滾齒法是現今製作外齒輪最常見的加工方式。強力刮齒法為一種新型且更高效的齒輪加工法。但目前缺乏這兩種刀具的數學模式。本文提出一個簡單的方法來設計滾刀及強力刮齒刀，所提出的數學模式具有以下特點:
(1)由齒條與齒輪的嚙合計算外齒輪的數學模式，使用4×4齊次座標轉換矩陣將直線齒型纏繞在圓柱上設計滾齒刀，再以共軛理論求解滾齒刀的切削刃曲面方程式，而由此曲面方程式可知滾刀的設計變數，如此便可簡化刀具設計過程。
(2)由於強力刮齒刀的切削刃並非一簡單的函數，故吾人定義其為一1×4的矩陣，再使用4×4齊次座標轉換矩陣建立強力刮齒刀與被加工齒輪的位姿，以建立強力刮齒刀的數學模式，而後再以共軛理論求解強力刮齒刀的切削刃曲面方程式，由此曲面方程式可知強力刮齒刀的設計變數，如此便可簡化刀具設計過程。
(3)吾人建立一具有六個自由度的4×4齊次座標轉換矩陣，並配合滾齒法及強力刮齒法的加工流程，由齊次座標轉換求解此六變數，可計算出滾齒刀及強力刮齒刀加工時的位姿。

SUMMARY

The hobbing method is the most common way of making external gears today. The powerful skiving method is a new and more efficient gear machining method. But the current lack of mathematical models of these two kinds of tools. This paper presents a simple method to establish the mathematical models of hobbing and power skiving tools, and the proposed mathematical model with the following characteristics.
The first is that the mathematical model of the external gear is calculated by the meshing of the rack and the gear. The 4 × 4 homogeneous coordinate transformation matrix is used to design the hobbing cutter on the cylinder, and then the conjugate theory is used to solve the cutting edge surface equation of the hobbing cutter. Thus the surface equation can be seen hatch design variables, it will help to simplify the tool design process.
The second feature is that, since the cutting edge of the powerful skiving tool is not a simple function, we define it as a 1 × 4 matrix. Then use 4 × 4 homogeneous coordinate transformation matrix to establish the position of the power skiving tool and the gear. So mathematical model of the powerful skiving tool will be established, and then the conjugate theory is used to solve the cutting edge surface equation of the powerful skiving tool. The parameter of the surface equations of power skiving tool design will be helpful to simplify the design process of the tool.
Finally, We have established a 4 × 4 homogeneous coordinate transformation matrix with six degrees of freedoms. With the machining processing of hobbing method and the power skiving method, then we will ues the homogeneous coordinate conversion to solve the six variables. So the positions of hobbing and power skiving method while machining external gears will be found.

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