Abstract In this study, a first attempt is made to develop an up-to-date symplectic superposition method for some new analytic free vibration solutions of side-cracked rectangular thin plates that were not obtained by conventional semi-inverse methods. In contrast with the classical Lagrangian system and Euclidean space, the present method is implemented within the Hamiltonian-system framework and symplectic space. The solution procedure involves expressing the problems in the Hamiltonian system and dividing a side-cracked plate into several sub-plates that are analytically solved by the symplectic superposition method, where the imposed quantities are determined by the plate boundary conditions, free edge conditions along the crack, and interfacial continuity conditions between the sub-plates. In the analytic solution of a sub-plate, specifically, the symplectic eigenvalue problems are formulated, followed by the symplectic eigen expansion. The integration of the solutions of the sub-plates yields the final solution of a side-cracked plate. The rigorous mathematical techniques, without predetermination of solution forms, qualify the present method as an unusual approach for exploring more analytic solutions. Comprehensive natural frequency and mode shape solutions of the side-cracked plates under three representative boundary constraints are provided and well validated by other methods. The new analytic solutions obtained may serve as benchmarks for other potential solution methods.

中文翻译：

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侧裂矩形薄板新解析自由振动解的辛叠加法

摘要 在这项研究中，首次尝试开发一种最新的辛叠加方法，用于解决一些新的侧裂矩形薄板的解析自由振动解，这些解是通过传统的半逆方法无法获得的。与经典的拉格朗日系统和欧几里得空间相比，本方法是在哈密顿系统框架和辛空间内实现的。求解过程包括在哈密顿系统中表达问题，并将侧裂板分成几个子板，这些子板通过辛叠加法解析求解，其中施加的量由板边界条件、沿边界条件的自由边缘条件确定。子板之间的裂纹和界面连续性条件。在子板的解析解中，具体来说，辛本征值问题被公式化，然后是辛本征展开。子板解的积分产生侧裂板的最终解。严格的数学技术，无需预先确定解的形式，使本方法成为探索更多解析解的不寻常方法。提供了三种代表性边界约束下侧裂板的综合固有频率和振型解，并通过其他方法得到了很好的验证。获得的新解析解可以作为其他潜在解法的基准。严格的数学技术，无需预先确定解的形式，使本方法成为探索更多解析解的不寻常方法。提供了三种代表性边界约束下侧裂板的综合固有频率和振型解，并通过其他方法得到了很好的验证。获得的新解析解可以作为其他潜在解法的基准。严谨的数学技术，无需预先确定解的形式，使本方法成为探索更多解析解的不寻常方法。提供了三种代表性边界约束下侧裂板的综合固有频率和振型解，并通过其他方法得到了很好的验证。获得的新解析解可以作为其他潜在解法的基准。