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76-GT-88

High Performance Epicyclic Gearsfor Gas Turbines

D. E. IMWALLE

Manager of Engineering,The Cincinnati Gear Company,Cincinnati, OhioAssoc. Mem. ASME

The history and explanation of the Cinti-BHS load equalization system for epicyclic gears isdiscussed in detail. Equations for relating gearbox size and weight are presented for parallelshaft and epicyclic gears. A derivation for the maximum ratio for the number of planets isalso included.

Contributed by the Gas Turbine Division of The American Society of Mechanical Engineers forpresentation at the Gas Turbine and Fluids Engineering Conference, New Orleans, La., March 21-25, 1976.Manuscript received at ASME Headquarters December 19, 1975.

Copies will be available until December 1, 1976.

THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS, UNITED ENGINEERING CENTER, 345 EAST 47th STREET, NEW YORK, N.Y. 10017

Copyright © 1976 by ASME

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High Performance Epicyclic Gearsfor Gas Turbines

D. E. IMWALLE

INTRODUCTION

The first epicyclic gear according to the and load equalization was accomplished by flexi-concepts of Dr. Wilhelm Stoeckicht was developed ble suspension of the annulus.by The BHS Company in West Germany. This gear In 1940, a floating sun arrangement waswas delivered to The Daimler Benz Company in perfected for an airplane propeller drive, While

December 1932. The gearing in this unit was single floating the sun to achieve load equalization

helical. The sun gear was mounted on bearings among the planets may seem obvious today, it was

NOMENCLATURE

C = center distanceCF = centrifugal force

Cf = surface condition factor

CH = hardness ratio factor

CL = life factor

Cm = load distribution factor

C o = overload factorC = elastic coefficient

CR = factor of safety

C s = size factor

CT = temperature factor

C, = dynamic factor

Dop = planet outside diameter (in.)D = planet pitch diameter (in.)

D s = sun pitch diameter (in.)Dg = gear pitch diameter (in.)

DR = Annulus pitch diameter (in.)

F = face width (in.)H = gearbox height (in.)

HMAX = maximum gear tooth hardness R"C"I = durability geometry factor

L = number of load cycles for a gear

L = planet length (in.)

LIFE = required design life (hr)

N = number of planets

N = number of planet teethNR = number of annulus teethNs = number of sun teeth

Rp = planet pitch radius (in.)

RR = annulus pitch radius (in.)

R s = sun pitch radius (in.)

R = reliability

SAC = allowable contact stress (psi)Sb = root bending stress (psi)

S c = Hertz contact stress (psi)

S = arc length of planet on planet mean

center diameter (in.)

ST = arc length between planet centers on

planet center diameter (in.)VpL = pitch line velocity (fpm)

Wp = planet weight (lb)Wt = tangential tooth load (lb)W = gearbox width (in.)

dp = pinion pitch diameter (in.)

e = radius of planet mean diameter (in.)

= gearbox length (in.)mG = gear ratiomN = load sharing ratio

nc = carrier speed (rpm)

np = pinion speed (rpm)

n1/4 = relative speed of sun with respect to

carrier (rpm)

n 2/4 = relative speed of planet with respectto carrier (rpm)

n3/4 = relative speed of annulus with respectto carrier (rpm)

p = power (hp)

r = epicyclic gear ratio r = NR/NS + 1R = overall le coefficient

= mounting quality coefficient/3A2 = gear quality coefficient/311 = hardness range coefficient

gm = material coefficientgrs = size coefficient

cZt = operating transverse pressure angle

Subscripts

1 = sun

2 = planet

3 = annulus

B = bending stress reliability

H = Hertz contact stress reliability

2


12 ---- --

. 500,0

,OR GAS.; :,,,1

.:040

Fig, 1

considered, at that time, a revolutionary featin epicyclic gearing.

The rapid progress of turbines led to in-creased demands for higher speed and power. In

1951, BHS introduced for the first time a BHS

Stoeckicht gear with double helical gearing for

turbine drives. Its success was immediately

recognized, and since then, nearly 20,000 units

have been successfully introduced in numerous

applications throughout the world.At the present time, new generation marine

gas turbines are being developed. These new

turbines range from 6000 to 50,000 hp. The new

turbines show promise in the areas of specific

fuel consumption, specific weight, and nigher

reliability. Fig, 1 shows the gas turbine power

spectrum under development. The survey of Fig. 2indicates some of the potential gearbox ratios

for marine and industrial applications for theseturbines.

While the gas turbine power is high, it isnecessary to reduce the speed with turbine gears

for actual applications such as generators.

Present applications of Cinti-BHS epicyclic gearsrange from input torque of 20,000 lb-in, up to

8,500,000 lb-in, For a 3.5:1 reduction gearbox,

OutputInput - G35 Turbine (HO)

35000 27500 16500

High Speed Slow Speed

DeviceOperationllSpeed (R 5 ,) 6000 6000

Generator

1800 2.0 1.89 2.63 6.11 4.00

1500 2.4 2.27 3.15 7.33 4.60

1200 3.0 2.83 3.94 9.16 6.00

1000 3.6 3.40 4.73 11.00 7.20

Materjet 4.0 4.0 4.25 6.60 4.30

High SpeedCRP

Propeller6.6 - 10.0 4.3 - 6.6

Low Speed

CRPPropene, 100 - 20018 - 36 17 - 34 24 47 35 - 110 36 - 72

Fig, 2 Required gearbox ratios for Marine gasturbine drives

While it would have been possible to

incorporate the entire ratio in a single stage of

epicyclic gearing, a small offset was required

in order to provide service access to the gasturbine. The waterjet is located as close to hull

line as possible by using the epicyclic gear inthe high torque final stage. Design data for the

epicyclic gear is shown in Fig. 4,

DESCRIPTION OF EPICYCLIC GEAR FOR CREW BOAT

The design has a rotating planet carrier.The stationary annuli are flexibly connected tothe gear casing by means of a double-tooth-type

coupling. The input sun gear is free of bearings;

it is connected to the parallel shaft gearbox

by means of a double-tooth-type couplin. Input

and output shaft rotate in the same direction.

Load equalization is obtained through self-adjust-

ment of the floating sun wheel and flexible annuli.The low-speed shaft, which is the planet carrier,

is supported on both sides in journal bearings.

The planet gears run on spindles, whose bearingsurfaces consist of several layers of different

materials for high reliability. Journal bearings

and gears are lubricated by forced lubrication.

Additional auxiliary drives are provided on thegear cover. The gear casing is split on thehorizontal centerline and thus permits simple

the output torque would be approximately 29,000,000 inspection without requiring re-alignment,

lb-in. With 43 years of epicyclic gear application, The gears are double helical type. As a

these gears can match the specific weight and high result, the noise level and the required number

reliability features of the new gas turbines in of thrust bearings are reduced. Sun wheel, planet

single reductions, wheels, high-speed coupling flange with its

The latest gear is in the main propulsion coupling sleeve are nitrided. Carburized, hardened,

drive system for a new high performance crew boat, and ground gears have also been used in other

Fig. 3 shows a section view of the gearbox applications. The two separate annuli are made

arrangement, An Allison 501 gas turbine provides of high quality heat-treated alloy steel. The

the input power. A waterjet is the propulsion gear teeth are shaved to AGMA 12 precision,

device. Compared to parallel shaft gears, the

3


5 Reliability: Theoretical framework for

evaluating reliability is provided.

Size

The first approximation to gearbox size can

be determined from the Hertz contact stress on the

gear teeth. The fundamental durability equation

for gears according to AGMA 215.01 is as follows:

S c . W t C o x Cs xCpCv

[ dpF

K2 = [ST - [SAC

Cp C P

K 1 = Fdp

Wt = p 126000 np dp

Cm Cf.]

C L C H]

CT CR

2

K3 = cosOt sinq't 2 (5)

F ,..:L'7,...__, ...

.."'".... , ..

-,-,---„if/'''' ' .,„,-74:/•• • -:,

•...,-,..

A.5-

P,

•

„.„....„,_,........,........, ,...„... ,......, ...... -.

Fig, 3

following features of epicyclic gears are evident:

1 Co-axial input and output shaft2 Division of power transmitted over more gear

tooth meshes

3 Small gear dimensions4 Slower speed and, therefore, lower pitch line

velocities

5 Lower dynamic loads

6 Lower weight and smaller rotary masses

7 Compact design

8 No high-speed bearings

9 Complete load equalization.

OBJECTIVE OF PAPER

The objective of this paper is to provide

information on the following topics:

1 Size and weight comparison between parallel

shaft and epicyclic gears. For those morefamiliar with parallel shaft gearing, it isdifficult to relate to the size changes and

weight possibilities of epicyclic gearing.

2 Centrifugal force: For epicyclic gears with

fixed annulus, the understanding of this

phenomena is essential.

3 Load equalization: Various theoreticalpossibilities are presented and the actual

practical solution for high-speed gears is

emphasized.

4 Number of planets: The number of planets in

epicyclic gears varies with the ratio.

4

Power

4500 HP

Input Speed

5650 RPM

Ratio 3 40

Type Simple Planetary Gear

Pitch Line Velocity . . . 7245 fpm

Planet Bearings Babbitted Spindle

Gearing Precision Double HelicalNitrided Gears

Weight 1590 lbs.

Lubricant MIL-L-23699

Housing Construction . . . Steel . Weldment

Number of Planets 5Fig, 4 Design data on crew boat epicyclic gear


( 7 )

(1 5)

I = K, mG

mN mG ± 1WCYL = n/4 x .283 x D 2 x Z = .222D 2 1 (14)

D = [Ks] 1/2 DR

DescriptionHelixAngle

Hid RaTio

Parallel Si.aftGears

EpicycliCGears

Double Helical PinionMeshing with One Geer 30 2.20 --

Double Helical PinionMeshing with Tao GearsDiametrically Opposed 26 2.6 2.6 - 3.6

Single Helical PinionMeshing with One or TwoGears li 1.1 1.S • 1.50

Spur Gear. 0 1.10 1.25 - 1.65

MOM The eon pinion is only subject to torsional twisting; whereas parallel shaftpinion has combined bending and torsional loading. la all cases near U.

f/d ratio, the d•ilectioas should be checked.

He.Treatment Material

Relative AllowableRoot Tensile Stress

Relative AllowableHertz Contact Stress

Qi.enched andTempered 4140 .0

ProgressiveInductionHardened(Tooth by Tooth) 4145 .80 .86

GasNltrided slcamovv, .00 .17

CarburizedandHardened

•900 1.00 1.00

•6121 Cla••ifiretioa

Fig. 5 Typical maximum face to diameter ratios Fig. 6 Relative design stresses for gearsfor gearing

Weight

1/K, = 126000 C o C s Cm Cf The weight of an epicyclic gear can beCv (6) estimated by modeling the gear after a solid

circular cylinder.

Rewriting equation (1)

K2 X_p 1 x 1 k mN ( mG ± 1)-—np dp

TK,K,K 4 mG

For parallel shaft gearbox designs, size isdependent upon center distance C.

2C = d p + mGd p = (mG + 1) dp(9)

dp 3 = sc3 (10)

(mG + 1) 3

Substitution in equation (8) yields for parallelshaft gearboxes

11/3 [ 1/3

C = [MN ( MG 1)6 (11)8 K 1 K 2 K 3 K 4 1nG np

or for equal hardness and ratio, center distanceis proportional to the cube root of the input

torque.

C [21n p

1/3

(12)

For epicyclic gears, gearbox size is dependent

upon annulus pitch diameter DR ; thus, the annuluspitch diameter is proportional to the cube root

of the input torque.

/ = K6 F

WEG = .222 K 5 K 6 K 7 FD 2 R K D 3 R

where K7 is a packing factor which indicates howmuch space is required by the parts in the volume

of the cylinder. The K constant in equation (17)is a function of D R.

For single reduction parallel shaft gears,the weight can be estimated by modeling the gear

after a solid rectangular prism.

WPS = WU WL (18)

The weight in the upper half of the box isestimated as follows:

Wu = .283 ZWH (19)

W (2C) Ke (20)

Note the significance of the center distance inthis equation.

= K9 F

= K 10 DG2

Wu = .283 K 8 K9K10K11 (FCD G )

(8)

(16)

(17)

(21)

(22)

(23)

DR 01 [pi V'n p

(13)

where K11

is a packing factor.

Wu = .283 K,K9K12K13 (FCDG) (24)

5


Coln.onent

Percent of TotalGearbox Weight

All Steel)

PossibleWeight SavingBy Material

Substitution t

Gears and Couplings 19 - 24 --

Planet Carrier 22 - 26 13

Housing 2S - 40 14

Bearings 8.5 - 10.1 --

Fig. 7 Total weight reduction possibility by

material substitution is approximately 27 percent wt

Wt

With Centrifugal Force

where K13 is a packing factor. K12 differs from

K10 since it is dependent on how much oil sump

is required, if any, below the gear.Rather than finding expressions for the

upper and lower half of a gearbox, it is possibleto find the weight by:

W = K (C) 3(25)

where K is a function of C. If lighter densitymaterials are used in the design, it is possible

to adjust the density coefficient. If the weightfor a given design is reduced by lightening the

components, the packing factor can be adjusted.

wt Wt.

Fig. 8 Planet bending stress--no centrifugalforce (Note: Reaction forces within gear areapproximately parabolic)

Example: Modern lightweight double helical

for applications requiring high

long design life (in excess of

data for the gearbox series is

Epicyclic Parallelgear shaftgears are required

reliability and30,000 hr).

The designas follows:

Pitch line velocity (fpm) 6145 10,600

Annulus pitch diameter (in.) 19.6 - - -

Center distance (in.) -- 21.1Relative weight 0,350 1.0Relative cost 1.01 1.0

Application RatioInput Speed,

rpm

Input Power,hp The center distance, annulus pitch diameter and

actual weights were determined from empirical12

4

4

4800

4800

8,250

16,500equations of existing gearboxes.

For Application 2, the following designswere determined.

The long design life and reliability requirements

dictate a choice of journal bearings for thegearbox. The modern, lightweight requirement

implies the selection of case hardened gears. A

comparison can now be made between a parallel

shaft gearbox and an epicyclic gear.

For Application 1, the following designswere determined:

Epicyclic Parallelgear shaft

Pitch line velocity (fpm) 7737 13,361

Annulus pitch diameter (in.) 24.7 - - -

Center distance (in.) 26.6

Relative weight 0.38 1.00

Relative cost 0.88 1.00

6


Ratio r

r • 'il • 1

V,

Annulus Pitch Diameter DR (inches)

2S 10 11 40 50 CO 00 SO 100

3 1465 7511 4019 6000 11718 20203 32156 48000 23744

4 2317 4360 6357 9474 1852i 32000 50705 70690 104923I

S 2749 4746 7542 11250 21072 17909 60292 90110 175776

6 3000 5184 8232 12228 24040 41472 65856 96104 192000

7 3160 5469 6687 12974 20124 41760 69505 101919 202592

S 3280 1663 9000 13414 26239 41341 72000 107476 229912

Fig. 9 Centrifugal force factor K11

Fig. 10 Planetary gear design without loaddistribution

This example shows that there are weight, sound,

and cost advantages for epicyclic gears at the

higher horsepower level. The noise level of a

gearbox is a function of its pitch line velocity. spindle. The centrifugal force also increases

Fig. 5 shows typical face to diameter ratio the planet bending stress. This stress is not

comparison between parallel shaft and epicyclic to be confused with the gear teeth stress. Fig.

gears. Since the sun is only subject to torsional 8 shows the planet bending stress free body

twist, the face width can be made larger than the diagrams.

pinion face in a parallel shaft design. The equation for centrifugal force is as

The design example was worked out for case-

hardened gears. Fig. 6 indicates the advantagethat case-hardened gears have over through-hardened

gears. There are also other advantages. Case-

hardened gears are smaller; thus, it is possible

to manufacture more accurate gears. More accurate

gears are subject to lower dynamic loads.

The design example was worked out forparallel shaft and epicyclic gearing without

resorting to material substitution or extensivelightening of individual components. Lighterdensity materials, such as aluminum and titanium,

have been used in parallel shaft and epicyclic

gear designs with success. Fig. 7 shows the typ-

ical weight reduction for a planetary gear bymaterial substitution. In fact, a 40,000-hpepicyclic gear with a 4:1 ratio and 4100-rpminput speed could be designed at a weight approxi-mately 6000 lb. This high horsepower design

would be approximately half the weight of a

standard steel construction epicyclic gearbox.

CENTRIFUGAL FORCE CONSIDERATION

For epicyclic gears with rotating carriers,the centrifugal force load on the planet gear must

be considered. It has a direct influence on the

performance of the planet bearing and the control

of proper planet gear tooth contacts. Too muchload on the planet will cause excessive spindledeflection, endanger gear teeth bearing, andreduces the margin of safety for oil film thick-

ness for planet spindle bearing. The increased

load also will pump more oil through the planet

follows: CF = .284 W pe(nc ) 2 (26)

100

'Wp = .128 D 2 p Lp For P1 (27)

For Planets with Spindles

e D s r ( 28 )

4

K11 = D2p D sr (29)

Substitution of equations (27), (28), and (29)

into equation (26) yields

CF = .009 K 11 Lp (nc )100

Fig. 9 shows a table for K11 values which

show the effect of increasing annulus pitch

diameter and ratio on centrifugal force. For

any fixed gear ratio, doubling the annulus pitchdiameter increases the centrifugal force effectby a minimum factor of eight. Also, for a fixed

annulus pitch diameter, increasing the ratio in-creases the centrifugal force.

The K11 centrifugal force factor is beyond

the designer's control since the application

usually specifies the torque and ratio require-ments. The designer has some freedom in theselection of planet width (L p ). For double

helical type of gears, it is essential to minimize

the groove width between right- and left-hand

helix. Some designers split the gear into

2

(30 )

7


/1);7777S1777I, /.771,/,'"/77.17J77/7F:;;AT,N , PLA,, Fl/eT1N; GE R,

PLANET A.ND CARRIER

SUN PINION WITHSINGLE GEAR TYPECOUPLING

DOUBLE. HELICAL GEARING

FLEXIBLE ANNUIOSMOUNT I NG ARAANGIAANT.NO1 L 111E I N FEEL: .NTSUPPORT OF EACH Ift.L I X.

%///, /////,/,

c

I

FLOA7, ,,PLANE: wH E EL

./7777,,,,,777;ELCA7NE1

JSIOC,X.

FLEXIBLE ANNULUSMOUNTING ARRANGEMENT

Fig. 11 Possibilities for radial load equalization

in epicyclic gear designs. (Black outlined parts—radial displacement constrained; white outlined

parts--radial displacement permissible; cross-

hatched parts--gear case: See footnote 1 for moredetail.)

Fig. 12 Spur and single helical gearing

separate halves and then join the different hand

parts either mechanically or by electron beam

welding. While the electron beam welding looks The following errors in Fig. 10 can be measured

promising, these techniques require rigorous con- from various manufactured components of the drive:

trol to assure structural integrity.

The centrifugal force consideration should A Eccentricity of the output carrier shaft in

be emphasized in the initial design selection. relation to the mean planet diameter of all

Star type of gear is not subject to centrifugal three planet bores, d

force on the planet gears and spindles. However, B Eccentricity between the output carrier

with increased rotational speed, the annulus of shaft and the bearing (c to b)

a star gear tends to behave as a rigid ring rather C Eccentricity between the central hole of

than as a flexible ring. A flexible annulus ring the housing cover plate and housing pilot

assists the load equalization process. diameter on both the high- and low-speed

LOAD EQUALIZATION

One of the fundamental advantages ofplanetary gears in comparison to parallel shaft

designs is that the power can be carried on more

than one shaft—generally, on three, four, or

five shafts via the sun gear. Fig. 10 shows a

planetary gear without load equalization. The

sides of the gearbox (e to c)

D Eccentricity between planet spindle diameter

and the bores for the spindle in the carrier

(f to g)E The runout error of the gears.

Since effective load equalization is

essential to successful operation, it is desirable

to review the various possible solutions.

planet wheels can be distinguished by their cross- Fig. 11 shows some of the major possibilities

hatching in the figure. Unequal circumferential of radial load equalization for epicyclic

forces on the planet gears arise from manufacturing These load equalization systems differ according

errors in the gear case, the planet carrier, and

differences between planet gears. All of theplanets are mounted on bearings. Depending uponmanufacturing accuracy, it is possible that asingle planet might carry all the power.

Unless the loads are equalized by some

system, the advantage of planetary gears is lost.

to speed. An effective system for low-speed gears

with large masses may not be feasible for high-

speed turbine gears. In Fig. 11, dark outlined

parts are fixed radially while white outlined

parts are free to move radially. At (A), there

is a schematic of an epicyclic gear in which onlythe sun gear is allowed to float in the radial

8


Fig. 13

Fig. 14

direction. By floating, it is meant that thecomponents equalize forces and stabilize their

relative positions. With three planets, the sun

gear will always shift so that the gear separating

forces are equal at all three sun-planet gear

tooth meshes. Since the separating forces are

equalized, the tangential forces are also

balanced. Thus, the position of the sun gear is

statically determined. Since the tangentialforces must balance for equilibrium at the planetspindle, it follows that the same tangential loadsare present at the three planet-annulus meshes.This simple measure leads to theoretically

perfect load equalization. In many cases, it ispossible to obtain this load equalization withvery large clearances in the sun gear bearings.However, this system is only applicable for slowspeed gears. Any eccentricity of the annulus,

for example, a single radial displacement, has to

be compensated. This eccentricity gives rise to

an oscillatory moment at each planet gear

completely exclusive of sun gear oscillation for

each revolution of the planet carrier. In thisregard, the solution according to (D), which hasfloating sun and annulus, can be used in almost

any situation and is ideal for turbine gears.

The solution according to (B), which has a

radially adjustable annulus, is similar in princi-

ple to (A). It is, however, more difficult to work

out the design details than (A).Solution (C) relies upon radially adjustable

planets. It is simple to build and is relatively

low in cost. Fig. 14 shows how the details of

this equalizing system can be worked out. Details

on this solution can be found in Reference (3). 1The load equalizing scheme according to (E)

is suitable more for slow speed than high speed.Here, the planet carrier and the planet gear are

free to adjust radially. However, it is necessary

that the entire, relatively heavy carrier, must

take part in the equalizing movement. In order

to float the carrier, the carrier must be

connected to the slow speed shaft either by

mechanical or elastic couplings.Another system is according to (F), where

the planet carrier, planet gear, and sun gear are

free to float together and are located by theannulus. For this system to work properly, it is

necessary to make the planet carrier weight as lowas possible in order to keep the inertial forces

low.Another load equalizing possibility, namely

where the carrier and annulus float and the sun

is fixed, is not shown in Fig. 11. This solution

is inefficient and also has unfavorable results

for the high-speed sun gear bearing.The author's solution for epicyclic gear

is based upon (D) in Fig. 11. Fig. 12 shows how

this theoretical concept is worked out in actual

design. Fig. 13 shows the annuli for the

previously mentioned crew boat epicyclic gear,

It is heat-treated steel and flexible. The ring

thickness is calculated carefully to provide thenecessary flexibility for the expected gear tooth

error.

1 Underlined numbers in parentheses desig-nate References at end of paper.

9


Gear Ratio.tb (nas)

Number of PlanetsN

Ratio of Are BetweenPlanets to Alc betweenPlanet Centers 5n/S1

12 3 .059

5.77 4 .093

4.10 5 .144

3.38 6 .196

2.99 7 .248

Ns

Fig. 15 Recommended maximum ratio for number of

Sp (Rp + Rs ) (20) = Rs re (36)

SIN (0) = Rp = (r - 2)(Rs + Rp) (37)

ST = (Rs + Rp) 2nN (38)

SIN I r - NSp/ST = Rs rON = ON = r (39)

(Rs + Rp) 2,r 180n

S S = 1 - Sp/S TST

planets (standard addendum gears)

NUMBER OF PLANETS

Since the capacity of an epicyclic gear isbased upon the number of planets, it is importantto consider what would be the maximum number ofplanets for a given ratio. For any given ratio,the outside diameters of the planets must not

interfere with each other. Also, there must be

enough room for the planet carrier support to

pass by the planet gear from low speed to high

In order to prevent possiblebetween planets, it is necessary

terms of outside diameter

N = 360 °

2

meshing

of planet,

interference

to define 0. in

(40)

(41)

(42)

29

SIN(0)

0 =

= NP +(Ns +

N p +

Np)

2 1Ns + Np

-4

speed side of the carrier,

In the following derivation, the S s/ST

Redefine equation (40)

N = 180variable is related to the carrier support require-ment. The calculation for N, the number ofplanets, considers the potential meshing inter-ference. The results are summarized in Fig. 15. Substitution

SIN - 2

(34) and

(43)

(35)

li■lp

Ns Np]

of equationsFor final design, a layout should be made.

For planetary type of epicyclic gearN = 180

r = NR + 1SIN - I [r - 2 + 4/N31 (44)

Ns (31)

Since the problem is whether large planets mightFor equally spaced gears in a plane

interfere with each other, design safety can beprovided with a small sun; Ns = 13

NR + NS = IntegerN (32) N 180 or r = 1.692

For uncorrected gearsSIN'' - 1.692 1 - SIN 180 (45)

r

Ns + 2Np = NR (33) The calculated N value must be reduced to the next

Rearranging equation (33) and combiningwith equation (31)

Np = NS (r - 2)2

lowest integer.For non-standard gears, with addendum

correction, the following expression is valid:

N = 180SIN - I {Pop]

2C(34)(46)

Also from equation (34) RELIABILITY

NP + NS NS r —2

(35)Bodensieck (2) has recently presented a

stress-life reliability system for rating gear

10


life. While the formulas are complex for hand

calculations, they can be readily programmed fora computer. This new reliability model hasincorporated into it all the important variables

and the results compare favorably with theory.

The Bodensieck model can also be adapted to each

manufacturer's quality level.In the preliminary design stage, it is now

possible to compare different designs. Thetheoretical framework is obvious for parallel shaft

gears. For epicyclic gears, the following set of

equations applies:

R1 R10 R 1H (47)

R 2 = (R2B P211 R' 211)

(48)

where the prime notation refers to the planet-

annulus mesh for the planet

R3 = (R3B R3H ) (49)

R = R 1 (R 2 ) N R3 ( 50)

The reliability for the gearing can be determined

by solving the following equations for R. See

Reference (2).

For Bending:

C 2 - 432 (HMAX)1.5

(51)

=1+1+1+ 1 a am 6 H BS 8A1 8A2

(52 )

Reference (2) had detailed information for ie

determination

C 3 = LN R + .00001

LN .001 (53)

C4 = 1.43 + .8 tanh (4.403263 - 1.8 LN LN L) (54)cs [Vp Li 2

225

C3 =[Sh C 4 C 5]

C 4 C 2

Then solve for R from equation (53). Note for

planet gears, use

C2 = 317 (HMAX )1.5

C 5 =

For Durability:

C6 = 1.3 + .9 tanh (8.7472 - 3 LN LN L)

C7 = 1060 (I mAx ) 1 • 5 (60)

cs = Sc C 6 C 7

(61)

Then solve for R from equation (53) by substitutingC8 for C3.

The number of load cycles is as follows:

L 1 = n 1/4 • 60. LIFE•N (6 2)

L 2 = n 214 • 60• LIFE

(63)

L 3 = n 314 • 60•LIFE•N

(64)

CONCLUSION

An analytical set of equations was developed

for evaluating the gearing reliability of

epicyclic gears.

A table of maximum gear ratios was determinedfor the number of planets. This should be auseful preliminary design tool.

The role of centrifugal force in simple

planetary type ,ears was emphasized. It is

important that the designer correctly compensatesfor this force in his desin.

A method for approximating gear size andweight for both epicyclic and parallel shaft

gears was presented. This is a useful preliminary

design tool. Since empirical equations for costing

gearboxes can be determined from price lists, it

is possible to include economical considerations

in the initial design selection.The importance of the proper load equaliza-

tion for epicyclic gears cannot be neglected. For

high-speed gears with little margin for error,

it is almost the exclusive consideration for drive

selection.

Depending upon cost, weight, and spacerequirements, there are two major gear systems-epicyclic and parallel shafts which should be

considered during the initial design stage.

REFERENCES

1 Ehrlenspiel, K., "Planetengetriebe -

Lastausgleich and Konstruktive Entwicklung," VDI

Report No. 105, pp. 57-67, 1967.2 Bodensieck, E. J., "A Stress-Life Reli-

ability Rating System for Gear and Rolling-Element

Bearing Compressive Stress and Gear Root Bending

Stress," AGMA Paper No. 229.19, 1974.

3 Imwalle, D. E., "Load Equalization inPlanetary Gear Systems," ASME Paper No. 72-PTG-29,1972.

(55)

(56)

VpL 2

318

(57)

(58)

(59)

1 1


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