隨著3D繪圖軟體以及有限元素軟體發展成熟，齒輪的應力分析可藉由CAD軟體(Solidworks、Pro/E)和有限元素分析軟體(ANSYS、ABAQUS)來達成。直傘齒輪於設計時為了降低誤差敏感度而將齒面修整成圓弧鼓形，以便能夠吸收裝配誤差，在3D繪圖軟體建立的齒形僅是近似於實際齒輪齒形。修整後的齒形需經過繁雜的數學模式才能求得其外形，經由3D軟體建模後轉換檔案格式(.SAT)再匯入有限元素軟體進行應力分析，以評估齒輪強度。有限元素軟體的功能眾多，在齒輪應力分析上只需結構分析的部分，因市售的套裝有限元素軟體為泛用型且成本昂貴，有鑒於此，本論文利用Mathematica數值軟體開發單一個直傘齒輪有限元素程式，並建立齒面負載分佈數學模式，計算單齒和雙齒接觸時的齒面負載分佈情形，以單一個直傘齒輪的齒面負載應力分析模擬一對直傘齒輪的嚙合接觸分析。最後將本論文開發的程式與ANSYS軟體進行數據比對，為節省分析時間，使用APDL參數化設計語言的操作方式將分析資料輸出成APDL腳本檔，再由ANSYS軟體讀取並自動執行分析。

Thank to mature development of the 3D CAD design software and the finite element analysis software, the stress analysis of gears can be simulated by the 3D CAD design software, like SolidWorks as well as Pro/Engineering, and the finite element analysis software, such as ANSYS and ABAQUS. To reduce the assembly sensitivity to the flank deviations and assemble errors, the tooth profile is often modified to the crowning flank during the straight bevel gear design process. However, the tooth surface of straight bevel gear is modeled using the function provided by the 3D CAD design software which is an approximate one. The modified tooth surface can only be obtained by importing from external calculation based on the complicated mathematical models. Therefore, the design gear geometry can be imported into the finite element analysis software to make the stress analysis via the data exchange file such as SAT file.
In the market, the finite element analysis softwares provide numerous analysis modules, so they are expensive. Considering that only the structure analysis is necessary in gear strength analysis, we develop the finite element analysis for gears strength. Therefore, in this thesis, the finite element analysis program for the single straight bevel gear is developed using the numerical computing software, Mathematica. And the mathematical model of the load distribution on the single gear is established. According to the input torque, the load distribution gives the forces exerted on one tooth and double teeth of the single gear to simulate contact of the gear pair. Finally, the results of the program developed by this research are compared with those from the ANSYS software. To reduce the time during the ANSYS analysis, the APDL parametric design language is used. The analysis data are output into the script file of APDL, and then read by ANSYS to analysis automatically.

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