本研究提出Laplace解析方法，隨著阻尼係數之變化，解析單自由度具1/2階微分阻尼系統之自由振動特性。研究根據Caputo之分數階微分定義，並以Laplace方法作為解析之工具，運算過程中將其關鍵方程式設為特徵方程式並討論，發現特徵方程式之四根將會隨著阻尼係數而改變，且會影響整體系統之振動情形，研究也顯示出若系統解拆成幾個部分，有些部分之函數會呈振盪模式，有些部分則是純衰減。其次再與系統之數值解作比較，驗證其解析解之正確性，最後提出等效模型，在阻尼係數較小時，可以簡化大量之計算達並到一定的精確度。

This research use Laplace transform method analysis the system. With the damping coefficient changing, we can find the properties of vibration for the simple model of a single degree of freedom which with fractional derivative damping of order 1/2. According to the fractional derivative definition of Caputo and use Laplace transform method as a tool to analysis .In the process of counting , we find its key equation ,and assume that as characteristic equation , then we can discover the roots of the characteristic equation will change with the damping coefficient ,and vibration of all system will be impacted. If we departed the solution of system , some of that will oscillate, and the other will decay . Then compared Laplace transform method with the numerical method ,to prove whether that solution is correct or not. Finally ,we propose the equivalent model for small damping to simplify the huge counting procedure ,and keep the accurateness of the solution of system.

1.Miller K.S., Ross B. An introduction to the fractional calculus and fractional differential equations (1993).
2.Rudolf Scherer.2011.The Grunwald-Letnikov method for Fractional differential equations. Computers and Mathematics with Applications 62:902-917
3.Shy-Der Lin ,Chia-Hung Lu "Laplace transform for solving some families of fractional differential equations and its applications." Advances in Difference Equations (2013).
4.Sakakibara Poperties of Vibration with Fractional Dervative Damping of Order 1/2 ,Int J of the Japan Society of Mechanical Engineers: Series C, 1997, 40 (3):393-399.
5. Murray R.Spiegel , Jhon Liu. "Mathematical Handbook of Formulas and Tables."
6.陳鈺承，"分數階微分系統之動力數值分析與討論"，碩士論文，國立台灣科技大學營建工程所(102)