臺灣博碩士論文加值系統

　　本文運用微分轉換法求解Bernoulli-Euler樑及非均勻柱體之振動問題。利用Hamilton原理推導系統之統御偏微分運動方程和時變彈性邊界條件。文中首先介紹微分轉換理論的基本定義、性質及演算方法，然後介紹此法在各種問題上的應用，並以Sturm-Liouville問題為例來說明如何應用此一理論配合符號運算求解其特徵值及特徵函數，進而就Bernoulli-Euler樑及柱體，應用微分變換法分析不同條件下之振動問題。
　　研究結果顯示使用微分轉換法進行模擬求解與其他方法相比較所需的運算處理時間較為短暫。有別於以往運用積分運算求解的方法，以微分轉換法求解各類問題省去了冗長及繁雜的變換程序，不僅顯得簡易且將更具系統性。而由微分轉換法所得到的結果，與解析解做比較，其誤差為約只有10-4之極小值，由此可見微分轉換法是求解線性或非線性微分方程式的有力工具。

　　This study introduces a method using differential transforms to solve the vibration problems of Bernoulli-Euler beam and non-uniform column. Applying Hamilton’s principle, the governing equations and boundary conditions of Bernoulli-Euler beam and non-uniform column are derived. The basic definitions and properties of the differential transformation method were introduced briefly and the applications of this method on the many problems were displayed later, and the Sturm-Liouville problem is considered to demonstrate how to find eigenvalues and eigenfunction by using the differential transformation method and symbolic computations. Moreover, considering Bernoulli-Euler beam and non-uniform column, differential transforms are applied to solve the vibration problem of these structures under different conditions.
　　The results of this research show that applying the method on the simulation procedure consumes less CPU time as compared with other methods. Unlike other integral transform methods, using the differential transform to solve many problems leaves out the copious and complex transform procedure and appears not only brief but also more systematical. The errors between the simulation results and the analytical solutions were very small, hence, the differential transformation method is a useful tool in solving linear and nonlinear equations.

摘 要 I
Abstract II
誌 謝 III
目 錄 IV
表 目 錄 VIII
圖 目 錄 IX
符號說明 XIII
第一章 緒論 1
1.1研究動機及目的 1
1.2文獻回顧 2
1.3本文架構 4
第二章 微分轉換法 5
2.1前言 5
2.2微分轉換的數學原理 5
2.3微分轉換的運算 8
2.4 微分轉換法在初始值問題之應用 14
2.5 T譜儲存法 16
2.6 微分轉換法在特徵值問題之應用 18
第三章 非均勻Bernoulli-Euler樑之振動問題 21
3.1 非均勻Bernoulli-Euler樑統御方程及邊界條件 21
3.2 分析非均勻樑自由振動情形 24
3.2.1 無因次化統御方程式與邊界條件 25
3.2.2 微分轉換法 26
3.3 分析非均勻樑強制振動情形 30
3.3.1 無因次化統御方程式與邊界條件 30
3.3.2 微分轉換法 31
3.4 均勻懸臂樑問題之分析 33
3.4.1 微分轉換法 33
3.4.2 解析解 37
3.5 非均勻懸臂樑問題之分析 38
3.5.1 自由振動情形 38
3.5.2 強制振動情形 43
第四章 儲水高塔之振動問題 59
4.1 非均勻柱統御方程及邊界條件 59
4.2 分析無受力非均勻柱之情形 61
4.2.1 統御方程及邊界條件 61
4.2.2 無因次化統御方程式與邊界條件 62
4.2.3 微分轉換法 63
4.2.4 均勻問題分析 64
4.3 分析受風力影響之非均勻柱情形 69
4.3.1 統御方程及邊界條件 70
4.3.2 無因次化統御方程式與邊界條件 71
4.3.3 微分轉換法 72
4.3.4均勻問題分析 73
4.4 分析受地震力影響之非均勻柱情形 78
4.4.1 統御方程及邊界條件 78
4.4.2 微分轉換法 78
4.4.3 均勻問題分析 78
4.5 分析受風力及地震力影響之非均勻柱情形 82
4.5.1統御方程及邊界條件 82
4.5.2 微分轉換法 82
4.5.3 均勻問題分析 83
第五章 結論與建議 101
5-1結論 101
5-2建議 102
參考文獻 103
附錄A 均勻Bernoulli-Euler樑統御方程及邊界條件之推導 106
A.1統御方程及邊界條件之推導 106
A.2自然頻率與振動模態 108
自 述 112

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