臺灣博碩士論文加值系統

本論文提出考慮橫向、軸向、側向的位移、剪力和扭轉變形，且包含初始曲率(curvature)和扭率(torsion)效應之三節點每節點6個自由度，圓截面平面曲樑元素，應用於分析具圓截面之複合材料圓柱形螺旋彈簧。其中假設彈簧是以不同疊層角度的單方向纖維加強複合材料板材堆疊製成的。為了瞭解所提出之曲樑元素的初始扭率和初始曲率對於模擬螺旋彈簧靜、動態特性的影響，本文中將原始所推導之平面曲樑元素另再簡化為不含扭率的平面曲樑和直樑兩種元素，藉以比較不同種類元素模擬彈簧之差異。
為確認所推導元素與所發展程式的可靠性，文中首先用此三種元素分析等向性材料彈簧的靜態特性，並將結果與利用虛功法所求得的解析解比較，發現這三種元素所得的數值皆較解析解稍小，但與解析解之間的差異最大不超過8%。其次，以此三種元素和ANSYS對等向性材料彈簧的振動特性作分析，發現此三種元素分析所得的模態與自然頻率和ANSYS分析的結果都還相符。
為瞭解單方向纖維加強複材板材之疊層角度η的影響，於本文實例中分析單疊層角度的複材實心螺旋彈簧(簡稱單層複材實心彈簧)及多疊層角度的複材中空螺旋彈簧(簡稱疊層複材中空彈簧)。由單層複材實心彈簧的靜態分析，發現在η=〖15〗^o~〖45〗^o間各種元素分析結果的差異較大，但其趨勢似乎都顯示η愈接近〖45〗^o時彈簧常數值愈大，其原因可能是彈簧變形時同時伴隨著彎曲與扭曲，而當纖維角度η=〖45〗^o時，複合板材的彎曲與扭曲的變形也有最大耦合的效應。由動態分析則顯示各種元素分析所得之對應模態的自然頻率與η關係趨勢大致上相似，但在某些纖維角度會有少數模態不完全對應(例如振盪方向有些微變動)或頻率差異變大的現象發生。最後，將這三種元素對疊層複材中空彈簧作靜、動態分析比較，發現其趨勢與單層複材實心彈簧的情形類似，但其間模態的差異相較於單層複材實心彈簧會減少，而自然頻率間的差異也有變小。這可能是因為其他不同疊層角度板材的變形牽制作用，以及有較多η=0^o疊層的緣故。

In this thesis, a three-node, 6 degree of freedom per node, curved beam element including the effect of initial curvature and torsion of beam is proposed. In this curved beam element, the lateral and axial displacements, as well as the deformation due to both twisting and shearing of beam are also considered. The curved beam element is employed to analyze circular helical springs with circular cross section, where the springs are made of unidirectional fiber-reinforced composite layers having a preselected set of lamination angles. To study the initial torsion and initial curvature effects in the above curved beam element, the current element is further simplified into other two types of element, a curved beam element without initial torsion and a straight beam element. Using these three types of elements, the effectiveness of curved and straight beam elements to simulate both static and dynamic characteristics of springs then can be evaluated.
To ensure the elements being derived and the computer program being developed are correct, first, springs made of an isotropic material are analyzed and the results expressed as the spring constants are compared with the analytical solution derived using the virtual work method. It is found that for the problem being analyzed all three types of elements yield a little bit smaller spring constant values than the analytical one, and the difference of the results between analytical method and any of three types of element is at its largest not exceeding 8%. Next, the vibration characteristics of springs made of an isotropic material are analyzed using these three types of element. The results are compared with those of ANSYS and they are found in good agreement.
To study the effect of lamination angle η on the static and dynamic characteristic of springs made of unidirectional fiber-reinforced composite material, composite solid springs with a single lamination angle (or composite solid springs) and laminated composite hollow springs with a particular set of lamination angles (or laminated composite hollow springs) are analyzed. Static analyses of composite solid springs indicate that discrepancies among the three types of elements are more conspicuous when η=〖15〗^o~〖45〗^o, and the values of spring constant seem attain their largest values as η approaches 〖45〗^o. This could be attributed by the bending and twisting deformation coupling of the off-axis composite layer as the coupling effect is strongest when 〖η=45〗^o and one knows that the spring deformed accompanied by both bending and twisting. From dynamic analysis, one finds that all three types of element yield similar trend of natural frequency to η relation. There are a few modes don’t match exactly (like the modes swing directions are different), or have a little bit greater natural frequency discrepancies at certain η values. Finally, the analyses of laminated composite hollow springs show almost the same trends of results as the composite solid springs. But the discrepancies among the mode shapes and natural frequencies become smaller. This might be due to the deformation coupling of layers of the different lamination angles, as well as more layers with 〖η=0〗^o.

Abstract i
中文摘要 iii
目錄 v
符號說明 vii
圖目錄 xiii
表目錄 xvi
第一章 緒論 1
1.1前言 1
1.2文獻回顧 1
1.3研究目標及內容 3
第二章 有限元素推導 5
2.1單層纖維加強複材板的本構方程式 5
2.2空間曲梁元素的應變位移關係 7
2.3位移轉置矩陣推導 10
2.3.1第一類平面曲樑元素位移轉換矩陣 13
2.3.2第二類平面曲樑元素位移轉換矩陣 14
2.3.3直樑元素位移轉換矩陣 17
2.4 形元素位移場之推導 18
2.4.1元素長度之推導 18
2.4.2 形狀函數與元素位移場 18
2.5質量矩陣 20
2.6勁度矩陣 22
2.7全域勁度與質量矩陣推導 26
第三章 實例分析和討論 28
3.1 等向性材料彈簧的靜態特性 28
3.2等向性材料彈簧的振動特性 30
3.2.1 等向性材料彈簧動態收斂性 30
3.2.2等向性材料彈簧的振動特性 34
3.3複合材料彈簧的分析例與討論 39
3.3.1單層複材彈簧的靜態性質 40
3.3.2單層複材彈簧的振動性質 41
3.3.3 疊層複材彈簧的靜態性質 51
3.3.4 疊層複材彈簧的振動性質 52
第四章 結論與未來展望 59
4.1 結論 59
4.2 未來展望 60
參考文獻 61
附錄A以虛功法推導實心等向性材料彈簧之力量位移關係 64
附錄B證明直梁元素切線向量和中間節點螺旋角對應螺線位置的密切面法向量垂直 67
附錄C1不同數量的元素各元素模式動態分析的自然頻率 68
附錄C2 本論文的元素以及ANSYS的等向性材料彈簧之相對應模態對照圖
72
附錄D 本論文提出的元素分析單層複材彈簧的模態比對圖 101
附錄E使用論文提出元素模體疊層複材空心彈簧模態對照圖 125

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