摘 要
平面漸開線於齒輪系統中應用廣泛的主要原因為當轉軸之中心距發生誤差時，其轉速比並不會改變。然而將平面漸開線應用於空間歪斜轉軸之螺旋漸開齒輪時卻失去對誤差不敏感之優點，因構成其齒形之曲線依舊為平面漸開線。相對於平面漸開線，Philips提出空間漸開線理論，由空間漸開線形成之空間齒輪具有與平面漸開線正齒輪相同之特性與優點，且此一空間漸開線可退化為平面漸開線。本文旨在建立空間漸開線的數學模型，並探討其共軛曲線。
首先，本文由Philips所建構之空間漸開線幾何模型建立數學參數式，再利用Litvin之座標轉換與嚙合理論方法，合成出空間漸開線的共軛曲線，確認其為空間漸開線，並討論嚙合方程式之兩組解所得之兩組共軛空間漸開線與其嚙合範圍。其次，本文找出共軛空間漸開線之節點與節圓，計算其共軛齒條直線之平移方向與平移量，再以空間漸開線合成出對應之共軛直線參數式。接著，本文討論Philips之參數設計步驟與其原因，並根據其數據合成兩組共軛空間漸開線與其共軛直線，此兩組共軛空間漸開線具有相同轉軸、轉速比與對稱之共軛齒條直線。最後，本文討論共軛空間漸開線退化至平面漸開線的特殊情況。
本文依據litvin之數學理論與Philips所提出之空間漸開線，建立一套產生空間漸開線與合成其共軛曲線之數學方法。本文對空間漸開線數學模型之研究將可作為空間漸開齒輪系統研究之基礎。

Abstract
Planar involute curves are widely used as tooth profiles of gears. The main advantage of using involute tooth profiles is that the constant-velocity ratios can be maintained when center distances between gears axes are slightly changed. However, this property does not exist in helical involute or other spatial gearing systems. In 2000, Philips proposed spatial involute gears, which are formed by spatial involute curves and possess the same error-insensitive property as planar involute gears. The methods used by Philips are geometrical. This thesis aims to construct the mathematical model of a spatial involute curve and its conjugate curve.
First, we establish the parametric equation of the spatial involute curve via Philips’ geometric definition of the curve. Second, we use coordinate transformations and conjugate theories to obtain its conjugate curve, which is shown to be a spatial involute curve too. Third, we locate the position and direction of rack that is conjugate to a spatial involute curve, and we derive the corresponding straight-line profile of the rack. Fourth, we investigate how a set of spatial involute gears was constructed in Philips’ design. Then we use numerical examples to confirm his design. Finally, we present how a spatial involute curve can degenerate to a planar involute curve.
This thesis provides a mathematical foundation for the synthesis of conjugate spatial involute curves. The material presented in this thesis will enable further investigations of spatial involute gears.

參 考 文 獻
[1]Litvin, F. L., 1968, Theory of Gearing, 2nd edition, Nauka, Moscow.
[2]Litvin, F. L., 1994, Gear Geometry and Applied Theory, Prentice-Hall Inc., New Jersey.
[3]Philips, Jack, 1999, ‘‘Some Geometrical Aspects of Skew Polyangular Involute Gearing,” Mechanism and Machine Theory, 34, pp. 781-790.
[4]Philips, Jack, 2000, ‘‘From the Trailed Disc Plough with Ball to General Involute Gearing,” Ball Symposium, July 9-12, Cambridge, U.K.
[5]Philips, Jack, 2003, General Spatial Involute Gearing, Springer-Verlag, Berlin.
[6]Waldron, K. J., and Kinzel, G. L., 2004, Kinematics, Dynamics, and Design of Machinery, John Wiley & Sons Inc., Hoboken.
[7]劉柏村，2006，空間漸開齒面之合成與接觸分析，國立成功大學機械工程研究所碩士論文。