Reissner板弯曲的辛求解体系  

Symplectic Solution System for Reissner Plate Bending

作　　者：姚伟岸[1] 隋永枫[1] 

机构地区：[1]大连理工大学工业装备结构分析国家重点实验室,大连116023

出　　处：《应用数学和力学》2004年第2期159-165,共7页Applied Mathematics and Mechanics

基　　金：国家自然科学基金资助项目(10172021);教委博士点专项基金资助项目(20010141024)

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。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 Hamiltonian 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 eigenfunction vectors for zero eignevalue are formed.It is showed that the all eigensolutions for zero eigenvalue 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.

关 键 词：REISSNER板 HAMILTON体系 辛几何 分离变量 

分 类 号：O343[理学—固体力学]