本研究考慮一伯努力-歐拉樑(Bernoulli-Euler beam)在兩端絞接(hinged - hinged)之邊界條件下，同時受到拉伸效應(Stretching Effect)及空氣動力(Aerodynamic)影響之振動情形。而文中也將提到樑因為拉伸效應所導致的非線性振動現象，即討論非線性簡支樑(nonlinear hinged-hinged beam)之振動行為，並在非線性簡支樑下方掛載一可調質量減振器(Tuned Mass Damper (TMD))或具非線性彈性係數之非線性可調質量減振器(nonlinear Tuned Mass Damper)，以分析該TMD對樑之減振效果。本研究使用多尺度法(Method of multiple scales (MOMS))分析系統於穩態固定點(Fixed Points)時各模態之頻率響應(Frequency response)，之後吾人將以數值方法求得一閉迴路(closed loop)耦合數值解，並全面分析非線性簡支樑附加線性TMD或非線性TMD時各項參數(如TMD之質量比、掛載位置、阻尼係數、彈性係數、非線性彈性係數)對樑之影響，同時吾人將繪製3D MAP(3D maximum amplitude plots)與3D MACP(3D maximum amplitude contour plots)以利相互對照及分析。而文中除了以時間響應(Time Response)圖驗證3D MACP之正確性外，也使用因次分析法(Dimensional Analysis)解得非線性TMD運動方程之解析解， 並將該解所求得之開迴路(open loop)解析解與先前提的閉迴路數值解相互比較，以獲得進一步的驗證，同時吾人也將給出非線性TMD振動率之明確關係式。最後吾人將以RK4(Fourth-Order Runge-Kutta)求出系統之Floquet Transition Matrix並搭配F.M.(Floquet Multipliers)判定法繪製BOA(Basin of Attraction)圖以探討系統在空氣動力作用下之穩定性。此外，吾人也使用Poincaré Map驗證BOA圖之正確性。本研究將提供最佳TMD的質量比、掛載位置、彈性係數、非線性彈性係數，並分析在系統受到空氣動力作用時須注意之項，以做為產學界未來研究之參考。

This study considers the Bernoulli-Euler beams with hinged-hinged boundary conditions which is influenced by both the stretching effect and the aerodynamic vibration at the same moment. Linear Tuned Mass Damper (TMD) or nonlinear Tuned Mass Damper is installed under the beam to analyze the damping effect of TMD on the beam which is discussed before. we use the method of multiple scales (MOMS)which was in order to solve the nonlinear equations, and analyze the system at the fixed point of each mode in frequency response. Then a comprehensive analysis of the beam with linear TMD or nonlinear TMD parameters(Such as TMD mass ratio, mounting position, damping coefficient, elasticity coefficient, nonlinear elastic coefficient). At last we will draw 3D MAP(3D maximum amplitude plots) and 3D MACP(3D maximum amplitude contour plots) for comparison and analysis. In addition, we use similarity solution to further verify the correctness of this calculation. We will also explicitly put forward nonlinear TMD vibration rate solution in the last part .We will use RK4 to find Floquet Transition Matrix. After that, we use F.M.(Floquet Multipliers) to draw BOA then explore the stability of the beam under aerodynamic action. This study will provide the best TMD mass ratio, mounting position, elasticity coefficient, nonlinear elasticity coefficient, and analyze items that should be noted when the system is subjected to aerodynamic action.

摘要.........................................I
英文摘要....................................II
目錄.......................................III
圖目錄.......................................V

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 系統理論模式建立......................9
二、1 非線性運動方程之推導與無因次化.........9
二、2 MOMS法................................12
第三章 內共振條件分析.......................14
三、1 無 TMD 之非線性橫樑運動方程式.........14
三、2 無TMD系統之內共振條件.................14
三、3 無TMD系統之頻率響應...................16
三、4 內共振之驗證..........................18
第四章 附加線性TMD之減振分析................20
四、1 非線性簡支樑附加線性TMD方程式之分析...20
四、2 非線性簡支樑附加線性TMD之頻率響應.....21
四、3 非線性簡支樑附加線性TMD之3D MAP.......25
第五章 非線性TMD之減振分析..................27
五、1 非線性簡支樑附加非線性TMD之方程式分析.27
五、2 非線性TMD之解析解.....................34
五、2-1 非線性TMD之頻率響應.................34
五、2-2 非線性TMD之時間域分析...............42
五、3 非線性TMD解析解之對照.................42
五、4 3D MACP 分析..........................43
第六章 系統之穩定性分析.....................47
第七章 結論.................................51
參考文獻....................................54
附錄一......................................57

 
圖目錄
圖 1具減振器之主體架構與邊界條件................................58
圖 2激擾第一模態之第一模態Fixed Point Plot......................58
圖 3激擾第一模態之第二模態Fixed Point Plot......................59
圖 4激擾第一模態之第三模態Fixed Point Plot......................59
圖 5激擾第二模態之第二模態Fixed Point Plot......................60
圖 6激擾第二模態之第一模態Fixed Point Plot......................60
圖 7激擾第二模態之第三模態Fixed Point Plot......................61
圖 8激擾第三模態之第三模態Fixed Point Plot......................61
圖 9激擾第三模態之第一模態Fixed Point Plot......................62
圖 10激擾第三模態之第二模態Fixed Point Plot.....................62
圖 11線性TMD之3DMAP.............................................63
圖 12線性TMD之3DMAP.............................................64
圖 13線性TMD之3DMAP.............................................65
圖 14線性TMD之3DMAP.............................................66
圖 15線性TMD之3DMAP.............................................67
圖 16-(a)線性TMD之3DMAP截面圖...................................68
圖 17-(a)線性TMD之3DMACP........................................69
圖 18 非線性彈性係數0及0.5時不同阻尼係數之頻率響應圖............70
圖 19非線性彈性係數為0及0.5，之時間響應圖.......................70
圖 20非線性彈性係數為0及0.5，之時間響應圖.......................71
圖 21非性TMD之3DMAP.............................................72
圖 22非性TMD之3DMAP.............................................73
圖 23非性TMD之3DMAP.............................................74
圖 24非性TMD之3DMAP.............................................75
圖 25非性TMD之3DMAP.............................................76
圖 26-(a) 非線性TMD之3DMAP截面圖................................77
圖 27-(a)線性TMD之3DMACP........................................78
圖 28-(a) 無TMD之Basin of Attraction............................79
圖 29-(a) 具線性TMD之Basin of Attraction........................80
圖 30-(a)具線性TMD之Basin of Attraction.........................81
圖 31  時各系統之Basin of Attraction............................82
圖 32  時各系統之Basin of Attraction............................82

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