臺灣博碩士論文加值系統

本文建立一維三節點具有23自由度的直樑元素來建立具矩形截面之旋轉環的有限元素模式。此模式含有橫向剪力變形、扭轉、翹曲、弦向曲率和側向位移效應等。於元素中，除了扭轉變形使用拉格朗治-赫米特混合型內插函數外，其餘皆使用拉格朗治內插函數。採用上述改善樑理論，應較簡易樑理論更能準確描述旋轉複合材料環具有的振動特性。

在推導運動方程式時，首先將旋轉環的變形以直樑元素來近似，其中直樑元素採用文獻[10]的位移場。根據此位移場配合本構方程式，求出直樑的動能及應變能，再應用漢米爾頓原理結合有限元素法，推導出環上不同位置直樑元素的運動方程式。接著參考文獻[13]直樑元素間之位移轉換矩陣，推導出旋轉環系統的運動方程式。利用上述有限元素模式探討不同寬厚比、轉速等對等向性和疊層複合材料旋轉環，以及不同纖維角對單層複合材料旋轉環之動態特性的影響。

The objective of this thesis is to develop a finite element model of a rotating ring with rectangular cross-section using one-dimensional three-node straight beam element with twenty-three degrees of freedom. The straight beam element has included structural effects such as transverse shear deformation, twisting, warping, chordwise curvature, and sidewise bending. In the elements, the mixed Lagrangian-Hermite type of interpolation functions are used to represent the twisting deformation, while Lagrangian interpolation functions are used for other displacement variables. As a result, the finite element model developed here is able to predict more truly the deformation of vibrational behaviors of rotating composite rings than those using simple beam model.
To develop the finite element model of the rotating ring, first, the straight beam finite elements are used to approximate the deformation of the ring. The displacement field of beam in Ref. [10] is adopted. The kinetic energy and the strain energy of the straight beam element at a typical location on the ring are found. Then, by employing Hamilton’s principle together with the finite element method, the equations of motion of straight beam element at such a typical ring’s location are derived. Next, the displacement transform matrix between elements similar to that of Ref. [13] is derived. With the displacement matrix and beam element’s matrices, the finite element equations of motion of rotating composite rings are obtained.
Based on the above ring finite element model, the natural frequencies and mode shapes of rotating rings are studied. The influence of the width to thickness ratios, rotating speeds, material types: isotropic material, single-layered and laminated composite material, as well as the fiber angles are investigated.

致謝 i
摘要 ii
ABSTRACT iii
章節目錄 iv
表目錄 vi
圖目錄 viii
符號索引 x
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 研究架構 2
第二章 理論推導 3
2.1 單層纖維加強複合材料板之本構方程式 3
2.2 直樑元素之位移場假設 7
2.3 有限元素內插函數與位移場關係之推導 14
2.4 運動方程式 19
2.4.1 系統之應變能 19
2.4.2 系統之動能 20
2.4.3 漢米爾頓原理 20
2.5 轉換矩陣之推導 23
2.6 旋轉複合材料環之自然振動頻率 30
2.7 模態繪製之處理 31
第三章 數值驗證及實例分析 33
3.1 數值驗證 33
3.1.1 靜態環收斂性分析 33
3.1.2 等向性環之數值驗證 36
3.1.3 複合材料環之數值驗證 44
3.2 實例分析 52
3.2.1 旋轉環收斂性分析 52
3.2.2 等向性環在不同轉速下之無因次化自然頻率分析 56
3.2.3 複合材料環在不同轉速下之自然頻率分析 63
第四章 結論與未來展望 73
4.1 結論 73
4.2 未來展望 73
參考文獻 74
附錄A：直樑元素之質量、迴旋、勁度矩陣 76
附錄B：有限元素數目收斂趨勢圖 87
附錄C：3.1.3節實例補充之模態圖 90

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