本研究係採取非線性的物理模式來模擬整個振動體之運動，非線性的物理模式係將彈簧模擬成三次方的變化量 (cubic spring) ，以使振動變成非線性的模式。非線性的問題雖然已被研究多年，但非線性的模式再輔之以非線性或非穩態空氣動力，同時再加入減振裝置，並探討各參數之影響，則尚待分析。本論文第一部分為利用牛頓法推導加入減振器之二維三自由度非線性振動系統的運動方程式。本研究減振器係採用一個質量、彈簧及阻尼裝置作為減振裝置，其安裝的位置及材料特性將視系統之彈性軸、質心、風速及外力頻率而定。

    第二部份簡化吾人之運動系統與前人之研究結果比對之後，第三部份則利用“古典振顫分析法” (Classic Flutter Analysis) 求出系統各自由度振幅在頻率域的運動方程式；並使用 Multiple Scales Analysis 求得時間域動態解析解。上述非線性系統之解析解，將與頻率域的數值解作比對，以驗證本論文之結果。    

    本研究中之二維振動系統可視為翼剖面 (airfoil)，亦可視為吊橋；而減振器可視為一般之派龍架，亦可視為吊橋下之減振裝置。吾人在考慮不變動大部分的設計之下，改變減振器位置以及材料特性，達到最經濟的減振效果。

There has been a long history in research and development of structural vibration worldwide. In early stage, scholars only emphasized on the design of the structure. With the improvement of technologies, bridges and buildings are more complex today. Aerodynamics and fluid mechanics are also incorporated into the design for further studies.

    The Newton’s second law is used to analyze a system of nonlinearly vibrational rigid body, doing pitching and plunging in conjunction with an unsteady nonlinear aerodynamic force and an absorber. The cubic spring is employed to simulate a nonlinear vibration system. Several analytic methods are employed to get the solutions of this nonlinear aeroelastic system. The“Classic Flutter Analysis”is utilized to get the amplitude-frequency function of this system. The“Multiple Scales Analysis”is used to get the analytic solution in time domain. The analytic solutions will be correlated with the numerical results to ensure the nonlinear model and the effects of the absorber.

    By using the mathematical model presented in this study, one can adjust the position or design of the absorber to reduce vibration, without changing the main configuration or original design. In conclusion, this model has an enormous applicable value towards applied engineering and related fields.

目 錄
中文摘要.....................................................................................................I
英文摘要..................................................................................................III
目錄..........................................................................................................III
圖表目錄.................................................................................................VII

第一章	緒 論..........................................................................................1
    ㄧ、1  研究動機..............................................................................1
    ㄧ、2  文獻回顧…………………………………………………..2
    ㄧ、3  研究方法…………………………………………...……...6

第二章 系統理論模式之建立…………………………………………..8
    二、1  機翼與減振裝置…………………………………………..8 
    二、2  力的分解…………………………………………………..8
    二、3  減振器之受力…………………………………………….11
    二、4  二維機翼與減振器之運動方程………………………….11
    二、5  二維機翼與減振器之無因次化運動方程……………….15

第三章 線性減振器對系統影響之討論………………………………51
    三、1  線性減振器………………………………………………51
    三、2  移除減振器之系統………………………………………53

第四章 系統之解析解…………………………………………………59
    四、1  具減振器之系統運動方程式之推導……………………59
    四、2  具減振器之系統運動方程式之解析展開………………61

第五章 成果與討論…………………………………………………..101
    五、1  無減振器之非線性系統分析及結果…………………..101
    五、2  具減振器之非線性系統分析及結果…………………..102

第六章 結論與未來研究方向………………………………………..107
參考文獻…………………………………………………………..…..110







圖表目錄
表2.1 無因次與有因次之關係及表………………………………....112
表2.2 運動方程式各項係數表示符號(上下移動)………………....114
表2.3 運動方程式各項係數表示符號(俯仰) ………………….......116
表2.4 運動方程式各項係數表示符號(減振器) …………………...118
表3.1 系統與減振器相關參數設定………………………………....119
表3.2 俯仰自由度在gs=0.5下之頻率響應……………………......120
表3.3 俯仰自由度在gs=1.5下之頻率響應……………………......121
表3.4 俯仰自由度在gs=3.0下之頻率響應……..............................122
表4.1 運動方程式各項係數表示符號(上下移動) ………………...123
表4.2 運動方程式各項係數表示符號(俯仰) ……...........................124
表4.3 運動方程式各項係數表示符號(減振器) …………………...125
表5.1 系統與減振器相關參數設定………………………………....126
表5.2 俯仰自由度在gs=0.2下之頻率響應……………………......127
表5.3 系統與減振器相關參數設定………………………………....128
表5.4 具減振器系統俯仰自由度之頻率響應……............................129



圖2.1  系統與減振裝置圖……………………………………..........130
圖2.2  翼剖面受力圖……………………………………………......131
圖2.3  減振器受力圖……………………………………………......131
圖3.1  俯仰自由度在不同外力下之頻率響應圖……………..…....132
圖5.1  俯仰自由度之相位圖( )……………………………....133
圖5.2  俯仰自由度收歛後之相位圖( )……............................133
圖5.3  俯仰自由度之相位圖( )……………………………....134
圖5.4  俯仰自由度收歛後之相位圖( )……………………....134
圖5.5  俯仰自由度之相位圖( )……………………………....135
圖5.6  俯仰自由度收歛後之相位圖( )……………………....135
圖5.7  俯仰自由度之相位圖( )……………………………....136
圖5.8  俯仰自由度收歛後之相位圖( )……………………....136
圖5.9  俯仰自由度之相位圖( )……………………………....137
圖5.10  俯仰自由度收歛後之相位圖( )…………………......137
圖5.11  俯仰自由度之相位圖( )…………………………......138
圖5.12  俯仰自由度收歛後之相位圖( )…………………......138
圖5.13  俯仰自由度之相位圖( )……………………………..139
圖5.14  俯仰自由度之相位圖( )……...............................139
圖5.15  俯仰自由度之頻率響應圖(gs=0.2) ………………….......140
圖5.16  具減振器之俯仰自由度頻率響應圖……………………....141
(安裝於距離翼前緣0.3之位置)
圖5.17  具減振器俯仰自由度收歛後之相位圖( )………......142
圖5.18  具減振器俯仰自由度收歛後之相位圖( )………......142
圖5.19  具減振器俯仰自由度收歛後之相位圖( )………......143
圖5.20  具減振器俯仰自由度收歛後之相位圖( )………......143
圖5.21  俯仰自由度之頻率響應圖……...........................................144
(減振器分別安裝於0.1 0.3 0.5 0.9之位置)
圖5.22  俯仰自由度之頻率響應圖....................................................145
(頻率由0~0.55)
圖5.23  俯仰自由度之頻率響應圖....................................................146
(頻率由0.55~1)
圖5.24  俯仰自由度之頻率響應圖....................................................147
(頻率由1~1.53752)
圖5.25  減振器位置與俯仰自由度振幅關係圖................................148
( )
圖5.26  減振器位置與俯仰自由度振幅關係圖................................149
( 、 、 、 )

圖5.27  系統自然振動頻率曲線圖....................................................150
(減振器分別安裝於0.1 0.3 0.5 0.8之位置)
圖5.28  速度與外力頻率與俯仰振幅之關係圖................................151
(減振器安裝於0.3 之位置)
圖5.29  速度與外力頻率與俯仰振幅之三維圖................................152
(減振器安裝於0.3 之位置)
圖5.30  數值解與解析解對照圖........................................................153
(減振器安裝於0.3 之位置)

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