Abstract Vibration analysis of nominally axisymmetric plate structures with either imperfections or geometric asymmetries due to practical motivations is of interest in designing and developing some mechanical structures. Semi-analytical methods to model such structures suffer from either choosing inappropriate admissible functions or both plausible convergence issues and additional computations owing to employing the addition theorem of Bessel functions. Therefore, the present study aims at developing a new mathematical method to analyze the vibrational behavior of circular plates with geometric asymmetries. The suggested approach makes use of the separation of variables to determine general solutions of the partial differential equation of the plate transverse displacement while defining multiple polar coordinate systems each of which offers a formulation of the plate deformation. Moreover, closed-form geometric equations and the chain rule for determining derivatives are implemented to move from one coordinate system to the other to satisfy boundary conditions without any need for the cumbersome transformation involved in using the addition theorem. A finite element model is also constructed to evaluate the validity of the proposed method before studying the effects of the cutout location and size on natural frequencies and mode shapes of eccentric annular plates.

中文翻译：

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一种分析偏心环形板自由振动的新数学方法

摘要 由于实际动机具有缺陷或几何不对称的名义轴对称板结构的振动分析是设计和开发一些机械结构的兴趣所在。由于采用了贝塞尔函数的加法定理，对此类结构进行建模的半解析方法要么选择不合适的容许函数，要么面临可能的收敛问题和额外计算的问题。因此，本研究旨在开发一种新的数学方法来分析具有几何不对称性的圆板的振动行为。建议的方法利用变量的分离来确定板横向位移偏微分方程的一般解，同时定义多个极坐标系，每个极坐标系都提供了板变形的公式。此外，闭式几何方程和用于确定导数的链式法则被实现为从一个坐标系移动到另一个坐标系以满足边界条件，而无需使用加法定理所涉及的繁琐变换。在研究切口位置和尺寸对偏心环形板的固有频率和振型的影响之前，还构建了一个有限元模型来评估所提出方法的有效性。封闭形式的几何方程和用于确定导数的链式法则被实施以从一个坐标系移动到另一个坐标系以满足边界条件，而无需使用加法定理所涉及的繁琐变换。在研究切口位置和尺寸对偏心环形板的固有频率和振型的影响之前，还构建了一个有限元模型来评估所提出方法的有效性。封闭形式的几何方程和用于确定导数的链式法则被实施以从一个坐标系移动到另一个坐标系以满足边界条件，而无需使用加法定理所涉及的繁琐变换。在研究切口位置和尺寸对偏心环形板的固有频率和振型的影响之前，还构建了一个有限元模型来评估所提出方法的有效性。