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系統識別號 U0002-0402201016252900
中文論文名稱 二維端點支撐之非線性振動系統研究
英文論文名稱 A Two Dimensional Vibration Analysis with Nonlinear End Struts
校院名稱 淡江大學
系所名稱(中) 航空太空工程學系碩士班
系所名稱(英) Department of Aerospace Engineering
學年度 98
學期 1
出版年 99
研究生中文姓名 林漢翔
研究生英文姓名 Han-Shiang Lin
學號 697430188
學位類別 碩士
語文別 中文
口試日期 2010-01-18
論文頁數 108頁
口試委員 指導教授-王怡仁
委員-洪健君
委員-蕭富元
中文關鍵字 振動  非線性  減振器  內共振  振顫速度 
英文關鍵字 Vibration  Nonlinear  Absorber  Internal Rensonance  Flutter Speed 
學科別分類 學科別應用科學航空太空
中文摘要 本文研究一極細平面剛體剖面振動系統,包含主體及其下方附加減振器之上下位移與俯仰旋轉共四個自由度。並將主體兩端之支撐假設為三次方非線性彈簧與二次方非線性阻尼,以分析系統之非線性振動行為。另外吾人更進一步假設此系統含有3 to 1內共振之特性並考慮系統受一空氣動力之影響,再利用牛頓定律推導出四自由度之運動方程組。而後藉由多尺度法(Method of Multiple Scales)求其穩態之頻率響應並分別激發第一、二自由度,與內共振等值線圖(Internal Resonance Contour Plot)相互比較並觀察內共振對於系統之影響性。而後吾人將利用古典振顫分析的觀念繪製Eigen Plot,找出系統的振顫速度,並與相位圖做比對,再藉由特徵值分析及Floquet Theorem來判斷系統之穩定性。
最終吾人藉由內共振等值線圖與振顫速度等值線圖(Flutter Speed Contour Plot)相互對照,嘗試找出減振器最佳之擺放位置,以達到不變動主體架構原則之下,僅改變減振器位置之最佳減振效果。本文發現,在系統設計時,除了振顫速度是必須注意的項目外,內共振對於系統之影響性也是不容小覷,某些飛行狀況之下,在未達到振顫速度時,可能發生內共振現象,且伴隨極大之振幅及不穩定現象,因此,必須將兩者皆列入設計上考慮的因素,缺一不可,如此一來將能使系統擁有更高的安全性。
英文摘要 This research used a two-dimensional rigid body plate section as the main body, which contains pitch and plunge motion. Both ends of this main body were supported by cubic spring and quadratic damper acted as the support to simulate main body’s nonlinear vibration. A rigid body vibration absorber was suspended underneath the main body, and was mounted with linear spring and damper at both ends. This study also took aerodynamic influence into consideration to form an aeroelastic system. The 3:1 internal resonance of the main body’s plunge and pitch modes was included in this research. The Method of Multiple Scales was employed to analyze the fixed points solutions of this system.
We made an Internal Resonance Contour Plot to get an overview of this 3:1 internal resonance system. This research found that simply adjust the absorber position can significantly reduce internal resonance. We also used the concept of Classical Flutter Analysis to make an eigen-plot to analyze the aerodynamic effects on this system. The phase plots and Flutter Speed Contour Plots were also correlated to verify the system flutter speed.
Finally, the Floquet Theorem was employed to get the Basin of Attraction of this system and the system stability information was concluded. In the case studied, we found that the internal resonance may play an important role in the aeroelastic system. It could result large vibration amplitude before the flutter speed. We also found that in the internal resonance-flutter coupled system, just change the absorber position can also significantly reduce vibration amplitude and increase flutter speed.
論文目次 摘要................................................I
英文摘要............................................II
目錄................................................III
表目錄..............................................V
圖目錄..............................................VI
第一章、緒論........................................1
1-1 研究動機........................................1
1-2 文獻回顧........................................3
1-3 研究方法........................................7
第二章、系統理論模式之建立..........................9
2-1 非線性運動方程式之推導..........................9
2-2 二維主體之氣體彈性運動方程式....................14
第三章、非線性系統之近似解析解......................19
3-1 非線性運動方程式之近似解析解....................19
3-2 解耦之頻率響應解................................23
第四章、系統之穩定性分析............................29
4-1 系統之穩定性....................................29
4.2 系統之振顫速度及分歧現象........................32
第五章、結果與討論..................................34
第六章、結論與未來研究方向..........................45
參考文獻............................................47
附錄(一) 系統之各項無因次化參數設定.................49
附錄(二) 運動方程式各項係數.........................50
附錄(三) 系統之外力展開項...........................52
附錄(四) 造成系統發散之外力項.......................57
論文簡要版..........................................101

表目錄

表1. 振動系統之各項參數設定值........................................................69
圖目錄

圖1 主體與減振裝置.............................................................................70
圖2 主體受力自由體圖.........................................................................70
圖3 減振器受力自由體圖.....................................................................71
圖4 具非線性阻尼與線性阻尼之穩態頻率響應比較圖.....................71
圖5 內共振立體示意圖.........................................................................72
圖6 內共振等值線圖.............................................................................72
圖7 U =7.0,m/M =0.1,之內共振剖面圖.............................................73
圖8 U =7.0,m/M =0.1,減振器位置0.12C激發第一自由度之FIXED POINT 圖(第一自由度)......................................................................................73
圖9 U =7.0,m/M =0.1,減振器位置0.12C激發第一自由度之FIXED POINT圖(第二自由度)......................................................................................74
圖10 U =7.0,m/M =0.1,減振器位置0.12C激發第二自由度之FIXED POINT圖..................................................................................................74
圖11 U =7.0,m/M =0.1,減振器位置0.12C激發第二自由度之FIXED POINT圖..................................................................................................75
圖12 U =7.0,m/M =0.1,激發第二自由度之POINCARE MAP(第一自由度) ............................................................................................................75
圖13 U =7.0,m/M =0.1,激發第二自由度之POINCARE MAP(第二自由度) ............................................................................................................76
圖14 U =7.0,m/M =0.3,激發第二自由度之POINCARE MAP(第一自由度) ............................................................................................................76
圖15 U =7.0,m/M =0.3,激發第二自由度之POINCARE MAP(第二自由度) .............................................................................................................77
圖16 U =7.0,m/M =0.3,之內共振剖面圖.............................................77
圖17 U =7.0,m/M =0.3,減振器位置0.0C激發第二自由度之
FIXED POINT圖......................................................................................78
圖18 U =7.0,m/M =0.3,減振器位置0.62C激發第二自由度之
FIXED POINT圖......................................................................................78
圖19 U =7.0,m/M =0.3,減振器位置0.0C激發第二自由度之相位圖(第一自由度).....................................................................................................79
圖20 U =7.0,m/M =0.3,減振器位置0.0C激發第二自由度之相位圖(第二自由度).....................................................................................................79
圖21 U =5.08,m/M=0.3,減振器位置0.62C,sigma=8,激發第二自由度之相位圖(第一自由度)............................................................................80
圖22 U =5.08,m/M=0.3,減振器位置0.62C,sigma=8,激發第二自由度之相位圖(第二自由度)........................................................................................80
圖23 振顫速度等值線圖.........................................................................81
圖24 m/M=0.3,減振器位置0.88C之EIGEN PLOT (虛部) ...................81
圖25 m/M=0.3,減振器位置0.88C之EIGEN PLOT (實部)....................82
圖26 U =6.9,m/M=0.3,減振器位置0.88C激發第一自由度之
相位圖(第一自由度),T=0~10000.........................................................82
圖27 U =6.9,m/M=0.3,減振器位置0.88C激發第一自由度之
相位圖(第一自由度)................................................................................83
圖28 U =6.9,m/M=0.3,減振器位置0.88C激發第一自由度之
相位圖(第二自由度)................................................................................83
圖29 U =6.932,m/M=0.3,減振器位置0.88C激發第一自由度之
相位圖(第一自由度)................................................................................84
圖30 U =6.932,m/M=0.3,減振器位置0.88C激發第一自由度之
相位圖(第二自由度)................................................................................84
圖31 U =6.9,m/M=0.3,減振器位置0.88C激發第一自由度之
POINCARE MAP(第一自由度)..............................................................85
圖32U =6.9,m/M=0.3,減振器位置0.88C激發第一自由度之
POINCARE MAP (第二自由度)..............................................................85
圖33U =6.932,m/M=0.3,減振器位置0.88C激發第一自由度之
POINCARE MAP (第一自由度) .............................................................86
圖34 U =6.932,m/M=0.3,減振器位置0.88C激發第一自由度之
POINCARE MAP (第二自由度) .............................................................86
圖35 U =6.932,m/M=0.3,減振器位置0.48C激發第一自由度之
相位圖(第一自由度).................................................................................87
圖36 U =6.932,m/M=0.3,減振器位置0.48C激發第一自由度之
相位圖(第二自由度)..................................................................................87
圖37 U =6.932,m/M=0.3,減振器位置0.48C激發第二自由度之
相位圖(第一自由度)..................................................................................88
圖38 U =6.932,m/M=0.3,減振器位置0.48C激發第二自由度之
相位圖(第二自由度)................................................................................88
圖39 U =6.932,m/M=0.3,減振器位置0.48C激發第一自由度之
POINCARE MAP (第一自由度) ..............................................................89
圖40 U =6.932,m/M=0.3,減振器位置0.48C激發第一自由度之
POINCARE MAP (第二自由度) ..............................................................89
圖41 U =6.932,m/M=0.3,減振器位置0.48C激發第二自由度之
POINCARE MAP (第一自由度) .............................................................90
圖42 U =6.932,m/M=0.3,減振器位置0.48C激發第二自由度之
POINCARE MAP (第二自由度) .............................................................90
圖43 U =0.7,m/M=0.1,減振器位置0.12C之BASION OF
ATTRACTION.........................................................................................91
圖44 U =0.7,m/M=0.1,減振器位置0.44C之BASION OF
ATTRACTION.........................................................................................91
圖45 U =0.7,m/M=0.3,減振器位置0.44C之BASION OF
ATTRACTION.........................................................................................92
圖46 U =0.7,m/M=0.3,減振器位置0.88C之BASION OF
ATTRACTION.........................................................................................92
圖47 U =2.0,m/M=0.1,減振器位置0.7C之BASION OF
ATTRACTION.........................................................................................93
圖48 U =2.24,m/M=0.1,減振器位置0.7C之BASION OF
ATTRACTION.........................................................................................93
圖49 U =2.3,m/M=0.1,減振器位置0.7C之BASION OF
ATTRACTION.........................................................................................94
圖80 U =8.65,m/M=0.1,減振器位置0.7C之BASION OF
ATTRACTION.........................................................................................94
圖51 U =8.7,m/M=0.1,減振器位置0.7C之BASION OF
ATTRACTION.........................................................................................95
圖52 U =6.932,m/M=0.3,減振器位置0.88C之BASION OF
ATTRACTION.........................................................................................95
圖53 U =6.932,m/M=0.3,之內共振剖面圖..........................................96
圖54 U =6.932,m/M=0.3,減振器位置0.1C之BASION OF
ATTRACTION.........................................................................................96
圖55 U =6.932,m/M=0.3,減振器位置0.48C之BASION OF
ATTRACTION.........................................................................................97
圖56 U =6.932,m/M=0.3,減振器位置0.5C之BASION OF
ATTRACTION.........................................................................................97
圖57 U =13.8,m/M=0.3,減振器位置0.42C激發第二自由度之
相位圖(第一自由度)................................................................................98
圖58 U =13.8,m/M=0.3,減振器位置0.42C激發第二自由度之
相位圖(第二自由度)................................................................................98
圖59U =14.0,m/M=0.3,減振器位置0.42C激發第二自由度之
相位圖(第一自由度)................................................................................99
圖60 U =14.0,m/M=0.3,減振器位置0.42C激發第二自由度之
相位圖(第二自由度)................................................................................99
圖61 U =14.1,m/M=0.3,減振器位置0.42C激發第二自由度之
相位圖(第一自由度)..............................................................................100
圖62 U =14.1,m/M=0.3,減振器位置0.42C激發第二自由度之
相位圖(第二自由度)..............................................................................100
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