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系統識別號 U0002-1707201400575000
中文論文名稱 動態減振器對非線性彈性樑之影響
英文論文名稱 Effects of a Dynamic Vibration Absorber On Nonlinear Hinged-free Beam
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 102
學期 2
出版年 103
研究生中文姓名 郭庭宏
研究生英文姓名 Ting-Hung Kuo
學號 602430034
學位類別 碩士
語文別 中文
口試日期 2013-06-20
論文頁數 105頁
口試委員 指導教授-王怡仁
委員-陳蓉珊
委員-馮朝剛
中文關鍵字 內共振  振動模態  動態減振器  非線性幾何形變  非線性慣量 
英文關鍵字 Internal Resonance  modeshape  Dynamic Vibration Absorber (DVA)  nonlinear geometry  nonlinear inertia 
學科別分類 學科別應用科學航空太空
中文摘要 本研究考慮一彈性樑,其下方以非線性彈簧支撐主體,以模擬彈性樑置於 Winkler type 彈性基座(Elastic Foundation)的振動行為。在本研究中,彈性樑的一端為扭矩彈簧搭配鉸接邊界支撐之,另一端則是自由端,而在彈性樑上掛載動態減振器(Dynamic Vibration Absorber (DVA)),利用DVA達到避開內共振及減振之效果。由於動態減振器放置於自由端邊界時,此邊界條件改變為時變邊界條件。因此吾人採用 Mindlin-Goodman 法分析此問題,並藉由多項式移位函數 (Shifting Polynomial Function) 將非齊次性邊界條件轉換為齊次性邊界條件。本文並使用多尺度法 (Method of Multiple Scales (MOMS) ) 解析此非線性系統,發現在彈性基座某彈性係數的環境下,系統中之第一模態(Mode)及第二模態存在一對三(1:3) 的內共振情形。本研究並繪製各模態之穩態固定點圖 (Fixed Point Plots),藉由振幅及振動模態觀察其非線性內共振現象(Internal Resonance),並藉由此解釋非線性幾何形變(nonlinear geometry)與非線性慣量(nonlinear inertia)兩者之間的關係,以及加裝DVA後,DVA之最佳彈性係數與質量比的組合,使得此系統避開內共振及達到最佳減振效果。本研究最後以數值法及簡易實驗模型驗證結果之正確性。
英文摘要 This study considered a slender hinged-free nonlinear beam embedded in a Winkler type elastic foundation simulated using nonlinear cubic springs. This also created multiple possibilities for mode coupling and internal resonance. The objective of this study was to avoid internal resonance within this system and achieve effective vibration damping. We added a time-dependent boundary dynamic vibration absorber (TDB DVA) that was suspended at the free end of the beam to reduce vibration and prevent internal resonance. The Mindlin-Goodman method was used to analyze this time dependent boundary condition problem. The internal resonance condition based on the ratio of the elastic foundation frequency to the beam frequency of the main structure was obtained. The vibration reduction effects of other positions of the DVA were also 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. DVAs with various locations and spring constants were considered and the optimal mass range for the DVA to reduce vibration in the main structure was also proposed. The results of this study were verified using numerical simulation, which, in addition to confirming the accuracy by through comparison, established the applicability in this study.
論文目次 目錄
摘要………………………………………………………………………I
英文摘要………………………………………………………………II
目錄……………………………………………………………………III
表目錄…………………………………………………………………VI
圖目錄………………………………………………………………VII
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………………………………………………8
第二章 系統理論模型之建立 ……………………………………10
二、1運動方程式之推導……………………………………10
二、2無因次化之運動方程式……………………………12
二、3無因次化動態減振器之運動方程式………………14
二、4多尺度法……………………………………………16
二、5 DVA位於自由端之分析……………………………18
二、6減振器之時間域動力方程式………………………24
第三章 系統內共振之條件……………………………………………29
三、1 無減振器之系統……………………………………29
三、2 內共振之條件分析…………………………………29
三、3 系統之頻率響應解分析……………………………36
三、3、(A) 激擾第一模態…………………………………37
三、3、(B) 激擾第二模態…………………………………41
第四章 具有減振器之系統分析………………………………………45
四、1 減振系統內共振之分析……………………………45
四、2 自由端減振器之系統頻率響應分析………………48
四、2、(A) 激擾第一模態…………………………………48
四、2、(B) 激擾第二模態…………………………………52
四、3減振器不位於端點之系統頻率響應分析…………55
四、3、(A) 激擾第一模態…………………………………56
四、3、(B) 激擾第二模態…………………………………58
第五章 結果與討論……………………………………………………63
五、1 討論…………………………………………………63
五、2 數值法驗證…………………………………………69
五、3 簡易實驗……………………………………………70
第六章 結論……………………………………………………………73
參考文獻 ……………………………………………………………75
附錄(一) ……………………………………………………………78
附錄(二) ………………………………………………………………79
論文簡要版……………………………………………………………98


表目錄
表1 四種方法對於激擾第一模態之第一模態最大振幅影響………80
表2四種方法對於激擾第一模態之第二模態最大振幅影響………80
表3 四種方法對於激擾第二模態之第一模態最大振幅影響………81
表4 四種方法對於激擾第二模態之第二模態最大振幅影響………81


圖目錄
圖1 具減振器之主體架構與邊界條件………………………………82
圖2 減振器於邊界之主體架構與邊界條件…………………………82
圖3 不具減振器之主體架構與邊界條件……………………………83
圖4 主體樑之第一、二模態振動示意圖……………………………83
圖5 激擾第一模態之第一、二模態Fixed point圖(無減振器) …84
圖6 激擾第二模態之第一、二模態Fixed point圖(無減振器) …85
圖7 與減振器質量比的關係圖…………………………………86
圖8 與減振器質量比的關係圖…………………………………86
圖9 非線性慣量與非線性幾何形變對系統之影響………………87
圖10 四種方法對於激擾第一模態之第一模態之總振幅圖…………88
圖11 四種方法對於激擾第一模態之第二模態之總振幅圖…………88
圖12 四種方法對於激擾第二模態之第一模態之總振幅圖…………89
圖13 四種方法對於激擾第二模態之第二模態之總振幅圖…………89
圖14 時激擾第一模態之第一模態時間域、Poincare Map驗證圖…………………………………………………………………90
圖15 時激擾第一模態之第一模態時間域、Poincare Map驗證圖…………………………………………………………………90
圖16 時激擾第一模態之第一模態時間域、Poincare Map驗證圖………………………………………………………………91
圖17 激擾第一模態之第一模態之四種方法時間域、Poincare Map驗證圖………………………………………………………………92
圖18 激擾第一模態之第二模態之四種方法時間域、Poincare Map驗證圖………………………………………………………………93
圖19 主體樑之結構圖………………………………………………94
圖20 主體樑與固定零件之組合圖…………………………………94
圖21 實驗系統之實體圖……………………………………………95
圖22 實驗流程圖……………………………………………………95
圖23激擾 時減振器位置與樑之振幅關係圖……………………96
圖24 激擾 時減振器位置與樑之振幅關係圖……………………96
圖25 激擾 時減振器位置與樑之振幅關係圖……………………97
圖26減振器位置與樑之振幅平均值關係圖………………………97
參考文獻 [1] A.H. Nayfeh and D.T. Mook, Nonlinear Oscillations, Wiley-Interscience, New York, 1979.
[2] A.H. Nayfeh, Perturbation Methods, Wiley, New York, 1973.
[3] A.H. Nayfeh, Introduction to Perturbation Techniques, Wiley, New York, 1981.
[4] A.H. Nayfeh, and S.A. Nayfeh, “On nonlinear modes of continuous systems,” Transactions of the ASME, Journal of Vibration and Acoustics, Vol. 116, 1994, pp.129–136.
[5] S.W. Shaw, and C. Pierre, “Normal modes for non-linear vibratory systems,” Journal of Sound and vibration, Vol. 164, 1993, pp.85–124.
[6] S.W. Shaw, and C. Pierre, “Normal modes of vibration for non-linear continuous systems,” Journal of Sound and vibration, Vol. 169, 1994, pp.319–347.
[7] E. Pesheck, and C. Pierre, “A new galerkin-based approach for accurate non-linear normal modes through invariant manifolds,” Journal of Sound and vibration, Vol. 249, No.5, 2002, pp. 971–993.
[8] W.T. Van Horssen, and G.J. Boertjens, “On Mode Interactions for a Weakly Nonlinear Beam Equation,” Nonlinear Dynamics 17, 1998, pp.23–40
[9] R.D. Mindlin, and L.E. Goodman, “Beam Vibration with Time-Dependent Boundary Conditions,” ASME Journal of Applied Mechanics, Vol.17, 1950, pp.377–380.
[10] C.A. Rossit, and P.A.A. Laura, “Free vibrations of a cantilever beam with a spring–mass system attached to the free end,” Journal of Ocean Engineering, 28, 2001, pp.933–939
[11] Y.R. Wang, and T.H. Chen, “The vibration analysis of a nonlinear rotating mechanism desk system,” Journal of Mechanics,” 24, 2008, pp.253–266.
[12] Y.R. Wang, and M.H. Chang, “On the vibration of a Nonlinear Support Base with Dual-shock-absorbers,” Journal of Aeronautics, Astronautics and Aviation, Series A, 42, 2010, pp.179–190
[13] Y.R. Wang, and H.S. Lin, “Stability analysis and vibration reduction for a two-dimensional nonlinear system,” International Journal of Structural and Dynamics (IJSSD), 13, No.5, 2013, 1350031-1~1350031-30
[14] Y.R. Wang, and C.M. Chang, “Elastic beam with nonlinear suspension and a dynamic vibration absorber at the free end,” Transaction of the Canadian Society for Mechanical Engineering (TCSME), Vol. 38, No.1, 2014, pp.107-137.
[15] L. Zuo, and S.A. Nayfeh, “Minimax optimization of multi-degree-of-freedom tuned-mass dampers,” Journal of Sound and Vibration, Vol. 272, 2004, pp. 893-908.
[16] J.J. Wu, “Study on the inertia effect of helical spring of the absorber on suppressing the dynamic responses of a beam subjected to a moving load,” Journal of Sound and Vibration, 297, 2006, pp. 981–999.
[17] F.S. Samani, and F. Pellicano, “Vibration reduction on beams subjected to moving loads using linear and nonlinear dynamic absorbers,” Journal of Sound and Vibration, 325, 2009, pp. 742–754.
[18] F.S. Samani, and F. Pellicano, “Vibration reduction of beams under successive traveling loads by means of linear and nonlinear dynamic absorbers,”Journal of Sound and Vibration, Vol. 331, 2012, pp.2272–2290.
[19] A.H. Nayfeh and P.F. Pai, “Linear and Nonlinear Structural Mechanics,” Wiley-Interscience Publication, Ch.4 and 5, New York, 2004.
[20] H.S. Shen, “A Novel Technique for Nonlinear Analysis of Beams on Two parameter Elastic Foundations,” International Journal of Structural Stability and Dynamics, Vol. 11, No.6, 2011, pp.999–1014.
[21] V.I. Babitsky and A.M. Veprik, “Damping of Beam Force Vibration by a Moving Washer,” Journal of Sound and Vibration, 166, No.1, 1993, pp.77–85
[22] E.C. Miranda and J.J. Thomsen, “Vibration Induced Sliding: Theory and Experiment for a Beam with a Spring-Loaded Mass,” Nonlinear Dynamics 16, 1998, pp.167–186
[23] C.S. Cai, W.J. Wu, and X.M. Shi, “Cable Vibration Reduction with a Hung-on TMD System. Part I: Theoretical Study,” Journal of Vibration and Control, 12, No.7, 2006, pp.801–814.
[24] C.S. Cai, and W.J. Wu, “Cable Vibration Reduction with a Hung-on TMD System. Part II: Parametric Study,” Journal of Vibration and Control, 12, No.8, 2006, pp.881–899.
[25] http://www2.taipei-101.com.tw/ch/DB/damper.asp
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