二階行星齒輪傳動機構效率分析

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王晨曄(Chen-Ye Wang) 查詢紙本館藏

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機械工程學系

論文名稱

二階行星齒輪傳動機構效率分析

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摘要(中)

行星齒輪機構相對於傳統平行軸齒輪系，擁有輸入輸出同軸線、緊密設計、高速比、高功率密度等特性，廣泛運用於各類型傳動機構之中，如車用齒輪箱、風機傳動、產業增減速機、伺服馬達減速齒輪箱等。而多階行星齒輪機構因多樣性的連接關係，具有更廣泛之速比範圍與功率流動方式。但少許特殊連接關係之機構，會因內部功率迴流產生內部功率放大，導致功率傳遞時損耗也隨之增加，使機構造成自鎖。
本論文之目的係探討於齒面摩擦影響下，四種兩階複合式與耦合式行星齒輪機構之傳動效率。而其中複合式行星齒輪機構以所謂2K-H型與3K型「悖論行星齒輪機構」做為案例。此兩類型齒輪機構具有兩個齒差之太陽齒輪或環齒輪在同中心距下與同一行星齒輪嚙合之特色。若使用傳統標準齒形與移位方式進行齒形設計，並無法得到合適齒形設計參數。因此本論文對此兩案例採用直接齒形設計方法，以齒冠高以及工作節圓齒厚作為設計參數，利用互補齒形與節距關係，得到機構中齒對間幾何參數之相依關係。在齒頂干涉限制、尖齒限制之齒形幾何限制條件下，限制出有效之幾何參數範圍。之後根據各齒對齒根、齒面強度決定機構中最佳承載能力之齒形幾何參數。耦合式行星齒輪機構則包括以第一階托架與第二階環齒輪連結設計(第一型)以及兩階環齒輪連結設計(第二型)兩種案例。
在效率分析方法上，藉由行星齒輪機構轉速、扭力關係，可推得機構功率流動方向，藉以辨識機構是否產生功率迴流。並導入嚙合效率，求得機構整體平均效率，同時亦能確認機構是否產生自鎖。為進一步求得行星齒輪機構之瞬時效率變化，在論文中將單階行星齒輪組齒對撓度法之負載分析模型拓展成二階並聯式行星齒輪機構之分析模型，以求得各嚙合齒對在摩擦影響下之分配負載，進而計算出嚙合瞬間之齒對嚙合效率，並求出各種行星齒輪機構於嚙合過程中之傳動效率變化。
在對行星齒輪機構傳動效率分析結果可以發現，行星齒輪機構齒數配置若為非因數型式，即非等相位嚙合，在傳動過程中嚙合效率跳動幅度會降低。而論文中複合式行星齒輪機構會產生功率迴流現象；而耦合式則無，在差速段會形成功率分流與匯流。複合式行星齒輪機構中2K-H悖論行星齒輪機構內部迴流功率約為外部功率19倍。故嚙合效率約大於5%以上時，便會產生自鎖現象。而3K悖論行星齒輪機構內部功率迴流約為外部功率7倍。差速分流耦合式行星齒輪機構中，在磨擦係數為0.15的狀況下，第二型傳動效率約為94.1%高於第一型85.7%，且瞬時效率跳動幅度也為第一型之三分之一。
由本論文所提出的解析計算模型，可以用於各種複合式以及耦合式行星齒輪機構，分析不同摩擦係數下傳動效率之變化，並可判斷機構於嚙合過程中是否發生自鎖，可有效提供此類型行星齒輪機構基本設計之參考。

摘要(英)

Compared with the traditional parallel axis gear train, the planetary gear drives have the characteristics of coaxial input or output, compact design, high speed ratio, high power density, etc. Therefore the planetary gear drives are widely used in various types of transmission application, such as automotive gearboxes, wind turbines, increase and reduction gear drives, servo motor reduction gearboxes, etc. The multi-stage planetary gear drives have a wider range of speed ratios and power flow type due to the various connection relationship. However, some drives with special connection relationship will have characteristics that internal circulation power will be generated and as a result power loss will increase during power transmission. Such the characterics might cause the drives to self-locking.
The purpose of this paper is to investigate the transmission efficiency of four two-stage compound and coupled planetary gear drives under the influence of tooth surface friction. The study cases for compound planetary gear drives in this paper are so-called the 2K-H type and the 3K type paradox planetary gear drives. These two type paradox planetary gear drives having tooth number differences engage with the same planetary gear under the same center distance. If the thooth profile is designed using the traditional gear desigh method of standard datum rack and profile shifting, suitable tooth profile design parameters can not be obtained. Therefore, this paper adopts the direct tooth profile design method, where the addendum and tooth thickness on the working pitch circle are used as the design parameters, and the complementary tooth profile and pitch relation of the gear are also applied to obtain the essential relation of the geometrical parameters of the tooth pairs. The effective geometric parameter range can be thus constraint by the geometrical limits of tooth tipping and interference. The gearing parameters for the optimal load capacity of the gear drive can be further determined according to the root or surface strength of each gear pair. The coupled planetary gear drives include two cases: type Ⅰ where the first-stage carrier connects with the second-stage ring gear, and type Ⅱ where the ring gears of the two stages connect with each other.
In order to determine the efficiency, the relations for the speed and torque of the planetary gear drives at first established. The power flow direction of the drives can be thus deduced, so as to idebtify whether the drives generates internal circulation power. Because of the revolution motion in the planetary gear train, the meshing efficiency is introduced to obtain the average efficiency od the drives. In oder to further obtain the instantaneous efficiency of the planetary gear drives, the load analysis model of one-stage planetary gear set based on stiffness method is extened to analyze two-stage planetary gear trains with considering the influence of the friction on teeth. The meshing efficiency of each tooth pair ateach meshing step can be calculated, and the variation of the transmission efficiency of various planetary gear drives during meshing can be represented accordingly.
From the analysis results of the transmission efficiency of the planetary gear drives, it can be found that the amplitude of the instantaneous meshing efficiency will be reduced during the transmission, if the number of teeth of the planetary gear drives are configured as non-factorizing, namely the meshing is in non-equal phase angle. The compound planetary gear drives in the paper will produce internal circulation power flow. On the other hand, the coupled planetary gear will not generate such the phenomenon, but power splitting or summation in the differential stage. The internal circulation power flow of the 2K-H paradox planetary gear drive is about 19 times the external power. Therefore, the self-locking will occur, when the meshing efficiency is greater then 5% or more. However, the internal circulation power flow of 3K paradox planetary gear drive is about 7 times the external power. In the differential type of coupled planetary gear drives, the transmission efficiency of the type Ⅱ is about 94.1% under the friction coefficient of 0.15, which is higher than 85.7% of the type Ⅰ. The fluctuation amplitude of the instantaneous efficiency is also one-third of the type Ⅰ.
The analytical calculation model proposed by this paper can be used for various compound and coupled planetary gear drives to, analyze the variation of transmission efficiency under different friction coefficients. The results can be used to determine whether the planetary gear drives are self-locking during the meshing process. This apprach can provide an effective reference for basic design of such the types of planetary gear drives.

關鍵字(中)

★ 複合式行星齒輪機構
★ 耦合式行星齒輪機構
★ 2K-H悖論行星齒輪機構
★ 3K悖論行星齒輪機構
★ 差速分流行星齒輪機構
★ 直接齒形設計
★ 齒面摩擦
★ 瞬時傳動效率

關鍵字(英)

★ Compound planetary gear drive
★ Coupled planetary gear drive
★ 2K-H paradox planetary gear drive
★ 3K paradox planetary gear drive
★ Differential type planetary gear drive
★ Direct gear design
★ Tooth face friction
★ Instantaneous transmission efficiency

論文目次

摘要........................................... i
Abstract....................................... iii
謝誌........................................... vi
目錄........................................... vii
圖目錄.......................................... x
表目錄.......................................... xvi
符號對照表...................................... xix
第1章前言................................... 1
1.1 研究背景................................ 1
1.2 文獻回顧................................ 5
1.3 研究目的................................ 7
1.4 論文架構................................ 8
第2章二階行星齒輪機構之結構與齒形設計.......... 10
2.1 行星齒輪機構類型........................ 11
2.2 行星齒輪機構之結構設計................... 13
2.2.1 悖論行星齒輪機構........................ 13
2.2.2 差速分流行星齒輪機構..................... 14
2.3 悖論行星齒輪機構齒形設計................. 15
2.3.1 齒對齒形幾何參數關係..................... 16
2.3.2 直接齒形設計方法基本關係................. 17
2.3.3 齒根圓角設計............................ 25
2.3.4 齒形幾何限制............................ 30
2.3.5 等效齒條刀具參數設計..................... 33
2.4 悖論行星齒輪機構齒形參數最佳化設計........ 35
2.4.1 齒根強度................................ 35
2.4.2 齒面強度................................ 40
2.4.3 最佳承載能力............................ 41
2.5 悖論行星齒輪機構案例之齒形參數設計........ 43
2.5.1 2K-H悖論行星齒輪機構.................... 43
2.5.2 3K悖論行星齒輪機構...................... 48
第3章行星齒輪組受載齒面接觸分析............... 53
3.1 行星齒輪組齒對接觸分析................... 53
3.1.1 行星齒輪組齒對接觸位置關係............... 53
3.1.2 各案例之理想接觸位置..................... 59
3.2 單階行星齒輪組受載齒面接觸分析模型........ 63
3.2.1 變形-位移關係........................... 63
3.2.2 力平衡關係.............................. 66
3.2.3 單階行星齒輪組負載計算矩陣............... 70
3.3 各案例行星齒輪機構負載計算矩陣............ 72
3.3.1 2K-H型悖論行星齒輪機構：................. 73
3.3.2 3K型悖論行星齒輪機構：................... 73
3.3.3 差速分流行星齒輪機構：................... 74
第4章行星齒輪機構瞬時效率計算模型.............. 77
4.1 行星齒輪機構轉速關係..................... 77
4.1.1 單階行星齒輪組轉速關係................... 77
4.1.2 各案例之轉速計算矩陣..................... 79
4.2 行星齒輪機構扭力關係..................... 81
4.2.1 單階行星齒輪組扭力關係................... 82
4.2.2 各案例之扭力計算矩陣..................... 82
4.3 行星齒輪機構功率關係..................... 84
4.3.1 含嚙合損耗之扭力計算矩陣................. 86
4.3.2 瞬時效率計算............................ 89
第5章 2K-H悖論行星齒輪機構分析................. 91
5.1 嚙合效率分析............................ 92
5.2 平均效率分析............................ 94
5.3 瞬時傳動效率分析........................ 97
第6章 3K悖論行星齒輪機構分析................... 99
6.1 嚙合效率分析............................ 100
6.2 平均效率分析............................ 104
6.3 瞬時傳動效率分析........................ 108
第7章差速分流行星齒輪機構分析................. 110
7.1 幾何參數決定方法........................ 111
7.2 嚙合效率分析............................ 115
7.3 平均效率分析............................ 123
7.4 瞬時傳動效率分析........................ 128
第8章結論與未來展望.......................... 130
8.1 結論................................... 130
8.2 未來展望................................ 131
參考文獻........................................ 133
附錄 A 案例各元件之轉速與扭矩關係............... 136

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指導教授

蔡錫錚(Shyi-Jeng Tsai)

審核日期

2022-9-28

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