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研究生： 張凱評
Chang, Kai-Ping
論文名稱： 非均勻變曲率樑的異平面振動分析
Out of Plane Vibrations of Non-uniform Beams with Variable Curvature
指導教授： 李森墉
Lee, Sen-Yung
學位類別： 碩士
Master
系所名稱： 工學院 - 機械工程學系
Department of Mechanical Engineering
論文出版年： 2011
畢業學年度： 99
語文別： 英文
論文頁數： 51
中文關鍵詞： 異平面振動 、曲樑 、變曲率 、非均勻樑元素 、自由振動
外文關鍵詞： out-of-plane vibration, curved beam, variable curvature, non-uniform beam, free vibration
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    • 於非均勻變曲率樑的異平面振動分析中，首先利用漢米爾頓原理求得兩個耦合的微分方程式，進而藉由兩個具有物理意義的參數簡化原來兩個耦合的微分方程式以便於分析。經由消去扭轉旋轉角後，兩個耦合的微分方程式非耦合化，而成為一個以彎曲位移為因變數的六階微分方程式。扭轉旋轉角亦可表示成以彎曲位移為因變數的關係式。如果非均勻變曲率樑的材料及幾何變化可利用多項式的形式表示，那麼異平面非均勻變曲率樑的振動真確解即可獲得。最後再以一些常用的例子來說明推導的正確性，並討論邊界條件、曲樑錐度、曲樑細長比、變曲率及曲樑中心角對自然頻率的影響。

    The two coupled governing differential equations for the out-of-plane vibrations of non-uniform beams with variable curvature are derived via the Hamilton’s principle. Two physical parameters are introduced to simplify the analysis. By reducing the order of differential operator acting on the torsional angle, one uncouples the two governing characteristic differential equations with variable coefficients and reduces them into a sixth-order ordinary differential equation with variable coefficients in terms of the flexural displacement parameter for the first time. The explicit relations between the flexural displacement and the torsional angle 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 out-of-plane vibrations of non-uniform beams with variable curvature can be obtained. Finally, the influence of the boundary conditions, the taper ratio, the slenderness ratio, the curvature parameter and the arc angle parameter on the curved beams is explored.

    中文摘要 .............................................. I Abstract ............................................. II 誌謝 .................................................. III Contents ............................................. IV List of Tables ....................................... VI List of Figures ...................................... VII Nomenclature ......................................... X Chapter 1 Introduction ....................................... 1 1.1 Preface ......................................... 1 1.2 Literature Review ............................... 1 1.3 Research Motive and Method ...................... 4 1.4 Scope ........................................... 5 2 Coupled Governing Equations ........................ 7 3 Out-of-Plane Vibrations ............................ 13 3.1 Uncoupled Governing Differential Equations in Terms of the Flexural Displacement ............................ 14 3.1.1 Non-uniform Beams with Variable Curvature ...... 14 3.1.1.1 Coupled Differential Equations in terms of W* and Ф*..................................................15 3.1.1.2 Uncoupled Governing Differential Equations in terms of W*............................................... 16 3.1.1.3 Explicit Relations ........................... 18 3.1.2 Non-uniform Beams with Constant Curvature ...... 18 3.1.3 Uniform Beams with Variable Curvature .......... 19 3.1.4 Curved Uniform Bernoulli-Euler Beams ........... 20 3.2 Exact Fundamental Solutions ...................... 20 4 Numerical Results and Discussion ................... 23 5 Conclusions ........................................ 30 Reference ............................................ 46 Appendix ............................................. 50 Vita ................................................. 51

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