1、06FTM03Detailed Procedure for the Optimum Design ofan Epicyclic Transmission Using Plastic Gearsby: I. Regalado and A. Hernndez, CIATEQTECHNICAL PAPERAmerican Gear Manufacturers AssociationDetailed Procedure for the Optimum Design of anEpicyclic Transmission Using Plastic GearsIsaias Regalado, Ph.D.
2、 and Alfredo Hernndez, CIATEQThe statements and opinions contained herein are those of the author and should not be construed as anofficial action or opinion of the American Gear Manufacturers Association.AbstractThis paper shows all the steps suggested to get an optimum (volume based) design for an
3、 epicyclic(planetary) transmission using plastic materials. The design was developed using the tooth proportionsproposed in ANSI/AGMA 1006-A97, Tooth Proportions for Plastic Gears and taking into account therecommendations given in AGMA 6023-A88, Design Manual for Enclosed Epicyclic Gear Drives andA
4、NSI/AGMA2101-C95,FundamentalRatingFactorsandCalculationMethodsforInvoluteSpurandHelicalGears taking into account the effect of changing the number of planets, the bending fatigue and contactstrength of the plastic materials, and the temperature effects on the size of the gears.The design procedure s
5、tarts with a preliminary analysis of the performance of the gears in a proposed (notoptimized)transmission,goingstepbysteptoanoptimumdesignforthegivenloadconditionsandexpectedminimum life.Copyright 2006American Gear Manufacturers Association500 Montgomery Street, Suite 350Alexandria, Virginia, 22314
6、October, 2006ISBN: 1-55589-885-81Detailed Procedure for the Optimum Design of an Epicyclic Transmission UsingPlastic GearsIsaias Regalado Ph. D. and Alfredo Hernndez, CIATEQIntroductionThe design of an epicyclic transmission involvessomespecialconsiderations likeassemblingpossi-bility, factorizing,
7、and idler planets considerationsnot necessary for a conventional parallel axlestransmission. If additionally we include the expan-sion characteristic of a plastic material, the designofaplasticepicyclictransmissionmaybecomeveryinvolving.Byfollowinganexamplesystematically,wewillcov-eralltheconsiderat
8、ionsneededfor thedesignofanoptimum epicyclic transmission.NomenclatureNSNumber of teeth in the sunNPNumber of teeth in the planetNRNumber of teeth in the rimSRotating speed of the sunPRotating speed of the planetRRotating speed of the rimCRotating speed of the carrierMg Overall gear ratioNP Number o
9、f equally distributed planetsCR Ratio operating/theoretical center distanceFWOperating face widthProblem DescriptionType of transmission PlanetaryGear ratio 4 (3%)Rim pitch diameter 70 mm (Max)Face width MinimizeSun input speed 600 RPMSun torque (Max.) 6.25 NmBending durability 2000 HrCarrier materi
10、al AluminumAmbient temperature 25COperating temperature 75CLubrication Oil lubricatedGeneral Design GuidelinesThe basic speed equations for an epicyclic trans-mission may be found in the literature 1:SNS= RNR+ CNS+ NR(1)SNS= CNS+ PNP(2)From eqn. 2, in the planetary:R= 0; Therefore,SC=NS+ NRNS(3)And
11、P=S CNSNP(4)Inaddition, for equally distributedplanets withstan-dard center distances, the assembly constraintsgiven by eqns. 5 and 6 must be accomplished:NR= NS+ 2 NP(5)NS+ NRNP= Integer (6)In practice, designs with non-standard center dis-tances and dropped tooth planets are commonlyused; in that
12、case, eqn. 5 may be ignored.MaterialsTheselectionofthetypeofplastictobeusedforthegears may be informationfor afull articleand is be-yond the scope of this paper. For this example, theselection of the material was based in the recom-mendations of the plastic vendor, who suggestedtwo possibilities:An
13、unfilled high molecular weight acetal copolymerfor maximum toughness (HMW POM) and a 25%glass reinforced acetal copolymer (GF POM). Oneimportant consideration is that for sliding operationlikegears,itisagoodpracticetocombinetwodiffer-ent materials or grades in both members. In thiscase, due to the a
14、pplication, it is also important tohaveaverystrongcage(rimgear);therefore,basedin their strength, the materials were assigned asfollows:2Rim GF POMPlanet HMW POMSun GF POMFromtheinformationprovidedbytheplasticvendor,the properties listed in Table 1 are relevant duringthe gear design.As may be observ
15、ed, the Youngs modulus andstrength of the plastics reduces with the tempera-tureinagreementwiththebehavior statedinAGMA2 and shown in Fig. 1. Therefore, for a conserva-tive calculation, we use the material properties atthe highest operating temperature highlighted inTable 1.Table 1. Relevant propert
16、ies of the materialsProperty MaterialHMW POM GF POMTemperature (deg C) 40 75 40 75Youngs Modulus(MPa)2414.5 1339.33 2700 1500Poissons Ratio 0.35 0.35 0.35 0.35Endurance limit 107cycles (MPa)61 34.92 55 47.97Contact stress dry(MPa)19 15.9 19 15.9Contact stress lubri-cated (MPa)63.2 52.5 63.2 52.5Line
17、ar thermal exp,coeff.1.2E-04 1.2E-04 3.0E-05 3.0E-05Aluminum L.T.E.C. 2.4E-05ProcedureTheAGMA 3 suggest thatfor plasticgears theba-sic rack must have the proportions shown in Fig. 2and summarized as follows:Addendum coefficient 1.33Dedendum coefficient 1.00Tip radius coefficient 0.43032 (full)Normal
18、 pressure angle 20Allthecalculationsshowninthispaperarebasedonthis tooth geometry with a non-undercut constrain.Additionally, although the minimum recommendedtooththicknessatthetipofthetoothis0.275M4,inorder toconsider as many options as possiblewith-out a pointed tooth, during the development of th
19、isstudy a minimum of 0.1M was used.Using eqn. 3 with the nominal gear ratio and themaximum pitch diameter of the rim, we get the op-tions given in Table 2 iterating from NS=11to30.Figure 1. Effect of strain rate and temperatureon stress-strain curves (from AGMA 2).Figure 2. AGMA PT basic rack (from
20、AGMA 2).3Table 2. Maximum allowable moduleNS11 12 13 14 15 16 17 18 19 20NR33 36 39 42 45 48 51 54 57 60NP11 12 13 14 15 16 17 18 19 20M 2.12 1.94 1.79 1.67 1.56 1.46 1.37 1.3 1.23 1.1721 22 23 24 25 26 27 28 29 3063 66 69 72 75 78 81 84 87 9021 22 23 24 25 26 27 28 29 301.11 1.06 1.01 0.97 0.93 0.9
21、 0.86 0.83 0.8 0.78Ithasbeenshown5thattheperformanceofagearset may be greatly improved by using nonstandardcenter distances and tooth proportions in the gearswithout changing the basic rack geometry; there-fore, a preliminary analysis about how to improvethe performance of the sun-planet and planet-
22、rimsets using non standard tooth proportions andcenter distances is recommended.Considerations for Stress CalculationsAlthough the scope of AGMA 2001-C956 is limit-edtometallicgears,thestresscalculations methodhas been used to determine the bending and con-tact stresses in the gears taking a unitary
23、 value forallthederatingfactorsandcalculatingthegeometryfactors according to AGMA 7. With this methodol-ogy, the author developed a computer program toexplore the nonstandard design space defined byCR and XP used to generate the contour plotsshown in this article.Preliminary AnalysisTaking now for e
24、xample the combination NS= 25,NR=75andNP=25fromTable2,andanalyzingtheperformanceforthesun-planet,set,wemaygettheplots shown from Fig.3 to Fig. 7.From Fig.3itis observedthat fromthecontactratiopoint of view, thebestoptionistheuseof areducedcenter distance(CR1)mustbeused.Fig. 4shows that in order to g
25、et an acceptablespe-cific sliding, we must use a standard or extendedcenter distance (CR 1).Fig. 5andFig. 6showthat for a better performancein pitting of the gear-set and bending of the pinion(sun),apositiveaddendumshiftingmust beappliedto the sun. Fig 7 on the other hand, shows that inorder to impr
26、ove the bending of the gear (planet), anegativetoolshiftingmustbegiventothesunandapositive one to the planet.Because the planet-rim set has a better confor-manceof their contactingsurfaces, andthe teethinan inner gear tend to be stronger to that of externalgears, we only have to analyze the bending
27、perfor-mance of the planet in the planet-rim set shown inFig. 8. From this figure, it may beseenthat inorderto improve the bending of the planet in the planet-rim set, a positive tool shifting in the planet isrequired.Figure 3. Performance for transverse contact ratio for the sun-planet set.4Figure
28、4. Performance for specific sliding for the sun-planet set.Figure 5. Performance for pitting for the sun-planet set.Figure 6. Performance for bending of the pinion (sun) for the sun-planet set.5Figure 7. Performance for bending in the gear (planet) for the sun-planet set.Figure 8. Performance for be
29、nding in the pinion (planet) for the planet-rim set.BasedintheconclusionsdrawnfromFig.3toFig.8,we conclude that in order to improve the perfor-mance of the sun-planet set and the planet in theplanet-rim set, we must use an extended centerdistance in the sun-planet and a positive tool shift-inginthep
30、lanet.Todothis,wereducethenumberofteeth in the planet from the theoretically calculatedand make the sun-planet set to operate in an ex-tendedcenter distancewhile theplanet-rim set op-erates at a standard one.Number of PlanetsFortheselectionofthenumberofteethandplanets,AGMA 8 recommends the applicati
31、on of the hunt-ingtoothandnon-factorizingcriteria. Thisstandardalso state that due to the idler like operation of theplanets, for analyzing their bending strength the al-lowable bending stress number must be multipliedby 0.7. Therefore, this must be taken into accountfor a balanced performance of th
32、e sun and planet,Prior totheapplicationoftheaboveconsiderations,wemustdeterminethemaximumnumberofplanetsvalid for the desired gear ratio. A good approxima-tion is obtained if we iterate for the number of plan-ets andconsideringaregular polygonwithas manysidesasnumberofplanetsinscribedinacircleofdi-a
33、meterNS+ NR2. If the length of the side in thispolygon is bigger than NP+ 2, then the number ofplanets may be valid. In our example, lets takeNS= 20, NR= 60 and NP= 20; then the diameter of6thecircletoinscribethepolygonsisD= 40,iteratingforthenumberofplanets,wefoundthatamaximumoffiveplanetsmightbeus
34、edasshowninFig9.Notethat at least two planets may always be used.Figure 9. Determination of maximum numberof planetsIteratingnowfortheoptionsgiveninTable2andap-plyingthehuntingandfactorizingcriteria,as wellastheconstraint of anextended center distance inthesun-planet set, we get the number of option
35、s givenin Table 3.Table 3. Available options with differentconstrainsNumber of plan-tTotal Optionsets1to5 4or5Free 157 125Hunt Sun 93 67Hunt Rim 82 60Hunt Both 64 48N.F. Sun 124 93N.F. Rim 124 93Constrained 52 40Note that the options for four or five planets arehighlighted;thisisbecausethesewillgive
36、thesmall-er volume for the transmission as will be shown inthe following section.Effect of the Number of PlanetsApplyingeqns. 1 being in thiscase a 20% of reduction, with a proportional reduc-tion in volume and weight. Therefore, we may con-7clude that in order to reduce the total volume of aplaneta
37、ry transmission we must use as many plan-ets as we could.Somedesignershavetheideathatthemoreplanetswe have, the more friction losses, but here we canprove this is not true. First, we know the frictionforce is given by eqn. 7, and that in the case ofgears, the normal force WNis given by eqn. 8.f = m
38、N (7)WN=WTcos ()(8)Now, if weconsider thetangential loadis carriedbyNP planets, then the tangential force in each planetwill beWTNPand therefore the normal load in each ofthe planets will beWNNPand then the total frictionforce will be f =m WNNP NP independent of thenumber of planets.Please notethat
39、theanalysis has been donebasedonly in the friction losses. However, in an oil lubri-cated transmission, the temperature increase duetotheoilmovementmay behighiftheoillevelisnotadequate,andathermalstudymaybeneededlead-ing in some cases to the conclusion that the highernumber of planets may not be pra
40、ctical.Goingbacktoourexample, thevaliditerationsfromthatindicatedinTable3areshowninTable7,wherea rounding of the modulus to a commercial valuehas been performed.Now we must calculate the required face width forevery sun-planet combination in Table 7 based onthe design stresses shown in Table 4 and T
41、able 5.The results of thevalid options after eliminatingun-dercut or pointedteethsolutions areshowninTable8.The procedure for getting the required face widthwas to optimize the sun-planet set for bending intheplanet andthenmakinganadjustmentof there-quired face width for the planet-rim set, being ne
42、c-essary from 7% to 75% increment in face width inthe planet for the planet-rim set compared to thesun-planet set. This is because the rotating speedintheplanet is 450rpm as has beenshowninTable4andTable5. Therefore, therequiredface widthiscontrolled by the planet-rim set.Table 7. Basic data for the
43、 available options.During the development of the calculations, it wasobserved that the limiting condition is the bendingstressintheplanetcalculatedfortheplanet-rimset;infact,thebendingstressofthesunandrimarewellbelow its design stress as may be seen in Fig. 11and Fig. 12. Please note that we may get
44、 a morebalanced bending stress between the sun and theplanet,butitwouldimplyanextremeundercutinthesun or a pointed tooth in the planet, which in thiscase was not permitted.8Table 8. Required face width, stesses andestimated volume.Fig. 10 shows a plot of the total volume; it is ob-servedthat ingener
45、al, for allthecases withcompa-rablevalueofnormalmoduleandCR,thevolumeforatransmissionwith4planetsisbiggertothatof fiveplanets. It is alsoobservedthat thebigger thevalueof CR, the smaller required face width.It may be observedthat for a normalmodule of 1.0,there are some cases where the total volume
46、issmaller with four planets than with five planets; thisoverlap corresponds to four planet transmissionswith CR 1 in the sun-planet set versus five planettransmissions withCR = 1. This confirm the needofanextendedcenterdistanceinthesun-planetsetinorder to improve the performance of the transmis-sion
47、 and reduce the required face width.Figure 10. Total volume for all the iterationsFigure 11. Bending stress in the sun for allthe iterationsFigure 12. Bending stress in the rim for allthe iterationsFig. 11 shows the sun bending stresses versusmoduleforalltheiterations.Itmaybeobservedthatin all the c
48、ases, the bending stress in the sun withfour planets is smaller than that for five planets and9well below the design limit. In other words, the sunwill be more over-designed with four planets thanwith five planets.As may be seen in Fig. 13 and Fig. 14, the contactstress inthesun-planetset wasalways
49、higherthanin the planet-rim set (from 15% to 90%).Figure 13. Contact stress in the sun-planetset for all the iterations.Figure 14. Contact stress in the planet-rimset for all the iterations.It is observed that the minimum volume is obtainedwiththefifth solutionofTable8;itisalsoevidentthetrend to increase the volume as the iterationsprogress.Please note that In this case, all the possible itera-tions are shown for the sake of verification of themethod; however, in a real problem, the
