螺旋傘齒輪與直傘齒輪和正齒輪相比之下有著更佳的傳遞效能與低噪音等優點，因此在工業界上有更廣泛的使用。以切削加工而言，面銑式和面滾式切製法為螺旋傘齒輪主要製造方法，然而這些切製法需要製作專用的刀具與在專用的切齒機上實施，對於生產少量或中等量齒輪的廠商來說這樣的經濟效益不符合成本。然而近年來五軸加工機的興起，許多廠商紛紛尋求使用五軸加工機生產的可能性，美國格里森公司與海勒公司合作提出於五軸加工機上以鐘型刀具製造螺旋傘齒輪的切製方法，但由於商業利益的考量，所以鐘型刀具切製法之螺旋傘齒輪齒面數學模式推導並未公開。
本論文目的是發展應用於五軸加工機之螺旋傘齒輪鐘型刀具切製法，首先以螺旋傘齒輪的齒面數學模式當作標準齒輪，該標準齒輪為格里森公司的SGDH切製法所建立。接著利用假想產形輪製造傘齒輪的概念推導泛用型搖台機之鐘型刀具切製法之螺旋傘齒輪齒面數學模式，最終基於五軸加工機上來推導鐘型刀具切製法之螺旋傘齒輪齒面數學模式，此泛用型搖台機之鐘型刀具切製法之螺旋傘齒輪齒面數學模式包含三個模組，分別為鐘型刀具、假想產形輪，以及工件齒輪與假想產形輪之間的相對運動。本論文所推導出的鐘型刀具切製法之螺旋傘齒輪齒面數學模式確保能接近標準齒輪，因此，將其與標準齒輪進行齒面拓樸誤差分析，以檢驗與標準齒輪之誤差量。為了利用五軸加工機當作製造的機台，本論文推導泛用搖台機至五軸加工機之機械設定，最後利用VERICUT模擬加工NC路徑來驗證推導之數學模式的正確性。

Compared with straight bevel gears, spiral bevel gears have been more widely used in industry due to their better transmission performance and lower noise. Spiral bevel gears are generally produced by cutting methods, such as face-milling and face-hobbing methods. However, the cutting methods need to use special cutting tools and implement on dedicated cutting machines. For the production of small or moderate amount of gear manufacturers that isn’t meet the cost. Considering cost and flexible manufacturing of bevel gears in small and medium batch size, five-axis machine center is another good option compared with the dedicated bevel gear cutting machine. Company Heller in cooperation with company Gleason offered a new solution for bevel gear cutting on its five-axis machine tool. Here, a bell-type milling cutter with indexable inserts replaces the special face-milling cutterhead. However, it did not disclose the details of such application due to commercial considerations.
The main goal of this paper is to establish a mathematical model of spiral bevel gear using a bell-type milling cutter on a five-axis machine center. First, a mathematical model of spiral bevel gear with the Gleason SGDH cutting method is established as a standard gear, and a virtual machine is proposed to provide movements between the cutter and the work gear. The generating gear purely rolls with the work gear that will obtain the tooth surface. This model contains three modules: a cutter, an imaginary generating gear, and the relative motion between an imaginary generating gear and work gear. And then, conversion from the virtual machine enables derivation of the nonlinear coordinates of the CNC five-axis machine. Finally, flank topographic errors are evaluated by compared with the theoretical tooth surface, and the five-axis tool paths is confirmed using VERICUT NC simulation software.

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