本文主要針對安裝誤差對於轉子系統的影響做一探討，此外並提出一創新之主動式平衡器機構，內容主要可分為三大部分。首先是以Bernoulli-Euler beam theory為理論基礎建立轉子系統之模型。在建立轉子系統模型中，針對不同的支撐條件，探討將軸承視為剛性支撐與線彈性支撐時轉子系統動態上的差異。並以理論發現在軸承剛性假設為線性彈簧時，安裝誤差對於轉子的動態影響為一不隨時間變化之外加的靜變形，也因此間接證明安裝誤差對於轉子動態的影響在於支撐剛性的非線性所引起。以上結果証明安裝誤差對於轉子的動態行為與非線性的支撐剛性有密切的關係以驗證非線性軸承模型建立的必要性。另外亦從理論的推導過程中發現，連續轉子在線彈性支撐下之渦動頻率，除了和轉子工作轉速有關外，亦與沿著轉子本身的位置有關，即代表不同部分的轉子會有不同的渦動頻率。接著是建立斜角滾珠軸承模型並推導其剛性矩陣。利用由De Mul所提出的方法，以Hertz contact theory描述斜角接觸滾珠軸承內部各滾珠的接觸剛性，進而得到滾珠軸承在各方向上的剛性矩陣，作為爾後分析搭載滾珠軸承之轉子系統的輸入。最後綜合整理了目前市面上各種主動式線上平衡機構並進而提出了一利用超音波馬達的特性之主動式線上平衡機構的驅動設計概念。

This thesis mainly discusses the effect of misalignment on the rotor system, and also proposes a generative mechanism of the active balancing balancer. The content can be divided into three parts. In the beginning, we use Bernoulli-Euler beam theory as the foundation of the entire research to model the rotor system. After this, we discuss the difference on the dynamics of the rotor when the bearing supports are considered as the rigid or linear elastic. Besides, through the establishment of the theory, the result shows the effect of misalignment on the dynamic of the rotor is the additional static deflection when the stiffness of the bearing support is considered as linear elastic. This phenomenon also shows the effect of the misalignment on the rotor dynamic is mainly due to the nonlinear bearing stiffness and verifies the importance of the establishment of the nonlinear bearing stiffness model. The another discovery is that the whirling frequency of an continuous rotor on the linear bearing stiffness supports not only have the relation to the rotating speed of the rotor ,but also the different parts of the rotor. It means the whirling frequency of the middle part of the rotor will be different from the end part under the same rotating speed.
The secondary part is the modeling of the angular contact ball bearing and derivation of the stiffness matrix. In this part we use the method proposed by De Mul et al, using the Hertz contact theory to describe the contact phenomenon happening in the angular ball bearing. And find the stiffness matrix in all direction. It can be used as the input when we want to analyze dynamic of the rotor system equipped with the ball bearings as the supports in the future.
In the last part, we introduce many kinds of the online active balancing system on the market, and propose a new concept of the online active balancing system adopt the supersonic motor as the driving mechanism.

1.http://www.mmsonline.com/
2.http://www.gmn.de/
3.Jones, A. B.,”The Mathmatical Theory of Rolling Element Bearings,” Mechanical Design and Systems Handbook, Section 13, McGraw-Hill, 1985, Rothbarth, M. A.,ed.
4.Jones, A. B., “ A General Theory for Elastically Constrained Ball and Radial Roller Bearings under Arbitrary Load and Speed Conditions,” ASME Journal of Basic Engineering, June 1960, pp.309-320
5.Andreason, S., “Load Distribution in a Taper Roller Bearing Arrangement Considering Misalignment,” Journal of Tribology, June 1973, pp.84-93.
6.Liu, J. Y, “Analysis of Tapered Roller Bearings Considering High Speed and Combined Loading,” ASME Journal of Lubrication Technology”, Oct.1976, pp564-574.
7.De Mul, J. M., Vree, J. M., Maas. D.A. “Equilibrium and Associated Load Distribution in Ball and Roller Bearings Loaded in Five Degrees of Freedom While neglecting Friction-Part 1: General Theory and Application to Ball Bearings” ASME Journal of Tribology, Vol.111, Jan.1989, pp.143-148.
8.Bishop, R. E. D., “On the Possibility of Balancing Rotating Flexible Shafts” Journal Mechanical Engineering Science, Vol.24, No.4, 1982, pp.215-220.
9.Van De Vegte, J., “Continuous Automatic Balancing of Rotating Systems”, Journal of Mechanical Engineering Science, Vol.6, No.3, pp.264-269.
10.Gosiewski, Z., “Automatic Balancing of Flexible Rotors, Part I: Theoretical Background” Journal of Sound and Vibration, Vol.100, No,4, pp.551-567
11.Dyer, S. W., Ni, J.,“ Adaptive Influence Control of Single-Plane Active Balancing Systems for Rotating Machinery” Journal of Manufacturing Science and Engineering., Vol. 123, May.2001, pp.291-298
12.Jeffcott, H. H., “The Lateral Vibration of Loaded Shafts in the Neighborhood of a Whirling Speed, Philosophical Magzine,304-314,1919
13.Hertz, H., “Über die Berührung fester, elastischer Körper.” J. für die reine und angewandte Mathematik, 1881, No.92, 156-171.
14.Brandlein, J.,“Ball and Roller Bearings : Theory, Design, and Application”pp.99-108, 1985,John Wiley
15.Harris, Tedric A.,“Rolling Bearing Analysis”John Wiley,1991
16.Wowk, Victor.,“Machinery Vibration: Balancing”McGraw-Hill,1995
17.Neale, Michael.,“Couplings and Shaft Alignment”, 1991, Mechanical Engineering Publications
18.Piotrowski, John,“Shaft alignment handbook”, 1995 ,M. Dekker
19.ISO 1940-1: Mechanical Vibration - Balance Quality Requirements of Rigid Rotors - Determination of Permissible Residual Unbalance.
20.www.irdbalancing.com
21.Ali M. Al-Shurafa.,“ Determination of Balancing Quality Limits”, Saudi Electricity Company- Ghazlan Power Plant
22.Toshiiku Sashida ,Takashi Kenjo.,“An introduction to Ultrasonic Motors”, 1993,Oxford University Press
23.Ueha, S. & Tomikawa, Y. et al ed.,“Ultrasonic Motors: Theory and Applications”, 1993,Oxford University Press
24.Uchino, Kenji .,“Piezoelectric Actuators and Ultrasonic Motors”, 1997, Kluwer Academic Press.
25.Stephen W. Dyer, Zhenhu Zhuang, Jun Ni, Jianjun Shi.,“Auto-Tuning Adaptive Supervisory Control of Single-Plane Active Balancing Systems’’, 2000, Society of Manufacturing Engineers
26.David P.Fleming, J.V.Poplawski.,’’Transient Vibration Prediction for Rotors on Ball Bearing Using Load-Dependent Non-Linear Bearing Stiffness’’, 2002,NASA Glenn Research Center
27.David P.Fleming, J.V.Poplawski., “Unbalance Response Prediction for Rotors on Ball Bearings Using Speed and Load-Dependent Non-Linear Bearing Stiffness’’, 2003,NASA Glenn Research Center
28.Toshio Yamamoto, Yukio Ishida., “ Linear and Nonlinear Rotordynamics :a Modern Treatment With Applications’’, Wiley series in nonlinear science
29. Lee, Y.-S., Lee, C.-W. “ Modelling and vibration analysis of misaligned rotor-ball bearing systems’’, Journal of Sound and Vibration, v 224, n 1, Jul 1, 1999, p 17-32
30.Deng, C.-S., Huang, J.-C., Chin, J.-H. “Effects of support misalignments in deep-hole drill shafts on hole straightness’’, International Journal of Machine Tools and Manufacture, v 41, n 8, June, 2001, p 1165-1188