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研究生： 吳志祥
Wu, Jyh-Shyang
論文名稱： 可拉伸之非均勻曲樑的平面振動
In-Plane Vibration of Extensional Curved Non-uniform Beams
指導教授： 李森墉
Lee, Sen-Yung
學位類別： 博士
Doctor
系所名稱： 工學院 - 機械工程學系
Department of Mechanical Engineering
論文出版年： 2009
畢業學年度： 97
語文別： 英文
論文頁數： 76
外文關鍵詞： extensional, curved non-uniform Timoshenko beam, free vibration
相關次數： 點閱：76  下載：1 
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    • 於非均勻 Timoshenko 曲樑的同平面振動分析中，首先利用漢米頓原理求得三個耦合的微分方程式。進而藉由三個具有物理意義的參數簡化原來三個耦合的微分方程式以便於分析。經由消去軸向位移及橫向位移後，三個耦合的微分方程式非耦合化，而成為一以彎曲旋轉角為因變數的六階微分方程式。軸向位移及橫向位移亦可表示成以彎曲旋轉角為因變數的關係式。如果非均勻曲樑的材料及幾何變化可利用多項式的形式表示，那麼同平面振動非均勻Timoshenko 曲樑的真確解即可獲得。最後再以一些極端的例子來說明推導的正確性，並討論曲樑錐度、及曲樑長度對第一、二自然頻率的影響。

    The three coupled governing differential equations for the in-plane vibrations of curved non-uniform Timoshenko beams are derived via the Hamilton’s principle. Three physical parameters are introduced to simplify the analysis. By eliminating all the terms with the axial displacement parameter, then reducing the order of differential operator acting on the flexural displacement parameter, one uncouples the three governing characteristic differential equations with variable coefficients and reduces them into a sixth-order ordinary differential equation with variable coefficients in term of the angle of the rotation due to bending for the first time. The explicit relations between the axial and the flexural displacements and the angle of the rotation due to bending are also revealed. It is shown that if the material and geometric properties of the beam are in arbitrary polynomial forms, then the exact solutions for the in-plane vibrations of the beam can be obtained. Several limiting studies are illustrated. Finally, limiting cases are studied and the influence of the taper ratio and the arc length on the first two natural frequencies of the beams is explored.

    中文摘要 I Abstract II 誌謝 III Contents IV List of Tables VII List of Figures VIII Nomenclature XI Chapter 1 Introduction 1 1.1 Literature Review 1 1.2 Scope 5 2 Free Vibration of Extensional Curved Non-uniform Timoshenko Beams 8 2.1 Coupled Governing Equations 8 2.2 Uncoupled Governing Equation in Terms of the Angle of Rotation due to Bending 11 2.2.1 Curved Non-uniform Timoshenko Beams 11 2.2.1.1 Coupled Differential Equations in terms of V and  12 2.2.1.2 Uncoupled Differential Equations in terms of  13 2.2.1.3 Explicit Relations 15 2.2.2 Curved Uniform Timoshenko Beams 17 2.2.3 Curved Non-uniform Bernoulli-Euler Beams 18 2.2.4 Curved Uniform Bernoulli-Euler Beams 18 2.2.5 Straight Non-Uniform Timoshenko Beams 19 2.2.6 Uniform Straight Timoshenko Beams 20 2.3 Exact Fundamental Solutions 21 2.4 Numerical Results and Discussion 22 2.5 Conclusions 26 3 Free Vibration of Extensional Non-uniform Bernoulli-Euler Beams with Variable Curvature 37 3.1 Coupled Governing Equations 37 3.2 Uncoupled Governing Differential Equation in Terms of the Flexural Displacement 41 3.2.1 Extensional Non-uniform Beams with Variable Curvature 41 3.2.1.1 Uncoupled Governing Differential Equations in Terms of V* 43 3.2.1.2 Explicit Relations 44 3.2.2 In-extensional Non-uniform Beams with Variable Curvature 45 3.2.3 Extensional Non-uniform Beams with Constant Curvature 47 3.2.4 Extensional Uniform Beams with Variable Curvature 48 3.2.5 Extensional Uniform Beams with Constant Curvature 48 3.2.6 Straight Non-Uniform Bernoulli-Euler Beams 49 3.2.7 Straight Uniform Bernoulli-Euler Beams 50 3.2.8 Straight Non-Uniform Bars 50 3.2.9 In-extensional Non-uniform Beams with Constant Curvature 51 3.2.10 In-extensional Uniform Beams with Variable Curvature 51 3.2.11 In-extensional Uniform Beams with Constant Curvature 52 3.3 Exact Fundamental Solutions 52 3.4 Numerical Results and Discussion 54 3.5 Conclusions 56 4 Conclusions 64 Reference 67 Appendix 75 Vita 76

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