齒面接觸分析包含齒面間接觸的路徑、角度傳遞的誤差，以及齒面上的接觸齒印(橢圓)，而在現有的文獻中對於齒面接觸分析（TCA）已經有推導出相當不錯且有系統的數學模式，且其分析方法在齒輪產業中也被廣泛的使用著。然而，當齒輪對一旦發生邊界接觸的情況，就會因大小齒輪齒面邊界點上的法向量沒有重合導致非線性方程組在求解的過程中發散，造成解值時的不穩定；此外文獻中現有的分析方法是根據兩接觸齒面的主曲率大小與方向並利用橢圓形去趨近齒輪對嚙合時的接觸區域，但其接觸橢圓的誤差是會隨著接觸齒面間相對歪斜的程度增加而上升。
因此，本論文提出一種新的齒面接觸分析方法的數學模型。透過最佳化方法中的黃金比例法求解出齒面的接觸點位置，此方法不需使用到大小齒輪齒面法向量共線的條件，因此可以避免掉上述文字中所提及之解值不穩定的情況發生；再者，本論文是以搜索接觸邊界的方式來找出接觸齒印，因此不會因接觸齒面間相對歪斜的程度變化來造成接觸齒印的誤差。
最後，本研究還建立了具齒長與齒形兩方向修形之球形漸開線齒形直傘齒輪、標準漸開線齒形圓柱齒輪，和面銑式螺旋傘齒輪數學模型，並以現有和本論文之兩種方法來評估齒輪對的齒接觸分析，並驗證了本研究所提出之齒面接觸分析方法數學模型的正確性。

The mathematical models for tooth contact analysis (TCA) of a gear pair have been well derived in the literature. TCA includes evaluation of contact path, angle transmission error, and elliptical contact pattern. The existing method is systematic, efficient, and commonly applied in gear industry. However, it has a drawback in unstable solution when the edge contact happens. Normal vectors of tooth surfaces of the pinion and gear at the edge point do not collinear that leads to a divergence in solving nonlinear equations. In addition, an ellipse is adopted to approach the contact pattern according to curvatures of two tooth surfaces. The deviations of contact pattern get larger while their curvatures change greatly.
The paper therefore develops a new mathematical model of TCA. Here, the optimization method of Golden ratio is adopted to directly determine the contact points. The proposed method avoids divergences in solving equations due to avoidance of involving the condition of same normal vectors between the mapped tooth surfaces. Moreover, a searching method for the boundary of contact pattern is applied to find the real contact pattern. The paper also establishes the mathematical models of straight bevel gears, cylindrical gears and spiral bevel gears with double crowning tooth surfaces. Their tooth surfaces are based on a spherical involute, standard involute and face-milled surfaces, respectively. Two methods, the existing and proposed methods, are adopted for evaluating the tooth contact analyses of gear pairs. The results confirm the correctness of the proposed mathematical models.

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