臺灣博碩士論文加值系統

本文應用Laplace Adomian 分解法解決均勻與非均勻樑之自由振動與強迫振動的問題。首先應用Laplace Adomian分解法將樑之統御方程式轉換成迭代方程，並在頻率方程式上做簡單代數計算，得到Euler-Bernoulli樑的自然頻率與振動模態，透過改變參數包括移動彈簧模數、旋轉彈簧模數、樑之錐度比與軸向張力強度，探討各參數對於動態系統之自然頻率的影響。接著利用Laplace Adomian 分解法探討樑在受外力作用下，樑的變形情形、共振現象與後挫曲行為。
研究結果得知，自然頻率隨著移動彈簧模數、旋轉彈簧模數、軸向張力強度的增加而提升，而錐度比的增加，第一自然頻率會越小，而第二與第三自然頻率會有提升，又以第三自然頻率提升較為顯著。當結構的自然頻率和強迫振盪的頻率相吻合時，會產生共振。共振產生時，即使外力很小，若以相同頻率持續施加於此系統，仍會使結構體被破壞。產生挫曲後，集中力強度提升偏轉角會繼續增加，且集中力角度越大，偏轉角越多。
本文中所得之計算結果與文獻及解析結果相當吻合，故Laplace Adomian 分解法是一種比其他分析方法更簡單、直接又快速的方法。

In this study, the Laplace Adomian decomposition method (LADM) is used to analyze the free vibration and force vibration problems of uniform and non-uniform beams. First, to obtain the natural frequency and mode shape of the Euler-Bernoulli beam, we transformed the governing equation to the algebraic equation by LADM and used simple algebraic operations on the frequency equations afterwards. Moreover, investigating the effect of the natural frequency of the dynamic system by physical parameters including translational spring constant, rotational spring constant, taper ratio of beams and the magnitude of axial tensile. Furthermore, the LADM is applied to analyze the dynamic behavior including deformation, resonance and post-buckling of beams under the external forces.
The results of this study show that the natural frequency upper by the upper translational spring constant, the upper rotational spring constant and the upper magnitude of axial tensile. However, the upper taper ratio would make the first frequency lower and the second frequency and the third frequency upper. When resonance occurs, the very little force can still bring about the collapse of the structure. After buckling occurs, it also indicates the deflection angular upper by the upper magnitude of concentrated force and the upper angle of concentrated force.
The results of this study is consistent with analytical and numerical results given in the literature.Therefore, the LADM is simpler, faster and more straightforward than other methods.

摘要 I
Extended Abstract II
誌謝 X
目錄 XI
表目錄 XV
圖目錄 XVII
符號說明 XXI
第一章 緒論 1
1-1 前言 1
1-2 文獻回顧 2
1-2-1 樑之振動問題研究回顧 2
1-2-2 Adomian分解法 3
1-3 本文架構 5
第二章 Laplace Adomian 分解法 (LADM) 6
2-1 Adomian 分解法 7
2-2 Adomian 多項式 11
2-2-1 非線性(nonlinear)多項式 11
2-2-2 三角(trigonometric)函數多項式 13
2-3 修正Adomian分解法 17
2-3-1 修正ADM（一） 17
2-3-2 修正ADM（二） 22
2-4 Laplace Adomian分解法 27
2-4-1 運算法則 27
2-4-2 LADM分解法在特徵值問題之應用 35
2-4-3 殘值定理(Residue Theorem) 47
2-4-4 Padé近似法 50
2-4-5 結論 52
第三章 樑之自由振動分析 64
3-1 模型建構 64
3-2 均勻Euler-Bernoulli樑 66
3-2-1 統御方程式與邊界條件 66
3-2-2 LADM分解法求解左固定右彈性拘束樑之自然頻率 68
3-2-3 LADM分解法求解左簡支右彈性拘束樑之自然頻率 74
3-2-4 結論 79
3-3 非均勻Euler-Bernoulli樑 80
3-3-1 統御方程式與邊界條件 80
3-3-2 LADM分解法求解楔形懸臂樑之自然頻率 83
3-3-3 LADM分解法求解錐形懸臂樑之自然頻率 89
3-3-4 結論 90
3-4 承受軸向力之非均勻Euler-Bernoulli樑 91
3-4-1 統御方程式與邊界條件 91
3-4-2 LADM分解法求解楔形懸臂樑之自然頻率 95
3-4-3 LADM分解法求解錐形懸臂樑之自然頻率 102
3-4-4 結論 102
3-5 總結 103
第四章 樑之強迫振動分析 130
4-1 橫向外力為分佈負荷之小撓度變形 130
4-1-1 統御方程式與邊界條件 130
4-1-2 LADM解題程序 132
4-1-3 數值結果與討論 136
4-2 橫向外力為集中負荷之大撓度變形 137
4-2-1 統御方程式與邊界條件 137
4-2-2 LADM解題程序 138
4-2-3 數值結果與討論 141
4-3 總結 142
第五章 結論與建議 149
5-1 結論 149
5-2 未來研究方向與建議 151
參考文獻 152

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