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52 'Gear Technology

Fig. ::I (Lelt) - Bevel gears. These shafts intersecl ala.right angle, although bevel gears may also be usedbetween shafts that intersect at larger or smallerangles. (Courtesy Mobil Oil Corporation.]

This article is an excerpt from Machine Design Fun-damentals, John Wi12y & SOrl5, Inc., Publisher.

Transmission of power between non-parallel shafts is inherently more difficultthan transmission between parallelshafts, but is justified when it saves spaceand results in more compact, morebalanced designs. Where axial space islimited compared to radial space, angulardrives are preferred despite their higherinitial cost. For this reason, angular gearmotors and worm gear drives are usedextensively in preference to parallel shaftdrives, particularly where couplings,brakes, and adjustable mountings add tothe axial space problem of parallel shaftspeed reducers.

In angular drives, the gears not onlyrotate in different planes, but contact isfrequently diagonal across the Face ofmating teeth (fig. 1). Such gears aregenerally more difficult to design,manufacture. and install and thus costmore than equivalent spur and parallel-axis helical gears. They are also moresensitive to mounting and manufactur-ing errors. In addition, mounting onoverhanging shafts makes some typessensitive to shaft deflections. As withspur and helical gearing, optimum designbecomes a compromise. Two generalclassifications cover the right-angle gears:coplanar types, which have intersectingaxes, and offset types, which havenon intersecting or skew axes (their axesdo not lie in a common plane).

Bevel GearsBevel gears provide the most efficient

means of transmitting power between in-tersecting shafts. The related frictionwheels are frustrums of cones, and the

Gears For Nonparallel Shafts

gears developed on these conical surfacesare called bevel gears. If teeth are cutstraight across the faces of conicalblanks, the gears are called straJ:ghtbevel;when the t.eetharetwisted along a curvedpath, thegears are termed spiral bevel(Figs. 1 and 2.) The involute tooth fonnis used.

Customarily, tooth dimensions aredetennined in a transverse plane {perpen-dicular to the common dement of thepitch cortes) at the large end of the teeth(Fig. 3a). Intersection of tooth surfaceswith this plane gives a tooth profile asshown in fig.3b. Fig. 3aa]so shews thatthe shaft angle equa]s the sum of thepitch angles, Thus

~'" C. + r,

where

I(sigmal ; shari angle: degC,,(gamma) = pilch angle of pinion: degr(;'lgamma)= pitch angle of gear: deg

fig.3c indicates that the followingrelationship exists for E = 90 deg,

-- r' d NpIan; -;--p 0 Nc;

_ 0 NGtan rG = - ::::.-::-::---- d N"

where

d = pitch diameter of pinion; mm, in.D = pitch diameter of gea..r; mm, in.

Np ,= number of teeth in pinionNG = number of 'teeth in gear

Nomenclature and symbolScemmonlyused for straightbevel gears are shown

byDr. Uffe Hindhede

Black Hawk CollegeMoline,. IL

in Fig. 4. The similarity to spur gearingshould be noted. With few exceptions,the nomenclature also applies to spiralbevel gears ..

Equivalent spur gears is a term com-monly used in connection with bevelgears. Two bevel gears roll on each otherin the same manner as a pair of spurgearswlth pitch diameters equal to thoseef the bevel gears. In. Fig..4 the equiv.a1entspur gear would have a. pitch diameterDc-

AlITHOR:

DR. um HINDHIDE is al1 Associa.!e Pr0-fessor in the Engineen'ng Related Tech11ologyDepartment at .Black Hawk Coll'ege. Moline..ntinois. He is a graduate of the TechnicalUniversity of Denmark and the UniveTSilyo!Illinois, Hindkede has authored severaltechnical papers and articles. fie is the prin-cipal author of Machine Design' Fundamen-tals and 50le writer of the dlRpter On ,Gearsfor Non-Parallel Shafts.

(1)

(2)

Flg, 2 (Belqw) - Spiral bevlilgears. (CoUJ'tesyMobil. Oil Ccrporation.)

(3)

NomenclatureC = center distanced = pitch diameter of bevel

pinion and wormD = pitch diameter of bevel

gear or worm gearDG = pitch diameter of worm

gearDw = pitch diameter of worm

e = efficiencyL = lead of worm

m moduleme - speed ratio

tic, I1w - speed of gear andworm, respectively

Np number of teeth inpinion

Ne = number of teeth in gearNw = number of teeth in

wormp,,-axial pitchPc - circular pitchPd - diametral pitchTR thermal rating

X{lambda) lead angleI'(gamma) = pitch angle.l:(sigma) shaft angle

1,l-(psi) helix angleO(theta) friction angle

The difference between spiral andstraight bevel gears is that spiral. teethhave a gradual pitch line contact and alarger number of teeth in contact. Theirteeth, instead of engaging in a full linecontact.at once, engage with one anothergraduaUy. This continuous contactmakes it possible to obtain smootheraction than is possible with straightbevels.

ArrangementBevel gears are widely used where a

right-angle change in direction of shaft-ing is required, although the shaftsoccasionally may intersect at acute orobtuse angles [Fig.5). 'Alhenof equal sizeand mounted on shafts at right angles,they are referred to as miter gears (Fig..Sb). A bevel gear with a right (90 deg)pitch angle is a crown gear (Fig. Se). In-ternal bevel gearing, like internal spur orhelical gearing, is sometimes used inplanetary or internal gear arrangements(Fig..sf).

ApplicationStraight bevel gears, like spur gears,

54 Gear Technology

t

«til_in..... pWni

t

Fig. 3 - Basic bevel ge.ar sections. (Courtesy General Motors Corporation.)

Fig. 4-Nomendatu~e for bevel gears. (Courtesy General Motors Corporation.)

are well suited for manual and low-speedoperations, such as in small hoists,valves, gates, or doors. \iVhen greaterspeed and more power are required,spiral bevel gears are preferable.

ratio of tooth numbers. For bevel gears,it is also the ratio of corresponding pitchradii. or diameters of the pitch cones(Fig. 4).

Design of Bevel GearsBevelgears are simple modifications of

spur gears. Beam straight and surfacedurability determine size and surfacehardness. The AGMA formuJ'asused aretherefore simple modifications of the for-mulas used for calculating spur gears.

MountingBevel gears require larger shaft

diameters and heavier bearings becausethey impose high reaction loads onbearings.

Speed Ratio.As in spur gears, the speed ratio is the

Example 1A pair 'of straight-toothed bevel gears

It I t -eo_ "'c - " mi_ gMn

~C.IJ:>IO_~ ...

pftdI ..... r. -eo ....rJ ,.eo ..

III J: > 110_ inunwl ....

liig. oS-Bevel-gear arrangements, (Courtesy General Motors Ccrporetion.I

LIlli. CO.TA.Cr

are to be designed for a shaft angle of 90deg and a reduction ratio of roughly 3:1.If the pinion is to have aminimum of }'7

teeth, find pitchangles and the numb rof teeth in the gear.

SolutionN - (3)(17) + 1 - 52 leads to a hun-

ting ratio equal Ito 3.06:1. Assuming amodule m, we obtain for the equivalentspur gears:

0'= m (17 mm) D = m (52 mm)

I' N" 1.7 Ir --Ian , " = N~ = 52 - p =UI •• del

Nc; 52tan, fr =' - = - --- rc:; = 71 9 d -

-y N~ 17 ' .-

Check:

.L, = F, + fa = 18.1 deg + 71.9deg

=90deg

Hypo,id GearsHypoid gears closely resemble spiral

bevel gears except thatthe pinion doesnot meet the ring gear at Us center. Itmeets it at a lower point (Fig. 6). Thepitch surfaces are hyperboloids fromwhich the term "hypoid" was derived.Curved teeth contribute to smooth,noiseless operation even at high speed.

Hypoid gears grew out of a need forsilent automotive diHerentials that wouJdalso allow the drive shaft ,to be placedwell below the cente:rline of therear axle,thus contributing to a [ower body design.Because the two shafts do not intersect,m 'two rear axles, instead 'of one, can besuccessively driven from the same trans-mission shaft. and (2) bearings can bemounted on both sides of the pinion.Although the altered tooth shape resultsin more sliding, lower efficiency, and thneed for special. lubricants,. it provid s aperfectly smooth drive and solves atmajor auecmotive problem-noise.

Helical GearingHelical. gearing is a 'term applied to alltypes of gears whose teeth are of helicalform .. Helicalgears work equally well 'to,connect parallel shafts and nonparallel,nonintersecting shafts. In the latter ea ,however, a distinction. must be mad be-

Fill. 6 (Bottom l.efll- Hypoid 81ar and pinian.These gears transmll motion between noninl:e'r-:sectil'lg .shafts ,crossing t right angl . The pilch, urfaces are hyperbolic in, form. (Courtesy MobilOiJ Cotparation.l

September/October l,986 55

Fig..7 - Crossed helical gears. These gears transmitmotion between nonlntersecting shafts crossing alan acute angle. The teeth are developed on cylin-drical pitch surfaces. SInce oru~point contact ex-ists between the gears, this arrangement is rarelyused to transmit loads of any magnitude. (Courtesyof Mobil Oil Corporaticn.)

Itween a gea:rand a worm, even thoughboth have helical Iteeth. As seen in Fig.7, the teeth on this sear make only a frac-tion of a revolution on the base cylinder.That the toethcurvatnre is helical is noteven obvious, What the teeth lack inlength, however, they make up for innumber, which always exceeds 10 ..

Crossed. Helical Gearingfig. 7 shows this form of gearing. For

what they can. do kinematically, crossed

.56 Gear Technology

helical gears are the acme of simplicity.Tooth contact, however, is only a pointwhich greatly limits their power 'trans-mission capability.

Worm GearingWorm gear drives are used on right-

angle applications with nonintersectingshafts. They provide smooth, quiet ac-tion and maximum reduction ratios forat given center distance. These favorablecharacteristics are obtained by using aworm gear combination. As seen in Fig.8, the gear has teeth inclined at the sameangle as the threads in the worm, Forspeed reduction or torque amplification,the worm is the d.river.

In a worm, the number of teeth rarelyexceeds 10 (one to four teeth is common,as shown in Figs. 8 and 9). Each tooth,however, makes at least one revolutionon the base cylinder, U only one toothis used. it winds around the base cylinderseveral times like a screw thread.

Worm gearing derives its character-istics .from the two simple machines ofwhich it is composed: the screw and thelever, From the screw it obtains a largemechanical advantage but a somewhatlower efficiency because of larger frictionforces.. The conjugate tooth action isidentical to that of a large spur gear andrack (Fig. 8). As the worm revolves, the

thread form advances along its axis andthe worm gear rotates ill correspondingamount. The presence of a scr~ insteadof a rack ensures ill vastly greater outputtorque on the gear shaH. The net ,effectis a torque converter of superior capa-city but, because of inherent slidingaction, one of reduced e.fficiency,

Worm gear terms are shown in Fig. 9.In this particular case the lead, thedistance advanced during one [ievolution,is three times greater than the pitch,Triple-threaded worms are more efficientthan single-threaded worms (due to lessfriction), but their reduction ratio is onlyone-third that of the single~th"eadedworm,

Fig, 10 shows the basic difference be-'tween single- and double-thread wO.MnS.

The slope or helix angle of the double-thread wonn is tunce that of the single-thread wonn, as is the lead. Thus, forone revolution the double-theead wormwill advance or tum. its mating gea:r anangle twice that 'of the single-threadworm ..

Tooth breakage hom. bending actionis not prevalent in.worm gear sets. Withrelatively high sliding velocities, thedesign criteria are usually based on scor-ing and pitting. Scoring is weal' resultingfrom failure of the lubricant film due 'tolocalized overheating of the mesh, per-

Fig. 8 - Worm gear. In this drawing the worm isrepresented as an endless rack. The· resulting pilchsurfaces an! a plane and a cylinder. [Courtesy MobilOil Corporation.)

Fig. 9-Worm gear terms. (Courtesy MobU OilCo.rporation. )

miuing metal-to-metafccnract ..Becausemore teeth are ineentact simultaneously,worm. gearing provides smoother opera-Han than involute gearing. The contactarea is also larger. Thus load capacity ishigh despite sliding action and fine con-tact. Entering side wedgesare producedon modem worm. gears to generate aload-supporting oil film ..

Theadvantagti of worm gearing com-pared to spur and helical gearing are:

1. A more compact design for the samereduction ratio or power capacity.

2. Much greater speed reduction and tor-que amplification in. a. single step.

3. Smooth and silent operation that canwithstand higher shock loads andhigher momentary loads.

The disadvantages compared with Of-

dinary gearing are:

1.. Lower and varying ,efficiency..2:. Greater axial forces requlnng costlier

bearings.3,. Overheating that limits duration and

capacity for power transmission.

Worm 'Gear Terminologyand Ki~e:matics

Fig. 11 shows worm gear terminologyand development of a worm thread. Forreasons of clarity, a triple-threadedworm. was used. A worm can be single,double, triple, quadruple, or multi-'threaded, plus left or righthanded.Threads in excess of 10 are rarelyadvantageous.

Axial pitch (p,.) of a worm (fig. 11) isthe distance measured axially from aPoint on one thr-·- -- ad to til -,colT""'""n.:l:""~a e IJ_ --~~ ... -_c""''6

•.............

•Fig. 10- The difference between a single- and a double-thread wonn.

(Courtesy Bureau of Naval Personnel.)

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September/October 1,986 51

Fig. 11- Wo.rmgear terminology and development of a worm thread. (For clarity, a multithreaded wormwas used.)

point on the next. thread. For propermesh, Pa must equal the circular pitch p',of the gear.

Lead (L) is the distance L that a threadadvances in one tum of the worm. Thus

where N is the number ·of threads on theworm; e.g., Nw = 2 for a double-threaded worm.

Lead angle t'Xu,) is the angle betweena tangent to the thread at the pitchdiameter and a plane normal to theworm axis.

Velocity ratio (mG) is the ratio ofpitch circumference of gear to lead ofworm, which equals tooth ratio. It is alsothe ratio of worm speed to gear speed.Thus

p4Nc; NG"" Deme =T= N...= nc = D. tan A. (5)

For worm. and gear to mesh properly,the lead angle of the worm must equalthe helix angle of the gear (Xu, = VtG),and axial pitch of the worm must equalcircular pitch of the gear (pQ= Pc).Since the circumference of the pitch cir-de of the gear can be expressed as 1tDGOr p,NG, this leads to

58 Gear Teel1nology

Substitution into Eq. 5 gives

(4)as another expression of the speedrelationship.

Development of one turn. of the wormleads to a triangle, as shown in Fig. 11.This triangle shows that

Center distance is 0,5 (Ow + Dd,which can. also be 'expressed as

LC =217 (me + cot A",) (9)

(6)

Thermal Ratings\lVhen worm gear teeth slide across the

surfaces of mating worm threads, farmore heat is generated than when thesame load is carried by any other typeof gearing. These thermal conditionsraise the operating temperature of the oil,thereby reducing its load-carryingcapacity. To keep oil temperatures belowcritical levels, manufacturers publishthermal ratings for each unit that indicatethe maximum power input that producesa safe rise in oil temperature. Because

t G.II-----1f-7iL---t-.>"''-!---+----::~1-t

;.1I--:H'i+-t

fig. 12-Efficiencyas a function of lead angle andcoefficient of friction, -

(8)

common gear lubricants deterioraterapidly at temperatures above 90°C(19S°F), operating levels are usually keptat or below 75°C (170°F). dearly, designof worm gearing must include tempera-ture effects, as well as strength and wear,as a limiting factor. Of thethree, over-heating is the controlling parameter. Forexample-a worm gear may have a ther-mal raring of 5 kW and a mechanicalrating of 7 kW for the higher speedranges. This means that the gearing iscapable of transmitting more than 5 kWas far as strength and wear are con-cerned - but not without overheating ..Recently, however, the use of computershas improved worm geometry, therebynarrowing the gap between thermal andmechanical capacity ..

Efficiency of Worm Ge.aringWhile spur and helical gearing exhibit

very high and virtually constant efficien~cies (0'.98 to 0.99),. those of worm gear-ing may range from a low of 0.50' to ahigh of around 0.98. General~y, ,effici-ency varies inversely with speed ratioprovided the coefficient of friction doesnot change. The various parametersdetermining efficiency do not have alinear relationship (!Fig, 12). Instead, ef-ficiency e increases with decreasing coef-ficients of friction. Efficiency is greatlyinfluenced by the lead angle for smallvalues, but less and less as ). increases ...A maximum is reached for). = 45 deg ..

For a well-designed, well-lubricatedunit, the following isa fair approxima-

Fig. 13 (Right) - Typical worm gear reduction unit.(Courtesy Bodine Electric Company.)

tion to efficiency.

tall JIi.,(!=----

Ian (A. + 6) (10)

where

(J = arctan f = friction angle

Back-driving is the tenn used when thegear drives 'the wonn. In this reverse ac-tion speed is increased at the expense offorce. Back-driving can thus be used toadvantage in speedup drives for cen-trifuges and turbochargers .. The corre-sponding expression for efficiency is

lan('\ - 8)e= (11)

tan A

Theoretically, back-driving is possibleonly for .>. >IJ, that is, when the lead .angIeis greater than the friction angle. Inreality, this reverse action occurs forhigher values of 8. Vibration, present inmost mecharueal equipment, effectivelylowers friction, thereby reducing "self-locking" to "a mechanical fringe benefit."Only a brake can effectively preventback-driving.

Optimum DesignEven though worm gear dnves are

among the oldest mechanisms, they areone of the least understood. Because oftheir inherent complexides, preciseanalytkali methods have not evolved, sodesign relies heavHyon trial-and-errortesting .. Worm gears have, however,reached a high degree of perfection.Thus, by analyzing the expression for ef-ficiency, we may single out major designparameters and compare them with theindustrial end product

Low coefficients of friction are ob-'tamed by musing dissimilar metals forworm and gear, (2) providing smoothtooth surfaces, and (3) ensuringadequat'elubrication. For instance, hardened andground steel worms are used with gearsof phosphor bronze or cast iron. Speciallubricants are available for wormgearing.

large lead angles are also desirable,butare obtained at the expense of lower

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speed ratios. Consider the followingequation.

L p.N;tan A ::: -. "" -- (U)

TrD"" TrDw

A large efficiency requires at large lead;therefore the worm should bemultithreaded, Consequently, whenworm gearing is designed primarily fortransmitting power, it should be multi-threaded, as is common practice.

'10 obtain a given ratio, some numberof worm wheel teeth divided by somenumber of worm threads must equal theratio. Thus, if the ratio is 6:

6 12 18 24 30 36 42I1Ia =:- =: - = -=- = -== -;;;;; -=

1234567

Any of these combinations may be used.The numerators represent the number ofworm wheel teeth, and the denominatorsare the number of worm threads. As thetotal. number of teeth increases, so doesefficiency,. but only at higher initial cost.

The expression for A also indicates thata small worm. diameter D"" is mostdesirable because it Jowers rubbingvelocity. As can be seen in Fig. 13, the

worm diameter is small relative to thethread height.

Example .2A right-angle speed reducer has a

triple-threaded worm and a 41-toothgear. find the speed ratio, lead, leadangle, helix angle, pitch diameter of gear,center distance, efficiency, power output,and transmitted force for the followingdata.Pc = 32 mm P1 = 900 rpm

Dw = 44mm

f = 0.05 p = 0.75 kW (input)

96mm= arctan (A.'" ) =: 34.78 dqJrr .... mrm

For proper mesh, the lead angle of theworm must equal the helix angle of thegear. Thus ,pG = Aw = 34.78· deg

As can be seen in Fig. 12

IP.. = 90 deg - A", = 55.22 deeFrom Eq, 6:

"" (32 mm)41= 417.62 mm1T'

From Eq. 9:

C ;;;;;!- (me; + col A,..)2rr

96mm - - .= --= (13.67 +- cot 3AI.78 deg)21T'

= 13(J.86 mm

From Eq. 10:

tan AN'e=--""";;-tan (A", + l!)

60' Gear Technolo,l'Y

Will the unit back-drive?

Solution

No 41ma = - ;;;- = 13.67Np 3 .

From Eq. 4:

L = N...pg =: 3(32 mm) =: 96 tom

From Eq.8:

LA .. "" arctan-

1TfJ'.,

CU~CLE A~115ON REA!DER REPLY CAIRD

Fig. 15- Typical medium-size worm gear speed reducer. (Courtesy of Morse Industrial Corp., subsidiary of Emerson Electric Co.l

September/October 1986 61

Ibl UranU' ring mou".td on h ....'b by mL"lmof ,I prttliir

Shah (input)

IIIWer plug

H""bltuill,orUD won)

-.-.- ~1,.1) Fllngtd r.ng mounf,1d on tht: 'hub b.- mMi'\l, 0' IhlIi' b01tl

Roller bearingradial tnrust

Shaft (output)

Fig. l4-Steering mechanism. ate that only a. gear sector is needed. (CourtesyThe Timken Company.I

Fig. Hi-Design of a large worm gear.

'Worm gear (bronze)

Rolling element bearing

'-----Oil seal

(continued on page 63)

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.. . ;.:::;,' ,.-

---

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62 Gear Technology

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(continued from page 60)

_~tan__.·_,_34_._78_· _dc-,,&:.....__ = 0.'90tan(34.78 deg + 2.86 deg)

Output: P = 0.90(0.75 kW) = 0.675 kW

The unit wiJ] back-drive because thelead angle of the worm tAw =34.78deg.) is much greater than the frictionangle e where

e =ardan f

= arctan 0.05 = 2.86 deg. «34.78 deg,

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GEARS-SPLINESIDE.SIGNIAN'D T'OOUNG

• Custom gear design including non-standard pressure angles for morest,rength. . -

• Programs to enter tooling data in-to computer files and search for ,ex-isting cutters to. cut a new gear orspline,

• Gearing computer software forsale. -

• Consulting services for gear andspline problems.

VAN GEFlPEN·FlEECE ENGINEERING1502 Grand Blvd,

Cedar Falls, Iowa 50613(319) 2664674

CIRCLE A.-36 ON READER REP,lY CARD

reduction ratios, but at the expense of ef-ficiency. In contrast, 8. multithread speedreducer will have high ,efficiency, but a.somewhat lower reduction ratio, Thisfact leads to three ma.jor areas of ,applica-tion for wonn gearing.

1. .Intermittent, infrequent operationswherea small, low-cost motor movesa heavy load, as in small hoists. Effi-ciency, of minor importance, is thus

(continued on page 64)

WANTED,:GEAR IMANUFACTUFlERWl1iHIFOULOWING CREDENTIALS'

• Certifiable Quality contrailS~'stemGM,SlIear 2.Ford Q2, mil spec 45208 or better,

2) Current use of SPC/SPS techniques,

3) Capability to manufacture gears to class 11 orbetter,

4) Experience in prototype as well: as high volumeruns,

5) Ability to assimilate large volume of wolik In a6-12 month period,

- Beveland spiral bevel capability a plus but notrequired.

- Respondents should be willing to enter into arepresentanen agr'eement.

Several ml1l1on dollars at' gear workavailable to' qualified manufac'urer.

Please respond promptly and in confidence to:

Suits 3054217 Highlandl Road • Pontiac. M148054

CIRQE A·36 ONI READER' REPLY CAR,D

'GEAR TOOTH GRINDING& HON/N,G ,ONL.Y

IF'mduction 'Ouantmes3/4" F'..O, to 27.5" P.O..;3.5 D.IF'.and 11" Face

We have no tumlng, hobblng orshaping capability

ALILIEGHIENY GEAIR CORP.23 Dick Road

Depew, NY 14043716-684·3811

CIRCLE A·17 ONI READER REPLY CARD

September/OCtoberl'986 6.3

IMPROVED GEAR LIFE .(continued from page 18)

References1. FUCHS, H., Shotpeening Effects and Specifications, ASTM Special

Technical Publication No. 196-1962.2. DALY, J" "Boosting Gear Life Through Shot Peening," Machine Design,

May 12, 1977.3. "Shot Peening of Metal Parts," MIL-S-13165B Amendment I!, June 25,

1979.4.. Shot Peening Applications, Metal Improvement Company, Sixth Edi-

lion, 1980. Figs. 2, 6, 8 and 9.5. McCORMACK, DOUG, "Shot Peen Gears for Longer Life," Design

Engineering, July, 1981.6. FUCHS, H.O., 'The Effect of Self Stresses on High Cycle Fatigue,"

JTEVA Vol. 10, No.4, July, 1982.7. LAWERENZ, MARK, Shot Peening and its Effect on Gearing. SAE

paper #841090.8. SEABROOK. JOHN B. and DUDLEY. DARLE W., "Results of Fifteen

Year Program of Flexural Fatigue Testing of Gear Teeth," 1963.9. TOWNSEND, DENNIS P. and ZARETSKY, E.V .. "Effect of Shot Peen-

ing on Surface Fatigue Life of Carburized and Hardened A151. 9310Spur Gears," Lewis Research Center, Cleveland, Ohio, NASA 2047.

10. Lloyd's Register of Shipping, June 6, 1975, Letter.

Reprinted with permission of the A merican Gear Manufacturer's Association.First presented at AGMAs Fall Technical Meeting, October. 1985.

The opinions, statements and conclusion presented in this paper are those ofthe Autho1:5ami in rro way represent the posiiion or opinion of the AMERlCANG£4R N1ANUFACTURERS ASSOCIATION.

PRACTICAL ANALYSIS OF HIGHLY LOADED ...(continued from page 26)

ConclusionBased on the DIN/ISO formulae for scoring capacity, a

simplified method adopting a modified scoring index has beendeveloped. As can be seen from typical applications, thismethod works with sufficient accuracy.

The calculation of scoring capacity will become more andmore important in parallel with an increasing demand intransmitted power per gear volume. The practical experiencewith highly-loaded gears with regard to scoring will give moresafety in the application of this calculation method and willpossibly permit a reduction of the safety margins used today.

References

1. HlRT, M. "An Improved Method to Determine the Scoring Resistanceof High Power High-Speed Gearing," Proc. of 1984 TurbomachinerySympOSium. Texas A&M University, Houston

2. AGMA 217.0, "Gear Scoring Design Guide for Aerospace Spur andHelical Power Gears." - -

3. BLOK, H., "Theoretical Study of Temperature Rise at Surface of Ac-tual Contact Under Oiliness," Lubricating Conditions. Proc. Gen. Disc,Lubric, IME London Bd 2(1937) Sh. 225.35,

4. DIl\I3990, Part 4: "Basic Principles for the Calculation of Load Capacityof Spur and Helical Gears, Calculation of Scuffing Load Capadty,' Draft(1980).

5. DUDLEY, D., Gear Handbook, McGraw Hill (1962).

6. HIRT, M .. "Design and Calculation Methods for High Speed Gears ofAdvanced Technology". Proceedings of the Twelfth TurbomachinerySymposium, Texas A&M University, College Station, Texas (1983).

7. NIEMANN, G. and WINTER, H., Maschinenelemente Part II, Springer(1983).

8.. WINTER, H. and MICHAELIS. K. "Scoring Load Capacity of GearsLubricated with EP-Oils," AGMA Paper 219.17 (Fall. 1983).

64 Gear TechnOlogy

GEARS FOR NONPARALLEL SHAITS ...(continued from page 62)

sacrificed to obtain a large mechanical advantage. Typicalapplications are standby pumps, large valves, and gates.

2. Intermittent, manual operations requiring a largemedtarucal advantage, such as in steering mechanisms andopening and closing of valves and gates by means of hand-wheels (Fig. 14).

J. Motorized, nearly continuous operations where worm gear-ing competes with gear reduction units. \!Vhen space is ata premium, as in machine tools, packaged, motor-drivenworm reduction units are used in preference to. gear reducers(Fig..13). Depending on size and application, theunit maybe self-contained or built integrally with an electric motor.Because of silent operation, such units are preferred inmachine tools and also in elevators, These units all requiremultithreaded worms and ratios not exceeding 1:18. Largerratios are achieved by connecting two units in series.

Design Detail of Worm GearingThe unit shown in Fig. 15 is a typical, medium-size worm

gear speed reducer. Smaller units of this type usually havehousings of cast aluminum alloys for maximum thennal rating,For larger units the preferred material is cast iron. The wormis case-hardened and ground alloy steel of integral shaft. design.The gear is cast bronze with generated teeth and keyed to theoutput shaft. Larger worm gears are often composed of a ringof bronze mounted on a center Of hub of less expensivematerial. A common design utilizes a flanged rim mountedon the hub by means of shear belts (Fig. 16a). Equally com-mon is mounting by means of a press fit (Fig. 16b) assistedby a pin connection. The output shaft is high-quality, medium-carbon steel, ground to dose tolerances. The worms and out-put shafts are frequently mounted en roller bearings. All. shaftextensions are equipped with lip style, synthetic oil seals.

LubricationGenerally, oil is contained within the housing and directed

by splash to. the bearings and to. the zone of tooth and threadcontact. Natural splash may be augmented by flingers, scrap-ers, and cups attached to the gear. Channels or ribs may befurnished inside the ho.USing to. help direct OiJIto. the bearings.

SummaryDespite higher initial cost, gears for nonparallel shafts are

justified because they often save space and lead to. a betterdesign. Kinematically, these gears all perform the very difficulttask of changing the plane of rotation. With the exception ofcrossed helical gears, all have reached a high degree of perfec-tion and a long, useful life of transmitting power. Hypoid gearsfor automotive differentials, for instance, rarely fail during thelife of a car. The versatility of worm gearing is due to the in-verse relationship of efficiency to torque and reduction ratio.Table 1 summarizes comparative characteristics of speedreducer gear families.

References1. BUCKINGHAM:, E. K. 'Taking Guesswork Out of Wonn-Gear Design."

Machine Design, March 20, 1975.2. MacFARlAND. W. C. "Straight Talk About Speed Reducers." Machine

Design. September 18. 1975-0ctober 2, 1975.3. WILL, R. J. "Selecting Speed Reducers." Machine Design,

September 8, 1977.