本研究以一非線性Bernoulli-Euler Beam 為主體，將此非線性彈性樑以鋼纜懸掛之，且鋼纜將以非線性彈簧及線性阻尼的組成來模擬其運動。此懸吊系統可模擬一般吊橋之振動方式，若將此懸吊之非線性彈簧與阻尼倒掛置於彈性樑之下，又可模擬此彈性樑置於 Winkler Type 彈性基底的振動型態，因此本研究極具應用價值。本研究中，非線性彈性樑的兩端皆為鉸接，而樑主體上掛載一動態減振器(Dynamic Vibration Absorber (DVA))。本文使用多尺度法(Method of Multiple Scales (MOMS))分析系統於穩態固定點 (Fixed Points) 各模態之頻率響應，且藉由振幅及振動模態觀察其內共振現象。本研究將分析Mass-Spring DVA (MSDVA)的質量、彈性係數及置放於樑上的位置對於此非線性樑的減振效益。再利用Fixed Point plots(Frequency Resp.)並輔之以3D振幅投影圖(3DMACP)全面性的求出此非線性系統之最佳MSDVA的減振組合。

Beam vibration has always been a concern for researchers and engineers and vibration within nonlinear systems is particularly problematic. This study considered a slender hinged-hinged nonlinear elastic beam with suspension cables simulated using nonlinear cubic springs and linear dampers to allow greater amplitude in the transverse direction. The model in this study could be applied to the engineering of structures with nonlinear suspension systems. In addition, inverting the system, we could simulate the beam placing on a Winkler-type elastic foundation. Therefore, there is a wide range of applications for this system. The primary objective of this study was to add a mass spring dynamic vibration absorber (MSDVA) on the beam to avoid internal resonance within this beam and achieve effective vibration damping. The internal resonance condition based on the ratio of the elastic foundation frequency to the beam frequency of the main structure was obtained. The influence of stretching effect and the location of the mass-spring were also taken into account. We employed the method of multiple scales (MOMS) to analyze this nonlinear problem. The Fixed point plots (steady state frequency response) were obtained.  MSDVA with various locations and spring constants were considered and the optimal mass range for the MSDVA to reduce vibration in the main structure was also proposed by using the novel concept of 3-dimensional maximum amplitude contour plots (3D-MACP). 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.

目錄
摘要………………………………………………………………………I
英文摘要………………………………………………………………II
目錄……………………………………………………………………IV
圖表目錄………………………………………………………………VI
第一章	 緒論……………………………………………………………1
一、1 研究動機…………………………………………………1
一、2 文獻回顧…………………………………………………2
一、3 研究方法…………………………………………………5
第二章 系統理論模式建立……………………………………………7
      二、1非線性運動方程式之推導與無因次化…………………7
      二、2 MOMS法…………………….………………………...…10
第三章 內共振條件的分析…………………………………………12
      三、1無MSDVA之非線性樑運動方程……………………12
      三、2 系統內共振之條件………………………………………12
      三、3系統頻率響應之分析……………………………………18
三、4內共振現象之驗證………………………………………23
三、5數值法驗證……………………………………………….23

第四章 MSDVA固定於定點之減振分析…………………………25
      四、1 減振器分析………………………………………………25
      四、2系統頻率響應之分析……………………………………27
第五章 成果與討論………………………………………..…………33
五、1 MSDVA 系統之內共振分析…………………………..33
五、2 MSDVA減振效益分析…………………………………33
五、3數值法驗證之分析……………………………………....37
第六章  結論…………………………..………………………………39
參考文獻………………………………………………………………40
附錄(一) ………………………………………………………………42
附錄(二) ………………………………………………………………43
附錄(三) ………………………………………………………………44
      


 

 
圖目錄
圖1具減振器之主體架構與邊界條件………………………………45
圖2無減振器之主體架構與邊界條件………………………………45
圖3激擾第一模態之第一模態Fixed Points圖 (無減振器) ………46
圖4激擾第一模態之第三模態Fixed Points圖 (無減振器) ………46
圖5激擾第一模態之Fixed Points圖 (無減振器) …………………47
圖6激擾第三模態之第三模態Fixed Points圖 (無減振器) ………47
圖7激擾第三模態之第一模態Fixed Points圖 (無減振器) ………48
圖8激擾第三模態之Fixed Point圖 (無減振器) ……………………48
圖9激擾第一模態之第一模態Fixed Points圖 (具減振器) ………49
圖10激擾第一模態之第三模態Fixed Points圖 (具減振器) ………49
圖11激擾第三模態之第一模態Fixed Points圖 (具減振器).………50
圖12激擾第三模態之第三模態Fixed Points圖 (具減振器) ………50
圖13 3D MACP, mode,   …………………………….……51
圖14 3D MACP, mode,  ……………..……………………51
圖15 3D MACP, mode,  ……………..…………………….52
圖16 3D MACP, mode,  ……………..………………….…52
圖17 3D MACP, mode,  ………...……..……………………53
圖18 3D MACP, mode,  …………..…...…………………53
圖19 3D MACP, mode,  ……………..……………………54
圖20 3D MACP, mode,  ……………....……………………54
圖21 (a)為針對  之數值驗證(b)為針對 之數值驗證……………..………………………55
圖22 (a)為針對 之數值驗證(b)為針對 之數值驗證………..……………….……56
圖23 (a)為針對 之數值驗證(b)為針對 之數值驗證………………..……………………57
圖24 (a)為針對 之數值驗證(b)為針對 之數值驗證..………..…………………58

 
表目錄
表1 Max. Amp. Resp. 1st mode……...………………………………….59
表2 Max Amp. Resp. 3rd mode.……………..………………………….60

[1] A. H. Nayfeh and D. T. Mook, Wiley-Interscience, Nonlinear Oscillations, New York, 1979.
[2] Z. Ji and J. W. Zu, Method of multiple scales for vibration analysis of rotor-shaft systems with non-linear bearing pedestal model, Journal of Sound and Vibration, 218 ,1998, pp.293-305. 
[3] 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.
[4] S. W. Shaw, and C. Pierre, “Normal modes for non-linear vibratory systems,” Journal of Sound and vibration, Vol. 164, pp.85-124, 1993.
[5] 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.
[6] 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.
[7] C. P. Chao, C.K Sung and C. C. Wang, Dynamic analysis of the optical disk drives equipped with an automatic ball balancer with consideration of torsional motions, Journal of Sound and Vibration, 246, 2001, pp.115-135.
[8] A. H. Nayfeh and B. Balachandran, Wiley-Interscience, Wiley-Interscience, Applied Nonlinear Dynamics, New York, 1995, pp.158-172.
[9] A. H. Nayfeh and P. F. Pai, New York, Linear and Nonlinear Structural Mechanics, 2004.
[10] W.T. Van Horssen, and G.J. Boertjens, “On Mode Interactions for a Weakly Nonlinear Beam Equation,” Nonlinear Dynamics, Vol.17, No.4, 1998, pp.23-40.
[11] J. S. Mundrey, Tata McGraw-Hill, Railway Track Engineering, New Delhi, 2000.
[12] Y. M. Fu, J. W. Hong and X. Q. Wang, “Analysis of nonlinear vibration for embedded carbon nanotubes,” Journal of Sound and Vibration, Vol. 296, 2006, pp.746-756.
[13] 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.
[14] Y. R. Wang, and T. H. Chen, “The vibration reduction analysis of a nonlinear rotating mechanism deck system,” Journal of Mechanics, Vol. 24, No. 3, 2008, pp.253-266.
[15] Y. R. Wang, and M. H. Chang, “On The Vibration Reduction of a Nonlinear Support Base with Dual-shock-absorbers,” Journal of Aeronautics, Astronautics and Aviation, Series A, Vol.42, No.3, 2010, pp.179-190.
[16]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, 331, 2012, pp. 2272–2290. 
[17] Y. R. Wang, and H. S. Lin, “Stability analysis and vibration reduction for a two -dimensional nonlinear system,” International Journal of Structural Stability and Dynamics (IJSSD), Vol.13, No.5, 2013. pp.1350031-1~1350031-30.
[18] 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.