本文主要探討銑削製程系統之特徵值與暫態響應，釐清穩定葉瓣圖上各區域之系統響應特徵。首先以平均力模式，針對對稱結構獲得系統特徵方程。進一步建立包含動態力之銑削系統模型，並以辛普森法獲得對應主軸轉速與軸向切深平面的離散特徵值，獲得z平面特徵值與s平面特徵值之映射關係式，進而建立轉速軸深平面之穩定裕度圖及顫振頻率之關係圖。經分析發現顫振頻率隨軸向切深增加而增加。同一軸向切深下，高轉速區之穩定性較低轉速區差，且高轉速區的穩定裕度對軸向切深變化較敏感。於葉瓣交集區的左右兩側其特徵值主要為各自葉瓣之貢獻，交集區相鄰葉瓣貢獻之特徵值都於單位圓外，故會有兩組顫振頻率。另外每一葉瓣內，從低頻區至高頻區系統特徵值實部由小變大再變小，呈現一山丘狀之變化。觀察z平面系統特徵值之運動軌跡，當固定主軸轉速逐漸增加軸向切深時，於穩定葉瓣圖上之半島區時，系統特徵值經過實軸(-1,0)離開單位圓稱為Flip分叉，其它離開單位圓方式則為二階Hopf分叉。最後以時域數值模擬獲得系統暫態響應，從顫振頻率以及振幅驗證本文建立之特徵值分析結果之正確性。

This study investigates the characteristics values and the transient response of the milling system to clarify the relationship between the vibration frequency, stability index (SI), spindle speeds and depth of cuts. With the assumption of axis-symmetric dynamics, zero-order-analysis (ZOA) is applied to obtain the characteristic equation of the milling system. Further, the dynamic milling model considering the dynamics forces is constructed and a discretization method based on Simpson method is proposed to calculate the characteristics values in z-domain. The formula for mapping the characteristic values from z-domain to s-domain is proposed. Stability lobe diagrams of SI are then utilized to discuss the relationship between vibration frequency, depth of cut, and spindle speed. It is shown that the vibration frequency arises as the depth of cut increases. At the same depth of cut, the stability margin at lower spindle speed is better than at higher spindle speed. The stability of the system is much sensitive to the changes of the depth of cut at higher spindle speed. The both sides of the intersection of the adjacent lobes, their own lobes contribute to the dominant characteristic values. However, in the intersection area, both the characteristic values due to the contribution of the adjacent lobes are outside of the unit circle, so there are two vibration frequencies. In addition, in each lobe, the contour of spindle speed, depth of cut, and SI is convex. Two distinct types of instabilities are illustrated by the characteristic values trajectories: (1) flip bifurcation occurs in the flip lobes for which the characteristic values passes through (-1, 0) while leaving the unit circle; and (2) a secondary Hopf bifurcation. Finally, the presented model for prediction characteristic values is verified by the time domain simulation.

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