相交軸螺旋錐形齒輪對移位嚙合設計與受載齒面接觸分析

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邱鈺婷(Yu-Ting Chiu) 查詢紙本館藏

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機械工程學系

論文名稱

相交軸螺旋錐形齒輪對移位嚙合設計與受載齒面接觸分析
(Profile-Shifted Transmission Design and Loaded Tooth Contact Analysis of Intersecting Helical Conical Gear Pairs)

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摘要(中)

漸開線螺旋錐形齒輪為漸開線圓柱齒輪之特殊型式，不僅加工容易，亦可以與各種漸開線圓柱齒輪形成多種可能空間齒輪對組合。由於錐形齒輪對組裝誤差敏感度低，而且可由調整錐形齒輪軸向位置來控制背隙，在漸開線圓柱齒輪中多應用在小軸交角場合。然而相交軸螺旋錐形齒輪對由於為點接觸型態，使得接觸應力過高而無法提高此類傳動機構之齒面承載能力。
本研究先從齒輪幾何設計著手，運用「移位嚙合設計」概念以控制接觸點位置在齒面寬中間，使接觸位置之曲率半徑加大，得以降低齒面接觸應力。另一方面為了使齒輪齒面數學模型符合實際加工狀況，本論文納入創成式蝸桿砂輪加工法，以直線邊做為蝸桿砂輪修整輪輪廓，推導出螺旋錐形齒輪齒面方程式。經與錐形齒輪理論漸開線齒面比較，發現兩者齒廓之偏差量極小。
由於齒輪對之齒面接觸情形會影響到傳動效能與接觸應力狀況，因此在本研究中分別建立出齒輪對無負載與受載下之齒面接觸分析模型。螺旋錐形齒輪對在無負載下的齒面接觸分析，係以兩齒輪齒面軸線以及接觸法線在具誤差下的空間關係式為基礎，藉由漸開線幾何特性發展出齒面接觸點幾何關係式，以簡化齒面接觸點位置的求解。在分析中比較標準設計和移位嚙合設計下的齒輪對在偏位、軸交角與軸向等組裝誤差以及偏心誤差狀況下，接觸點軌跡與傳動誤差變化；其中亦納入蝸桿砂輪加工錐形齒輪之分析。由分析結果可以看出移位嚙合設計下的接觸點軌跡確實較標準設計下的齒輪對偏往大端且落在齒面寬中間。而各種組裝誤差對接觸點軌跡的影響程度，以軸交角誤差的影響最大，軸向誤差最小。而在各種組裝誤差下，無論何種設計下皆無傳動誤差；僅有偏心誤差會產生正弦變化曲線型式的傳動誤差。而以直線邊修整輪為基礎所建立的蝸桿砂輪加工錐形齒輪對，分析得到的接觸點軌跡與理論漸開線錐形齒輪對的差異極小
本研究之受載齒面接觸分析模型係以影響係數法為基礎，並納入齒面赫茲接觸、輪齒撓曲、軸彎曲以及軸扭轉等各種變形影響，可求得嚙合齒面之接觸斑與應力分佈，以及變形位移。分析結果顯示，在以錐形齒輪為基準，右旋螺旋錐形齒輪對在右齒腹側接觸與左旋錐形齒輪對在左齒腹側接觸的應力分析結果是相同；另一旋向亦有類比關係。而右旋螺旋錐形齒輪對在左齒腹嚙合時，容易產生具邊緣應力集中之接觸型態，反之在右齒腹嚙合狀況下，齒面負載情形較佳。而以右旋--右齒腹之組合來比較標準設計與移位嚙合設計齒輪對接觸狀況，可以見到移位嚙合設計下的齒輪對由於接觸斑偏向大端，最大應力值較標準設計下齒輪對為低，同時受載傳動誤差高低變化值也較小。而在組裝誤差影響下，若接觸斑偏往大端時，應力值會下降，反之亦然。
經由分析結果可驗證本研究針對相交軸螺旋錐形齒輪對所建立之移位嚙合設計的方法、齒輪對的嚙合分析模型以及齒面接觸應力計算模型，確實可做為螺旋錐形齒輪對移位嚙合設計與接觸分析之工具，以提高傳動機構之齒面承載能力。

摘要(英)

Helical conical involute gear as a special type of cylindrical involute gear is not only easily manufactured, but also is able to apply for various spatial gear pair with any cylindrical involute gear. Conical involute gears are often used in the application of the drives with a small shaft angle because of low sensitivity to assembly error, and the controllable backlash by adjusting the axial position of the conical gear. However, the intersecting helical conical gear has high contact stress due to point contact. Consequently, the surface durability cannot be increased for heavy power transmission.
The concept of profile-shifted transmission is at first applied for geometrical design, in order to locate the contact point on the middle of tooth width. The contact stress on flanks can be therefore reduced due to increasing the radius of curvature on contact position. On the other hand, a mathematical model for the tooth surface of the conical gears is established considering the actual manufacturing condition. The tooth surfaces of the helical conical gear are generated by using the grinding worm, which is formed by a dressing conical disc with straight line. Comparing the calculated tooth surface from this model with the theoretical involute flank, the profile error is very small.
Because the tooth contact condition of gear pairs influences the performance of the transmission and the contact stress, tooth contact analysis (TCA) model under unloaded condition and loaded tooth contact analysis (LTCA) model are established in the paper one after another. Based on the properties of involute gearing, the spatial relations with two axes of the gear tooth surfaces and the common contact normal line are established in order to simplify the calculation. The locus of contact points and the transmission error of the gear pair which are influenced by the error of the offset, shaft angle, axial mounting and eccentric errors are compared for design with standard and profile-shifted transmission respectively. From the results, the locus of contact points of the gear pair with profile-shifted transmission is close to the heel of the conical gear and indeed as expected locates on the middle of the tooth width. Among the various errors, the shaft angle error has the largest influence on the performances, and the axial mounting error has the less influence. The eccentric error causes the transmission error to perform as a sinus curve. The locus of contact points of the gear pair manufactured by continuous gear grinding has little difference from that of the gear pair with theoretical involute surface.
LTCA model in the study is based on the influence coefficient method with considering the deformation of Hertzian contact, tooth bending deflection, shaft bending and shaft torsion. The corresponding contact patterns, the contact stress distribution and the angular displacement due to deformation of the engaged teeth can be simulated. The result of LTCA for the working flank on right side of gears with right hand helix angle (with respect to the conical gear) and that for the left flank of gears with left hand helix angle are the same, vice versa. Considering the conical gear with right hand helix angle, the tooth contact on right flank side is better than on left flank side because the concentrated stress on the edge occurs easily on the left flank. The contact stress of the gear drives with profile-shifted transmission is better than with the standard design because the contact patterns is close to the heel and the contact stress is reduced. And the same condition is also valid for the loaded transmission error where the peak-to-peak value with profile-shifted transmission is smaller.
It can be verified from the analysis result that the proposed approach for profile-shifted transmission, tooth contact analysis and load tooth contact analysis in this thesis is indeed an efficient design tool for helical conical gear drives with intersecting axes to increase the tooth surface durability.

關鍵字(中)

★ 螺旋錐形齒輪
★ 漸開線齒輪
★ 蝸桿砂輪加工
★ 移位嚙合設計
★ 組裝誤差
★ 偏心誤差
★ 受載齒面接觸分析
★ 傳動誤差

關鍵字(英)

★ helical conical gear
★ involute gear
★ threaded grinding wheel
★ profile-shifted transmission
★ assembly error
★ eccentric error
★ load tooth contact analysis
★ transmission error

論文目次

摘要 i
Abstract iii
謝誌 vi
目錄 vii
圖目錄 xi
表目錄 xvii
符號說明 xviii
第1章前言 1
1.1 研究背景 1
1.2 文獻回顧 7
1.2.1 基本幾何特性 7
1.2.2 齒面數學模型 7
1.2.3 點接觸情形之改良方法 8
1.2.4 齒面接觸分析 9
1.2.5 齒面受載接觸分析 9
1.3 研究目的 10
1.4 論文架構 11
第2章螺旋錐形齒輪齒面數學模型 13
2.1 漸開線螺旋錐形齒輪之基本幾何特性 13
2.1.1 漸開線齒輪與基準齒條之關係 13
2.1.2 漸開線齒面特點 15
2.1.3 齒面寬的幾何限制 16
2.2 漸開線螺旋錐形齒輪理論齒面方程式 18
2.3 創成式蝸桿砂輪加工之齒輪齒面數學模型 22
2.3.1 機台介紹 23
2.3.2 蝸桿砂輪齒面數學模型 24
2.3.3 機台設定 28
2.3.4 蝸桿砂輪加工之齒輪齒面方程式 32
2.3.5 齒形偏差量 36
第3章螺旋錐形齒輪對移位嚙合設計 39
3.1 基本嚙合關係 39
3.1.1 作用線特性 39
3.1.2 齒輪對誤差下齒面軸線關係 41
3.1.3 接觸點決定 46
3.2 移位嚙合齒輪對齒輪設計參數決定 48
3.2.1 齒面寬預期接觸點位置之規劃 48
3.2.2 作用線空間位置 49
3.2.3 螺旋錐形齒輪基圓相關參數關係建立 52
3.2.4 螺旋錐形齒輪軸向位置 53
3.3 接觸率 53
第4章受載齒面接觸分析模型 56
4.1 基本分析模型 56
4.1.1 齒對接觸應力模型 56
4.1.2 齒面接觸赫茲變形影響係數 58
4.2 輪齒懸臂梁撓曲影響係數 59
4.2.1 彎曲變形撓度 60
4.2.2 剪切變形撓度 61
4.2.3 Lagrange多項式曲線擬合 62
4.2.4 撓曲影響函數定義 62
4.3 軸彎曲變形影響係數 71
4.3.1 奇異函數 71
4.3.2 軸變形對齒面變形之影響 72
4.4 軸扭轉變形影響係數 74
4.4.1 軸扭轉變形定義 74
4.4.2 齒面變形之影響係數 76
4.5 螺旋錐形齒輪對齒面間距 77
4.6 傳動誤差定義 80
第5章螺旋錐形齒輪對分析案例 81
第6章嚙合齒對接觸分析 85
6.1 無誤差 85
6.1.1 不同案例之漸開線螺旋錐形齒輪對 85
6.1.2 蝸桿砂輪加工之螺旋錐形齒輪對 88
6.2 偏位誤差 88
6.3 軸交角誤差 90
6.4 軸向誤差 92
6.5 偏心誤差 94
第7章嚙合齒對齒面受載接觸分析 99
7.1 螺旋旋向與工作齒腹側對受載接觸之影響 99
7.1.1 嚙合過程之齒面接觸斑變化 99
7.1.2 嚙合過程之負載分配率 106
7.1.3 嚙合過程之最大應力變化 108
7.2 誤差對嚙合過程之齒面接觸斑之影響 110
7.2.1 一般齒面寬設計之接觸斑 110
7.2.2 無誤差 113
7.2.3 軸交角誤差 116
7.3 嚙合過程之負載分配率 117
7.3.1 無誤差 117
7.3.2 軸交角誤差 118
7.4 嚙合過程之最大應力變化 120
7.4.1 無誤差 120
7.4.2 軸交角誤差 122
7.5 受載傳動誤差 125
7.5.1 無誤差 125
7.5.2 軸交角誤差 126
第8章結論與展望 128
8.1 結論 128
8.2 未來展望 133
參考文獻 134
附錄 140

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指導教授

蔡錫錚(Shyi-Jeng Tsai)

審核日期

2018-7-20

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