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

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

鲍四元¹,
邓子辰²

¹苏州科技学院土木工程学院工程力学系, 江苏苏州 215011;²西北工业大学工程力学系, 西安 710072

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
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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

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

BAO Si-yuan¹,
DENG Zi-chen²

¹Department of Engineering Mechanics, School of Civil Engineering, Suzhou University of Science and Technology, Suzhou, Jiangsu 215011, P.R.China;²Department of Engineering Mechanics, Northwestern Polytechnical University, Xi’an 710072, P.R.China

Funds: The National Natural Science Foundation of China（11202146）

摘要

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.
- Hamilton体系 /
- 薄板 /
- 自由振动 /
- 挠度 /
- 频率 /
- 对称性
Abstract: For free bending vibrations of thin plates, based on the variable separation method, the analytical solutions of deflection functions were obtained in the Hamiltonian system, and the eigenvalue equations about the 2 coordinate axes were established. Then the vibration frequencies were solved as parametrical variables, and the mode shapes of different orders were got. The analytical form of the deflection function was in fact a high-precision approximate solution satisfying the displacement boundary conditions. From the approximate frequency values calculated with the Rayleigh-Ritz method, it is found that the previous analytical method had great precision, thus the effectiveness was demonstrated. In addition, for clamped or simply supported boundary conditions, different points were properly selected as the coordinate origins. The unified forms of the frequency equations were given. These forms were used to discuss the rectangular plates with 4 edges clamped, or 4 edges simply supported, or some clamped and the other(s) simply supported, and so on. The correctness of the frequency equation forms was verified with the symmetry of the plates’ deformations. The linkage and transformation between the frequency equations under different boundary conditions were obtained.
- Hamiltonian system /
- thin plate /
- free vibration /
- deflection /
- frequency /
- symmetry

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参考文献(18)

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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

doi: 10.21656/1000-0887.370005

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）．

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

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留言板

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

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）．

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出版历程

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

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出版历程

目录

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）．