中 文 摘 要
端銑刀是機械切削加工最常用的一種刀具，其幾何形狀深深地影響著切削性能與工件品質，尤其在曲面加工方面，螺旋槽式球端銑刀扮演極重要之角色。然而螺旋槽式球端銑刀之幾何形狀非常複雜，研磨製造與重新磨銳需要專門技術工人研磨與高精度之研磨工具機為之。為了克服這些缺點遂有日本Aoyama等人提出無螺旋槽幾何外型簡單之橢圓刃球端銑刀之技術文獻，本文第三章與第四章則在建立此型刀具的數學模式。此型刀具的特點在於僅需兩道研磨步驟就可研磨出此刀具，而且用鈍的刀刃僅需一個研磨步驟就可以產生新的刀刃。吾人依據國際標準(ISO)定義刀具角度，並運用齊次轉換矩陣建立了此型刀具的刀刃曲線、斜角、隙角及楔角之數學公式，最後使用CNC工具機實際研磨此型刀具，並量測研磨後之刀具，以驗證這些公式。
本文的第五章以微分幾何做為數學工具，建立了非等截面螺旋刀具之幾何數學模式，並以階梯端銑刀作例子推導其母線、刀刃曲線及刀具角度之數學公式。第六章則採用RPY矩陣以使得在空間中的研磨輪可適應各種可能的研磨方位，建構了研磨階梯端銑刀之位姿矩陣，接著第七章建立瑞士Ewag六軸CNC工具磨床研磨階梯端銑刀之功能函數矩陣， 聯立第六章之位姿矩陣與第七章之功能函數矩陣導出NC方程式。

ABSTRACT
End mills are widely used in various machining. Its geometry of cutting edges have critical effects on the milling processes and product qualities. Especially, helical ball-end-mills are in the widest use to create a variety of curved surfaces. However, the geometry of a helix ball-end-mill is complicated. Therefore, its manufacturing or re-sharpening usually needs high-skilled technician and precision tool-grinding machine. To overcome this difficulty, an unique geometry of ball-end-mill without helix flute is introduced by Aoyama, et. al. and modeled theoretically in this paper. It takes only one simple step for grinding or re-sharpening. Its elliptical cutting edges enable the end mill to meet the demanding requirements in terms of easy manufacturing, thereby enabling high-accuracy and high-performance cutting for machine parts. Its rake angle, clearance angle, wedge angle and cutting edge angle are then investigated according to International Standard Organization (ISO) by using homogeneous transformation matrices. To validate the developed algorithms, an elliptic ball-end-mill is designed and produced on an universal tool grinding machine. This design eliminates the needs of precision tool-grinding machine in manufacturing or/and re-sharpening processes, and enables excellent tool accuracy to be attained.

In section IV of the study, a geometric shape of the step ball end mill is established by using differential geometric method; section V covers the normal vector of the face and flank of the cutting curve of the step ball end mill. In section IV, the required position and posture of the wheel for grinding the face and flank of the mill is determined, and since the cutting tools shape discussed in this article is from a unique generatrix, it takes the transformation of RPY matrix to get the required degrees of freedom of the position and posture as well as the matrix of the grinding wheel and the step ball end mill. Based on the matrix of the position and shape of the end mill deduced in section VI, it is concluded that the six-axis CNC machine tool of Ewag Corporation, Switzerland, is used to grind the step ball end mill. The main reference source of mathematical model of such six-axis CNC machine tool is the article uses the results to infer the NC program of mills with different forms of generatrix end. The step ball end mill is taken as an example.

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