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研究生:杜忠興
研究生(外文):Toh, Chowshing
論文名稱:繞性磁浮軸承之建模與控制
論文名稱(外文):Modeling and Control of Magnetic Flexible Rotor Bearing System
指導教授:陳世樂陳世樂引用關係
指導教授(外文):Chen, Shyhleh
口試委員:蔡孟勲鄭志鈞鄭銘揚
口試委員(外文):M. S. TsaiC. C. ChengCheng, Mingyang
口試日期:2011-06-29
學位類別:碩士
校院名稱:國立中正大學
系所名稱:機械工程學系暨研究所
學門:工程學門
學類:機械工程學類
論文種類:學術論文
論文出版年:2011
畢業學年度:99
語文別:英文
論文頁數:86
中文關鍵詞:磁浮軸承繞性轉子
外文關鍵詞:Magnetic BearingFlexible Rotor
相關次數:
  • 被引用被引用:1
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本論文主要在探討繞性轉子於全磁浮軸承系統之建模與控制,而所提及的建模方法包括:廣義多項式展開法以及特徵模態法,強健控制器的部份則是利用積分滑動模式。在本文中,廣義多項式展開法被利用來求出一組軸向不均勻轉子的數值解,進而再將此組數值解當成特徵模態法中的基底函數。最後,我們可以將整個系統動態表示式進行降階的動作,而基底函數彼此之間的正交性也顯得相當的重要,它會直接影響降階後的系統動態表示式之準確性,本文是假設在前三個模態彼此之間的正交性存在的情況下。
在控制器設計的過程中,我們先利用回授線性化的方式,將原本非線性的三極全磁浮系統動態轉變成線性系統,但是此線性系統表示式只存在於理想狀況。所以,我們可以合理地假設實際的系統表示式是由此象徵性的線性系統再加上一個不確定的系統動態所組成。最後,我們透過積分滑動模式的控制法則來設計控制器,其能允許的不確量之範圍也相當大,且也可有效地降低穩態誤差。
本文中,也針對了不同形式的不確定量進行模擬,其中包括:系統參數的不確定性、隨時間呈梯形變化的轉速以及高階動態的不確定性,從模擬的結果,我們可以看出積分滑動模式控制器的強健性。雖然,系統建模的準確性無法由此看得出來,但特徵模態法也提供了將來做系統鑑別的一種方法,甚至有可能大大地降低即時控制的難易度。

The two ways of developing a model of flexible rotor in AMBs which established in this study are generalized polynomial expansion method (G.P.E.M) and the so-called eigen mode method. In eigen mode method, a set of basic functions for non-uniform rotating rotor are required, and Generalized Polynomial Expansion Method is applied to compute numerical solutions. At last, we able to represent the overall system by a lower order model, and the orthogonality of basic functions are higher priority, it will cause performance of model reduction directly.
In issue of controller design, feedback linearization and Integral Sliding Mode Control (ISMC) are applied to overcome the effect of nonlinearity and uncertainty. The feedback linearization is based on perfect models, which are almost impossible to exist in practice. A more reasonable assumption is that real system model is nominal one plus a bounded uncertain part. In this study, an integral sliding mode controller will be designed to allow for large uncertainties and to achieve good steady-state accuracy.
Simulations are carried out to verify the proposed modeling method and robust controller for magnetic flexible rotor bearing system. The main concerns are the performances in various influences of uncertainties caused by the variation of system parameters, time-varying rotating speed, and unmodeled dynamics.
In addition, some important issues must be emphasized, including time effort of computation for control algorithms, because it is related to difficulty of real time control. It is obvious that the numbers of equations based on the modeling approach of eigen mode method are always smaller than that of finite element method (F.E.M).

CONTENTS I
LIST OF TABLES III
LIST OF FIGURES IV
NOMENCLATURE VI
1. INTRODUCTION 1
1.1. Motivation of Research 2
1.2. Literature Review 3
1.3. Outline of Thesis 5
2. MODELING AND DYNAMIC ANALYSIS OF THE ROTOR SYSTEMS WITH ACTIVE MAGNETIC BEARING 9
2.1. Magnetic Circuit Analysis and Magnetic Force 9
2.2. Total Kinetic Energy, Potential Energy and Work 14
2.3. Generalized Polynomial Expansion Method 16
2.4. Eigen Mode Method 18
2.5. Critical Speed and Campbell Diagram 19
3. NONLINEAR CONTROL OF A CURRENT-CONTROLLED TOTAL 3-POLE ACTIVE MAGNETIC BEARING SYSTEM 29
3.1. State Space Model 29
3.2. Feedback Linearization 30
3.3. Bias Virtual Control Current 33
3.4. Robust Control 35
4. SIMULATION RESULTS AND DISCUSSIONS 40
4.1. Descriptions of magnetic flexible rotor bearing system 40
4.2. Computation of mode shapes by G .P .E .M 42
4.3. Case 1 : Uncertainties for unmodeled higher order dynamics 44
4.4. Case 2 : Uncertainties for variation of system parameters 46
4.5. Case 3 : Uncertainties for various rotating speed 47
5. CONCLUSIONS AND FUTURE WORKS 63
A. Derivation for Kinetic Energy of shaft 66
REFERENCES 71


[1]C.T. Hsu and S.L. Chen, “Optimal Design, Modeling and Control of a 3-Pole Active Magnetic Bearing System,” PhD Thesis of National Chung Cheng University, 2002, pp. 1-50
[2]L.W. Chen and D.M. Ku, “Finite Element Analysis of Natural Whirl Speeds of Rotating Shafts,” Computers & Structures, Vol.40, No. 3, 1991, pp. 741-747
[3]H. D. Nelson and J. M. McVaugh, “The Dynamics of Rotor-Bearing Systems Using Finite Elements,” Transactions of the ASME on Journal of Engineering for Industry, MAY 1976, pp. 593-600
[4]H. D. Nelson, “A Finite Rotating Shaft Element Using Timoshenko Beam Theory,” Transactions of the ASME on Journal of Mechanical Design, October 1980, Vol. 102, pp. 793-803
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[6]C.C. Cheng and J.K. Lin, “Modelling a rotating shaft subjected to a high-speed moving force,” Journal of sound and vibration, 2003, Vol.261, pp.955-965
[7]U.C. Gu and C.C. Cheng, “Vibration analysis of a high-speed spindle under the action of a moving mass,” Journal of Sound and Vibration, 2004, Vol.278, pp. 1131-1146
[8]S.Y. Yoo, C.H. Park, B.C. Park and Myounggyu D. Noh, “Stability Analysis of a Magnetically-Levitated Large Flywheel Using Flexible Rotor Modeling,” IEEE international Conference on Industrial Technology, 2009
[9]S.L Lei and A. Palazzolo, “Control of flexible rotor systems with active magnetic bearings,” Journal of Sound and Vibration, Vol. 314, 2008, pp. 19-38
[10]I. Arredondo, J. Jugo and V. Etxebarria, “Modeling and control of a flexible rotor system with AMB-based sustentation,” ISA Transactions, Vol.47, 2008, pp. 101-112
[11]N. Tanaka, T. Watanabe and K. Seto, “Levitation and Vibration Control of a Flexible Rotor by using Active Magnetic Bearing,” 11th International Symposium on Magnetic Bearings, August, Japan, pp. 26-29
[12]Y. Okada, K. Shimizu and S. Ueno, “Vibration Control of Flexible Rotor by Inclination Control Magnetic Bearings With Axial Self-Bearing Motor,” IEEE/ASME transactions on mechatronics, December 2001, Vol. 6, No. 4, pp. 521-524
[13]S.Y. Yoo, C.H. Park, S.K. Choi, and M. Noh, “Flexible Rotor Modeling for a Large Capacity Flywheel Energy Storage System,” 11th International Symposium on Magnetic Bearings, August 26-29, Nara, Japan, pp. 238-242
[14]H. Choi, G. Buckner and N. Gibson, “Neural Robust Control of a High-Speed Flexible Rotor Supported on Active Magnetic Bearings,” Proceedings of the 2006 American Control Conference, USA, June, pp. 14-16
[15]B. Shafai, S. Beale, P. LaRocca and E. Cusson, “Magnetic Bearing Control Systems and Adaptive Forced Balancing,” IEEE Control Systems, 1994, pp. 4-13
[16]M.J. Jang, C.L. Chen and Y.M. Tsao, “Sliding mode control for active magnetic bearing system with flexible rotor,” Journal of the Franklin Institute, Vol. 342, 2005, pp. 401-419
[17]Nathan S. Gibson and Gregory D. Buckner, “Real-time Adaptive Control of Active Magnetic Bearings Using Linear Parameter Varying Models,” Proceedings of IEEE SoutheastCon 2002, pp. 268-272
[18]Y.C. Chuang and T.N. Shiau, “Dynamic Analysis and Parameter Identification of Rotor System with Nonlinear Active Magnetic Bearing,” Master Thesis of National Chung Cheng University, 2002, pp. 18-30
[19]V. Jayanth, H. Choi and G. Buckner, “Identification and Control of a Flexible Rotor Supported on Active Magnetic Bearings,” Proceedings IEEE SoutheastCon 2002, pp. 273-278

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