隨著科技的進步，電腦計算能力的發展，有各式各樣的零件和機構可以藉由有限元素法配合電腦強大的計算能力去估算其應力分析，其中也包含著各種類的齒輪，如螺旋傘齒輪。而一般的分析流程不外乎先透過複雜的數學模式求得齒輪的外形，將齒輪外形實體建模，匯入有限元素軟體(如ANSYS和ABAQUS)進行應力分析。而因為螺旋傘齒輪有複雜的曲面結構，在網格建立時往往建立的不盡理想並耗費許多計算時間，而不理想的網格後續計算出的應力結果也會較不準確。有鑑於此，為了縮短分析的時長與得到較佳的分析結果，本論文建立有限元素法(Finite element method)之螺旋傘齒對強度分析數學模式。首先根據螺旋傘齒輪齒面數學模式建立網格，使用齒面接觸分析(Tooth contact analysis)評估一對齒輪之齒面負載分佈情形，依此設定單一齒輪的負載，最後設定限制條件後，完成有限元素法分析應力。除此之外，利用ANSYS分析單個和一對齒輪之應力分佈，以驗證建立數學模式之正確性。為節省ANSYS繁雜的分析步驟與計算時間，採用APDL (ANSYS parametric design language)參數化設計語言進行分析，以Mathematica數學工具軟體自動產生分析流程之APDL腳本檔，再由APDL讀取匯出分析結果以做比較。

With the rapid development of science and technology and computing capabilities, the finite element method (FEM) is a powerful tool to analyze the stresses of many parts and mechanisms, including various types of gears such as spiral bevel gears.
For general FEM analysis of spiral bevel gear, the tooth surfaces of gear first are calculated through the complicated mathematical model, the 3D solid model is then built by CAD design software, for example, SolidWorks, Inventor and UG. Finally, the solid model is imported into a commercial package software (ANSYS or ABAQUS) to determine the stresses of gear using FEM. Commercial FEM software are expensive. The provided meshing tools are for a general-purpose, therefore, the accuracy of meshing is not enough in bevel gear analysis.
The paper therefore develops a new mathematical model of stress analysis of spiral bevel gear using FEM. An automatic meshing program is developed based on the determined tooth surfaces of gear. Unload tooth contact analysis is adopted to evaluate the load distribution of a gear pair. Moreover, for shorten the analysis time, FEM model of a single gear (pinion or gear) is established instead of a gear pair. After giving constraints and loads, the stresses of gear are determined using the finite element programs developed by our lab. The ANSYS analysis results of two cases, a gear pair and a single gear, used to confirm the correctness of the proposed mathematical models. The APDL parametric design language is adopted for reducing the time of preprocessor.

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