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Article(id=1246023207329842128, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1246023204117005194, articleNumber=null, orderNo=null, doi=10.19636/j.cnki.cjsm42-1250/o3.2023.037, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1690819200000, receivedDateStr=2023-08-01, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1775004682618, onlineDateStr=2026-04-01, pubDate=1713974400000, pubDateStr=2024-04-25, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1775004682618, onlineIssueDateStr=2026-04-01, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1775004682618, creator=13701087609, updateTime=1775004682618, updator=13701087609, issue=Issue{id=1246023204117005194, tenantId=1146029695717560320, journalId=1241755870837649424, year='2024', volume='45', issue='2', pageStart='145', pageEnd='288', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1775004681852, creator=13701087609, updateTime=1775004747143, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1246023478026027853, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1246023204117005194, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1246023478026027854, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1246023204117005194, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=238, endPage=252, ext={EN=ArticleExt(id=1246023207581500379, articleId=1246023207329842128, tenantId=1146029695717560320, journalId=1241755870837649424, language=EN, title=Forced Vibration Analysis of Arbitrary Shells Based on 3D Continuous Shell Theory and Meshless Method, columnId=1244229834482757770, journalTitle=Chinese Journal of Solid Mechanics, columnName=Research Paper, runingTitle=null, highlight=null, articleAbstract=

Shell structures are widely used in engineering, especially in aerospace and civil engineering. As a result, the study on dynamic behaviors of shell structures is crucial for engineering applications. Over the years, shell theory has undergone continuous improvement and development, leading to analytical solutions for specific shell structures. However, solving analytical solutions for complex shell structures with intricate shapes becomes highly challenging and even unattainable. Therefore, numerical methods like the finite element method (FEM) and the meshless method are employed for further solutions. The meshless method is a powerful complement to FEM, relying solely on nodal information for the formation of shape functions and enabling easy construction of higher-order smooth approximations. Consequently, it naturally holds an advantage in the numerical analysis of plate shell structures. Based on the 3D continuous shell theory and the moving least-squares (MLS) approximation, a meshless model for arbitrary shells is established in this paper. The MLS approximation is used not only for geometric surface interpolation, but also for displacement field approximation. The meshless equation governing the forced vibration of arbitrary shells is derived under Hamilton's principle and solved using the time-domain implicit Newmark method. The full transformation method is used to impose the essential boundary conditions. The code for the proposed method is developed in the MATLAB platform and used to compute several representative shell examples, obtaining the first ten natural frequencies of each shell type and the time history deflection at the center point under different pulse loads, considering both with and without damping. The calculated results are compared with ABAQUS solutions to verify the effectiveness and accuracy of the presented method. When using the meshless method to solve the forced vibration of arbitrary shell structures, it does not rely on grid partitioning. This enables effective analysis of various shell structures with different shapes, showcasing its strong adaptability and vast potential for applications.

, correspAuthors=Wei Chen, authorNote=null, correspAuthorsNote=null, copyrightStatement=null, copyrightOwner=null, extLink=null, articleAbsUrl=null, sourceXml=null, magXml=null, pdfUrl=null, pdf=null, pdfFileSize=null, pdfExtLink=null, richHtmlUrl=null, mobilePdfUrl=null, reviewReport=null, pdfFirstPage=null, abstractGraph=null, abstractGraphContent=null, abstractVideo=null, citation=null, cebUrl=null, magXmlContent=null, mapNumber=null, authorCompany=null, fund=null, authors=null, authorsList=Linxin Peng, Jianfei Zhang, Wei Chen), CN=ArticleExt(id=1246023221410120230, articleId=1246023207329842128, tenantId=1146029695717560320, journalId=1241755870837649424, language=CN, title=基于3D连续壳理论和无网格法的任意壳受迫振动分析, columnId=1241831201896469478, journalTitle=固体力学学报, columnName=研究论文, runingTitle=null, highlight=null, articleAbstract=

板壳结构是航空航天和建筑水利等工程领域中最常见的基本构件，研究板壳受迫振动问题对工程应用具有重要意义. 本文基于3D连续壳理论和移动最小二乘近似建立了任意壳的无网格模型，其中移动最小二乘近似不仅用于几何曲面插值，还用于位移场近似. 利用Hamilton原理导出描述任意壳受迫振动的无网格控制方程，并采用时域隐式Newmark方法求解该方程，采用完全转换法来施加本质边界条件. 最后，通过MATLAB编制无网格程序计算了几个具有代表性的壳体算例，并将计算结果和ABAQUS有限元解进行比对，验证了本文方法求解任意壳受迫振动的有效性及准确性. 结果表明，无网格法不依赖网格划分，适应性较强，所提方法可以有效地求解各种不同形状的板壳结构受迫振动问题，具有广阔的应用前景.

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caption=波纹板在脉冲荷载作用下有阻尼中点挠度, figureFileSmall=QwpVqEtDOEJcAyoHICOTPQ==, figureFileBig=TzT0gIx78phHeZt2i1GryQ==, tableContent=null), ArticleFig(id=1246023226107740845, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246023207329842128, language=EN, label=Table 1, caption=

Comparison of the first ten orders of frequency of the flat plate (Hz)

, figureFileSmall=null, figureFileBig=null, tableContent=
阶数ABAQUS本文解（11×11）误差阶数ABAQUS本文解（11×11）误差
1405.1405.10.01%61288.21289.10.07%
2772.7773.20.06%71534.21532.10.14%
3772.7773.20.06%81534.21532.10.14%
41083.01083.40.03%91543.21544.80.11%
51276.01277.20.09%101543.21544.80.11%
), ArticleFig(id=1246023226191626927, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246023207329842128, language=CN, label=表1, caption=

平板前十阶频率对比（Hz）

, figureFileSmall=null, figureFileBig=null, tableContent=
阶数ABAQUS本文解（11×11）误差阶数ABAQUS本文解（11×11）误差
1405.1405.10.01%61288.21289.10.07%
2772.7773.20.06%71534.21532.10.14%
3772.7773.20.06%81534.21532.10.14%
41083.01083.40.03%91543.21544.80.11%
51276.01277.20.09%101543.21544.80.11%
), ArticleFig(id=1246023226279707313, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246023207329842128, language=EN, label=Table 2, caption=

Frequency comparison of the first ten orders of the inclined plate (Hz)

, figureFileSmall=null, figureFileBig=null, tableContent=
阶数ABAQUS本文解（11×11）误差阶数ABAQUS本文解（11×11）误差
1303.7303.80.03%61161.81160.00.16%
2501.9501.60.06%71316.01312.50.26%
3756.2755.00.16%81511.61510.50.07%
4921.9920.90.11%91514.91510.90.26%
51038.01035.80.21%101613.41608.30.32%
), ArticleFig(id=1246023226351010483, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246023207329842128, language=CN, label=表2, caption=

斜板前十阶频率对比（Hz）

, figureFileSmall=null, figureFileBig=null, tableContent=
阶数ABAQUS本文解（11×11）误差阶数ABAQUS本文解（11×11）误差
1303.7303.80.03%61161.81160.00.16%
2501.9501.60.06%71316.01312.50.26%
3756.2755.00.16%81511.61510.50.07%
4921.9920.90.11%91514.91510.90.26%
51038.01035.80.21%101613.41608.30.32%
), ArticleFig(id=1246023226409730741, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246023207329842128, language=EN, label=Table 3, caption=

Comparison of the first ten orders frequency of the roof shell (Hz)

, figureFileSmall=null, figureFileBig=null, tableContent=
阶数ABAQUS本文解（15×15）误差阶数ABAQUS本文解（15×15）误差
1137.7137.40.28%6455.9459.30.74%
2174.1176.11.14%7621.1616.40.75%
3343.5342.00.44%8669.0671.10.32%
4395.7395.50.05%9678.6675.40.47%
5447.2446.00.27%10867.1865.80.15%
), ArticleFig(id=1246023226485228216, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246023207329842128, language=CN, label=表3, caption=

屋顶壳前十阶频率对比（Hz）

, figureFileSmall=null, figureFileBig=null, tableContent=
阶数ABAQUS本文解（15×15）误差阶数ABAQUS本文解（15×15）误差
1137.7137.40.28%6455.9459.30.74%
2174.1176.11.14%7621.1616.40.75%
3343.5342.00.44%8669.0671.10.32%
4395.7395.50.05%9678.6675.40.47%
5447.2446.00.27%10867.1865.80.15%
), ArticleFig(id=1246023226552337082, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246023207329842128, language=EN, label=Table 4, caption=

Comparison of the first ten orders of frequency of hyperbolic flat shell (Hz)

, figureFileSmall=null, figureFileBig=null, tableContent=
阶数ABAQUS本文解（15×15）误差阶数ABAQUS本文解（15×15）误差
1475.6476.00.08%6802.9804.40.18%
2521.1521.40.06%7939.9941.20.14%
3521.1522.50.27%8939.9942.00.23%
4676.9678.00.17%91147.31149.10.16%
5781.5783.00.19%101147.31151.10.33%
), ArticleFig(id=1246023226632028860, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246023207329842128, language=CN, label=表4, caption=

双曲扁壳前十阶频率对比（Hz）

, figureFileSmall=null, figureFileBig=null, tableContent=
阶数ABAQUS本文解（15×15）误差阶数ABAQUS本文解（15×15）误差
1475.6476.00.08%6802.9804.40.18%
2521.1521.40.06%7939.9941.20.14%
3521.1522.50.27%8939.9942.00.23%
4676.9678.00.17%91147.31149.10.16%
5781.5783.00.19%101147.31151.10.33%
), ArticleFig(id=1246023226703332030, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246023207329842128, language=EN, label=Table 5, caption=

Comparison of the first ten orders of frequency of corrugated plate (Hz)

, figureFileSmall=null, figureFileBig=null, tableContent=
阶数ABAQUS本文解（15×15）误差阶数ABAQUS本文解（15×15）误差
1242.7246.61.59%61321.01321.70.06%
2487.3489.20.40%71615.51609.70.36%
3837.4840.70.40%81680.11675.00.30%
41101.51107.00.50%91830.01825.50.25%
51274.11277.40.26%101840.71833.50.39%
), ArticleFig(id=1246023226778829504, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246023207329842128, language=CN, label=表5, caption=

波纹板前十阶频率对比（Hz）

, figureFileSmall=null, figureFileBig=null, tableContent=
阶数ABAQUS本文解（15×15）误差阶数ABAQUS本文解（15×15）误差
1242.7246.61.59%61321.01321.70.06%
2487.3489.20.40%71615.51609.70.36%
3837.4840.70.40%81680.11675.00.30%
41101.51107.00.50%91830.01825.50.25%
51274.11277.40.26%101840.71833.50.39%
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基于3D连续壳理论和无网格法的任意壳受迫振动分析
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彭林欣 1, 2 , 张鉴飞 1 , 陈卫 3, **
固体力学学报 | 研究论文 2024,45(2): 238-252
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固体力学学报 | 研究论文 2024, 45(2): 238-252
基于3D连续壳理论和无网格法的任意壳受迫振动分析
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彭林欣1, 2, 张鉴飞1, 陈卫3, **
作者信息
  • 1广西大学土木建筑工程学院，南宁，530004
  • 2广西大学工程防灾与结构安全教育部重点实验室，南宁，530004
  • 3南华大学土木工程学院，衡阳，421001

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** E-mail：.
Forced Vibration Analysis of Arbitrary Shells Based on 3D Continuous Shell Theory and Meshless Method
Linxin Peng1, 2, Jianfei Zhang1, Wei Chen3, **
Affiliations
  • 1School of Civil Engineering and Architecture, Guangxi University, Nanning, 530004
  • 2Guangxi Key Laboratory of Disaster Prevention and Engineering Safety, Guangxi University, Nanning, 530004
  • 3School of Civil Engineering, University of South China, Hengyang, 421001
出版时间: 2024-04-25 doi: 10.19636/j.cnki.cjsm42-1250/o3.2023.037
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摘要
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板壳结构是航空航天和建筑水利等工程领域中最常见的基本构件，研究板壳受迫振动问题对工程应用具有重要意义. 本文基于3D连续壳理论和移动最小二乘近似建立了任意壳的无网格模型，其中移动最小二乘近似不仅用于几何曲面插值，还用于位移场近似. 利用Hamilton原理导出描述任意壳受迫振动的无网格控制方程，并采用时域隐式Newmark方法求解该方程，采用完全转换法来施加本质边界条件. 最后，通过MATLAB编制无网格程序计算了几个具有代表性的壳体算例，并将计算结果和ABAQUS有限元解进行比对，验证了本文方法求解任意壳受迫振动的有效性及准确性. 结果表明，无网格法不依赖网格划分，适应性较强，所提方法可以有效地求解各种不同形状的板壳结构受迫振动问题，具有广阔的应用前景.

3D连续壳理论  /  任意壳  /  受迫振动  /  无网格法  /  完全转换法

Shell structures are widely used in engineering, especially in aerospace and civil engineering. As a result, the study on dynamic behaviors of shell structures is crucial for engineering applications. Over the years, shell theory has undergone continuous improvement and development, leading to analytical solutions for specific shell structures. However, solving analytical solutions for complex shell structures with intricate shapes becomes highly challenging and even unattainable. Therefore, numerical methods like the finite element method (FEM) and the meshless method are employed for further solutions. The meshless method is a powerful complement to FEM, relying solely on nodal information for the formation of shape functions and enabling easy construction of higher-order smooth approximations. Consequently, it naturally holds an advantage in the numerical analysis of plate shell structures. Based on the 3D continuous shell theory and the moving least-squares (MLS) approximation, a meshless model for arbitrary shells is established in this paper. The MLS approximation is used not only for geometric surface interpolation, but also for displacement field approximation. The meshless equation governing the forced vibration of arbitrary shells is derived under Hamilton's principle and solved using the time-domain implicit Newmark method. The full transformation method is used to impose the essential boundary conditions. The code for the proposed method is developed in the MATLAB platform and used to compute several representative shell examples, obtaining the first ten natural frequencies of each shell type and the time history deflection at the center point under different pulse loads, considering both with and without damping. The calculated results are compared with ABAQUS solutions to verify the effectiveness and accuracy of the presented method. When using the meshless method to solve the forced vibration of arbitrary shell structures, it does not rely on grid partitioning. This enables effective analysis of various shell structures with different shapes, showcasing its strong adaptability and vast potential for applications.

3D continuous shell theory  /  arbitrary shell  /  forced vibration  /  meshless method  /  full transformation method
彭林欣, 张鉴飞, 陈卫. 基于3D连续壳理论和无网格法的任意壳受迫振动分析. 固体力学学报, 2024 , 45 (2) : 238 -252 . DOI: 10.19636/j.cnki.cjsm42-1250/o3.2023.037
Linxin Peng, Jianfei Zhang, Wei Chen. Forced Vibration Analysis of Arbitrary Shells Based on 3D Continuous Shell Theory and Meshless Method[J]. Chinese Journal of Solid Mechanics, 2024 , 45 (2) : 238 -252 . DOI: 10.19636/j.cnki.cjsm42-1250/o3.2023.037
正文
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0　引言
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板壳结构是固体力学中最基本的结构形式，其外形厚度方向上的尺寸远比其他两个方向的尺寸小得多. 平分板壳厚度的中面若为平面，则此结构为板；反之若该中面为曲面则为壳体[1]. 板壳是航空航天、建筑水利等工程领域中最常见的基本构件，研究壳体在不同荷载类型作用下的受迫振动响应问题对工程应用具有重要意义.
针对板壳结构的分析起源于18世纪，Euler[2]最先对板的弯曲问题进行了研究探索.1850年，Kirchoff[3]提出了薄板弯曲理论.1874年，Aron[4]做了薄壳的分析工作. 由于经典板壳理论未考虑剪切变形的影响，Mindlin[5]和Reissner[6]分别将Timoshenko梁理论运用到中厚板壳的分析中，提出了一阶剪切变形的概念，并得到广泛应用.1970年由Ahmad[7]提出的用于一般壳体分析，从三维弹性实体单元蜕化而来的退化壳单元，也可称为3D连续壳单元，它是基于Mindlin-Reissner板壳理论，其位移和转动都是独立的场函数，只需要满足C0连续性即可. 经过多年的研究探索，如今的壳体理论已经日臻完善. 针对壳体的受迫振动，也有了不少的研究文献，Christoforou等[8]利用解析法分析了两端简支圆柱壳在径向冲击载荷的振动响应问题. 王宇等人[9]针对受径向谐波激励或径向冲击激励的薄壁圆柱壳构件，基于Love壳体理论开展了其受迫振动下的响应特征分析. 庞福振等[10]利用一种半解析法，基于区域能量法和Reissner-Naghdi薄壳理论，对柱壳结构进行了受迫振动特性分析. 贾菊芳等[11]基于Reissner薄壳理论，利用辛解析方法求解弹性圆柱壳受迫振动方程解析表达式. Qi等[12]利用变分法研究了受经典边界条件约束的连接圆柱-球面壳的自由振动和受迫振动. Ma等[13]采用修正Fourier-Ritz方法对具有任意边界条件的锥形-圆柱形耦合壳体进行了自由和强迫振动分析.
关于壳体的振动分析，仅在极少数特殊情况下能够得出解析解，同时限于实验设备、技术以及成本控制的影响，很多实际问题也无法进行实验处理，因此多数情况下我们需要利用数值方法来进行弹性壳体的振动特性分析. 目前常见的数值方法包括且不限于有限单元法[14]、能量法[15]以及无网格法[16]等.
无网格法可以看作有限元法的补充与改进，在理论和形式上具有诸多相似之处. 相比较于有限元法，无网格法形函数不依赖于网格划分，只依据节点的信息，克服了有限元法对网格的依赖，在涉及网格畸变、网格移动等问题中具有明显的优势[17]. 在无网格法中，可以通过提高基函数的阶次，构造高阶次的形函数，从而有效避免剪切和薄膜自锁现象的发生[18]. 无网格法发展至今，用于研究壳结构的文献并不多见，已有的文献主要关注于壳体的线性弯曲和自由振动[19-25]，研究壳的受迫振动鲜见于文献.
综上分析，采用无网格法研究任意壳的受迫振动问题在国内外并不多见. 本文首先介绍了基于3D连续壳理论的壳几何模型，基于一阶剪切变形理论和移动最小二乘近似建立的壳体位移场近似表达式，接着推导了笛卡尔坐标系下各向同性壳的位移-应变关系，以及笛卡尔坐标系与随动坐标系下的应力-应变转换关系式，然后根据哈密顿原理推导出任意壳受迫振动的控制方程. 最后，文末通过几个算例来验证了本文方法的收敛性及准确性.
1　壳的伽辽金无网格模型
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1.1　几何模型
如图1所示，任意壳均可先在随动坐标系中创建，然后将节点信息映射到三维笛卡尔坐标系当中. 无网格节点设置在壳体中面上，两个坐标系的节点值具有一一对应关系.
根据Mindlin板壳理论，壳上任意一点的位置矢量和位移矢量参数可以表示为：
式中，X=（x，y，z）是在笛卡尔坐标系中的位置矢量，r=（r1，r2，r3）是随动坐标系中的位置矢量. Xmid是壳中面的点在笛卡尔坐标系中的位置矢量，h为壳厚度（等厚度），r1、r2是壳中面内的参数，r3是厚度方向的参数. umid（u，v，w）是壳中面的点在笛卡尔坐标系中的位移矢量，θ1，θ2分别为绕壳中面正交矢量V1，V2的转角. Vi是壳中面某一点的正交单位矢量，V3是垂直壳中面的单位法向矢量，V1，V2，V3满足以下关系：
式中，ei是笛卡尔坐标系中正交单位基矢量，Gi是协变基矢量，且有：
1.2　移动最小二乘形函数
移动最小二乘（MLS）近似最初被数学家用来进行数据拟合和表面构造，现被广泛用于构造无网格法形函数[26]. 设定义域Ω上的函数u（r）可以被函数在子域上局部近似[27]：
式中，是基函数，m为单项式基函数的个数. a（r）是为该子域内与相对应的待求系数.
为了求解局部近似函数的待求系数，通常取局部近似函数在局部子域的离散节点处的加权离散范数L取得极小值：
其中，WI（r）为离散节点rI处的权函数在计算点r处的值，权函数WI（r）只在节点rI周围的有限区域ΩI中大于零，这个有限区域ΩI叫做权函数WI（r）的支撑域.
求解代求系数并回代式（5），可以得到在整个求解域上的全局近似，即：
式中，u是节点参数，
1.3　位移场近似
将壳上任意一点的位置矢量和位移矢量用离散点来近似表示为：
改写为矩阵形式：
式中，[uI，vI，wI，θ1I，θ2I]T=UI是壳中面上第I个节点的节点参数；uI，vI，wI是壳中面上I节点沿着笛卡尔坐标系x，y，z三个方向的位移；θ1I，θ2I是壳中面I节点绕I节点正交矢量V1，V2的转角，n是壳中面节点个数.
为了完成笛卡尔坐标系与随动坐标系的映射关系，需要推导协变基矢量：
将式（11）代入式（15），则有：
整理得到雅可比转换矩阵：
1.4　应变-位移关系
利用移动最小二乘近似，壳上任意一点的向量Vi可以表示为：
式中，VIi表示第I个节点的Vi矢量.
由式（14）取位移u（r）分别对随动坐标r1，r2，r3求偏导.
当j=1，2时，
当j=3时，
式中计算如下：
式中，I为3×3的单位矩阵.
小位移分析应变张量为：
将式（18）、（20）、（21）代入式（24）得：
式中：；；中的（i，j）表示雅克比的逆中元素所在的行和列.
1.5　应力-应变关系
分析壳体时，假定垂直于中面的壳体正应力等于零，因此需要求出随动坐标系的应力与应变. 需要注意的是，壳体中面的法线方向与笛卡尔坐标系的z轴并不一致，而且法线的方向是随着点的位置不同而变化的.
壳上任意一点在随动坐标系下及笛卡尔坐标系下的应变转换关系可写为
其中：
式中，ε′为随动坐标系下的应变矢量，ε为笛卡尔坐标系下的应变矢量. li，mi，ni分别是局部正交单位矢量Vi（i=1，2，3）的三个分量.
因为在随动坐标系下及笛卡尔坐标系下的两组应力表示同一个应力状态，根据虚功原理，可以推出笛卡尔坐标系下的应力-应变关系为：
式中，随动坐标系下的弹性矩阵D′为：
式中，Ks=6/5是剪切修正因子，E是弹性模量，ν是泊松比.
2　受迫振动控制方程本质边界条件处理
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2.1　受迫振动控制方程
采用移动最小二乘近似，壳在外部径向动荷载q（t）作用下产生的外力功可写为：
其中：
受迫振动控制方程根据Hamilton原理导出，其数学表达式为：
式中，外力功W是关于时间t的函数，包括激振力和阻尼力的功.
进一步可以得到各向同性壳在动荷载作用下的控制方程：
忽略阻尼时有：
式中，C为阻尼矩阵，一般可视为质量矩阵和刚度矩阵的线性组合，即：
式中，系数α、β与体系的固有频率和阻尼比有关. 设ωi、ωj分别为第i个和第j个固有频率，ξi、ξj分别为第i个和第j个振型的阻尼比，当ξi=ξj=ξ时，α、β可表示为：
ωi、ωj取系统的前两阶固有频率.
2.2　Newmark法
采用时域隐式Newmark平均常加速度法对控制方程式进行数值求解. 由此，动态问题在每一步都将转化为静态问题求解. 在t～t+Δt的时间区域内，由加速度和速度的定义式：
其中，0≤γ≤1，0≤β≤0.5，Δt根据解的精度确定时间步长. 本文取γ=0.5，β=0.25，Δt=1×10-5 s.
联立求解得到、，再代入t+Δt时刻的运动方程得：
由此可计算出t+Δt时刻的位移，进而求出t+Δt时的速度及加速度.
2.3　完全转换法
由于基于移动最小二乘近似的无网格法中节点未知量是节点参数而非真实节点位移，不能直接施加边界条件. 为解决这一难题，研究学者提出了几种不同的方法，较典型的有：Lagrange乘子法，罚函数法，修正变分原理以及完全转换法等. 本文选用由Chen等[28]提出的完全转换法（Full transformation method）对任意壳的受迫振动控制方程进行修正.
考虑到由移动最小二乘近似给出的近似位移uh（r），当离散节点足够多时，便会收敛于真实位移，即：
式中，u1为节点参数，
由式（40）得：
此时，忽略阻尼、以真实位移为未知量的受迫振动控制方程便可写为，
式中，、、.
3　数值算例
收起
本文所有算例均采用圆形影响域，取dmax=5，采用高斯积分为3×3×2，基函数取m=6，权函数选择三次样条函数. 材料属性为钢材，其弹性模量E=210 GPa，泊松比ν=0.3，密度ρ=7850 kg/m2. t1=0.006 s. 文中考虑了无阻尼和有阻尼（0.05）两种情况下平板、斜板、屋顶壳、双曲扁壳及波纹板五种经典板壳形式的受迫振动，给出了矩形和三角形脉冲荷载的一个时程曲线，荷载的相应表达式如下：
矩形脉冲荷载
正弦脉冲荷载
ω1为基础频率（第一阶），单位是赫兹.
3.1　平板
平板是壳结构形式的一种特殊形式. 如图2所示一块长a=2 m，宽b=2 m的四边固支方板，表面承受横向均布荷载150 MPa. ABAQUS建模选用8节点壳单元（S8R），网格划分为400个单元，共计1281个节点. 研究平板中点挠度ω在矩形、正弦脉冲荷载作用下随时间的变化. 采用本文方法与有限元法计算得到的平板前十阶自振频率及二者误差如表1所示，脉冲荷载作用下平板中点挠度值的计算与比较如图3、图4所示. 从计算结果来看，本文解法求解平板的受迫振动问题具有计算精度高，收敛速度快特点. 当11×11无网格节点时，计算结果已经收敛，并且本文解法所获得的时程曲线与有限元解数据十分吻合，证明了本文方法求解平板受迫振动问题具有良好的收敛性及准确性.
3.2　斜板
如图5所示，四边铰接的斜板a=2 m， m，倾斜角ϕ=45°，厚度h=0.2 m. 表面承受横向荷载150 MPa. ABAQUS建模选择8节点壳单元（S8R），共计15409个节点. 本文方法与有限元法的求解对比计算结果分别绘制于表2、图6、图7. 由计算结果可知，采用本文方法所获得的斜板前十阶频率与有限元结果误差均小于0.5%，且时程曲线吻合良好，说明了本文方法求解斜板受迫振动问题的有效性.
3.3　屋顶壳
如图8所示，曲边固支的屋顶壳R=2 m，弧度角θ=24°，厚度h=0.1 m，长度L=3 m. 表面承受径向均布荷载150 MPa，讨论屋顶壳中点挠度在矩形、正弦脉冲荷载作用下随时间的变化. ABAQUS建模选择8节点壳单元（S8R），共计4357个节点数，相关计算结果如表3、图9、图10所示. 由图表可知，本文解与有限元解的波纹板自振频率、时程曲线均十分吻合，说明本文方法适用于求解柱壳受迫振动问题，且具有较高的计算精度.
3.4　双曲扁壳
如图11所示，四边固支的双曲扁壳，其长度L=2 m，厚度h=0.2 m. 表面承受径向均布荷载150 MPa. ABAQUS建模选择8节点壳单元（S8R），共计3605个节点. 表4展示了本文解法与有限元法计算双曲扁壳自振频率以及二者的相对误差，双曲扁壳在脉冲荷载作用下随时间变化的中点挠度值ω如图12-13所示. 经和有限元计算结果对比，证明了本文方法求解双曲扁壳受迫振动问题具有良好的收敛性及准确性.
3.5　波纹板
如图14所示，直边固支的正弦波波纹板，直边板长L=0.8 m，曲边宽W=0.8 m，板厚h=0.03 m，半波长C=0.4 m，共1个完整波纹，幅值F=0.05 m. 表面承受径向均布荷载15 MPa，讨论波纹板中点挠度在矩形、正弦脉冲荷载作用下随时间变化. ABAQUS建模时选择8节点壳单元（S8R），共计29193个节点，计算结果分别绘制于表5、图15、图16所示. 由图表可知，采用本文方法求解的波纹板前十阶频率与有限元计算误差在2%以内，波纹板中点挠度时程曲线与有限元解结果也比较吻合. 随着幅值的增大，本文方法可以通过设置更多的节点数来保证计算结果的精度.
4　结论
收起
本文基于3D连续壳理论，利用映射技术将三维参数化曲面转换成二维无网格模型. 基于移动最小二乘近似和一阶剪切变形理论给出任意壳的位移场. 采用完全转换法处理本质边界条件，利用时域隐式Newmark法求解了任意壳受迫振动的控制方程，并将计算结果与ABAQUS有限元计算结果对比，研究表明：
（1）本文方法计算精度比较高，计算结果随着节点数的增加迅速收敛.
（2）本文方法计算所得几个经典板壳形状的前十阶自振频率与有限元解误差大部分均在2%以内，说明了本文方法求解任意壳自振频率的准确性.
（3）本文方法计算得到的中点挠度时程曲线与有限元解吻合良好，证明了本文方法求解任意壳受迫振动的有效性. 整体来看不考虑阻尼计算得到的挠度值要比考虑阻尼计算得到的挠度值更接近于有限元解.
（4）运用无网格法计算前处理比较简单，相对于有限元而不依赖于网格的划分，可以有效地计算各种不同形状的壳体结构形式，适应性比较强，具有广阔的应用前景.
基金
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  • 国家自然科学基金项目(12162004)
  • 国家重点研发计划项目(2019YFC1511103)
  • 广西重点研发计划(桂科AB22036007)
文献
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doi: 10.19636/j.cnki.cjsm42-1250/o3.2023.037
  • 接收时间：2023-08-01
  • 首发时间：2026-04-01
  • 出版时间：2024-04-25
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  • 收稿日期：2023-08-01
基金
国家自然科学基金项目(12162004)
国家重点研发计划项目(2019YFC1511103)
广西重点研发计划(桂科AB22036007)
作者信息
    1广西大学土木建筑工程学院，南宁，530004
    2广西大学工程防灾与结构安全教育部重点实验室，南宁，530004
    3南华大学土木工程学院，衡阳，421001

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2种不同金属材料的力学参数
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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
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