在本論文中，我們實作了一個多關節錘吊式系統，記錄下3個自由度的質點的水平位移，並利用系統識別的原理，以簡單線性的數學模型來表示整個系統。我們使用的系統識別方法為特徵系統實現演算法 (ERA)，此演算法是由Jer-Nan Juang 與Pappa, R. S.在1985所提出的，ERA發展於NASA Langley研究中心，並大量的應用控制的設計、模擬、以及識別。此外，我們有對多關節錘吊式系統做理論分析，透過理論值，我們可以推得系統的理論特徵值、初始值、以及虛擬的阻尼，並由特徵值我們可以得出理論的自然頻率及模態形狀。
由於實驗過程中會有雜訊或是計算上的誤差，透過ERA所識別出來的結果其維度會比理論上大得多。因此，我們需要想辦法將雜訊從我們識別的結果刪除。而我們使用五種指標:TMAC， MPC，MPC2，奇異值指標，以及特徵值指標，我們會在文章裡一一詳細介紹，並將其結果做出討論與分析。最後得出奇異值指標是最佳的方法，利用奇異值指標所識別的結果其自然頻率及模態形狀與理論相比是幾乎相同的，此外奇異值指標的計算量也是所有方法裡最低的。因此，我們可以使用特徵系統實現演算法 (ERA) ，再使用奇異值指標來識別出多關節錘吊式系統。

This thesis discusses the application of the Eigensystem Realization Algorithm (ERA) for modeling of a coupled pendulum system, which is designed for the study of the heliogyro solar sail blade. Eigensystem Realization Algorithm (ERA) was developed by Dr. Jer-Nan Juang for the modal parameter identification and model reduction of NASA large space structures. This thesis also presents a study of using five indicators, Total Modal Amplitude Coherence (TMAC), Modal Phase Collinearity (MPC), MPC2, singular value indicator, and eigenvalue indicator, to generate the reduced-order ERA models. The study is based on the comparison of the reduced-order models with the experimental data and the theoretical model. In this thesis, a technique is applied to generate a theoretical model with the added damping, which can be obtained from the identified model. The identified natural frequencies and mode shapes of the reduced-order ERA model based on various indicators are compared with those of the theoretical model. The investigation demonstrates that the reduced-model based on the singular value indicator gives the best results with the least computational effort.

[1] J.-N. Juang, Pappa, R. S., “An Eigensystem Realization Algorithm for Modal Parameter Identification and Model Reduction”. Journal of Guidance, Control, and Dynamics, Vol. 8, Sept.-Oct. 1985, pp. 620-627.
[2] J.-N. Juang, Pappa, R. S., “Effects of Noise on Modal Parameters Identified by the Eigensystem Realization Algorithm”. Journal of Guidance, Control, and Dynamics, Vol. 9, NO. 3, May.-June. 1986, pp. 620-627.
[3] J.-N. Juang, C.-H Hung, and W. K. Wilkie, “Dynamics of a slender spinning membrane,” in Jer-Nan Juang’s Astrodynamics Symposium, June 2012.
[4] R. H. MacNeal, “The Heliogyro: An Interplanetary flying machine,” tech.rep., Astro Reaearch Corporation, March 1967.
[5] John R. Taylor, Classical Mechanics, University Science Books 2005,
[6] J.-N. Juang, Applied System Identification, Prentice Hall, Inc., Englewood Cliffs, New Jersey 07632, 1994, ISBN 0-13-079211-X.
[7] J.-N. Juang, Phan, M. Q., Identification and Control of Mechanical System, Cambridge University Press, New York, NY 10011-4211, 2001, ISBN 0-521-78355-0.
[8] Pappa, R. S., Elliott, K. B., and Axel Schenk “A Consistent-Mode Indicator for the Eigensystem Realization Algorithm.” Journal of Guidance Control and Dynamics, Vol. 16, No. 5,1993 ,pp. 852-858.
[9] Y. Huang, J.-N. Juang, C.-H Hung, and W. K. Wilkie, “Dynamics of a Coupled Pendulum Model of a Heliogyro Membrane Blade”, 3rd International Symposium on Solar sailing, June 2013.