In this paper, the natural vibration characteristics of elastically supported functionally graded material plate are investigated using the dynamic stiffness method (DSM). Power-law functionally graded (P-FG) plate, the material properties of which vary smoothly along the thickness direction following the power-law function, that has been used for the analysis. Classical plate theory and Hamilton’s principle are used for deriving the governing differential equation of motion and associated edge conditions for P-FG plate supported by elastic foundation. During the formulation of dynamic stiffness (DS) matrix, the concepts of rotary inertia and neutral surface are implemented. Wittrick–Williams (W-W) algorithm is used as a solving technique for the DS matrix to compute eigenvalues. The results thus obtained by DSM for the isotropic, P-FG plate, and the P-FG plate with elastic foundation compare well with published results that are based on different analytical and numerical methods. The comparisons indicate that this approach is very accurate. Furthermore, results are provided for elastically supported P-FG plate under four different considerations in order to see the differences in frequencies with the inclusion or exclusion of neutral surface and/or rotary inertia. It is noticed that the inclusion of rotary inertia and neutral surface influences the eigenvalues of P-FG plate, and that cannot be discounted. The study also examines the influence of plate geometry, material gradient index, edge conditions, and elastic foundation modulus on the natural frequency of P-FG plate.

中文翻译：

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弹性支撑功能梯度板面外自然振动的动态刚度公式

本文采用动态刚度法（DSM）研究了弹性支撑功能梯度材料板的自振特性。幂律功能梯度（P-FG）板，其材料特性沿厚度方向平滑变化，遵循幂律函数，已用于分析。经典板理论和Hamilton原理用于推导弹性地基支撑的P-FG板的运动控制微分方程和相关的边缘条件。在动态刚度（DS）矩阵的制定过程中，实现了转动惯量和中性面的概念。Wittrick-Williams (WW) 算法用作 DS 矩阵的求解技术以计算特征值。因此 DSM 获得的各向同性 P-FG 板的结果，弹性基础的 P-FG 板与基于不同解析和数值方法的已发表结果进行了很好的比较。比较表明这种方法非常准确。此外，在四种不同的考虑因素下提供了弹性支撑的 P-FG 板的结果，以查看包含或排除中性表面和/或旋转惯性的频率差异。注意到转动惯量和中性面对P-FG板特征值的影响是不可忽视的。该研究还检查了板几何形状、材料梯度指数、边缘条件和弹性地基模量对 P-FG 板固有频率的影响。