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研究生： 梁文宇
Liang, Wen-Yu
論文名稱： 橫斷面等向性圓板之軸對稱自由振動
Axisymmetric Free Vibration of Transversely Isotropic Circular Plate
指導教授： 譚建國
Tarn, Jian-quo
學位類別： 碩士
Master
系所名稱： 工學院 - 土木工程學系
Department of Civil Engineering
論文出版年： 2010
畢業學年度： 98
語文別： 中文
論文頁數： 59
中文關鍵詞： 狀態空間法 、自由振動 、圓板 、Mindlin板理論
外文關鍵詞： State space approach, free vibration, circular plate, Mindlin plate theory
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    • 本文擬採用狀態空間法，解析橫斷面等向性圓板之軸對稱自由振動。由三維彈性力學基本方程式出發，在圓柱座標下，建立橫斷面等向性圓板軸對稱自由振動之狀態空間方程式。文中將各應力與位移分量皆以兩組狀態方程式之特別解組合表示，含適量之待定係數，可於圓板任意邊界情況下，求得相對之自由振動頻率。數值結果以等向性材料與參考文獻數值比較，探討線性剪力變形之Mindlin板理論適用之厚度。

    On the basis of the state space approach, the axisymmetric free vibration of transversely isotropic circular plate is analyzed. The state equation of axisymmetric free vibration of transversely isotropic circular plate is established from three-dimensional basic equations of elasticity in the cylindrical coordinates. By using separation of variables, the components of displacements and stresses are expressed as a superposition of two sets of particular solutions of the state equation with enough coefficients to satisfy arbitrary boundary conditions and determine the frequencies for the free vibration of plate. The results for the transversely isotropic and the isotropic plates are presented, and the latter are compared with those based on the linear shear-deformation Mindlin plate theory.

    摘要 ....................................................... I Abstract .................................................. II 致謝 ..................................................... III 目錄 ...................................................... VI 表目錄 ................................................... VIII 圖目錄 .................................................... IX 符號表 .................................................... XI 第一章 緒論 .............................................. 1 第二章 圓柱座標系下之狀態空間方程式 ........................ 5 2-1 基本方程式 ............................................. 5 2-2 狀態空間方程式 ......................................... 6 第三章 狀態空間方程式之解 ................................. 10 3-1 第一類型特別解 ........................................ 12 3-2 第二類型特別解 ........................................ 15 3-3 兩特別解之組合 ........................................ 17 3-4 邊界條件之滿足 ........................................ 19 3-4-1 頂底兩面邊界條件之滿足 .................... 19 3-4-2 圓板周邊各邊界條件之滿足 .................. 22 (一) 固定邊界 .............................. 22 (二) 簡支邊界 .............................. 26 (三) 自由邊界 .............................. 30 (四) 輥支邊界 .............................. 32 第四章 數值結果與探討 ..................................... 35 4-1 簡化矩陣式 ............................................ 35 4-2 圓板之材料性質 ........................................ 36 4-3 無因次化 .............................................. 36 4-4 數值解析方法 .......................................... 36 4-5 無窮級數取有限項時之收斂性質 .......................... 37 4-6 理論驗證 .............................................. 38 4-7 數值列表 .............................................. 38 4-7-1 無窮級數收斂性與穩定性 .................... 38 4-7-2 等向性材料之解與文獻比較 .................. 44 第五章 結論 ............................................... 47 參考文獻 .................................................. 48 附錄A ..................................................... 50 附錄B ..................................................... 53

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