在土木工程議題中，橋梁的振動是非常重要的一塊領域，而橋梁又可以理想地簡化為尤拉梁進行分析。在正算尤拉梁問題時處理手段有許多種，而當處理欲反求外力的尤拉梁問題(源識別)時，計算量與困難度將會增加許多。本文使用邊界積分方程法(BIEM)，搭配伴隨Trefftz測試函數之概念，引入振態的概念做為基底，並發展出新的閉合係數展開識別法。由於過程之中沒有重積分及大尺度的反矩陣運算，且求出之解為閉合解，因此能夠得到誤差小、精度高的結果。論文中將以數值算例，實際求解尤拉梁的外力反算問題，其中包含各種不同邊界條件的梁以及其對應之振態做為基底，另外並加入噪音進行分析，最後與精確解進行比對，分析數值結果。

In the field of Civil engineering, the forced vibration of a bridge is a big issue. When discussing the behavior of a bridge, we can apply the Euler-Bernoulli beam theory to analyze it simply. There are several methods to analyze the Euler-Bernoulli beam equation under an external force; however, without knowing the external force, it becomes an inverse source problem which is much more complicated.In this thesis, we adopt the boundary integral equation method (BIEM) with the mode shapes as adjoint test functions. Then, we can develop a non-iterative method to recover a space-dependent load on the Euler-Bernoulli beam named closed-form expansion coefficients method. Finally ,we give some numerical examples to demonstrate the efficiency and accuracy of the proposed new method.

目錄
誌謝 I
摘要 III
ABSTRACT IV
圖目錄 VII
表目錄 IX
第一章 緒論 1
　 1.1 前言 1
　 1.2 文獻回顧 1
　 1.3 研究動機與目的 2
　 1.4 論文架構 2
第二章 理論基礎 4
　 2.1 尤拉梁理論 4
　 2.2 自我伴隨運算子(Self-Adjoint Operators) 6
　 2.3 廣義格林第二恆等式(Generalized Green’s Second Identity) 7
　 2.4 伴隨Trefftz測試函數(Adjoint Trefftz Test Functions) 9
　 2.5 擬時間積分法 11
　 2.6 高斯-克朗羅德法(Gauss-Krontod Quadrature Formula) 12
　 2.7 閉合係數展開識別法(Closed-Form Expansion Coefficients Method) 13
第三章 邊界積分方程法(BIEM) 18
3.1 等斷面尤拉梁 18
　 3.2 各邊界條件之尤拉梁振態推導 20
第四章 數值算例 29
　 4.1 數值算例一 30
　 4.2 數值算例二 32
　 4.3 數值算例三 34
　 4.4 數值算例四 35
　 4.5 數值算例五 36
　 4.6 數值算例六 38
　 4.7 算例表 40
　 4.8 算例圖 46
第五章 結論與未來工作 62
參考文獻 64

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