System No. U0002-2407201617391700 含飽和流體多孔彈性圓板之脈衝響應 Impulse Responses of Fluid-Saturated Poroelastic Circular Plates 淡江大學 機械與機電工程學系碩士班 Department of Mechanical and Electro-Mechanical Engineering 104 2 105 陳正輝 Jheng-Huei Chen 603350058 碩士 Traditional Chinese 2016-06-03 86page advisor - 蔡慧駿 co-chair - 柯德祥 co-chair - 葉豐輝 co-chair - 蔡慧駿 Biot多孔彈性理論 古典板理論 多孔彈性圓板 脈衝響應 有限元素頻域分析 無因次分析 Biot's Poroelastic Theory Classical Plate Theory Poroelastic Circular Plates Impulse Responses Finite Element Frequency Domain Analysis Dimensionless Analysis ```本文應用Biot多孔彈性理論與古典板理論，推導含飽和流體多孔彈性板的彎曲振動統御方程組。再於頻域中應用Galerkin型態有限元素法推導多孔彈性板三角形與四邊形元素的剛性矩陣及荷重向量。並藉由脈衝負荷作用與彈性邊界限制完成多孔彈性板的彎曲振動有限元素頻域分析，探討含飽和流體多孔彈性圓板的脈衝響應。 透過分析彈性板的無因次頻率係數及驗證前人發表的多孔彈性板數值與實驗結果，顯示有限元素頻域分析確可準確模擬多孔彈性圓板因材質、負荷及邊界限制等變化影響的彎曲振動行為。含飽和流體多孔彈性圓板具有流體與固體架構交互作用的動態消散特性，由多孔圓板撓度振幅的衰減可發現流體黏滯係數愈大其振幅衰減愈顯著，流體體積模數增加亦顯著提升多孔圓板的基礎自然頻率，藉由流體的改變可調整多孔彈性圓板的基礎自然頻率與撓度。最後研究應用無因次有限元素頻域分析探討各無因次參數變異於邊界固定支撐多孔彈性圓板脈衝響應的影響。結果顯示無因次固體有效密度及固體與流體的耦合有效密度分別對基礎自然頻率及撓度振幅的影響最為顯著。``` ```In this study, Biot’s poroelastic theory and classical plate theory are applied to derive the governing equations of flexural vibrations of fluid-saturated poroelastic plates. The Galerkin type finite element approach is applied to derive the stiffness matrices and load vectors of triangular as well as quadrilateral poroelastic plate elements in the frequency domain. After applying the impulsive loadings and adjusting the elastic restraints, the finite element frequency domain analysis of flexural vibrations of poroelastic plates can thus be accomplished. The impulse responses of fluid-saturated poroelastic circular plates are explored. Upon examining the results of the non-dimensional frequency parameters of elastic plates as well as the numerical and experimental results of poroelastic plates published by other researchers, it is validated that the finite element frequency domain analysis can obtain accurate results which are influenced by material properties, loadings and boundary restraints for the flexural vibrations of poroelastic circular plates. A fluid-saturated poroelastic circular plate can present a dynamic dissipation effect owing to the interactions of the fluid and the solid skeleton. Upon examining the reduction in deflection amplitude of poroelastic circular plates, it is found that the dissipation effect is an increasing function of fluid’s viscosity, and the fundamental natural frequency is an increasing function of the bulk modulus of the fluid. Accordingly, the fundamental natural frequencies and the deflections of poroelastic circular plates can be adjusted by changing the properties of the saturated fluid. At the end of this study, the dimensionless finite element frequency domain analysis is applied to explore the influence of dimensionless parameters on the impulse responses of clamped poroelastic circular plates. The results indicated that the value of dimensionless effective mass of the solid has a pronounced effect on the fundamental natural frequency, and the value of dimensionless mass coupling parameter between the fluid and the solid has a pronounced effect on the deflection amplitude.``` ```目錄 中文摘要 I 英文摘要 II 目 錄 IV 圖目錄 VI 表目錄 IX 第一章 緒論 1 1.1 前言 1 1.2 研究動機 3 1.3 文獻回顧 4 1.4 研究內容 11 第二章 含飽和流體多孔彈性板理論 13 2.1 多孔彈性力學理論 13 2.2 多孔材料係數 15 2.2.1 孔洞率 15 2.2.2 多孔材料有效密度 15 2.2.3 消散係數 16 2.2.4 空氣體積模數 16 2.2.5 彈性係數與材料係數的關係 16 2.3 多孔彈性板彎曲振動統御方程組 17 2.3.1 應力、應變與位移關係 18 2.3.2 古典板理論 21 2.3.3 古典板理論─能量法 24 2.3.4 平面應力理論 28 第三章 有限元素頻域分析 30 3.1 多孔彈性板三角形元素 31 3.2 多孔彈性板四邊形元素 35 第四章 結果與討論 40 4.1 彈性板無因次頻率係數分析 40 4.2 多孔方板有限元素頻域分析 46 4.3 多孔圓板有限元素頻域分析 52 第五章 無因次參數變異分析 60 5.1 多孔彈性板無因次方程式 60 5.2 多孔圓板無因次參數變異分析 61 5.2.1 無因次彈性係數變異的影響探討 66 5.2.2 無因次有效密度變異的影響探討 68 5.2.3 無因次消散係數變異的影響探討 70 5.2.4 無因次均佈負荷與流體壓力差變異的影響探討 70 第六章 結論與未來展望 72 5.1 結論 72 5.2 未來展望 75 參考文獻 76 符號索引 82 圖目錄 圖2-1 多孔彈性圓板承受脈衝負荷示意圖 18 圖2-2 多孔彈性板彎曲後x-z平面示意圖 21 圖2-3 多孔彈性板彎曲後y-z平面示意圖 21 圖2-4 多孔彈性圓板受彈性邊界限制示意圖 25 圖3-1 多孔彈性板三角形元素及其節點位移示意圖 31 圖3-2 三角形元素面積座標系示意圖 32 圖3-3 四邊形元素自然座標系示意圖 35 圖3-4 多孔彈性板四邊形元素及其節點位移示意圖 38 圖4-1 彈性方板四邊形板元素分割示意圖 42 圖4-2 彈性方板三角形板元素分割示意圖 43 圖4-3 彈性圓板三角形板元素分割示意圖 43 圖4-4 40×40個多孔板四邊形元素使用4×4個高斯積分點與40×40個多孔板矩形元素板中心點的撓度頻率響應誤差分析 48 圖4-5 40×40×2個多孔板三角形元素使用7個高斯積分點與40×40個多孔板矩形元素板中心點的撓度頻率響應誤差分析 49 圖4-6 40×40及50×50個多孔板四邊形元素使用4×4個高斯積分點板中心點的撓度頻率響應收斂性分析 49 圖4-7 四邊固定支撐含飽和空氣G-Foam方板受0.1 Pa均佈脈衝負荷板中心點的撓度頻率響應驗證 50 圖4-8 四邊固定支撐含飽和空氣Y-Foam方板受0.1 Pa均佈脈衝負荷板中心點的撓度頻率響應驗證 50 圖4-9 四邊固定支撐含飽和空氣Coustone方板受0.1 Pa均佈脈衝負荷板中心點的撓度頻率響應驗證 51 圖4-10 2768及3616個多孔板三角形元素使用7個高斯積分點板中心點的撓度頻率響應收斂性分析 54 圖4-11 邊界固定支撐含飽和空氣G-Foam圓板受0.1 Pa均佈脈衝負荷板中心點的撓度頻率響應驗證 54 圖4-12 邊界固定支撐含飽和空氣Y-Foam圓板受0.1 Pa均佈脈衝負荷板中心點的撓度頻率響應驗證 55 圖4-13 邊界固定支撐含飽和空氣Coustone圓板受0.1 Pa均佈脈衝負荷板中心點的撓度頻率響應驗證 55 圖4-14 邊界固定及簡支撐含飽和空氣G-Foam圓板受0.1 Pa均佈脈衝負荷板中心點的撓度頻率響應 56 圖4-15 邊界固定及簡支撐含飽和空氣Y-Foam圓板受0.1 Pa均佈脈衝負荷板中心點的撓度頻率響應 57 圖4-16 邊界固定支撐含飽和流體G-Foam圓板受0.1 Pa均佈脈衝負荷板中心點的撓度頻率響應 57 圖4-17 邊界固定支撐含飽和流體Y-Foam圓板受0.1 Pa均佈脈衝負荷板中心點的撓度頻率響應 58 圖4-18 邊界固定支撐含飽和流體G-Foam圓板受0.1 Pa均佈脈衝負荷板中心點的撓度時間響應 58 圖4-19 邊界固定支撐含飽和流體Y-Foam圓板受0.1 Pa均佈脈衝負荷板中心點的撓度時間響應 59 圖5-1 無因次彈性係數 變異於多孔圓板撓度頻率響應的影響 66 圖5-2 無因次彈性係數 變異於多孔圓板撓度頻率響應的影響 67 圖5-3 無因次彈性係數 變異於多孔圓板撓度頻率響應的影響 67 圖5-4 無因次有效密度 變異於多孔圓板撓度頻率響應的影響 68 圖5-5 無因次有效密度 變異於多孔圓板撓度頻率響應的影響 69 圖5-6 無因次有效密度 變異於多孔圓板撓度頻率響應的影響 69 圖5-7 無因次消散係數 變異於多孔圓板撓度頻率響應的影響 70 圖5-8 無因次均佈負荷 變異於多孔圓板撓度頻率響應的影響 71 圖5-9 無因次流體壓力差 變異於多孔圓板撓度頻率響應的影響 71 表目錄 表2-1 多孔彈性板彎曲振動統御方程組中抗撓剛度的差異 29 表4-1 四邊固定支撐彈性方板的無因次頻率係數 44 表4-2 四邊簡支撐彈性方板的無因次頻率係數 44 表4-3 邊界固定支撐彈性圓板的無因次頻率係數 45 表4-4 邊界簡支撐彈性圓板的無因次頻率係數 45 表4-5 含飽和流體多孔彈性板材料係數[23] 48 表4-6 四邊固定支撐含飽和空氣G-Foam方板的自然頻率 51 表4-7 四邊固定支撐含飽和空氣Y-Foam方板的自然頻率 51 表4-8 四邊固定支撐含飽和空氣Coustone方板的自然頻率 52 表4-9 邊界固定支撐含飽和空氣G-Foam圓板的自然頻率 56 表4-10 邊界固定支撐含飽和空氣Y-Foam圓板的自然頻率 56 表4-11 邊界固定支撐含飽和空氣Coustone圓板的自然頻率 56 表5-1 含飽和流體多孔彈性板材料係數[23][43][44] 62 表5-2 Biot多孔彈性係數、多孔材料有效密度及消散係數 63 表5-3 含飽和流體多孔彈性板無因次參數值 64 表5-4 無因次參數值變異範圍 65``` ```參考文獻 1. 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