本研究設計一單擺調質減振器Pendulum Tuned Mass Damper (PTMD)，懸掛於二維振動剛體下方，此減振器係以一扭矩彈簧及拉伸彈簧與主體連接，提供該單擺減振器兩個自由度的運動，以抑制主體之振動。本系統之主體 (振動剛體) 兩端係以非線性彈簧支撐，用以模擬非線性振動。本研究設計的PTMD，除了其質量、兩個自由度的彈簧係數之外，還有減振器擺放的位置也一併考慮之，以達到系統之最佳減振效果。吾人利用時間多尺度法 (Method of Multiple Scales) 將非線性運動方程式，分成兩個不同的時間尺度，找出系統各自由度之振動頻率，以尋求發生內共振的條件。並藉由本文提出之3-D內共振等值線圖 (Internal Resonance Contour Plot) 的觀念及Fixed Points 頻率響應 (Fixed Points Frequency Response) 的分析，同時分析PTMD的多種參數，以找出避開內共振且大幅減少主體振幅的PTMD組合。最後並以數值模擬驗證本研究的正確性。

This study examines the effects of pendulum tuned mass damper (PTMD) on a nonlinear vibration system. A novel approach is proposed in which an internal resonance contour plot (IRCP) is used for the analysis of nonlinear dynamic stability. This study considers a vibration system with a planar rigid-body, including plunge and pitch vibration. Both ends of the body are supported by cubic nonlinear springs. The vibration absorber attached beneath is a pendulum tuned mass damper (PTMD) and the mass and position of the absorber are adjusted to optimize vibration reduction. The PTMD is attached to the main rigid body by a set of cubic transverse vibration spring and nonlinear torsional spring. The effects of the hinged springs parameters are also considered. The internal resonance condition for the main rigid body is studied by eigen-analysis. The method of multiple scales (MOMS) is employed to obtain a fixed point solution. This study also uses the Poincare Maps and numerical simulations for comparison. Finally, IRCP with Fixed Point plots are cross-referenced to provide guide-lines to identify the optimal location and the mass for the PTMD with regard to stability and reduced vibration magnitude without the need to alter the framework of the main body.

摘要………………………………………………………………………I
Abstract…………………………………………………………………II
目錄……………………………………………………………………III
表目錄………………………………………………………………… V
圖目錄…………………………………………………………………VII
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 非線性模擬系統模型之建立…………………………………8
2.1 非線性運動方程式之推導……………………………………8
2.2 無因次之非線性運動方程式………………………………12
第三章 非線性系統之分析……………………………………………16
3.1具減振器之非線性運動方程式分析…………………………16
3.2無減振器主體之非線性運動方程式分析……………………19
3.3 內共振條件之分析…………………………………………...20
3.4系統之頻率響應分析…………………………………………22
3.5無減振器主體之頻率響應分析………………………………29
第四章 結果與討論……………………………………………………30
4.1無PTMD主體內共振分析……………………………………30
4.2 具PTMD之系統振動分析…………………………………32
4.2.1有效避開內共振之分析………………………………32
4.2.2有效減振之減振器擺放位置…………………………37
4.2.3 減振器質量比 ( ) 及其他參數的影響……………41
第五章 結論……………………………………………………………46
參考文獻………………………………………………………………48
附錄 (一) ………………………………………………………………51
附錄 (二) ………………………………………………………………52
附錄 (三) ………………………………………………………………54
附錄 (四) ………………………………………………………………55
附錄 (五) ………………………………………………………………68
附錄 (六) ………………………………………………………………69
論文簡要版……………………………………………………………103

[1]王怡仁， 郭蕙蘭， “旋轉機構基座之減振最佳化設計，” 2004中華民國機械學會，機械工程學術研討會，高雄 國立中山大學， 民國九十三年十一月。
[2]Wang, Y. R., and Lin, H. S., “Stability Analysis and Vibration Reduction for A Two-Dimensional Nonlinear System,” International Journal of Structural Stability and Dynamics, Vol.13, No.6, 2013.
[3]Den Hartog, J.P., Mechanical Vibration, McGraw–Hill, 1947.
[4]Vakakis, A.F., and Paipetis, S.A. “The effect of a viscously damped dynamic absorbers on a linear multi-degree-of-freedom system,”  Journal of Sound and Vibration, Vol. 105, 1986, pp. 49–60.
[5]Asustek Computer Inc., “Dynamic damping system for disk drives,” Japan Patent, No. 2, 951, 943, Sep. 1999 (in Japanese).
[6]Zuo, L., and Nayfeh, S. A., “Minimax optimization of multi-degree-of-freedom tuned-mass dampers,” Journal of Sound and Vibration, Vol. 272, 2004, pp. 893–908. 
[7]Almazan, Jose L. et al.“A bidirectional and homo-geneous tuned mass damper: A new device for passive control of vibrations,” Engineering Structures, Vol. 29, 2007, pp. 1548–1560. 
[8]Eissa, M. and Sayed, M., “Vibration reduction of a three DOF non-linear spring pendulum,“ Communications in Nonlinear Science and Numerical Simulation, Vol. 13, 2008, pp. 465–488. 
[9]Nagasaka, I., Ishida, Y., Koyama, T., and Fujimatsu, N. “Vibration suppression of a helicopter fuselage by pendulum absorbers: rigid-body blades with aerodynamic excitation force,” Journal of System Design and Dynamics, Vol. 2, (6), 2008, pp.1230–1238.
[10]Matta, E. and Stefano, A. De Robust design of mass-uncertain rolling-pendulum TMDs for the seismic protection of buildings, Mechanical Systems and Signal Processing, Vol. 23, (1), 2009, pp.127–147.
[11]Wu, S. T., “Active pendulum vibration absorbers with a spinning support,” Journal of Sound and Vibration, Vol. 323, 2009, pp. 1-16.
[12]Wu, S. T., Chen, Y. R., and Wang, S. S., “Two-degree-of-freedom rotational-pendulum vibration absorbers,” Journal of Sound and Vibration, Vol. 330, 2011, pp. 1052-1064.
[13]Wu, S. T. and Siao, P. S. “Auto-tuning of a two-degree-of-freedom rotational peudulum absorber,” Journal of Sound and Vibration, Vol. 323, 2012, pp.1-16.
[14]Pinot, P. and Geneves, G., “Numerical simulation for designing tuned liquid dampers to damp out double-pendulum oscillations,” Measurement Science And Technology, Vol. 22, 2011, pp. 1-14.
[15]Nayfeh, A. H., Introduction to Perturbation Techniques, Wiley- Interscience, New York, 1981, pp. 107-131. 
[16]Hassan, A. “Use of transformations with the higher order method of multiple scales to determine the steady state periodic response of harmonically excited non-linear oscillators,” Journal of Sound and Vibration, Vol. 178, 1994, pp. 21-40.
[17]Ji, Z. and Zu, J. W., “Method of multiple scales for vibration analysis of rotor–shaft systems with non-linear bearing pedestal model,” Journal of Sound and Vibration, Vol. 218, 2, 1998, pp. 293-305. 
[18]Chao, C.P., Sung, C.K., and Wang, C.C., “Dynamic analysis of the optical disk drives equipped with an automatic ball balancer with consideration of torsional motions,” Journal of Sound and vibration , Vol. 2001, 246(1), pp. 115-135.
[19]Vyas, A. and Bajaj, A.K. “Dynamics of autoparametric vibration absorbers using multiple pendulums,” Journal of Sound and Vibration, Vol. 246, 2001, pp. 115-135. 
[20]Nayfeh, A. H. and Mook, Dean T., ”Nonlinear Oscillations” John Wiley & Sons, Inc., New York, 1979, pp. 417-424.