本研究主要目的為利用解析解與數值解兩種方法分析雙軸三支撐式移動質量結構的振動情形。文中利用相關假設推導出系統動態方程式的解析解並運用數值方法計算數值解，比較兩者計算分析產生的差異，最終得到解析解的適用條件，幫助快速求得分析結果。當求得系統之振動情形後，將軸的最大振幅當作減振目標，在軸上懸掛一彈簧、質量系統作為避振器，同樣運用解析解與數值解兩種方法進行分析，並對兩者運用相同的設計流程，首先利用黃金比例搜尋法找出設計之速度條件，接著利用圖形最佳化方法作圖找出避振器設計參數，從分析結果可以得知兩個方法所求得的避振器參數十分接近，接著將避振器參數代入模型，證實設計之避振器可在一速度區間下產生減振效用。最後討論解析解與數值解的差異，數值解可適用的分析範圍較廣、所得結果較為精準，但須花費較長的時間，而解析解在滿足一定條件下，可快速分析求得與數值解相近的答案，減少計算時間。

This study analyzes the dynamic system with a moving mass using three supports on parallel simply supported beams. The method to design a vibration absorber mounted on the lower beam of the system is proposed to reduce the vibration of the beam excited by the moving mass. The governing equations of the system are constructed based on the Euler–Bernoulli beam theory. Both analytical and numerical methods are used to analyze the dynamic system to solve the motion equations for the moving mass, vibration absorber, upper beam, and lower beam. The design objective of the vibration absorber is to minimize the maximum amplitude of the lower beam. The golden section search method is used to identify the critical speed of the moving mass which can excite the maximum amplitude of the lower beam. Then the graphical optimization method is used to search for the optimal design parameters (stiffness and location) of the vibration absorber to minimize the maximum amplitude at the critical speed. The results show that the optimized design of the vibration absorber can minimize the oscillation of the system.

[1] J. M. Biggs and B. Testa, 1964, Introduction to structural dynamics, vol. 3, ed: McGraw-Hill New York.
[2] Y.-H. Lin and M. W. Trethewey, 1990, "Finite element analysis of elastic beams subjected to moving dynamic loads," Journal of Sound and Vibration, vol. 136, pp. 323-342.
[3] M. Olsson, 1991, "On the fundamental moving load problem," Journal of sound and vibration, vol. 145, pp. 299-307.
[4] 陳彥樺, 2007, "移動質量與荷載作用下之剛架結構動力行為分析," 土木工程研究所碩士論文, 國立中央大學.
[5] 邱培倫, 2009, "隨機參數吸振器於承載移動質量梁之減振設計," 機械工程研究所碩士論文, 國立台灣科技大學.
[6] Y.-B. Yang, C. Lin, and J. Yau, 2004, "Extracting bridge frequencies from the dynamic response of a passing vehicle," Journal of Sound and Vibration, vol. 272, pp. 471-493.
[7] 陳志偉, 2003, "以有限元素法分析軌道結構於輪-軌互制作用下之反應," 土木工程研究所碩士論文, 國立成功大學.
[8] 伍華永, 2011, "軌道不平整效應之實驗與模擬," 土木工程研究所碩士論文, 國立中央大學.
[9] J. Yang and R. Duan, 2013, "Modelling and Simulation of a Bridge interacting with a moving Vehicle System," M.S. thesis, Department of mechanical engineering, Blekinge Institute of Technology Karlskrona, Sweden.
[10] 林益煌和丁祥軒, 2012, "承受移動動態負載之高速精密定位平台的動態分析與性能改善," 中華民國振動與噪音工程學會論文集, pp. 225-232.
[11] 賴士程, 2007, "桿件承受移動式質量塊之振動模擬與穩定性分析," 機械工程研究所碩士論文, 國立台灣科技大學.
[12] S. Heinlein, 1994, “Optimal design of beams for moving loads with a deflection constraint,” Journal of Non-Linear Mechanics, vol. 29, pp. 205-216.
[13] P. Walsh and J. Lamancusa, 1992, "A variable stiffness vibration absorber for minimization of transient vibrations," Journal of sound and vibration, vol. 158, pp. 195-211.
[14] H.-C. Kwon, M.-C. Kim, and I.-W. Lee, 1998, "Vibration control of bridges under moving loads," Computers & Structures, vol. 66, pp. 473-480.

[15] J.-J. Wu, 2006, "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, vol. 297, pp. 981-999.
[16] R. Faal, M. Amiri, A. Pirmohammadi, and A. Milani, 2012, "Vibration analysis of undamped, suspended multi-beam absorber systems," Meccanica, vol. 47, pp. 1059-1078.
[17] P. Museros and M. Martinez-Rodrigo, 2007, "Vibration control of simply supported beams under moving loads using fluid viscous dampers," Journal of sound and vibration, vol. 300, pp. 292-315.
[18] M. Febbo and S. Vera, 2008, "Optimization of a two degree of freedom system acting as a dynamic vibration absorber," Journal of vibration and acoustics, vol. 130, pp. 11-13.
[19] F. S. Samani and F. Pellicano, 2009, "Vibration reduction on beams subjected to moving loads using linear and nonlinear dynamic absorbers," Journal of sound and vibration, vol. 325, pp. 742-754.
[20] J. S. Arora, 2012, "Chapter 3 - Graphical Optimization and Basic Concepts," Introduction to Optimum Design (Third Edition), ed:Academic Press Boston, pp. 65-94.
[21] R. Clough and J. Penzien, 1975, "Structural dynamics," ed:McGraw Hill New York.
[22] N. M. Newmark, 1959, "A method of computation for structural dynamics," Journal of the engineering mechanics division, vol. 85, pp. 67-94.
[23] S. Hugul, 2005, "Vibration analysis of systems subjected to moving loads by using the finite element method," M.S thesis, Department of mechanical engineering, Dokuz Eylül University, Turkey.
[24] 葉乃昇, 2012, "利用移動彈簧-阻尼-質量元素來進行承受移動負載," 輪機工程研究所碩士論文, 國立高雄海洋科技大學.
[25] J. S. Arora, 2012, "Chapter 10 - Numerical Methods for Unconstrained Optimum Design," Introduction to Optimum Design (Third Edition), ed:Academic Press Boston, pp. 411-441.