滾珠自動平衡裝置在自轉陀螺機構的應用

Abstract

滾珠型自動平衡裝置已成功的應用在碟狀剛性轉子的自動平衡，關於單平面上滾珠型自動平衡裝置的理論與特性也有詳細的研究。然而對於滾珠型自動平衡裝置在消除剛性長轉子偏心振動上的效能，迄今未有詳細的討論。本文的目的即為評估滾珠自動平衡裝置對消除長轉子偏心振動的可行性。為了簡化問題，我們探討自轉陀螺機構的偏心振動：假設剛性長轉子的一端固定，另一端的支撐系統可由等向性的彈簧和黏滯阻尼代表，在支撐端加上滾珠型自動平衡裝置。首先利用Lagrange’s equations推導運動方程式。在小變形的條件下，我們針對完全平衡和部分平衡分別選擇適當的座標系以得到非時變的運動方程式，並證明這些運動方程式為一階近似。接著求出系統的平衡解，探討系統參數對於平衡解的穩定性影響。理論分析結果顯示在適當條件下滾珠自動平衡裝置能有效消除自轉陀螺機構的振動。

Ball-type automatic balancers have been successfully employed to balance rotors of the rigid disk type. The theoretic basis and properties of automatic balancing on a single plane have been investigated in detail. However, little research has been done regarding the effects of ball-type automatic balancers on the balancing of long rigid rotors. The purpose of this thesis is to evaluate the feasibility of using ball-type automatic balancers to eliminate imbalanced vibrations of long rigid rotors. In order to simplify the problem, we study spinning-top type mechanisms. The main part of the mechanism is a long rigid rotor spinning at a constant speed. One end of the spinning axis is fixed and the other end is connected to the ground via a suspension system. The suspension system can be treated as a set of isotropic linear springs and viscous dampers. A ball-type balancer is installed on the suspension end of the rotor. Lagrange’s equations are used to derive the governing equations. When the deformation of the rotor is small, in order to get time-invariant equations, suitable coordinate systems are employed for the perfect balancing and partial balancing situations, respectively. We can show that these sets of governing equations are first order approximations to each other. From the governing equations, we can determine the equilibrium positions. The influence of system parameters on the stability of the equilibrium positions are studied in detail. The results, which are confirmed by numerical analysis, indicate that ball-type automatic balancers can effectively reduce the imbalance vibration of long rigid rotors.