傘齒輪的製造方式主要分為面銑式切製法以及面滾式切製法，由於面銑式切製法適用於研磨加工，導致使用其切製法生產出來的齒輪精度較高，因此工業界較常選擇以面銑式切製法的加工方式來加工齒輪。在近十年裡，面銑式切製法使用的是五軸加工機來進行切削加工，但由於五軸加工機的加工路徑為較複雜的非線性加工路徑，因此為了避免在切削過程中機台產生碰撞，需要預先透過切削模擬來確認刀具路徑、NC加工碼以及被加工齒輪齒面的正確性。
早期大多數切削模擬軟體是使用體素法(Voxel)配合MC(Marching Cube)演算法的形式來當作切削模擬軟的運算核心，然而MC(Marching Cube)演算法中需要使用到的布林(Boolean)運算會大大的延緩切削模擬軟體的模擬時間。為了改善模擬時間過長的問題，近期許多的論文選擇以射線(Dexel-Ray)的形式來加快切削模擬的模擬時間，而本論文則是提出一套以三維環線取代體素切削的傘齒輪切削模擬，透過三維環線來建構的齒胚，並根據切削位置將三維環線與刀具面數學模式的交點求出，最後透過三角網格鋪面的方式來建構切削的齒輪。此切削模擬的方式適用於面銑式切製法中的成形法加工以及創成法加工，並可以透過此方法來計算成形法加工以及創成法加工時的體積移除率，最後將切削完成的齒面與理論齒面進行齒面誤差比對，以驗證切削模擬方法的正確性。

Face milling and face hobbing are mainly cutting methods for spiral bevel and hypoid gears. Because the face milling method can be used in grinding, the produced gear has higher precision. Therefore, the face milling method is widely used in the industry. In decades, this cutting method has been implemented on a five-axis CNC machine tool. Its cutting path with five nonlinear coordinates is complicated. Therefore the NC codes need to be checked before actual cutting to avoid a collision. Moreover, the produced tooth surfaces are required to be inspected.
Early, the Voxel method and marching cube algorithm were adopted to simulate the bevel gear cutting process. However, it is time-consuming due to Boolean operation. To overcome this problem, recently, a new Dexel-ray method has been introduced in the literature. In this research, a three-dimensional-circle method is proposed to simulate bevel gear face-milling cutting. The gear blank is constructed by three-dimensional circles instead of Dexel rays. The interfering part between the tool and the gear blank is determined according to their cutting positions. And a set of triangles is used to represent the surface of the produced gear. This simulation method can be applied in plunge and generating processes of face milling. Moreover, the material removal rates of plunge and generating processes are calculated based on this method. Finally, the tooth flank deviations are evaluated between the produced and theoretic tooth surfaces. The results confirm the correctness of the proposed method.

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