Higher-order Mindlin Plate Equations And Free Vibrations Of Circular Elastic Plates

Circular Elastic plates are typical elements of structures which are extensively used in engineering applications and analyzed with the classical plate theory. The classical plate theory, however, will not be enough when we analyze the vibrations of thicker plates or the thicknessshear and higher-order overtone modes of plates. In such situations, we need to use the Mindlin and Lee higher-order plate theories, which already have a complete, analytical procedure through the research of the high frequency vibrations of rectangular plates in Cartesian coordinates. In this paper, we will derive the Mindlin higher-order plate equations of the circular elastic plates in polar coordinates, which will also be truncated, modified, and simplified for the analysis of rectangular plates.In this study, we considered high frequency vibrations of infinite plates and obtained the exact dispersion relations in rectangular and cylindrical coordinates, respectively. Then, we extended the systematic analysis of high frequency vibrations of plates from rectangular type to circular plates and derived the two-dimensional equations of circular plates by expanding the displacements in an infinite series of powers of the thickness-coordinate of elastic plates in polar coordinates. After truncating, modifying, and simplifying the higher-order equations, we obtained the first-order Mindlin plate equations, which can be degenerated to the classical plate equations by following the procedure of Mindlin in rectangular coordinates. The first-order equations can be used to analyze the thickness-shear vibrations of circular plates by comparing with the exact dispersion relations. Finally, we calculated the spectra and mode shapes of axially and non-axially symmetric free vibrations of isotropic, circular plates, which were in good agreement with Mindlin’s earlier studies.In conclusion, we derived the higher-order Mindlin plate equations of circular elastic plates and studied thickness vibrations of isotropic, circular plates. With these results, we will continue to develop higher-order plate equations of anisotropic material, such as quartz crystal, following the experience and method in the analysis of isotropic, circular plates. The frequency spectra and mode shapes of coupled thickness-shear vibrations are calculated. The method and results from such analysis of anisotropic plates will provide theoretical guidance for the design optimization of circular quartz crystal resonators.