本論文研究目標致力於探討以Duffing-like模型識別遲滯動態之發展性。在機械及材料系統中，例如用於隔震和減震的磁流變阻尼器，鐵磁性材料元件等等，其輸入輸出關係常見遲滯迴圈，但造成遲滯迴圈的因素各不相同。現有文獻中，常見以Bingham和Bouc-Wen模型擬合遲滯曲線。由於這些模型多半由不連續函數、片段連續函數或者不確定函數組成，致使進一步分析系統特性相對困難。
本文以Duffing-like模型擬合遲滯曲線，同時嘗試提出標準識別流程法。瞭解Duffing方程式的基本形式，且與文獻中從物理系統推導出的系統方程式對比，理解Duffing方程式各項參數的物理意義。進一步說明從Duffing模型發展成Duffing-Like模型的過程，同時介紹Duffing-Like模型。根據非線性理論方法分析穩定性及係數，然後從模擬結果中定義Duffing-Like 方程式之係數物理特性，及影響遲滯迴圈的因果變化，最後歸納Duffing-like的各項參數，嘗試提出識別系統中遲滯動態，擬合遲滯迴圈的標準識別流程法。
藉由本文提出的標準識別流程，擬合磁阻感測器、鐵磁性材料和磁流變阻尼器三種系統所形成的遲滯迴圈，將擬合結果與原模型實驗結果的遲滯曲線比較，討論Duffing-like模型之實務應用性與可靠度。

The purpose of this study is to identify the parameters of hysteresis dynamics using the duffing-like model. In the mechanical and material systems, hysteresis curves often behave in the input-output graph, such as magnetorheological(MR) damper and ferromagnetic material components. However, the factors that cause hysteresis in these systems are different. In the literature, Bingham model and Bouc-Wen model are used to identify hysteresis curve, these models have complex mathematical equations including piecewise, nondeterministic, and discontinuous functions that make system analysis difficult.
Therefore, this study uses the Duffing-like model to identify hysteresis curve, and provides the new standard operating procedures of hysteresis system identification. The Duffing-like model was developed based on the Duffing equation, showing a scenario of continuous nonlinear ordinary differential equation. Then, this study applies the Lyapunov theorem to show the stability analysis of Duffing-like model and obtain the operating conditions related to the Duffing-like equation parameters. Next, Duffing-like model is built in Simulink for simulation. The simulation results show the relationship between the parameters of Duffing-like model and hysteresis curve. The series of simulation results provide the information about establishing the procedure to fit the hysteresis curve.
Finally, this study identifies hysteresis curve with the three systems, magnetoresistive sensor, ferromagnetic material, and magnetorheological damper with the provided procedure and using Duffing-like model. Comparing identification results to the experimental results, and discusses the practical applicability and reliability of hysteresis dynamic identification using Duffing-like model.

1. Jiles, D.C. and Atherton, D.L. Theory of ferromagnetic hysteresis. in Proceedings of the Twenty-Ninth Annual Conference on Magnetism and Magnetic Materials, 8-11 Nov. 1983. 1984. USA.
2. Morris, K.A., What is Hysteresis? Applied Mechanics Reviews, 2012. 64(5):
3. Prabakar, R.S., Sujatha, C., and Narayanan, S., Optimal semi-active preview control response of a half car vehicle model with magnetorheological damper. Journal of Sound and Vibration, 2009. 326(3-5): p. 400-420.
4. Jr., B.F.S., Dyke, S.J.,Sain, M.K.,and Carlson, J.D., Phenomenological model for magnetorheological dampers. Journal of Engineering Mechanics, 1997. 123(3): p. 230-238.
5. Pang, L., Kamath, G.M., and Wereley, N.M. Dynamic characterization and analysis of magnetorheological damper behavior. in Smart Structures and Materials 1998: Passive Damping and Isolation, 2-3 March 1998. 1998. USA: SPIE-Int. Soc. Opt. Eng.
6. Dyke, S.J., Spencer, B.F., Jr., Sain, M.K., and Carlson, J.D. Seismic response reduction using magnetorheological dampers. in Proceedings of the 13th World Congress. Vol.L: Systems Engineering and Management, 30 June-5 July 1996. 1997. Oxford, UK: Pergamon.
7. Yang, G., Spencer Jr, B.F., Carlson, J.D., and Sain, M.K., Large-scale MR fluid dampers: Modeling and dynamic performance considerations. Engineering Structures, 2002. 24(3): p. 309-323.
8. Choi, S.B., Lee, S.K., and Park, Y.P., A hysteresis model for the field-dependent damping force of a magnetorheological damper. Journal of Sound and Vibration, 2001. 245(2): p. 375-383.
9. Tu, J.-Y., Lin, P.-Y., and Cheng, T.-Y., Continuous hysteresis model using Duffing-like equation. Nonlinear Dynamics, 2015. 80(1-2): p. 1039-1049.
10. Duffing, G., Erzwungene Schwingungen bei veränderlicher Eigenfrequenz und ihre Technische Bedeutung. 1918.
11. Moon, F.C. and Holmes, P.J., Magnetoelastic strange attractor. Journal of Sound and Vibration, 1979. 65(2): p. 275-296.
12. Albisetti, E., Petti, D., Cantoni, M., Damin, F., Torti, A., Chiari, M., and Bertacco, R., Conditions for efficient on-chip magnetic bead detection via magnetoresistive sensors. Biosensors and Bioelectronics, 2013. 47: p. 213-217.
13. Chang, L.-T., Wang, C.-Y., Tang, J., Nie, T., Jiang, W., Chu, C.-P., Arafin, S., He, L., Afsal, M., Chen, L.-J., and Wang, K.L., Electric-field control of ferromagnetism in Mn-doped ZnO nanowires. Nano Letters, 2014. 14(4): p. 1823-1829.
14. Cheng, T.-Y., Using Duffing Equation to Model Magnetorheological Damper Dynamic. Master thesis, Power Mechanical Engineering, Nationa Tsing Hua University, 2014.