博碩士論文 105322013 詳細資訊




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姓名 林珈均(Chia-Chun Lin)  查詢紙本館藏   畢業系所 土木工程學系
論文名稱 薄殼結構非線性運動之向量式有限元分析法
(Motion Analysis of Thin Shell Structure with Large Displacement by the VFIFE method)
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摘要(中) 為了分析薄殼結構的大變位問題,本研究以向量式有限元以及運動解析作為薄殼
結構分析之基本理論。利用滿足歐拉運動定理之有限廣義質點,與藉由運動解析理論
中的概念:點值描述,途徑單元和移動基礎架構,以材料力學的微變形概念,確保變
形座標系統中,元素的內力滿足平衡方程中,將薄殼結構的非線性運動行為化成小變
形問題,且利用 DKT 板元與 CST 薄膜元所結合的殼元素分析運動行為。此外,逆向
剛體運動採用四元數的旋轉向量,並引用有限旋轉的運動公式,使其能用合理的數學
理論得出有限轉動之運動方程式,並透過薄殼結構非線性運動之數值分析與實驗結果
比較,驗證 VFIFE-DKT 元程序的精確性。
摘要(英) In order to simulate the large displacement of a thin shell structure, a novel shell element
based on the vector form intrinsic finite element (VFIFE) method is presented. The motion
of the shell structure is characterized by the motions of finite particles. The motion of each
particle satisfies the Law of Mechanics. In addition, three key processes of the VFIFE
method such as the point value description, path element, and connected material frame are
adopted. Ensuring that the internal force of the element in the deformation coordinate
system is satisfied. In the equilibrium equation, the nonlinear motion of the thin shell
structure is transformed into a small deformation problem, and the shell behavior of the DKT
plate element combined with the CST thin film element is used to analyze the motion
behavior. A fictitious reversed rigid body motion is used to separate the rigid body motion
by the quaternion rotation theory and the deformations of the VFIFE-DKT element within
each path element. Besides, the finite rotation theory is also applied in the analysis of the
motion of each particle. Through the numerical analyses of the benchmark structures and
experiments undergo extremely-large displacements and rotation during motion, this novel
shell element of the VFIFE method demonstrates its outstanding accuracy and efficiency.
關鍵字(中) ★ 殼元素
★ 大變位分析
★ 向量式有限元(VFIFE)
★ 逆向剛體運動
★ 變形座標
關鍵字(英) ★ shell element
★ large displacement
★ vector form intrinsic finite element (VFIFE)
★ fictitious reversed rigid body motion
★ deformation coordinates
論文目次 目錄

中文摘要 ............................................................................................................................. I
Abstract .............................................................................................................................. II
致謝 .................................................................................................................................. III
目錄 .................................................................................................................................. IV
圖目錄 ........................................................................................................................... VIII
表目錄 ........................................................................................................................... XIV
符號說明 ........................................................................................................................ XV
第一章、緒論 .................................................................................................................... 1
1.1 研究方向 ............................................................................................................. 2
1.2 研究方法及步驟 ................................................................................................. 2
第二章、文獻回顧 ............................................................................................................ 3
2.1 薄殼元素發展 ..................................................................................................... 3
2.2 向量式有限元之研究與發展近況 ..................................................................... 7
2.2.1 改善分析模式 .......................................................................................... 8
2.3 文獻回顧整理與研究方法 ................................................................................. 9
第三章、運動解析之理論 .............................................................................................. 13
3.1 質點運動控制方程式 ....................................................................................... 13
3.2 途徑單元與點值描述 ....................................................................................... 14
3.3 剛體運動估算 ................................................................................................... 16
3.4 逆向轉動和變形座標 ....................................................................................... 20
第四章、向量式有限薄殼元 .......................................................................................... 24
4.1 向量式有限薄殼元(VFIFE-DKT)之推導 ........................................................ 24
4.2 VFIFE-DKT 薄殼元之質量矩陣 ...................................................................... 37
4.2.1 三角平板式轉動慣量矩陣 .................................................................... 38
4.2.2 細棒式轉動慣量 .................................................................................... 41
4.2.3 圓柱式轉動慣量 .................................................................................... 43
4.2.4 係數式轉動慣量 .................................................................................... 44
4.3 運動方程式 ....................................................................................................... 45
4.4 薄殼元外力計算 ............................................................................................... 47
4.5 VFIFE-DKT 元靜力分析模式 .......................................................................... 52
4.6 變形座標之定義程序 ....................................................................................... 56
第五章、三維空間有限轉動 .......................................................................................... 61
5.1 有限轉動與無限小轉動 ................................................................................... 61
5.1.1 有限轉動張量 ......................................................................................... 61
5.1.2 無限小轉動 ............................................................................................. 66
5.2 李群與李代數 ................................................................................................... 68
5.2.1 旋轉群 SO(3)與其李代數 so(3) ............................................................. 68
5.2.2 一般的參數化旋轉 ................................................................................. 73
5.2.3 指數映射 ................................................................................................. 74
5.2.4 三維空間之變換關係 ............................................................................. 80
5.3 大轉動驗證 ....................................................................................................... 87
5.3.1 轉角計算 ................................................................................................ 87
5.3.2 轉角累積誤差 ........................................................................................ 87
5.3.3 L 型板之合成旋轉 .................................................................................. 89
5.4 VFIFE 之轉動應用 ............................................................................................ 90
5.4.1 逆向剛體轉動 ........................................................................................ 91
5.4.2 變形彎角之處理 .................................................................................... 91
第六章、數值算例 .......................................................................................................... 93
6.1 VFIFE-DKT 之大轉動驗證 .............................................................................. 93
6.1.1 剛性板受自重之單擺運動行為 ............................................................ 93
6.1.2 壓克力板之單擺運動 ............................................................................ 95
6.2 VFIFE-DKT 之動力分析 ................................................................................ 101
6.2.1 扭曲梁之小變位振動分析 .................................................................. 101
6.2.2 半圓柱型頂板受集中力作用後之動力失穩行為分析 ...................... 104
6.2.3 板梁受集中力作用後之小應變分析 .................................................. 108
6.2.4 旋轉拉力板之正向應變行為分析 ...................................................... 110
6.3 VFIFE-DKT 之大變位靜力分析 .................................................................... 114
6.3.1 簡支圓柱受線性均布力作用之行為分析 .......................................... 116
6.3.2 圓柱受集中力作用之行為分析 .......................................................... 117
6.3.3 方型薄膜受靜壓下之行為分析 .......................................................... 120
6.4 VFIFE-DKT 之薄殼應用 ................................................................................ 123
6.4.1 圓環型薄板受集中力後之大變位行為分析 ...................................... 123
6.4.2 L 形懸臂梁之小應變振動行為分析 .................................................... 130
第七章、結論與建議 .................................................................................................... 142
7.1 結論 ................................................................................................................. 142
7.2 建議 ................................................................................................................. 143
參考文獻 ........................................................................................................................ 144
附錄 A、DKT 板元形狀函數 ....................................................................................... 151
附錄 B、三角板之轉動慣量積分 ................................................................................ 154
附錄 C、四元數與空間旋轉 ........................................................................................ 168
C1 四元數基本原理與運算 ................................................................................. 168
C2 羅德里格旋轉公式 ......................................................................................... 171
C3 合成旋轉 ......................................................................................................... 178
附錄 D、剛體定點轉動的微分方程 ............................................................................ 180
D1 剛體的質量幾何 ............................................................................................. 180
D2 剛體繞定點轉動的運動微分方程 ................................................................. 184
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指導教授 王仲宇(Chung-Yue Wang) 審核日期 2018-7-19
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