BACK TO BASICS ...'Spur Gear Fundamentals

Fig. ] - Friction wheels on parallel shafts. (CourtesyMobil Oil Corporanon.)

Fig. 2-Spur gears. The teeth of these gears aredeveloped from blank cylinders. (Courtesy Mobil OilCorporation.)

Gears are toothed wheels used primar-ily to transmit motion and power between.rotating shafts. Gearing is an assembly oftwo or more gears. The most durable of allmechanical drives, gearing can transmithigh power at efficiencies approaching 0.99and with long service life. As precisionmachine elements gears must be designed,

36 Sear Technology

UHe HaidhedeBlack Hawk College

Moline, n

manufactured and installed with great careif they are to function properly.

Relative shaft position - parallel, in-tersecting or skew - accounts for threebasic types of gearing, each of which can bestudied by observing a single pair. This art-iclewill discuss the fundamentals, kinemat-its and strength of gears in terms of spurgears (parallel shafts), which are the easiesttype to comprehend. Spur gears composethe largest group of gears, and many oftheir fundamental principles apply to theother gear types.

function and DesignProbably the earliest method of trans-

mitting motion from one revolving shaft toanother was by contact from unlubricatedfriction wheels (Fig. 1). Because they allowno 'control over slippage, friction drivescannot be used successfully where machineparts must maintain contact and constantangular velocity. To transmit power with-out slippage, a positive drive is required, acondition that can be fulfilled by properlydesigned teeth. Gears are thus a logical ex-tension of the friction wheel concept(Fig.2).

Gears are spinning levers capable of per-forming three important functions. Theycan.provide a positive displacement coupl-ing between shafts, increase. decrease ormaintain the speed of rotation 'Withaccom-panying change in torque, and change thedirection of rotation and/ or shaft arrange-ment (orientation).

To function properly, gears assume var-ious shapes to accommodate shaft orien-tation. If the shafts are parallel, the basicfriction wheels and gears developed fromthem assume the shape of cylind 1'5 (Fig.3.a). When the shafts are intersecting, thewheels become frustrums of cones/andgears developed. on these conical surfacesare called bevel gears (Fig. 3.b). When theshafts cross (skew, one above the other),the friction wheels may be cylindrical or of

hyperbolic cross section (Fig. 3c). In addi-tion, a gear is sometimes meshed with atoothed bar called a rack (Fig. 3.a), whichproduces linear motion. Besides shaft posi-tion and tooth fonn, gears may be classi-fied according to:

System of Measurement. Pitch (EU)or module (S1).Pitch. Coarse or fine.Quality. Commerical, precision andultraprecision or tolerance dassifica-tionsper AGMA 390.03.

Law of GearingGears are provided with teeth. shaped so

that motion is transmitted in the manner ofsmooth curves rolling together withoutslipping. The rolling curves are called pitchcurves because on them the pitch or toothspacing is the same for both engaginggears. The pitch curves are usually circlesor straight lines, and the motion trans-mitted is 'either rotation or straight-linetranslation at a constant velocity.

Mating tooth profiles, as shown i:n.l1g..4, are essentiall y a pair of cams .in con-tact (back to back). For one cam to driveanother cam with a constant angular dis-placement ratio, the common normal atthe point of contact must at all times in-tersect the line of centers at 'the pitch point.This fixed point is the point of tangency ofthe pitch circles. To ensure continuous con-tact and the existence 'of one and only onenormal at each point of contact, the cam-like tooth profiles must be continuous dif-ferentiable curves.

AlITHOR:

urnHINDHf:DE is an Associate Professorin the Enginefflllg Related Technology Depart-ment at Black Hawk College. Molille, Illinois.He is a graduate of the Technical University ofDenmark and the University of Illinois. Hind-hede has authored seuem.l technl'ca/ papers andarticles and he is the principal author of MachineDesign Fundamentals.

"""foJ P.... II.' "'.fts.

$trl1.gh~ he! ical

fbJ IntorMC1.ng ""fts.

(c) Non ....parlne~. non - inmfllKt'":g: hklW~ mlfts.

Fig. J -Important types of gears.

Mtringbone Fig. 4 - Two cams showing the law of gearing. Thecommon normal must, for all useful positions, gothrough the fixed point P.

Mating cam profiles 'that yield CIi con-stant angular displacement ratio are termedconjugate. Although an infinite number ofprofile curves will satisfy the la.w o.f gear-ing, only the cycloid and the involute havebeen standardized .. The involute has sev-eral advantages; 'the most important is itsease of manufacture and the fact that thecenter distance between two involute gearsmay vary without changing the velocityratio.

a = addendumb = dedendumB = backlash, linear measure

along pitch circleC = center distance

Co = velocity factorD = pitch diameter of geard = pitch diameter of pinion

Db = base diameter of geardb = base diameter of pinionDo = outside diameter of geardo = outside diameter of pinionDr = root diameter of geard. = root diameter of pinionF = face widthh. = effective height (paraholi.c

tooth)h, = whole depth (tooth height)L = lead (advance of helical. gear

tooth in 1revolution)Lp(Lc;) = lead of pinion (gear) in helical

gears

NomenclatureM = measurement over pinsm = module

mf = face contact ratiomG = gear or speed ratio

(mG = NGIN,,)mn = normal modulemp = profile contact ratiomt = total contact ratio

N,,(NcJ = number of teeth in pinion(gear)

N, = critical number of teeth for noundercutting

nine) = speed of pinion (gear)Pa = axial pitchp& = base pitch (equals normal

pitch for helical gears)Pc = circular pitchPd = diametral pitchPn = normal diametral pitch of

helical gearp.. = normal circular pitch of

helical gear

R = pitch radius gearr = pitch radius pinion

'b(R,,) = base radius of pinion(gear)

r,,(Ra> = outside radius of pinion(gear)

Se = endurance limittc = circular tooth thickness

(theoretical)t = tooth thickness at rootv = pitch line velocity

W = tooth load, totalWa = axial loadW. = radial loadWt = tangential load

y = tooth form. factorZ = length of action~ = pressure angle

~11 = pressure angle in normalplane

'" = helix angle

January jFebruary 1989 37'

·In.volute Gear Principles"An involute curve is generated by a

point moving in a definite relationship toa circle, called the base circle. Two prin-ciples are used in mechanical involutegeneration. Figs. Sa and Sb show the prin-ciple of the fixed base circle, Inthis methodthe base circle and the drawing plane inwhich the involutes are traced remainfixed, This is the underlying principle of in-volute compasses and involute dressers(where the motion of a diamond tool givesan involute profile to grinding wheels),

The second principle, that of the revolv-ing base circle, is used in generating in-volute teeth by hob bing, shaping, shavingand other finishing processes (Fig. se). Thismethod is employed primarily where thegenerating tool and the gear blank areintended to work with each other liketwo gears in mesh for purposes of gearmanufacturing.

Cord Method. In using this method theinvolute path is traced by a taut, inexten-sible cord as it unwinds from the cir-cumference of the fixed base circle (Fig. 6).The radius of curvature starts at zerolength on the base circle and increases

steadily as the cord unwinds, After onerevolution, the radius of curvature equalsthe cirClllIlference of the base circle (11'Dol.It is significant that the radius of curoatureto (my point is always tangent to the basecircle and normal to one and only onetangent on the involute.

From Fig. 6 it can be seen that the fullinvolute curve is a spiral beginning at thebase circle and having an Lnfinite numberof equidistant coils (distance 1fDl:J How-ever, only a small part of the innermostcord is used in gearing. The character of theinvolute near the base reveals the existenceof a cusp at point 0 and a second branch ofthe involute going in the opposite direction(shown as a dash-dot curve). The secondbranch serves, as we will see later, to form.the back side of the gear tooth after a spaceis left for a meshing tooth.

Properties of the Involute. The follow-ing properties can be seen in Fig. 6,

1. Any tangent to the base circle is alwaysnormal to the involute.

2. The length of such a tangent is the radiusof curvature of the involute at thatpoint. The center is always located on

Ihi Rolling beam met,",odHbeed base errelel

lei Ae'Volving base crrcles, thl!' b9m '00 basecircle roll with eacn other without iJhpo

Fig, 5 - Generation of the involute.

FOf mil poslliiQn the ge'l"Ief'3llng radius rD~Is eqUI' to Ihe baY Circle cirr:umferenc-eo

fig. 6 - Generation of an involute by the cordmethod.

38 Gear Technologv

Fig. 7 - Geometric similarity of all involutes explainswhy the teeth of a large gear can mesh properly withthose of a small gear.

the base circle.3. For any involute there is only one base

circle.4 .. For any base circle there is a family of

equivalent involutes, infinite in number,each with a different starting point.

5 ...A11 involutes to the same base circle aresimilar (congruent) and equidistant; forexample, the distance of any two of suchinvolutes in normal direction is constant-o, (Fig. 6),

6,. Involutes to different base circles aregeometrically similar (Fig. 7). That is,corresponding angles are equal, whilecorresponding lines, curves or circularsections are in the ratio of the base cir-cle radii ..Thus, when the radius of thebase circle approaches infinity, the in-volute becomes a straight line, Geomet-ric similarity explains why the teeth ofa large gear can mesh properly withthose of a small gear.

Involutes in Contact. Mathematically,the involute is a continuous, differentiablecurve; that is, it has only one tangent andonly one normalat each point. Thus, twoinvolutes in contact (back to back) haveone common tangent and one commonnormal (Fig. 8). This common normal, fur-thermore, is a common tangent to the basecircles. Since this normal for all positionsintersects the centerline at a fixed point,conjugate motion is assured. Thus con-jugate motion is the term used to describethis important characteristic of involutegear action.

The action portrayed is that o.ftwo over-size gear teeth in contact. The oversizedteeth are the result of using too large acenter distance. This situation is presentedonly for reasons of clarity,

When two involute gear teeth move incontact, there is a positive drive impartedto the two shafts passing through the basecircle centers, thus ensuring shaft speedsproportional to the base circle diameters.This is equivalent to a positive drive im-parted by an inextensible connecting cordas it winds onto one base circle and un-winds from the other. It is analogous to apulley with a crossed belt arrangement.Note that the surfaces of both involutes atthe point of contact are moving in the samedirection .

• The material. in the following three sect-ions was inpart extracted hom a gear manual formerly used atInternational Harvester, Farmall Works. courtesyRobert Custer and Frederick Brooks.

A rack is a.gear with its center at infinity.It is a simplified gear in which all circlesconcentric with the base circle and all invo-lutes have become straight lines. A racktherefore has a baseline and a linear toothprofile.

Relationship of Pitch and Base Circles.Referring to Fig. 8, we find that trianglesQC10] and QC20Zare similar. Therefore

rb Rbcos 0 = --;= Ii

where

rr" Rb = base circle radius of pinion andgear, respectively; mm, in.

r, R = pitch circle radius IQf pinion andgear, respectively; mm, in.

Also,

db Dbcos~ = d = 0

where

db, Db = base diameters of pinion andgear, resepectively: mm, in.

d,D = pitch diameters of pinion andgear, respectively; mm, in.

(1)

Fig. 9 shows the smaller of the two gearsin Fig. 8 meshing with a rack (obtainedby moving the center of the larger gear 'toinfinity). The rack is represented by asingle tooth that can move horizontally,as shown. If the involute is turned coun-terclockwise the "rack" will move to theright because of a horizontal force compo-nent. The motion of rack and pinion isconjugate because the pitch point has notchanged and the normal to the rack toothgoes through this point.

Pitch circle diameter ~ d

P,lch Circle drameter lJ

CenterD,stance

(2)

The Mechanics of Involute TeethEffect of Changing Center Distance. Fig. 10shows the same two involutes as in Fig 8brought into contact through appropriaterotation on a reduced center distance (2").

Fig. 8-Curved involutes in contact.

Consequently:• A new pitch point was established.• The pressure angle was reduced from

70 to 50° (still large by normalstandards).

• The line of action was shortened.• The pitch diameters were reduced

(halved), but their ratio remained un-changed (similar triangles).The speed ratios are not affected by

altering center distance because they arefunctions of base radii only . The twotriangles (crosshatched) remain similar,regardless of center distance alterations.Furthermore, two corresponding sides, thebase radii, do not change; hence, the ratioof pitch radii cannot change.

Rolling and Sliding Action BetweenContacting Involutes ..Fig. 11a returns thetwo involutes from Fig. 8 to their former,larger center distance (4"), but in a differentrelative position - the only position forwhich corresponding arc lengths are equal(arc 12 = arc 12'). By the belt analogy, anequal length of cord has been exchangedbetween the smaller base cylinder of the pi-

Fig. 9- Curved involute containing a flat surface(rack tooth).

.Fig.10- Effect of changing center distance.

JanuaryjFebru.ary 1989 39

I---~'--,..-Corrt'spond'mgtenqrhs of arc,

III\\\\,

\,"""

(u)

Fig. lla - Posit jon of minimum sliding is the one shown here where arc 12 equals arc 12', For rotation in eitherdirection, sliding action will increase until the contact point reaches the base circle.

PilCh (nell!

'hi

tOOl" spacing onbulh PIU!; cv-ves

Fig. I1Il - Rolling and sliding action between two in-volutes on fixed centers. Contact between arcs 10 and10' involves much sliding, since arc 10 is almost one-third longer than arc 10'. Fig. 12 - Gear geometry and terminology.

40 GearTechnol'ogy

19'

nion and the larger base cylinder of thegear. The angular but equal displacementsof the gear are, therefore, smaller thanthose of the pinion by a ratio of 1:1.5.

For clarity, the angular increments of thepinion were chosen at 22.50

, making thoseof the gear 15° The length of arc corre-sponding to each pair ofincrements will bein contact during rotation. However, sinceeach pai:r varies in length, the rolling mo-tion of one involute on another inevitablymust be accompanied by sliding becausethe time elapsed to cover correspondingbut unequal lengths is the same.

For counterclockwise rotation of thepinion, the lengths of arc 11, 10, 9 and soforth, will be in synchronized contact withthe arc 11',10'.,9' and so on (Fig. lIb). Theformer steadily increase in length, but thelatter steadily decrease in length, makingsliding inevitable. Rotation in oppositedirection produces the same .effect, hut thistime gear teeth have the greater surfacespeed. Maximum sliding takes place whenthe point of contact is dose to either basecircle. Thus, in gearing, only a small sec-tion of any involute is useful if sliding is tobe minimized.

Gear TerminologyBasic Terminology. To make further

discussion more meaningful, geometricquantities resulting from involute contactwill now be defined, discussed and as-signed nomenclature, as shown in Figs.12-14.

Pinion. A pinion is usually the smaller oftwo mating gears. The larger is often calledthe gear.

Center distance (C). The distance be-tween the centers of the pitch or basecircles.

C = 0.5(D + d) (3)

where

D = pitch diameter of geard = pitch diameter of pinion

Base circle. The circle from which an in-volute tooth curve is developed.

Base Pitch. (pJ. The pitch on the basecircle (or along the line of action) cor-responding to the circular pitch.

Pitch Circle. Since the pitch point isfixed, only two circles, each concentricwith a base circle, can be drawn throughthe pitch point. These two imaginarycircles, tangent to each other, are the pitch

circles. They are visualized as rolling oneach other, without sliding,. as the basecircles rotate in conjugate motion.

The ratio of pitch diameters is also thatof the base diameters (similar triangles).Because the pitch circles are tangent to eachother, they are used in preference to thebase circles in many of the calculations.Note that pitch circles must respond to anycenter distance variation for a meshing gearpair by enlarging or contracting. In con-trast, the base circles never change size,

Circular Pitch (pJ. The identical toothspacing on each of the two pitch circles.

Pressure Angle (¢). The pressure anglelies between the common tangent to thepitch circles and the common tangent tothe base circles, shown exaggerated inFig. 8. The pressure angle is also the acuteangle between the common normal and thedirection of motion, when the contactpoint is on the centerline ..Since a pair ofmeshing gear teeth is, in essence, a pair ofcams in 'contact, the pressure angle of gear-ing is identical to the one encountered in

de .'. Th ... r ~~~1 f· T t-cam eSIgI1... epresswe ... 'O'eo. conacinginvclotes, as opposed to the one oncams, is constant throughout its entirecycle, a feature of great practical im-portance.

Line of Action. This is the commontangent to the base circles. Contact be-tween the involutes must be on this line togive smooth operation. Force is transmit-ted between tooth surfaces along the line ofaction. Thus a constant force generates aconstant torque.

VelD city Ratio. (me;>. This ratio.alsecalled speed ratio, is the angular velocity ofthe driver divided by the angular velocityof the driven member. Because the line ofaction cuts the line of centers into therespective pitch radii, the speed ratio.becomes the inverse proportion of thosedistances and related quantities (e.g., baseand pitch diameters).

Because most gears are designed forspeed reduction, one generally finds the pi-nion driving the gear; from now en we willassume that this is the case. Therefore

r!p D NGmG = riG = d = Np (4)

whe,,;e

mG = speed ratio (gear ratio)np = speed of pinion; rpm

Rg. IJ-SpuT gear geometry. (G. W. Michalec. Precision Gearing, Wiley, 1966.)

TQQ\'" 'II.....k

BOl1om ItndTop ~lrld

FUIet nlCh'!A

Tooth Rltfaee'-- __ P.,"'1'i- CI'l'Ctll!i

~---ROO1't:I"'[le

.fig. 14- Tooth parts ofspur gears.

riG = speed of gear; rpmD = pitch diameter of gear; mrn, in.d = pitch diameter of pinion; mm,

in.NG = number of teeth in the gearNp = number of teeth in the pinion

In practice, speed ratios are determinedprincipally from ratios of tooth numbers

Fig. IS -Measurement over pins.

because they involve whole numbersonly.

Tooth Parts. The following tooth partsare shown in Figs. 13 and 14.

Addendum. Height of tooth abovepitch circle(Fig. 13).

Bottom land. The surface of the gearbetween the flanks of adjacent teeth(Fig. 14).

Dedendum. Depth of tooth below the

January/February 1989 41

Ag. 16 - Involutegear teeth are generated by a series of symmetrical involutes oriented alterna! Iy in a clockwiseand counterclockwise direction. For the two gears to mesh properly, they must have the same base pitch.

The,equ;"'Mm'1 tooth IS asingle tooth o~ ev-er·incrl!!a~r.,length wl,h contact at 7'

Base pitch

j

I'-'-'rI

/1,,/ r

Pinion

"\,\,

I "I\\\

\II

II

Rg. 17-A succession of short symmetrical involutes gives continuous motion to a rack in either direction,

pitch rude (Fig. 13).Face widtll (F). length of tooth in axial

direction (Fig. 14).Tooth face. Surface between th pitch

line element and top of tooth (Fig. 14).Tooth fillet. Portion of tooth flank join-

ing it to the bottom land. (Fig. 13).Tooth flank. The surface between the

pitch line element and the bottom land(Fig. 14).

Tooth surface. Tooth face and flankcombined (Fig. 14).

Top land. The surface of the top of thetooth (Fig. 14).

Circular tooth thickness (t~). Thisdimension is the arc length on the pikhcircle subtending a single tooth. For equaladdendum gears the th oretical thicknessis half the circular pitch (Fig. 14).

Overpirrs measurements (M). The pitchcircle is an imaginary circle; hence, pitchdiameters cannot be measured directly.However, indirectly the pitch diam terofspur gears can be measured by th pinmethod .\Nhen spur gear sizes are checkedby this method, cylindrical pins of knowndiameter are placed in diametrically op-posite tooth spaces; or, if the gear has anodd number of teeth, the pins are locatedas nearly opposite one another as possible(Fig 15).The measurement M over thesepins is then checked by using any suffi-ciently accurate method of measurement.

Involute Gear TeethSo far we have considered only two

profiles in contact. However, successiverevolutions are merely successive contactsof two profiles. fig. 16 shows how a seriesof symmetrical involute profiles, alter-nately clockwise and counterclockwiseand with a tooth space allowed formeshing, will produce a complete set ofpointed teeth. By using only th portionnear the base circle, mating gear toothprofiles can be formed with two or rnorteeth in contact at all times, thus permit-ting continuous rotation in eitherdirection.

Fig. 17 shows in greater detail thedevelopment of curved and straight teeth.Continuous motion of a rack necessitatesa series of short, symmetrical, equallyspaced involute teeth on the base circle cir-cumference. The pitch, in this case, isnamed base pitch (Pb)' Point. 1 is the pointof tangency for the line of action. Thus thedistances 1-2,2-3 and the like, measuredalong the base circle, are all equal: by

definition, this is the base pitch. Frompoints 2 to 7 involutes have been extendeduntil they intersect the line of action,dividing it into distano s equal to the basepitch. From the equidistant points l' to7', lines have been drawn perpendicularto the line of action. The full line portionrepresents one side of the rack.

As the gear rotates, the gear teeth willcontact successive rack teeth in a con-tinuous, overlapping motion. The forcewill be exerted along the line of action,causing the rack to move horizontally.The pressur angle, as shown, is the acutangle between the directions of force andmotion.

When the gear rotates in the directionshown, the surface of an involute geartooth contacts the nat-surfaced racktooth. As rotation continues, the contactpoints move down the line of action awayfrom the base circle. This continues untilthe tooth surfaces lose contact at the upperend of the line ofaetion represented by thefull line. Before contact is lost, anotherpair of teeth come into contact, thus pro-viding continuous motion. This tooth ac-tion is equivalent to the action of a singletooth of ever-increasing length contactingone ever-increasing Rat surface along anever-increasing line of action. The out-ward motion of the involute originating atpoint 7 mirrors the equivalent ingle toothaction. For the position shown. contact isat point 7 ' . Rack and pinion, like meshinggears, have two pressure lines and, hence,permit motion in both directions.

Summary of Involute Gears. The sim-plicity and ingenuity of involute gearingmay be summarized as follows.• Involute profiles fulfill the law of gearing

at any center distance.• AU involute g ars of a given pitch and

pressure angle can be produced from ontool and are compl tely interchangeabl .

• The basic rack has a straight tooth pro-file and therefore can be mad accuratelyand simply.

Standard Spur 'GearsSpur gears can be made with greater

precision than other gears because they arethe least sophisticated geometrically. Allteeth are cut across the faces of the gearblanks parallel to. the axis, a procedure thatgreatly facilitates manufacturing and ac-counts for the relatively low cost of spurgears compared to other types. Spur gearsare therefore the most widely used meansof transmitting motion and are found in

everything from watches to drawbridges.Pitches and Medules. The base pitch

(Pb) is the distance between successive in-volutes of the same hand, measured alongthe base circle. It is th base circl cir-cumference divided by the number ofteeth.

Mating teeth must have the same basepitch (figs. 16 and 17).

The circular pitch (Pc) is the distancealong the pitch circle betweencorrespond-ing points of adjacent teet h. Meshing teethmust have the same circular pitch (fig B).The pitch circle circumference is thus thecircular pitch times the number of teeth.

p)J = 1fD

7lDPc = N

Module: D mmm = N tooth (definition)

Diametral pitch:N teeth

Pd = D· -.- (definition) (8)m.

By substituting Db = 0 cos cp intoEquation 2, we obtain

Pb = Pc cos cp

Diametral pitch is related to the module asfollows.

mPd = 25.4 (10)

Module, the amount of pitch diameterper tooth, is an index of tooth size. Ahigher module number denotes a largertooth, and vice versa, Because module isproportional to circular pitch, meshinggears must have the' sam module.

p~ = 1frn (ll)

Diarnetral pitch, the number of teethper inch of pitch diameter, is also an indexof tooth size. A large diametral pitchnumber denotes a. small tooth, and viceversa. Because diametral pitch is inverselyproportional to circular pitch, meshing

gears must. have the same diametral pitch.

(12)

(5)

The diametral pitch is the number of't,eethper inch of pitch diameter and is nota pitch. A misnomer, it is easily conlusedwith base and circular pitch. To avoidconfusion, the word "pitch," when usedalone from now on, refers solely '10diametral pitch.

(6)

Standard Tooth Pmpod:iofl5of Spur Gears

Gears are standardized to serve thosewho want the convenience of stock gearsor standard tools for cutting their owngears. To meet these needs, however, gearstandards must provide users with suffi-cient latitude to.cover their requirements.Optimum design requires a wide range ofpitches and modules, but only a fewpressure angles. There should also beanextensive choice in the number of teethavailable. A practical range or stock gearsis from 16 to 120 teeth with suitable in-cremental steps. The correspondi.ng ratiosvary from 1:1 to 7.5:1.

Pressure Angle. The preferred pressureangle in both systems- modul and pitch- is 20", followed by 2'5", 22.5", and14.5". The 20" angle is a. good com-promise for most. power and precisiongearing. Increasing the pressure angle, forinstance, would improve tooth strengthhut shorten the duration of contact,Decreasing the pressure angle on standardgears requires more teeth in th pinion 10avoid undercutting of the teeth.

Diametral Pitch System. This systemapplies to most gears made in the UnitedStates and is covered by AGMA stand-ards ..These standards are outlined in 65technical. publications available fromAGMA. For gear systems we have201.02-1968: Tooth Proportions forCoarse-Pitch Involute Spur Gears.

Despite the rapid transition ItoS[ by themechanical industries, the change to themodule system wiJl probably be slower.The reasons are:

(7)

(9)

.' AGMA has yet to complete its SIstandards.

.' Many existing gear hobs (tools. for mak-ing gears), for reasons of economy, willbe kept in service and not be replacedunri] worn out.

• The need for repair of older gears willcontinue for several decades.

Janucry/Februcry 1989' 43

Thus, future gear reduction units may beall metric except for the pitch system.

Selection of pitch is related to load andgear size. Optimum design is achieved byvarying the pitch, but rarely the pressureangle; hence, there isa wide selection of"preferred values" (Table 1). Small pitchvalues yield large teeth; large pitch values

~iJr;». T

I

Fig.1S-Contact ratio, mI"

Lccahzed ~COr!"9

fig. 19 - Tip and root interference. This gear showsdear evidence that the tip of its mating gear has pro-ducedan interfence condition in the toot section.Localized scoring has taken place, causing rapidremoval in !he root section. Cenerally, all interferenceof this nature causes considerable damage if not cor-rected. (Extracted from AGMA Standard Nomen-clature of Gear Tooth Failure Modes (AGMA110.04), with permission of the publisher, the Amer-ican Gear Manufacturers Association.)

44 'Gear techno'iogy

yield small teeth. Table 2 gives involutetooth dimensions based on pitch.

Module System. Tooth proportions formetric gears are specified by the Interna-tional Standards Organizations (ISO).They are based on the ]50 basic rack (notshown) and the module m. A wide varietyof modules is available to cover everytooth size required from instrument gearsto gears fer steel mills. Table 3 shows onlythe preferred values ranging from 0.2 to. 50mm. Speciflc tooth dimensions are ob-tained by multiplying the dimension of therack by the module (Table 4).

Because of the simple relationship bet-ween pitch and module (mPd = 25.4),metrication of gearing does not seemoverly difficult. However. the transitionfrom pitch to module rarely yields stan-dard values. Thus module gears are not in-

terchangeable with pitch gears. Herein liesthe difficulty of metric conversion ingearing.

limitations on SpW' GearsTwo spur gears wi!] mesh properly,

within wide limits, provided they have thesame pressure angle and the same diame-tral pitch or module. Umitations are set bymany factors, but two in particular are im-portant: contact ratio and interference. Toobtain the contact ratio, the length of ac-tion must first be introduced. The length ofaction (Z) or length of contact is thedistance on an involute line of actionthrough which the point of contact movesduring the action of the tooth profiles. It isthe part of the line of action located be-tween the two addendum circles or outsidediameters (Fig. 18).

TABlEl National Pitch SystemCoarse Pitch

0.5 0.75 1 1..5 2 2.S 3 3..5 4 5 67 8 9 10 11 12 13 14 15 16 18

Fine Pitch

20 22 24 28 30 32 36 40 44 4850 64 72 80 96 120 125 150 180 200

Stub

TABLE 2 Involute Gear Tooth Dimensions Based on Pitch-Coarse

Full Depth

Pressure angle, 0 degAddendum, CI

Dedendurn, bTooth thickness, te(theoret ical)

200.801P1.00/ P7r/2P

Full Depth Stub

20liP1.2SIP7rI2P

2S0.801P1.00IP7rI2P

2SliP1.251 P1I'12P

TABLE 3 Preferred Values for Module m (mrn)

0.2 0.6 0.9 1.75 2.75 3.75 5 7 14 24 420.3 0.7 1.0 2.00 3 4 5.5 8 16 30 SO0.4 0.75 l..2S 2.25 3.25 4.50 6 10 18 360.5 0.80 1.50 2.50 3.50 4.75 6.5 12 20

TABLE 4 Involute Gear Tooth Dimensions Based on Module m (rnrn)

Stub Full_Depth Stub Full Depth

Pressure angle, 0'"Addendum, aDedendum, b

20 20 25 25

Tooth thickness, tc(theoretical)Circular pitch, P;

O.8m m O.8m m1.25m 1.25m 1.25m 1.25mO.51rm 0.5l1"m O.5'11'm O.57rm

1l'm 1rn1 7r17l 1rn1

I '

Contact ratio (mn). As two gears rotate,smooth, continous transfer of motion fromone pair of meshing teeth to the followingpair is achieved when contact of the firstpair continues until the following pair hasestablished initial contact. In fact, con-siderable overlapping is necessary to com-pensate for contact delays caused by toothdeflection, errors in tooth spacing, andcenter distance tolerances.

To assure a smooth transfer of motion,overlapping should not be less than 20%.In power gearing it is often 60 to 70 % .Contact ratio, mp' is another, more com-mon means of expressing overlappingtooth contact. On a time basis, it is thenumber of pa:irs of teeth simultaneouslyengaged. If two pairs of teeth were in con-tact all the time, the ratio would be 2.0,corresponding to 100% overlapping.

Contact ratio is calculated as length ofcontact Z divided by the base pitch Pb

(Fig. 13-18).

Zmp = Pb

~ R5 - Rf, + ~?o - ri - C sin cp

Pc cos cp(13)

wherePc = circular pitch; mm, in.Ro = outside radius, gear; mrn, in.R/;> = base cirde radius, gear; mm, in.ro = outside radius, pinion; mm, in.rb = base radius, pinion; mrn, in.C = center distance; mm, in.cp = pressure angle; deg

Note that base pitch Pbequals thetheoretical minumum path of contactbecause mp = 1.0 for Z = Pb'

Contact ratios should always becalculated to avoid intermittent contact.Increasing the number of teeth anddecreasing the pressure angle are bothbeneficial. but each has an adverse sideeffect such as increasing the probability ofinterference.

Interference. Under certain conditions,tooth profiles overlap or cut into eachother. This situation, termed interference,should be avoided because of excess wear,vibration or jamming. Generally, it in-volves contact between involute surfacesof one gear and noninvolute surfaces of themating gears (Fig. 19).

Fig. 20 shows maximum length of con-tact being limited to the full length of the

common tangent. Any tooth addendumextended beyond the tangent points T andQ, termed interference points, is uselessand interferes with the root fillet area of themating tooth. To operate without profileoverlapping would require undercut teeth.But undercutting weakens a tooth (inbending) and may also remove part of theuseful involute profile near the base circle(Fig. 21).

Interference is first encountered during"approach," when the tip of each gear toothdigs into the root section of its mating pin-ion tooth. During "recess" this sequence isreversed. Thus we have both tip and rootinterference as shown in Fig. 19. Becauseaddenda are standardized (a = m), the in-terference condition intensifies as thenumber of teeth on the pinion decreases,The pinion in Fig. 21 has less than 10 teeth.The minimum number of teeth Nc in apinion meshing with a rack to avoid under-cut is given by the expression

The minimum number of teeth varies in-versely with the pressure angle. By in-creasing the pressure angle from 14.5° to20 0, the limiting number drops from 32 to17. The corresponding increase in themaximum speed ratio potential indicatesone of several reasons why the 200 pres-sure angle is preferred in power gearing.

Interference can be avoided if:

Ro ~ ~ Ri + C2 sin2 cp (15)

r < ~/rt, + C2 sinz A. (16)O_'1b 'I'

For a given center distance, an increasein pressure angle, with the resultingdecrease in base radius.Jengthens the in-volute curve between the base and pitchcircles, thereby diminishing interference(Fig. 22).

When stock gears to suit a specific ratioare selected, it may not be sufficient toprovide gears of the same module,pressure angle and width. A pair must alsohave an acceptable contact ratio and meshwithout interference.

Other limitations on spur gears are setby speed and noise level, VVhenstandardspur gears mesh, overlapping is less than100%. The transmitted load is thereforebriefly carried by one tooth on each gear.The sudden increase in load causes deflec-

tion of both meshing teeth and thus affectsgear geometry adversely. The ideal con-stant velocity is no longer achieved. Atlow speed, this is not a serious factor but,as speed and load increase, deformationand impact may cause noise and shockbeyond acceptable limits. Consequently,spur gears are seldom used for pitch linevelocities exceeding 50 mls (10,000 fpm).

Modifications of Spur Gears(Nonstandard)

The teeth of a pinion will always be

(continued Or! page 48)

(14)

Fig..20 - Interference sets a geometrical limitation ontooth profiles. For standard tooth forms interferencetakes place for contact to the right of point T and tothe left of point Q.

Fig. 21- To operate without interference, either thepinion must be undercut or the gear must have stubteeth. Although interference is avoided. intermittentcontact persists as Pb is greater than Z.

January/February 1989 45

SPUR GEAR FUNDAMENTALS ...(continued from page 45)

weaker than those of the gear when stand-ard proportions are used. They are nar-rower at the root and are loaded more

often. If the speed ratio is three, each pin-ion tooth will be loaded three times asoften as any gear tooth. Furthermore, ifthe number of teeth is less than the theo-retical minimum, undercutting - with itsresulting loss of strength -cannot be

\\\

Fig. 22 - EHeel of changing the pressure angle. Interference and contact ratio vary inversely with the pressureangle. When the pressure angle increases from 01 to O2, the involute section between the pitch line and the baseline lengthens. tending to alleviate interference. The path of contact, however, shortens, thereby effectivelylowering the contact ratio. (Only the path of approach is shown.)

Fig. 23- tong and short addenda.

48 Gear Technology

Fig, 24 - Backlash,

avoided. These adverse conditions can becircumvented by specifying nonstandardaddenda and dedenda.

long and Short Addenda or ProfileShift Gears. In order to strengthen thepinion tooth, avoid undercutting and im-prove the tooth action, its dedendum maybe decreased and the addendum increasedcorrespondingly. In practice. this is doneby retracting the gear cutter a predeter-mined distance from its standard settingprior to cutting. Each pinion toothbecomes thicker and, therefore, stronger(Fig. 23). For such pinions to mesh pro-perly with the driven gear, on the samecenter distance, the addendum of eachdriven tooth is correspondingly decreasedand its dedendum increased. Although thegear teeth have thus become weaker, thenet effect has been one of equalizing toothstrengths. The increased outer diameter ofthe pinion and decreased outer diameterof the gear have been achieved withoutchanging the pitch diameters,

Extended Center Distance. In this ar-rangement a modified pinion is meshedwith a standard gear, Pinions with de-creased dedenda and increased addendahave thicker teeth than equivalent standardgear teeth. They also provide less space forany mating gear tooth. Consequently, pro-per mesh requires a larger center distance.

Both modifications are widely usedbecause they can be achieved by means ofstandard cutters. A different setting of thegenerating tool is all that is required.

Backlash (B)(tooth thinning), in general,is play between mating teeth (Fig.24). It oc-curs only when gears are in mesh. In orderto measure and calculate backlash, it isdefined as the amount by which a toothspace exceeds the thickness of an engagingtooth. The general purpose of backlash isto prevent gears Irom jamming together(making contact on both sides of their teethsimultaneously). Backlash also compen-sates for machining errors and beat expan-sion. It is obtained by decreasing the tooththickness and thereby increasing the toothspace or by increasing the center distancebetween mating gears.

These modifications will improveprimarily the kinematics of spur gears.

Acknowledgement:Reprinted from Hindhede/Zimrnerman/Hopkins/Ersrrum/Hull I Lang, Machine Design Fundametltais:A Practical Approach, c 1983, Excerpted by perrnis-sian of Prentice-Hall. Inc., Englewood Cliffs, NJ.