考慮間隙及摩擦力影響下之K–H–V型擺線行星齒輪機構受載接觸分析

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張甯喬(Ling-Chiao Chang) 查詢紙本館藏

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機械工程學系

論文名稱

考慮間隙及摩擦力影響下之K–H–V型擺線行星齒輪機構受載接觸分析
(Loaded Contact Analysis of K–H–V Type Cycloid Planetary Gear Drive Considering Clearances and Friction)

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摘要(中)

在常見擺線行星齒輪機構中，以K-H-V型與2K-H型為兩種主要型式。而其中K-H-V型，即習稱之Cyclo型式擺線行星齒輪機，已有長期的發展歷史，並且廣泛運用於小體積、高減速比的傳動機構中，如生產線輸送系統、自動導引車、化學工業攪拌裝置等各種產業傳動場合。然而隨著對精度的需求增加，分析機構中各元件在嚴苛條件下，如加工、組裝誤差，擺線輪修整齒廓、背隙，以及曲軸軸承間隙等，對於傳動誤差及接觸負載等傳動性能的影響則至關重要；特別是支撐擺線盤的曲軸軸承因受到負載最大，是機構中承載能力最弱之元件。另一方面，在永續減碳的趨勢下，傳動機構效率也必須事先掌握。在考慮這些眾多因素的情況下，以平面機構進行分析時，擺線盤因曲軸軸承具有間隙，被視為具有三個自由度。然而由於擺線盤同時與多個元件接觸，導致接觸分析變得更加複雜。
本研究的目的即是在考慮軸承間隙、擺線齒廓修整、元件誤差以及摩擦影響的情況下，建立K-H-V型擺線齒輪機構的完整分析模型，以分析各個接觸對的受力狀況。本論文分析的接觸對包含擺線齒輪–銷齒接觸對、軸承滾子–內外環接觸對以及爪銷孔–爪銷接觸對等三個類型。本研究共建立考慮有或無軸承間隙之接觸分析模型，以利進行不同目標之分析。在無軸承間隙接觸分析模型中，以各接觸對在給定外部負載下之變形–位移關係以及力與力矩平衡方程式，建立基於影響係數法之矩陣方程式，迭代計算求解出各接觸對之分佈負載、擺線盤位移以及曲軸轉角位移。此計算模型亦可應用於擺線齒廓修整設計。然而在考慮軸承間隙時，各接觸對受載變形前的位置並無法僅透過運轉角度關係確定；因此本研究採假設擺線盤三個位移以及曲軸轉動角位移的方式，推導出考慮間隙下的受載齒面接觸分析模型。首先在給定前述的位移下，計算出各元件最終位置以及各接觸對干涉或間隙，並據此以影響係數法所建立的剛性圖計算出各接觸對負載。再由納入摩擦影響之力與力矩平衡關係式，作為分析模型迭代求解的收斂條件。而迭代計算所需的下一步位移猜值則以前位置下的正切剛性矩陣求得。
本研究使用一個實際單齒差擺線減速機案例作為分析模型。在案例分析中，探討軸承間隙、摩擦、曲軸變形以及誤差等因素對接觸特性的影響，並評估比較三個不同軸承間隙的影響。分析結果顯示，軸承間隙對爪銷的負載影響比對軸承、銷齒的影響更為顯著，且與ADAMS模型的分析結果相差不大，驗證了數值分析模型的可行性。在摩擦的影響下，曲軸輸入力矩會增加以達到固定的輸出力矩，機構的平均效率在習用條件下皆為86.5%以上。曲軸變形會導致受載傳動誤差和各接觸對負載產生大幅度波動，但軸承間隙可補償曲軸變形影響。
在誤差影響分析方面，由誤差相關性分析結果顯示，元件偏心誤差以及爪銷切向誤差對受載傳動誤差有極大的影響。在銷齒負載分析結果，發現軸承間隙可補償誤差的影響。在頻譜分析中，單一誤差對於銷齒第一倍嚙合頻的受載傳動誤差振幅值皆大於爪銷第一倍頻。最後，綜合誤差條件分析比較了兩種不同擺線齒廓修整的設計。結果顯示，除了平均機構效率有較明顯的差異外，齒廓修整對接觸負載特性並無太大影響。
由案例分析結果顯示，本論文所提出的擺線行星齒輪分析模型，除能夠解決軸承間隙下的負載分析問題，亦可以模擬各種主要元件的誤差以及摩擦對接觸特性的影響，同時也能評估機構效率。這一分析方法不僅能夠有效地模擬擺線行星齒輪機構在各種實際運轉狀況下的傳動效能，也可以做為整體機構的性能評估和最佳化設計的實用工具。

摘要(英)

In typical cycloidal planetary gear mechanisms, the K-H-V type and 2K-H type are two primary configurations. Among them, the K-H-V type, commonly known as the Cyclo type cycloidal planetary gear mechanism, has been developed for a long time and is widely utilized in transmission systems requiring compact volume and high reduction ratios. These transmission mechanisms are commonly found in various industrial applications ,including assembly line conveyors, automated guided vehicles, and industrial chemical mixers. As the demand for precision increases, the analysis of the transmission mechanism becomes more complex. Under strict conditions, such as machining and assembly errors, modified tooth cycloid profile, backlash, and bearing clearances, the impact on transmission performance and contact loads between components become crucial factors to consider. Especially, the roller bearings supporting the cycloidal discs bear the maximum load and are the weakest components in the mechanism. On the other hand, under the trend of sustainable decarbonization, the efficiency of transmission mechanisms must be considered. When analyzing these numerous factors, the cycloidal disc, treated as a planar mechanism, is regarded as having three degrees of freedom due to the bearing clearance. However, since the cycloidal disc simultaneously contacts multiple components, the contact analysis becomes even more complex.
The aim of the dissertation is thus to establish a comprehensive analytical model for the K-H-V type cycloid planetary gear mechanism, taking into account the influence of bearing clearance, errors, flank modification, and friction. The model includes meshing analysis of various contact pairs under different output conditions and a loaded tooth contact analysis model (LTCA), based on the influence coefficient method. These four contact pairs analyzed in this dissertation include: cycloid–pin, bearing roller–inner race and outer race, and cycloid–pinshaft. The contact analysis model considers both the presence and absence of bearing clearances, allowing for analysis of different objectives. In the absence of bearing clearance, deformation-displacement relationships of each contact pair under given external loads, as well as force and torque balance equations, are formulated using the influence coefficient method. These equations can be assembled into a matrix form and then iteratively solved to obtain the distributed loads of each contact pair, displacements of the cycloid disc, and angular displacement of the crankshaft. This calculation model is also applied to the design of cycloid tooth profile modification. However, when accounting for bearing clearance, the positions of each contact pair cannot be determined solely through angular relationships. Therefore, this study assumes three directions of displacement of the cycloid disc and the rotational displacement of the crankshaft to derive a loaded tooth contact analysis model considering clearances. Initially, the final positions of each component and the interference or clearance of each contact pair are calculated based on the given displacements. Subsequently, the loads of each contact pair are computed using the stiffness map established by the influence coefficient method. The frictional influence is then incorporated into the force and moment balance equations as the convergence condition for the iterative calculation. Displacement guess value for the next step needed is obtained by solving the tangent stiffness matrix based on the previous position.
The analysis model is then validated through a practical case study of a cycloidal speed reducer with a single-tooth difference. In the case study, the effects of bearing clearance, friction, crankshaft deformation, and errors on contact characteristics are investigated, and the impact of three different clearance values is compared. The analysis results indicate that bearing clearance has a significant impact on the load of the cycloid–pinshaft, and the findings are consistent with the ADAMS model, validating the feasibility of the numerical analysis model. Additionally, the frictional influence increases the input torque to achieve a constant ouptut torque, with an average mechanical efficiency of over 86.5% under normal operating conditions. Crankshaft deformation leads to significant fluctuations in loaded transmission error and loads of various contact pairs. However, bearing clearances have the capability to compensate for the effects of crankshaft deformation.
In the error analysis, the results reveal that the component eccentricity error and tangential error of pinshafts have a significant impact on loaded transmission error. The bearing clearances can compensate for the effects of errors in the load analysis of pin-wheel. In spectral analysis, the loaded transmission error values of each error for the first meshing frequency of pin-wheel are greater than those for the first meshing frequency of pinshaft. Finally, a comparative analysis of two different cycloid modification profiles in the context of all error conditions was conducted. The results show that, apart from noticeable differences in average mechanical efficiency, cycloid profile modifications have minimal influence on the contact load characteristics.
The results of the case study demonstrate that the cycloid planetary gear analysis model proposed in this dissertation not only solves the load analysis problem under bearing clearances but also simulates the impact of different primary component errors and friction on contact characteristics, and evaluates the mechanical efficiency. This analytical approach effectively simulates the transmission performance of the cycloid planetary gear mechanism under various actual operatinbg conditions. Consequently, it serves as a practical tool for assessing performance and optimizing the design of the entire mechanism.

關鍵字(中)

★ K-H-V型擺線齒輪機構
★ 受載齒面接觸分析模型
★ 加工組裝誤差
★ 軸承間隙
★ 效率
★ 頻譜分析
★ 摩擦
★ 剛性圖

關鍵字(英)

★ Cycloid planetary gear drive
★ loaded tooth contact analysis model
★ manufacturing and assembly errors
★ bearing clearance
★ efficiency
★ spectral analysis
★ friction
★ stiffness map

論文目次

摘要 ii
Abstract iv
誌謝 vii
目錄 viii
圖目錄 xii
表目錄 xxii
符號說明 xxiii
第 1 章緒論 1
1.1 研究背景 1
1.2 文獻回顧 3
1.2.1 擺線齒輪嚙合分析及誤差影響 3
1.2.2 擺線齒輪之受載齒面接觸分析模型 5
1.2.3 元件間隙影響 7
1.2.4 擺線行星齒輪機構效率研究 8
1.3 研究目的及論文架構 14
第 2 章擺線行星齒輪機構分析模型基本理論 16
2.1 擺線行星齒輪機構簡介 16
2.2 機構運動關係 17
2.2.1 速比關係 17
2.2.2 相對擺線盤之運動關係 18
2.3 機構功率與效率關係 20
2.3.1 機構元件扭矩與功率 20
2.3.2 機構效率關係-功率流分析方法 22
2.3.3 機構效率關係-耗損功率方法 25
2.4 擺線齒輪齒面數學模型與修形方法 30
2.4.1 理論齒面方程式 30
2.4.2 修形齒面方程式 32
2.5 接觸對等效摩擦係數 33
2.5.1 擺線齒—銷齒接觸對 33
2.5.2 爪銷孔—爪銷接觸對 35
2.5.3 軸承滾子接觸對 38
第 3 章擺線行星齒輪機構加工及組裝誤差 39
3.1 誤差總覽 39
3.2 擺線盤相關誤差 40
3.2.1 爪銷孔半徑與分佈圓半徑誤差 40
3.2.2 爪銷孔位置誤差 40
3.2.3 爪銷孔偏心誤差 40
3.3 爪銷相關誤差 41
3.3.1 爪銷之銷半徑誤差與分爪銷佈圓徑誤差 41
3.3.2 爪銷之銷-位置誤差 42
3.3.3 爪銷-偏心誤差 43
3.4 可組裝條件 43
第 4 章無間隙及無摩擦下接觸分析模型 45
4.1 接觸計算分析 45
4.1.1 擺線齒輪—銷齒接觸對 45
4.1.2 爪銷孔—爪銷接觸對 48
4.1.3 軸承滾子接觸對 52
4.2 傳動誤差計算方法 53
4.3 受載齒面接觸分析模型基本理論 55
4.4 K-H-V型受載齒面接觸分析模型 57
4.4.1 接觸對位移關係式 57
4.4.2 接觸對間隙關係 60
4.4.3 力與力矩平衡關係式 62
4.4.4 完整受載齒面接觸分析模型 67
4.4.5 受載傳動誤差計算 68
4.5 誤差相關性分析 68
4.6 擺線行星齒輪機構嚙合頻 69
第 5 章考慮間隙及誤差下接觸分析模型 72
5.1 分析模型假設 72
5.2 修正受載齒面接觸分析模型 73
5.3 元件最終位置計算 74
5.3.1 曲軸為固定座標系 74
5.3.2 擺線盤為固定座標系 78
5.4 接觸對間隙計算 79
5.4.1 擺線齒輪—銷齒接觸對 79
5.4.2 爪銷孔—爪銷接觸對 80
5.4.3 軸承滾子接觸對 81
5.5 力與力矩平衡關係式 85
5.5.1 軸承滾子 85
5.5.2 擺線盤 88
5.5.3 輸出力矩關係式 94
5.5.4 輸入力矩關係計算 96
5.6 剛性圖法 97
5.6.1 軸承滾子接觸對 97
5.6.2 三次樣條差值法 97
5.6.3 擺線齒輪–銷齒接觸對 98
5.7 分析收斂條件及猜測位移計算 100
5.8 模型計算流程 102
第 6 章擺線行星齒輪機構不同因素影響下結果分析比較 103
6.1 分析案例 103
6.2 齒廓修整係數探討 104
6.3 無誤差具不同間隙下修整齒面接觸分析結果 106
6.3.1 受載傳動誤差 106
6.3.2 接觸負載結果 107
6.4 具摩擦力下不同間隙之分析結果 113
6.4.1 受載傳動誤差 113
6.4.2 接觸負載結果 114
6.4.3 機構效率 116
6.4.4 機構效率計算方法差異性 117
6.5 具曲軸變形影響下不同間隙之分析結果 120
6.5.1 受載傳動誤差 120
6.5.2 接觸負載結果 121
6.6 無誤差條件下CAE軟體分析結果驗證 125
6.6.1 ADAMS分析模型與設定 125
6.6.2 更改設定及分析條件 126
6.6.3 受載傳動誤差 127
6.6.4 接觸負載比較結果 128
第 7 章擺線行星齒輪機構誤差影響 131
7.1 誤差與受載傳動誤差之相關性分析 131
7.1.1 誤差分類 131
7.1.2 相關性影響結果 132
7.2 分析案例之誤差條件 134
7.3 單一誤差下修整齒面接觸分析結果 135
7.3.1 不同偏心誤差影響 135
7.3.2 爪銷切向誤差 (ER 5) 影響 148
7.3.3 各項單一誤差分析結果比較 155
7.4 綜合誤差影響 161
7.4.1 受載傳動誤差 162
7.4.2 接觸對負載結果 162
7.4.3 機構效率 167
7.4.4 頻譜分析 168
第 8 章結論與未來展望 170
8.1 結論 170
8.1.1 單一因素對於不同軸承間隙影響 170
8.1.2 單一誤差條件下影響 171
8.1.3 綜合誤差影響 172
8.1.4 在綜合誤差下，不同擺線齒廓修整曲線影響 172
8.2 未來展望 172
參考文獻 174
附錄A. CAE結果比較 183
簡歷 188
著作目錄 190

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指導教授

蔡錫錚(Shyi-Jeng Tsai)

審核日期

2024-7-26

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