臺灣博碩士論文加值系統

橋梁的振動在土木工程中是一個非常重要的議題，而橋梁可以簡化為尤拉梁模型來作為分析。本文提供的分析方法為邊界積分方程法(BIEM)，其搭配了伴隨Trefftz測試函數為基底作係數的展開，而伴隨Trefftz測試函數本身是滿足齊性控制方程式和邊界條件的，因此能夠消除吉布斯現象和避免矩陣運算，也就是說能夠在誤差極小的情況下得到數值解。邊界積分方程法能將難以求得解析解的微分方程式問題轉換成依靠邊界條件來描述整個場的等效積分方程式問題。最後由數值算例可以知道邊界積分方程法在追求高精度、高效率的情況下是可行的。

In this thesis we numerically solve the direct Euler-Bernoulli beam problems by using a boundary integral equation method(BIEM) which is based on the generalized Green’s second identity and the self-adjoint operators. In the BIEM, we choose a set of adjoint Trefftz test functions which can be obtained by the method of separation of variables. In the numerical algorithm, we can expand a trial solution by using the bases satisfying the homogeneous governing equation and the boundary conditions simultaneously. To satisfy the above two properties of the bases, we use the adjoint Trefftz test functions as the bases and impose the specified boundary condition. By using these bases, moreover, we can eliminate the Gibbs phenomenon and avoid the matrix computations. Finally, there are several numerical examples to validate the effectiveness of the proposed scheme in this thesis and the results show that the BIEM is a highly accurate numerical method.

口試委員會審定書i
誌謝i
摘要iii
ABSTRACTiv
目錄v
圖目錄viii
表目錄x
第一章 緒論1
1.1 前言 1
1.2 文獻回顧 2
1.3 研究動機與目的 3
1.4 論文架構 4
第二章 理論基礎 5
2.1 尤拉梁理論(Euler-Bernoulli Beam Theory) 5
2.2 自我伴隨運算子(Self-adjoint Operators) 7
2.3 廣義格林第二恆等式(Generalized Green’s Second Identity) 8
2.4 伴隨Trefftz測試函數(Adjoint Trefftz Test Functions) 10
2.4.1 簡支梁(simple beam) 12
2.4.2 懸臂梁(cantilever beam) 13
2.4.3 兩端固定梁(clamped beam) 15
2.4.4 一端固定，一端簡支(clamped-pinned beam) 16
2.4.5 一端固定，一端導向支承(clamped-guided beam) 18
2.4.6 一端簡支，一端導向支承(pinned-guided beam) 19
2.5 擬時間積分法(The Fictitious Time Integration Method) 21
2.6 振態正交性(Mode Orthogonality) 26
2.7 振態疊加法(Mode Superposition Method ) 27
2.8 常微分方程式的數值解法(Numerical Methods for Ordinary Differential Equation ) 28
2.9 共軛梯度法(Conjugate Gradient Method) 30
2.10 全局適應高斯-克朗羅德積分(Global Adaptive Quadrature Using Gauss-Kronrod) 31
2.11 傅立葉分析(Fourier Analysis) 34
2.12 瑞利-里茲法(Rayleigh-Ritz method) 37
第三章 邊界積分方程法(BIEM) 39
3.1 等斷面尤拉梁(齊性邊界條件) 39
3.1.1 試驗解(Trial Solution) 41
3.2 等斷面尤拉梁(非齊性邊界條件) 44
3.2.1 簡支梁的非齊性邊界條件 44
3.2.2 一端固定一端簡支的非齊性邊界條件 46
3.2.3 非齊性邊界條件下BIEM搭配伴隨Trefftz測試函數總結 47
3.3 非均勻斷面尤拉梁(齊性邊界條件) 48
第四章 數值算例 53
4.1 數值算例一 53
4.2 數值算例二 55
4.3 數值算例三 56
4.4 數值算例四 57
4.5 數值算例五 57
4.6 數值算例六 58
4.7 數值算例七 58
4.8 數值算例八 60
4.9 數值算例九 61
4.10 數值算例十 63
4.11 算例表 64
4.12 算例圖 69
第五章 結論與未來工作 86
參考文獻 88
附錄A 91
附錄B 92
附錄C 93
附錄D 95

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