Parametric Study of Periodic Solutions of a Long Rigid Rotor Equipped with Two Autobalancers

由於對工作效率的要求日增，各類機具的轉速也隨之提昇。為了避免運轉時產生過大的偏心振動，高速旋轉機械需經正確的動態平衡校正。當轉子的偏心量隨工作情況改變時，可因應調整以減少旋轉振動的自動平衡裝置是極有助益的。滾珠型自動平衡裝置已證實可有效的抑制單平面上的偏心振動，並應用於光碟機等設備的減振。除了單平面制振外，分別配置在剛性長轉子兩端面的滾珠平衡裝置也可以抑制長轉子的偏心振動。在適當條件下，兩端面上的滾珠會定位於特定位置，因而抵消轉子的固有偏心量。這個特定的位置稱為完全平衡位置。然而，除了讓系統進入沒有偏心振動的完全平衡位置外，自動平衡裝置也可能引發滾珠相對於轉子振盪或持續旋轉的週期性運動，同時伴隨有劇烈的振動。為了確保系統能夠進入完全平衡位置，必需深入了解週期性運動的特性。然而這一方面的研究較為缺乏。本論文即探討配備滾珠自動平衡裝置的長轉子系統的週期解的特性。首先建立系統的理論模型，利用Lagrange方程式推導系統的統御方程式。採用新近開發的修正漸近諧和平衡法（modified incremental harmonic balance method，MIHB法）求取週期解，並依兩端面上滾珠相對於轉子的運動型態加以分類；週期解的穩定性則由Floquet理論決定。接著在參數平面上搜尋各類週期解的穩定區域，並和完全平衡解的穩定區域相比較；探討系統的穩態行為隨歷程變化的情形。最後架設實驗裝置，實際量測系統的動態特性，和數值分析的結果相比對。Due to the growing demand of high efficiency, the operational speeds of various kinds of machines have been increased significantly. To avoid large imbnalance vibrations, high speed rotational machines have to be balanced precisely in advance. When the imbalance varies with the working conditions, it is desirable to have a balancer that can suppress rotational vibrations automatically. Ball-type automatic balancers can effectively reduce in-plane imbalance vibrations and have been widely used in commercial equipments, e.g., optical-disk drives. Moreover, two sets of auto-balancers at two terminal planes of a long rigid rotor can also effectively suppress rotational vibrations the rotor. Under proper conditions, the balls at each terminal plane will move to specific positions so that the rotor is counter-balanced perfectly. This particular equilibrium position is called the perfect balancing position. However, instead of leading to the perfect balancing position, the auto-balancers may also induce various types of periodic motions where the balls oscillate in or circulate around the orbit and result in large vibrations. To ensure the system will approach the perfect balancing position in the steady state, a clear understanding of the properties of the periodic solutions is essential. However, little research has been conducted on this topic. This thesis aims to study the properties of periodic solutions of the long rigid rotor and autobalancer system. A theoretic model for the system is constructed first. The governing equations of the system are derived from Lagrange’s equations. Periodic solutions are determined using the newly developed modified incremental harmonic balance method and classified according to the motions of the balls relative to the rotor. The stability of the periodic solution is determined by the Floquet theory. Stable regions of different types of periodic solutions in parameter planes are identified and compared with those of the perfect balancing position. The relation between the steady state behavior and the history of the system is examined. We also set up experimental apparatus to measure the dynamical characteristics of the system and compare the results with those of the numerical analysis