臺灣博碩士論文加值系統

本論文使用兩種樑元素研究矩形截面螺旋彈簧的振動特性。第一種元素是3節點18自由度曲樑元素，第二種為3節點23自由度直樑元素，用於模擬分析等向性材料與纖維增強複合材料彈簧。兩種元素皆考慮軸向與橫向位移變形以及由於剪切和扭轉引起的變形，另在直樑元素中，考慮樑具有寬度方向上的柔性彎曲變形與橫截面的翹曲。

矩形截面的曲樑元素參考文獻的推導[14]的圓形截面彈簧，直樑元素則採用文獻[18]的位移場，並參考文獻[4]推導出直樑元素間的位移變換矩陣。文中使用Frenet-Serret區域座標來描述圓柱形彈簧變形，再以漢米爾頓原理配合有限元素法導出曲樑和直樑元素的質量和勁度矩陣。曲樑元素矩陣可直接組合出全域質量和勁度矩陣。直樑元素則須先以位移變換矩陣進行轉換，才能組合得全域質量和勁度矩陣。

在數值驗證中，首先以等向性材三圈螺旋彈簧為例，模擬分析寬厚比b/h = 1, 3, 5自由-自由和自由-固定振動頻率和模態，並與ANSYS比較。由分析結果發現，當b/h=1或接近1時，曲樑元素與ANSYS的分析結果相當相似。而直樑元素當b/h=1時儘管不盡理想，但在寬厚比b/h=3, 5較大時，不同於曲樑元素能獲得較接近ANSYS的模擬結果。論文中也經由實例探討蒲松比與複合材料纖維角的影響。

In this thesis the free vibration of rectangular cross-section helical springs is studied using two types of beam element. The first type element is the 3 node, 18 degree-of-freedom, curved beam element and the second, the 3 node, 23 degree-of-freedom, straight beam finite element. The springs are assumed to be made of isotropic materials or fiber-reinforced composite materials. In both elements, the axial and lateral displacement, as well as the deformation due to both shearing and twisting of beam are considered, while in straight beam element, additional flexural deformations of beam’s flexibility in its width direction and the warping of cross section are included.

The curved beam element of rectangular cross section is adapted from Ref. [14] where springs of circular cross section are treated. The straight beam element are derived based on displacement field of Ref. [18] and by following the same procedure of obtaining the displacement transformation matrix considered by Ref. [4]. The circular cylindrical spring deformation is described in terms of Frenet-Serret local coordinates. The Hamilton’s principle together with finite element method is used to derive the element’s mass and stiffness matrices for both curved and straight beam elements. For the case of curved beam element matrices, they can be directly assembled to obtain the corresponding global mass and stiffness matrices. In contrast, the straight beam element matrices must be transformed first using the displacement transform matrix before being assembled to obtain the global ones.

In the numerical examples, natural frequencies and mode shapes of isotropic helical springs of 3 turns and the width to thickness ratio b/h=1, 3, 5 are studied and compared with those of ANSYS for the free-free and fix-free boundary conditions. It is found that the present curved beam element performs well when b/h = 1 or close to one, while the straight beam element although doesn’t work well when b/h = 1, do obtain much better results than curved beam element comparing with those of ANSYS for the studied cases b/h = 3, 5. Also studied in the numerical examples are the effects of Poisson’s ratio, fiber angles of composite laminate.

第一章 緒論 1
1.1前言 1
1.2文獻回顧 1
1.3研究目的與內容 3
第二章 理論推導 4
2.1單層複合材料板材(lamina)的本構方程式 4
2.2十八自由度曲樑元素 8
2.2.1曲樑元素的運動方程 8
2.2.2曲樑元素形狀函數與其微分的推導 10
2.2.3曲樑元素勁度矩陣 12
2.2.4曲樑元素質量矩陣 17
2.2.5 曲樑元素座標轉換矩陣 20
2.3二十三自由度直樑元素 24
2.3.1直樑元素位移場假設 24
2.3.2直樑元素勁度矩陣 30
2.3.3直樑元素質量矩陣 34
2.3.4直樑元素轉換矩陣 35
第三章 實例分析 42
3.1收斂測試 42
3.2三圈矩形截面彈簧與ANSYS比較(等向，單層) 47
3.2.1曲樑元素與ANSYS比較 (單層等向性彈簧) 47
3.2.2直樑元素與ANSYS比較 (單層等向性彈簧) 51
3.2.3忽略蒲松比之曲樑元素 (單層等向性彈簧) 55
3.2.4忽略蒲松比之直樑元素 (單層等向性彈簧) 60
3.3 不同寬厚比，直、曲樑元素與ANSYS之比較(等向性材疊層彈簧) 64
3.4不同纖維角與寬厚比單層複材彈簧之分析 67
3.4.1無拘束彈簧(單層，纖維角 0°~90°) 67
3.4.2頭端固定彈簧(單層，纖維角 0°~90°) 71
3.5不同疊層複材彈簧分析 75
第四章 結論與未來展望 84
4.1結論 84
參考文獻 86
附錄A-1 3.2.1節中無拘束彈簧振形(曲樑元素b/h=1) 89
附錄A-2 3.2.1節中無拘束彈簧振形(曲樑元素b/h=3) 93
附錄A-3 3.2.1節中無拘束彈簧振形(曲樑元素b/h=5) 96
附錄A-4 3.2.1節中頭端固定彈簧振形(曲樑元素b/h=1) 98
附錄A-5 3.2.1節中頭端固定彈簧振形(曲樑元素b/h=3) 102
附錄A-6 3.2.1節中頭端固定彈簧振形(曲樑元素b/h=5) 105
附錄B-1 3.2.2節中無拘束彈簧振形(直樑元素b/h=1) 107
附錄B-2 3.2.2節中無拘束彈簧振形(直樑元素b/h=3) 110
附錄B-3 3.2.2節中無拘束彈簧振形(直樑元素b/h=5) 113
附錄B-4 3.2.2節中頭端固定彈簧振形(直樑元素b/h=1) 116
附錄B-5 3.2.2節中頭端固定彈簧振形(直樑元素b/h=3) 118
附錄B-6 3.2.2節中頭端固定彈簧振形(直樑元素b/h=5) 120
附錄C-1 3.3節中等向疊層彈簧無拘束彈簧振形 (b/h=1) 122
附錄C-2 3.3節中等向疊層彈簧無拘束彈簧振形 (b/h=3) 124
附錄C-3 3.3節中等向疊層彈簧無拘束彈簧振形(b/h=5) 126

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