由於科技發展快速，諧波齒輪已不再是當年昂貴的零組件，得利於此，諧波齒輪又比以往更廣泛的運用在各個科技領域，也因為這個原因，研究諧波齒輪變得更有其價值。
先前研究的常微分方程，是求解剛輪和柔輪的接觸點方程式，用以決定剛輪齒型，其中假設發波器與柔輪齒接觸在正中央。而在本文中是為了研究，當改變發波器必接觸在柔輪齒中心點的這個假設，諧波齒輪的運動會有如何的相對應改變，採用原本的常微分方程，加入分析方法求取精確的接觸點，用此結果調整常微分方程，再以Euler數值方法和second order Runge-Kutta法求數值解，具體的將改動結果表示出來。
經過反覆的驗證，這個改動確實是可靠的。諧波齒輪使用在包含精密科技等各個領域，在工業精度不斷進步的時代，假以時日，這個不算太大的誤差，在更小的尺度下，必定也會產生關鍵性的影響。也給日後要繼續優化諧波齒輪一個方向。

As the rapid development of science and technology, harmonic gear is no longer expensive components. Benefit from this, harmonic gears are more widely used in various fields of science and technology than ever before, and for this reason, the study of harmonic gears becomes more valuable.
The previous study of the ordinary differential equation solved the contact point equation of the circular spline and the flex spline, used to determine the gear shape of the circular spline, which assumes that the and the flex spine contact in the center.In this paper. We cancel the hypothesis that the wave generator must be contact the center of the flex spline, but to calculate the exact contact point to replace this hypothesis. According to this change, we interest in how does the motion of the harmonic gear change. Follow the original ordinary differential equation, add the analytic method to solve the exact contact points, with this result to adjust the original ordinary differential equation. Moreover, we use Euler method and second order Runge-Kutta method to get numerical solution, which can express the changes more specifically.
After repeated verification, this change is indeed reliable. Harmonic gears are used in various fields, including precision technology, robots, and etc. In this fields, the precision of industrial manufacturing is getting higher and higher. One day, the error which is not too much must have a critical impact in a smaller scale.

[1]Robert L. Norton and謝慶雄.機構學.高立圖書有限公司, 2005.第三版.
[2]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.
[3]司光雄.諧波齒輪傳動.國防工業出版社,北京, 1978.
[4]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.
[5]Harmonic drive llc. http://www.harmonicdrive.net/ .
[6]林琦焜.向量分析.滄海書局, 2012.
[7]Kincaid Cheney.Numerical Analysis:Mathematics of Scientific Computing. Flo-rence, Kentucky, U.S.A.: Brooks/Cole Pub Co, third edition edition, 2002.
[8]洪維恩.C語言教學手冊.旗標, 2016.第四版.