The analytic free vibration solutions of triangular plates are important for both rapid analyses and preliminary designs of similar structures. Due to the difficulty in solving the complex boundary value problems of the governing high-order partial differential equations, the current knowledge about the analytic solutions is limited. This study presents a first attempt to explore an up-to-date symplectic superposition method for analytic free vibration solutions of right triangular plates. Specifically, an original problem is regarded as the superposition of three fundamental subproblems of the corresponding rectangular plates that are solved by the symplectic eigenexpansion within the Hamiltonian-system framework, involving the coordinate transformation. The analytic frequency and mode shape solutions are then obtained by the requirement of the equivalence between the original problem and the superposition. By comparison with the numerical results for the right triangular plates under six different combinations of clamped and simply supported boundary constraints, the fast convergence and high accuracy of the present approach are well confirmed. Within the current solution framework, the extension to the problems of more polygonal plates is possible.

三角板的解析自由振动解对于相似结构的快速分析和初步设计都非常重要。由于难以解决控制高阶偏微分方程的复杂边值问题，因此目前关于解析解的知识有限。这项研究提出了探索直角三角板解析自由振动解的最新辛叠加方法的首次尝试。具体而言，原始问题被视为相应矩形板的三个基本子问题的叠加，这些子问题由哈密顿系统框架内的辛本征展开解决，涉及坐标变换。然后，根据原始问题和叠加之间的等价要求，获得解析的频率和模式形状解。通过与在六个不同组合的夹紧和简单支撑的边界约束下的直角三角形板的数值结果进行比较，可以很好地确认本方法的快速收敛性和高精度。在当前的解决方案框架内，可以扩展更多多边形板的问题。