本文旨在應用Adomian分解法與Adomian修正分解法來求解特徵值問題以及處理樑之自由振動問題。
首先利用Adomian分解法(ADM)求出Sturm-Liouville問題之特徵值與正規化特徵函數，將統御微分方程式轉換成遞迴型式之代數方程式，邊界條件轉換成簡單的代數型式之頻率方程式，藉由符號運算之使用以及在頻率方程式上做簡單代數計算，可得到第 個特徵值與第 個特徵函數之閉合級數解；其次，探討均勻Euler-Bernoulli樑在不同邊界條件下之自由振動問題。之後，再利用Adomian修正分解法(AMDM)分析置於彈性基座上、承受軸向力作用之非均勻Euler-Bernoulli樑，此樑兩端彈性拘束且附著具有轉動慣性與偏心之端點質量；藉由求得之某些數值結果來說明物理參數對動態系統之自然頻率的影響。最後，探討非均勻Timoshenko樑之自由振動問題。
本文中所得之計算結果與文獻中解析結果及數值結果是相當吻合的；此結果表示本文之分析是精確的，而且提供一個比其它分析方法更簡單、更直接、更具有統一性、系統性之方法。

The paper solves the eigenvalue problems and deals with the free vibration problems by using the Adomian decomposition method (ADM) and Adomian modified decomposition method (AMDM).
First, using the ADM, the eigenvalues and normalized eigenfunctions for the Strum-Liouville eigenvalue problem are solved, and the governing differential equation becomes a recursive algebraic equation and boundary conditions become simple algebraic frequency equations which are suitable for symbolic computation. Moreover, after some simple algebraic operations on these frequency equations any th natural frequency, the closed form series solution of any th mode shape can be obtained. Second, the free vibration problems of Euler-Bernoulli beam under various supporting conditions are discussed. Third, using the AMDM, the free vibration problems of an elastically restrained non-uniform Euler-Bernoulli beam with tip mass of rotatory inertia and eccentricity resting on an elastic foundation and subjected to an axial load are proposed. Some numerical results are given to illustrate the influence of the physical parameters on the natural frequencies of the dynamic system. Finally, this paper deals with free vibration problems of non-uniform Timoshenko beams.
In this paper, the computed results agree well with those analytical and numerical results given in the literature. These results indicate that the present analysis is accurate, and provides a unified and systematic procedure which is simple and more straightforward than the other analyses.

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