本研究以一非線性彈性樑放置於非線性彈簧基座，並考慮流體流經非線性橫樑(Fluid-Conveying Nonlinear Beam)上，以模擬置放於彈性基底上之流體輸送管系統(Fluid-Conveying Tube)，如海底輸油管、海底電纜、微機電系統的水冷式散熱管系統，甚至人體內主動脈之振動模型。除了流速以外，吾人也考慮系統受一均勻分佈力之影響，其中包含流體與彈性樑之重量以及彈性樑所承受之壓力，此外，由於本模型兩端為鉸接(Hinge)且系統包含流場，因此吾人也將拉伸效應(Stretching Effect)及流固耦合效應(Fluid-Structure Interaction)考慮在內。本研究使用Hamilton’s principle推導出此系統之流固耦合運動方程式，再利用多尺度法(Method of Multiple Scales (MOMS))分析系統於穩態固定點(Fixed Point)各模態之頻率響應(Frequency Response)，以探討是否有內共振之現象，並以數值法模擬其時間域之振動情形，相互驗證之。此外，為了達到減振效益，吾人在此系統掛載一線性調質減振器(Tuned Mass Damper (TMD))，分別探討在不同TMD之質量、彈性係數、阻尼係數以及擺放位置對於主體系統振動之影響，同時吾人也繪製3D MAP(3D Maximum Amplitude Plot)與3D MACP(3D Maximum Amplitude Contour Plot)觀察TMD之最佳組合，以達到本系統之最佳減振目的。另外，為了模擬現實生活中不同之流體輸送管系統，吾人更全面探討流場於不同流速之情況，並分析不同參數條件對系統振動之影響。最後，吾人將以四階Runge-Kutta法求出系統之Floquet Transition Matrix，並搭配Floquet Multipliers (F.M.)之判定法繪製出系統之Basin of Attraction (BOA)圖，藉此分析系統在不同流速作用下以及附加TMD後之穩定性，以提供學術及實務參考之用。

Studies of flow-induced vibration have always been a concern for researchers and engineers because the fluid inside the pipe dynamically interacts with the pipe motion, possibly causing the pipe to vibrate. This study considers a slender nonlinear fluid-conveying elastic beam with hinged-hinged boundary conditions to simulate the fluid-conveying tube systems. We assume that the system is placed on the nonlinear elastic foundation and the beam is subjected to the distributed load which includes the weight of the beam and fluid. There is a wide range of applications in this field such as submarine cables, offshore oil pipes, water cooling radiator in micro-electromechanical systems, and even artery or aorta vibration in human bodies. The primary objective of this study is to find if there is any internal resonance in this system and achieve the effective vibration reduction. The influence of fluid-structure interaction and stretching effect were taken into account as well. We applied Hamilton’s principle to derive the nonlinear equation of motion of this couple system and employed the method of multiple scales (MOMS) to analyze this nonlinear problem. The Fixed point plots (steady state frequency response) of each mode were obtained. In order to simulate various fluid-conveying tube system in real life, we comprehensively examined the influence of different fluid velocity and parameters on this system. The 1:3 internal resonance was found in the 1st and 3rd mode under a specific combination of parameters  and fluid velocity. Also, we added a linear tuned mass damper (TMD) suspended under the beam to reduce vibration and prevent internal resonance. TMD with various locations, mass, spring constants, and damping coefficients were fully analyzed and the optimal combination for TMD to achieve the vibration damping was also proposed by observing 3D maximum amplitude contour plots (3D MACP). Finally, the fluid velocity was included to investigate the stability of this system. We intriduced the Floquet theory and Floquet multipliers to analyze the stability. The basin of attraction charts was also made to verify the effects of TMD on system stability.

目錄
摘要 I
英文摘要	II
目錄III
表目錄VI
圖目錄VII
第一章 緒論 1
一、1 研究動機 1
一、2 文獻回顧 2
一、3 研究方法 6
第二章 理論模式之建立與分析 9
二、1 流體輸送管系統之建立 9
二、2 非線性流固耦合運動方程式之推導與無因次化 10
二、3 與相關研究比較本模式之架構 15
二、4 多尺度法 18
二、5 彈性樑之模態分析 19
第三章 系統內共振之分析 22
三、1 無TMD之非線性流固耦合運動方程式 22
三、2 內共振條件之分析 22
三、3 系統頻率響應之分析 25
三、4 內共振現象之分析 33
第四章 附加TMD之系統分析 35
四、1 附加TMD之理論模式 35
四、2 附加TMD系統之頻率響應分析 37
第五章 不同參數條件對系統之影響 45
五、1 內共振現象之分析 45
五、2 附加TMD系統之頻率響應分析 46
第六章 系統穩定性之分析 47
第七章 結果與討論 50
七、1 流體流速對系統影響之探討 50
七、2 附加TMD系統之減振效益分析 51
七、3 系統穩定性之分析 58
第八章 結論 62
參考文獻	65
附錄(一) 系統之各項無因次化參數定義 68
附錄(二) 時間項通解之表示式 69
附錄(三) 各係數之參數定義	70
論文簡要版 122



表目錄
表1 不同參數條件對於系統振幅之影響(I=0.01) 71
表2 不同參數條件對於系統振幅之影響(I=0.05) 73
表3 不同參數條件對於系統振幅之影響(I=0.1) 75
表4 流體流速對系統振幅之影響 77
表5 附加減振器激擾第一模態之第一模態(gs=0.1) 78
表6 附加減振器激擾第一模態之第一模態(gs=0.5) 79
表7 附加減振器激擾第一模態之第一模態(gs=0.9) 80
表8 附加減振器激擾第三模態之第一模態(gs=0.1) 81
表9 附加減振器激擾第三模態之第一模態(gs=0.5) 82
表10 附加減振器激擾第三模態之第一模態(gs=0.9) 83
表11 附加減振器激擾第三模態之第三模態(gs=0.1) 84
表12 附加減振器激擾第三模態之第三模態(gs=0.5) 85
表13 附加減振器激擾第三模態之第三模態(gs=0.9) 86
表14 不同Case情況下各模態之減振效益 87



圖目錄
圖1 物理模式：流體輸送管系統 88
圖2 理論模式：流體輸送非線性彈性樑	88
圖3流體輸送非線性彈性樑之單一Element 89
圖4 Freq. Ratio與kw之關係曲線 89
圖5 激擾第一模態之Fixed Point圖(無減振器)	90
圖6 激擾第三模態之Fixed Point圖(無減振器)	90
圖7 各kw情況下激擾第三模態之第一模態Fixed Point圖(無減振器)	91
圖8 各kw情況下激擾第三模態之第三模態Fixed Point圖(無減振器)	92
圖9 激擾第一模態之第一模態Fixed Point圖與數值驗證圖 93
圖10 激擾第三模態之第三模態Fixed Point圖與數值驗證圖 93
圖11 Case(G-1)~(G-3)：激擾第三模態之Fixed Point圖(無減振器)94
圖12 激擾第一模態之Fixed Point圖(具減振器) 95
圖13 激擾第三模態之Fixed Point圖(具減振器) 95
圖14 各kw情況下激擾第三模態之第一模態Fixed Point圖(具減振器)96
圖15 各kw情況下激擾第三模態之第三模態Fixed Point圖(具減振器)97
圖16 Case(G-1)~(G-3)：激擾第三模態之Fixed Point圖(具減振器)98
圖17 激擾第一模態之第一模態3D MAP (gs=0.1) 99
圖18 激擾第一模態之第一模態3D MAP (gs=0.5) 99
圖19 激擾第一模態之第一模態3D MAP (gs=0.9) 100
圖20 激擾第三模態之第一模態3D MAP (gs=0.1) 100
圖21 激擾第三模態之第一模態3D MAP (gs=0.5) 101
圖22 激擾第三模態之第一模態3D MAP (gs=0.9) 101
圖23 激擾第三模態之第三模態3D MAP (gs=0.1) 102
圖24 激擾第三模態之第三模態3D MAP (gs=0.5) 102
圖25 激擾第三模態之第三模態3D MAP (gs=0.9) 103
圖26 激擾第一模態之第一模態3D MACP (fs=9)	104
圖27 激擾第三模態之第一模態3D MACP (fs=9)	105
圖28 激擾第三模態之第三模態3D MACP (fs=9)	106
圖29 Mode Shapes of a Hinged-Hinged Beam 107
圖30 激擾第一模態之第一模態3D MACP (fs=9, gs=0.1)	108
圖31 激擾第三模態之第一模態3D MACP (fs=9, gs=0.1)	109
圖32 激擾第三模態之第三模態3D MACP (fs=9, gs=0.9) 110
圖33 Case(G-1):激擾第三模態之第一模態3D MACP(fs=9, gs=0.1)111
圖34 Case(G-1):激擾第三模態之第三模態3D MACP(fs=9, gs=0.9)112
圖35 Case(G-2):激擾第三模態之第一模態3D MACP(fs=9, gs=0.1)113
圖36 Case(G-2):激擾第三模態之第三模態3D MACP(fs=9, gs=0.9)114
圖37 Case(G-3):激擾第三模態之第一模態3D MACP(fs=9, gs=0.1)115
圖38 Case(G-3):激擾第三模態之第三模態3D MACP(fs=9, gs=0.9)116
圖39 微弱內共振Case之Basin of Attraction(無減振器)-放大z軸117
圖40 微弱內共振Case之Basin of Attraction(無減振器) 118
圖41 微弱內共振Case之Basin of Attraction(具減振器) 119
圖42 內共振Case(G-1)之Basin of Attraction(無減振器) 120
圖43 內共振Case(G-1)之Basin of Attraction(具減振器) 121

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