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 摘要 本文在不考慮樑的軸向延展性、質量慣性矩、科氏力（Coriolis force）影響的情況下，利用Hamililton’s principle推導出，一彈性拘束的非均勻、非對稱樑的三條bending-bending-torsion的統御方程式及邊界條件。以半解析方法求得統御特性微分方程式（governing characteristic differential equation）的轉移矩陣（transition matrix），再配合邊界條件而求得系統的頻率方程式，及推導出特徵函數（eigenfunction）的正交條件，再以疊加法得到系統的強迫振動解。本文探討了warping對系統的影響、截面幾何參數、樑長度比例及均勻預扭對自然頻率的影響。
 ABSTRACT The governing differential equations and twelve boundary conditions for the bending-bending-torsion vibration of a nonuniform and asymmetry beam with the elastic boundary conditions are derived by using Hamilton’s principle. The frequency equation of the system is derived and expressed in terms of the transition matrix of the transformed vector characteristic governing equation. An efficient algorithm for determining the semi-analytical transition matrix of the system is derived. The orthogonality condition for the eigenfunctions of the system with elastic boundary conditions is also derived. Using the method of mode superposition derives the analytical solution of force vibration of thin-walled beam. The influence of the warping function, the geometric parameter of the cross-section of the beam, the parameter between the cross-section and length of the beam and the uniform pretwists on the natural frequencies are investigated.
 目 錄摘要 ……………………………………………………………………. Ⅰ英文摘要 ………………………………………………………………. Ⅱ誌謝 ……………………………………………………………………. Ⅲ目錄 ……………………………………………………………………. Ⅳ表目錄 …………………………………………………………………. Ⅵ圖目錄 …………………………………………………………………. Ⅵ符號說明 ………………………………………………………………. Ⅷ第一章 緒論 ………………………………………………………….. 1 1.1 前言 …………………………………………………….…….. 1 1.2 文獻回顧 ………………………………………………….….. 1 1.3 研究方向及目的 ………………………………………….….. 4第二章 推導統御方程式 ………………………..………………..…. . 5 第三章 無因次化 ……………………………………………………. 11 3.1 無因次化參數 ..………………………………………………11 3.2 無因次化的統御方程式及邊界條件、初始條件 …….…… 13 第四章 解法 ..……………………………………………….………. 174.1 以線性系統理論分析自由振動 ……..…………….…..…… 17 4.1.1 將統御特性微分方程式變換向量 …………..……… 18 4.1.2 系統的轉移矩陣 …………………………………..… 22 4.1.3 系統的頻率方程式 ………………………………….. 23 4.2 強迫振動分析 ……….…..…………..………….………….. 26 4.2.1 正交條件 ………………………..…………………… 26 4.2.2 以模態疊加法求得系統的解 ……..………………… 28第五章 數值結果與討論 …..……………………………………….. 30 5.1 數值參數 ……………………………………………………. 30 5.2 數值收斂情形與數值比較 …………………………………. 30 5.3 數值結果與討論 ……………………………………………. 35 第六章 結論 …………………………………………………..……... 38 參考文獻 …………………………………….…………………………60 附錄 ………………………………………………………………….… 63 自述 ………………………………………………………………….… 69表目錄表1. 以本文方法求得前四個無因次自然頻率的收斂表...…..….… ..32表2. 前四個薄壁樑位於z軸對稱位置上的無因次自然頻率與純彎曲解的數值比較…………………………………………………...32表3. 前三個薄壁樑位於z軸對稱位置上，有無考慮warping的無因次自然頻率的數值比較…………………………………….……..33表4. 用本文方法求得對稱截面懸臂式預扭樑的前四個自然頻率與參考文獻的數值比較…………….………………………………..34圖目錄圖1. 預扭薄壁樑的幾何及截面座標系統圖…………………………40圖2. 樑在z軸對稱上，在不同截面夾角α的情況下，圓弧截面半徑變化率λ2對薄壁樑的自然頻率之影響 …………………………41圖3. 樑在z軸對稱上，在不同端點截面夾角α的情況下，圓弧截面夾角變化率λ3對薄壁樑的自然頻率之影響 ……………………42圖4. 樑在z軸對稱上，在固定截面夾角α的情況下，樑的截面半徑與樑的長度比值η2對薄壁樑的自然頻率之影響 ……………43圖5. 樑在z軸對稱上，在固定截面夾角α的情況下，樑的截面厚度沿著x軸做λ1 的線性變化對薄壁樑的自然頻率之影響 ……44圖6. 樑在z軸對稱上，在固定截面夾角α的情況下，樑的圓弧截面半徑變化率λ2對薄壁樑的第一模態位移之影響 ……………45圖7. 樑在z軸對稱上，在固定截面夾角α的情況下，樑的圓弧截面半徑變化率λ2對薄壁樑的第二模態位移之影響 ……………46圖8. 樑在z軸對稱上，在固定截面夾角α的情況下，樑的圓弧截面半徑變化率λ2對薄壁樑的第三模態位移之影響 ……………47圖9. 樑在z軸對稱上，在一端點截面夾角α=60°的情況下，圓弧截面夾角變化率λ3對薄壁樑的第一模態位移之影響 …………48圖10. 樑在z軸對稱上，在一端點截面夾角α=60°的情況下，圓弧截面夾角變化率λ3對薄壁樑的第二模態位移之影響 …………49圖11. 樑在z軸對稱上，在一端點截面夾角α=60°的情況下，圓弧截面夾角變化率λ3對薄壁樑的第三模態位移之影響 …………50圖12. 在不考慮剪力中心下，不同均勻預扭角對均勻預扭懸臂式薄壁樑的自然頻率之影響……………………………………………51圖13.不同均勻預扭角對均勻預扭懸臂式薄壁樑的自然頻率之響……52圖14. 在固定預扭角及截面夾角α的情況下，樑的截面半徑與樑的長度比值η2對均勻預扭懸臂式薄壁樑的自然頻率之影響……………………53圖15.在固定均勻預扭角的情況下，樑的截面夾角α對均勻預扭懸臂式薄壁樑的自然頻率之影響……………………………………54圖16.在固定截面夾角α及預扭角φ的情況下，樑的截面厚度沿著x軸做λ1 的線性變化對均勻預扭懸臂式薄壁樑的自然頻率之影響…………………………………………………………………55圖17.在固定截面夾角α及預扭角的情況下，y方向的移動彈簧係數β1對均勻預扭懸臂式薄壁樑的自然頻率之影響 ……………56圖18.在固定截面夾角α及預扭角的情況下，z方向的移動彈簧係數β2對均勻預扭懸臂式薄壁樑的自然頻率之影響 ……………57圖19.在固定截面夾角α及預扭角的情況下，y方向的旋轉彈簧係數β3對均勻預扭懸臂式薄壁樑的自然頻率之影響 ……………58圖20.在固定截面夾角α及預扭角的情況下，z方向的旋轉彈簧係數β4對均勻預扭懸臂式薄壁樑的自然頻率之影響 ……………59
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Cranch, “A Theory of Torsional and Coupled Bending Torsional Waves in Thin-Walled Open Section Beams,” Journal of Applied Mechanics, Vol. 34, 1967, pp. 337-343.8.Y. Y. Yu, “Variational Equation of Motion for Coupled Flexure and Torsion of Bars of Thin-Walled Open Section Including Thermal Effect,” Journal of Applied Mechanics, Vol. 38, 1971, pp. 502-506.9.S. Dubigeon and C. B. Kim, “A Finite Element for The Study of Coupled Bending-Prevented Torsion of a Straight Beam,” Journal of Applied Mechanics, Vol. 81, 1982, pp. 255-270.10. A. W. Leissa, “Vibrations of Shells,” NASA SP-288, 1973.11. T. Irie, G. Yamada and K. Tanaka, “Free Vibration of A Thin-Walled Beam-Shell of Arc Cross-Section,” Journal of Sound and Vibration, Vol. 94, No 4,1984, pp. 563-572.12. G. A. Gunnlaugsson and P. T. Pederson, “A Finite Element Formulation for Beams with Thin Walled Cross-Sections,” Comput. Struct. Vol. 15, 1982, pp. 691-699.13. T. 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