- 學位論文
利用諧和平衡法分析滾珠自動平衡裝置的旋轉週期解
Pure-Rotary Periodic Motions of Auto-Balancer Systems by Harmonic Balance Method
摘要
在各式機具轉速日益提升的現今，為了避免旋轉時產生的偏心振動，需要精確的動態平衡校正。然而隨著工作情況的不同，轉子的偏心量也可能產生差異，此時可以減少偏心振動的自動平衡裝置是極有益處的。其中滾珠型自動平衡裝置在適當條件下對平面或長轉子系統皆可有效的達到制振效果。在抵銷轉子偏心量時，滾珠會定位於特定位置，稱為完全平衡位置。除了完全平衡位置，滾珠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，並伴隨有劇烈的振動。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。這一方面的研究一般使用修正漸近諧和平衡法（modified incremental harmonic balance method，MIHB法）求取週期解。但MIHB法計算上需要長時間迭代運算，同時在某些條件下會無法收斂，本論文即探討如何利用諧和平衡法(harmonic balance method，HB法)更有效率的求取平面以及長轉子系統旋轉週期解的近似解。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。觀察旋轉週期解的特性以提出適當的假設並得到近似解的形式，接著利用諧和平衡法求取週期解，週期解的穩定性則由Floquet理論判別。將結果與MIHB法比較，了解HB法所得結果準確性與適用範圍。最後架設實驗裝置，驗證HB法所得的結果。
並列摘要
Nowadays the speed of rotary machinery is keeping increasing to meet the demand for higher efficiency. To avoid imbalance vibrations at high rotational speeds, precision dynamic balancing of the rotor is required. However, the imbalance of the rotor may change during the working process. In this case, it is beneficial to have an auto-balance instrument that can counterbalance the varying imbalance. Ball-type automatic balancers have been employed successfully in planar and long rigid rotor system for suppressing the imbalance vibration. When the rotor is completely counterbalanced, the balls will position themselves at specific locations. The corresponding equilibrium configuration of the system is referred to as the perfect balancing position. Instead of the perfect balancing position, the system may settle to a periodic motion and suffers from large vibrations. To ensure the system settles to the perfect balancing position, it is essential to have a full understanding of the periodic motions. The modified incremental harmonic balance (MIHB) method has been successfully applied to the determination of the periodic motions of auto-balance systems. However, a lot of iteration calculations are required for the MIHB method and the iteration process may diverge sometimes. This thesis studies the feasibility of using the harmonic balance (HB) method for an efficient determination of the periodic solution of an auto-balancer system. First, Lagrange’s equations are employed to derive the governing equations for the mathematical model of the auto-balancer system. Then, the MIHB method is used to determine periodic solutions under different sets of parameter values. On the basis of the essential characteristics of the periodic solution, mathematic forms for the approximate solution for the periodic solution are proposed and the HB method is used to determine the unknown coefficients. The stability of the periodic solutions is determined by the Floquet theory. The results of the HB method are compared with those of the MIHB method. Finally, experiments are conducted to verify the results of the HB method.