.. .,GEAIR FUNDAMENTALS _

The Basics of Gear Metrologyand Terminology Part I

Edward lalwson,.Senior A:pplications IEngineer, Mahr C'orporation

[ is very common for mose working in the gear manu-·iacluring iJldusll'y to have only a. limited understand-ing of the fundamental principals of involute helicoidgear metrology. ihe tend ney being to leave the topic

vide numerical information pertaining ·10 given elementalparameter : pecificarions of a production piece. Thi type oftest data often serve as the ba i for acceplJreject decision .However, since analytical te ling i unavoidably ba ed onsample type data ,(the number of teeth te ted, number of te t

traces per tooth). it. could fail. to detect anomalous errors sucha . nieki or hard pots. Composite action te ling. whichinclude observation of all surfaces of all teeth. would be 8.

more reliable melhod for detecting such errors which. thoughnot y uematic, ,eouM adversely affect product performance.

Analytical testing is. generally quantitative rather thanqualitative and is usually the mol valuable source of procecontrol Infermation since proces variables 1I ually relatemore directly to elemental parameter . Like functional test-ing, analytical testiag ob ervation can bebaed upon either. Ingle flank or double Hank inspection practice. AGMA tol.-erances are provided for involute profile, teothalignment(formerly called lead), pitch and pitehline ru IlOUI.parameters.All are single flank parameters except pitchline runout,

Dto specialists ill me gear lab. It is wen known that quiet, reli-able gears can only be made using the information gleanedborn proper gear metrology.

Pari I: 'Gear lospedionGears are one of the rna t. common devices within the

world of engineering, offering an elegant solution to the prob-lern of effechve power iran mi ion. Modern gear drivedesigns must provide quiet, reliable service at high powerdensities, which can only be achieved by u ing gears whichaccurately embody a geometry like the involute helicoid sys-tem. Gear metrology may be divided into two SUbtopics,functional gaging and analytical te Ling. The e two categorieof gear in pection provide fundamentally different type ofinformation, each with its advantage and di advantage ..They can each be: further divided into ingle flankand doubleflank testing procedure . [t is important to understand thecapabilitie and limitation of!hese categorie because mi -conception about the proper m DtI1ingand 1I age of !he infor-mation they provide are very common.

The funetional gaging type of gearinspeetion can be char-aeterized as an. "attribute inspection," meaning that it deter-mines if a given production pieee willi function as intended ;i,nthe product, It does not determine whether the various clemen-tali specifications affecting functional perfonnance are in toler-ance or control since lIch elements often combine in either acumulative QIj compen atory fa hion, Functional. gagitng is,therefore, m re qualitative than qllantiUltive. More sophl tic' t-

ed versions of gear functionel gaging instruments can providean assortment of numerica! te t data. However, since most ofthis information is based upon a fundamentally compositeobservation. it is usually best applied [0 processperformanceraring exercises rather than to control of process variables asthese relate more di.rec!ly to elemental test parameter data.

FunctionW gaging observation can be based upon either sin-gIe Hank '0 double flank meshing configurations of the masterand production gears. The ingle flank version provides a directobservation of transmission errol'S, while the double flanJl: ver-sion provides observation of variation of center distance.

The analytical te ling type of gear in pection would becharacierizedas a "variable inspection," meaning thaI. it. pro-

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____ GEAR FUNDAMENiTAtS ..II Single Flank.vs. Double .Flank

Single flank testing provides. observations (analytical orfunctional) of gear geometric quality involving onJy one flankata time. The data provided is tangential rather than radial indirection, thereby offering information about the way the gearoperates-an advantage over double flank testing 'Operations.A single flank composite testi~g instrument (see Fig. 1) pro-vides two spindles. to carry !:he master and production gears,mounted in fixed locations en the instrument to simulate themounting of the gears at their proper center distance with back-lash. Each spindle is fitted with a high-precision angularencoder as well as a means to apply a braking load to one ofthe gears as they are rotated through mesh, thereby maintainingcontact en the loaded flallk. The gears are placed on the spin-dies, brought into single flank contact with backlash and rotat-ed through at [east one revolution of the production gear,During that rotation, variation in the relative rotational veloci-ties of the gears is observed. This procedure is based upon theassumption that two-perfect gears would produce zero variationin rotational velocity, or IlO transmisslon error.

Double flank testing provides observations (analytical orfunctional) of gear geometric quality involving both flanksimultaneously. It provides radial rather than tangential data.

information related only indirectly to the way the gear oper-ates. The double flank composite testing instrument (Fig, 2)

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CIRCLE 15242 GEAR TECH"'O~OGY

the instrument and the other is mounted all a linear slidewhich is arranged to permit tile center distance berweenjhetwo gears to vary. A means is also provided to apply a. load tothe slide mechanism which will serve to maintain. zero back-lashbetween the gears. In operation, the gears are mountedon the spindles, brought. into zero backlash mesh and rotatedthrough at least one revolution of the production gear. Duringthat rotation, variation ill center distance between the gears is'Observed. This procedure assumes that two perfect gears thustested would produce zero variation in center distance,

Double flank compo ite action test data can reveal radialeccentricity or out-of-round errors thai can produce gear trans-mission error. ]t cannot, however. reveal angular tooth positionerrors which also produce transmission errors. Certain manu-facturing processes (i.e, shaving) often produce gears with sig-nificant angular errors that cannot be detected by double flanktesting. It is also not possible with this testing method to direct-ly relate large tooth-to-tooth errors to gear function. includingnoise problems. It can. however. find non-systematic errorssuch as nicks, burrs or hard spots and it does offer an idealmeans for evaluating funcsionaltooth thickness based uponobservations of the average center distance during testing witha calibrated. master gear. Occasionally. 'Oneof the spmdles isfitted with. a gimble mounting to permit tilting in response toline of contact errors in the production gear, For spur gears thisobservation relates very well with tooth alignment errors.However, for helical gears the rib ervation is equally and in ep-arably affected. by both lead and profile errors.

... ,GEABFUNDAM'ENITAlS _

IErrors Detected by Composite Action TestingIf the 'error ob erved during either a Ingle flank compo ite

action test, or a double flank compoite action te l. is plotred,the re ulting trace willl.ypicaUy consist of'long term andhart term error components,

The long term component is composed of two categoriesof error. the most common occuring in a simi oidal patternonce per revolution of the production gear and relating to theeccentricity of its pitchline, Thesecond category relates toerrors of the gear' shape or roundness. For example, a thin-walled ring gear which has been held ill a three-jawchuckwith exce ive force could display a long term error of threecycles per revolution.

The hort term component is normally observed al a fre-quency of one cycle per mesh cycle. lin a single flank. test, thistype of error relates to errors of tooih geometry and is directlyrelated to noise problems, which may only be inferred fromdouble flank tests. Frequency spectrum analysis of singleflank test data usually correlates very well with the noise pat-terns generated by problem gears. Also, because short. termerrors oceuring in regions of substantial slope on the long termcomponent are affected accordingly, some standard permitthe removal of the long term component from the test databefore observations of the short term component proceed.

Occasionally. another pattern of short term error isobserved mthe single flank composite action test which does Fig'.3 -I'nvolute Promo lesting Probe. @ ANSIIAGMA2IJOI)..,A88"

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not correlate with 'the me rung cycle frequency. Commonlyreferred to asghost harmonies.these patterns rypically relateto kinematic errors in the associated machining operation .

Testing Mo.chinesThe classic method of testing an involate i to employ a

base circle disk made 10, the same diameter as the base circle ofthe gear to be tested ..That disk is mounted on a spindle whichcan also carry the gear. The device must also provide a linearslide arranged so as to 'Operate in a direction tangemialro thebase circle disk. The slide carries a straight edge which is heldin firm contact with the di k, A ensitive measurement probe isalso carried by 'the slide, The probe is placed so that .il will con-tact one of the gear teeth w.ithin the plane of action.

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CIRCLE 1411·44· GEIIA TECHNOLOGY

____ GEA'R FUNDAMENTALS I

The device is moved through a course of motion whichwill cause the probe to traverse the gear profile from root totip following all involute path (Fig. 3). Because of thearrangement of the inspection device, rotating the gear withthe disk and carrying the probe with the straight edge, this isautomatic. As the disk and straight edge roll past one another,the straight edge and probe travel a linear distance equal tothe circumferential distance upon the base circle disk associat-ed with the angle through which the disk and gear have beenrotated. During this motion, the probe is carried along withinthe plane of action by the straight edge. If the cam (the invo-lute gear tooth) is an accurate involute. the sensitive probewill measure no error during the motion.

There are several different kinds of test machines. Theinvolute test instrument uses the method described above. Thestraight edge mounted on the slide rolls tangentially with thedisk mounted on the spindle. The probe is carried within theplane of action while contacting the gear tooth that is carriedalong with the disk. A related device uses a master involutecam on the spindle instead of the base circle disk. Thi camdrives a follower on the slide which carries the probe. Thegear tooth's involute profile can also be inspected using acoordinate measurement machine (CMM). This method con-siders the involute helicoid surface in rectilinear coordinates,a considerably more complex procedure than the classic gen-erative methods described above am! not very common.

The most common category of involute test instrument todayis the CNC tester. These devices employ a rotary axis and linearslides that are not. kinematically connected to one another.Instead, each axis is fitted with a high resolution scale so thai itsmovements can be controlled! by a CNC module. Typically, anaxis radial 10 the rotary axis positions the measurement probe tocontact the involute tooth flank wit.hin theplane of action, Therotary axis and tangential linear slide are then commanded tomove at constant velocities such that the linear distance travelledby the slide is equal. to the circumferential distance upon the the-oretical base circle of !he gear associated with the angle throughwhich the rotary rods travels.

Another type of CNC tester uses the computer controlledaxes in a fundamentally different way to inspect the involute.Such instruments move the measurement probe in a radialdirection only. while the rotary axis is moved in a nonlinearrelationship according to the given involute, This practicelowers the cost of the instruments due to the lack oflhe tan-gential measurernenraxis normally used to generate the invo-lute according to the constant rise cam principal ..

Part D: The Involute ProfileAn ideal gear would provide both the smooth running

properties of friction disks and the positive power transmit-ting qualities of teeth. Thi can be accomplished by usingteeth with a geometry which conforms toothe law of gearing:"In order for two gears to transmit uniform rotary motion,the common normal of the mating profiles must passthrough the same point on their line of centers at every pointof contact." Such gears exhibit conjugate action, which is to

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46 GEA!'! TECHNO~OGY

____ GlEAR FUNDAMENTALS 1 _

say they exi1ibilprogressive contact. between mating teeth

while their p.itch circles rotate together at uniform, propor-

tionate velocities.There are limide geometries which will satisfy the law of

gearing. but DO geometry accomplishes this task wi.th the ele-gant simplicity of tile involute helicoid. system. Since it. mesh-

es with a straight sided theoretical r-ack, it can be accuratelymanufactured with relatively simple, straight-sided cutting

tools. Tile involute helicoid geometry also has lite property of

operating on varied center distances w.ithout affecting its abil-ity to provide unifonn transmission of rotational motion at the

same ratio. All other geometries are conjugate only at theirdesign center distance.

Involute Geometry

An involute can be defined as the locus of III point on a linerolling on its base circle or, inthree dimensions, a base cylin-

der. It can be imagined like this: envision a lin can as the basecylinder. Fa ten a string to some point on the can and wrap itpart way around while hokling it taught. The string now rep-

re ent III line or planetangentto the base cylinder. This i theline (or plane) of action. Focus on a point where the string lie

wrappednext to the can and begin to lowly rai e the tringaway. keeping utanght all the time (Fig. 4). A i1 ri S, thepath foUowed by the point on the string a . it move up and

away from the can will be an involute At first, the point will

move nearly straight up from the surface of the can. [t willthen quickly begin to follow a curved path similar to an

Archimedes spiral. That path is the involute curve.Two things are important to note here. The firs I. is !:hal. this

only occurs within dle plane of rotation. perpendicular to theaxis of the base cylinder. It is alsoimponam t.o note that, at anypoint you wish to consider along the involute curve, the string(line/plane of acl:i.on) is perpendicalar to the involute. Further,the curvature radius of the involute is always equal to the lengthoflhe string (line/plane of action) fromih involute to the point

of tangency of the string with the can (base circle/cylinder).Consider now the same base circle (Fig. 5) with a single line

of action tangent to it. In this case, the base circle can rotate

about. its axis when Lliteline of action is pulled to the left Twoequally spaced points along the line of action have generatedtwoparalJel involutes as the line of action was pulled left. The

distance between the involutes along the line of action is equal.

liig.4- Circ·lewitflltangent lines d'escribing an invoMe curve.Courtesy of A:GMA

_-----GEAlR FUNU~MENTALS _to the distance along the circumference oli'the base circle. Thisistrue because the am pain onlhe line of acaon (string) thatha e generated these Iwoinvolute were once re ting upon thesurface of me base circle (can) before the involur were gener-ated by puUing aw.ay the~lring.

The Involute CamThi ,0 servation give rise to til ma t .impattanl. property

oflhe involute" that it will erve as a con rant rise cam to. afollower that is constrained 10 move within the plane tangentto the base cylinder of that involute. Imaginethat such a fol-lower ha been po itioned wilhinthe plane of action and incontact with the involute curve to the right. Now considerwhat will occur if the base circle i rotated counterclockwisethrough the angle required to move the involute at the right tothe position of the involute atthe left. The follower will bedriven to a position within rhc plane of acnon contacting theleft involute. It has been driven a distance along this Lineequal to the length of the base circle circumference swept bythe couotercleckwi e rotatlon of the base circle. Any suchrotation of lite base eire! and it involute "cam " will pro-duce such a eonsiant ¢Ii placement of such a follower,

Figure 6. showstwo base circle with it Ingle line 'Of actionLangent to both. Involute 'profile from born base circle inter-act with !he tring as we have een before but now, with tnetwo circle arranged to . hare this single line of contact, wecan al 0 see the tWOB ociated sets of lnvol.utes interacting. IIcan be een that these involute profiles only contact oneanother within the plane of action. This condition also existfor all involute helicoid gear, ts, where the mating toothflanks only contact one another within the plane of actionwhich is tangent to the base cylinders of both gears ..

Observe the interaction of these mating constantrise invo-lute cams. Begin w.ilh the point of contact labeled 2 near theba e circle of the upper gear (driver) and at the on of thelower gear (driven) in Figure 6. Now, rotate the driver COUD-

terclockwise !hrQugh an angle ne e _aT)' to move its involuteto location 3. A this move proceeds from location 2 to 3, thepoint of contact with the mating involute is driven along theline of action, carrying (he driven profile al 0 to location 3.The di lance travelled alonglhe line of contact is equal to thecircumferential di 'lance along both base circle swept bytheir rotation which wiU be in proportion tojheir diameters,

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CIRCLE 146

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_------------GEARFUNDAMENTAILS-------------Therefore. any rotation of the driver will cause an exactly pro-portionate rotation of the driven gear according to their diam-eter ratios.

PronJe errors - SymptomsInvolute profile errors can result in gear noise. strength

problems associated with dynamic load promoting fatigueand durability problems associated! with localized contactstress. Gear noise invariably relates to transmission error orinconsistent rotational velocities caused by geometry errors.Profile errors have a particularly troublesome effect upon

IFig'.7 - The Constant Rise Com praducea constant transmissionof rotational motion. Courtesy o. AGMA.

transmission error because they tend to be consistent in theteeth of the gear. This causes transmission errors at mesh fre-quency which is usually in the range of human audio acuity.Also, since theerror is typically consistent for all teeth, theassociated mesh frequency transmission error is usually ofconsistent amplitude throughout the rotation of the gear. Thisis perceived by the ear as a pure tone which is much moreobjectionable than a noise source of equal average amplitudethat exhibits a modulated frequency of amplitude.

Involute profile error can also adversely affect thestrength and durability of a gear. Tooth strength ratings arecalculated assuming that the torque load will be appliedconsistently and that loads applied at the critical regionnear the tooth tip will be shared by adjacent teeth enteringmesh. Profile errors can increase dynamic loading and tiploading conditions thereby promoting fatigue and prema-ture failure. Localized tooth contact stresses are alsoincreased by profile errors that accelerate pitting and simi-lar durability failures.

Profile errors-s-CansesIt is clear the primary contributors to involute profile errors

are cutting tool accuracy and mounting errors. Cutting toolgeometry errors typically transfer consistently to the gear pre-file on a one-to-one basis. Since other influences can alsoaffect the gear profile. it is important that the cutting tools aresignificantly more accurate than the gears they are expected

CIRCLE 11848 GEAR TECHNOLOGY

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CIRCLE 125!!iO GEAFI TECH"IOlOGV

____ IIGEAR FUND'AMENTALS _

10 produce. Mounting errors can be even more problematic. Aperfect cutting tool mounted inaccurately will perform no bel-ter than a low grade, inaccurate cutting ·1001. The use of bentor diny 1001 arbors and failure to check truing diameters arepossibly the rnosteommon and co tly ins occurring in gearcutting operations. Errors ill cutting 1001 accuracy or mount-ing tend to produce involute errors that are consistent whenone observes teeth located at various positions around thegear, Errors in gear blank accuracy or mounting rend to pro-duce involute errors that vary when one observes teethlocated at variou po itions around 11rIegear.

Gear blank accuracy and mounting errors are ,compo edof two categerle • eccentricity and out-of-round.Eccentricity conditions produce ainu oidal pattern of vari-arion in the lope trend of the involute test trace taken atvarious positions around the gear. Out-of-rouad conditionsproduce deformation ofmvolute trace a. cording to thegiven roundness error pattern. Radial runout of a. gearcaused by either an eccentric blank or an eccentric mountingof a good blank will cause a characteristic error pattern inwhich the profiles will display a slope error that varies in asinu oidal pattern around the gear), This category of appar-ent profile error w.ill not adverselyaffect the strength ordurability of a gear or contribute to the generation of noise.Runout can be the source of several type of problem thataffect gear performance. However, the sinusoidal pattern ofprofile slope errors it creates is not one of those prob\ems.

It is possible to produce a gear with a substantial out-of-round condition thaI would exhibit proper conjugate actionwith a mate. Such gears are sometimes preduced when acyclic acceleration/deceleration is de ired in a mechani m.However, when a gear Ihal is intended to be round i deformedinto an out-oi-round condition during manufacturing, it cannotbe expected to operate without an a soeiared detrimentaleffect, as would be the case with simple eccentriciry,

Recognizing the absence of detrimental effects fromeccentricity-based apparent profile error, procedures havebeen employed tllal. tolerance only averaged profile errors.This practice is inadequate because averaging may alsoremove the detrimental effects of out-of-round conditions.Rather, it is only correct to adjust test results according to thegears simple eccentricity condition which has fi.rsl. beendetermined by analysis of its radial runout or by II more com-plex geometry-based analysis of !he profile traces. Watch forthe conclusion in the next issue of Gear Technology. 0

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