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薄板弯曲自由振动问题的高精度近似解析解及改进研究

鲍四元 邓子辰

鲍四元, 邓子辰. 薄板弯曲自由振动问题的高精度近似解析解及改进研究[J]. 应用数学和力学, 2016, 37(11): 1169-1180. doi: 10.21656/1000-0887.370005
引用本文: 鲍四元, 邓子辰. 薄板弯曲自由振动问题的高精度近似解析解及改进研究[J]. 应用数学和力学, 2016, 37(11): 1169-1180. doi: 10.21656/1000-0887.370005
BAO Si-yuan, DENG Zi-chen. High-Precision Approximate Analytical Solutions for Free Bending Vibrations of Thin Plates and an Improvement[J]. Applied Mathematics and Mechanics, 2016, 37(11): 1169-1180. doi: 10.21656/1000-0887.370005
Citation: BAO Si-yuan, DENG Zi-chen. High-Precision Approximate Analytical Solutions for Free Bending Vibrations of Thin Plates and an Improvement[J]. Applied Mathematics and Mechanics, 2016, 37(11): 1169-1180. doi: 10.21656/1000-0887.370005

薄板弯曲自由振动问题的高精度近似解析解及改进研究

doi: 10.21656/1000-0887.370005
基金项目: 国家自然科学基金(11202146);江苏省青蓝工程基金
详细信息
    作者简介:

    鲍四元(1980—),男,副教授,博士(E-mail: bsiyuan@126.com);邓子辰(1964—),男,教授,博士生导师(通讯作者. E-mail: dweifan@nwpu.edu.cn).

  • 中图分类号: O343

High-Precision Approximate Analytical Solutions for Free Bending Vibrations of Thin Plates and an Improvement

Funds: The National Natural Science Foundation of China(11202146)
  • 摘要: 对于薄板弯曲自由振动问题,已有如下方法:在Hamilton(哈密顿)体系下基于分离变量法得到挠度的解析形式,并建立自振频率联立方程组,给出求解振动频率和振型函数的方法.笔者指出该方法中所用挠度函数的解析式实际上是一种满足位移边界条件的高精度近似解,基于Rayleigh-Ritz(瑞利-里茨)法再次求近似频率后发现,原方法的近似解的精度很高.另外,对于含有固支、简支等不同的边界形式,恰当地选取不同位置作为坐标系的原点,得到含有频率的方程组的统一形式,且较为简洁.这些形式可基于四边固支、四边简支等边界条件的矩形板研究,依照板变形的对称性可验证频率方程组形式的正确性,并得到不同边界条件下频率方程形式之间的联系与转化.
  • [1] 黄炎. 弹性薄板理论[M]. 长沙: 国防科技大学出版社, 1992.(HUANG Yan. Theory of Elastic Thin Plates [M]. Changsha: National University of Defense Technology Press, 1992.(in Chinese))
    [2] 曲庆璋, 章权, 季求和, 梁兴复. 弹性板理论[M]. 北京: 人民交通出版社, 2000.(QU Qin-zhang, ZHANG Quan, JI Qiu-he, LIANG Xing-fu. Theory of Elastic Plates [M]. Beijing: China Communications Press, 2000.(in Chinese))
    [3] 付宝连. 弹性力学混合变量的变分原理及其应用[M]. 北京: 国防工业出版社, 2010.(FU Bao-lian. Variational Principles With Mixed Variables in Elasticity and Their Applications [M]. Beijing: National Defence Industry Press, 2010.(in Chinese))
    [4] 钟万勰. 应用力学对偶体系[M]. 北京: 科学出版社, 2002.(ZHONG Wan-xie. The Dual System of Applied Mechanics [M]. Beijing: Science Press, 2002.(in Chinese))
    [5] Yao W A, Zhong W X, Lim C W. Symplectic Elasticity [M]. Singapore: World Scientific, 2009.
    [6] 杨有贞, 葛修润, 王燕昌. 哈密顿辛对偶体系理论与梁、板结构弯曲模拟[M]. 北京: 科学出版社, 2012.(YANG You-zhen, GE Xiu-run, WANG Yan-chang. An Analysis Method of Symplectic Dual System for Beams/Plated Structures [M]. Beijing: Science Press, 2012.(in Chinese))
    [7] 钟万勰, 姚伟岸. 板弯曲求解新体系及其应用[J]. 力学学报, 1999,31(2): 173-184.(ZHONG Wan-xie, YAO Wei-an. New solution system for plate bending and its application[J]. Acta Mechanica Sinica,1999,31(2): 173-184.(in Chinese))
    [8] 鲍四元, 邓子辰. 哈密顿体系下矩形薄板自由振动的一般解[J]. 动力学与控制学报, 2005,3(2): 10-16.(BAO Si-yuan, DENG Zi-chen. A general solution of free vibration for rectangular thin plates in Hamilton system[J]. Journal of Dynamics and Control,2005,3(2): 10-16.(in Chinese))
    [9] 鲍四元, 邓子辰. Hamilton体系下中厚板静力弯曲和自由弯曲振动问题的一类模型及通解[J]. 动力学与控制学报, 2012,10(2): 121-126.(BAO Si-yuan, DENG Zi-chen. A type of model and ordinary solutions for static bending and free vibration problems of moderate thick in Hamilton system[J]. Journal of Dynamics and Control,2012,10(2): 121-126.(in Chinese))
    [10] Lim C W, Lü C F, Xiang Y, Yao W. On new symplectic elasticity approach for exact free vibration solutions of rectangular Kirchhoff plates[J]. International Journal of Engineering Science,2009,47(1): 131-140.
    [11] 姚伟岸, 蔡智宇, 胡小飞. 矩形正交各向异性薄板弯曲受迫振动问题的分析解[J]. 动力学与控制学报, 2011,9(1): 12-17.(YAO Wei-an, CAI Zhi-yu, HU Xiao-fei. Analytical solution for the forced vibration of orthotropic rectangular thin plates[J]. Journal of Dynamics and Control,2011,9(1): 12-17.(in Chinese))
    [12] 钟阳, 李锐, 田斌. 四边固支矩形薄板自由振动的哈密顿解析解[J]. 应用力学学报, 2011,28(4): 323-327.(ZHONG Yang, LI Rui, TIAN Bing. Analytic free vibration solution of fully clamped rectangular thin plate[J]. Chineses Journal of Applied Mechanics,2011,28(4): 323-327.(in Chinese))
    [13] XING Yu-feng, LIU Bo. New exact solutions for free vibrations of rectangular thin plates by symplectic dual method[J]. Acta Mechanica Sinica,2009,25(2): 265-270.
    [14] Xing Y F, Liu B. New exact solutions for free vibrations of thin orthotropic rectangular plates[J]. Composite Structures,2009,89(4): 567-574.
    [15] 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.(XU Teng-fei, XING Yu-feng. Exact solutions for free vibrations of rectangular thin plates on the Winkler foundation[J]. Engineering Mechanics,2014,31(5): 43-48.(in Chinese))
    [16] 邢誉峰, 刘波. 板壳自由振动问题的精确解[M]. 北京: 科学出版社, 2015.(XING Yu-feng, LIU Bo. Exact Solutions for Free Vibrations of Plates and Shells [M]. Beijing: Science Press, 2015.(in Chinese))
    [17] Bahrami A, Bahrami M N, Ilkhani M R. Comments on “New exact solutions for free vibrations of thin orthotropic rectangular plates”[J]. Composite Structures,2014,107: 745-746.
    [18] LI Rui, WANG Peng-cheng, TIAN Yu, WANG Bo, LI Gang. A unified analytic solution approach to static bending and free vibration problems of rectangular thin plates[J]. Scientific Reports,2015(5): 17054. doi: 10.1038/srep17054.
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出版历程
  • 收稿日期:  2016-01-04
  • 修回日期:  2016-06-30
  • 刊出日期:  2016-11-15

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