基于Hamilton体系的功能梯度矩形板自由振动问题的解析解

张继超; 钟心雨; 陈一鸣; 石越卿; 郭程洁; 李锐

doi:10.21656/1000-0887.440279

基于Hamilton体系的功能梯度矩形板自由振动问题的解析解

doi: 10.21656/1000-0887.440279

1. 大连理工大学力学与航空航天学院 工业装备结构分析优化与CAE软件全国重点实验室，辽宁大连 116024;
2. 西安交通大学航天航空学院多尺度力学医学交叉实验室，西安 710049

基金项目:

国家自然科学基金（12372067

12022209

11972103）

详细信息

作者简介:
张继超（1997—），男，硕士生(E-mail: 690484356@mail.dlut.edu.cn)；钟心雨（1999—），女，硕士生(E-mail: zhongxinyu@stu.xjtu.edu.cn)；陈一鸣（1999—），男，硕士生(E-mail: YimingChen@mail.dlut.edu.cn)；石越卿（1995—），男，博士生(E-mail: shiyueqing@mail.dlut.edu.cn)；郭程洁（1997—），女，博士生(E-mail: 1784971935@mail.dlut.edu.cn)；李锐（1985—），男，教授，博士，博士生导师(通讯作者. E-mail: ruili@dlut.edu.cn).

通讯作者:
李锐（1985—），男，教授，博士，博士生导师(通讯作者. E-mail: ruili@dlut.edu.cn).

中图分类号: O302
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出版历程
- 收稿日期: 2023-09-20
- 修回日期: 2023-11-05
- 网络出版日期: 2024-09-30

Hamiltonian SystemBased Analytical Solutions to Free Vibration Problems of Functionally Graded Rectangular Plates

1. State Key Laboratory of Structural Analysis, Optimization and CAE Software for Industrial Equipment, School of Mechanics and Aerospace Engineering, Dalian University of Technology, Dalian, Liaoning 116024, P.R.China;
2. Laboratory for Multiscale Mechanics and Medical Science, School of Aerospace Engineering, Xi’an Jiaotong University, Xi’an 710049, P.R.China

Funds:

The National Science Foundation of China(12372067

12022209

11972103)

摘要

摘要: 对于功能梯度矩形板的自由振动问题，寻求既满足高阶偏微分控制方程又满足各种非Lévy型边界条件的振型函数十分困难，这使得利用传统方法难以解析求解该类问题.该文拓展了近年来发展的基于Hamilton体系的辛叠加方法，将其成功应用于功能梯度矩形板自由振动问题的解析求解.求解方案将原问题拆分成子问题，并引入物理中性面消除了由于横向材料不均匀产生的拉弯耦合效应，采用在传统Lagrange体系中无法使用的分离变量、辛本征展开等数学方法对子问题进行求解，最后通过叠加获得了原问题的解答.辛叠加方法的优点是不需要事先假定解的形式，克服了传统半逆解法的限制，能够获得更多复杂问题的解析解.将该方法的求解结果与数值解进行对比，证明了其正确性，并在此基础上进行了定量的参数分析，研究了不同边界条件、材料分布和长宽比对板固有频率的影响.
- 功能梯度板 /
- 自由振动 /
- Hamilton体系 /
- 解析解
Abstract: Finding vibration mode functions satisfying both high-order partial differential governing equations and various non-Lévy-type boundary conditions is extremely challenging for the free vibration problems of functionally graded rectangular plates, making it difficult to analytically solve the problem with traditional methods. Herein, the newly developed Hamiltonian system-based symplectic superposition method was extended and successfully applied to analytical solutions to the free vibration problems of functionally graded rectangular plates. For the solution methodology, the original vibration problem was divided into sub-problems and the physical neutral plane was introduced to eliminate the stretching-bending coupling effect caused by the transversely non-uniform materials. The sub-problems were analytically solved with some mathematical techniques, i.e., the variable separation and the symplectic eigenvector expansion, which are not applicable in the traditional Lagrangian system. The final solution to an original vibration problem was obtained through the superposition of sub-problems. The symplectic superposition method has the advantage of not requiring the pre-defined solution forms, which overcomes the limitations of traditional semi-inverse methods and allows for obtaining analytical solutions to more complex problems. Comparison of the obtained solutions with the numerical solutions proves the accuracy of the presented method. On this basis, quantitative parameter analyses on the natural frequencies were conducted to reveal the effects of boundary conditions, material distributions and aspect ratios.
- functionally graded plate /
- free vibration /
- Hamiltonian system /
- analytical solution

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