本文應用Biot多孔彈性理論，於尤拉樑及平面應力假設下推導多孔樑與多孔板之彎曲振動統御方程組。並於拉普拉斯域下推導多孔樑、多孔板以及多孔介質元素的剛性矩陣。再藉由衝擊負荷與彈性支撐邊界限制進行多孔樑、多孔板、多孔介質之有限元素頻域分析。探討多孔樑、多孔板、多孔介質、加肋多孔板、多孔樑與多孔介質耦合等耦合系統之動態行為，以期契合實際之應用狀態。
多孔樑、多孔板以及多孔介質因內含之流體與固體架構交互作用而有特殊之動態消散特性。由含飽和多孔樑與多孔板模態振幅衰減之現象可發現流體黏滯係數越大時其消散特性影響相對增加，而流體體積模數主要影響多孔樑和板之模態頻率。因此藉由流體之改變可調整多孔樑與多孔板之模態頻率與振幅。由加肋多孔板模態頻率變動可明顯觀察到，孔洞率提高時消散係數及模態頻率也相對增高。故藉由孔洞率及流體改變可精確的調整加肋多孔板之動態反應。另多孔樑與空間聲場耦合分析結果顯示空間聲場內可同時觀察到多孔樑及聲場耦合的模態頻率且空間聲場內模態頻率和振幅均有顯著改變。

Under the assumptions of Euler beam and plane stress, this study formulates the governing equations of flexural vibrations for the porous beam and plate using Biot’s poroelastic theory.  Then, the stiffness matrices of the porous beam, plate and medium elements are derived in Laplace domain.  Thereafter, using the impulsive loading and the elastic boundary conditions, the finite element frequency domain analyses are performed to study the dynamic behaviors of porous beams, plates, and mediums.  In order to match the application condition, the dynamic behaviors of stiffened porous plates (porous beam coupled with porous plate) and porous beam coupled with porous medium are also evaluated.
The porous beam, plate, and medium present a typical dissipation effect due to the interaction between the saturated fluid and the solid skeleton.  Upon examining the reduction of modal amplitudes of the saturated porous beam and plate, the dissipation effect is found growing with the increase of the fluid’s viscosity, and the bulk modulus of the fluid has major effect on their modal frequencies.  Therefore, by changing of the saturated fluid, the modal frequency and amplitude of the porous beam and plate can be adjusted.  From the modal frequency fluctuations of the stiffened porous plate, the increase of both dissipation coefficient and modal frequency are clearly observed with the raise of porosity.  Hence, the dynamic behavior of the stiffened porous plate can be precisely adjusted by the changes of the porosity and the saturated fluid.  In addition, the analysis results of the coupling of a porous beam with an acoustic field show that the coupled modal frequencies of the porous beam and the acoustic field can be simultaneously observed, as well as the remarkable changes on the modal frequencies and amplitudes.

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	1
1.3 文獻回顧	2
1.4 研究內容	4
第二章 多孔樑與多孔板彎曲振動統御方程組	5
2.1 Biot多孔彈性理論	5
2.1.1應力、應變及位移	5
2.1.2 Biot多孔彈性係數	7
2.2多孔材料參數	7
2.2.1孔洞率	8
2.2.2多孔材料有效密度	8
2.2.3流體體積模數	9
2.2.4消散係數	9
2.3多孔樑與多孔板之彎曲振動	9
2.3.1固體與流體之應力與應變關係	11
2.3.2流體壓力差	15
2.3.3動能與消耗能	16
2.3.4應變能與功	16
2.3.5彈簧能	18
2.3.6多孔樑與多孔板彎曲振動統御方程組	19
第三章 有限元素頻域分析	23
3.1多孔樑與多孔板有限元素頻域分析	23
3.1.1多孔樑與多孔板直角坐標系元素	24
3.1.2多孔介質直角坐標系矩形元素	28
第四章 有限元素頻域分析結果比較	32
4.1彈性樑與彈性板以及加肋彈性板動態行為驗證	32
4.2多孔樑與多孔板之有限元素頻域分析	38
4.3橫向加肋板參數變異之影響	53
4.4多孔吸音樑與多孔介質耦合分析	55
4.4.1二維聲場之聲響特性	55
4.4.2含多孔吸音樑二維聲場之聲響特性	56
第五章 結論與未來展望	59
5.1結論	59
5.2 未來展望	60
參考文獻	62
 
圖目錄
圖2-1   多孔樑受分佈壓力負荷示意圖	10
圖2-2   多孔板受分佈壓力負荷示意圖	11
圖2-3   多孔樑彈性邊界限制示意圖	18
圖2-4   多孔板彈性邊界限制示意圖	19
圖3-1   直角坐標系多孔樑元素示意圖	25
圖3-2   直角坐標系多孔板矩形元素示意圖	26
圖4-1  加肋彈性板示意圖	38
圖4-2  兩邊固定之含飽和水砂岩樑受1Pa均佈衝擊壓力負荷後
樑中心點撓度頻域響應圖	40
圖4-3  兩端固定之含飽和空氣泡棉樑受1Pa均佈衝擊壓力負荷
後樑中心點撓度頻域響應圖	40
圖4-4  兩邊簡支撐之含飽和空氣砂岩樑受1Pa均佈衝擊壓力負
荷後樑中心點撓度頻域響應圖	41
圖4-5  兩邊簡支撐之含飽和水砂岩樑受1Pa均佈衝擊壓力負荷
後樑中心點撓度頻域響應圖	41
圖4-6  兩邊簡支撐之含飽和砂岩樑受1Pa均佈衝擊壓力負荷後
樑中心點撓度頻域響應圖	42
圖4-7  兩邊簡支撐之含飽和空氣泡棉樑受0.1Pa均佈衝擊壓力
負荷後樑中心點撓度頻域響應圖	42
 
圖4-8  兩邊簡支撐之含飽和空氣泡棉樑受點衝擊力10N後樑(x=0.15m)點上撓度頻域響應圖	43
圖4-9  兩邊簡支撐之含飽和水樑受0.1Pa均佈衝擊壓力負荷後
樑中心點撓度頻域響應圖	43
圖4-10 懸臂砂岩－泡棉耦合樑示意圖	44
圖4-11 含飽和空氣砂岩-泡棉樑於懸臂下受1Pa均佈衝擊壓力負
荷後樑中心點與端點撓度頻域響應圖	44
圖4-12 含飽和水砂岩-泡棉樑於懸臂下受1Pa均佈衝擊壓力負荷
後樑中心點與端點撓度頻域響應圖	45
圖4-13 含飽和空氣泡棉-砂岩樑受1Pa均佈衝擊壓力負荷後樑中
心點與端點撓度頻域響應圖	45
圖4-14 含飽和水泡棉-砂岩樑受1Pa均佈衝擊壓力負荷後樑中心
點與端點撓度頻域響應圖	46
圖4-15 四邊固定之含飽和水砂岩板受0.1Pa均佈衝擊壓力負荷後
板中心點撓度頻域響應圖	48
圖4-16 四邊固定之含飽和空氣泡棉板受0.1Pa均佈衝擊壓力負荷
後板中心點撓度頻域響應圖	48
圖4-17 四邊簡支撐之含飽和空氣砂岩板受1400Pa均佈衝擊壓力
負荷後板中心點撓度頻域響應圖	49
圖4-18 四邊簡支撐之含飽和水砂岩板受1400Pa均佈衝擊壓力負
荷後板中心點撓度頻域響應圖	49
圖4-19 四邊簡支撐之砂岩板受1400Pa均佈衝擊壓力負荷後板中
心點撓度頻域響應圖	50
圖4-20 四邊簡支撐之含飽和空氣泡棉板受0.1Pa均佈衝擊壓力負
荷後板中心點撓度頻域響應圖	50
圖4-21 四邊簡支撐之含飽和空氣泡棉板受10N點衝擊力負荷後(x=0.15m, y=0.1m)點上撓度頻域響應圖	51
圖4-22 四邊簡支撐之泡棉板受0.1Pa均佈衝擊壓力負荷後板中心
點撓度頻域響應圖	51
圖4-23 懸臂砂岩-泡棉耦合板示意圖	52
圖4-24 含飽和空氣砂岩-泡棉板於懸臂下受1Pa均佈衝擊壓力負荷
後板中心點與端點撓度頻域響應圖	52
圖4-25 孔洞率變異於動態消散係數的影響圖	54
圖4-26 二維聲場之網格及邊界條件圖	55
圖4-27 二維聲場之位移頻域響應圖	56
圖4-28 含多孔吸音樑二維聲場之網格及邊界條件圖	57
圖4-29 含多孔吸音樑二維聲場之位移頻域響應圖	57
圖4-30 二維聲場側邊含多孔吸音樑之位移頻域響應圖	58
 
表目錄
表4-1  含飽和水砂岩與含飽和空氣泡棉之材料性質	33
表4-2  砂岩彈性樑邊界受簡支撐限制之模態頻率	34
表4-3  砂岩彈性樑邊界受固定支撐限制之模態頻率	34
表4-4  砂岩彈性樑邊界為懸臂態之模態頻率	34
表4-5  泡棉彈性樑邊界受簡支撐限制之模態頻率	35
表4-6  泡棉彈性樑邊界受固定支撐限制之模態頻率	35
表4-7  泡棉彈性樑邊界為懸臂態之模態頻率	35
表4-8  砂岩彈性板四邊受簡支撐限制之模態頻率	35
表4-9  砂岩彈性板四邊受固定支撐限制之模態頻率	36
表4-10 砂岩彈性板對邊受簡支撐之模態頻率	36
表4-11 泡棉彈性板四邊受簡支撐限制之模態頻率	36
表4-12 泡棉彈性板四邊受固定支撐限制之模態頻率	36
表4-13 泡棉彈性板對邊受簡支撐之模態頻率	37
表4-14 加肋砂岩方板四邊受固定支撐限制之模態頻率
		                           37
表4-15 加肋砂岩方板四邊受簡支撐限制之模態頻率	37
表4-16 不同孔洞率固定支撐加肋水砂岩方板之模態頻率	53
表4-17 不同孔洞率簡支撐加肋水砂岩方板之模態頻率	54

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