齒輪機構在工業產品中廣泛應用，其設計與製造水平直接影響到工業產品的品質。本研究從兩齒輪之間的接觸條件，到齒輪運動之常微分方程式，再利用數值方法求解來求其運動，並對特定齒形之齒輪加以探討與研究。
假設兩齒輪在空間中作剛體運動，主動齒輪接觸被動齒輪而後旋轉，兩傘齒輪齒形為球面上的兩條平滑曲線，此兩曲線可分別對不同軸旋轉，利用其接觸條件可導出兩齒輪的運動常微分方程式。
利用上述推導的常微分方程式，給定一個已知解的特例，以兩種數值方法（Euler數值方法與四階Runge-Kutta數值方法）寫入程式中求得數值解，再與正確解作比較，分析結果與誤差，結果顯示該運動之常微分方程正確無誤，但數值法中的曲線內插函數需使用更高階的方法。

Gear mechanisms are indispensable components used in the industrial, the design and level of manufacturing directly affect the quality of industrial products. In this paper, we describe the contact conditions of two gears to derive formulas of ordinary differential equations, use numerical methods for observing their motion, and discuss for the specific tooth shape gear.
Suppose that two gears do the rigid body motion in space, the driving gear contact the driven gear and rotate each other, the tooth of two bevel gears could be regarded as the curve of the spherical surface, the two curves could rotate with the different axis, we use the contact condition to derive formulas of ordinary differential equations.
We use ordinary differential equations and give a special case of the known solution and input the program to solve it by Euler method and Runge-Kutta method, then we compare it with the analytic solution and analysis results and errors, the results showed that the movement of ordinary differential equations is correct, but the interpolation functions of the curve need to use more advanced methods.

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