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系統識別號 U0026-1008201011333000
論文名稱(中文) 橫斷面等向性圓板之軸對稱自由振動
論文名稱(英文) Axisymmetric Free Vibration of Transversely Isotropic Circular Plate
校院名稱 成功大學
系所名稱(中) 土木工程學系碩博士班
系所名稱(英) Department of Civil Engineering
學年度 98
學期 2
出版年 99
研究生(中文) 梁文宇
研究生(英文) Wen-Yu Liang
學號 N6697118
學位類別 碩士
語文別 中文
論文頁數 59頁
口試委員 指導教授-譚建國
口試委員-陳東陽
口試委員-宋見春
中文關鍵字 狀態空間法  自由振動  圓板  Mindlin板理論 
英文關鍵字 State space approach  free vibration  circular plate  Mindlin plate theory 
學科別分類
中文摘要 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。
英文摘要 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.
論文目次 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
參考文獻 1. Civan, F.: Solving multivariable mathematical models by the quadrature and cubature methods. Numerical Method for Partial Differential Equations 10, pp.545-567 (1994).
2. Ebenezer, D.D., Ravichandran, K., Chandramouli Padmanabhan: Forced vibrations of solid elastic cylinders, Journal of Sound and Vibration 282, pp.991-1007 (2005).
3. Hutchinson, J.R.: Vibrations of solid cylinders, Journal of Applied Mechanics 47, pp.901-907 (1980).
4. Hutchinson, J.R.: Vibrations of free hollow circular cylinders, Journal of Applied Mechanics 53, pp.641-646 (1988).
5. Leissa, A.W.: Vibration Of Plate, US Govt Printing Office, Washington, D.C. (1969).
6. Lusher, C.P., Hardy, W.N.: Axisymmetric free vibrations of a transversely isotropic finite cylindrical rod, Journal of Applied Mechanics 55, pp.855-862 (1988).
7. Liew, K.M., Han, J.B., Xiao, Z.M.: Vibration analysis of circular Mindlin Plates using the Differential Quadrature Method. Journal of Sound and Vibration 205(5), pp.617-630 (1997).
8. Mindlin, R.D.: Influence of rotatory inertia and shear on flexural motion of isotropic, elastic plates. Transactions of the American Society of Mechanical Engineers, Journal of Applied Mechanics 18, pp.31-38 (1951).
9. Tarn, J.Q.: A state space formalism for anisotropic elasticity. Part II: Cylindrical anisotropy, International Journal of Solids and Structure 39, pp.5157-5172 (2002b)
10. Tarn, J.Q., Chang, H.H., Tseng, W.D.: Axisymmetric Deformation of a Transversely Isotropic Cylindrical Body: A Hamiltonian State-Space Approach, Journal of Elasticity 97(2), pp.131-154 (2009).
11. Wei, X.X., Chau, K.T.: Finite and transversely isotropic elastic cylinderunder compression with end constraint induced by friction. International Journal of Solids and Structures 46, pp.1953-1965 (2009).
12. 武藍和, 李向國, 王立彬.: 圓板軸對稱自由振動的微分容積法.振動與衝擊 22(3), pp.75-77 (2003).
13. 范家讓:強厚度疊層板殼的精確理論, 科學出版社, 北京 (1996).
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