滾珠平衡系統週期解分析

Abstract

為了降低光碟機的振動，各式的減振機構陸續被提出，其中以滾珠型自動平衡系統較為廣泛使用，也有詳盡的分析。但是目前關於滾珠自動平衡機構的研究大都集中在系統的局部動態特性。為了進行完整的全域分析，有必要探討滾珠平衡系統的週期解。滾珠常見的週期解可分為兩種形式：有限範圍的來回振盪和持續繞圈的旋轉運動。目前並無有效的方法可以求解旋轉週期運動。本文主要貢獻為推廣Incremental Harmonic Balance（IHB）法使其除了振盪週期運動外也可以有效的求解旋轉週期運動。利用這個方法，我們對滾珠平衡系統進行全域的動態分析：搜尋可能的週期解的形式；以格點掃描方式在 平面尋找各種形式週期解的存在區域，判別其穩定性並與平衡解穩定區域作相互對照；在狀態空間求得各吸引子的吸引區域。我們發現在某些參數條件下，完全平衡位置的吸引區域的邊界非常複雜，使得平衡裝置的效能對起始條件非常的敏感。

Many devices have been proposed for the reduction of the rotational vibrations of optical disk drives. Among these vibration-reduction devices, the ball-type automatic balancer is most popular and has been analyzed comprehensively. However, most of the studies on the automatic balancers to date focused on the local dynamical characteristics of the system about the equilibrium positions. In order to have a global analysis of the system, one has to determine the periodic solutions of the system first. The periodic motions of the balls can be classified into two types: oscillatory and rotary periodic motions. Little attention has been paid to the determination of rotary periodic motions of a dynamic system. In this paper, we modify the incremental harmonic balance method such that the rotary periodic motions can be determined efficiently as well as the oscillatory periodic motions. Using this method, we perform global dynamic analysis of the automatic balancer system. We scan the parametric space for possible periodic motions and determine the associated existence regions. The stable regions of the periodic motions are compared with those of the equilibrium positions. Where there exist multiple attractors, the domains of attraction of the attractors in the state space are determined. Under some conditions, the boundary of the basin of attraction of the perfect balancing positions is quite complicated. In this case, the performance of the automatic balancer is sensitive to the initial states of the system.