Diem-Fe scheme, refAbstract=null), Reference(id=1243220412935549669, tenantId=1146029695717560320, journalId=1242844143416819734, articleId=1243220383927743297, doi=null, pmid=null, pmcid=null, year=2019, volume=35, issue=4, pageStart=513, pageEnd=525, url=null, language=null, rfNumber=[18], rfOrder=26, authorNames=Liu D S, Chen Y W, journalName=Journal of Mechanics, refType=null, unstructuredReference=Liu D S, Chen Y W. Application of Craig-Bampton reduction technique and 2D dynamic infinite element modeling approach to membrane vibration problems[J]. Journal of Mechanics, 2019, 35(4): 513-525., articleTitle=Application of Craig-Bampton reduction technique and 2D dynamic infinite element modeling approach to membrane vibration problems, refAbstract=null), Reference(id=1243220413027824363, tenantId=1146029695717560320, journalId=1242844143416819734, articleId=1243220383927743297, doi=null, pmid=null, pmcid=null, year=2018, volume=18, issue=10, pageStart=null, pageEnd=null, url=null, language=null, rfNumber=[19], rfOrder=27, authorNames=Wu J C, Wang D D, Lin Z, journalName=International Journal of Structural Stability & Dynamics, refType=null, unstructuredReference=Wu J C, Wang D D, Lin Z. A mesh free higher order mass matrix formulation for structural vibration analysis[J]. International Journal of Structural Stability & Dynamics, 2018, 18(10): 1850121., articleTitle=A mesh free higher order mass matrix formulation for structural vibration analysis, refAbstract=null), Reference(id=1243220413149459184, tenantId=1146029695717560320, journalId=1242844143416819734, articleId=1243220383927743297, doi=null, pmid=null, pmcid=null, year=2023, volume=182, issue=null, pageStart=1, pageEnd=15, url=null, language=null, rfNumber=[20], rfOrder=28, authorNames=Ren H, Fan W, journalName=Thin-Walled structures, refType=null, unstructuredReference=Ren H, Fan W. An adaptive triangular element of absolute nodal coordinate formulation for thin plates and membranes[J]. Thin-Walled structures, 2023, 182: 1-15., articleTitle=An adaptive triangular element of absolute nodal coordinate formulation for thin plates and membranes, refAbstract=null), Reference(id=1243220413271094006, tenantId=1146029695717560320, journalId=1242844143416819734, articleId=1243220383927743297, doi=null, pmid=null, pmcid=null, year=2010, volume=8, issue=3, pageStart=202, pageEnd=206, url=null, language=null, rfNumber=[21], rfOrder=29, authorNames=林文静, 陈树辉, journalName=动力学与控制学报, refType=null, unstructuredReference=林文静, 陈树辉. 平面薄膜自由振动的有限元分析[J]. 动力学与控制学报, 2010, 8(3): 202-206., articleTitle=平面薄膜自由振动的有限元分析, refAbstract=null), Reference(id=1243220413363368696, tenantId=1146029695717560320, journalId=1242844143416819734, articleId=1243220383927743297, doi=null, pmid=null, pmcid=null, year=2010, volume=8, issue=3, pageStart=202, pageEnd=206, url=null, language=null, rfNumber=[21], rfOrder=30, authorNames=Lin W J, Chen S H, journalName=Journal of Dynamics and Control, refType=null, unstructuredReference=Lin W J, Chen S H. Free vibration analysis of plane membranes by finite element method[J]. Journal of Dynamics and Control, 2010, 8(3): 202-206 (in Chinese)., articleTitle=Free vibration analysis of plane membranes by finite element method, refAbstract=null)], funds=[Fund(id=1243220409173258850, tenantId=1146029695717560320, journalId=1242844143416819734, articleId=1243220383927743297, awardId=51709194, language=CN, fundingSource=国家自然科学基金(51709194), fundOrder=null, country=null)], companyList=[AuthorCompany(id=1243220403452227922, tenantId=1146029695717560320, journalId=1242844143416819734, articleId=1243220383927743297, xref=null, ext=[AuthorCompanyExt(id=1243220403460616531, tenantId=1146029695717560320, journalId=1242844143416819734, articleId=1243220383927743297, companyId=1243220403452227922, language=EN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=School of Civil Engineering, Suzhou University of Science and Technology, Suzhou 215011, Jiangsu, China), AuthorCompanyExt(id=1243220403477393748, tenantId=1146029695717560320, journalId=1242844143416819734, articleId=1243220383927743297, companyId=1243220403452227922, language=CN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=苏州科技大学　土木工程学院，江苏　苏州　215011)])], figs=[ArticleFig(id=1243220405868147171, tenantId=1146029695717560320, journalId=1242844143416819734, articleId=1243220383927743297, language=EN, label=Fig.1, caption=Common planar triangular elements, figureFileSmall=amFYJV1g9X9PPIpTMyH4gQ==, figureFileBig=slw1591/q8s9+ZkfapsOmw==, tableContent=null), ArticleFig(id=1243220406144971240, tenantId=1146029695717560320, journalId=1242844143416819734, articleId=1243220383927743297, language=CN, label=图1, caption=常见平面三角形单元, figureFileSmall=amFYJV1g9X9PPIpTMyH4gQ==, figureFileBig=slw1591/q8s9+ZkfapsOmw==, tableContent=null), ArticleFig(id=1243220406405018105, tenantId=1146029695717560320, journalId=1242844143416819734, articleId=1243220383927743297, language=EN, label=Fig.2, 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journalId=1242844143416819734, articleId=1243220383927743297, language=CN, label=图5, caption=周边固定的矩形薄膜自由振动的前4阶模态对比, figureFileSmall=mSZZkTkfHcIxKyTiTuHtJA==, figureFileBig=Ph8mL4IXEmgWWkBZGLN3Hw==, tableContent=null), ArticleFig(id=1243220407529091610, tenantId=1146029695717560320, journalId=1242844143416819734, articleId=1243220383927743297, language=EN, label=Fig.6, caption=Isosceles right triangle membrane fixed at the boundary, figureFileSmall=roDW9rnoJs1bZbYTprSUUg==, figureFileBig=KVPOlvvYvXCZX6VDj//ZhA==, tableContent=null), ArticleFig(id=1243220407663309341, tenantId=1146029695717560320, journalId=1242844143416819734, articleId=1243220383927743297, language=CN, label=图6, caption=周边固定的等腰直角三角形薄膜, figureFileSmall=roDW9rnoJs1bZbYTprSUUg==, figureFileBig=KVPOlvvYvXCZX6VDj//ZhA==, tableContent=null), ArticleFig(id=1243220407772361251, tenantId=1146029695717560320, journalId=1242844143416819734, articleId=1243220383927743297, language=EN, label=Fig.7, caption=Comparison of the modes of the free vibration of an isosceles right-angled triangular membrane fixed around the periphery, figureFileSmall=jozFgo4ri41T1zgNreThIw==, figureFileBig=1xMXnk03qim2BsSWK17h1g==, tableContent=null), ArticleFig(id=1243220407881413160, tenantId=1146029695717560320, journalId=1242844143416819734, articleId=1243220383927743297, language=CN, label=图7, caption=周边固定的等腰直角三角形薄膜自由振动的模态对比, figureFileSmall=jozFgo4ri41T1zgNreThIw==, figureFileBig=1xMXnk03qim2BsSWK17h1g==, tableContent=null), ArticleFig(id=1243220408049185326, tenantId=1146029695717560320, journalId=1242844143416819734, articleId=1243220383927743297, language=EN, label=Tab.1, caption=

First eight natural frequencies of the periphery-fixed rectangular membrane vibrating freely using different elements

, figureFileSmall=null, figureFileBig=null, tableContent=
(m n)Theoretical solution/HzAnsys simulation results/HzThe present numerical solution/Hz
120 elements DOF=102270 elements DOF=2262502 elements DOF=2 35026 elements DOF=9752 elements DOF=202542 elements DOF=2 338
(1 1)23.50622.260 (5.30%)22.546 (4.08%)23.140 (1.56%)23.401 (0.45%)23.389 (0.50%)23.506 (0%)
(2 1)29.73328.554 (3.97%)28.786 (3.19%)29.375 (1.20%)29.653 (0.27%)29.557 (0.59%)29.732 (0.003%)
(3 1)37.90236.898 (2.65%)37.047 (2.26%)37.491 (1.08%)37.842 (0.16%)37.660 (0.64%)37.900 (0.005%)
(1 2)43.34344.597 (2.89%)43.111 (0.54%)43.145 (0.46%)43.407 (0.15%)43.280 (0.15%)43.340 (0.007%)
(2 2)47.01248.901 (4.02%)46.679 (0.71%)46.602 (0.87%)46.889 (0.26%)46.910 (0.22%)47.008 (0.008%)
(4 1)47.01253.054 (12.85%)47.548 (1.14%)46.941 (0.15%)47.554 (1.15%)47.069 (0.12%)47.008 (0.008%)
(3 2)52.56163.890 (21.55%)54.285 (3.28%)52.488 (0.14%)54.050 (2.83%)52.731 (0.32%)52.556 (0.009%)
(5 1)56.61064.767 (14.41%)57.536 (1.64%)56.324 (0.13%)57.439 (1.46%)56.558 (0.09%)56.605 (0.009%)
), ArticleFig(id=1243220408158237239, tenantId=1146029695717560320, journalId=1242844143416819734, articleId=1243220383927743297, language=CN, label=表1, caption=

不同单元数目情况下周边固定的矩形薄膜自由振动的前8阶固有频率

, figureFileSmall=null, figureFileBig=null, tableContent=
(m n)Theoretical solution/HzAnsys simulation results/HzThe present numerical solution/Hz
120 elements DOF=102270 elements DOF=2262502 elements DOF=2 35026 elements DOF=9752 elements DOF=202542 elements DOF=2 338
(1 1)23.50622.260 (5.30%)22.546 (4.08%)23.140 (1.56%)23.401 (0.45%)23.389 (0.50%)23.506 (0%)
(2 1)29.73328.554 (3.97%)28.786 (3.19%)29.375 (1.20%)29.653 (0.27%)29.557 (0.59%)29.732 (0.003%)
(3 1)37.90236.898 (2.65%)37.047 (2.26%)37.491 (1.08%)37.842 (0.16%)37.660 (0.64%)37.900 (0.005%)
(1 2)43.34344.597 (2.89%)43.111 (0.54%)43.145 (0.46%)43.407 (0.15%)43.280 (0.15%)43.340 (0.007%)
(2 2)47.01248.901 (4.02%)46.679 (0.71%)46.602 (0.87%)46.889 (0.26%)46.910 (0.22%)47.008 (0.008%)
(4 1)47.01253.054 (12.85%)47.548 (1.14%)46.941 (0.15%)47.554 (1.15%)47.069 (0.12%)47.008 (0.008%)
(3 2)52.56163.890 (21.55%)54.285 (3.28%)52.488 (0.14%)54.050 (2.83%)52.731 (0.32%)52.556 (0.009%)
(5 1)56.61064.767 (14.41%)57.536 (1.64%)56.324 (0.13%)57.439 (1.46%)56.558 (0.09%)56.605 (0.009%)
), ArticleFig(id=1243220408384729658, tenantId=1146029695717560320, journalId=1242844143416819734, articleId=1243220383927743297, language=EN, label=Tab.2, caption=

Natural frequencies of a fully clamped rectangular membrane using two types of 10-node triangular elements

, figureFileSmall=null, figureFileBig=null, tableContent=
(m n)Theoretical solutionStandard T10The presentStandard T10The presentStandard T10The present
26 elements (DOF=97)52 elements (DOF=202)542 elements (DOF=2 338)
(1 1)23.50623.508 (0.009%)23.517 (0.04%)23.506 (0%)23.510 (0.02%)23.506 (0%)23.505 (0.004%)
(2 1)29.73329.746 (0.04%)29.757 (0.08%)29.735 (0.007%)29.740 (0.02%)29.732 (0.003%)29.731 (0.007%)
(3 1)37.90237.917 (0.04%)37.959 (0.15%)37.906 (0.01%)37.926 (0.06%)37.902 (0%)37.900 (0.005%)
(1 2)43.34343.573 (0.53%)43.566 (0.51%)43.457 (0.26%)43.375 (0.07%)43.342 (0.002%)43.341 (0.005%)
(2 2)47.01247.093 (0.17%)47.117 (0.22%)47.022 (0.02%)47.037 (0.05%)47.011 (0.002%)47.010 (0.005%)
(4 1)47.01247.182 (0.36%)47.235 (0.47%)47.029 (0.04%)47.057 (0.10%)47.011 (0.002%)47.010 (0.004%)
(3 2)52.56152.734 (0.33%)52.842 (0.53%)52.590 (0.06%)52.637 (0.14%)52.558 (0.006%)52.560 (0.002%)
(5 1)56.61056.632 (0.04%)56.788 (0.31%)56.743 (0.23%)56.675 (0.11%)56.606 (0.007%)56.609 (0.002%)
), ArticleFig(id=1243220408569279043, tenantId=1146029695717560320, journalId=1242844143416819734, articleId=1243220383927743297, language=CN, label=表2, caption=

两种10节点三角形单元下周边固定矩形薄膜的固有频率

, figureFileSmall=null, figureFileBig=null, tableContent=
(m n)Theoretical solutionStandard T10The presentStandard T10The presentStandard T10The present
26 elements (DOF=97)52 elements (DOF=202)542 elements (DOF=2 338)
(1 1)23.50623.508 (0.009%)23.517 (0.04%)23.506 (0%)23.510 (0.02%)23.506 (0%)23.505 (0.004%)
(2 1)29.73329.746 (0.04%)29.757 (0.08%)29.735 (0.007%)29.740 (0.02%)29.732 (0.003%)29.731 (0.007%)
(3 1)37.90237.917 (0.04%)37.959 (0.15%)37.906 (0.01%)37.926 (0.06%)37.902 (0%)37.900 (0.005%)
(1 2)43.34343.573 (0.53%)43.566 (0.51%)43.457 (0.26%)43.375 (0.07%)43.342 (0.002%)43.341 (0.005%)
(2 2)47.01247.093 (0.17%)47.117 (0.22%)47.022 (0.02%)47.037 (0.05%)47.011 (0.002%)47.010 (0.005%)
(4 1)47.01247.182 (0.36%)47.235 (0.47%)47.029 (0.04%)47.057 (0.10%)47.011 (0.002%)47.010 (0.004%)
(3 2)52.56152.734 (0.33%)52.842 (0.53%)52.590 (0.06%)52.637 (0.14%)52.558 (0.006%)52.560 (0.002%)
(5 1)56.61056.632 (0.04%)56.788 (0.31%)56.743 (0.23%)56.675 (0.11%)56.606 (0.007%)56.609 (0.002%)
), ArticleFig(id=1243220408678330953, tenantId=1146029695717560320, journalId=1242844143416819734, articleId=1243220383927743297, language=EN, label=Tab.3, caption=

First eight natural frequencies of periphery-fixed isosceles right-angled triangle membrane vibrating freely using different elements (or freedoms)

, figureFileSmall=null, figureFileBig=null, tableContent=
(m n)Theoretical solution/HzAnsys simulation results/HzThe present numerical solution/Hz
184 elements DOF=140254 elements DOF=2021538 elements DOF=1 41237 elements DOF=13351 elements DOF=196327 elements DOF=1 384
(1 1)60.69258.312 (3.92%)58.663 (3.34%)59.692 (1.65%)60.719 (0.04%)60.694 (0.003%)60.689 (0.004%)
(2 1)85.83284.029 (2.10%)84.081 (2.04%)84.666 (1.36%)85.901 (0.08%)85.854 (0.03%)85.829 (0.003%)
(1 2)97.86396.453 (1.44%)95.947 (1.96%)96.705 (1.18%)97.996 (0.14%)97.891 (0.03%)97.860 (0.003%)
(3 1)111.911111.550 (0.32%)110.570 (1.20%)110.940 (0.87%)112.312 (0.36%)111.980 (0.06%)111.911 (0%)
(2 2)121.384123.600 (1.83%)122.650 (1.04%)120.490 (0.74%)121.867 (0.40%)121.500 (0.10%)121.382 (0.002%)
(1 3)135.712142.350 (4.89%)138.280 (1.89%)135.130 (0.43%)136.332 (0.46%)135.923 (0.16%)135.712 (0%)
(4 1)138.399142.840 (3.21%)140.850 (1.77%)138.230 (0.12%)139.415 (0.73%)138.628 (0.17%)138.400 (0.001%)
(3 2)146.166152.460 (4.31%)148.500 (1.60%)144.530 (1.12%)147.273 (0.76%)146.504 (0.23%)146.167 (0.001%)
), ArticleFig(id=1243220408783188560, tenantId=1146029695717560320, journalId=1242844143416819734, articleId=1243220383927743297, language=CN, label=表3, caption=

不同单元数目（自由度）情况下周边固定的等腰直角三角形薄膜自由振动的前8阶固有频率

, figureFileSmall=null, figureFileBig=null, tableContent=
(m n)Theoretical solution/HzAnsys simulation results/HzThe present numerical solution/Hz
184 elements DOF=140254 elements DOF=2021538 elements DOF=1 41237 elements DOF=13351 elements DOF=196327 elements DOF=1 384
(1 1)60.69258.312 (3.92%)58.663 (3.34%)59.692 (1.65%)60.719 (0.04%)60.694 (0.003%)60.689 (0.004%)
(2 1)85.83284.029 (2.10%)84.081 (2.04%)84.666 (1.36%)85.901 (0.08%)85.854 (0.03%)85.829 (0.003%)
(1 2)97.86396.453 (1.44%)95.947 (1.96%)96.705 (1.18%)97.996 (0.14%)97.891 (0.03%)97.860 (0.003%)
(3 1)111.911111.550 (0.32%)110.570 (1.20%)110.940 (0.87%)112.312 (0.36%)111.980 (0.06%)111.911 (0%)
(2 2)121.384123.600 (1.83%)122.650 (1.04%)120.490 (0.74%)121.867 (0.40%)121.500 (0.10%)121.382 (0.002%)
(1 3)135.712142.350 (4.89%)138.280 (1.89%)135.130 (0.43%)136.332 (0.46%)135.923 (0.16%)135.712 (0%)
(4 1)138.399142.840 (3.21%)140.850 (1.77%)138.230 (0.12%)139.415 (0.73%)138.628 (0.17%)138.400 (0.001%)
(3 2)146.166152.460 (4.31%)148.500 (1.60%)144.530 (1.12%)147.273 (0.76%)146.504 (0.23%)146.167 (0.001%)
), ArticleFig(id=1243220408917406293, tenantId=1146029695717560320, journalId=1242844143416819734, articleId=1243220383927743297, language=EN, label=Tab.4, caption=

Natural frequencies of a fully clamped isosceles right triangular membrane using two types of triangular elements

, figureFileSmall=null, figureFileBig=null, tableContent=
(m n)Theoretical solution/HzStandard T10The presentStandard T10The presentStandard T10The present
37 elements DOF=13337 elements DOF=13351 elements DOF=19651 elements DOF=196327 elements DOF=1 384327 elements DOF=1 384
(1 1)60.69260.694 (0.003%)60.719 (0.04%)60.692 (0%)60.694 (0.003%)60.692 (0%)60.689 (0.004%)
(2 1)85.83285.878 (0.05%)85.901 (0.08%)85.835 (0.003%)85.854 (0.03%)85.831 (0.001%)85.829 (0.003%)
(1 2)97.86397.925 (0.06%)97.996 (0.14%)97.773 (0.09%)97.891 (0.03%)97.861 (0.002%)97.860 (0.003%)
(3 1)111.911112.157 (0.22%)112.312 (0.36%)111.942 (0.03%)111.980 (0.06%)111.909 (0.002%)111.911 (0%)
(2 2)121.384121.749 (0.30%)121.867 (0.40%)121.539 (0.13%)121.500 (0.10%)121.383 (0.001%)121.382 (0.002%)
(1 3)135.712136.284 (0.42%)136.332 (0.46%)135.793 (0.06%)135.923 (0.16%)135.710 (0.001%)135.712 (0%)
), ArticleFig(id=1243220409026458202, tenantId=1146029695717560320, journalId=1242844143416819734, articleId=1243220383927743297, language=CN, label=表4, caption=

两种三角形单元下周边固定的等腰直角三角形薄膜固有频率

, figureFileSmall=null, figureFileBig=null, tableContent=
(m n)Theoretical solution/HzStandard T10The presentStandard T10The presentStandard T10The present
37 elements DOF=13337 elements DOF=13351 elements DOF=19651 elements DOF=196327 elements DOF=1 384327 elements DOF=1 384
(1 1)60.69260.694 (0.003%)60.719 (0.04%)60.692 (0%)60.694 (0.003%)60.692 (0%)60.689 (0.004%)
(2 1)85.83285.878 (0.05%)85.901 (0.08%)85.835 (0.003%)85.854 (0.03%)85.831 (0.001%)85.829 (0.003%)
(1 2)97.86397.925 (0.06%)97.996 (0.14%)97.773 (0.09%)97.891 (0.03%)97.861 (0.002%)97.860 (0.003%)
(3 1)111.911112.157 (0.22%)112.312 (0.36%)111.942 (0.03%)111.980 (0.06%)111.909 (0.002%)111.911 (0%)
(2 2)121.384121.749 (0.30%)121.867 (0.40%)121.539 (0.13%)121.500 (0.10%)121.383 (0.001%)121.382 (0.002%)
(1 3)135.712136.284 (0.42%)136.332 (0.46%)135.793 (0.06%)135.923 (0.16%)135.710 (0.001%)135.712 (0%)
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一种新型三角形单元及其在薄膜振动中的应用
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鲍四元 , 任一璇 , 沈峰
力学季刊 | 2025,46(3): 684-694
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力学季刊 | 2025, 46(3): 684-694
一种新型三角形单元及其在薄膜振动中的应用
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鲍四元 , 任一璇, 沈峰
作者信息
  • 苏州科技大学　土木工程学院，江苏　苏州　215011

    • 鲍四元，教授．研究方向：结构新型有限元及应用研究．

通讯作者:

鲍四元．E-mail：
A New Triangular Element and Its Application in Vibration Analysis of Membranes
Siyuan BAO , Yixuan REN, Feng SHEN
Affiliations
  • School of Civil Engineering, Suzhou University of Science and Technology, Suzhou 215011, Jiangsu, China
出版时间: 2025-09-25 doi: 10.15959/j.cnki.0254-0053.2025.03.010
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摘要
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薄膜的自由振动分析在工程结构中具有重要意义，尤其在薄膜结构的设计与优化过程中．本文提出一种新型三角形单元，旨在提高薄膜自由振动分析中的计算精度．传统的三节点三角形单元在处理薄膜振动时，通常依赖于多项式形函数，但这种方法在复杂的振动模式和高阶频率计算中存在精度不够理想的问题．为了解决这一问题，本文构建了一种形函数包含三角函数的10节点三角形单元，所提三角形单元由三个角点、三条边的两个三等分点及重心点组成，其形函数通过面积坐标表示．推导所提单元的刚度矩阵和质量矩阵，计算薄膜自由振动的频率和模态，从而研究其动力学特性．为了评估该单元的效果，本文选取矩形薄膜和三角形薄膜自由振动作为典型算例．算例中给出本文结果、Ansys三节点单元的计算结果及理论解．结果之间的对比表明，10节点三角形单元在结构划分为较少计算单元的情况下，能够逼近理论解，且本文10节点单元与标准10节点单元的精度相当．从而说明所提有限单元在分析膜自由振动问题时具有高精度，具有进一步研究和推广的潜力．

薄膜自由振动  /  有限元分析  /  三角形膜单元  /  形函数

The free vibration analysis of membranes is of significant importance in engineering structures, especially in the design and optimization of membrane structures. This paper presents a new type of triangular element, aiming to improve the computational accuracy in free vibration analysis of membranes. Traditional 3-node triangular elements in membrane vibration analysis typically rely on polynomial shape functions, but this method often suffers from insufficient accuracy in complex vibration modes and high-order frequencies. To address this issue, this paper constructs a 10-node triangular element with the shape function incorporating trigonometric functions. The proposed 10-node triangular element consists of three corner nodes, two points of trisection for every edge, and the centroid node with its shape functions derived using the area coordinate method. The stiffness matrix and mass matrix are derived, and the frequencies and modes for free vibration of the membrane are computed, thereby the dynamic characteristics can be studied. To evaluate the effectiveness of this element, several typical examples are chosen, including the free vibration analysis of rectangular membrane and triangular membrane. By comparing with theoretical solutions and the 3-node element calculations in Ansys, the obtained results show that the 10-node triangular element can approximate the theoretical solutions with few computational elements. And the precision of the presented 10-node element is similar with that of the standard 10-node triangular element. The high precision of the proposed element is demonstrated in analysis of free vibration of membrane, which has the potential of further research and promotion.

free vibration of membrane  /  finite element analysis  /  triangular membrane element  /  shape function
鲍四元, 任一璇, 沈峰. 一种新型三角形单元及其在薄膜振动中的应用. 力学季刊, 2025 , 46 (3) : 684 -694 . DOI: 10.15959/j.cnki.0254-0053.2025.03.010
Siyuan BAO, Yixuan REN, Feng SHEN. A New Triangular Element and Its Application in Vibration Analysis of Membranes[J]. Chinese Quarterly of Mechanics, 2025 , 46 (3) : 684 -694 . DOI: 10.15959/j.cnki.0254-0053.2025.03.010
正文
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薄膜的自由振动作为材料科学和工程领域中的一个重要研究课题，不仅具有显著的理论价值，而且在许多工程应用中展现了广泛的潜力，如通过施加预应力实现薄膜结构复杂多变的造型．薄膜结构的力学行为具有典型的强非线性特征，且因其轻量化的特性，在风荷载作用下容易产生风振，成为破坏的主要原因．因此，深入理解薄膜的振动行为对于保障结构安全至关重要．
学者们使用不同方法研究了薄膜振动特性，骆江南[1]基于无网格法对薄膜振动特性进行研究．在薄膜振动方面，赵凤群等[2]研究了四边简支矩形运动薄膜在两对边均有均匀分布张力作用时的非线性振动特性；周纪晨等[3]将一维弦振动问题推广到二维，得到四周固定的均匀矩形薄膜在受迫振动下的解析解．郑志刚[4]对矩形薄膜的振动问题进行了研究，并以四周固支的矩形薄膜为研究对象，研究薄膜结构自由振动动力学中的非线性特性．Kang等[5-6]对非对称梯形和带弯曲界面的复合矩形薄膜进行了自由振动分析．
有限元方法作为一种数值方法，在理论和工程应用中都获得了广泛而深远的应用．岑松等[7]介绍了形状自由的高性能有限元方法的研究进展，指出高性能有限元方法在已有的杂交应力元法和非对称有限元法基础上，综合利用了解析试函数法、新型自然坐标方法、广义协调方法等先进技术，并取得了重要进展．傅向荣等[8]提出以单元模型的平衡性与协调性进行分类，并对弹性力学平面问题中的经典单元进行了分析归类，总结出了协调元、非协调元和超协调元的协调性方法，以及基于解析试函数法的平衡型方法．袁驷等[9]提出二维自由振动问题的自适应有限元方法，通过引入EEP超收敛计算和自适应求解技术，可得到满足精度要求的自振频率和满足用户给定误差限的振型．
本研究旨在探讨三角形薄膜的自由振动，重点研究应用有限元方法时如何提高固有频率的计算精度问题．目前，平面薄膜自由振动问题存在理论解析解大多限于矩形、三角形、圆形[10-11]等规则几何形状的情况．对于更复杂形状的薄膜，数值分析成为不可或缺的手段．Houmat[12-13]基于三角函数的p型有限单元法分析了薄膜结构的自振特性．Buchanan等[14-15]则采用极坐标单元研究了圆形和椭圆形膜的自由振动特性．Fantuzzi等[16]研究了不同形状和密度的薄膜结构的自振特性．Liu等[17-18]结合了2D无限元法与有限元法，分析了多孔薄膜结构，并进一步将这两种方法结合应用于薄膜结构的自振特性分析．Wu等[19]提出了一种基于高阶质量矩阵的无网格方案，并将其应用于杆结构和薄膜结构的振动分析．Ren等[20]则为薄板和薄膜结构开发了一种基于绝对节点坐标法的自适应三角形单元．由于Ansys软件的限制，目前其用于计算薄膜自振特性的单元仅限于三节点的三角形单元或四节点的矩形单元，导致计算精度有待提高．因此，开发新的单元类型以提高有限元计算的精度成为必要．林文静等[21]的研究表明，六节点三角形单元在自由振动频率的计算上与理论解更为接近，其结果与Ansys采用的三节点单元相比有显著提升．因此，通过构建新型单元来提高薄膜自由振动频率的计算精度，使其更加接近理论解，具有重要的研究意义．
在矩形膜自由振动问题中，解析解往往出现正弦或者余弦的形式．这是因为薄膜在振动时满足波动方程，波动方程的特征解中包括三角函数．传统的有限元方法中通常采用多项式作为位移插值函数，但是低次多项式描述复杂振型的能力有限．基于以上思路本文提出了一种形函数包含三角函数的10节点三角形单元．由于该单元形函数包含三角函数，可以更好地捕捉复杂结构的振动模式，因此能够显著提高计算精度．在算例部分研究矩形薄膜和等腰直角三角形薄膜，并将本文方法的计算结果与理论解进行比较，以验证所提方法的正确性和精度．
1　一个10节点三角形单元
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研究薄膜的振动特性时，常采用平面3节点三角形单元和6节点三角形单元．前者是由3个角点确定的平面三角形，如图1（a）所示，其位移模式一般采用一次完全多项式．后者单元在前者的基础上增加了三个边的中点，如图1（b）所示，其位移模式可采用完整的二次多项式．
为了提高计算精度，本文构造一种10节点三角形单元，其节点包括三个角点、三条边的两个三等分点，以及重心处的节点．采用面积坐标进行计算，各节点的直角坐标为（xi，yi）（i=1，2，3，…，10），节点的面积坐标如图2所示．
由于单元内一点的面积坐标之和为1，将面积坐标定义为ξ，η和1-ξ-η．其中面积坐标与直角坐标的关系为
三角形区域内一点的面积坐标可以通过线性插值的方式由其直角坐标（x，y）表示，具体公式如下
其中，i=1，2，3，Ae为单元的面积．ai、bi、ci是与坐标相关的系数，定义如下
10节点三角形单元的位移场模式基于帕斯卡三角形并添加正弦和余弦项表示如下
其中，v为竖直方向上的位移，，，…，是待定参数．与式（5）对应，推导可得10节点三角形单元中位移场基于形函数的插值表达式为
其中，we为竖直方向的位移列阵；N为形函数矩阵，由形函数N1到N10组成．根据有限元方法，将每个单元中节点的面积坐标代入式（5），得到十个节点位移条件，解出系数后，整理可得到Ni（i=1，2，…，10）的表达式如下
而N3由式得到．上述形函数表达式中c1、c2…c12表示常数，具体值如下
通过验证，形函数在其对应节点处的值为1，而在其他节点处的值为0．实际运算中，上述形函数一般采用小数形式，其形式能够显得简洁．
位移模式（6）分别对ξ，η求导，得
其中，，．形函数对局部坐标的导数转换成对全局坐标的导数，其关系如下
其中，Rx，Ry分别对应形函数在x，y方向上的导数，J为雅可比矩阵．
薄膜自由振动过程中，薄膜的势能为
其中，T为膜单位长度上的张力．而薄膜的动能为
其中，ρ为单位面积上的质量．
由式（7）和式（20），可得10节点单元体的势能为
其中，Ke称为单元刚度矩阵，形式如下
由式（7）和式（21）求出节点单元的动能，并推导出单元质量矩阵Me的表达式为
2　固有频率及模态
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薄膜自由振动的方程可以表示为
其中，K，M分别为薄膜的总体刚度矩阵和总体质量矩阵，w为薄膜总位移列阵．假设解的形式为简谐振动：w（t）=φeiωt，其中φ是振型（模态向量），ω是固有频率．式（25）可转化为
这是一个矩阵广义特征值问题，其中解得的特征值为对应于结构固有频率的平方，而求得的特征向量代入式（7）可得结构振型函数．
3　算例
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算例1：计算图3所示的周边固定矩形薄膜的固有频率，其长度为A=2.0 m，宽度为B=1.0 m，单位长度上的张力T=13 800 N/m，单位面积质量ρ=7.805 kg/m2．使用Ansys有限元软件计算薄膜自由振动的频率时，膜厚取0.001 m，材料弹性模量是E=7.0×109 N/m2，泊松比为0.4．
四边固定的矩形薄膜自由振动问题具有理论解[21]，其固有频率ωmn表达式为
固有频率对应的模态函数Wmn如下
其中，Amn为振幅，m，n为正整数．
为了验证包含三角函数的10节点三角形单元在位移模式中的精度．首先采用粗网格进行计算．初步使用26个单元，如图4（a）所示，计算周边固定的矩形薄膜自由振动的固有频率，并将结果与Ansys模拟结果以及矩形薄膜自由振动的理论解进行对比．然后细分网格，如图4（b）所示，研究随着网格逐渐变得精细，矩形薄膜自由振动频率计算值的收敛趋势，以进一步评估该单元的计算精度．
表1给出四周固定的矩形薄膜有限元计算自振频率结果、Ansys结果及理论解．其中，括号内的数字表示有限元结果与理论解之间的相对误差．另外，本算例还研究了标准10节点三次三角形单元（记为标准T10单元），其形函数为完备的三次多项式．表2为本文10节点三角形单元与标准T10三角形单元计算结果的对比情况．
Ansys计算薄膜自由振动频率所采用的shell181单元是一种4节点有限应变壳单元，可使用退化三角形单元进行网格划分，此时shell181单元中各节点具有平动和横向自由度（Degree of freedom，DOF）．
从表1可以看出，当使用较少量自由度计算时，本文有限元计算在前5阶频率上的结果与理论解非常接近，相对误差控制在1%以内．例如，使用97个总自由度（26个计算单元）时，矩形薄膜自由振动的前5阶频率接近理论解，误差较小（最大0.45%），而后3阶频率的误差则有所较大（最大2.83%）．计算总自由度增至202（单元数量为52）时，前4阶频率的误差略有增大（但仍在0.5%左右），且后4阶频率的结果逐渐趋近于理论解．
在相近的总自由度条件下，对Ansys模拟结果与本文所提三角形单元的计算结果比较知，本文结果的误差更小．例如表1中Ansys使用120个计算单元（总自由度数为102），而本文采用52个10节点三角形单元（总自由度数为97），无论是前5阶频率还是第6~8阶频率，Ansys模拟结果在精度上都明显弱于本文基于10节点三角形单元的有限元计算结果．如前5阶固频而言，Ansys结果相对误差为2.65%~5.30%，而本文结果的相对误差为0.15%~0.45%．当Ansys使用2 502个计算单元（总自由度数为2 350），对应本文的情况是542个10节点三角形单元（总自由度数为2 338）．Ansys软件所得自振频率结果的精度均大于本文三角形单元的有限元计算结果，其中前3阶自振频率的相对误差本文结果都比Ansys软件结果低1%以上．
表2对比了两种10节点三角形单元计算的精度表现，标准T10三角形单元和本文10节点三角形单元在不同网格下的计算结果，其中相对误差以理论解作为基准．数据显示，标准单元在97个自由度（26单元）时精度较高（大部分误差≤0.36%），而同种情况下本文单元所得结果的相对误差略大，但均不大于0.53%；在202个自由度（52单元）时，本文单元所得固有频率结果的较大相对误差为0.11%和0.14%，普遍小于标准10节点三角形单元相对误差所对应的0.23%和0.26%，其他阶次固有频率的精度接近．在2 338个自由度（542单元）时本文结果逼近理论解，最大相对误差0.007%，与标准10节点单元的最大相对误差相等．这表明两种单元精度相当，部分网格划分情况时本文10节点单元所得固频结果的精度更高．且在个人电脑上运行同一网格所用时间接近，如542个单元时分别为148.65 s（标准T10单元）和160.27 s（本文T10单元）．
图5分别展示本文方法与Ansys所得的四周固定矩形薄膜的模态．图5中左侧为Ansys所得的第1至第4阶模态图，右侧为本文方法所得的前4阶模态图，通过比较图形知两者结果吻合，从而验证本文方法的准确性．
算例2：研究图6（a）所示的周边固定的等腰直角三角形薄膜的自由振动的频率以及相应的振型，其中等腰直角三角形的直角边长度A=1 m，单位长度上的张力T=23 000 N/m，单位质量面积ρ=7.805 kg/m2，膜材数据同算例1．
四周固定时等腰直角三角形薄膜的自由振动问题有理论解[21]，其固有频率ωmn和振型φmn分别为
其中，m和n为正整数．
图6（b）~（c）展示等腰直角三角形作有限元分析时的网格划分情况，即分为37个和327个单元．表3给出四周固定的等腰直角三角形薄膜固有频率的本文数值计算结果、理论解以及Ansys有限元计算结果．括号内的数字表示有限元计算结果和Ansys计算结果相对于理论解的相对误差．
从表3中可以看出，当本文10节点三角形单元使用较少的总自由度计算时，例如采用133个总自由度（37个单元），前8阶频率与理论解之间的相对误差都在0.76%以内．随着计算总自由度的适度增加，本文方法的计算结果逐渐趋近于理论解．例如，当总自由度增加至196时，本文的有限元结果与理论解非常接近，相对误差最大只有0.23%．另外，继续增加总自由度，如为1 384时，前8阶固有频率的相对误差可以保持在0.004%以内．从表中结果可以看出，对比Ansys软件中的shell181单元，结构总自由数相近或略少时，所提三角形单元所得固有频率结果的相对误差更小，故所提单元在精度上优于Ansys有限元软件的模拟结果．
表4中对比本文所提10节点三角形单元和标准T10三角形单元的数值结果．可以看出，两种计算单元都有较高的精确性．如总自由度同为133时，本文单元与标准T10单元所得的前6阶固有频率相对误差都在0.5%以内；总自由度同为196时，两种单元前6阶固频结果的相对误差都在0.2%以内；而总自由度同为1 384时，两种单元所得前6阶固频结果的相对误差都在0.004%以内．因此可认为两种单元的性能相当．
图7展示了本文方法与Ansys所得的四周固定等腰直角三角形薄膜的前3阶模态．图中左侧各图为Ansys软件所得第1~3阶模态图，右侧自上而下为本文方法所得的第1~3阶模态图．对比发现，两种不同单元所得的模态图一致，说明了本文所提方法的正确性和有效性．
4　结论
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本文提出了一种新的有限元单元，其形函数中包含三角函数，能够高效地计算平面薄膜的自由振动固有频率及振型．与Ansys软件中的shell181单元相比，该三角形单元在总自由度较少的情况下，也能得出较高精度的计算结果．
所提三角形单元具备高精度的优势，其性能优于shell181单元，与标准的10节点三角形单元性能相当，因此具有进一步研究和推广的潜力．为了提高效率，后续研究中在求解所提单元质量矩阵或刚度矩阵时拟应用数值积分方法对其进行深入的研究．
基金
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  • 国家自然科学基金(51709194)
文献
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参考文献 引证文献
排序方式：
[1]
骆江南. 基于无网格法的薄膜振动特性研究[D]. 西安: 西安理工大学, 2014.
Luo J N. Study on membrane vibration characteristics based on meshless method[D]. Xi'an: Xi'an University of Technology, 2014 (in Chinese).
[2]
赵凤群, 王忠民. 运动矩形薄膜的非线性振动分析[J]. 机械科学与技术, 2010, 29(6): 768-771.
Zhao F Q, Wang Z M. Nonlinear vibration analysis of a moving rectangular membrane[J]. Mechanical Science and Technology for Aerospace Engineering, 2010, 29(6): 768-771 (in Chinese).
[3]
周纪晨, 郭琴. 矩形薄膜受迫横振动问题的求解及可视化[J]. 大学物理, 2022, 41(3): 56-60.
Zhou J C, Guo Q. Solution and visualization of forced transverse vibration of rectangular thin films[J]. College Physics, 2022, 41(3): 56-60 (in Chinese).
[4]
郑志刚. 空间薄膜结构自由振动非线性效应研究[D]. 哈尔滨: 哈尔滨工业大学, 2019.
Zheng Z G. Research on nonlinear effect of free vibration of membrane structures in space[D]. Harbin: Harbin Institute of Technology, 2019 (in Chinese).
[5]
Kang S W, Lee J M. Free vibration analysis of an unsymmetric trapezoidal membrane[J]. Journal of Sound and Vibration, 2004, 272(1-2): 450-460.
[6]
Kang S W. Free vibration analysis of composite rectangular membranes with a bent interface[J]. Journal of Sound and Vibration, 2004, 272(1-2): 39-53.
[7]
岑松, 尚闫, 周培蕾, 等. 形状自由的高性能有限元方法研究的一些进展[J]. 工程力学, 2017, 34(3): 1-14.
Cen S, Shang Y, Zhou P L. Advances in shape-free finite element methods: A review[J]. Engineering Mechanics, 2017, 34(3): 1-14 (in Chinese).
[8]
傅向荣, 陈璞, 孙树立, 等. 弹性力学有限元分析中的平衡与协调理论[J]. 工程力学, 2023, 40(2): 8-16.
Fu X R, Chen P, Sun S L, et al. Research on equilibrium and conforming theory of the finite element method in elasticity[J]. Engineering Mechanics, 2023, 40(2): 8-16 (in Chinese).
[9]
袁驷, 孙浩涵. 二维自由振动问题的自适应有限元分析初探[J]. 工程力学, 2020, 37(1): 17-25.
Yuan S, Sun H H. A preliminary study on adaptive finite element analysis of two-dimensional free vibration problems[J]. Engineering Mechanics, 2020, 37(1): 17-25 (in Chinese).
[10]
黄涛, 何泽青, 宋林. 圆柱薄膜振动特性[J]. 上海交通大学学报, 2024, 58(4): 428-437.
Huang T, He Z Q, Song L. Vibration characteristics of cylindrical membrane[J]. Journal of Shanghai Jiaotong University, 2024, 58(4): 428-437 (in Chinese).
[11]
林文静, 陈树辉, 李森. 圆形薄膜自由振动的理论解[J]. 振动与冲击, 2009, 28(5): 84-86.
Lin W J, Chen S H, Li S. Analytical solution of the free vibration of circular membrane[J]. Journal of Vibration and Shock, 2009, 28(5): 84-86 (in Chinese).
[12]
Houmat A. Free vibration analysis of membranes using the h-p version of the finite element method[J]. Journal of Sound and Vibration, 2005, 282(1-2): 401-410.
[13]
Houmat A. Free vibration analysis of arbitrarily shaped membranes using the trigonometric p-version of the finite element method[J]. Thin-Walled Structures, 2006, 44(9): 943-951.
[14]
Buchanan G R. Vibration of circular membranes with linearly varying density along a diameter[J]. Journal of Sound and Vibration, 2005, 280(1-2): 407-414.
[15]
Buchanan G R, Peddieson J J. A finite element in elliptic coordinates with application to membrane vibration[J]. Thin-Walled Structures, 2005, 43(9): 1444-1454.
[16]
Fantuzzi N, Tornabene F, Viola E. Generalized differential quadrature finite element method for vibration analysis of arbitrarily shaped membranes[J]. International Journal of Mechanical Sciences, 2014, 79: 216-251.
[17]
Liu D S, Lin I H. Vibration analysis of the multiple-hole membrane by using the coupled Diem-Fe scheme[J]. International Review of the Red Cross, 2016, 32(2): 163-173.
[18]
Liu D S, Chen Y W. Application of Craig-Bampton reduction technique and 2D dynamic infinite element modeling approach to membrane vibration problems[J]. Journal of Mechanics, 2019, 35(4): 513-525.
[19]
Wu J C, Wang D D, Lin Z. A mesh free higher order mass matrix formulation for structural vibration analysis[J]. International Journal of Structural Stability & Dynamics, 2018, 18(10): 1850121.
[20]
Ren H, Fan W. An adaptive triangular element of absolute nodal coordinate formulation for thin plates and membranes[J]. Thin-Walled structures, 2023, 182: 1-15.
[21]
林文静, 陈树辉. 平面薄膜自由振动的有限元分析[J]. 动力学与控制学报, 2010, 8(3): 202-206.
Lin W J, Chen S H. Free vibration analysis of plane membranes by finite element method[J]. Journal of Dynamics and Control, 2010, 8(3): 202-206 (in Chinese).
2025年第46卷第3期
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doi: 10.15959/j.cnki.0254-0053.2025.03.010
  • 接收时间：2025-03-11
  • 首发时间：2026-03-24
  • 出版时间：2025-09-25
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  • 收稿日期：2025-03-11
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国家自然科学基金(51709194)
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    苏州科技大学　土木工程学院，江苏　苏州　215011

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鲍四元．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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