臺灣博碩士論文加值系統

本論文採用兩節點，每節點有九個自由度的直樑元素，發展出一模擬具矩形截面螺旋形彈簧動態特性的有限元素模式，其中螺旋彈簧材料可為等向性材料或複合材料。所採用的直樑元素包含橫向剪力變形、扭轉、截面翹曲、弦向曲率、以及軸向、橫向和側向位移等效應。
本文採用劉大成[11]的直樑元素運動方程，其藉由漢米爾頓原理配合有限元素推導得到。為了應用此直樑元素分析螺旋彈簧動態特性，需先求得彈簧幾何形狀與直樑元素所採用區域座標之間關係。為此目的，本文利用 Frenet-Serret formulas [13]，導出每一元素區域座標與全域座標間的座標轉換矩陣，再利用此座標轉換矩陣並考慮相鄰兩直樑元素之間共同節點位移的連續性，找出位移轉換矩陣。藉此位移轉換矩陣以及直樑元素的質量與勁度矩陣，推導出分析具矩形截面螺旋形彈簧動態特性的有限元素方程的全域質量與勁度矩陣。
本文實例中首先分析等向性鋼材螺旋彈簧，並與套裝軟體 ansys比較，發現在兩端無拘束和一端固定都是在寬厚比值為 3 時，本文分析值與 ansys 結果相差較少。其次，本文分析了兩端為自由端(ff)和一端為固定端(cf)的單層與疊層複合材料螺旋彈簧，探討不同寬厚比(b/h)、纖維角度、疊層角度對系統自然頻率和模態之影響。由本文分析結果顯示，單層複材彈簧在不同寬厚比，其自然頻率隨纖維角度之變化並沒有規律性。分析複材疊層彈簧時，僅考慮 b/h=3 的情形。對疊層[0/θ/0/-θ]s而言，不管彈簧是無拘束或頭端拘束，自然頻率最大值都會出現在θ值接近 45 度時。當彈簧頭端拘束時，疊層[0/θ/0/-θ]s與[90/θ/0/-θ]s於第二與第三模態會出現 w 與 ϕ的耦合振型。當彈簧無拘束時，這兩種疊層於第七或第八也出現此 w 與ϕ的耦合模態。

In this thesis, a finite element model based on two-noded straight beam finite elements, with each node of nine degrees of freedom, is developed to study the dynamic characteristics of helical springs having rectangular cross sections. The helical springs could be made of isotropic materials or fiber-reinforced composite materials. The straight beam element being adopted has included the effects of the transverse shear deformation, torsion, cross-section warping, chordwise curvature, axial, transversal and lateral displacements.
The equation of motion of the straight beam element developed in [11] is used, which is derived by employing Hamilton’s principle together with the finite element method. In order to apply the straight beam element to the analyses of the dynamic characteristics of the helical spring, the relation between the spring geometry and the local coordinate systems chosen for straight beam elements must be obtained first. For this purpose, Frenet-Serret formulas [13] are used to derive the coordinate transformation matrix between local coordinate systems of beam elements and global coordinate system. By using the obtained coordinate transformation matrices and considering the continuity of the displacement of the common node between two adjacent straight beam elements, the displacement transformation matrices for these two beam elements are found. With displacement transformation matrices and the mass and stiffness matrices of the straight beam elements, the global mass and stiffness matrices of helical springs are then derived.
In the numerical examples, first, the steel helical springs free of constraints and with one of its ends fixed are analyzed. The results are compared with those obtained from commercial software Ansys. It is shown that in both boundary conditions the natural frequencies obtained from the present model and those of Ansys are better in agreement when the width to thickness ratio (b/h) of spring is three. Next, the single-layered and laminated composite helical springs are studied, in which the influences of the width to thickness ratio (b/h), the fiber angle, and the angles of lamination on the natural frequencies and mode shapes are investigated. It is found that for different b/h ratios, natural frequencies of the single-layered composite spring vary irregularly with fiber angles. When analyzing composite laminated springs, only cases with b/h = 3 are considered. For springs with lamination [0/θ/0/-θ]s regardless of being free or fixed at one end, the maximum values of the natural frequencies appear at the angle θ close to 45 degrees. When the spring is fixed at one end, both 2nd and 3rd modes of composite springs with laminations [0/θ/0/-θ]s and [90/θ/0/-θ]s are coupling mode of w and ϕ. When the springs are free, springs of both laminations also have this coupling mode to appear either at 7th or 8th mode.

誌謝………………………………………………………………………I
中文摘要…………………………………………………………………II
ABSTRACT…………………………………………………………………IV
目錄………………………………………………………………………VI
符號說明…………………………………………………………………VIII
圖目錄 ……………………………………………………………………X
表目錄……………………………………………………………………XIV
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…………………………………………………………5
2.1 單層複合材料板的本構方程式………………………………………5
2.2 直樑元素的運動方程……………………………………………10
2.2.1 直樑元素的位移場..…………………………………………10
2.2.2 內插函數………………………………………………………14
2.2.3 運動方程式……………………………………………………18
2.3 轉換矩陣推導……………………………………………………21
2.3.1 座標轉換矩陣…………………………………………………21
2.3.2 位移轉換矩陣…………………………………………………27
2.4 轉換後直樑元素之運動方程……………………………………33
第三章 實例分析………………………………………………………34
3.1 數值驗證……………………………………………………………34
3.2 實例分析…………………………………………………………79
3.2.1 無拘束彈簧……………………………………………………79
3.2.2 頭端固定彈簧…………………………………………………85
3.2.3 不同疊層角度下的螺旋彈簧振動頻率………………………91
第四章 結論與未來展望…………………………………………………100
4.1 結論……………………………………………………………………100
4.2 未來展望…………………………………………………………101
參考文獻…………………………………………………………………102
附錄 A:直樑元素的質量與勁度矩陣[11]……………………………104
附錄 B:第三張一些實例的模態圖……………………………………113

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