本文建立了一套延伸的Reissner-Mindlin(RM) 離散層理論，且將其用來分析一個四邊簡支承之功能性壓電材料板的自然振動問題。在本文提出的理論中，該板將被人工分為NL個個別層；一套R-M型態的位移場結合線性近似的電位勢被應用到每一個組合層板的離散層中。將各個相鄰層界面之位移以及電位勢連續條件視為一種束制條件，且應用拉格朗日乘數法 (the method of Lagrange multipliers) 引入板能量方程式中。使用延伸的漢彌爾頓原理 (the extended Hamilton’s principle) 推導出一套板運動方程式以及相關合理的邊界條件。應用分離變數法，將場變數展開為滿足邊界條件的函數型式；其一是以面內座標表示的雙重富立葉級數 (the double Fourier series)，另一部份是時間變數組成的週期函數。依此特定場量函數，求解該組運動方程式，則壓電板的自然頻率以及相對應的各模態場量變數沿厚度方向的分佈形狀，均可依序求解。本文最後亦針對分層數量以及材料性質梯度指標對自然頻率的影響作進一步的討論。

An extended discrete layer Reissner-Mindlin (RM) theory is proposed and developed for the free vibration analysis of simply-supported, functionally graded (FG) piezoelectric plates. In the present analysis, the plate is artificially divided into NL individual layers. A set of RM-type displacements combined with the linear approximation for the electric potential is assigned to each discrete layer constituting the plate. The displacement and electric potential continuity conditions at interfaces between adjacent layers are regarded as the constraints and introduced in the energy functional by means of the method of Lagrange multipliers. A set of motion equations and associated possible boundary conditions of the present theory are derived using the extended Hamilton’s principle. By means of the method of separation of variables, the field variables are expanded as the forms of double Fourier series in the in-surface coordinates and harmonic function in the time variable where the edge boundary conditions are satisfied. The natural frequencies of FG piezoelectric plates can then be determined using the specific field variables. The distributions of corresponding modal field variables through the thickness coordinate are also presented. A parametric study for the total number of artificial layers and the material property gradient index on the natural frequencies of FG piezoelectric plates is conducted.

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