滾珠自動平衡機構應用於二維及三維系統偏心制振之研究

Study of the Application of Ball-type Automatic Balancers to the Vibration Suppression of 2D and 3D Rotors

王明正(Ming-Cheng Wang)

指導教授 ： 盧中仁

國立臺灣大學/工學院/機械工程學研究所/博士(2010年)

https://doi.org/10.6342/NTU.2010.00361

摘要

隨著微精密加工的需求，唯有高速平穩的機具運轉才能提供定位尺寸的精度與細緻的表面粗度，而不平衡的迴轉振動除了將誘發擾人的噪音、降低機械運轉的精確度更甚而導致機具的損壞。因而抑制迴轉機具的不平衡振動，進而提升高速運轉的可靠性，成了眾多研究人員與設計者的目標。
為了能消除轉子在加工製造時無可避免的偏心誤差，傳統偏心轉子的校正方法是在初始裝配使用之前，先透過迴轉量測，再藉由安置平衡配重或是移除局部質量的方式來平衡轉子的固有偏心量。但是，由於長時間運轉所導致的熱變形或材料磨耗等因素，使得轉子質量的分布可能隨時產生變化，因此何時、何處將再次引發偏心振動就變得難以估測，嚴重時則必須再次針對轉子的偏心情況實施量測與校正。因此，若能於轉子上裝配自動平衡機構，則不僅能避免傳統平衡校正的問題，更能隨時因應轉子偏心的變化，即時抑制偏心振動。
滾珠型自動平衡機構是常見的自動平衡裝置之ㄧ，目前已成功地應用於如光碟機（二維）與洗衣機（三維）等迴轉機具。滾珠型自動平衡機構乃由單一或數個軌道及可沿軌道自由移動的數顆滾珠所組成。該機構於過臨界轉速等速迴轉時，在適當的參數條件下，滾珠將移至適當位置以抑制系統的振動。
本論文主要研究滾珠型自動平衡機構應用於二維及三維系統的偏心制振行為與特性。針對不同的系統建構相對應的理論模型，再利用Lagrange’s equations推導各系統的統御方程，同時解出其平衡解並分析各平衡位置的存在區間與條件，接著以Routh-Hurwitz criterion解析平衡解的穩定性，並且探討不同系統參數對穩定區域的影響。首先針對二維雙滾珠自動平衡系統平衡解的穩定性提出解析結果。接著將雙滾珠系統的固定軌道改以徑向彈簧取代，同樣以解析的方式探究此雙滾珠-彈簧系統的穩定性，並比較兩者完全平衡解穩定區域的差異。然後探討剛性偏心長轉子三維制振系統的行為與特性，並於文中提出兩端面裝置雙滾珠自動平衡機構的偏心剛性長轉子系統在靜態不平衡(static unbalance)與力偶不平衡(couple unbalance)狀態下的解析解。最後則架構三維制振系統的實驗設施，同時另考量定子的影響，藉以修正原有的基本理論模型，使理論與實驗相互驗證。於結論中統整全文的結果，並提出滾珠型自動平衡機構應用於二維與三維系統設計的一般性準則。

關鍵字

偏心振動 ； 剛性長轉子 ； 碟片 ； 自動平衡裝置 ； 穩定性

並列摘要

In many industrial applications mass imbalance is the primary cause of vibrations in rotating machines, which generates undesirable noises and compromising the efficiency of machines and eventually causes damages. Therefore, the balancing of rotors is important for the high-speed rotating machines. To eliminate the imbalance vibrations due to the unavoidable manufacture tolerance, rotors are typically balanced by adding or subtracting correction masses in two planes at an initial stage. However, if the unbalance is varied depending on working conditions, such as thermal deformation and material erosion etc, it is hard to predict when and where the imbalance will occur so that the balancing procedure may have to be repeated.
One method to counterbalance the variety imbalances is to equip rotors with ball-type automatic balancers. A ball-type automatic balancer is a simple device characterized by having several balls moving freely on a single or several circular orbits. Under proper working conditions, the balls will move to the other side of the imbalance mass and suppress the imbalance vibrations.
This study aims to investigate the application of ball-type automatic balancers to the vibration suppression of two and three dimensions rotors. A model of the 2D system which consists of an eccentric disk packed with an automatic balancer and suspension system is introduced first. A modified ball-type balancer composed of several ball-rod-spring units will be presented. We also propose a 3D theoretical model, where two automatic balancers are used to limit the vibrations of an eccentric long rigid rotor. In order to investigate the effects of the automatic balancers, the governing equations of the theoretical models for these cases mentioned above are derived using Lagrange’s equations. The closed-form formulas and the existence conditions for the equilibrium positions are presented. Moreover, the stability of each equilibrium position is determined by Routh-Hurwitz criterion. Finally, experiments are conducted to verify the theoretical results of the 3D system. To this end, conclusions and general guidelines are presented.

並列關鍵字

imbalance vibration ； rigid rotor ； disk ； automatic balancer ； stability

參考文獻

[31] 蔡震星與郭立華，大小滾珠式之碟片自動平衡裝置，源興科技股份有限公司，中華民國專利公告號398671，2000。

[32] 蔡震星與郭立華，多極磁石式之碟片自動平衡裝置，源興科技股份有限公司，中華民國專利公告號414357，2000。

[1] Lee, J. and Moorham, W. K. V., 1996, “Analytical and Experimental Analysis of a Self-Compensating Dynamic Balancer in a Rotating Mechanism,” ASME Journal of Dynamic Systems, Measurement and Control, 118, pp. 468-475.

[2] Rajalingham, C., Bhat, R. B., and Rakheja, S., 1998, "Automatic Balancing of Flexible Vertical Rotors Using a Guided Ball," International Journal of Mechanical Science, 40(9), pp. 825-834.

[3] Chao, P.C.P., Sung, C.-K., Huang, C.-L., and Huang, J.-S., 2005, "Precision Repositioning of the Balancing Ball in an Auto-Balancer System Via a Fuzzy Speed Regulator Equipped with a Sliding-Mode Observer," IEEE Transactions on Control Systems Technology, 13(6), pp. 1107-1118.