AGMA 13FTM06-2013 High Gear Ratio Epicyclic Drives Analysis.pdf

1、13FTM06 AGMA Technical Paper High Gear Ratio Epicyclic Drives Analysis By Dr. A. Kapelevich, AKGears, LLC2 13FTM06 High Gear Ratio Epicyclic Drives Analysis Dr. Alex Kapelevich, AKGears, LLC The statements and opinions contained herein are those of the author and should not be construed as an offici

2、al action or opinion of the American Gear Manufacturers Association. Abstract Epicyclic gear stages provide high load capacity and compactness to gear drives. There is a wide variety of different combinations of planetary gear arrangements 1, 2. For simple epicyclic planetary stages when the ring ge

3、ar is stationary, the practical gear ratio range varies from 3:1 to 9:1. For similar epicyclic planetary stages with compound planet gears, the practical gear ratio range varies from 8:1 to 30:1. This paper presents analysis and design of epicyclic gear arrangements that provide extremely high gear

4、ratios. Using differential-planetary gear arrangements it is possible to achieve gear ratios of several hundred to one in one-stage drive with common planet gears and several thousand to one in one-stage drive with compound planet gears. A special two-stage planetary arrangement may utilize a gear r

5、atio of over one hundred thousand to one. This paper shows an analysis of such uncommon gear drive arrangements, defines their major parameters, limitations, and gear ratio maximization approaches. It also demonstrates numerical examples, existing designs, and potential applications. Copyright 2013

6、American Gear Manufacturers Association 1001 N. Fairfax Street, Suite 500 Alexandria, Virginia 22314 September 2013 ISBN: 978-1-61481-063-6 3 13FTM06 High Gear Ratio Epicyclic Drives Analysis Dr. Alex Kapelevich, AKGears, LLC Introduction Epicyclic gear stages provide high load capacity and compactn

7、ess to gear drives. There is a wide variety of different combinations of planetary gear arrangements 1, 2. For simple epicyclic planetary stages when the ring gear is stationary, the practical gear ratio range varies from 3:1 to 9:1. For similar epicyclic planetary stages with compound planet gears,

8、 the practical gear ratio range varies from 8:1 to 30:1. This paper presents analysis and design of epicyclic gear arrangements that provide extremely high gear ratios. Using differential-planetary gear arrangements it is possible to achieve gear ratios of several hundred to one in one-stage drive w

9、ith common planet gears and several thousand to one in one-stage drive with compound planet gears. A special two-stage planetary arrangement may utilize a gear ratio of over one hundred thousand to one. This paper shows: an analysis of such uncommon gear drive arrangements, defines their major param

10、eters, limitations, and gear ratio maximization approaches. It also demonstrates numerical examples, existing designs, and potential applications. One-stage arrangements There are one-stage differential-planetary arrangements that provide much higher gear ratios. In these arrangements the output sha

11、ft is connected to the second rotating ring gear instead of the carrier, like in the epicyclic planetary stages. In this case a carrier does not transmit torque and it is called a cage because it is used just to support planet gears. Figures 1a and 1b presents differential planetary arrangements wit

12、h compound planet gears. In the arrangement in Figure 1a the sun gear is engaged with a portion of the planet gear that is in mesh with the stationary ring gear. In this case the gear ratio is: 113a12b 3a2a 3bzzuzzzz(1) where u gear ratio; z1sun gear number of teeth; z2anumber of teeth the planet ge

13、ar engaged with the sun gear and stationary ring gear; a. Compound planet gears b. Compound planet gears c. Common planet gears Figure 1. Differential-planetary arrangements Key 1 Sun gear 2 Planet gear 2a, Two portions of 2b compound planet gear 3a Stationary ring gear 3b Rotating ring gear 4 plane

14、t cage 4 13FTM06 z2bnumber of teeth the planet gear engaged with the rotating ring gear; z3astationary ring gear number of teeth; z3brotating ring gear number of teeth. In the arrangement in Figure 1b, the sun gear is engaged with a portion of the planet gear that is in mesh with the rotating ring g

15、ear. In this case the gear ratio is: 113a 2b12a3a 2b3b 2azzzzuzzzz(2) If a gear ratio is negative, the input and output shaft rotation directions are opposite. All gear meshes in differential planetary arrangements have the same center distance. This condition allows for definition of relations betw

16、een the operating modules, mw, or diametral pitches, DPw. For the arrangement in Figure 1a they are: w12a 1 2a w2a3a 3a 2a w2b3b 3b 2bmzzm zzm zz (3) or 12a 3a2a 3b2bw12a w2a3a w2b3bzz zz zzDP DP DP (4) The relation between operating pressure angles in the gear meshes z1- z2aand z2a z3ais defined by

17、 equation 5: coscosw2a-3a 1 2aw1-2a 3a 2azzzz(5) where w1-2aoperating pressure angle in a mesh of the sun gear and the planet gear engaged with the stationary ring gear; w2a-3a operating pressure angle in the planet/stationary ring gear mesh. Similar to the arrangement in Figure 1b: w12b 1 2b w2b3b

18、3b 2b w2a3a 3a 2amzzm zzm zz (6) or 12b 3b2b 3a2aw12b w2b3b w2a3azz zz zzDP DP DP (7) The relation between operating pressure angles in the gear meshes z1 - z2b and z2b z3b is defined by equation coscosw2b-3b 1 2bw1-2b 3a 2bzzzz(8) where w1-2b operating pressure angle in a mesh of the sun gear and t

19、he planet gear engaged with the rotating ring gear; w2b-3b operating pressure angle in the planet/rotating ring gear mesh. In differential planetary arrangements with compound planet gears, operating pressure angles in the planet/stationary ring gear mesh and in the planet the planet/rotating ring g

20、ear mesh can be selected independently. This allows for balancing specific sliding velocities in these meshes to maximize gear efficiency, which could be 8090%, depending on the gear ratio 2. The maximum gear ratio in such 5 13FTM06 arrangements is limited by possible tip/tip interference of the nei

21、ghboring planet gears. In order to avoid this interference the following condition should satisfied: For the arrangement in Figure 1a sin 21sin12aw2awzhnzn(9) For the arrangement in Figure 1b sin 21sin12bw2bwzhnzn(10) where nwnumber of planets; h2a, h2boperating addendum coefficients of the planet g

22、ears z2aand z2baccordingly. Maximum gear ratio values for the differential-planetary arrangement with the compound planet gears (assuming h2a= h2b= 1.0) are shown in the Table 1. The assembly condition for these gear arrangements is: 3a 3bwintegerzzn (11) Two parts of a compound planet gear should b

23、e angularly aligned for proper assembly. This is typically achieved by aligning the axes of one tooth of each part of the compound planet gear, which makes its fabrication more complicated. Assembly of such gear drives requires certain angular positioning of planet gears. All these factors increase

24、the cost of this type of gear drive. Examples of differential planetary gear actuators with compound planet gears are shown in Figure 2. A simplified version of the one-stage differential planetary arrangement is shown in Figure 1c. This arrangement does not use the compound planet gear. The common

25、planet gear is engaged with the sun gear, and both the stationary and the rotating ring gears. This does not allow for specific sliding velocities in each mesh to be equalized, resulting in a reduction of gear efficiency of about 7084% 2. However, the assembly of such gear drives does not require ce

26、rtain angular positioning of planet gears and their manufacturing cost is significantly lower for drives with the compound planet gears. Example of the differential planetary gear actuator with common planet gears is shown in Figure 3. Table 1. Maximum gear ratio values for differential-planetary ar

27、rangements with compound planet gears Number of planets Sun gear tooth number Maximum gear ratio* 3 10 1579:115 2857:1 25 5183:14 10 144:115 273:1 25 518:15 10 49:115 80:1 25 162:1* Sign “+” if the input and output shaft rotation directions are the same, sign “-” if they are opposite. 6 13FTM06 a) C

28、ros section b) Cros section c) Photograph Figure 2. Differential-planetary gear actuators with compound planet gears Relations between operating pressure angles in the gear meshes are defined by equations 12 through 15: coscosw2-3a 12w1-2 3a 2zzzz (12) coscosw2-3b 12w1-2 3b 2zzzz (13) coscosw2-3b 3a

29、 2w2-3a 3b 2zzzz(14) where: w1-2 operating pressure angle in sun/planet gear mesh; w2-3a operating pressure angle in planet/stationary ring gear mesh; w2-3b operating pressure angle in planet/rotating ring gear mesh. 7 13FTM06 a) Drawing with cros section b) Gear drive assembly photo c) Gear drive a

30、ssembly photo Figure 3. Differential-planetary gear actuator A gear ratio is 113a13a3bzzuzz(15) Maximum gear ratio values for the differential-planetary arrangement with the common planet gears (assuming h2a= h2b= 1.0) are shown in the Table 2. 8 13FTM06 Table 2. Maximum gear ratio values for differ

31、ential-planetary arrangements with compound planet gears Number of planets Sun gear tooth number Maximum gear ratio 3 10 405:115 767:125 1432:1 4 10 59:115 101:125 198:1 5 10 20:115 32:125 70:1 In differential planetary arrangements (Figure 1) tangent forces applied to the planet gear teeth from the

32、 stationary and rotating ring gears are unbalanced, because they lie on different parallel planes and have opposite directions. A sturdy planet cage is required to avoid severe planet gear mesh misalignment. There are some gear drives that use the differential planetary arrangements with balanced pl

33、anet gear tangent forces (Figure 4). In this case, the triple-compound planet gears (Figure 4a and 4b) are used. They have identical gear profiles on their end portions that are engaged with the two identical stationary ring gears. The middle portion of such planet gears has a different profile than

34、 those on the ends and is engaged with the rotating ring gear. The arrangement in Figure 4c has common planet gears engaged with the sun gear, two stationary ring gears, and one rotating ring gear. These types of differential-planetary drives typically do not have the cage and bearings, because the

35、planet gear forces are balanced and planet gears themselves work like the roll bearings for radial support of the rotating ring gear. Two-stage arrangements In most conventional two-stage epicyclic arrangements the gear ratio usually does not exceed 100:1, although there are possible arrangements th

36、at allow a significant increase in the gear ratio 3. Figure 5 shows the planetary gear arrangement A with the sun gears of the first and second stages connected together and the compound cage supporting the planet gears of both first and second stages. a) and b) Triple compound planet gear c) Common

37、 planet gear Figure 4. Differential-planetary arrangements without planet gear cage Key 1 Sun gear 2 Planet gear 2a, Two portions of 2b triple compound planet gear 3a Stationary ring gear 3b Rotating ring gear 9 13FTM06 Figure 5. Two stage planetary (arrangement A) with connected sun gears of 1st an

38、d 2nd stages and compound cage a) Axial cross section b) Section I-I c) Section II-II Figure 6. Two stage planetary gearbox (arrangement A) with connected sun gears of 1st and 2nd stages and compound cage Key 1I, 1II Sun gears 2I, 2II Planet gears 3I Stationary ring gear of 1st stage 3II Rotating ri

39、ng gear of 2nd stage 4 Compound cage, indexes “I” and “II” are for 1st and 2nd stages accordingly Key i Central location angles between planet gears, indexes “I” and “II” are for 1st and 2nd stages 10 13FTM06 A sketch of the gearbox with arrangement A is presented in Figure 6. Both sun gears are con

40、nected to the input shaft and are engaged with the planet gears of the first and second stages accordingly. The first-stage ring gear is stationary and is connected with the gearbox housing. It is engaged with the first-stage planet gears. The compound cage practically contains the first- and second

41、-stage cages connected together. The ring gear of the second stage is engaged with the second-stage planet gears and connected to the output shaft. The gear ratio of arrangement A is: II I I31 3III I I13 1 3zz zuzz z z(16) where z1I, z1IInumbers of teeth of the sun gears of the 1st and 2nd stages; z

42、2I, z2II numbers of teeth of the planet gears of the 1st and 2nd stages; z3I, z3II numbers of teeth of the ring gears of the 1st and 2nd stages. Figure 7 shows the alternative gear arrangement B with the sun gears of both stages connected together and the ring gears of both stages also connected tog

43、ether. A sketch of the gearbox with the alternative arrangement B is presented in Figure 8. Both sun gears are connected to the input shaft and engaged with the planet gears of the first and second stages accordingly. The shafts supporting the first-stage planet gears are connected (pressed in, for

44、example) to the gearbox housing. Both ring gears are connected together and engaged with the planet gears of the first and second stages accordingly. The second-stage carrier is connected to the output shaft. The gear ratio of arrangement B is III I31 3II I I II13 13zz zuzz zz(17) The maximum gear r

45、atios of these two-stage planetary arrangements A and B are achieved when the denominator of Equations 16 and 17 is equal to 1 or 1. This condition can be presented as: 1III I I13 1 3zz z z (18) When this denominator is 1, the input and output shafts are rotating in the same direction. When it is le

46、ss -1, the input and output shafts are rotating in opposite directions. If a number of planet gears are more than 1 (nwI 1 and nwII 1), condition (18) requires irregular angular positioning of the planet gears in one or both planetary stages. This means that the central location angles ibetween plan

47、et gears in one or both stages are not identical (see Figures 6b 6c, 8b and 8c). A definition of the central angles with irregular angular positioning of the planet gears that provides proper assembly is described in 3. Figure 7. Two stage planetary arrangement B with sun gears and ring gears of 1st

48、 and 2nd stages connected together Key 1I, 1II Sun gears 2I, 2II Planet gears 3I Ring gears 4 2ndstage carrier, indexes “I” and “II” are for 1st and 2nd stages accordingly 11 13FTM06 a) Axial cross section b) Section I-I c) Section II-II Figure 8. Two stage planetary gearbox (arrangement B) with sun

49、 gears and ring gears of 1st and 2nd stages connected together The neighboring planet gears located at the minimum central angles must be checked for the possibility of tip/tip interference. Irregular angular positioning of the planet gears may result in an imbalance in the planetary stage. This must be avoided by carrier assembly balancing. Application of the two-stage planetary arrangemen