具齒面修整之行星齒輪組受載齒面接觸分析

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葉湘羭(Siang-Yu Ye) 查詢紙本館藏

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機械工程學系

論文名稱

具齒面修整之行星齒輪組受載齒面接觸分析
(Loaded Tooth Contact Analysis of Planetary Gear Sets with Flank Modification)

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摘要(中)

行星齒輪機構因具功率分流設計，與一般平行軸齒輪對比較，具有較高功率體積密度、輸出輸入軸同軸心與設計緊實等優點，因此廣泛運用於各種傳動中。但實際運轉中，行星齒輪組一方面會因受到加工、組裝誤差與嚙合剛性影響，造成行星齒輪間負載分配不均與嚙合過程負載、應力變化跳動；另一方面，因受組件變形影響，如托架、行星銷軸與太陽齒輪扭轉等變形，皆會造成嚙合齒面負載分佈不均。一般而言，在工業應用上會以齒面修整方式，以改善前述問題，進而提升齒輪機構承載能力與傳動效能。但修整後之齒面在受載下的接觸結果卻必須要能事先預估，始得以掌握齒面修整特性。因此本論文之研究目標即為提出一可分析行星齒輪組之受載齒面接觸分析模型(LTCA)，以模擬、並分析在齒面修整、組件變形、以及存在各種誤差等真實狀況下之嚙合齒面接觸狀況。

在本論文中先以行星齒輪組齒面修整模式：太陽齒輪採螺旋角修整，行星齒輪採齒線隆起修整以及齒形修整形式，建立成形磨加工之三維齒面方程式。為求得行星齒輪組之接觸狀況，本研究並根據齒輪組特性，納入修整齒面與誤差，建立齒面接觸分析模型(TCA)，計算嚙合齒面接觸點參數，以及分析在齒面修整與各種誤差下之傳動誤差、背隙以及齒間間隙變化。納入所發展的分析模型的誤差包括行星齒輪齒厚誤差，行星銷孔位置偏差，托架、太陽齒輪、行星齒輪之偏心以及行星齒輪銷軸偏斜等誤差。

行星齒輪組受載齒面接觸分析模型(LTCA)係以影響係數法為基礎建立而得，可用以分析各種齒面受載下接觸特性。此LTCA分析模型納入組件變形之影響，包括輪齒齒面接觸、彎曲、剪力變形，托架與行星銷軸變形，太陽齒輪軸扭轉變形等。模型亦納入由齒面接觸分析模型(TCA)計算之接觸齒面幾何關係，可模擬各種赫茲、非赫茲接觸狀態，以及分析各項組件之加工、組裝誤差對受載接觸之影響。

為了驗證理論模型之可靠度，本研究以有限元素分析軟體計算所探討之行星齒輪組機構之受載下齒面應力與行星齒輪間負載分配，並與分析模型比較。結果顯示兩者應力變化趨勢相近且最大值差距少於10%，負載分配差距少於1%，證明理論分析模型具有相當的可信度。

本論文並以一實際之行星齒輪組為分析案例。首先以發展之TCA模型分析在有、無誤差以及齒面修整下，行星齒輪組之傳動誤差、背隙變化。再以LTCA模型分析在組件變形以不同誤差影響下，行星齒輪組嚙合過程之應力、負載與受載傳動誤差之變化，以及齒面上接觸斑大小與應力分佈情況。論文中亦包括模擬因變形造成之齒頂邊緣接觸以及因行星銷軸偏斜誤差所造成齒端邊緣接觸等之非赫茲接觸應力狀況，以及在具太陽齒輪浮動設計下，誤差對行星齒輪間負載分配與太陽齒輪中心軌跡。

分析結果顯示環齒輪與行星齒輪嚙合齒對因托架剛性影響，齒面負載分佈不均程度較大。行星齒輪組齒面經雙隆起修整後各嚙合齒對之齒面應力分佈可呈現左右對稱且最大應力落於齒面寬中間，嚙合過程應力與負載皆連續變化。在考慮相同誤差量影響下，托架偏心對行星齒輪負載分配與受載傳動誤差影響最大。

本論文所提出之行星齒輪組分析模型，可提供設計者有效分析嚙合過程齒面受載情況以及行星齒輪組齒面修整參數設計之依據，並可評估行星齒輪組之組件加工、組裝精度對傳動誤差與背隙之影響範圍，以及對行星齒輪間負載分配與受載傳動誤差之影響。

摘要(英)

Compared with the parallel axis gear drive, the planetary gear drive, as a power-split mechanism, has the advantages of high power density, coaxial input and output shaft, as well as compact design. Therefore, it is widely applied in various transmissions. But in actual operation, the time-variant mesh stiffness as well as manufacturing and assembly errors will cause the uneven load sharing among multiple planet gears and the jump in variation of contact stress and shared load of contact tooth pairs of the planetary gear set. On the other hand, the deformation of the components in the planetary gear set will also result in uneven load distribution on the tooth flanks. The effective solution in the practice is to apply flank modification to enhance the load capacity and also to improve transmission performance. But the loaded contact characteristics of the drive have to be simulated in advance, so as to proceed suitable flank modification. The goal of the dissertation is thus to propose a loaded tooth contact analysis model for planetary gear drives, so as to simulate and analyze the contact characteristics of the engaged teeth under the real condition.

The equations of the three-dimensional modified flanks, manufactured by profile grinding, are at first derived in the dissertation. Considering the flank modification and the errors, a tooth contact analysis model (TCA) is developed for calculating the variables of the contact points. The transmission errors, the backlashes and the clearances in the planetary gear sets can be analyzed accordingly. The errors, involved in the TCA model, include the tooth thickness error of the planets, the position errors of the pin-hole, the eccentric errors of the carrier, the sun gear and the planet gears, as well as the angular misalignments of the planet pin.

The proposed loaded tooth contact analysis (LTCA) model for planetary gear sets, is based on the influence coefficient method, and involves the influences of the deformation of the teeth, the carrier, the sun gear and the planet pins. The topological conditions of the contact teeth calculated by the TCA model are also involved in the analysis, so as to simulate the Hertzian and Non-Hertzian tooth contact and to analyze the influences of the various manufacturing/assembly errors on the loaded tooth contact.

In order to verify the reliability of the proposed LTCA model, the analyzed results are compared with the finite element method (FEM). The results include the contact stress on the tooth flank along the face-width and the load sharing among each planet gear with the non-floating and floating sun gear. The comparative results show the same trend of contact stress and the difference with less than 10%. This indicates the model is in a good agreement with FEM.

A planetary gear set used in the practice is analyzed by the proposed TCA and LTCA model. The variation of the transmission errors and the backlashes of the planetary gear set is analyzed by applying the developed TCA approach, considering various cases with and without flank modification and errors. The variation of shared loads, contact stress and loaded transmission errors stress during gear meshing, as well as the contact pattern and stress distribution on each engaged tooth flank are analyzed by using the LTCA model. The non-Hertzian contact stresses of tip corner edge contact due to deformation and the face-end edge contact due to misalignment of the planet pin are also simulated. The influence of the errors on the load sharing among planets and the trajectory of the floating sun gear are also analyzed in the dissertation.

The analysis results show that tip corner contact occurs more likely in the planet-annulus gear pairs than sun-planet gear pairs. The contact stress on the flanks of planet-annulus gear pairs distributes more uneven due to the stiffness of the carrier. The stress distributions of the engaged tooth pairs are even and symmetric after flank modification. On the other hand, the eccentric error of the carrier has a larger influence on the load sharing among planet gear and loaded transmission errors.

The proposed LTCA model for planetary gear sets in the dissertation can simulate effectively and efficiently contact characteristics of engaged teeth with or without loading during gear meshing. The results can provide good evidence for designing flank modification of a planetary gear set. It can serve as the model for evaluating the influence of the manufacturing/assembly accuracy on loaded or unloaded transmission errors, backlash, as well as load sharing among planet gears.

關鍵字(中)

★ 行星齒輪組
★ 加工誤差
★ 組裝誤差
★ 負載分配
★ 負載分佈
★ 均載機構
★ 齒面修整
★ 有限元素分析

關鍵字(英)

★ planetary gear set
★ manufacturing error
★ assembly error
★ load sharing
★ load distribution
★ load balancing mechanism
★ flank modification
★ finite element method

論文目次

摘要 i
Abstract iii
謝誌 vi
目錄 vii
圖目錄 xi
表目錄 xix
符號對照表 xx
第1章緒論 1
1.1 行星齒輪組之設計問題 1
1.1.1 機構設計特點 1
1.1.2 承載能力計算問題 2
1.1.3 行星齒輪組加工、組裝誤差之影響 3
1.1.4 均載機構之應用 4
1.2 研究背景 5
1.3 文獻回顧 7
1.3.1 行星齒輪組齒面之接觸分析TCA 7
1.3.2 行星齒輪組相關負載分析研究 8
1.3.3 受載齒面接觸分析方法 12
1.3.4 齒面修整研究 14
1.4 研究動機與目的 15
1.5 研究方向 17
1.6 論文架構 18
第2章漸開線正齒輪齒面數學模型 20
2.1 漸開線正齒輪齒面方程式 20
2.1.1 正齒輪齒面方程式與法向量 20
2.1.2 環齒輪齒面方程式與法向量 21
2.2 行星齒輪齒面修整齒面方程式 22
2.2.1 齒面修整規劃 23
2.2.2 齒形修整方程式 24
2.2.3 齒線修整方程式 28
第3章行星齒輪組接觸分析模型 32
3.1 偏差與偏心誤差類型 32
3.1.1 偏心誤差 33
3.1.2 行星齒輪軸偏斜誤差 34
3.1.3 行星齒輪之旋轉中心偏心誤差 35
3.2 行星齒輪組之組件座標系 36
3.2.1 太陽齒輪座標系 36
3.2.2 行星齒輪座標系 37
3.3 齒輪對之嚙合分析 39
3.3.1 環齒輪與行星齒輪之接觸分析 39
3.3.2 太陽齒輪與行星齒輪接觸分析 43
3.3.3 嚙合齒對數目計算 46
3.4 行星齒輪組嚙合特性 47
3.4.1 齒隙 47
3.4.2 背隙 48
3.4.3 傳動誤差 50
第4章行星齒輪組受載齒面接觸分析模型 52
4.1 受載齒面接觸分析基本模型 52
4.1.1 單齒對接觸模型 53
4.1.2 多齒對接觸模型 56
4.2 行星齒輪組之負載變形 57
4.2.1 負載下變形之影響層級 57
4.2.2 多齒對耦合接觸模型 59
4.2.3 多齒輪對耦合接觸模型 60
4.2.4 負載計算矩陣方程式 64
4.3 影響係數建立 66
4.3.1 齒面接觸、輪齒彎曲變形、剪切變形 66
4.3.2 太陽齒輪軸扭轉變形 67
4.3.3 托架變形 68
4.3.4 托架銷軸彎曲變形 71
4.4 齒面間距計算 74
4.5 行星齒輪組具太陽齒輪浮動均載機構分析 75
第5章分析案例 78
5.1 分析案例概述 78
5.2 分析之誤差參數設計 80
5.3 分析內容與條件設定 80
5.4 托架剛性分析 82
5.5 分析模型離散網格收斂分析 83
第6章行星齒輪組受載齒面接觸分析模型與有限元素法分析結果比較 86
6.1 有限元素模型 86
6.2 行星齒輪組分析結果與比較 88
第7章行星齒輪組嚙合接觸分析 92
7.1 無誤差下行星齒輪組接觸特性 92
7.1.1 傳動誤差與齒隙關係 92
7.1.2 齒面修整量對傳動誤差之影響 94
7.1.3 齒面修整量對背隙之影響 94
7.2 行星齒輪組具誤差下之傳動誤差 96
7.2.1 太陽齒輪偏心誤差 96
7.2.2 托架偏心誤差 101
7.2.3 行星齒輪偏心誤差 105
7.2.4 行星齒輪銷軸偏斜誤差 108
7.2.5 單一誤差下傳動誤差高低差值比較 110
7.2.6 綜合誤差 111
7.3 行星齒輪組具誤差下之背隙 116
7.3.1 太陽齒輪偏心誤差 116
7.3.2 托架偏心誤差 118
7.3.3 行星齒輪偏心誤差 120
7.3.4 行星齒輪銷軸偏斜誤差 122
7.3.5 綜合誤差 124
第8章行星齒輪組受載齒面接觸分析 126
8.1 無誤差無齒面修整下受載齒面接觸分析 126
8.1.1 嚙合過程負載變化 127
8.1.2 嚙合過程應力變化 128
8.1.3 特定嚙合位置齒面接觸應力分佈 130
8.1.4 嚙合過程傳動誤差 134
8.2 無誤差下齒面修整參數對受載齒面接觸之影響 135
8.2.1 行星齒輪齒線隆起修整 135
8.2.2 太陽齒輪螺旋角修整 137
8.2.3 行星齒輪齒端修整 138
8.2.4 齒線修整後應力分佈 140
8.2.5 行星齒輪齒形修整 141
8.3 齒面修整下之受載齒面接觸分析 147
8.3.1 特定嚙合位置與誤差齒面接觸應力分佈 148
8.3.2 嚙合過程負載變化 151
8.3.3 嚙合過程應力變化 153
8.3.4 嚙合過程傳動誤差 154
8.3.5 行星齒輪間之負載分配 158
8.4 誤差下行星齒輪間之負載分配 159
8.4.1 太陽齒輪偏心誤差 159
8.4.2 托架偏心誤差 161
8.4.3 行星齒輪偏心誤差 162
8.4.4 行星齒輪軸偏差誤差 164
8.4.5 綜合誤差 166
8.4.6 小結 167
8.5 誤差下之受載傳動誤差 168
8.5.1 太陽齒輪偏心誤差 168
8.5.2 托架偏心誤差 170
8.5.3 行星齒輪偏心誤差 171
8.5.4 行星齒輪軸偏差誤差 173
8.5.5 綜合誤差 174
8.5.6 小結 174
8.6 行星齒輪組具太陽齒輪浮動受載齒面接觸分析 175
8.6.1 行星齒輪切向銷孔位置誤差 175
8.6.2 托架偏心誤差 177
8.6.3 行星齒輪偏心誤差 178
第9章結論與未來展望 181
9.1 結論 181
9.2 未來展望 184
參考文獻 185
附錄組件誤差規範 195
作者簡介 198

參考文獻

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49 A. N. Montestruc and P. E. “Influence of planet pin stiffness on load sharing in planetary gear drives,” ASME Journal of Mechanical Design, Vol. 133, pp. 0145011-0145017, 2011.
50 吳思漢，”近似線接觸型態之歪斜軸漸開線錐形齒輪對齒面接觸強度之研究”，國立中央大學機械工程學系博士論文，2009。
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指導教授

蔡錫錚(Shyi-Jeng Tsai)

審核日期

2016-8-23

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