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系統識別號 U0002-2608200801255900
中文論文名稱 端點非線性支撐之彈性樑振動研究
英文論文名稱 A Study of Beam Vibration with Nonlinear Boundaries
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 96
學期 2
出版年 97
研究生中文姓名 方志緯
研究生英文姓名 Zhi-Wei Fang
學號 695430164
學位類別 碩士
語文別 中文
口試日期 2008-07-22
論文頁數 52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  端點非線性支撐  彈性樑 
英文關鍵字 Vibration  Nonlinear boundaries  Elastomeric beam 
學科別分類 學科別應用科學航空太空
中文摘要 在本研究中,吾人將研究彈性樑在兩端點為非線性支撐的條件下,承受簡諧外力時的頻率響應分析。其中,彈性樑假設為Bernoullis-Euler beam,兩端以三次方彈簧的支撐以模擬非線性邊界條件,再利用傅立葉展開式(Fourier expansion)描述樑的運動行為。如此,以建立吾人之偽非線性振動模型。在運動方程建立完成後,利用貝索函數(Bessel function)來模擬解析函數的形式,其次再以漢克轉換(Hankel transform)來求其特解。觀察研究結果可發現,在非線性頻率響應分析當中會有跳躍現象(jump phenomenon)產生,其現象不只在單一頻率時發生,而是在一頻率域之中發生。由此現象可得知物體振動時的共振頻率及不穩定頻率域,而線性邊界條件假設則無法捕捉此現象。因為線性邊界條件狀況下只能觀察到物體的共振頻率。此一模型及本文所提之新的分析方式可運用在廣泛的工程問題當中,如吊橋、高速鐵路以及各種機械裝置。根據研究結果,在探討振動問題時,除了要關心共振頻率外,在非線性運動狀況下,不穩定頻率域的觀察也是極其重要的。
英文摘要 In this research, the beam vibration with nonlinear boundary conditions sustaining simple harmonic loads is studied. The elastic beam is modeled by the Bernoullis-Euler beam theory. The two ends are supported by cubic springs allowing nonlinear boundaries. The Fourier expansion is applied to the linear motion of the beam alone. However, cubic spring forces give the nonlinear constrains on the boundaries. This quasi-nonlinear analytic model for the beam vibration equation is established. The Bessel function is used to formulate the beam vibration. The Hankel transform is applied to obtain the solution. The results discover the nonlinear jump phenomenon in several frequency ranges. This gives additional information of the unstable behavior for the beam vibration, which cannot be predicted from linear approximations. This model and the new analytic technique can be applied in a wide range of engineering problems, such as suspension bridge, high speed rail road, and other mechanical devices. According to the results, one should not only concern the resonant frequency, but also the unstable frequency ranges for the nonlinear motion.
論文目次 中文摘要………………………………………………………………I
英文摘要………………………………………………………………II
目錄……………………………………………………………………IV
圖目錄…………………………………………………………………VI
表目錄…………………………………………………………………VII
第一章、緒論…………………………………………………………1
ㄧ、1 研究機…………………………………………………………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 彈性樑模型建立………………………………………………8
二、2 彈性樑之運動方程……………………………………………8
第三章、研究成果……………………………………………………29
三、1 前言…………………………………………………………29
三、2 成果探討……………………………………………………29
第四章、結論…………………………………………………………32
參考文獻………………………………………………………………33
附錄……………………………………………………………………44

圖目錄
圖 1 線性支撐(節錄自F.Rudinger[4]文中)……………………36
圖 2 非線性支撐(節錄自Akihiro Suzuki等人[11]文中) ……36
圖 3 彈性樑與邊界條件…………………………………………37
圖 4 彈性樑第二模態圖…………………………………………37
圖 5 圓盤振動模態圖……………………………………………38
圖 6 彈性樑頻率響應圖(X=0.3,β=4.5 )…………………38
圖 7 彈性樑頻率響應圖(X=0.5,β=4.5 )…………………39
圖 8 彈性樑頻率響應圖(X=0.7,β=4.5 )…………………39
圖 9 跳躍現象(Jump Phenomenoe)頻率響應圖…………………40
圖 10 線性邊界彈性樑頻率響應圖 ………………………………40
圖 11 彈性樑頻率響應圖(X=0.5,β=2.0 )…………………41
圖 12 彈性樑頻率響應圖(X=0.5,β=9.0 )…………………41
圖 13 彈性樑頻率響應圖(X=0.5,β=2.0 )…………………42










表目錄
表 1 β=4.5╳109…………………………………………………43
表 2 X=0.5………………………………………………………43


參考文獻 [1] Almazan, Jose L., De la Llera, Juan C., Inaudi, Jose A., Lopez-Garcıa, Diego and Izquierdo, Luis E., “A Bidirectional and Homogeneous Tuned Mass Damper: A New Device for Passive Control of Vibrations,”Elsevier, Engineering Structures, 29, pp. 1548–1560, (2007).
[2] Hijmissen, J. W. and Horssen, W. T. van, “On Aspects of Damping for A Vertical Beam with A Tuned Mass Damper at the Top,”Springer Netherlands, Nonlinear Dynamics, 50, pp. 169–190, (2007).
[3] Zhang, Xin, Brownjohn, James M. W., Wang, Yi, and Pan, Tso-Chien, “Direct Observations of Non-Stationary Bridge Deck Aeroelastic Vibration in Wind Tunnel,” Elsevier, Journal of Sound and Vibration, 291, pp. 202–214, (2006).
[4] Rüdinger, F., “Response Spectral Density for Oscillators with Nonlinear Damping,”ASCE, Journal of Engineering Mechanics, 133, pp. 278–289, (2007).
[5] Ibrahim, R.A. and Somnay, R.J., “Nonlinear Dynamic Analysis of An Elastic Beam Isolator Sliding on Frictional Supports,” Elsevier, Journal of Sound and Vibration, 308, pp. 735–757, (2007).
[6] Chio, Y. S., Jen, K. C. and Mckenna, P. J., “The Structure of the Solution Set for Seriodic Oscillations in A Sispension Bridge Model,”IMA, Journal of Applied Mathematics, 47, pp. 283–306, (1991).
[7] Mindlin, R. D. and Goodman, L. E., “Beam Vibration with Time-Dependent Boundary Conditions,”ASME, Journal of Applied Mechanics, 17, pp. 377–380, (1950).
[8] Grant, D. A., “Beam Vibration with Time-Dependent Boundary Conditions,” Elsevier, Journal of Sound and Vibration, 89, pp. 519–522, (1983).
[9] 黃志誠, “邊界會隨時間變化的非均勻Timoshenko樑的振動分析.” (1997).
[10] Yang, Jann N., Pan, Shuwen, and Lin, Silian, “Least-Squares Estimation with Unknown Excitations for Damage Identification of Structures,”ASCE, Journal of Engineering Mechanics, 133, pp. 12–21, (1983).
[11] Suzuki, Akihiro, Kamiya, Keisuke, and Yasuda, Kimihiko, 2006,“Identification Technique for Nonlinear Boundary Conditions of A Circular Plate,” Elsevier, Journal of Sound and Vibration, 289, pp. 130–147, (2006).
[12] Shueei, M. L., 1998,“Pretwisted Nonuniform Beams with Time-Dependent Elastic Boundary Conditions,” AIAA, AIAA Journal, 36, pp. 1516–1522, (1998).
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