中文摘要：

根据平面弹性与薄板弯曲问题的相似性原理,极坐标系板弯曲的弯矩函数被引入作为原变量,并通过恰当的辛内积定义建立了环扇形薄板弯曲问题的一个辛几何空间.然后应用类Hellinger-Reissner变分原理,导出了辛几何空间的对偶方程,从而将环扇形薄板弯曲问题导入到辛对偶求解体系.于是,分离变量和本征展开的有效数学物理方法得以实施,给出环扇形薄板弯曲问题的一个分析求解方法.具体讨论了两弧边简支和两弧边一边固支一边自由薄板的本征问题,分别导出它们对应的本征值超越方程和本征向量,并给出原问题本征展开形式的通解.最后,给出了两个算例的分析解并与已有文献或数值方法的解进行了对比,结果表明该方法有很好的收敛性和精度.

英文摘要：

The bending of annular sector plates is investigated with the symplectic solution in this paper. On the analogy between plane elasticity and plate bending theory, moment functions of plate bending in polar coordinates are introduced as original variables to establish a symplectic space in circumferential direction for the bending of annular sector plate with an appropriate definition of symplectic inner product. With the application of the Pro-Hellinger-Reissner variational principle, dual equations in the symplectic space are yielded to present an analytical method for the bending of annular sector plates. Moreover, two cases of the symplectic eigenvalue problems are presented： one is for the plate simply supported on the two arc sides, the other for the arc cantilever plate.. The transcendental equations of the eigenvalues and their eigenvectors are obtained. The comparison between the theoretical results in the paper and in other literatures shows a good convergence and accuracy of the symplectic solution.