齒輪機構一直以來是工業上不可或缺的要件，齒輪運動的準確度會影響到工業產品的品質。本研究從兩直齒齒輪在空間中做剛體運動開始描述，在兩直齒齒輪曲線平滑的條件下，利用其接觸條件推導出兩直齒齒輪的常微分方程式，接著說明兩直齒齒輪剛體運動中球面漸開線的旋轉方法。
在數值方法中，我們利用高階內插法使節點位置向量、切線向量與曲率求得更精確，並且給定一個已知解特例，以Euler與Runge-Kutta兩種數值方法寫入程式求解，以高階內插法分析結果，依誤差情形，判別是否達到良好的齒輪運動。

關鍵字：剛體運動、接觸條件、常微分方程、球面漸開線、高階內插法

SUMMARY
A gear has always been an inevitable component of the industrial sector. The
precision of the transmission gear affects the quality of industrial products. This study first describes two straight bevel gears doing spatial rigid body motion. Under the condition that the tooth of two gears could be regarded as the curve of a spherical surface, the contact condition is used to derive formulas of ordinary differential equations. Then, the study explains the behavior of spherical involute gears in rigid body motion.
In the numerical method, we use the high order interpolating function to derive
a more precise nodal position, tangent, and curvature. A special case of the known solution is also given. We use the Euler method and Runge-Kutta method to solve the equation, and then we analyze the results and errors with high order interpolating function, so as to determine whether the movement of the gears works smoothly.
Key words: Rigid body motion, Contact condition, Ordinary differential equations, Spherical surface, High order interpolating function

[1]Robert L. Norton著，謝慶雄譯，機構學，第三版，高立圖書有限公司，2005.
[2]黃泰翔，齒輪繪製及標示法，中華民國職業訓練研究發展中心，2001.
[3]小原齒車工業株式會社，齒輪技術入門篇，小原齒車工業株式會社，2006.
[4]陳怡婷，傘齒輪運動分析─使用常微分方程法，國立成功大學應用數學所碩士
論文，2012.
[5]陳長城，向量分析導論，滄海書局，2000.
[6]黃子嘉，線性代數及其應用，鼎茂，2000.
[7]石伊蓓，機械系統設計與實務-齒輪應用，臺灣科技大學機械工程學系，2010.
[8]Richard L. Burden, J. Douglas Faires , Numerical Analysis, 7 th edition, CA :
Brooks/Cole, 2001.