三維長轉子暨自動平衡裝置系統的平衡解

Abstract

由於對工作效率的要求日增，各類機械裝置的轉速也隨之提升。然而在高轉速的情形下，即使微小的偏心質量也會造成過大的偏心振動，因此高速旋轉機具的平衡裝置越來越重要。在適當的條件下，滾珠型自動平衡裝置可以有效抑制偏心質量所產生的偏心振動，且已經成功應用於二維平面的轉子系統。然而，對於在三維長轉子偏心制振的研究相對較為缺乏。本論文探討剛性偏心長轉子於兩端面個別裝置滾珠平衡裝置之系統的平衡解，首先建構由滾珠型自動平衡裝置、剛性長轉子、偏心質量、等向性線性彈簧以及黏滯阻尼器所組成系統的理論模型，接著利用Lagrange方程式推導統御方程式。我們發現三維長轉子-自動平衡裝置系統共有16種平衡解。在某些特定條件下，可以得到其中幾種型式平衡解的解析解。至於一般情形下，則需仰賴數值解。我們詳細探討各個平衡解的存在區間及穩定性，展示各個平衡解下系統的振幅隨轉速變化的情形，並以實驗定性檢驗數值解的正確性。

Due to the demand of higher and higher efficiency, the rotational speed of rotary machinery has been increased continually. At high rotational speeds, even a small amount of unbalance may induce significant vibrations. Therefore, precise balancing of high-speed rotors has become more and more important. The ball-type automatic balancer, under proper conditions, can suppress arbitrary amount of unbalance and has been successfully applied to two dimensional planar rotor systems. However, relatively few studies have been conducted on its application to three dimensional long rotor systems. This paper investigated equilibrium configurations of a long rigid rotor equipped with an automatic balancer on each end. The rotor was supported at each end by isotropic linear springs and viscous dampers. First, the mathematical model of the system consisting of the long rigid rotor, auto-balancers and the suspension was constructed. Second, the governing equations were derived by Lagrange's equations and the equilibrium equations were presented. There are totally sixteen different types of equilibrium configurations. Under specific conditions, some equilibrium configurations can be determined analytically. However, in general, equilibrium configurations can only be determined numerically. We investigated the dependence of equilibrium configurations on system parameters comprehensively. For each equilibrium configuration, the variation of the vibration amplitude of the system with the rotational speed was studied in detail. The existence and stable rotational speed regions of the equilibrium configurations were identified. Finally experiments were conducted to verify the numerical results.