On the Transient Contact Behaviors of Automatic Ball Balancers

滾珠型自動平衡裝置具有在高於系統自然振頻的大範圍轉速下平衡振動的能力，因此在不定負荷的旋轉機械中已有廣泛的應用，然而仍一直有性能表現一致性不佳的缺點。本研究以探討雙滾珠自動平衡裝置之暫態行為特性為目標，藉由赫茲接觸理論與有限元素法所建構之接觸參數模型，來建立滾珠與滾珠之間、滾珠與軌道之間的接觸關係；並經由改變滾珠初始位置所得的暫態模擬結果，來計算系統收斂時間及最大振幅之變異情況，以觀察系統性能表現的一致程度。再經由特性參數之敏感度分析，顯示等效定子質量及軌道黏滯液阻尼係數對系統收斂時間的影響較大，系統等效阻尼係數及等效定子質量則對最大振幅有較大的影響，而碰撞次數的最大影響參數則為軌道黏滯液阻尼係數，其結果期望能提供設計工程師在設定系統參數數值及改良時的參考。Being capable of balancing vibration at speeds higher than the natural frequency, automatic ball balancers have been widely used in rotary machinery with uncertain imbalance. However, it suffers from inconsistent performance in real operations. The purpose of this research is to analyze transient contact behaviors of two-ball balancers. Using Hertz contact theory and finite element analysis, contact models are established for collisions between balls and those between balls and the race. Consistency of system performance is investigated by observing the variation of convergent time and the largest vibration amplitude by transient simulation under changing initial conditions of balls. Based on the results, the effects of system parameters on transient behaviors are realized. It is found that mass of the stator and race damping have significant influence on convergent time, whereas system damping and mass of the stator have strong effect on the largest amplitude of system vibration. Changing race damping are influential on the number of collision. The results are expected to facilitate the settings of system parameters design improvement.口試委員會審定書.................................................................................... i 誌謝........................................................................................................... ii 中文摘要.................................................................................................. iv 英文摘要................................................................................................... v 目錄.......................................................................................................... vi 第一章 緒論 1 1.1 研究背景與目的.......................................................................... 1 1.2 文獻回顧...................................................................................... 2 1.2.1 自動平衡裝置簡介............................................................... 2 1.2.2 專利....................................................................................... 4 1.2.3 穩態響應之研究................................................................... 4 1.2.4 暫態響應之研究................................................................... 5 1.3 研究方法與流程……………………………………………….. 7 1.4 論文架構……………………………………………………….. 7 圖片…………………………………………………………………. 8 第二章 系統概述及研究方法 12 2.1 系統概述.................................................................................... 12 2.2 運動方程式之建構法則............................................................ 12 2.2.1 D’Alembert原理............................................................... 13 2.2.2 Lagrange方程式............................................................... 13 2.3 碰撞模型介紹............................................................................ 14 2.3.1 接觸剛性之型式................................................................. 15 2.3.2 接觸阻尼之型式................................................................. 16 2.4 系統穩定性判別法.................................................................... 17 圖片………………………………………………………………... 19 第三章 系統模型之建立與驗證 21 3.1 模型建構.................................................................................... 21 3.1.1 系統運動方程式................................................................. 22 3.1.2 碰撞關係判別式................................................................. 26 3.1.3 接觸阻尼值之估算............................................................. 29 3.2 平衡位置分析............................................................................ 32 3.2.1 平衡位置之推導................................................................. 33 3.2.2 平衡位置之穩定性分析..................................................... 33 3.3 模型之暫態行為驗證................................................................ 35 表格………………………………………………………………... 37 圖片………………………………………………………………... 42 第四章 時域暫態模擬分析 50 4.1 前言…………………………………………………………… 50 4.1.1 分析指標............................................................................. 50 4.1.2 主軸加速度曲線................................................................. 51 4.2 暫態行為之基礎特性分析........................................................ 52 4.2.1 滾珠質量的作用................................................................. 52 4.2.2 滾珠間碰撞關係式的作用................................................. 53 4.2.3 滾珠初始位置之作用......................................................... 54 4.3 敏感度分析................................................................................ 56 4.3.1 參數範圍............................................................................. 56 4.3.2 結果與討論......................................................................... 57 4.4 成果檢討與設計建議................................................................ 65 表格………………………………………………………………... 67 圖片………………………………………………………………... 68 第五章 結論 83 5.1 綜合結論.................................................................................... 83 5.2 未來展望與研究方向................................................................ 84 參考文獻 85 附錄一 89 A1.1 平衡位置之推導過程.............................................................. 89 附錄二 95 A2.1 系統之線性化賈可比矩陣...................................................... 9