Symplectic Solution System for Reissner Plate Bending
-
摘要: 基于Reissner板弯曲问题的Hellinger-Reissner变分原理,通过引入对偶变量,导出Reissner板弯曲的Hamilton对偶方程组.从而将该问题导入到哈密顿体系,实现从欧几里德空间向辛几何空间,拉格朗日体系向哈密顿体系的过渡.于是在由原变量及其对偶变量组成的辛几何空间内,许多有效的数学物理方法如分离变量法和本征函数向量展开法等均可直接应用于Reissner板弯曲问题的求解.这里详细求解出Hamilton算子矩阵零本征值的所有本征解及其约当型本征解,给出其具体的物理意义.形成了零本征值本征向量之间的共轭辛正交关系.可以看到,这些零本征值的本征解是Saint-Venant问题所有的基本解,这些解可以张成一个完备的零本征值辛子空间.而非零本征值的本征解是圣维南原理所覆盖的部分.新方法突破了传统半逆解法的限制,有广阔的应用前景.
-
关键词:
- Reissner板 /
- Hamilton体系 /
- 辛几何 /
- 分离变量
Abstract: Based on the Hellinger-Reissner variatonal principle for Reissner plate bending and introducing dual variables,Hamiltonian dual equations for Reissner plate bending were presented.Therefore Hamiltonian solution system can also be applied to Reissner plate bending problem,and the transformation from Euclidian space to symplectic space and from Lagrangian system to Hamilt onian system was realized.So in the symplectic space which consists of the original variables and their dual variables,the problem can be solved via effective mathematical physics methods such as the method of separation of variables and eigenfunction-vector expansion.All the eigensolutions and Jordan canonical form eigensolutions for zero eigenvalue of the Hamiltonian operator matrix are solved in detail,and their physical meanings are showed clearly.The adjoint symplectic orthonormal relation of the eigen-function vectors for zero eignevalue are formed.It is showed that the all eigensolutions for zero eigen-value are basic solutions of the Saint-Venant problem and they form a perfect symplectic subspace for zero eigenvalue.And the eigensolutions for nonzero eigenvalue are covered by the Saint-Venant theorem.The symplectic solution method is not the same as the classical semi-inverse method and breaks through the limit of the traditional semi-inverse solution.The symplectic solution method will have vast application.-
Key words:
- Reissner plate /
- Hamiltonian system /
- symplectic geometry /
- separation of variable
-
[1] 钟万勰.弹性力学求解新体系[M].大连:大连理工大学出版社,1995. [2] 姚伟岸、钟万勰.辛弹性力学[M].北京:高等教育出版社,2002. [3] YAO Wei-an,YANG Hai-tian.Hamiltonian system based Saint Venant solutions for multi-layered composite plane anisotropic plates[J].International Journal of Solids and Structures,2001,38(32):5807—5817. doi: 10.1016/S0020-7683(00)00371-1 [4] 钟万勰.应用力学对偶体系[M].北京:科学出版社,2002. [5] 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. [6] YAO Wei-an,ZHONG Wan-xie,SU Bin.New solution system for circular sector plate bending and its application[J].Acta Mechanica Solida Sinica,1999,12(4):307—315. [7] 姚伟岸,苏滨,钟万勰.基于相似性原理的正交各向异性板弯曲Hamilton体系[J].中国科学,E辑,2001,31(4):342—347. [8] 钟万勰,姚伟岸,郑长良.Reissner板弯曲与平面偶应力模拟[J].大连理工大学学报,2002,42(5):519—521.
计量
- 文章访问数: 2797
- HTML全文浏览量: 139
- PDF下载量: 1044
- 被引次数: 0