臺灣博碩士論文加值系統

在梁的分析模型中，通常使用尤拉－伯努力梁方程理論，在正算尤拉梁問題其解決方法則不計其數。然而當問題的待求項變為系統參數之一時，其複雜度則非一般正算問題可比擬。本論文介紹了解決非齊性的尤拉梁方程反問題的一種數值方法，其目的為在梁上找回其外力(源識別問題)。本篇論文將邊界積分方程方法應用至尤拉梁上，以其振態作為伴隨測試函數，再以我們假設的試解帶入積分方程，以數值方法解此代數方程組，即可得到外力源之數值解。在論文中將以數值算例實際求解尤拉梁的反問題，其中包含四種不同邊界條件的梁以及使用傅立葉級數與振型函數兩種試解之基底，並分析其數值結果。

Euler-Bernoulli beam theory is a typical beam theory when discussing the behavior of beams. There are several methods to obtain the behaviors of the Euler-Bernoulli beam under an external force, but without knowing the external force, the problem becomes an inverse source problem which is the subject of this thesis. Different from the direct problems, the inverse problems are considered more ill-posed. In this thesis, the boundary integral equations method will be adopted to solve the Euler-Bernoulli beam problem, with its mode shape as an adjoint test function. Then, we assume the trail solution of the integral equation. Finally, we can obtain the numerical solution of the external force. Six examples of Euler beam are used to test the performance of the present method.

口試委員審定書 i
誌謝 ii
摘要 iv
ABSTRACT v
目錄 vi
表目錄 viii
圖目錄 ix
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 3
第二章 理論基礎 5
2.1 自我伴隨運算子(Self-Adjoint Operator) 5
2.2 Trefftz方法(Trefftz Method) 6
2.3 廣義格林第二定理(General Green''s Second Identity) 6
2.4 尤拉法(Euler Method) 8
2.5 辛普森法(Simpson’s Rule) 9
2.6 龍格－庫塔法(Runge-Kutta Method) 11
2.7 高斯－克朗羅德法(Gauss–Kronrod Quadrature Formula) 12
2.8 擬時間積分法(Fictitious Time Integration Method) 15
2.9 共軛梯度法(Conjugate Gradient Method) 16
2.10 傅立葉級數(Fourier Series) 19
第三章 尤拉梁的邊界積分方程 21
3.1 反問題 21
3.2 尤拉梁的邊界積分方程推導 21
3.3 簡支梁分析 24
3.4 懸臂梁分析 26
3.5 兩端固定梁分析 28
3.6 一端固定與一端簡支梁 30
第四章 數值算例 34
4.1 數值算例一 34
4.2 數值算例二 36
4.3 數值算例三 38
4.4 數值算例四 39
4.5 數值算例五 40
4.6 數值算例六 43
第五章 結論與未來工作 63
參考文獻 66

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