非均勻圓板之軸對稱振動的統御方程式為一個四階規則奇點變系數的常微分方程式。若該厚度為徑向變數的任意多項式，則可推導出四個線性獨立基本解的確切閉合解型式，其中必有兩個基本解的對徑向三次微分在圓板中心處其值必為奇異。該頻率方程式亦可以以它們來表示之。本論文研究重點會深入探討研究正規化基本解的相互關係，再配合格林函數來表示整個非均勻圓板在彈性邊界下的靜態撓度。其中格林函數可由已知的邊界條件和外在負載以及四個正規化基本解來獲得。最後，以頻率方程式的數值來和現有其他方法的文獻做比較，來驗證本文方法的正確性。

The governing characteristic differential equation for the axisymmetric vibrations of a non-uniform circular plate is a regular singular fourth-order ordinary differential equation with variable coefficients. If the variable radial thickness of the circular plate varies in arbitrary polynomial form, then the four fundamental solutions can be derived and expressed in exact closed forms. One also shows that the third derivatives of the two fundamental solutions are singular at the center of the circular plate. The natural frequency equation of the system is expressed in terms of the fundamental solutions. The exact general solution of the static deflection of a non-uniform plate with general elastically restrained boundary conditions is developed in closed integral form. This study is concerned with application of normalized fundamental solutions’ relation and Green’s function to the force vibration of circular plate with variable thickness. The Green’s function for the circular plate with homogeneous boundary conditions and various loading is presented and expressed in terms of four normalized fundamental solutions of the system. Finally, the numerical results are compared with those in the existing literatures.

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