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博碩士論文 etd-0904104-070859 詳細資訊
Title page for etd-0904104-070859
論文名稱
Title
樑的混合型平面應變有限元素振動分析
Mixed-type Plane Strain Finite Element Analysis of Beam Vibration
系所名稱
Department
畢業學年期
Year, semester
語文別
Language
學位類別
Degree
頁數
Number of pages
61
研究生
Author
指導教授
Advisor
召集委員
Convenor
口試委員
Advisory Committee
口試日期
Date of Exam
2004-06-23
繳交日期
Date of Submission
2004-09-04
關鍵字
Keywords
有限元素、樑、振動
Finite Element, Beam, vibration
統計
Statistics
本論文已被瀏覽 5686 次,被下載 46
The thesis/dissertation has been browsed 5686 times, has been downloaded 46 times.
中文摘要
本文主要是探討具有相當厚度之樑的自由振動行為。使用平面應變有限元素法,以彈性力學為理論基礎,將傳統位移變分原理( conventional displacement-type variational principle )加入Reissner’s principle 組合而成的一種新的混合型( mixed-type )有限元素法分析。藉由這種組合所產生的矩陣關係式,應力會如同位移一樣成為主要變數。而應力邊界條件也可以與位移邊界條件一樣,容易且正確的加上。
藉由此方法可以得到不同長寬比的樑在不同邊界條件下之自然振動頻率和模態。並且將所得的結果分別與尤拉樑理論、Timoshenko樑理論、高階樑理論及平面應變位移型(displacement-type)有限元素法來作比較,以了解本方法的二階理論與其他傳統一階樑理論之間的差異,以及滿足應力邊界條件比只考慮位移邊界條件所造成的影響。
Abstract
Free vibration of beam with moderate thickness is analyzed in the present study. Plane strain finite element is employed, which is based on 2-D elasticity. The conventional displacement-type variational principle is combined with Reissner’s principle and a mixed-type variational formulation is derived. With such formulation, stresses, as well as displacements, are the primacy variables and both boundary conditions can be imposed exactly and simultaneously.
Beams with various aspect ratios and boundary conditions are analyzed. Vibration frequencies and modes are obtained and compared to those by Euler’s beam theory, Timoshenko beam theory, higher-order theory and displacement-type plane strain finite element method to see the effects of 2-D elasticity beam analysis compared to traditional 1-D theories, and the satisfying of stress boundary conditions, in addition to the displacement ones.
目次 Table of Contents
【目錄】................................................I
【圖目錄】............................................III
【表目錄】..............................................V
【中文摘要】..........................................VII
【英文摘要】.........................................VIII

第一章 緒論............................................1
1.1 前言...........................................1
1.2 文獻回顧.......................................2
1.2-1 尤拉樑理論...............................2
1.2-2 Timoshenko樑理論.........................5
1.2-3 高階剪切變形理論.........................9
1.2-4 彈性力學................................11

第二章 樑的混合型平面應變有限元素振動分析.............13
2.1 前言..........................................13
2.2 理論推導......................................13

第三章 實例計算.......................................23
3.1 前言..........................................23
3.2 計算實例......................................23

第四章 緒論 ..........................................28
4.1 前言..........................................28
4.2 收斂試驗 .....................................28
4.3 本文結果討論與文獻之比較......................30

第五章 結論 ..........................................55

【參考文獻】...........................................56
【附錄】...............................................61

【圖目錄】
(圖1-1) : 尤拉樑理論假設下之自由體圖.................. 4
(圖1-2): Timoshenko樑理論假設下之自由體圖..............8
(圖1-3) : 高階樑理論假設下之自由體圖 ..................10
(圖1-4) : 一維樑理論無法表現的厚度變化.................12
(圖2-1) : 樑的尺寸標記和卡式系統之位移座標示意圖.......14
(圖3-1) : 位移邊界條件.................................25
(圖3-2) : 應力邊界條件.................................26
(圖3-3) : 八節點編號慣例...............................27
(圖4-1) : 在不同長寬比及邊界條件下前三個彎曲模態之無因次
自然振動頻率曲線圖 .....................................46
(圖4-2) : 長寬比:L/H=8,網格切割數:Mesh=16 4,邊界條件為
Clamped-Clamped的樑之第1~5個振動模態圖.................47
(圖4-3) : 長寬比:L/H=8,網格切割數:Mesh=16 4,邊界條件為
Clamped-Clamped的樑之第6~10個振動模態圖................48
(圖4-4) : 長寬比:L/H=8,網格切割數:Mesh=16 4,邊界條件為
Clamped-Pined的樑之第1~5個振動模態圖 ..................49
(圖4-5) : 長寬比:L/H=8,網格切割數:Mesh=16 4,邊界條件為
Clamped-Pined的樑之第6~10個振動模態圖..................50
(圖4-6) : 長寬比:L/H=8,網格切割數:Mesh=16 4,邊界條件為
Clamped-Free的樑之第1~5個振動模態圖....................51
(圖4-7) : 長寬比:L/H=8,網格切割數:Mesh=16 4,邊界條件為
Clamped-Free的樑之第6~10個振動模態圖 ..................52
(圖4-8) : 長寬比:L/H=8,網格切割數:Mesh=16 4,邊界條件為
Pined- Pined的樑之第1~5個振動模態圖....................53
(圖4-9) : 長寬比:L/H=8,網格切割數:Mesh=16 4,邊界條件為
Pined- Pined的樑之第6~10個振動模態圖...................54

【附錄】: Fortran程式流程圖 ...........................61

【表目錄】
(表4-1):平面應變混合型法和平面應變位移型法在不同邊界條件及網格下之樑的無因次化自然振動頻率(Mode 1)................32
(表4-2):平面應變混合型法和平面應變位移型法在不同邊界條件及網格下之樑的無因次化自然振動頻率(Mode 2)................35
(表4-3):平面應變混合型法和平面應變位移型法在不同邊界條件及網格下之樑的無因次化自然振動頻率(Mode 3)................38
(表4-4):本文混合型法與尤拉樑理論、Timoshenko樑理論、及位移型法之第一模態無因次化自然振動頻率之比較,邊界條件:
(Clamped-Clamped ) ......................................41
(表4-5):本文混合型法與尤拉樑理論、Timoshenko樑理論、及位移型法之第一模態無因次化自然振動頻率之比較,邊界條件:
(Clamped-Pined )........................................42
(表4-6):本文混合型法與尤拉樑理論、Timoshenko樑理論、及位移型法之第一模態無因次化自然振動頻率之比較,邊界條件:
(Clamped-Free ).........................................43
(表4-7):本文混合型法與尤拉樑理論、Timoshenko樑理論、及位移型法之第一模態無因次化自然振動頻率之比較,邊界條件:
(Pined - Pined )........................................44
(表4-8):本文混合型法與高階樑理論之第一模態無因次化自然振動頻率之比較。( 邊界條件: Clamped-Clamped )...............45
(表4-9):本文混合型法與高階樑理論之第一模態無因次化自然振動頻率之比較。( 邊界條件: Clamped- Pined )................45
參考文獻 References
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