Plates with rectangular cutouts is widely seen in the field of engineering structures. Therefore, it is crucial to examine analytical solutions for free vibration (FV) of these structures. Despite the existence of approximate/numerical methods, analytical solutions are lacking in the literature. In this study, we employ the symplectic superposition method to effectively analyze the FV problems encountered in plates with rectangular cutouts while integrating the domain decomposition technique. To address the issue of irregular geometry, the rectangular cutout plate is divided into multiple sub-plates. By dividing the problem into multiple sub-problems and solving them separately using variable separation and symplectic eigen expansion, we obtain analytical solutions. Finally, we combine the sub-problem solutions to resolve the initial issue. This solution method can be considered a logical, analytical, and systematic approach as it starts with the fundamental governing equation and is derived without assuming forms of solutions. The study presents a comprehensive set of numerical results that include mode shapes (MSs) and natural frequencies (NFs). The results are rigorously validated using the finite element method (FEM) and relevant literature. The symplectic superposition method demonstrates excellent convergence and precise accuracy, making it suitable for analytically modeling more complex mechanical problems of plates.

中文翻译：

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辛叠加法结合域分解技术解析矩形切口板的自由振动问题


具有矩形切口的板材在工程结构领域中随处可见。因此，检查这些结构的自由振动（FV）的解析解至关重要。尽管存在近似/数值方法，但文献中缺乏解析解。在本研究中，我们采用辛叠加方法，结合域分解技术，有效分析具有矩形切口的板中遇到的FV问题。为了解决不规则几何形状的问题，矩形切口板被分成多个子板。通过将问题划分为多个子问题并使用变量分离和辛特征展开分别求解它们，我们获得了解析解。最后，我们结合子问题的解决方案来解决初始问题。这种求解方法可以被认为是一种逻辑、分析和系统的方法，因为它从基本控制方程开始，并且在不假设解形式的情况下导出。该研究提供了一套全面的数值结果，包括振型 (MS) 和固有频率 (NF)。结果使用有限元法（FEM）和相关文献进行了严格验证。辛叠加方法表现出优异的收敛性和精确的精度，使其适合对更复杂的板力学问题进行分析建模。