Planetary Gear Drives

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Drives

'References ar e tabulated a t end of ar tic le ,

Basic Considerations: Certain general conceptsof gear performance will be useful in analysis ofplanetary gear systems and perhaps should be re-viewed.

of equilibrium of moments f ir st suggested by Mas-sot.1

Thi s a rt ic le presents a new approach to theproblem and offers a general method of solution,utilizing th e concept of an "equivalent" conven-tional gear train in conjunction with th e particularkinematic characteristics of planetary systems.Several representative planetary gear arrangements are analyzed to illustrate application ofth e principles discussed.

.''.

Planeta ,y GearBy Eugene I. RadzimovskyAssociate Professor of Mechanical EngineeringUniversity of II1inoisUrbana, 111.

A simplified approach tordetermining power losses andefficiency o f

I N A PLANETARY gear drive, th e power circulating within th e system elements may differappreciably from that being t ransmit ted. Thisunique characteristic, which is a result of th erelative motion of the gear members, i s presentin some degree in al l planetary g ea r t ra in s andha s a significant effect on th e performance andefficiency of sys tems transmitting power continuously. Power losses become a function of th epower developed in th e gearing ra ther than of th eactual power t ran sm it ted t hroug h the train.

Several methods fo r determination of powerlosses and efficiencies o f p la ne ta ry ge aring havebeen propo:sed, and expressions for a number o fspecific gear arrangements have been developed.Most of the se ana lyse s ar e based on th e principle

Nomenclatured = Pitch diameter of gear , in.e = Efficiency of planetary gear pairet =Overa ll e ff ic iency of planetary gea r t ra inF = Tangential force acting at th e pitch cir-cle of th e driving gear, IbL = Tooth mesh power loss in a planetary gear

pai r, f t- lb pe r se cL t = Total tooth mesh power loss in a planetarygear train, ft-Ib pe r secJ=T oo th m esh loss in a conventional gear

pair, f t- lb pe r secIt =Total t oo th mes h loss in a conventional

gear train, ft-Ib pe r secm = Speed ratio of gear pair

m p =Overall speed ratio of planetary gear trainn = Number o f meshing pairs in a gear trainPi, Po = Input and outpu t power, respectively, it-lb

pe r secR = Planetary velocity ratio defined by Equa-tion 6v = Pitch-line velocity of gear pai r, fps

V e=Velocity of engagement of a gear pair, fpsA=Ratio of tooth mesh loss t o i nput power ina gear pair

7/ =Efficiency of conventional gear pair7/t =Overall efficiency of conventional gear

train'" =Angular velocity, radians pe r sec

February 9, 1956 101


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Principal contributing sources of power losses

The tooth mesh loss in a pair of gears withfixed centers is

where subscript s 1 and 2 identify individualmeshing pairs in the train. I f = =~ = ~ , t h e n

in g ea r tr ai ns a re as follows:1. Sliding friction between meshing tooth surfaces.2. Oi l churning.3. Friction in shaft support bearings.Tooth mesh loss, as the first type will be canedhere, depends upon th e geometry of tooth profiles

and upon a "mean coefficient of friction" betweenthe mating sur faces. Thi s mean coefficient of friction is a function of the acting tooth forces, gearmaterial, contact surface condition, and the velocity of relative motion of these surfaces in thedifferent phases of engagement. The last factoris also a function of geometry of th e gearing.

Although analytical expressions have been developed fo r determination of tooth mesh loss,the relationships involved are rather complex andno t always reliable. They appear to be bestsuited fo r comparative evaluation of differentgear-tooth forms r at he r th an fo r calculation of

(3 )

(1 )

(2 )

(4 )

where symbols are defined in Nomenclature. Theefficiency of a drive system consisting of one suchpair of gears is

Po PI - l'It == ." = -- =--- = 1 - dP, PIFo r a train with more than one pair of gears,

'I = (1 - d ) " = .,,"

1.00.8.6.6 0.4 0.2 0 0.2 0.4...... - - - - - Internal gears IExternal gears - - - - - I . ~

Ratio of Numbers of TeetH (pinion/ geor)

0.8

, , ..... ..... - - -\ \ \ ',\l'\" "- ~ "'" ~ ~ ::: ::::::r- - -- I--- t - - I- --- t - - - t - - - r---t--- --'0. I - - - - ' -, " \ \ ' \ '" "- ...... '" '""io;I - - r-- I-- - r - - ""';;;;;;;:i'-.. i " ' - - t - - r---t - - r--i'.. '- r-- r-- '-1\ \ \ 1\ '" '" "- I':::: ~ --.. --r-- ~f'... !""--- r-..... I--.. r-- r-.. r--t - -\ \ \ " '" '" '"-......r-..... ......... r--... ........... r--...1\ f\ "- , t--.....\ f\ \ \ ' \ r" ............ i'--- .............. ~ -...... ~\. ~ ~ "-\ \ \ ' \ '" ........ r---..... ......f\ t\. l"- t--... ~\ \ ' \ .......... "" ~ f'-.." \. ............ .....1\ \ 1\ \ "\ "'" .........."'""\ \ \ , '" .......... .......... '-..- "\ 1\ 1\ "-'" ~\ \ \ '" '" .......... r--....\ \ ~ ~ ......'"1\ 1\ \ ~ '" "-\\ \ '\ ~ ~ r--..\ f\ ~ " ~, \ '\ '" " ~\80901.0

70

Ql.cEZ 60

.s::.+Ql~ 50-c:.2. 40c:

30

20

10

Fig. 1 - Efficiency losses f rom sliding fr ict ion between a pairof full.depth 20degree sr,ur gears of British standard form.Losses vary negligibly for different tooth forms and pressure anglesEfficiency Loss (per cent)

0.1 0.2 0.30.4 0.50.6 0.8 1.0 1.2 1.4 L6 1.8 2.0 2.2 2.4 2.6

102MACHINE D E S I G ~


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Fig. 2-Comparlsoo ofequivalent gear paksin, il , simplt planetarysystem 'and b, cooven-doDal gear train

103

(bJ

The essential difference between Equations 1 and5 is in th e magnitude of the velocity of engagement. In a conventional gear train with fixedaxes of gear rotation, v. is equal to pitch-line velocity v of th e gear, and th e maximum limit of th eproduct Fv is equal to th e input power o f the t ra in .In a p lane ta ry gea r t ra in , however, th e velocityof engagement i s a ffec ted by th e relative motionof the planet cage. Thus, velocity of engagement

PLANETARY GEAR DRIVES

6. Fo r practical purposes, th e loss in a pairof spur gea rs m ay be assumed to be proportionalto th e product of the pitch-line velocity and th etangential force acting at th e pitch circle of th edriving gear.

Merritt2 ha s developed a chart., Fig. 1, whichgives th e value of A f or ext erna l and int erna l gea rpairs as a funct ion of th e speed ratio of th e twogears. In this chart., th e abscissas represent th eratios between th e numbers of teeth on pairs ofgea rs ; t he numera to r of th e ra tio is always th enumber of teeth on the smaller gear. Range ofchart values includes pinions varying in size from10 to 90 teeth. Plots are based on the slidingf rict ion between a pair of spur gears of BritishS tandard form having 20-degree pressure angleand ful l depth . A mean coefficient of friction equalto 0.08 i s a ssumed . Fig. 1 may be used fo r a firstapproximation in calculation of efficiency of geartrains, i f more pertinent data are no t available.The chart values inc lude loss for ball or rollerbearings.

(6 )

(o j

February 9, 1956

L = Ii FV e .

Too th mesh loss of a pair of gears in a planeta ry tr ai n depends upon th e same factors as fora pair of gea rs w ith f ixed centers . Here. also, itcan be a ssumed that th e loss i n engagemen t isproportional to th e product of the tangential forceat th e pitch circle of th e driving gear and th evelocity of engagement :

- - - -Equ iva l en t System Concept-------

absolute values of gear losses. Available experimental data offer a more reliable source of info rmat ion on tooth mesh power losses in gearsthan do th e theoretical expressions. The following conclusions can be made:

1. Tooth mesh losses in intlilrnal gears are somewhat less than those in external gears.2. Pitch-line velocity has small influence on gearefficiency.

3. With the exception of light load conditions,It may be assumed that th e ratio, A, of toothmesh loss to input power does no t depend uponthis power.4. The difference in losses fo r sever al s tand ard tooth forms in common use Is so small thatthe same values of efficiency may be assumed fo rvarious tooth forms.5. Losses from oil ohurning ar e independent ofthe load t ransmi tted , but depend upon gear velocities, gear--drive design, and viscosity of lubricants. I t seems that these losses can be neglectedfor gear trains which transmit relatively largepower, bu t must be taken in to ac count in th etotal efficiency of high-velocity, low-power gearsystems.


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is not equal to pitch line velocity v; and prodt Fv . is no t equal to the input power at the

gear of th e pair.For comparison, two gear pairs will be conFig. 2: one in a planetary gear train, th e

in a conventional train. Assume that th esize, and other characteristics of th egears are the same. Tangential

F acti ng on both gear pairs ar e equal, ande pitch-line, or absolute, velocIty of the sun

in the planetary train is the same as thatr the corresponding gear in the conventionalThe ratio of tooth mesh losses i n the se "equivent" gear pairs isLl

AFv.!>.Fv v R (6 )

where R is defined here as th e planetary velocityratio. Therefore, th e t oo th mesh loss in a Pairof gears in a planetary train isL = lR == A FvR = Pi A R = PiR (1 - 1/) ..... (7 )where l is the loss in an equivalent gear pai r withfixed centers and the same input power as theplanetary pair, Fig. 2.The efficiency of t hi s pl an etary ge ar pair isgiven by

_ Po _ Pi - Le- - - - =1-AR=1-R(1-lI) (8 )Pi P i

This same method may be applied fo r determin_ing losses in any planetary gear pair and, consequently, for determining the efficiency of theentire planetary train.

Planetary Train--------

lola (1 __ )l o l ~ ~ 1RaB =-- = -------- ' '--- = 1 - -- . _. . (11)lola lola m p

. (10)ola (1 __ ) .m p

This result means that the velocity of engagementof gear A in the planetary t r a i ~ J Fig. 3a, is RABtimes the pitch-line velocity of this gear, and thepower developed by gear A is R AB times the powerat input shaft D.

With the planet cage stopped and plane t gearB serving as an idler, Fig. 3b, al l gears in thetrain have th e same pitch-line velocity. Therefore,R BO ' the p lane ta ry velocity ratio fo r gears B and0, is equal to RaB , and RaB = R BO = R is a constant value fo r a given planetary train.To derive an expression fo r the efficiency ofthe planetary train, it is important to know whichgear in th e t ra in with the stopped planet cage isth e driver. The fact that gear A is the driver inth e planetary train, Fig. 3a, does not necessarilymean that gear A' has th e same function in thestopped system, Fig. 3b.

In any gear train, an external torque appliedto the input shaft acts in the direction of rotationof t hi s sha ft . Therefore, th e p roduct o f torqueand angular velocity at the input shaft is alwayspositive. The externa l resi st ing torque act ing onthe outpu t shaf t, however, always acts in a di rac-

Velocity "'A f , which is th e pitch-line velocity ofgear A', is thus also the velocity of engagementof gear A in Fig. 3a. Since the value of mp forthis p lane ta ry train is always great er t han one,th e value of "'a' is always positive i f "'a is positive.The plane ta ry velocity ratio, RaB, for gears Aand B is

(9 )

subscripts identify the corresponding sysm elements. The value of mp is always positivethis a rrangement , and input and output shaft s

rotate in the same direction.Assume now that th e planet cage is "stopped"adding to the entire system an additional an

gular velocity, - "'H, equal in absolute value tothe angular velocity of th e output shaft, but

in direction. The relative motions ofl members of the planetary train ar e unchanged.

planet gear B now becomes an idler,and th e entire train may be considered as aconventional gear train with fixed axes of rotaion. Since th e external torque applied to inputhaft does not change, th e tangential forces acting on the gears are the same as those in th eactua l p lane ta ry t ra in . The only change is in th epitch-line velocities of th e gears; they are nowequal to th e velocities of engagement of the gearsiu the planetary train. This train with "stopped"planet cage is depicted in Fig. 3b.f In Fig. 3b, th e absolute angular velocity, "'A',o gear A' is given by

104

This method may be best illustrated by an exConsider th e simple planetary gear train,

2a, which is represented schematically in3a. This system is one of the most compact

gear arrangements having a wide rangef application possibilities in design. Design of

actual planetary train of t his t ype is shownFig. 4. In this train, Fig. 3a, planet gear Bages simultaneously with sun gear A, whichkeyed to inpu t shaft D, and with fixed internal

(ring) gear O. Planet gear B rotates freelyit s own axis on a shaft mounted to th ecage, G, which is keyed to and drives out

t shaft H.The speed ratio of this train is

MACHINE DESIGN


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(14)

(15)

Fig. 3--Simple planetary gear system showing, a, s c h ~ t i & at.

I ; ' a n g e m e ~ t of ~ s t e me1ements,and, b, eC:(uiv-alent system obtalDedby "stopping" theplanet cage. Prime not;ttions identify cor-responding members

Pi - Pi R (1 - 'It)Pi

0 '

(b)


et = 1 - (1 - 'It) ( 1 __1 _ ). mp1 - R (1 - 'It) .

From Equations 11 and 14

Ring gea r C is a lways larger than sun gear Aso that th e value of R fo r th is p lan eta ry trainis always smaller than one but la rg er than 0.5.The tooth mesh losses in this train, then, aresmaller than, but not less than half, the toothlosses in a conventional train consisting of th esame gea rs . Consequently, th e efficiency of thist ype o f p lane ta ry train is a lways higher t ha n theefficiency of a corresponding conventional trainbu t lower than th e efficiency of one pair of gearswith fixed centers and th e same unit tooth meshloss ratio, 6..

General ly speaking, t ll is efficiency may be considered as highly satisfactory fo r plane ta ry gear,(13)

(12)

Ring geor,C(fixed)PIonet gear 8Sun gear. AInput ~ t t . O H ' ~

777777T

(a)

Output shoft,H

The efficiency of a planetary train of this type is

where tj t = th e overall efficiency of a conventionalgear train consisting of th e same gears as th eplanetary train. Therefore, in th e planetary trainand it s equiva lent, wh ich have equal losses,

tion opposite to that of shaft rotation. Therefore,tlle p roduct o f externa l torque and angular velocity fo r th e output shaft is always negative.Fo r t he p re sent case, Fig. 3, le t it be assumedtbat OJD) wD', and the external torque acting onsbaft D' all have positive values. Thus, th e prod-uct of t orque and angul ar velocity' at shaft D' ispositive, and gear A' is the d river. In this in-stance, th e position of th e driving gear is th esarne in both trains, Figs. 3a and b.The power, PtJ.B', developed between gears A'and B' in th e equivalent train, Fig. 3b) is : PdB ' =

pJ?-. Equal power is developed by gears A andB, Fig. 3a. I f t he too th mesh loss ratio fo r gears,Af and B' is 6.1> and the fo r gears B' and C' is

th e power delivered at output shaft H' is

Fig. 4 - T y p i c a l . e l ~ e tary gear translWSSl.ondesign employing thebas i c auangementshown in Fig. 3a

February 9, 1956 105


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systems. However, th is type of train can be designed only for relat ively small speed rat ios.

W1) _ Wa _ dA- - - -1+--"'H


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rotate in the same direction. If mp i s nega! l ~ t h e Y revolve in opposite d.irections. The mostt : J v e ~ s t i n g peculiarity of this train is that a speeditl:o of any magni tude. even inttnity ("'0 = 0) ,(f/o theoretically be obtained.,cn;l1e efficiency of this train may be determinedthe methods discussed previollsly. Assume thatbY' t shaft H revolves in a clockwise (positive)iIlPu .. etion and dJidJ/dBdK< l so that mp>l and IS~ t i v e . To stop planet cage GJ l et t he ent ir e syspo be revolved at an angular velocity, -"'rr With=lanet cage stopped, th e absolute angular velocitY of gear K becomes


of planetary train and appears, at first, to be adriven member. Consequently, with th e planetcage fixed, gear A' becomes the driven member.Now the expression fo r efficiency may be derived.I f th e power at output shaft H of th e planetarytrain is P OJ th e power developed in the engagement of gears J and K is PoR. Since gear K' isconsidered as th e driver, the tooth mesh loss inthis train is

(22)

The value of R is 'negative. This result meansthat th e motion of gear K relative to gear J inthi s p lane ta ry train is opposite in direction tothe absolute velocity of gear K. However, thedirection of relative rotation does no t influencethe magnitude of too th mesh loss in t he gea r pair.Therefore, th e absolute value of R must be usedin the calculation of train losses and efficiency,Introducing this va lue in to Equation 23 gives

Planetary vel

(241

(23)

('/'IL-p - 1)

11 + R(1 - 'It>::::------

Efficiency of th e train is'P oe, == - - - - - - - -o + P.,R(1 - 'It)_ WD (1- _1 )

" ' / / 1npR = - - == - - - - - - - -WK(21)

WD, :;;:; "'/1 + ( - "'D) =-- - "' 0 ==. , j { m p

For the assumed conditions, th e value of wK' isnegative. Actua lly, in th e plane ta ry t ra in , Fig.7a J gear K has a positive direction of x:otation and,being connected to the outpu t shaft, 18 loaded byan external torque act ing in a direction oppositeto its motion. Thus, th e external torque actingon shaft H and gea rs K and K' is negative. Thel!roduct of negative angular velocity and negativetorque is positive. Therefore, gear K' is th e driverin the equivalent train obtained by stopping theplanet cage.This case is an example of an interesting peculiarity i n analysis . The gear that must be considered as the driver for analysis of trainformance is actually connected to the driven shaft

p ~ 5-Simple planetary Bear train inwhich the sun F is the fixed member. This tratn is a modificationof the arrangement shown in Fig. 3a Fig. 6 - Compound planetarygear train for large speed ratios

Ii'ebruary 9, 1956

Ring gear,C(tiKed I

107

RIng gear,C

Planet gear): iPlanet cage, G


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'GO 20020

o

80

""'Speed Ratio

Speed Ratia

fb)-2

/' I---';:;f"\1\ I\

JI \I \I 1\,

./

'"...V I"'--t-l- f- 10 - r - t---I--....- ~ -IGO -120 -8 0 -4 0

80

(bJ

80..a.~ Input shaft,D f 40&:;\&\>\"

~ ' ' ' ' ' L;j~ S u n geor,4 ~(fixedl(0 )

Fig. 8--Differential planetary gear.. system showing, a, arrangementof system elements and, b, relationship between efficiency andspeed ratio. This train is basically the same as the one shownin Fig. 7a e.'Ccept that direction of power f low has been reversed

Fig. 7-Basic differential planetary gear system showing, a, arrangementof system elements and, b, relationship between efficiency and speed ratio

100Plonel llear,S Plane! gear."

80

Planet coge, G :;; 600.OU!P@'i;gJ "-Input shoft.H uc//////// .. 40A /J/ / / /7 wSun gear, K 1\0Sun g)!ar,A(Iixed)

0-2 5

(o)

1et = ----------1 + (1 - lit) (r n p - 1)1 . .. . . . . . . (25)

In the planetary train, Fig. 7a, t he angular velocity of the gear K i s now nega ti ve, and the external r es is ting torque act ing on outpu t sha ft His positive.

When th e planet cage is stopped, th e angularvelocity of gear K becomes

MACHINE DESIGf'/

WK ' = - WK + ( - WD) = - (W K + WD) . _ ' . (26)

Therefore, th e efficiency of thi s t ra in , considering'tooth mesh loss only, is

which is a negative value. The product of externalt orque act ing on this gear and angular velocityi s n egat iv e. Therefor e, gear K' is a driven member in the equivalent train with a stopped planetcage. Gear A' is the driver in this case.

The power developed by gear K is PoR. Thepower of engagement of the driving gear A' isPA.B' = PoR/T/t. Tooth mesh los s in this train is

.................. (27)1L t = -- Po R (1 - 'Ill'It

Equation 25 shows that as th e speed ratio ofthis train increases, th e absolute value of Rand,consequently, th e tooth mesh losses in th e traininc rease, whi le efficiency decl'eases. For trainsof this type, having a large speed ratio, th e velocities of gear engagement ar e relat ively highand the efficiency relatively low. The power developed at th e gears in such trains is R timeshigher than in a conventional train having th esame size gears and the same speed and powerat th e output shaft.

Fo r th e train shown in Fig. 7a, assume that1


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(35)

(31)

..... (34\

o when

........... (33)'I t

'I t

'I t - (1 - '1t)R

'I t =1- '7t (_1_ - 1)'I1tp

'I t - (1 - ' I t) ( _ 1_ - 1 )'1ltp

-WK(_l - I) .1np

P,-LPi

I t is interest ing to note that et

and this mechanism becomes self-locking.Consider now a train of this type which has ratil}

dAdJ/dBdK > 1. Such a mechanism will increasespeed and reverse th e direction of ro tation when1 < dAdJ/dBdK < 2. I t will reduce speed and reverse th e direction of rotat ion if dAdJ/dBdK > 2.This latter case will be assumed fo r illustration .In this reduction drive, m p is negative. Angularvelocity of the planet cage is "'G = "'K/mp , andth e angular velocity of gear K with th e planetcage stopped becomesWK ' = WK + ( - ~ ) = WK ( 1- _1_) . .. (36)'1ltp '1ltp

Introducing th e absolute value of R into Equation 33 gives

Eff iciency of th e train may be expressed by

Fo r th e assumed conditions, th e value of "'K' isnegative. Therefore, th e product of positive torqueand negative angular ve loci ty i s negat ive , andgear K' will be a driven member in th e equivalent train with a stopped planet cage. Gear A'will thus be th e driver.If gea r A' is th e driver, PAB' = PiR/'Y)t and th e

t o o ~ h mesh loss in the planetary train is. 1L t == -- Pi R (1 - ' It) (32)'I t


greater than one i s theoretically impossible fo r - __t hi s t ra in , because it requires th e value of dAdJ/dBdK to be less than zero.To "stop" p lane t cage G) it will be necessary

to revolve th e en tir e system at an angular velocity, - "'G' Then, th e angular velocity of gearK becomes

where

WK"'K ' = "'K + (- wo) = WK - -- =1np

Because m p in this case has a negative value,"'K' is positive. The product of torque and velocity

(28)'I t'I t + ( l - ' 1 t )R

. . . . . . . . . . . . . . . . . . . . . . . . . , . (30)

The value of R for thi s train is given byWK - (WK + "'D)_ = = - (1 - 1n p ) (29)[l, =::::: WK ' "'K

et=: : : : : - 1Po + --PoR (1- 'It>

'I t

The value of m p i s po si ti ve or negative depending upon th e value of dAdJ/dBdK. Assume inputshaft H) connected to gear K) revolves clockwise.The externa l torque applied to this shaft actsin a positive direction. Assume also, that dAdJ/dBdK < 1 so that 1 > mp > O. This arrangementwill provide a speed increase without reversingth e direction of rotation. A positive speed ratio

Reversed Power Flow: If , in th e t ra in depictedin Fig. 7a, th e sun gear K is made th e inputmember and the shaft connected to the planetcage becomes th e output member, a differentmechanism will be obtained, Fig. 8a.The speed ratio of this modified train is

The absolute value of R must be used in Equation28 fo r efficiency calculations.Sample Calculations: A planetary differential. '

train. Fig. 7a) ha s th e following design data:djdJ/dBdK = 1.025 and 'Y)t = 0.97. From Equat ion 20, m p = - 40; from Equation 29, R = - 41;and from Equation 28, using th e absolute valueof R = 41, et = 0.4409. In a planetary gear trainof this type with Po = 1 hp, th e power developed by gears J and K is more than 41 hp whilethat developed by gears A and B is 41/0.97 =42.26 hp. These conditions must be taken intoaccount when designing gea rs f or th e planetarytrain.The relat ionship between efficiency and speed

ratio fo r th e differential pl an et ar y g ea r tra in ,Fig. 7a) is shown in Fig. 7b. The lef t-hand portion of th e curve represents th e cases where speedratio is negative, and th e right-hand port ion thosewith positive speed ra tio s. When m p = +1, th eentire system i s r evolving as one unit, withoutrelat ive motion of i ts p arts , and 6 t = 1 or 100per cent. Actually, however, such a ratio is no tpossible in practice because it requires thatd,idJ/dBdK = O. When, mp = - 1, th e absolutevalue of R = 2, and th e losses in t hi s t ra in ar eabou t twice as large as in a conventional trainconsisting of th e same gears.The chart indicates that efficiency of this typeof planetary train decreases rapidly with the in

creasing speed ratio, and that gears in trainshaving large speed ratios operate under extremely'severe service conditions. Apparently, trains ofthis type ar e most suitable fo r applications wherespeed ratio or power transmitted are relativelylow.

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Fig. 9-Medi.fied differendalsystem in which the sun gearsare internal gears

K is connected to .the output shaft . P lane tG is connected to inpu t shaft D. catfFor this train, as in th e arrangement ShowFig. 7a.. speed ratio is given py Equation 20. ' l ' ~ itt

two different ial systems also have th e sarne \ ~ e .pression.a fo r 6, and R. Here again, two pr ' e)(,.ll'l pa l cases must be considered: (1 ) diJ.dJ/dlJ{l CI.1 and (2) dA,dJ/dndK > 1. In th e first case "K' is the d riv er i n th e equivalent train Withp lanet c ag e stopped, and Equat ions 23, 24 e25 must be used for efficiency calculations :? d "Ot'th e second case, gear A' becomes th e drivel'Eguatioris 28 and 29 are pertinent. andThe only difference between th e two trainrangements, Figs. 7a and 9, lies in th e values flU\.or6.1, 6.2. and "It of th e train. The l os se s in internalgears, i f al l other factors remain th e same. aresomewhat lower than for comparable externalgear pairs. Therefore, th e value of "It may be Con-siderably smaller fo r th e internal gear arrange_ment, Fig. 9, than for the external gear syste1ll,Fig. 7a. Total efficiency fo r a planetary train ofthis type will thus be h igher f or a given speedratio if interna l gears ar e used.

REFERENCE81. M. P. :MU5.ol-"ln(luenco du Frot tement sur la FunCtionnernen tdes Trains ~ p l c ) ' c l o l d a u " , " 8001"t.II

Planetary Gear Drives

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