考量修形、變形與誤差影響下之擺線行星齒輪 機構受載接觸特性之研究

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姓名

黃勁儫(Ching-Hao Huang) 查詢紙本館藏

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光機電工程研究所

論文名稱

考量修形、變形與誤差影響下之擺線行星齒輪機構受載接觸特性之研究
(A Study on Loaded Contact Characteristics of Cycloid Planetary Gear Drives Considering the Influence of Profile Modification, Componet Deformations and Errors)

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摘要(中)

行星分流式擺線針輪減速機，即業界俗稱之RV®減速機，為結合漸開線行星齒輪組與擺線針輪行星減速機構的二階式減速機。此設計具有擺線針輪行星齒輪機構高減速比、高嚙合剛性之優勢以及漸開線行星組齒輪功率分流、製造成熟之優點。因此也多應用在高精度與大負載傳動場合。然而這也使得此類型減速機必須考量以下問題：擺線盤與漸開線齒廓之修形模式、擺線齒盤支撐軸承壽命、多接觸對之受載狀況，以及元件加工誤差與組裝誤差對傳動與受載接觸特性的影響。
為分析前述之問題，本論文提出了納入減速機主要負載元件，如漸開線齒輪、擺線針輪以及擺線齒盤支撐軸承等多接觸對的齒面嚙合分析(Tooth Contact Analysis, TCA)模型與受載齒面接觸分析(Loaded Tooth Contact Analysis, LTCA)模型。在TCA模型中，擺線針輪接觸對的嚙合分析在理想擺線輪廓狀況下，係利用瞬心法分析；以移距--等距修整組合產生的修整擺線輪廓，則使用軌跡圓法來進行分析。而漸開線齒輪接觸對則是使用漸開線齒面嚙合關係求解。如此可求得個別減速段受誤差下之傳動誤差與背隙變化，以及由運動關係進一步求出整體機構在誤差下的傳動誤差與背隙曲線。
LTCA模型則是利用影響係數法為基礎建立多接觸對計算模型，以分析減速機中主要受負載元件之相關接觸對的負載、接觸斑與接觸應力分布情形。此LTCA模式係納入TCA模型之接觸對輪廓在具誤差條件下之幾何關係，以及考慮赫茲變形、齒輪齒彎曲變形、軸彎曲變形與軸扭轉變形影響之情況；其中擺線盤支撐軸承則考慮實際圓柱滾子與曲軸、擺線齒盤軸承孔之接觸。同時亦應用Ioannides-Harris軸承壽命模型，根據滾子接觸應力分布來評估各種滾子輪廓之軸承壽命。
在本研究中使用一款市售減速機產品做為案例，探討齒廓修整與元件誤差對接觸特性之影響。齒廓修整係以正移距--正等距、負移距--正等距與負移距--負等距等三種修整組合；誤差則考慮擺線盤偏心誤差、銷輪節圓中心偏心誤差、曲軸相位角誤差、銷位置誤差等時變誤差，以及銷輪節圓徑誤差、銷徑誤差與曲軸偏心誤差等非時變誤差。
嚙合分析結果顯示在相近設計背隙條件下，以正移距--正等距修整擺線齒廓會得到較低的傳動誤差變化量。而在具有元件誤差狀況下，元件誤差對傳動誤差的影響大於修整形式，其中時變誤差中的偏心誤差影響最高，0.01 mm的銷輪偏心誤差會使傳動誤差峰對峰值增加至16.4 arcsec，背隙損失達49.5 arcsec。而正移距—正等距修整輪廓在偏心誤差下會造成接觸位置接近齒底的狀況，使得傳動誤差峰對峰值加劇變化；除此條件之外，元件誤差與輪廓修整並不會對傳動誤差造成影響。
而LTCA的分析分為理想擺線輪廓以及修整輪廓設計下之負載分析兩部分。擺線齒輪機構具理想輪廓之負載分析主要目的係求得在擺線盤理想輪廓以及無誤差狀況下的各種負載特性，包含銷負載與接觸應力變化曲線，軸承負載曲線，擺線盤與曲軸的扭矩曲線、漸開線齒對的負載曲線以及元件受載位移曲線與機構剛性，以做為分析的參考基準。從理論擺線輪廓設計下之負載結果顯示，單一擺線輪廓與銷接觸個數為銷數目的44%，最大接觸應力則發生在擺線輪廓曲率最大位置附近。軸承負載則會隨著輸出轉角有週期變化，在本案例中，最高可達21 kN，最低則僅為2.3kN。而軸承最大、最小負載發生位置會發生在曲軸上的特定位置，即垂直於曲軸軸線與軸承中心連線上，在此兩相差180°的位置會分別受到最大與最小負荷與應力。另外曲軸的軸彎曲變形亦會使得漸開線齒輪對的負載分佈不均，其齒面負載係數KHβ可達1.52。
而在具擺線輪廓修整與機構誤差之負載分析重點，則是探討輪廓與機構誤差對負載變化造成的影響。分析結果顯示，擺線輪廓修整主要影響銷輪接觸對負載，對其他負載特性幾乎沒有影響。正移距--正等距修整擺線齒廓因為具有較多的銷接觸個數，而有較低的銷負載與較佳的機構剛性，與負移距--負等距修整擺線齒廓相比，平均銷接觸數多出56%，負載峰值減少37 %，而且機構剛性高出8.5%。在誤差影響方面，以時變誤差中的偏心誤差影響最大，如在銷輪偏心誤差為0.01 mm @ 0°狀況下，增加了約43.9 %至71.2%的銷負載峰值；而且偏心誤差亦會使擺線盤扭矩分配與曲軸扭矩分配產生變化。例如正移距--正等距修整的擺線盤扭矩傳輸分配曲線，會因銷輪偏心誤差(0.01 mm @ 0°)產生扭矩30.3% 的曲線振幅變動。在曲軸傳輸扭矩方面，銷輪偏心誤差則會使兩個曲軸的平均傳輸扭矩上升12%，另一個曲軸減少19%。而非時變誤差僅對擺線盤接觸對產生較明顯的影響，其他則無。另一方面，漸開線齒對相關誤差僅對擺線盤扭矩分配與曲軸扭矩分配產生較明顯的影響。同時分析結果亦顯示修整輪廓與機構誤差的交互影響很輕微。
在擺線盤支撐軸承負載與壽命評估方面，軸承負載變化並不會因擺線輪廓修整形式而有明顯的不同，其中負移距--負等距修整擺線輪廓雖會使軸承負載略微降低，但僅有0.08 %的差距，可忽略。而滾子輪廓修整方面，本論文比較了無修整、兩種對數曲線修整與端面拋物線修整、以及廠商特定滾柱輪廓的應力分布變化。結果顯示每種修整皆能達到消除邊緣應力集中的效果，而端面拋物線修整會在修整起始處附近產生應力上升的現象，其上升量約為滾子中央應力的4~4.7 %。由滾子修整輪廓所得到的接觸應力分布，以Ioannides-Harris軸承壽命模型計算出軸承壽命顯示，廠商特定滾柱輪廓在無誤差下，壽命可達近12,900小時，比起其他修整輪廓高出70% 至300%不等之壽命。而以對數曲線滾柱輪廓在綜合誤差下所得到分析結果顯示，壽命會從無誤差狀況下7,420小時下降到2,950小時。
綜上所述，本論文所提出的LTCA模型提供了完整的行星分流式擺線針輪減速機分析能力，可以協助業界改善既有設計或是輔助開發新型減速機以提升傳動效能，提升產業競爭力。

摘要(英)

The power-split type cycloid planetary reducer, i.e., so-called RV® reducer, includes the involute planetary gear stage and the cycloid gear stage. This drive has the advantages of high gear ratio and high contact stiffness from the cycloid reducer, and power-split and the mature manufacturing technology from the involute planatary gear set. So it is usually applied in the transmission for high accuracy and heavy loading. However this type of reducers has some problems which must be considered for development: the profile modification of the cycloid disk and the involute gear, the fatigue life of the supporting bearing for cycloid disk, the loaded tooth contact characteristics under the multiple tooth contact pairs and the influence of the component machining error and assembly error.
In order to analyze those mentioned problems, a tooth contact analysis (TCA) model and a loaded tooth contact analysis (LTCA) model for multiple tooth contact pairs are proposed in the dissertation. The important contact pairs of the loaded components in the reducer are considered in the proposed models, such as the involute gears tooth pairs, the cycloid-pin pairs and the supporting bearing roller pairs for cycloid discs. In this dissertaion, TCA model for mesh analysis of cycloid gearing is based on the instant center method for ideal cycloid gear profile, and the trajectory circle method for modified cycloid profile. Flank modification of cycloid profile used in the study is achieved by combination of shifting offset modification and equidistant offset modification. On the other hand, the TCA model for the involute planetary gear stage is based on the geometrical relationship of the involute gearing. The TCA model can therefore provide the variation of the transmission errors and backlashes in presence of errors, either for each individual stage or for the complete reducer using kinematic relation.
LTCA model is a multi-tooth-contact-pair calculation model, which is based on the influence coefficient method to analyze the contact load, contact pattern and contact stress distribution of the loaded contact pairs of components. The proposed LTCA model incorporates the geometrical relations of the contact pairs in presence of erros from the TCA model, as well as the influences of Hertzian contact deformation, the gear tooth bending deformation, the shaft bending and twisting deformation on the contact loading. In this model, the contact pairs in the supporting bearings are consist of cycloid disk-roller pairs and crank-roller pairs to meet the real condition. The Ioannides-Harris model for bearing life calculation is also applied to evaluate the bearing lifes for the different roller profile based on the contact stress distribution of the roller contact pairs.
A reducer from industry is used as the study case to analyze the influence of the profile modification and the component machining errors in the dissertation. Three types of profile modification for cycloid gear are considered, namely the combination of positive shifting offset & positive equidistant offset, negative shifting offset & positive equidistant offset, and negative shifting offset & negative equidistant offset. The errors considered for analysis are divided into the time-variant errors and time-invariant errors. The time-variant errors include the cycloid profile eccentricity error, the pin-wheel eccentricity error, carrier eccentricity error, crank phase angle error, pin position error etc. The time-invariant errors include the pin wheel pitch circle diameter error, pin diameter error and the eccentricity value error of cranks.
The TCA results show that the modified profile with the combination of positive shifting offset & positive equidistant offset has lower peak to peak value of transmission errors under the similar level of backlash. The profile modification has less influence to TCA results than machining errors. The eccentricity error has the strongest influence among the errors. For example, the pin wheel eccentricity error with a value of 0.01 mm @ 0° causes the peak to peak value of transmission error from 0.1 up to 16.4 arcsec, and the drop value of backlash in 49.5 arcsec. The combination of positive shifting offset & positive equidistant offset profile modification enlarges the peak-to-peak value of the transmission errors due to tooth contact near the cycloid tooth bottom. Otherwise, the component errors and the cycloid profile modification have no additional interactive effect.
The LTCA for contact loaded characteristics are divided into two parts: the drive with ideal cycloid profile and with modified cycloid profiles in presence of errors. The analysis results of the ideal cycloid profile provides the loaded characteristics as the reference datum for analysis. The loaded characteristics include the variation of the shared loads and contact stress of pins, the variation of bearing loading, the transmitted torque variation in the cycloid disks and the cranks, shared load variation of the involute gear tooth pairs, the variation of the mechanism stiffness. The LTCA results of the case of the ideal cycloid profile show that the number of contact pairs between the cycloid tooth and the pin is about 44% of the number of pins, and the maximum contact stress occurs near the maximum curvature of the cycloid profile. The bearing loads vary periodically, and the maximum value is about 21kN and minimum about 2.3 kN in the study case. The maximum of minimum bearing load occurs on the specific position of the crank, i.e., perpendicular to the center line of the crankshaft axis and the center of the bearing. The bending and the twisting deformation of the crank shaft cause an uneven contact load distribution on the involute tooth, and the flace load factor KHβ is about 1.52 accordingly.
The loaded tooth contact analysis for the drives with modified cycloid profiles in presence of errors is foucused on the influences of the profile modification and errors under loading. The analysis results show that the types of profile modification affect the contact loads between cycloid tooth and pin larger then the errors. The drive with the modification combination of the positive shifting offset & positive equidistant offset has lower maximum pin load and the better mechanic stiffness due to more contact pin number than other combination. It has about 56% more contact pin number, 37 % lower maximum pin load and 8.5 % more mechanic stiffness than the other combinations. However, the profile modification has nearly no influences on other contact characteristics. The significant influence of the errors on loaded characteristics is the eccentricity errors, similarly to the TCA results. The pinwheel eccentricity error with a value of 0.01 mm @ 0° causes an increase of contact load of pins in 3 kN (about 43.9% to 71.2%). The transmitted torques of the cycloid disks and cranks are also affected accordingly, e.g., the peak to peak value of the transmitted torque of the cycloid disk increases about 30%. The average transmitted torque increase 12% in the two of three cranks and decrease 19 % in another. The time-invariant errors have only considerable influence on loading in cycloid contact pairs. The errors related to involute gears affect only the transmitted torque of the cranks. The LTCA results also show that the influences of the profile modification and of the errors on loaded characteristics are independant.
The results from the bearing load analysis and life evaluation show that cycloid profile modification has nearly no influence on the bearing loads, where the combination of the negative shifting offset & negative equidistant offset modification reduces bearing load slightly, but only 0.08%. In the dissertation, the ideal profile, two types of the logarithm curve modification, the parabola end relief and the specific profile of rollers are considered for contact stress analysis. The results show that the stress concentration effect on face-end of the rollers can be reduced in the cases of profile modified rollers. However the contact stress on the rollers with the parabola end relief has an about 4 % - 4.7 % increasement at the start point of the profile modification. The calculated bearing life using Ioannides-Harris model also show that the bearing using the specific roller profile has the longest life, about 70 % to 300 % more than the other profiles. The analysis result of the roller with logarithm profile shows that the life will be reduced from 7,420 hours without error to 2,950 hours under the combined error.
In summary, the proposed TCA and LTCA models in the dissertation can provide the complete analysis capacitiy of the power-split type cycloid planetary reducer. It can enhance the competitiveness by improving the proformance of the existing design or assist in development of the new reducers.

關鍵字(中)

★ 齒面嚙合分析
★ 受載齒面接觸分析
★ 擺線減速機
★ 輪廓修形
★ 影響係數法
★ 擺線行星齒輪機構
★ 軌跡圓法
★ 軸承壽命評估

關鍵字(英)

★ Tooth Contact Analysis
★ Loaded Tooth Contact Analysis
★ Cycloid Reducer
★ Profile Modification
★ Influence Coefficient Method
★ Trajectory Circle Method
★ Bearing Life Prediction Model

論文目次

摘要 I
ABSTRACT III
謝誌 X
目錄 XI
圖目錄 XVII
表目錄 XXX
符號對照表 XXXII
第 1 章緒論 1
1.1 背景 1
1.2 文獻回顧 3
1.2.1 擺線幾何、修整與齒面嚙合分析 3
1.2.2 擺線減速機構之誤差與影響 5
1.2.3 軸承輪廓修整與壽命模型 5
1.2.4 漸開線齒輪齒廓修整、齒對嚙合分析與誤差 6
1.2.5 受載齒面接觸分析模型 7
1.3 研究目的與研究架構 9
第 2 章行星分流式擺線針輪減速機構簡介 12
2.1 架構 12
2.2 運動與力流關係 13
第 3 章接觸對元件幾何輪廓數學模型與修形模式 17
3.1 擺線齒廓 17
3.1.1 理論擺線齒廓 17
3.1.2 擺線齒廓幾何特性 18
3.1.3 擺線齒廓修形模式 19
3.2 擺線盤支撐軸承滾子輪廓 21
3.2.1 對數曲線修整 21
3.2.2 端面曲線修整 22
3.3 漸開線齒廓 23
3.3.1 理論漸開線齒廓 23
3.3.2 漸開線齒廓修整模式 24
第 4 章機構誤差分類與定義 26
4.1 擺線盤相關加工誤差 26
4.2 銷輪相關加工誤差 27
4.3 曲軸相關誤差 28
4.4 托架相關誤差 29
4.5 漸開線行星齒輪對相關誤差 29
4.6 組裝誤差 30
第 5 章接觸對分析模型 31
5.1 基於瞬心法之擺線齒面嚙合分析模型 31
5.2 基於軌跡圓法之擺線齒面嚙合分析模型 32
5.2.1 基本原理 32
5.2.2 在誤差影響下之嚙合分析 37
5.2.3 傳動誤差與背隙計算法 38
5.3 軸承滾子嚙合分析 39
5.4 漸開線齒面嚙合分析模型 39
5.4.1 基本原理 39
5.4.2 背隙與傳動誤差計算法 40
5.5 機構傳動誤差與背隙 41
第 6 章受載齒面接觸分析模型 42
6.1 影響係數法之基本原理 42
6.2 行星分流式擺線針輪減速機構位移與負載關係 44
6.3 變形因子與影響係數 45
6.3.1 赫茲變形影響係數 47
6.3.2 輪齒彎曲變形影響係數 48
6.3.3 軸彎曲變形與軸扭轉變形 50
6.4 接觸對位移之幾何關係 53
6.4.1 銷-擺線盤接觸對 53
6.4.2 擺線盤-軸承滾子接觸對 54
6.4.3 軸承滾子-曲軸接觸對 55
6.4.4 漸開線齒輪接觸對 55
6.5 力平衡關係 56
6.5.1 銷輪與輸出扭矩負載關係 56
6.5.2 擺線盤力平衡 57
6.5.3 軸承滾子力平衡 59
6.5.4 曲軸力平衡 60
6.5.5 輸入軸扭力平衡 60
6.6 接觸對之間隙計算 61
6.7 計算分析模型 63
6.7.1 影響係數子矩陣A 63
6.7.2 位移係數子矩陣Q 70
6.7.3 力平衡子矩陣J 73
6.7.4 應力子向量P與位移子向量 74
6.7.5 間隙條件子向量H與力平衡條件子向量F 75
6.8 行星分流式擺線針輪減速機壽命估算模型 76
6.9 計算模型檢驗 77
第 7 章分析案例 79
7.1 設計參數 79
7.2 輪廓修整參數 80
7.3 分析案例之誤差條件 82
7.3.1 時變誤差 82
7.3.2 非時變誤差 83
7.3.3 漸開線齒輪對相關誤差 83
7.3.4 綜合誤差組合 83
7.4 電腦輔助計算效能 85
第 8 章理論齒廓受載齒面接觸分析結果 86
8.1 擺線盤與銷接觸對 86
8.1.1 銷負載與接觸應力變化 86
8.1.2 銷接觸對之接觸應力分布圖 87
8.2 擺線盤支撐軸承 88
8.2.1 軸承負載變化 88
8.2.2 扭力傳輸變化 89
8.2.3 軸承滾子接觸應力分布圖 90
8.3 漸開線齒輪對 93
8.3.1 齒對負載分配與接觸應力變化 93
8.3.2 齒對接觸應力分布圖 94
8.4 受載傳動誤差 96
8.5 機構壽命估算 98
第 9 章擺線齒廓修形對齒面接觸之影響分析 100
9.1 無誤差下之影響 100
9.1.1 齒面嚙合分析結果 100
9.1.2 受載齒面接觸分析結果 102
9.1.3 受載傳動誤差 106
9.2 單一時變誤差影響 107
9.2.1 齒面嚙合分析結果 107
9.2.2 受載齒面接觸分析結果 110
9.2.3 受載傳動誤差 119
9.3 單一非時變誤差影響 124
9.3.1 齒面嚙合分析結果 124
9.3.2 受載齒面接觸分析結果 126
9.3.3 受載傳動誤差 129
9.4 漸開線齒輪相關誤差影響 131
9.4.1 齒面嚙合分析結果 131
9.4.2 受載齒面接觸分析結果 132
9.4.3 受載傳動誤差 135
9.5 不同擺線盤接觸對誤差組合影響 137
9.5.1 齒面嚙合分析結果 137
9.5.2 受載齒面接觸分析結果 147
9.5.3 受載傳動誤差 165
9.6 小結 176
9.6.1 擺線輪廓修整影響 176
9.6.2 機構誤差影響 176
第 10 章綜合誤差情況下機構嚙合情況與負載表現 178
10.1 輪廓修整設定 178
10.2 齒面嚙合分析結果 178
10.2.1 背隙變化 178
10.2.2 傳動誤差變化 179
10.3 受載齒面接觸分析結果 181
10.3.1 擺線減速段負荷變化 181
10.3.2 擺線盤支撐軸承負荷變化 184
10.3.3 漸開線減速段負荷變化 189
10.3.4 受載傳動誤差變化 189
10.4 機構壽命估算 191
10.5 小結 192
第 11 章結論與未來展望 193
11.1 結論 193
11.1.1 擺線減速段接觸特性 193
11.1.2 漸開線減速段接觸特性與曲軸負載影響 195
11.1.3 軸承接觸對接觸特性 195
11.2 未來展望 196
參考文獻 197
作者簡歷 1

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指導教授

蔡錫錚(Shyi-Jeng Tsai)

審核日期

2021-1-28

推文