Circular and annular elastic plates have wide applications as essential elements in various engineering structures and products demanding accurate analysis of their vibrations. At higher frequencies, the analysis of vibrations needs appropriate equations, as shown by the Mindlin plate equations for rectangular plates with tailored applications for the analysis of quartz crystal resonators. Naturally, there are equally strong demands for the equations and applications in circular and annular plates with the consideration of higher-order vibration modes. By following the procedure established by Mindlin based on displacement expansion in the thickness coordinate, a set of higher-order equations of vibrations of circular and annular plates are derived and truncated for comparisons with classical and first-order plate equations of circular plates. By utilizing these equations, coupled thickness-shear and flexural vibrations of circular and annular plates are analyzed for vibration frequencies and mode shapes.

中文翻译：

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基于Mindlin板理论的圆形和环形板的高频振动

圆形和环形弹性板作为各种工程结构和产品中必不可少的元件，具有广泛的应用，需要对其振动进行精确分析。在更高的频率下，振动分析需要合适的方程式，如矩形板的Mindlin板式方程式所示，该应用程序专门用于分析石英晶体谐振器。自然，考虑到高阶振动模式，对圆板和圆板的方程式和应用也同样有强烈的要求。通过遵循Mindlin基于厚度坐标上的位移扩展建立的过程，推导出一组圆盘和圆盘振动的高阶方程，并将其截断，以便与圆盘的经典和一阶板方程进行比较。