Abstract The symplectic approach is used to establish the unified framework for solving governing equations of some thin plate problems in elasticity. By introducing appropriate functions, the given partial differential equation is first transferred into a separable Hamiltonian system. The completeness of the system of generalized eigenvectors of corresponding Hamiltonian operator matrix is proved, which serves as the theoretical foundation of symplectic approach. Utilizing the expansion theorems, the general solutions of the boundary value problem for partial differential equation under consideration are obtained. Moreover, the bending, buckling, and free vibration problems of fully clamped rectangular thin plates governed by the given partial differential equation are solved analytically by the technique of superposition. Numerical results for bending, buckling, and free vibration plates are presented to demonstrate the availability and validity of the approach by comparison with those available in the literatures.

中文翻译：

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一类由弹性引起的偏微分方程的完备辛方法

摘要 采用辛法建立了求解某些薄板弹性问题控制方程的统一框架。通过引入适当的函数，给定的偏微分方程首先转化为可分离的哈密顿系统。证明了对应哈密顿算子矩阵的广义特征向量系统的完备性，为辛方法的理论基础。利用展开定理，得到了所考虑的偏微分方程边值问题的一般解。此外，由给定的偏微分方程控制的完全夹紧的矩形薄板的弯曲、屈曲和自由振动问题通过叠加技术进行解析求解。