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中文論文名稱 應用有限元素頻域法於含飽和液體多孔薄板之彎曲振動分析
英文論文名稱 On the Flexural Vibration of Thin porous plate Saturated with Fluid by Finite Element Frequency Domain Analysis
校院名稱 淡江大學
系所名稱(中) 機械與機電工程學系碩士班
系所名稱(英) Department of Mechanical and Electro-Mechanical Engineering
學年度 97
學期 1
出版年 98
研究生中文姓名 陳央澤
研究生英文姓名 Yang-Ze Chen
學號 695371269
學位類別 碩士
語文別 中文
口試日期 2009-01-12
論文頁數 88頁
口試委員 指導教授-蔡慧駿
委員-戴念華
委員-柯德祥
委員-葉豐輝
委員-李經倫
中文關鍵字 多孔薄板  彎曲振動  彈性支撐  頻域  有限元素分析 
英文關鍵字 Thin Porous Plate  Flexural Vibration  Elastic Restraint  Frequency Domain  Finite Element Analysis 
學科別分類 學科別應用科學機械工程
中文摘要 本文使用Biot多孔彈性理論推導多孔薄板之彎曲振動統御方程組與頻域多孔薄板元素剛性矩陣,以探討多孔薄板之彎曲振動行為。
文中應用多孔彈性理論,於平面應力假設下推導多孔薄板之統御方程組,再於拉普拉斯域中推導多孔薄板三角與矩形元素之剛性矩陣,並藉由衝擊負荷作用與各式邊界條件限制完成多孔薄板之有限元素頻域分析。由多孔薄板統御方程組之比較驗證及多孔薄板有限元素頻域分析之模態頻率與模態行為結果與理論值之比較顯示,本研究建立之有限元素頻域分析確可準確模擬多孔薄板受特定及彈性邊界支撐限制之彎曲振動行為。
多孔薄板因內含之液體與固體架構耦合作用而有特殊之動態消散特性。由多孔薄板撓度頻率響應之模態振幅衰減結果顯示消散係數愈大其振幅影響愈顯著,同時增加液體體積模數也顯著提升多孔薄板之模態頻率。因此藉由飽和液體之改變將可調整多孔薄板之模態頻率與振幅,進而達到振動控制之目的。最後研究經無因次分析了解無因次參數變異於多孔薄板第一模態撓度頻率響應的影響。經分類發現無因次材料參數的影響大致可分為起始振幅、模態振幅及模態頻率三大部份。
英文摘要 In this study, Biot's poroelastic theory is used to derive the governing equations of flexural vibration of thin porous plates as well as the stiffness matrixes of the frequency domain thin porous plate elements.
First, the poroelastic theory is used to formulate the governing equations of flexural vibration of thin porous plates based on the plane stress assumptions. Then, the governing equations are transformed to Laplace domain and the Galerkin finite element approach is applied to derive the stiffness matrixes of triangular and rectangular porous elements. After applying impulsive loadings and boundary conditions, the finite element frequency domain analysis of thin porous plates can thus be accomplished. Upon examining the governing equations and the modal frequencies and mode shapes of thin porous plates with specified boundary conditions or elastic restraints, it is validated that the finite element frequency domain analysis can obtain good flexural vibration results for thin porous plates.
A thin fluid-saturated porous plate could present typical dissipation effects owing to the interaction of the fluid and the solid skeleton. Upon examining the reduction of the modal amplitudes after the increase of the fluid’s viscosity, it is learned that the more the increase in dissipation effects the more the reduction in modal amplitudes. It is also found if the bulk modulus of the saturated fluid is increased the plate’s modal frequencies are increased. Accordingly, the modal frequency and the modal amplitude can be adjusted by changing the properties of the saturated fluid, and the vibration control of thin porous plates can thus be achieved. In the end of this study, the influences of dimensionless coefficients on the first mode of a thin porous plate are examined and the results are discussed into three categories signifying the effects on the beginning amplitude, the modal amplitude, and the modal frequency.
論文目次 中文摘要 I
英文摘要 II
目 錄 IV
圖目錄 VII
表目錄 X
第一章 緒 論 1
1.1 前言 1
1.2 研究動機 2
1.3 文獻回顧 3
1.4 論文之構成 5
第二章 含飽和液體多孔薄板理論 8
2.1 Biot多孔彈性理論 8
2.1.1 應力、應變與位移關係 9
2.1.2 Biot多孔彈性係數 10
2.2多孔材料係數 11
2.2.1 孔洞係數 12
2.2.2 多孔材料有效密度 13
2.2.3 消散係數 13
2.2.4 空氣之體積模數 14
2.2.5 彈性係數與材料係數之關係 14
2.3 含飽和液體多孔薄板之彎曲振動理論 15
2.3.1 平面應力與固液體應變之關係 16
2.3.2 多孔薄板撓度與應變及應力之關係 17
2.3.3 液體壓力差 18
2.3.4 應變能與外力作功 19
2.3.5 動能與消耗能 20
2.3.6 多孔薄板彎曲振動統御方程組 21
2.3.7 彈性支撐邊界條件 26
第三章 有限元素頻域分析 29
3.1 二維有限元素法 29
3.1.1 二維三角形元素 30
3.1.2 二維矩形元素 34
3.2 頻域有限元素分析之邊界條件 37
第四章 有限元素頻域分析驗證 39
4.1 統御方程之差異 39
4.2 多孔薄板之材料參數 41
4.2.1 空氣之動態體積模數 42
4.2.2 動態消散係數 43
4.3 有限元素頻域分析與驗證 45
4.3.1 含飽和液體多孔板於簡支撐限制下之撓度頻率響應 45
4.3.2 消散特性於多孔薄板彎曲振動之影響 48
4.3.3 模態頻率與模態行為分析驗證 50
4.4 多孔薄板受彈性支撐邊界限制分析驗證 55
4.4.1 含飽和液體多孔薄板於彈性支撐限制下之撓度頻率
響應 56
4.4.2 彈性支撐限制下之模態頻率與模態行為分析驗證 58
第五章 無因次分析探討 60
5.1 材料參數變異於多孔薄板撓度頻率響應之影響探討 60
5.1.1 材質損失因子變異之影響 64
5.1.2 無因次彈性係數變異之影響探討 65
5.1.3 無因次有效質量密度變異之影響探討 67
5.1.4 無因次消散係數變異之影響探討 69
5.1.5 無因次液體壓力差與均佈壓力變異之影響探討 69
5.2 幾何形狀與彈性邊界限制變異之影響探討 71
第六章 結論與未來展望 75
6.1 結論 75
6.2 未來展望 79
參考文獻 81
符號索引 85
圖2-1:燒結金屬銅材料之孔洞分佈圖 12
圖2-2:多孔薄板受分佈壓力負荷示意圖 16
圖2-3:薄板彎曲後任意點P之位移示意圖 17
圖2-4:多孔薄板邊界條件示意圖 26
圖2-5:多孔薄板邊界受彈性支撐限制示意圖 27
圖3-1:矩形多孔薄板受均佈衝擊壓力負荷示意圖 30
圖3-2:直角座標系三角形元素示意圖 31
圖3-3:直角座標系三角形元素 32
圖3-4:直角座標系矩形元素示意圖 34
圖3-5:直角座標系矩形元素 36
圖3-6:含飽和液體多孔薄板之矩形元素受簡支撐邊界限制示意
圖 38
圖3-7:含飽和液體多孔薄板之矩形元素受固定支撐邊界限制示
意圖 38
圖4-1:空氣體積模數與頻率關係圖 43
圖4-2:含飽和水砂岩薄板之消散係數與頻率關係圖 44
圖4-3:含飽和空氣泡棉薄板之消散係數與頻率關係圖 44
圖4-4:多孔薄板四邊簡支撐限制示意圖 46
圖4-5:含飽和水砂岩薄板於四邊簡支撐受均佈衝擊壓力負
荷(q0=1400 Pa)後板中心點之撓度頻率響應圖 47
圖4-6:含飽和空氣泡棉薄板於四邊簡支撐分別受均佈衝擊壓力
負荷(q0=0.1 Pa)與點衝擊力(1 N, x=0.15 m﹐y=0.1 m)
後板中心點之撓度頻率響應圖 47
圖4-7:多孔薄板四邊受固定支撐邊界限制示意圖 49
圖4-8:含飽和液體砂岩薄板於四邊固定支撐受均佈衝擊壓力負
荷(q0=0.1 Pa)後板中心點之撓度頻率響應圖 49
圖4-9:含飽和液體泡棉薄板於四邊固定支撐分別受均佈衝擊壓
力負荷(q0=0.1 Pa)與點衝擊力(10 N, x=0.25 m, y=0.25 m)後板中心點之撓度頻率響應圖 50
圖4-10:多孔平板垂直x軸邊界受簡支撐限制示意圖 51
圖4-11:彈性鋼板之振動模態一(74.922 Hz) 53
圖4-12:彈性鋼板之振動模態二(152.853Hz) 53
圖4-13:彈性鋼板之振動模態三(152.853 Hz) 54
圖4-14:彈性鋼板之振動模態四(225.495 Hz) 54
圖4-15:彈性鋼板之振動模態五(274.019 Hz) 54
圖4-16:矩形元素之彈性支撐邊界限制示意圖 55
圖4-17:多孔薄板之彈性支撐邊界限制示意圖 56
圖4-18:含飽和水砂岩薄板於四邊固定支撐受均佈衝擊壓力負荷
(q0=0.1 Pa)後板中心點之撓度頻率響應圖 57
圖4-19:含飽和空氣泡棉薄板於四邊固定支撐受均佈衝擊壓力負
荷(q0=0.1 Pa)後板中心點之撓度頻率響應圖 57
圖5-1:損失因子 變異於多孔薄板第一模態撓度頻率響應之影
響 64
圖5-2:無因次彈性係數 變異於多孔薄板第一模態撓度頻率響
應之影響 65
圖5-3:無因次彈性係數 變異於多孔薄板第一模態撓度頻率響
應之影響 66
圖5-4:無因次彈性係數 變異於多孔薄板第一模態撓度頻率響
應之影響 66
圖5-5:無因次固體有效質量密度 變異於多孔薄板第一模態撓
度頻率響應之影響 67
圖5-6:無因次液體有效質量密度 變異於多孔薄板第一模態撓
度頻率響應之影響 68
圖5-7:無因次耦合有效質量密度 變異於多孔薄板第一模態撓
度頻率響應之影響 68
圖5-8:無因次消散係數 變異於多孔薄板第一模態撓度頻率響
應之影響 69
圖5-9:無因次液體壓力 變異於多孔薄板撓度頻率響應之影
響 70
圖5-10:無因次壓力負荷 變異於多孔薄板撓度頻率響之影響 70
圖5-11:無因次彈性支撐邊界示意圖 73
圖6-1:夾具與振動量測概念圖 80
圖6-2:簡支撐與固定支撐夾具設計圖 80
表4-1:Biot多孔彈性係數與Lamé彈性係數之對照表 40
表4-2:多孔砂岩與泡棉薄板尺寸與材料參數表 41
表4-3:砂岩與泡棉之有效密度 42
表4-4:砂岩與泡棉之靜態彈性係數 42
表4-5:零孔洞率砂岩彈性薄板受各式邊界限制之模態頻率 51
表4-6:零孔洞率泡棉彈性薄板受各式邊界限制之模態頻率 52
表4-7:彈性薄鋼板受固定邊界限制之模態係數模態頻率 53
表4-8:砂岩彈性薄板以彈性支撐模擬各式邊界限制之模態頻率 58
表4-9:泡棉彈性薄板以彈性支撐模擬各式邊界限制之模態頻率 59
表5-1:砂岩、陶瓷與泡棉之材料參數 62
表5-2:無因次參數值 63
表5-3:無因次參數變異範圍 64
表5-4:彈性薄板受各式邊界限制之模態係數 71
表5-4:彈性薄板受各式邊界限制之模態係數(續) 72
表5-5:彈性薄板受彈性邊界限制之模態係數 74

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