工程力学 ›› 2019, Vol. 36 ›› Issue (3): 24-32,39.doi: 10.6052/j.issn.1000-4750.2018.01.0006

• 基本方法 • 上一篇    下一篇

平行四边形加肋板自由振动分析的无网格法

覃霞1, 刘珊珊1, 吴宇1, 彭林欣1,2   

  1. 1. 广西大学土木建筑工程学院, 南宁 530004;
    2. 广西防灾减灾与工程安全重点实验室 工程防灾与结构安全教育部重点实验室, 广西大学, 南宁 530004
  • 收稿日期:2018-01-03 修回日期:2018-06-20 出版日期:2019-03-29 发布日期:2019-03-16
  • 通讯作者: 彭林欣(1977-),男,湖南武冈人,教授,博士,博导,主要从事结构力学研究(E-mail:penglx@gxu.edu.cn). E-mail:penglx@gxu.edu.cn
  • 作者简介:覃霞(1989-),女,广西河池人,硕士生,主要从事计算结构力学研究(E-mail:Sarah0901@yeah.net);刘珊珊(1993-),女,广西玉林人,硕士生,主要从事计算结构力学研究(E-mail:1654858312@qq.com);吴宇(1993-),男,陕西汉中人,硕士生,主要从事计算结构力学研究(E-mail:395937471@qq.com)
  • 基金资助:
    国家自然科学基金项目(11562001,11102044)

FREE VIBRATION ANALYSIS OF RIBBED SKEW PLATES WITH A MESHFREE METHOD

QIN Xia1, LIU Shan-shan1, WU Yu1, PENG Lin-xin1,2   

  1. 1. Civil Engineering and Architecture School, Guangxi University, Nanning 530004, China;
    2. Key Laboratory of Disaster Prevention and Structural Safety of China Ministry of Education, Guangxi Key Laboratory of Disaster Prevention and Engineering Safety of Guangxi University, Guangxi University, Nanning 530004, China
  • Received:2018-01-03 Revised:2018-06-20 Online:2019-03-29 Published:2019-03-16

摘要: 基于一阶剪切理论,提出一种求解平行四边形加肋板自由振动问题的无网格法,通过用一系列点来离散平板及肋条,得到加肋板的无网格模型。基于一阶剪切理论及移动最小二乘近似求出位移场,以梁模拟肋条,求出平行四边形加肋板总动能及总势能。再由Hamilton原理导出加肋板自由振动的控制方程,采用完全转换法引入边界条件,求解方程得出结构自振频率。以不同参数的加肋板为例,将该文解与ABAQUS有限元解进行比较分析。研究表明,该方法能有效地分析平行四边形加肋板自由振动问题,在肋条位置改变时,又避免了网格重构。

关键词: 无网格法, 平行四边形加肋板, 一阶剪切理论, 自由振动, 完全转换法

Abstract: Based on the first-order shear deformation theory (FSDT), a meshfree method for solving the free vibration problem of ribbed skew plates is proposed. The plates and ribs are discretized with a series of points to obtain a meshfree model of the stiffened plate. The FSDT and the moving least-squares approximation are used to establish the displacement field. The total dynamic energy and total potential energy of the stiffened plate are obtained by simulating the ribs as beams. The governing equation for the free vibration of the stiffened plate is derived by the Hamilton principle. The boundary condition is introduced by the full transformation method, and the free vibration frequencies are solved. Several examples are calculated, and the results given by the proposed method are compared with those from other researches or ABAQUS. The results show that the method can effectively analyze the free vibration problem of ribbed skew plates and can avoid the redistribution of plate nodes when the rib position changes.

Key words: meshfree method, ribbed skew plate, first-order shear deformation theory, free vibration, full transformation method

中图分类号: 

  • TU311.4
[1] 宋超. 薄板壳屈曲分析的埃尔米特无网格法[C]//中国计算力学大会2014暨第三届钱令希计算力学奖颁奖大会论文集. 北京:中国力学学会计算力学专业委员会, 2014:1. Song Chao. The ermit meshless method for buckling analysis of thin plate shell[C]//China Computational Mechanics Conference 2014 And The third Proceedings of Qian Lingxi's Computational Mechanics Awards. Beijing:China Mechanical Society of Computational Mechanics Professional Committee, 2014:1. (in Chinese)
[2] 殷宇. 层合板壳结构的无网格方法分析及其应用[D]. 苏州:苏州大学, 2013. Ying Yu. The analysis and application of meshless method for laminated plate and shell strctures[D]. Suzhou:Suzhou University, 2013. (in Chinese)
[3] 龚曙光, 曾维栋, 张建平. Reissner-Mindlin板壳无网格法的闭锁与灵敏度分析及优化的研究[J]. 工程力学, 2011, 28(4):42-48. Gong Shuguang, Zeng Weidong, Zhang Jianping. Study on numerical locking and sensitivity analysis and optimization of reissner-mindlin plate and shell with meshless method[J]. Engineering Mechanics, 2011, 28(4):42-48. (in Chinese)
[4] 彭林欣, 柏挺. 波纹夹层板线性弯曲分析的无网格伽辽金法[J]. 工程力学, 2011, 28(8):17-22. Peng Linxin, Bai Ting. Bending analysis of corrugatedcore sandwich plates by the element-free galerkin method[J]. Engineering Mechanics, 2011, 28(8):17-22. (in Chinese)
[5] 杨柳, 彭建设. 解平行四边形板弯曲问题的GD法[J]. 成都大学学报(自然科学版), 2014(3):230-233. Yang Liu, Peng Jiancheng. Necessary and sufficient conditions of metapositive definite matrices[J]. Journal of Chengdu University (Natural Science Edition), 2014(3):230-233. (in Chinese)
[6] 王克林, 李璐, 汤翔, 等. 有自由边的各向异性平行四边形板的弯曲、振动与屈曲的傅里叶分析[J]. 工程力学, 2008, 25(3):31-37. Wang Kelin, Li Lu, Tang Xiang, et al. Free vibration, buckling and bending analyses of anisotropic skew plates with free edges using fourier series[J]. Engineering Mechanics, 2008, 25(3):31-37. (in Chinese)
[7] 方电新, 李卧东, 王元汉, 等. 用无网格法计算平板弯曲问题[J]. 岩土力学, 2001, 22(3):347-349. Fang Dianxin, Li Wodong, Wang Yuanhan, et al. Calculating plate bending problem by meshless method[J]. Rock and Soil Mechanics, 2001, 22(3):347-349. (in Chinese)
[8] 谢根全, 刘行. 基于无网格Local Petrov-Galerkin法的变厚度薄板的弯曲分析[J]. 工程力学, 2013, 30(5):19-23, 62. Xie Genquan, Liu Xing. The bending of thin plates with varying thickness based on meshless local PetrovGalerkin method[J]. Engineering Mechanics, 2013, 30(5):19-23, 62. (in Chinese)
[9] 谭飞, 张友良. 弹性地基板弯曲的杂交边界点法[J]. 工程力学, 2013, 30(4):35-41. Tan Fei, Zhang Youliang. A hybrid boundary node method for the bending of plates on elastic foundation[J]. Engineering Mechanics, 2013, 30(4):35-41. (in Chinese)
[10] 曾祥勇, 张鹞, 邓安福. 双参数地基上Kirchhoff板计算的无网格自然单元法[J]. 工程力学, 2008, 25(5):196-201. Zeng Xiangyong, Zhang Yao, Deng Anfu. Natural element method for computation of Kirchhoff plate bending on two-parameter soil foundation[J]. Engineering Mechanics, 2008, 25(5):196-201. (in Chinese)
[11] 李顶河, 徐建新, 卿光辉. Hamilton体系下含弱粘接复合材料层合板的无网格求解方法[J]. 工程力学, 2012, 29(2):9-15. Li Dinghe, Xu Jianxin, Qing Guanghui. Meshless method of composite laminated plates with bonding interfacial imperfections in Hamilton system[J]. Engineering Mechanics, 2012, 29(2):9-15. (in Chinese)
[12] McGee O G. Flexural vibrations of clamped-free rhombic plates with corner stress singularities, Part 1:Review of research[J]. Journal of Vibration & Control, 2015, 21:2583-2603.
[13] McGee O G. Flexural vibrations of clamped-free rhombic plates with corner stress singularities, Part Ⅱ:comparison of solutions[J]. Journal of Vibration & Control, 2015, 21(13):2639-2660.
[14] Zhou L, Zheng W X. Vibration of skew plates by the MLS-Ritz method[J]. International Journal of Mechanical Sciences, 2008, 50(7):1133-1141.
[15] Jin C, Wang X. Weak form quadrature element method for accurate free vibration analysis of thin skew plates[J]. Computers & Mathematics with Applications, 2015, 70(8):2074-2086.
[16] 刘灿礼, 袁丽芸, 王俊鹏, 等. 传递矩阵法分析平行四边形板的自由振动问题[J]. 广西科技大学学报, 2017, 28(2):99-105. Liu Canli, Yuan Liyuan, Wang Junpeng, et al. Analysis of free vibration of paralled quadrilateral plate by transfer matrix method[J]. Journal of Guangxi University of Science and Technology, 2017, 28(2):99-105. (in Chinese)
[17] 李兴辉. 应用边界积分法求解弯曲厚矩形板的固有频率[D]. 秦皇岛:燕山大学, 2016. Li Xinghui. Applied to the boundary integral method to solve the natural frequency of bending thick rectangular plates[D]. Qinhuangdao:Yanshan University, 2016. (in Chinese)
[18] 曾军才, 王久法, 姚望, 等. 正交各向异性矩形板的自由振动特性分析[J]. 振动与冲击, 2015, 34(24):123-127, 143. Zeng Juncai, Wang Jiufa, Yao Wang, et al. Free vibration characteristics of orthotropic rectangular plates[J]. Journal of Vibration and Shock, 2015, 34(24):123-127, 143. (in Chinese)
[19] 彭林欣. 加肋板自由振动的移动最小二乘无单元分析[J]. 振动与冲击, 2011, 30(6):67-73. Peng Linxin. Moving-least-square meshless analysis on free vibration behavior of ribbed plates[J]. Journal of Vibration and Shock, 2011, 30(6):67-73. (in Chinese)
[20] Liew K M, Xiang Y, Kitipornchai S, et al. Formulation of Mindlin-Engesser model for stiffened plate vibration[J]. Computer Methods in Applied Mechanics and Engineering, 1995, 120(3/4):339-353.
[21] Belytschko T, Krongauz Y, Organ D, et al. Meshless methods:An overview and recent developments[J]. Computer Methods in Applied Mechanics and Engineering, 1996, 139(1/2/3/4):3-47.
[22] Li S, Liu W K. Meshfree and particle methods and their applications[J]. Applied Mechanics Reviews, 2002, 55(1):1-34.
[23] Liu G R. Mesh free methods:moving beyond the finite element method[M]. Boca Raton:CRC Press, 2002:10-12.
[24] Belytschko T, Lu Y Y, Gu L. Element-free Galerkin methods[J]. International journal for numerical methods in engineering, 1994, 37(2):229-256.
[25] Reddy J N. Theory and analysis of elastic plates and shells[M]. Boca Raton:CRC Press, 2006:23-25.
[26] Reddy J N. Theory and analysis of elastic plates[M]. London:Taylor & Francis, 1999:120-125.
[27] Chen J S, Pan C, Wu C T, et al. Reproducing kernel particle methods for large deformation analysis of non-linear structures[J]. Computer Methods in Applied Mechanics and Engineering, 1996, 139(1/2/3/4):195-227.
[28] Civalek Ö. A four-node discrete singular convolution for geometric transformation and its application to numerical solution of vibration problem of arbitrary straight-sided quadrilateral plates[J]. Applied Mathematical Modelling, 2009, 33(1):300-314.
[29] Wang X, Yuan Z. Buckling analysis of isotropic skew plates under general in-plane loads by the modified differential quadrature method[J]. Applied Mathematical Modelling, 2018, 56(1):83-95.
[1] 马建军, 聂梦强, 高笑娟, 秦紫果. 考虑土体质量的Winkler地基梁非线性自由振动分析[J]. 工程力学, 2018, 35(S1): 150-155.
[2] 王峰, 林皋, 周宜红, 赵春菊, 周华维. 非均质材料的扩展无单元Galerkin法模拟[J]. 工程力学, 2018, 35(8): 14-20,66.
[3] 杨永宝, 危银涛. 弹性基础上正交各向异性圆柱壳的自由振动[J]. 工程力学, 2018, 35(4): 24-32.
[4] 叶康生, 曾强. 结构自由振动问题有限元新型超收敛算法研究[J]. 工程力学, 2017, 34(1): 45-50,68.
[5] 陈旭东, 叶康生. 中厚圆柱壳自由振动的动力刚度法分析[J]. 工程力学, 2016, 33(9): 40-48.
[6] 陈万吉, 任鹤飞. 基于新修正偶应力理论的Mindlin层合板自由振动分析[J]. 工程力学, 2016, 33(12): 31-37,43.
[7] 王志芬, 李春光, 刘丰, 郑宏. 基于欧拉插值的最小二乘混合配点法在弹性力学平面问题中的应用[J]. 工程力学, 2015, 32(9): 27-33,48.
[8] 周凤玺,马强,宋瑞霞. 含液饱和多孔二维梁的动力特性分析[J]. 工程力学, 2015, 32(5): 198-207.
[9] 马文涛,许艳,马海龙. 修正的内部基扩充无网格法求解多裂纹应力强度因子[J]. 工程力学, 2015, 32(10): 18-24.
[10] 任丽梅,刘建民,肖玉柱. 基于镜像激励的结构动力学系统的设计点激励[J]. 工程力学, 2015, 32(10): 233-238.
[11] 周一波, 李晓叶, 王效贵. 压电与导体双材料界面端的奇异性研究[J]. 工程力学, 2014, 31(8): 209-216.
[12] 任勇生,代其义. 考虑剪切变形旋转运动复合材料薄壁梁的 动力学特性[J]. 工程力学, 2014, 31(7): 215-222.
[13] 王振, 孙秦. 基于共旋三角形厚薄通用壳元的几何非线性分析[J]. 工程力学, 2014, 31(5): 27-33.
[14] 徐腾飞, 邢誉峰. 弹性地基上矩形薄板自由振动的精确解[J]. 工程力学, 2014, 31(5): 43-48.
[15] 买买提明·艾尼,热合买提江·依明. 现代数值模拟方法与工程实际应用[J]. 工程力学, 2014, 31(4): 11-18.
Viewed
Full text


Abstract

Cited

  Shared   
  Discussed   
No Suggested Reading articles found!
X

近日,本刊多次接到来电,称有不法网站冒充《工程力学》杂志官网,并向投稿人收取高额费用。在此,我们郑重申明:

1.《工程力学》官方网站是本刊唯一的投稿渠道(原网站已停用),《工程力学》所有刊载论文必须经本刊官方网站的在线投稿审稿系统完成评审。我们不接受邮件投稿,也不通过任何中介或编辑收费组稿。

2.《工程力学》在稿件符合投稿条件并接收后会发出接收通知,请作者在接到版面费或审稿费通知时,仔细检查收款人是否为“《工程力学》杂志社”,千万不要汇款给任何的个人账号。请广大读者、作者相互转告,广为宣传!如有疑问,请来电咨询:010-62788648。

感谢大家多年来对《工程力学》的支持与厚爱,欢迎继续关注我们!

《工程力学》杂志社

2018年11月15日