生活中常見於齒輪的蹤跡甚至是工業中，舉例來說，家中的鐘錶係用於針齒輪，本研究主要探討針齒輪的運動分析及利用數值方法的模擬。
設定針齒輪的接觸條件後，利用偏導數或內積的方式找到該常微分方程，為了可以更加明瞭該常微分方程，舉出一個簡單的例子帶入其中，假定主動輪的初始位置向量與曲率，可以求得接觸較為平滑地轉動的從動輪。在計算常微分方程時，可以先手算出實際值，但當問題較具複雜性時，利用紙筆運算會花費許多時間，為了使計算過程更具有效率及快速，本研究採用Euler數值方法來計算出近似值，經由Visual C++算出每一個節點的數值後，再使用Matlab繪圖出來，模擬出二齒輪接觸的狀況，包含有二齒輪的角速度比關係；從動輪對主動輪的關係；Euler法中所取的間格數越多，則估計值能夠更佳地準確。為使得有更佳的數值解，研究發現所取的間隔數呈倍數關係時，誤差也將呈線性關係，如此一來可以找到更佳的估計值。除了Euler法外，數值方法也有不同的方法，其中Runge-Katta法也可以用來模擬函數的，能估計出更接近真實狀況的數值，找出較佳的數值解以利於針齒輪的轉動。

Gears used in industry. In this paper, we discussed the pin gears motion analysis with numerical methods.
Suppose that the contact conditions of pin gears, we found the ordinary differential equations by partial derivative and inner product. Then we make a easy example to show what we did. Suppose the initial position vector and curvature of the driving wheel. Then there is a driven wheel which smoothly rotated with driving wheel.
When calculating the ordinary differential equations, I calculated the real value in advance. When the problem is more complicated, the calculation will take a lot of time.
We calculated the approximation by Euler’s method. After I calculated the value of all nodes by Visual C++, I drew the nodes by Matlab. I simulated the contact conditions of the gears, including angular velocity ratio and the relation between the driving wheel and the driven wheel. The more space numbers I decided, the better the estimates were. The space numbers were 10 times of the previous one; then the errors were 0.1 times of the previous one. The error was the linear function. Then I could get the better estimate.
Besides the Euler’s method, we also used Runge-Katta method to simulate the ordinary differential equations. In this method, we can estimate the numerical solutions which closed to the real values. The better numerical solutions will facilitate the rotation of the pin gears.

[1] Manfredo P. do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1976.
[2] 袁步浚、林國靖 合著，機動學，第二版，大行出版社，1993
[3] Litvin, F. L., Gear Geometry and Applied Theory, Prentice-Hall Inc., New Jersey, 1994.
[4] C.F.,Gerald, P.O., Applied numerical analysis, Wheatly, 6th edition, 1999.
[5] Richard L. Burden, J. Douglas Faires, Numerical Analysis, eighth edition, Thomson Brooks/Cole, 2005.
[6] 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.
[7] 張充鑫 編著，機動學，第四版，全華圖書股份有限公司，2017.
[8] 陳政逸，齒輪運動分析—使用常微分方程，國立成功大學應用數學所碩士論文，2011.
[9] 陳怡婷，傘齒輪運動分析—使用常微分方程，國立成功大學應用數學所碩士論文，2013.
[10]程廷瑋，諧波齒輪運動分析，國立成功大學應用數學所碩士論文，2016.
[11]林佳慶，諧波齒輪運動分析—使用Runge-kutta法，國立成功大學應用數學所碩士論文，2017.