Designing

COMPO.UNDE'PICYCLIfor ...~ximuD.leHi~i.ncy. at lIi~;'.v.'ocity

A N IMPORTANT basic feature of epicyclicgear systems is the high velocity' ratiothat can be achieved with only a small

number of gears. With certain types' of "com-pound" arrangements, for example, ratios of 2000:1or greater are easily possible. Of particular inter-est in design is the simple four-gear co'mpoundepicyclic train, Fig. 1, which can be utilized toprovide high velocity ratios with reasonably smallnumbers of teeth. However, as velocity ratio in thetrain increases, efficiency may be substantiallyreduced. _ .

In this article, attention will be devoted to some-of the practical possibilities and limitations of thefour-gear epicyclic train in drive applications. A

NomenelatureF = Gear tooth load, or normal force reaction

at point of contact of two gears, IbhIe =Working depth of gear teeth, in.

ma = Velocity ratio of gear trainN =Number of teeth in gearP = Power, hp

Pi> Po = Input and output power, respectively, hpp = Circular pitch of gear teeth, in.

rb = Base radius of gear, in.s =Displacement, in.

Sp = Displacement of point P, in.T = Torque applied to gear, Ib-in.

W = Work, in-lbWI' = Work lost in friction, in.-lb

ti.t = Elapsed time, secf = Efficiency8 =Angular displacement of gear relative to

plane of gear axis, radiansp. = Coefficient of friction

Internal gear 0,....------:"...-:::::...-.,.

Planet gear C

Balance weightin carrier

._----_. -1--1--l-

Internal gear8

Planet gear A

101

'(200 - 2)p 198 Pd D = ----

7T' 11"

To equalize these center distances, the blankdiameter of gear A could be increased to dA =103 p/-rr, making the common center distance lAB =lCD = 50.5 p/7r. Alternatively, the blank diameterof gear C might be decreased to de = 100 p/7r'giving a common center distance of lCD = lAB =51 p/7r.

The teeth would be generated by standard cut-ters sunk into the blanks to slightly more thanfull standard depth to provide some backlash whenthese gears are meshed at the designed center dis-tance.

Power loss by tooth friction in a pair of con-ventionally mounted spur gears is usually lessthan one per cent of the transmitted power. Cor-responding loss in a pinion and internal gear com-bination is even less. Thus, it is not immediatelyobvious that the efficiency of a train of the typeshown in Fig. 1 can be extremely low. The basicreason for this condition is that tooth load is highcorresponding to the output torque delivered atthe low speed of the output shaft, while meshingspeed of the teeth is high due to the high speedof the input shaft.

Where a high velocity ratio is developed bysimple trains of gears in series, the heavy toothloads are confined to the low-speed gears, whilethe high-speed gears have small tooth loads.

and for gears B and A)lAB = ( 200 - 102 + 2 ) _P_ = 51.0 P

2 'iT 'iT

Center distance lCD for gears C and D) Fig. 1,with ,,:orking depth equal to 2p/-rr would be

( 198 - 101 + 2 ) _p_ = 50.5 PlOD = 2 'if rrr(2)

Fig. I-Four-gear compound epi-cyclic' train which can be used toobtain high velocity ratios.

d A =(100 + 2)p 102 P

----

'iT 'iT

dB =(202 - 2)p 200 P

----

'iT 'iT

do =(99 + 2)p 101 P

----

'iT 'TT

AR TRAINS

NA 2=--------

bc NAbNJ1c+------

n n

rna ~ 200(100) =10,000200(100) -202(99)

41 ~re, the internal gears are about twice as large110 /~eter as the planetary gears and there is

'l'blfflCulty in using a standard tooth form.Of t e Usual blank diameters, d, for these numberstOOt~e~h Would be, since addendum of a full-depth

1$ P/To,

If () == 1 and b n (n must be a whole numbergreater than 1), then from Equation 2, ma = N A 2orNJ1 = y(ma ).

fore generally, if b = en (the denominator of/I expressed as a fraction must be a factor of N A)'from Equation 2, ma = N A 2/e2 or N A = cy(ma)'

Suppose, for example, that a velocity ratio of"'0 == 10,000 is desired in such a train. If e = 1,It::::: 2 and b = 2, N A = y(10,000) = 100. Solv-~g for the other gears of the train gives: N D =00, NB == 202 and N C = 99. From Equation 1,

as a check on solution accuracy,

bere n is any factor that makes product nNA whole number, and band e are whole numbers.Then, from Equation 1,

From Equation 11 then, for conventional valuesof

1103

COMPOUND EPICYCLIC GEAR TRAINS

Design Example: To illustrate use of the equa-tions, it will be assumed that a compound epicyclictrain of the type shown in Fig. 1 is to be designedto provide a velocity ratio of 256 :1. As in theprevious discussion, factors 0) n} and b will be as-signed values of c = 1, n = 2 and b = 2. FromEquation 2, N A = y(256) = 16. Solving for theother gears of the train, based on the assignedfactor values, gives: N D = 2(16) = 32; N B =2(16) + 2 = 34; and No = 16 - 1 = 15. FromEquation 1, as a cheek on these calculations,ma = 256. Total tooth friction loss is, fromEquation 17: W /-'T = 3.5 WE or, in terms of power,P/-,T = 3.5 Po. Efficiency of the gear train, con-sidering only tooth friction losses, would be, fromEquation 18, e = 0.24 or 24 per cent.

This efficiency yalue may not be acceptable, inwhich case it will be necessary to find a means ofreducing the value of the quantity, liNA + liN0- 11NB -liND' An easy way is to multiply thenumbers of teeth already determined for the dif-ferent gears by any whole number, say 3. Thus,N A = 48, N B:.= 102, No = 45 and N D = 96. FromEquation 17,ltooth friction loss P/-'T = 1.17 Po andfrom Equation 18, e = 46 per cent.

A further possibility is to make N D and N 11more nearly equal by using a value of n between1 and 2, -say 1.25, with numbers of teeth in therange of 100. Under normal design circum-stances, load capacity of the gears tends to belimited by tooth breakage if the numbers of teethin pinions are much in excess of 80.

Using the relationship, b = on) gives: N 11 =cV(mG)' The value of c to make N A about 100is then c = 100/y(256) = 6.25 or, for conveni-ence, c = 6. Thus, N A = 6y(256) = 96.

To make b as nearly equal to c as possible, letn = 7/6, which gives b = 7. Solving for the othergears then, based on the new factor values, pro-vides: N D = 112, N B = 119 and No = 90. FromEquation 1, ma = 256. Tooth friction loss (Equa-tion 17) becomes P/-'T = 0.214 PO) and efficiency(Equation 18) is: e = 82 per cent.

This value of efficiency may be regarded assatisfactory, although a higher value can be ob-tained by using stilI greater numbers of teethand a somewhat lower ratio of NDINA

For a gear design of given material, diameterand width, increasing the number of teeth re-duces the circular pitch. Thus, this approach isrestrained by considerations of gear load capacity,as limited by tooth-breakage. Moreover, the num-bers of- teeth in a pinion and mating internal gearare limited by tooth interference problems toa certain minimum difference and a minimum ratio,depending on the ratios of the addenda of themating teeth to their pitch.

High Velocity Ratio': The facility with whichthe four-gear compound epicyclic train can be de-signed to give very high velocity ratios is tempt-ing. But, as demonstrated in the preceding ex-ample, it may give a low efficiency. For example,

(14)WpZ=--To

=--------------------

April 4, 1957

Total work lost in tooth friction is, from Equa-tions 15 and 16,

WAB =2- (_1_ - _1_) (rna - l)W,.. (16)., 5 N

AN

B

ing gears.If one of the gears is an internal gear, the cor-

responding value of N becomes a negative number.This result indicates that where an internal gearwith a large number of teeth meshes with a pinionnearly as big, tooth-friction loss is very small.Thus, it can be seen that a compound epicyclicgear train which is to be designed for a high ve-locity ratio with the highest practical efficiencymust embody internal gears with large numbersof teeth, and large planets. -

In the analysis of epicyclic gear trains it is nec-essary to give careful attention to the definitionof Z} Equation 5. Displacement sp is the displace-ment of the gear tooth along the line of pressurerelative to the line of centers of the mating gearsor, precisely, to the common plane of the axes ofthe. gears.

Thus, displacement sp = rb(} while tooth loadF = T Irb From Equation 5 then,

W OD = 2- (_1_ - _1_) (rna - I)W~ (15)P. 5 No N D r

and overall efficiency of the train becomes, there-fore,

where W'. OD is the work lost in tooth friction andW F is the output work delivered by shaft F.

If frictional loss in the spindle bearings of gearsA and a is neglected, tooth loads at these gearswill be nearly equal since their diameters are near-ly equal. Thus, the loss in tooth friction betweengears A and B is:

1( 1 1 1 1)W.T =5 Nil + No - N

B- N

D(rna -1)W,..

.(17)

where T represents the torque on either gear inthe mating pair and () is the angular displacementof the same gear relative to the common plane ofthe axes.

Consider now the pair of meshing gears, a andD) in Fig. 1. If the angular velocity of the outputshaft is lJ)F = 1, angular velocity of the inputshaft, and planetary spindle, is "'B = mG and an-gular velocity of gear D relative to the commonplane of the axes of gears a and D is "'D = mG - 1.Thus, the product of torque on gear D and itsangular velocity relative to the axes of gears aand D is equal to the output power multiplied bythe factor, mG = 1. From Equations 13 and 14then, the work lost in tooth friction between gearsOandDis:

COMPOUND EPICYCLIC GEAR TRAINS

a velocity ratio of about 2000:1 can be obtainedby making N A = 45, N B = 92, No = 44 and N D =90. Operating efficiency of the train is only 10 percent.

At the cost of an extra pair of gears to give aninitial reduction of about 8 :1, this ratio can beobtained (approximately) with an epicyclic train,such as the one in the example, having a velocityratio of 256 :1. The overall efficiency, neglectinglosses other than those due to tooth friction wouldthen be the product of about 0.97 for the extrareduction pair and 0.82 for the epicyclic gear train,giving a combined efficiency of about 79 per cent.

Importancl' of Efficiency: When the powertransmitted is small, as may be the case in drivingeven a large machine if the speed is low, the costof providing a larger motor and extra. power tomake up for the low efficiency of the gears maynot be important. On the other hand, dissipationof the power lost in friction may be difficult. Ifnatural cooling of the epicyclic gear unit by con-duction, convection and radiation is relied upon,the temperature rise in the gear train may be toohigh for satisfactory lubrication by an oil thatalso is satisfactory for starting at low temper-ature. Artificial cooling by fan, cooling coil orcirculation of oil to a separate cooling tank maythen be necessary.

Problems in Mechanical Design: Based on theprevious discussion, it can be seen that to mini-mize frictional losses in a compound epicyclictrain, three important points must be kept inmind in design:

1. Use internal gears as fixed and driven mem-bers.

2. Make diameters of planet gears more than halfas large as those of the internal gears.

3. Keep numbers of teeth in the 80 to 150 range.Of these practices, item 3 tends to limit load

capacity for given gear materials and diameters,while item 2 precludes the use of multiple planets,leading to a heavy resultant load on the driven in-ternal gear which is overhung from the bearingsof output shaft. Substantial mountings are there-fore necessary for the bearings and for the plane-tary spindle since any deformation causes de-parture from uniformity of distribution of loadacross the width of the gears. Part of this dif-ficulty is overcome by allowing the driven internalgear to act as the inner race of a large rollerbearing in which the outer race is carried in thegear case.

For reasons of this kind, the compound epicyclictrain tends to be expensive in relation to its loadcapacity as well as less compact than might atfirst seem possible.

Internal gears are not the easiest type of gearsto manufacture and so the possibility of replacingthem by external gears is naturally suggested.This substitution is kinematically possible but thechange may mean a serious drop in efficiency

104

since the basic relationship developed for internalgears, liNA + No - liNB - liNDJ must thenbe replaced by liNA. + liN0 + liNB + liND'In the previous design example, this substitutionwould act to reduce the efficiency from 82 to 33per cent. .

Even if this reduction in efficiency can be toler.ated, the four-gear train of external epicyclicgears has much lower load capacity within the sameoverall dimensions than the corresponding trainwith two internal gears. The inferior load ca-pacity may, however, be offset by using multipleplanetary pinions, but special arrangements arenecessary in design, manufacture and assembly toassure a reasonably close approach to uniformityof load sharing between the planets.

Estimating E:fficiency: The simplest method ofestimating the efficiency of an epicyclic gear trainis, first, to determine the angular velocity of everygear and carrier in the normal running condition.An output shaft torque is then assumed and thecorresponding torque on each gear is calculated,assuming that the torques exerted on each otherby a pair of meshing gears are proportional tothe numbers of teeth.

The work lost by tooth friction between a pairof gears is equal to the product of a "loss factor,"the torque on either gear and the angular displace-ment of that gear relative to the plane of the axesof the gears (Equation 14). The work input bya single shaft to an assembly of gears is equal tothe work output plus the sum of the losses at thevarious mesh points.

By this method, it is not necessary to distinguishbetween positive or negative torques, betweendriving or driven gears, or between directions ofso-called "flow of power."

If the total loss is found to be greater thanthe output work (that is, if the efficiency is lessthan 50 per cent), then the gears could not berotated by a torque applied to what is normallythe "output" shaft because otherwise the worklost in friction would exceed the input work. Suchan assembly of gears in a train is said to be "ir-reversible."

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MACHINE DESIGN

Designing Compound Epicyclic Gear Trains Tooth Friction LossesFig. 2-Analysis of tooth action...Design ExampleHigh Velocity RatiosImportance of EfficiencyProblems in Mechanical DesignEstimating Efficiency