本研究以一兩端鉸接之非線性彈性樑模擬工程橫樑之運動，考慮主體受一簡諧外力影響，相對一般於橫樑上放置減振器之減振方法，本研究以一致動器 (Actuator) 置於此樑之端點，嘗試以微小的激擾改變主體結構之自然頻率，進而影響橫樑之側向 (transverse) 振動模態，以期達到避開共振並可有效控制主體之振動。此系統類似Mathieu equation的系統，端點之致動器外力會影響整個系統的穩定性，因此本文將探討其振動情形與致動器之振幅及頻率對於系統穩定性之影響，首先利用牛頓定律推導其運動方程，接著採用多尺度法 (Method of Multiple Scales(MOMS)) 分析系統於穩態固定點 (Fixed Point Plot) 時各模態之頻率響應，並藉由振幅觀察其振動現象。為增加本模式的應用程度，吾人並加入空氣動力，討論不同風速對於此系統穩定性之影響，利用弗羅凱理論 (Floquet Theory) 搭配弗羅凱轉移矩陣 (Floquet Transition Matrix) 判斷其穩定性，藉由繪製系統在各個情況下之吸引域圖 (Basin of Attraction) ，觀察系統在不同風速與不同 Actuator 振幅的激擾下，不同起始位置及速度對系統穩定性之影響。本研究之模式可藉由調整 Actuator 之振幅大小及頻率或相位，以達到控制橫樑振動之振幅及穩定性，可用於一般振動主體之振動控制，亦可適用於振動能量 (Vibration Energy) 與電能 (Electric Power Generator) 之間之轉換，應用範圍廣泛。

This study considered a slender hinged-hinged nonlinear Bernoulli-Euler Beam which affected by a simple harmonic force with stretching effect. In contrast to the traditional vibration reduction, this study used an actuator attached to one’s tip of the beam to change the natural frequency of the main structure with small disturbance. The objective of this study was to avoid internal resonance within this system and achieve effective vibration control. This system is similar to the Mathieu equation system, the actuator’s external force will affect the stability of the entire system. Therefore, this research would explore its vibration situation and the effect of the amplitude or frequency of the actuator to the stability of the system. In the first place, we employed the Method of Multiple Scales to analyze this nonlinear system. The Fixed Point plots (steady state frequency response) were obtained and compared with the numerical results to verify the system’s internal resonance and observe the vibration phenomenon by the amplitude of this system. For the purpose of increasing the degree of application of this model, we also considered the aerodynamics to discuss the influence of different wind speeds with the stability of this system. The Poincaré Map was also utilized to identify the stability of this system with the frequency regions of the jump phenomenon. The Floquet Theorem was employed to get the Basin of Attraction of this system and the system’s information for stability was included. Hence, on condition that the excitation by different wind speeds and different Actuator amplitudes, we could observe the influence of different initial position and initial velocity about the stability of this system. The model of this study can be used to adjusted the amplitude, frequency or phase of the Actuator to control the beam’s vibration amplitude and its stability.

目錄
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	3
一、3 研究方法	10
第二章 理論模式之建立	12
二、1	非線性運動方程式之推導	12
二、2	非線性運動方程式之無因次化	14
二、3	多尺度法(Method of multiple scales，MOMS)	16
第三章 內共振條件分析	18
三、1	系統內共振條件之分析	18
三、2	系統之內共振分析	22
三、3	系統之頻率響應	24
第四章 橫樑與端點致動器系統之穩定性分析	31
四、1 橫樑與端點致動器系統之頻率響應	31
四、2 致動器 (Actuator) 振幅之影響	37
四、3 特徵值方程式之穩定分析	39
第五章 系統之穩定性分析	42
第六章 結論	47
參考文獻	49
附錄(一) 無因次化參數定義	53
論文簡要版	76

 
圖目錄
圖 1  hinged-hinged beam 加裝端點致動器(Actuator)之主體架構	54
圖 2 激擾第一模態之第一、二、三模態Fixed Point圖	55
圖 3 激擾第二模態之第一、二、三模態Fixed Point圖	56
圖 4 激擾第三模態之第一、二、三模態Fixed Point圖	57
圖 5 附加端點致動器時，激擾第一模態之Fixed Point圖	58
圖 6 附加端點致動器時，激擾第二模態之Fixed Point圖	59
圖 7 附加端點致動器時，激擾第三模態之Fixed Point圖	60
圖 8 致動器 (Actuator) 與橫樑第一模態之振幅響應圖	61
圖 9 致動器 (Actuator) 與橫樑第二模態之振幅響應圖	61
圖 10 致動器 (Actuator) 與橫樑第三模態之振幅響應圖	62
圖 11 致動器輸入振幅=4.9138之橫樑第一模態Fixed Point圖	62
圖 12 致動器輸入振幅=4.9398之橫樑第二模態Fixed Point圖	63
圖 13 致動器輸入振幅=4.9168之橫樑第三模態Fixed Point圖	63
圖 14 非線性彈性樑振動系統之eigenvalue圖	64
圖 15 Z=0.3，U=20~30m/s，psi=0~pi之Basin of Attraction	65
圖 16 Z=0.7，U=20~30m/s，psi=0~pi之Basin of Attraction	65
圖 17 Z=1.0，U=20~30m/s，psi=0~pi之Basin of Attraction	66
圖 18 U=20m/s，psi=0~pi，Z=0.3、0.7、1.0時之Basin of Attraction比較圖	67
圖 19 Z=0.3，U=20m/s，psi=0~pi時之穩定性分析圖	68
圖 20 Z=0.7，U=20m/s，psi=0~pi時之穩定性分析圖	69
圖 21 Z=1.0，U=20m/s，psi=0~pi時之穩定性分析圖	70
圖 22 U=24m/s，psi=0~pi，Z=0.3、0.7、1.0時之Basin of Attraction比較圖	71
圖 23 Z=0.3，U=24m/s，psi=0~pi時之穩定性分析圖	72
圖 24 Z=0.7，U=24m/s，psi=0~pi時之穩定性分析圖	73
圖 25 Z=1.0，U=24m/s，psi=0~pi時之穩定性分析圖	74
圖 26 U=20m/s，Z=0.3，psi=0~pi與-pi~0之Basin of Attraction比較圖	75
圖 27 U=24m/s，Z=0.3，psi=0~pi與-pi~0之Basin of Attraction比較圖	75

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