本研究以一非線性彈性樑為主體，其一端為固定端(fixed end)，另一端為自由端(free end)，以模擬機翼、一般飛行器或是海上鑽油平台的吊臂。首先，吾人以牛頓第二運動定律為基礎，推導出此非線性彈性樑之運動方程，再利用時間多尺度法(Method of Multiple Scales (MOMS))將非線性運動方程式，分成兩個不同的時間尺度，找出系統之振動頻率，並探討是否有內共振之現象。此外，為了達到減振效益，吾人在此彈性樑下方分別掛載線性及非線性調質減振器(Tuned Mass Damper (TMD))分析此懸臂樑系統，在不同的TMD質量、彈性係數、阻尼係數以及擺放位置對於整體振動之影響，並利用本系統最大振福的3D圖及投影之3D Maximum Amplitude Contour Plot (3D MACP)觀察TMD的最佳組合，達到本系統之最佳減振目的。最後，吾人以一簡單的空氣動力函數模擬氣流對於本彈性樑系統之阻尼的影響，藉由改變風速的大小，利用Floquet Theory 搭配 Floquet Multipliers (F.M.)判定法則來分析此系統之穩定性，吾人分別以附加線性及非線性TMD之系統之最佳減振組合與各種風速影響之下的Basin of Attration (BOA)圖形，觀察此系統在不同狀態下之穩定性，以獲得最後結論。

This study considered a slender fixed-free nonlinear beam subjected to distributed loads and unsteady aerodynamic forces. The objective of this study was to find if there is any internal resonance in this system and achieve effective vibration damping. We added a tuned mass damper (TMD) that was suspended under the beam to reduce vibration and prevent internal resonance. The effects of linear and nonlinear TMDs were studied. The influence of shortening effect (nonlinear inertia) and nonlinear geometry of this beam were taken into account as well. We employed the method of multiple scales (MOMS) to analyze this nonlinear problem. The Fixed point plots (steady state frequency response) were obtained. TMDs with various locations and spring constants were considered and the optimal mass range for the TMD to reduce vibration in the main structure was also proposed. The Poincaré Map was also utilized to identify the system instability frequency region of the jump phenomenon. The parameters of the added TMD were studied. The internal resonance can be avoid for the existence of the TMD. The optimal TMD mass and the spring constant were provided for best beam vibration reduction. Finally, the wind speeds and aerodynamic loads were included to investigate the stability of this system. The system stability was analyzed by Floquet theory and Floquet multipliers. The basin of attraction charts were made to verify the effects of the combinations of TMD’s mass and the spring constant at diverge speed.

目錄
摘要	I
英文摘要	II
目錄	II
表目錄	IV
圖目錄	V
第一章 緒論	1
一、1 研究動機	1
一、2 文獻回顧	1
一、3 研究方法	8
第二章 理論模式之建立與分析	10
二、1 運動方程式之推導	10
二、2 多尺度法	13
第三章 系統內共振之條件	16
三、1 內共振條件之分析	16
三、2 系統之頻率響應分析	17
第四章 非線性樑與線性 TMD 之振動系統分析	24
四、1 線性 TMD 方程之分析	24
四、2 非線性樑附加線性 TMD 之頻率響應分析	25
第五章 非線性樑與非線性 TMD 之振動系統分析	31
第六章 系統之穩定性分析	39
第七章 結果與討論	42
七、1 附加TMD系統之減振分析	42
七、2 系統之穩定性分析	45
第八章 結論	47
參考文獻	49
附錄(一)	53
附錄(二)	54
附錄(三)	55
論文簡要版	80

 
表目錄
表一 激擾w方向第一模態 k=0.1  =0.1  =0	57
表二 激擾w方向第一模態 k=0.5  =0.1  =0	57
表三 激擾w方向第一模態 k=0.9  =0.1  =0	58
表四 激擾w方向第一模態 k=0.1  =0.5  =0	58
表五 激擾w方向第一模態 k=0.5  =0.5  =0	59
表六 激擾w方向第一模態 k=0.9  =0.5  =0	59
表七 激擾w方向第一模態 k=0.1  =0.9  =0	60
表八 激擾w方向第一模態 k=0.5  =0.9  =0	60
表九 激擾w方向第一模態 k=0.9  =0.9  =0	61
表十 激擾w方向第一模態 k=0.1  =0.1  =0.1	61
表十一 激擾w方向第一模態 k=0.5  =0.1  =0.1	62
表十二 激擾w方向第一模態 k=0.9  =0.1  =0.1	62
表十三 激擾w方向第一模態 k=0.1  =0.5  =0.1	63
表十四 激擾w方向第一模態 k=0.5  =0.5  =0.1	63
表十五 激擾w方向第一模態 k=0.9  =0.5  =0.1	64
表十六 激擾w方向第一模態 k=0.1  =0.9  =0.1	64
表十七 激擾w方向第一模態 k=0.5  =0.9  =0.1	65
表十八 激擾w方向第一模態 k=0.9  =0.9  =0.1	65
圖目錄
圖1 具減振器之主體架構與邊界條件	66
圖2 激擾第一模態之各模態 Fixed point圖 (無減振器)	67
圖3 激擾第二模態之各模態 Fixed point圖 (無減振器)	68
圖4 激擾第三模態之各模態 Fixed point圖 (無減振器)	69
圖5 附加線性減振器激擾第一模態,  =0.1,  =0	70
圖6 附加線性減振器激擾第一模態,  =0.5 ,  =0	70
圖7 附加線性減振器激擾第一模態,  =0.9 ,  =0	71
圖8 附加非線性減振器激擾第一模態,  =0.1 ,  =0.1	71
圖9 附加非線性減振器激擾第一模態,  =0.5 ,  =0.1	72
圖10 附加非線性減振器激擾第一模態,=0.9 ,  =0.1	72
圖11 附加線性減振器激擾第一模態之 3D MACP	73
圖12 附加非線性減振器激擾第一模態之 3D MACP	74
圖13 Fixed-free beam mode shapes	75
圖14 無減振器之系統之 Basin of Attraction	76
圖15 附加線性TMD 之系統之Basin of Attraction	77
圖16 附加非線性TMD 之系統之Basin of Attraction	78
圖17 風速 =17 之 Basin of Attraction	79
圖18 風速 =17.9 之 Basin of Attraction	79

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