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博碩士論文 etd-0730101-132539 詳細資訊
Title page for etd-0730101-132539
論文名稱
Title
滿足應力邊界條件圓形板及環形板之三維振動分析
THREE-DIMENSIONAL VIBRATION ANALYSIS SATISFYING STRESS BOUNDARY CONDITIONS OF CIRCULAR AND ANNULAR
系所名稱
Department
畢業學年期
Year, semester
語文別
Language
學位類別
Degree
頁數
Number of pages
83
研究生
Author
指導教授
Advisor
召集委員
Convenor
口試委員
Advisory Committee
口試日期
Date of Exam
2001-06-29
繳交日期
Date of Submission
2001-07-30
關鍵字
Keywords
環形板、圓形板、混合型有限元素、振動
annular plate, circular plate, mixed finite element, vibration
統計
Statistics
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The thesis/dissertation has been browsed 5668 times, has been downloaded 1348 times.
中文摘要
本文擬針對圓形板及環形板之三維振動現象,以混合型之有限元素來加以分析。
此混合型有限元素是以位移及應力為主要變數,因此所有的位移及應力之邊界條件均可以如實加上,同時所採用的有限元素是根據三維彈性力學的軸對稱元素,再加以修改,因此可以得到最有涵蓋性的圓形板及環形板之三維軸對稱與非軸對稱的振動分析結果。
本文之結果將與傳統的位移型有限元素分析,Ritz method的分析以及series method的分析結果做比較,以顯示其準確性及有效性,尤其可以了解應力邊界條件對圓形板之非軸對稱振動的影響,此在文獻上尚無任何記載。
Abstract
In the proposed project,the three - dimensional vibration of circular and annular plates is analyzed by a mixed finite element。
Stresses,as well as displacements, are primary variables in the mixed finite element formulation,therefore,all the stress and displacement boundary conditions can be imposed exactly。Meanwhile,the proposed finite element is a modification of axisymmetric finite element which is based on three – dimensional elasticity,so general results of both axisymmetric and unaxisymmetric vibration of circular and annular plates can be obtained。
Results of the present project will be compared to those by conventional displacement – type finite element,Ritz method and series method to show the difference among these theories。Especially,the effect of satisfying the stress boundary conditions on the unaxisymmetric vibration analyses can be demonstrated,which is not available in the literature up to date。
目次 Table of Contents
摘要…………………………………………………………………… i
目錄 ……………………………………………………………… i i
表目錄 ……………………………………………………………… iv
圖目錄 ……………………………………………………………… vi
第一章 緒論 ……………………………………………………… 1
1-1 前言 …………………………………………………… 1
1-2 文獻回顧 ……………………………………………… 2
1-2-1 古典板理論 …………………………………… 2
1-2-2 Mindlin板理論 ……………………………… 3
1-2-3 三維彈性力學分析法 ………………………… 4
第二章 滿足應力邊界條件圓形板及環形板之三維振動分析 … 7
2-1 前言 …………………………………………………… 7
2-2 理論推導 ……………………………………………… 7
第三章 問題解析 ………………………………………………… 19
3-1 前言 …………………………………………………… 19
3-2 問題描述 ……………………………………………… 19
(1) 材料性質 …………………………………………… 19
(2) 邊界條件 …………………………………………… 20
(3) 幾何比例 …………………………………………… 20
(4) 元素選取 …………………………………………… 21
(5) 自然振動頻率之無因次化 ………………………… 21
第四章 結果與討論 …………………………………………… 26
4-1 前言 …………………………………………………… 26
4-2 收斂試驗 ……………………………………………… 26
4-3 符號說明 ……………………………………………… 28
4-4 文獻比較與結果之討論 ……………………………… 29
第五章 結論與建議 …………………………………………… 71
5-1 結論 …………………………………………………… 71
5-2 建議 …………………………………………………… 72

參考文獻 ………………………………………………………… 73
附錄 A ………………………………………………………… 77
附錄 B ………………………………………………………… 81





表目錄

表 4.1 圓形板在不同邊界條件及不同元素分割方式情況下的前五個無因次化自然振動頻率 (υ= 0.3 and n = 0 ) ……… 33
表 4.2 圓形板在不同邊界條件及不同元素分割方式情況下的前五個無因次化自然振動頻率 (υ= 0.3 and n = 1 ) ……… 36
表 4.3 圓形板在不同邊界條件及不同元素分割方式情況下的前五個無因次化自然振動頻率 與位移型態(DIS.)之比較( a/h = 5,υ= 0.3 and n = 0 ) ……………………………………… 39
表 4.4 圓形板在不同邊界條件及不同元素分割方式情況下的前五個無因次化自然振動頻率 與位移型態(DIS.)之比較( a/h = 5,υ= 0.3 and n = 1 ) ……………………………………… 42
表4.5 環形板在不同邊界條件及不同元素分割方式情況下的前五個無因次化自然振動頻率 與位移型態(DIS.)之比較(a/h = 10 , b/a = 0.5 , υ= 0.3 and n = 0) …………………………… 45
表4.6 環形板在不同邊界條件及不同元素分割方式情況下的前五個無因次化自然振動頻率 與位移型態(DIS.)之比較(a/h = 10 , b/a = 0.5 , υ= 0.3 and n = 1) …………………………… 48
表4.7 環形板在不同邊界條件及不同元素分割方式情況下的前五個無因次化自然振動頻率 與位移型態(DIS.)之比較(a/h = 10 , b/a = 0.5 , υ= 0.3 and n = 2) …………………………… 51
表4.8 環形板在不同邊界條件及不同元素分割方式情況下的前五個無因次化自然振動頻率 與位移型態(DIS.)之比較(a/h = 10 , b/a = 0.5 , υ= 0.3 and n = 3) …………………………… 54
表4.9 本文、位移型態列式(DIS.)、3-D Ritz 和 2-D Mindlin 圓形板無因次化彎曲自然振動頻率之比較,a/h=10 (mesh 20×2) and a/h=5 (mesh 10×2) ……………………………………… 57
表4.10 本文、位移型(DIS.)、3-D Ritz(3DR)、3-D Hutchinson’s series(3DH)和 2-D Mindlin(2DM)環形厚板無因次化彎曲自然振動頻率之比較,a/h=2.5,b/a=0.5(mesh 10X4) and a/h=1,b/a=0.5(mesh 8X8) ………………………………………… 58
表4.11 本文、位移型(DIS.)、3-D Ritz(3DR) 和3-D finite element(3DF)環形厚板無因次化彎曲自然振動頻率之比較,a/h=1.7,b/a=0.1762(mesh 6X4) ………………………… 60
表4.12 本文、位移型(DIS.)、3-D exact 和2-D Mindlin 夾支撐(Clamp)圓形板無因次化彎曲自然振動頻率之比較,(20X4 mesh for a/h=5 and 10X4 for a/h=2.5) ……………………………… 61

圖目錄

圖2.1 圓形板及環形板座標系及示意圖 …………………………18
圖3.1 圓形板邊界條件 ………………………………………… 22
圖3.2環形板邊界條件 ………………………………………… 24
圖4.1在不同邊界條件下,第一無因次自然振動頻率與幾何比例 a/h 之關係圖…………………………………………………… 62
圖4.2 無因次自然振動頻率與分割節點數之關係圖 ………… 63
圖4.3 典型radial straining mode ……………………………… 65
圖4.4 典型circumferentially vibrating mode …………………… 66
圖4.5 典型flexural vibrating mod ……………………………… 67
圖4.6 典型symmetric vibration mode ………………………… 68
圖4.7 典型antisymmetric vibration mode …………………… 69
圖4.8 混合型振動模態 ……………………………………… 70
參考文獻 References
[1] Leonard Meirovitch 1967,New York: Macmillan Publishing Co.,Inc. ”Analytical methods in vibration”.

[2] S.M.Vogel and D.W.Skinner 1965, Naturnal Frequencies of Transversely Vibrating Uniform Annular Plates. Journal of Applied Mechanics 32, p926-931.

[3] K.Vijayakumar and G.K.Ramaiah 1972, On The Use of A Coordinate Transformation For Analysis of Axisymmetric Vibration of Polar Orthotropic Annular Plates. Journal of Sound and Vibration 24(2), p165-175.

[4] K.Vijayakumar and G.K.Ramaiah 1973, Naturnal Frequencies of Polar Orthotropic Annular Plates. Journal of Sound and Vibration 26(4), p517-531.

[5] Vincent X. Kunukkasseril and A.S.J.Swamidas 1974, Vibration of Continuous Circular Plates. International journal of solids and structures 10, p603-619.

[6] Y.Narita 1984, Free Vibration of Continuous Polar Orthotropic Annular and Circular Plates. Journal of Sound and Vibration 93(4), p503-511.

[7] I.A.Minkarah and W.H.Hoppmann 1963, Flexural Vibrations of Cylindrically Aeolotropic circular Plates. The Journal of The Acoustical Society of America 36, p475-475.

[8] R.Lal and U.S.Gupta 1981, Axisymmetric vibrations of Polar Orthotropic Annular Plates of Variable Thickness. Journal of Sound and Vibration 83(2), p229-240.

[9] A.W.Leissa 1969, NASA SP-169Vibration of Plates. Washingtion, D.C.Office of Technology Utilization.
[10]A.W.Leissa 1977,. Recent Research in Plate Vibrations:classical theory. The Shock and Vibration Digest 9(10), p13-24

[11]A.W.Leissa 1981, Plate Vibration Research 1976-1980:classical theory. The Shock and Vibration Digest 13(9), p11-22.

[12]H.Deresiewicz and R.D.Mindlin 1954, Thickness-Shear and Flexural Vibrations of A Circular Disk Journal of Applied Physics 25, p1329-1332.

[13] H.Deresiewicz and R.D.Mindlin 1955, Axisymmetric Flexural Vibrations of A Circular Disk. Journal of Applied Mechanics 22, p86-88.

[14] W.R.Callahan and Jagjit Singh Bakshi 1965, Flexural Vibrations of A Circular Ring When Transverse Shear and Rotary Inertia are Considered. Journal of The Acoustical Society of America 40, p372-375.

[15] S.R.Soni and C.L. Amba-RaoLord 1975, On Radially Symmetric Vibrations of Orthotropic Non-Uniform Disks Including Shear Deformation. Journal of Sound and Vibration 40(1), p57-63.

[16] Lord Rayleigh 1877, Axisymmetric Vibrations of Polar Orthotropic Mindlin Annular Plates of Variable Thickness. Journal of Sound and Vibration 40(1), p57-63.

[17]S.S.Rao and A.S.Prasad 1975, Vibrations of Annular Plates Including The Effects of Rotatory Inertia and Transverse Shear Deformation. Journal of Sound and Vibration 42(3), p305-324.

[18]T.Irie,G.Yamada and S.Aomura 1979, Free vibration of Amindlin Annular Plate of Varying Thickness. Journal of Sound and Vibration 66(2), p187-197.

[19]G.M.L.Gladwell and D.K.Vijay 1975, Natural Frequencies of Finite-Length circular Cylinders. Journal of Sound and Vibration 42(3), p387-397.

[20]J.R.Hutchinson 1979, 4. Axisymmetric Flexural Vibrations of A Thick Free Circular Plate. Journal of Applied Mechanics 46, p139-14

[21]J.R.Hutchinson 1984, Axisymmetric Flexural Vibrations of A Thick Free Circular Plate. Journal of Applied Mechanics 51, p581-585.

[22]J.R.Hutchinson and J.A.El-Azhari 1986, Axisymmetric Flexural Vibrations of A Thick Free Circular Plate. Refined Dynamical Theories of Beams, Plates, and Shells and Their Applications, Proceedings of the Euromech-Colloquium 219, p102-111.

[23]A.W.Leissa and J.So 1995, Comparisons of Vibration Frequencies for Rods and Beams from One-Dimensional and Three-Dimensional Analyses. Journal of The Acoustical Society of America 98, p2122-2135.

[24] A.W.Leissa and J.So 1995, Accurate Vibration Frequencies of Circular Cylinders from Three-Dimensional analysis. Journal of The Acoustical Society of America 98, p2136-2141.

[25]A.W.Leissa and J.So 1998, Three-Dimensional Vibrations of Thick Circular and Annular Plates. Journal of Sound and Vibration 209(1), p15-41.

[26]C.F.Liu and G.T.Chen 1995, A Simple Finite Element Analysis of Axisymmetric Vibration of Annular and Circular Plates. International journal of mechanical sciences 37(8), p861-971.

[27]C.F.Liu and Y.T.Lee 2000, Finite Element Analysis of Three-Dimensional Vibration of Thick Circular and Annular Plates. Journal of Sound and Vibration 233(1), p63-80.

[28]C.F.Liu, T.J Chen and C.Y.Hwang 2001, Effect of Satisfying Stress Boundary Conditions in the Axisymmetric Vibration Analysis of Circular and Annular Plates. International Journal of Solids and Structures (to appear).

[29]T.Irie, G.Yamada and S.Aomura 1980, Natural Frequencies of Free Mindlin Circular Plates. Journal of Applied Mechanics 47, p652-655.

[30]J.R.Hutchinson 1986, On the Axisymmetric Vibrations of Thick Clamped Plates. Proceedings of the International Conference on Vibration Problem in Engineering, Xian, China, June 19-22, 75-81.
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