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研究生:王進猶
研究生(外文):Chin-Yu Wang
論文名稱:任意齒形嚙合元件設計與成型磨齒加工的研究
論文名稱(外文):Studies on the Design and Form Grinding of Meshing Elements with Arbitrary Tooth Profile
指導教授:陳朝光陳朝光引用關係
指導教授(外文):Cha'o-Kuang Chen
學位類別:博士
校院名稱:國立成功大學
系所名稱:機械工程學系
學門:工程學門
學類:電資工程學類
論文種類:學術論文
論文出版年:2002
畢業學年度:90
語文別:中文
論文頁數:215
中文關鍵詞:齒面接觸分析圓弧齒輪成型磨齒法傳動誤差誤差補償
外文關鍵詞:Tooth contact analysisCircular-arc helical gearForm grinding methodTransmission errorsError compensation
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本文討論平行軸傳動、不過切的嚙合元件的設計與加工。首先將常用的嚙合元件基本齒形母線,定義在三種不同截面上,並提出以圓的參數式法,表示此基本齒形母線,以此圓的參數式加上嚙合理論,與矩陣轉換理論,即可求得輪齒的齒面方程、接觸線方程、與磨輪的輪廓。齒輪的參數與輪磨參數的改變,均會改變接觸線的型態與輪磨效率,文中舉數個數值範例說明參數調整的重要性。最後並使用工研院機械所OPAL 50齒輪成型磨齒機,加工若干齒輪,再藉齒輪成型磨齒機附屬的量測功能,以驗證設計與加工後齒形的一致性。
輪磨加工參數與齒輪幾何參數的變動均會造成輪磨效率或被磨面精度的變化,這都和輪磨面和新齒面間的接觸線型態有關。實際上,選擇適當的磨輪安裝角與磨輪半徑,將使接觸線沿磨輪衝程方向距離最短,此時磨輪研磨效率最高、齒面精度最佳、接觸線保持連續、且兩側輪磨時受力最均勻等優點,以提高輪磨的加工品質與壽命。而維持齒面的一致性也是最佳化欲求解的重要指標之一,可以藉磨輪安裝角的改變以補償因磨輪的磨損所造成接觸線的改變。為驗證上述理論的正確性,舉一些法、端、軸截面已知的情況,計算其所對應的磨輪輪廓,及最佳輪磨加工參數。
本文並探討雙偏雙圓弧齒輪電腦模擬設計及嚙合分析等相關問題,首先將製造誤差與裝配誤差計入齒輪的嚙合中;其次,引用齒面接觸分析(tooth contact analysis)理論,探討兩嚙合齒面間接觸位置與傳動誤差(transmission errors)受裝配、製造誤差的影響;最後提出藉可調整軸瓦軸承,以補償偏位誤差與偏角誤差,除可將齒印調整回近理想位置外,並可將傳動誤差降低。文中並舉大陸GB-91型雙偏雙階雙圓弧齒輪為例,說明與驗證結果。

This thesis discusses the design and manufacture method for parallel axial meshing and without undercutting conditions. It presents for calculating the profile of a grinding wheel for finishing meshing elements with arbitrary tooth profiles on the three different sections. A general circular parameter method (CPM) is proposed to describe the geometry of a basic profile on the specific section. It can derive tooth surface, contact line and grinding wheel contour easily. Gear parameters and grinding variables that can improve contact line conditions and efficiency are also discussed. Finally, examples are presented in which the profiles of grinding wheels for finishing some gears in different cases are calculated numerically. These gears are then finished by an OPAL gear’s grinding machine. It is clearly shown that the gear profiles after wheel grinding are almost the same as the tooth profiles specified in the design stage.
 Final adjustment and control of the grinding process is very important for best grinding efficiency and gear quality. In order to control the grinding process, a simultaneous contact line between the grinding wheel and gear finishing must be considered. Practically, the length of the contact line along the grinding stroke direction should be minimized by optimum selection of the grinding wheel setting angle, which implies that the grinding stroke be as small as possible. Besides, when the length of the contact line is minimal, the contact line discontinuous phenomena are minimal, also. Those improve grinding efficiency, precise, and balances grinding forces in the wheel entry and exit zones for double flank grinding.
 A mathematical model of a stepped double circular-arc helical tooth profile with two center offsets is developed. The conditions of gear meshing that reflect manufacturing and assembly errors are simulated. The locations of bearing contact and contact path pattern of mating tooth surfaces are determined by Tooth Contact Analysis (TCA). By applying the proposed mathematical model and TCA, single error impact can be determined. To compensate offset and angular misalignment, the authors propose an adjustable bearing whereby transmission errors can be minimized. The investigation is illustrated with several numerical examples.

中文摘要I
英文摘要III
目錄V
表目錄X
圖目錄XI
符號說明XV
第一章 緒論1
1.1母線通用參數式求解的便利性2
1.2本演算法的全面性3
1.3本演算法的必要性3
1.4成型磨齒法的重要性3
1.5章節瀏覽4
1.6研究的範圍與架構10
第二章 文獻回顧與研究動機11
2.1圓弧齒形螺旋齒輪(圓弧齒輪) 12
2.1.1圓弧齒輪的發展12
2.1.2圓弧齒輪的特點13
2.1.3圓弧齒輪相關的研究14
2.1.4研究圓弧齒輪的動機15
2.2. 成型磨齒法16
2.2.1成型磨齒法的相關研究17
2.3其它嚙合元件的文獻19
2.3.1具月牙板設計的內齒輪泵19
2.3.2螺旋壓縮機轉子20
2.4研究動機21
第三章 理論基礎23
3.1 微分幾何23
3.1.1 空間曲線23
3.1.2 空間曲線之切向量、副法向量與主法向量24
3.2 曲面數學26
3.2.1 曲面的數學參數式26
3.2.2 空間切平面與法向量26
3.2.3 齒輪傳動基本定律27
3.2.4 曲面的二類基本式27
3.2.5 曲面上曲線的曲率及短程撓率29 
3.2.6 主曲率、主方向32
3.3 坐標轉換32
3.4確定共軛曲面的方法35
3.4.1相對靜止法37
3.4.2包絡法37
3.4.3運動學法37
3.5 常見的齒輪加工法38
3.5.1 滾齒法或法截面齒廓已知38
3.5.2 刨齒或端截面齒廓已知39
3.5.3 軸向齒廓已知41
第四章任意截面上任意齒廓螺旋齒輪的數學模型42
4.1端截面上的母線齒形已知42
4.2軸截面上的母線齒形已知52
4.3法截面上的母線齒形已知55
4.3.1單圓弧齒輪57
4.3.2基本齒條的建模57
4.3.3 齒面方程式的推導60
4.3.4 兩參數型的齒面方程式63
4.3.5 數值範例68
4.3.6磨輪截形的求法72
4.4漸開線螺旋齒輪75
4.5雙圓弧齒輪80
4.6 結果討論83
4.7本章架構與流程圖85
第五章 輪磨參數的最佳化設計86
5.1最佳化目標函數86
5.1.1設計變量與限制條件86
5.2 接觸線沿銜程方向長度最短為目標數88
5.2.1單圓弧凹凸齒輪對的最佳化問題89
5.2.2漸開線螺旋齒輪的最佳化問題98
5.3 考慮連續接觸線的最佳化問題104
5.3.1 數值範例111
5.4 結果與討論111
5.5最佳化程式流程圖及本章架構流程圖112
第六章 誤差分析與誤差補償116
6.1誤差的定義及來源117
6.2基本齒廓的齒面方程式120
6.3 齒面方程式123
6.4齒面接觸分析(TCA)125
6.5 接觸橢圓130
6.6 傳動誤差134
6.7 誤差分析135
6.8 電腦模擬誤差補償142
6.9 本章結論145
6.10本章架構與流程圖146
第七章 幾何模型與成型研磨148
7.1齒輪與輪磨的幾何建模148
7.2成型磨齒法160
7.2.1成型磨齒法的特色160
7.2.2 成型磨齒法的發展162
7.2.3成型磨輪機的功能162
7.3 成型磨齒加工與量測164
7.4 本章結論164
7.5成型磨齒法加工步驟流程圖165
第八章 結論與未來研究方向167
8.1 綜合結論167
8.2未來研究及展望168
參考文獻180
附錄A 解非線性方程式與方程組192
A.1非線性方程式192
A.1.1 Maller’s法192
A.1.2 Muller’s法的演算法194
A.2 非線性方程組的解法195
A.2.1 牛頓法195
A.2.2 最速下降法197
附錄B 解非線性方程式與方程組的程式199
B.1 以套裝軟體MAPPLE求解199
B.2以FORTRAN中IMSL程式庫,求解非線性方程組204
B.2.1 以Levenberg-Marquardt演算法及有限差分
法求解近似的Jacobian式204
B.2.2 以輸入JACOBIAN的方式,用IMSL求解
非線性方程組206
B.3 以FORTRAN程式求解208
B.3.1 用牛頓法求解208
B.3.2 用最速下降法求解210
附錄C 最佳化問題演算法211

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