本文旨在研究單自由度動力系統具分數階微分阻尼在穩態高斯白噪外力之隨機反應。將兩種不同方法分別為脈衝函數法與蒙地卡羅法一併納入比較，據以分析參數正規化下之位移平方期望值的反應及差異。
研究結果顯示當微分階數上升，系統的整體反應隨之下降且快速進入穩態，此顯示因有效阻尼之提升造成系統的消能能力增加。此外，當微分階數及分數階微分係數同時增加，系統的消能行為會更大幅放大。
本研究針對合理參數組合範圍進行探討，分析結果顯示脈衝函數法與蒙地卡羅法的位移平方期望值高度吻合，因而在線性分析時可互為取代。

This study investigates stochastic responses of single-degree-of-freedom systems with a fractional differential damping subjected to white noise excitations. Both the unit impulse response function method and the Monte Carlo method are used to analyze the response differences. The mean square of the response displacements are studied through the variation of parameters, including the damping ratio, the fractional differential order, and the fractional differential coefficient.
The results show that an increase in the fractional order will result in a response decrease with a faster rate in reaching steady states. This implies an enhancement of the energy dissipation capacity of the system due to a substantial increase in the effective damping. Moreover, when the fractional coefficient and fractional order are increased simultaneously, the response of the system would even decrease more significantly.
It was found that the results based on the impulse function method and Monte Carlo method agree with each other. It’s concluded that both methods can substitute each other when only a linear system is considered.

1. Andrew Gement. 1938. “On fractional differences.” Philosophical Magazine, 25 (1): 92-96.
2. Bagley, R. L. and Torvik, P. J. 1983. “Fractional calculus - A different approach to the analysis of viscoelastically damped structures.” AIAAJ, 21(5): 741-748.
3. Koh, C. G. and Kelly, J. M. 1990. Earthquake Engineering and Structural Dynamics. Application of Fractional Derivatives to Seismic Analysis of Based-Isolated Models, 19, 229-241.
4. Zhang, W. and Shimizu, N. 1998. “Numerical Algorithm for Dynamic Problems Involving Fractional Operators.” JSME International Journal, Series C, 41 (3): 365-370.
5. Li-Li Liu and Jun-Sheng Duan. 2015. “A detailed analysis for the fundamental solution of fractional vibration equation.” Open Mathematics 13.1. doi: 10.1515/2015/0077.
6. Shy-Der Lin and Chia-Hung Lu. 2013. Advances in Difference Equations. Laplace transform for solving some families of fractional differential equations and its applications.
7. Naber, M. 2010. “Linear fractionally damped oscillator.” International Journal of Differential Equations, 12, Article ID 197020. doi: 10.1155/2010/197020.
8. Davies, B. 2002. Integral transforms and their applications. 3rd ed., Springer-Verlag, New York.
9. Yang, C. Y. 1986. Random Vibration of Structures. John Wiley and Sons, New York.
10. 陳鈺承（2013）。分數階微分系統之動力數值分析與探討（碩士論文）。國立台灣科技大學營建工程所。
11. 喬建穎（2014）。分數階微分系統之穩態隨機過程分析與探討（碩士論文）。國立台灣科技大學營建工程所。
12. 王堯民（2015）。分數階微分系統之隨機振動分析與誤差探討（碩士論文）。國立台灣科技大學營建工程所。
13. 游俊傑（2016）。分數階微分系統自由振盪反應解析（碩士論文）。國立台灣科技大學營建工程所。