本文使用Biot動態統御方程組，於拉普拉斯域應用有限元素理論建立多孔介質三維六面體之剛度矩陣，並引用前人建立之多孔介質二維三角形及四邊形元素、多孔樑元素與多孔板二維四邊形元素等之剛度矩陣，最後伴隨多點拘束法完成流體位移剛度矩陣及耦合結構之有限元素頻域分析。
研究模擬時以多孔介質模擬聲場，建立加肋多孔板與聲場耦合之模型。藉由加肋多孔板與聲場耦合系統之分析結果驗證確立多孔樑、多孔板、加肋多孔板、聲場及加肋多孔板與聲場耦合之有限元素頻域分析之正確性。接著透過幾何及材料參數變異分析，探討影響耦合結構模態頻率和振幅之影響因子。分析結果顯示，除邊界限制對模態頻率有顯著影響外，加肋多孔板因內含之流體與固體架構耦合作用而有之動態消散特性與流體體積模數亦為影響系統模態頻率之主要因素。另外，消散特性亦影響聲場中各點之模態振幅。於此，可確立本文所建立之加肋多孔板與聲場耦合之有限元素頻域分析，確能達成特定區域聲響行為改善之目標。

This study applied Biot's dynamic governing equations in Laplace domain and used the finite element theory to build the stiffness matrix of a porous medium 3D hexahedron element.  Then, the matrices of porous medium 2D triangular element as well as quadrilateral element, porous beam, and porous plate 2D quadrilateral element derived by other researchers are adopted.  Finally, the multipoint constraint approach is applied to generate the stiffness elements that related to the fluid displacements and complete the Finite Element Frequency Domain Analysis (FEFDA) of a stiffened porous plate coupled with a porous medium.

This study used porous medium to simulate acoustic field and built a model of a stiffened porous plate coupled with an acoustic field.  Based on the analysis results of the system of a stiffened plate coupled with an acoustic field, the FEFDA results of the porous beam, the porous plate, the stiffened porous plate, acoustic field, and the coupling of the stiffened porous plate and the acoustic field were verified.  Then, through the geometric analysis and material parameter variance analysis, the influential factors of modal frequency and amplitude in the coupling structure were explored.  According to the analysis results, besides the significant influence of boundary restraints on modal frequencies, the dynamic dissipation effects caused by the coupling of the fluid with the solid skeleton and the bulk modulus of the fluid were also the main factors influencing the system modal frequencies.  In addition, the FEFDA of the stiffened porous plate coupled with the acoustic field conducted by this study could indeed achieve the goal of improving the acoustic behaviors of certain areas.

中文摘要                          	I
英文摘要                          	II
目  錄                              	IV
圖目錄                            	VI
表目錄                            	XI
第一章 緒論                       	1
1.1 前言	                                1
1.2 研究動機                      	1
1.3 文獻回顧                        	2
1.4 研究內容                      	5
第二章 多孔彈性結構理論               	7
2.1 多孔介質之動態統御方程組         	7
2.1.1 應力、應變及位移關係          	7
2.1.2 動能及耗損能量                	9
2.2 多孔樑與多孔板之彎曲振動統御方程組 	10
2.2.1 應力、應變及位移關係          	11
2.2.2 動能、耗損能量與負荷功	                14
2.2.3 彈性邊界位能	                        15
2.2.4 流體壓力差                  	17
2.2.5 多孔樑與多孔板彎曲振動統御方程組	17
2.3 多孔材料參數	                        18
第三章 有限元素頻域分析	                20
3.1 多孔介質二維有限元素頻域分析	        20
3.1.1 多孔介質二維三角形元素	                21
3.1.2 多孔介質二維四邊形元素         	24
3.2 多孔介質三維有限元素頻域分析      	26
3.2.1 多孔介質三維有限元素           	27
3.3 多孔樑與多孔板有限元素頻域分析      	33
3.3.1 多孔樑與多孔板元素            	34
第四章 結果與討論	                        39
4.1 多孔樑有限元素頻域分析	                39
4.2 多孔耦合樑有限元素頻域分析	        47
4.3 多孔板有限元素頻域分析	                49
4.4 多孔耦合板有限元素頻域分析	        57
4.5 多孔介質有限元素頻域分析	                59
4.6 多孔結構與多孔介質之耦合分析      	63
4.6.1 多孔樑與二維聲場耦合之模態頻率響應	63
4.6.2 多孔板與三維聲場耦合之模態頻率響應	65
4.7 加肋多孔結構與多孔介質之耦合分析	        69
4.7.1 加肋多孔結構有限元素頻域分析	        69
4.7.2 加肋多孔板與三維聲場耦合之模態頻率響應	71
第五章 結論與未來展望	                73
5.1 結論	                                73 
5.2 未來展望                      	75
參考文獻	                                76


圖目錄
圖2-1 多孔樑受均佈壓力負荷示意圖................................................10
圖2-2 多孔板受均佈壓力負荷示意圖..............................................10
圖2-3 多孔樑彈性邊界限制示意圖..................................................16
圖2-4 多孔板彈性邊界限制示意圖..................................................16
圖3-1 多孔介質三角形元素直角座標系示意圖..............................21
圖3-2 多孔介質三角形元素自然座標系示意圖..............................21
圖3-3 多孔介質四邊形元素直角座標系示意圖..............................24
圖3-4 多孔介質矩形元素自然座標系示意圖..................................25
圖3-5 多孔介質六面體元素直角座標系示意圖..............................27
圖3-6 多孔介質矩形體元素自然座標系示意圖..............................28
圖3-7 多孔樑元素直角坐標系示意圖..............................................34
圖3-8 多孔樑元素自然坐標系示意圖..............................................34
圖3-9 多孔板矩形元素直角坐標系示意圖......................................36
圖3-10 多孔板矩形元素自然座標系示意圖........................................37
圖4-1 含飽和流體砂岩樑於兩端簡支撐及1 Pa流體壓力差下受1 Pa均佈衝擊壓力負荷後樑中心點之撓度頻率響應收斂性驗證........................................................................................................41
圖4-2 含飽和流體砂岩樑於兩端簡支撐受1 Pa均佈衝擊力負荷後樑中心點之撓度頻率響應驗證[36].............................................42
圖4-3 含飽和流體泡棉樑於兩端簡支撐分別受0.1 Pa均佈衝擊力負
荷與點衝擊力(10N,x=0.15m)後樑中心點之饒度頻率響應驗
證[36]......................................................................................42
圖4-4 含飽和水砂岩樑於兩端固定受1 Pa均佈衝擊壓力負荷後樑中心點之撓度與流體位移頻率響應............................................43
圖4-5 含飽和水砂岩樑於兩端簡支撐受1 Pa均佈衝擊壓力負荷後樑中心點之撓度與流體位移頻率響應........................................43
圖4-6 含飽和空氣泡棉樑於兩端固定受1 Pa均佈衝擊壓力負荷後樑中心點之撓度與流體位移頻率響應........................................44
圖4-7 含飽和空氣泡棉樑兩端簡支撐受1 Pa均佈衝擊壓力負荷後樑中心點之撓度與流體位移頻率響應........................................44
圖4-8 含飽和流體砂岩樑於兩端簡支撐受1 Pa均佈衝擊壓力負荷後樑中心點之撓度頻率響應........................................................45
圖4-9 含飽和流體泡棉樑於兩端簡支撐受0.1 Pa均佈衝擊壓力負荷後樑中心點之撓度頻率響應....................................................45
圖4-10 含飽和水泡棉樑於兩端簡支撐受0.1 Pa均佈衝擊壓力負荷後樑中心點之撓度與流體位移頻率響應....................................46
圖4-11 含飽和水泡棉樑於兩端簡支撐受點衝擊力(10N,x=0.15m)後樑中心點之撓度與流體位移頻率響應....................................46
圖4-12 懸臂砂岩－泡棉耦合樑示意圖....................................47
圖4-13 含飽和空氣砂岩－泡棉耦合樑於懸臂下受1 Pa均佈衝擊壓力負荷後樑中心點及端點之撓度頻率響應....................................48
圖4-14 含飽和水砂岩－泡棉耦合樑於懸臂下受1 Pa均佈衝擊壓力負荷後樑中心點及端點之撓度頻率響應....................................48
圖4-15 含飽和流體砂岩板於四邊簡支撐受1400 Pa均佈衝擊力負荷後板中心點之撓度頻率響應收斂性驗證....................................51
圖4-16 含飽和水砂岩板於四邊簡支撐及1400 Pa流體壓力差下受1400 Pa均佈衝擊壓力負荷後板中心點之撓度頻率響應[10]....................................................................................................52
圖4-17 含飽和空氣泡棉板於四邊簡支撐分別受0.1 Pa均佈衝擊力負荷與點衝擊力(1N,x=1.5m,y=0.1m)後板中心點之撓度頻率響應[10]....................................................................................................52
圖4-18 含飽和水砂岩板於四邊固定受1400 Pa均佈衝擊壓力負荷後板中心點之位移頻率響應....................................53
圖4-19 含飽和水砂岩板於四邊簡支撐受1400 Pa均佈衝擊壓力負荷後板中心點之撓度與流體位移頻率響應....................................53
圖4-20 含飽和空氣泡棉板於四邊固定受0.1 Pa均佈衝擊壓力負荷後板中心點之撓度與流體位移頻率響應....................................54
圖4-21 含飽和空氣泡棉板於四邊簡支撐受0.1 Pa均佈衝擊壓力負荷後板中心點之撓度與流體位移頻率響應....................................54
圖4-22 含飽和流體砂岩板於四邊簡支撐受1400 Pa均佈衝擊壓力負荷後板中心點之撓度頻率響應圖....................................55
圖4-23 含飽和流體泡棉板於四邊簡支撐受0.1 Pa均佈衝擊壓力負荷後板中心點之撓度頻率響應....................................55
圖4-24 含飽和水泡棉板於四邊簡支撐受0.1 Pa均佈衝擊壓力負荷後板中心點之撓度與流體位移頻率響應....................................56
圖4-25 含飽和空氣泡棉板於四邊簡支撐受點衝擊力   後板中心點之撓度與流體位移頻率響應....................................56
圖4-26 懸臂砂岩－泡棉耦合板示意圖....................................57
圖4-27 含飽和空氣砂岩－泡棉耦合板於懸臂下受1 Pa均部衝擊壓力負荷後板中心點及端點撓度頻率響應....................................58
圖4-28 含飽和水砂岩－泡棉耦合板於懸臂下受1 Pa均部衝擊壓力負荷後板中心點及端點撓度頻率響應....................................58
圖4-29 阻抗量測系統照片....................................60
圖4-30 二維聲場模擬一維聲場之負荷及邊界限制圖....................................60
圖4-31 二維聲場模擬一維聲場之模態頻率響應....................................61
圖4-32 三維聲場模擬一維聲場之負荷及邊界限制圖....................................62
圖4-33 三維聲場模擬一維聲場之模態頻率響應....................................62
圖4-34 多孔樑與二維聲場耦合之負荷及邊界限制圖....................................64
圖4-35 兩端固定受0.1 Pa均佈衝擊負荷作用之含飽和空氣砂岩樑與二維聲場耦合後各觀測點與未耦合砂岩樑中心點之位移頻率響應比較....................................64
圖4-36 多孔板與三維聲場耦合之負荷及邊界條件圖....................................66
圖4-37 四邊固定受0.1 Pa均佈衝擊負荷作用之鋁板與三維聲場耦合後各觀測點與未耦合鋁板之中心點位移頻率響應比較....................................66
圖4-38 四邊固定受0.1 Pa均佈衝擊負荷作用之含飽和空氣砂岩板與三維聲場耦合後各觀測點與未耦合砂岩板中心點位移頻率響應比較....................................67
圖4-39 流阻量測儀照片....................................68
圖4-40 四邊固定受0.1 Pa均佈衝擊負荷作用之含飽和空氣氧化鋁板與三維聲場耦合後各觀測點與未耦合氧化鋁板之中心點位移頻率響應比較....................................68
圖4-41 加肋多孔結構示意圖....................................70
圖4-42 四邊固定受1 Pa均佈衝擊負荷作用之加肋砂岩板中心點與未耦合樑、板之中心點位移頻率響應比較....................................71
圖4-43 四邊固定受1 Pa均佈衝擊負荷作用之含飽和空氣砂岩加肋板與三維聲場耦合後各觀測點與未耦合砂岩加肋板之中心點位移頻率響應比較....................................72

表目錄
表4-1  含飽和水或空氣的砂岩與泡棉樑材料參數....................................41
表4-2  含飽和水或空氣的砂岩與泡棉板材料參數....................................51
表4-3  空氣之材料參數[31]....................................59
表4-4  二維和三維多孔介質頻域分析一維模擬之模態頻率比較....................................62
表4-5  含飽和空氣氧化鋁多孔板之材料參數....................................67
表4-6  加肋多孔結構模擬彈性材料之模態頻率比較....................................70

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