漸開線齒輪因為其容易加工製作且對裝配誤差不敏感的特點，已經被廣泛地使用了一百多年，但此漸開線僅是指在平面上形成的漸開線，裝配軸必須要互相平行才不會影響其嚙合特性。Philips於2000年對漸開線提出新見解，發現了空間漸開線，定義出在空間中的漸開線形式，可視為廣義的漸開線，且設計出可裝配在歪斜軸上的空間漸開線齒輪。由於Philips對空間漸開線齒輪，僅有幾何形式的定義，並無數學模型與其運動分析可供參考，因此本文嘗試建立空間漸開線齒面之數學模型。
本文一開始先由空間漸開線齒面之幾何構成設置其座標系，並由座標轉換推導出空間漸開線齒面之數學參數式。有了空間漸開線齒面的參數式後，再利用Litvin提出之齒輪嚙合理論，由一已知的空間漸開線齒面合成出與此空間漸開線齒面嚙合之共軛齒面。接著本文對正在嚙合中的齒面，以Litvin的齒面接觸分析方法做其接觸分析，最後並藉由空間漸開線齒面接觸分析的結果來驗證空間漸開線齒輪對之轉速比對裝配誤差不敏感。
本文建立空間漸開線齒面之參數式，並以嚙合理論與齒面接觸分析來探討空間漸開線齒面之合成，以及齒面在嚙合時的接觸分析，補強空間漸開線齒面之數學模型以作為往後對空間漸開線齒輪之研究與應用的基礎。

Planar involute gearing is popular for over a century because involute gears are easy to produce and, most importantly, insensitive to assembly errors. Nevertheless, to have this characteristic, the axes of planar involute gear pairs must be parallel. Philips proposed a new spatial involute gear in 2000, based on the idea of a slip track, which can be thought of as a spatial involute curve. Philips designed general spatial involute gears that can be used in skew-axis transmission applications. However, he defined the spatial involute gears based on geometric constructions only. In this thesis, we seek to build the mathematic models of an involute gear tooth surface and its conjugate surface.

First, we use coordinate transformations to build the parametric equation of a spatial involute gear tooth surface. Second, we use conjugate theories to synthesize its conjugate tooth surface. Finally, we employ TCA technique to analyze spatial involute gearing and to demonstrate that it is insensitive to assembly errors.

This thesis has built mathematical models of conjugate spatial involute gear pairs and conjugate rack-and-pinion systems. This research can serve as a foundation for further investigation of spatial involute gearing.

[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, UK.
[5] Philips, Jack, 2003, General Spatial Involute Gearing, Springer-Verlag, Berlin.
[6] 洪世杰，2006，共軛空間漸開線之研究，國立成功大學機械工程研究所碩士論文