鐘型刀具切製法是美國格里森公司(Gleason)與海勒(Heller)提出用於五軸工具機切削螺旋傘齒輪和戟齒輪的方案，其刀具設計靈活性高，非常適用於大型齒輪的少量生產。相較於傘齒輪專用機，五軸工具機之設備成本低，取得簡單，因此鐘型刀具切製法相當具有優勢。由於商業考量，鐘形刀具切製法數學模式並未公開。
本論文主要目的在建立基於任意傘齒輪齒面之鐘型刀具切製法數學模式。首先以曲面擬合方法趨近齒輪齒面，因此可獲得任意傘齒輪的齒面數學模式；並依照鐘型刀具之幾何外型建立鐘型刀具之數學模式；接著應用逆向運動學，推導混合型和工作台傾斜型兩種五軸工具機之機械設定。另外，提出齒面細化方法，讓被加工齒輪齒面殘留量能滿足粗糙度要求。我們亦開發傘齒輪鐘型刀具切製法之製造軟體，此軟體能根據五軸機械設定，自動規劃並產出西門子840D NC加工程式。在使用VERICUT NC模擬軟體確認刀具路徑無誤後，以工作台傾斜型五軸機進行大齒輪加工實驗，最後量測結果驗證了建立之數學模式的正確性。

The company Gleason and Heller released a new cutting method for spiral bevel gear and hypoid gears in 2013. This cutting method adopts a bell-type milling cutter on a general-purpose five-axis CNC machines. And it is flexible and suitable to manufacture large-size gears in small-scale production. General-purpose five-axis CNC machines are cheaper and easier to be obtained compared to the dedicated bevel gear cutting machines. However, the mathematical model of this method is not revealed under commercial considerations.
This research aims to establish a mathematical model of bell-type cutting method based on the target tooth surface. The tooth surface is first approximated using NURBS surface fitting method. And the mathematical model of bell-type cutter is established. The five-axis coordinates of head-table type and table-table type five-axis machines are then derived through inverse kinematics. In addition, fitted tooth surface are subdivided to reduce cusp during milling process, and let the surface of workpeice meet the roughness requirement. We develop a spiral bevel gear manufacturing software base on the bell-type cutting method. NC codes for Siemens 840D controller can be automatically generated according to the derived five-axis coordinates. After NC data and tool path are checked using VERICUT NC simulation software, a cutting experiment for producing the ring gear on the table-table type five-axis machine is done. The measurement result verifies the correctness of the proposed mathematical models.

[1] Gleason Works, "Calculation Instructions - Generated Spiral Bevel Gears, Duplex–Helical Method, Including Grinding," Rochester, NY, USA, 1971.
[2] Litvin, F. L., and Gutman, Y., (1980), "Methods of Synthesis and Analysis for Hypoid Gear-Drives of 'Formate' and 'Helixform', Part 1, 2 and 3," ASME J. Mech. Des., 80 -C2/DET-31.
[3] 曾韬，螺旋傘齒輪設計與加工，湖南，哈爾濱工業大學出版社，1989。
[4] Fong, Z. H., and Tsay, B. C. B., (1991), "A study on the tooth geometry and cutting machine mechanisms of spiral bevel gears," ASME J. Mech. Des., 113(3), pp. 346-351.
[5] Jerard, R. B., Drysdale, R. L., Hauck, K. E., Schaudt, B., and Magewick, J., (1989), "Methods for detecting errors in numerically controlled machining of sculptured surfaces," IEEE Computer Graphics and Applications, 9(1), pp. 26-39.
[6] Warkentin, A., Ismail, F., and Bedi, S., (1998), "Intersection approach to multi-point machining of sculptured surfaces," Comput. Aided. Geom. D., 15(6), pp. 567-584.
[7] Warkentin, A., Ismail, F., and Bedi, S., (2000), "Multi-point tool positioning strategy for 5-axis machining of sculptured surfaces," Comput. Aided. Geom. D., 17(1), pp. 83-100.
[8] Suh, S. H., Jih, W. S., Hong, H. D., and Chung, D. H., (2001), "Sculptured surface machining of spiral bevel gears with CNC milling," Int. J. Mach. Tool. Manu., 41(6), pp. 833-850.
[9] 徐振炫，五軸刀具路徑規劃應用於齒輪加工之研究，國立中正大學，碩士論文，2004。
[10] Yu, Y., Ding, F., Miao, Z., and Yang, D., (2011), "Application research of NC machining of circular-arc aligned gear," Advanced Materials Research, pp. 264-268.
[11] Han, L., Gao, X.-S., and Li, H., (2012), "Space cutter radius compensation method for free form surface end milling," Int. J. Adv. Manuf. Tech., 67(9-12), pp. 2563-2575.
[12] 林禎祥，以齒輪幾何量測點資料進行虛擬單腹檢技術之研究，國立中正大學，博士論文，2014。
[13] 賴癸澧，五軸工具機之面銑式螺旋傘齒輪製造，國立台灣科技大學，碩士論文，2015。
[14] Gleason Works, "Gleason Heller 5-Axis Machining Center Solutions," Rochester, NY, USA, 2013.
[15] 邱政民，五軸加工機之螺旋傘齒輪鐘形刀具切製法研究，國立台灣科技大學，碩士論文，2014。
[16] Deng, X.-z., Li, G.-g., Wei, B.-y., and Deng, J., (2014), "Face-milling spiral bevel gear tooth surfaces by application of 5-axis CNC machine tool," Int. J. Adv. Manuf. Tech., 71(5-8), pp. 1049-1057.
[17] Rogers, D. F. A., J.A.,, Mathematical Elements for computer Graphics, New York, McGrawHill, 1990.
[18] SIMENS Series SINUMERIK 840D sl/828D 基礎編程手冊。
[19] ANSI/AGMA ISO 17485-A08, Bevel Gears - ISO System of Accuracy, AGMA, 2008.
[20] ISO 4287:1997, Geometrical Product Specifications (GPS) - Surface texture: Profile method - Terms, definitions and surface texture parameters, ISO, 1997.
[21] 楊勝群，VERICUT7.0 中文版 數控加工仿真技術，北京，清華大學出版社，2010。
[22] Yong, W., Dong, J., Wang, T. Y., and Lin, Z., (2012), "NC machining simulation of spiral bevel gear based on VERICUT," Appl. Mech. Mater., pp. 376-380.
[23] Wang, Y., Zhao, L., and Kou, J. B., (2011), "Virtual machining of spiral bevel gear based on VERICUT," Advanced Materials Research, pp. 1885-1889.
[24] Tang, J. Y., Liu, Y. P., and Pu, T. P., (2011), "Simulation for tilt manufacturing of spiral bevel gears based on five-axes CNC machine tool," Journal of Beijing University of Technology, 37(4), pp. 487-493.