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_GEAR FUNDIAMENTAtS". - __
Involute Inspection Methodsand Interpretation ofInspection Results
Introdu HODWhat is so uniq-ue about gear manu-
facturing and in pection? Machining imostly associated with mabng either flator cylindrical shape . These hape canbe created by a machine' imple linear orcircular movements, but an involutecurve is neither a straight line nor a circle.In fact, each point of !he involure curvehas a different radius and center of curva-ture. Is it necessary to, go beyond simplecircular and linear machine movementsin. order '10 create an involute curve? Oneof the unique features of the involute :isthe fact lhat :it can be generated by l:irikingcircular and linear movement. Tbisuniqnene s has become fernie . oil, formany invention that have implifiedgear manufacturing and in peclion ..As isthe case with gear generating machine.Ithe 'traditional involute inspectionmachine take advantage of some of theinvolute properties. Even today. wilencomputers can sym:bromze axe for ere-ating any curve, taking advantage oflinvolute properties can be very helpful It
b
fig. 2 - Methods for gBnerating aD linvolute curvll: allh- string m thod; Ill,I balm rolling amundl.fixed base circle; cl balm IBndbase circle rolling with IIlch 'odler 'wltboUlllip.can impJify synchronization of machinemovements and reduce the number ofvariable ' to monitor.
Involute Definition.,Geometric Propertiesand Involute Function
The involute curve is a spiral begin-Ging at the base circle and! having an infi-nite Dumber of equidistant coils (Fig. O.OnJy a smaIl portion of the innermost. coilhas been utilized inpracticaI applications ..The easiest way to visualize this is bydescribing the way it can be generated,The involute curve can be generated byapoint on atightIy held. inextensible andextremely thin thread that is unwoundfrom a. fixed circle, called the base circle(Fig. 2a). This method is called the stringmethod, An i,nvolute can also be generat-led by a beam rellingarcund a. fixed basecircle (Fig. 2b} or by a beam and base cir-de rolling with each other without slip(Fig. 2~). All these principles are used ingear generating and! inspection machines.
Important. geometric properties of theinvolute curve can be derived from its
IB
O=-I8 = segmt"1 '8Q = 1.aIllo ,,~-I
Il1ig.3 - G omllme prloplIrtiB' of 1b.1 inV'olm.generation. Some of the e properties areused either for the inspection machinemovements or referenced in the inspec-tion re ults,
The line tangent 10' the base circle.drawn from any point 'of th involute, i.alway perpendicular to the involutecurve ( ee Fig. 3).
The egment of the tangent line, Q8,is the radiu of curvature of the involutefor the point Q. Points wher the stringseparate from the ba e eirel are instanta-neous centers of the involule curvature.
J U L'" '" U IJ U .T 1 gu 41'
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_------------OEAR:FUNDAMENTALS ---------- ..Involute Inspection Chart
Probe LDeflection ~--~---...,.-- _
Probe Trdvel (Length of RoU)o ABea r= ~
ProbeMovement
Fig. ,4- Basic meGIIllnics of an linvolute inspee-tion process l!lSingII probe. -
~ on-Traditional Involute Inspection
Gear Rotation
Fig. 5, - MechllniGlI ,of non"lradilonlll invDlute,inspection.
For any point on the involute, arclength A.B, contained by the beginning ofUte involute and the poinl of tangency, isequal to the length of the line ' egmeatQBlangent to the base circle.
Involute FnnctiioDLet the involute or polar angle be 6,
the pressure angle be 41 and the IOU anglebe e. Then let us assume for simplicity !hatth,ebase radius equals I unit of linear mea-surement. In this case, the length olam arcequals the angalas measurement usingradians as measuring units, Therefore, (J =e - 'J, where e = arcAB = segment BQ' =OS tan41 = t, ta.nq, = tan(p. The involutefunction can be derived by replacing Ewith its function of 41: e = tanq, - f/J.
Most analytical gear inspectionmachines use these involute geometryproperties:
Length of roll Q:B equals Ute lengthofare of roB AB.
The line tangent to the base circle is.always perpendicular to the involute curve.Principles of Traditional Mechan1cal
or CNC Invofllte Inspectionand Resulting Charts
The Mecllanics .of Maclline Move-ment Traditional involute inspectionconcepts are based on combining the lin-ear motion of the probe carrier and therotational motion of the gear. TIll com-bined movement generates an involutepath for the probe relative to the gearprofile. While the probe moves along thepath that is tangent to the base circle. the
~
:~~ooA : B-------~-----------------------~--------------------------------
SurfaceVariation
1C D Total
iFig"Ii, - :SUrfll til Yllr,illUon frem the horizDntal plane.
148 OE"R TECHNOLOGY
distance equal the length of roll a, andthe gear rotates the angle A (FigA).
One important beneficial distinctionof the traditional involute inspectionmethod i the unchanging probe contaetpoint. throughout the entire probe travel.(Fig. 4). Thi unchangiag contact pointsimplifies the inspection process byreducmg the number of variables thai.need to be monitored.
The probe deflection represents adeviation of tho gear profile from theinvolute curve. If the gear profile is a per-feet involute, the probe deflectien wouldstay consta:ntth.1'Ouglloul the entire move-ment, and the resulting inspection chanwould be astraight horizontal line. Adeviation from this straight line wouldconstitute the profile error.
Involute Inspection Charts. An invo-lute inspection chan. is scaled proportion-ately to the le.lIgth or angle of roll. The Xcoordinate (probe travel) representslength or angle of roll, The Y coordinaterepresent profile deviation from the per-fect involute in the direction normal tothe involute curvature.
It is important to reiterate that metra-dnional involute inspectionchart is notproportional to the diameter, nor is it pro-portional to the length of involute curve.Non- Ttamtional Involute InspectionWith the proliferation of coordinate
measuring machines, other involuteinspection methods have come intobeing, Some CMMs use the traditionalmethod, but some don't. Nevertheless,the inspection results are presented in theold fa hioned way-profile tracing iscaled proportionately to the length ofangle of roll, as shown in the upper sec-tion of Fig. 4.
Machines thai do not use thelrnciitiOIl-al method include
. CMMs without rotary tables. Theprobe contours a fixed gear .
CMMs with rotary tables, but willi-out tangential slides.
Tbe principle difference betweennon-traditional and traditional machineis the fact that Don-trad:itional machinehave three axes instead of four. A fewernumber of axes makes one part of themachine less expensive; however. italso creates an additional burden in
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III_-- GEARFUNDAMENIIAlS_--- another area of the machine. In bothnon-traditioaal case ,in addition to twomoving machine axes, the y tern Ita tokeep track of one extra variable-thecoruact point of the probe. Thu , [hemachinecannot take full advantage ofinvolute properties for reducing thenumber of variable to monitor duringinvolute in pection.
Fig. 5 depicts the involute inspectionprinciple for the machine without a tan-gential slide. X & Y eoerdinare of theprobe contact are continuously changingas the probe moves from ['COl to tip (SeeFig. 5). To make matters worse, in thec-ase of helical gears. X, Yand Z coordi-nate of the probe contact are continu-ou ly changing.
The advantages ofllon-t:Jraditiona]machines are that they have fewer axes,and their 3-d1mensional probe give themthe potential for adding non-gearinginspection capabitHties to the machine .
The di advantages of II ing these non-traditional machines include the need to,monitor the extra variables during invo-lute inspection, which can make the sys-tems either les accurate or more expen-sive 10 develop, and the requirement for3-dimensional probe which add a ilg-nificant co I to the apparatu .
Sflme Common PriDdple .ofSurface Evaluation.
How do people analyze, qualify andquantify the urface deviation fromde ired conditions? What do we mean by"profile error"? Is this the amount of erroror the shape of the error or both? Thereare situations in which one number fordefining involute error is not. ufficient toquantify and qualify the error.
Let's introduce three definitions: slopeerror, form error and total error. Thedrawings in F.ig. 6 help to illustrate thediffsrences between these three concepts.Fig. 6a shows surface variation From thehorizontal plane. Is i1 a lot or a LitLIe?Fora farmer it may be 101:, but fOI a.: kier, itmay not be enough, Fig. 6bhows a dif-ferem type of surface variation. Even forsam - skiers, it may be too milch. In real-ity however, people frequently deal! wilh acombination of the kinds of surface errors. hewn in a and b. The situation is motelike the one shown in Fig. 6c. The errorsmay have the arne value. but a differentappearance, a hown in the Fig. 6
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The breakdown of total error intoform and slope components is applicableto both involute and lead in pection,Because slope and form errors comefrom different sources, isolating andassigning a value to each error componentis very helpful for finding the largest con-tributor to the tooth surface inaccuracy.
Slope error sources include.' Lead. The wrong machine settings
for the helix angle .. Involute, Wrong hob pressure angle
or wrong rake angle induced during hob
sharpening.Form error sources include Lead. Excessive feed rate. Involute, Excessive hob runout
hob gash index error or excessive hob leaderror 'Orinsufficient Dumber of hob gashes.
The breakdown of the total error intothe slope and form error components canbe invaluable for determining exactmachine or tool adjustments, thus elimi-nating time-consuming trial and errortechniques. For example, lead or taperadiustment on a CNC bobbing machine
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can be determined accurately with theaverage lead slope errors.'Iooth Surface Patterns Created byVarIous Manufacturing ProcessesEvery gear manufacturing method
creates a certain gear topology that. dur-ing tile inspection.jranslates mostly intofonn errors. Some examples of tooth.topology are shown in Figs. Sa-d. Notethat all these examples display involuteand lead errors despite the fact that thesegears were manufactured under idealconditions of machine, tool. fixture andblank. These error are referred to asinherent errors introduced by the pro-cess principle.
Even the most accurate hobs andmachines can create greater than allowedlead and involute errors, Understandingthe tooth topology helps to differentiatebetween the inherent errors introducedby the process principle and errorsinduced by the process variables(machine, cutting tool, fixture and blankinaccuracies). Frequently, determiningthe error source is a bigger challengethan the elimination of that source,
Common. ~ofi.le ModificationCommonly gears are designed with
tip and/or root relief (Fig. 9). The numer-ical evaluation of such profiles becomesmore complicated. There are variouscomputerized techniques available forevaluation of profile modifications.Some people use .11 compari on with . anideal curve. Some evaluate various por-tions of the tracing separately; someevaluate crown; some evaluate hollow;
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G,EAR FUiNDAMiE,NITALS,
GroWIl Max CroWD Crown
b
valuationRange
EvaluationRange
a e
EvaluationRange
fi'g. 10 - Computiz~d IDchnJq,uesfor IcrOWI! 'llvalultion: III bast fit curve method: b:1 cDlII,parisonof both, end. 101tracing w:i1h higb poill\; ,e)compariSOI ofa line connecting tracing ends witll, hig'b poinl d some jrsejh K' h h .an. some use the . -enart tee mque, The answer depends on which inter-
Most people use some kind of comb ina- prctation method i used, An inspectiontion of these methods. tracing could be justified to a "plus"
Crown of tile Surface" There are material condition as shown on tile I,eflvarious computerized techniques avail-able for C1'Own evaluation. Some use abest-fit curve (Fig. lOa). Some compareboth ends of the tracing with a highpoint (Fig. lOb), and some compare aline connecting tracing ends with thebighe t point (Fig. toC)I.
Bollow. Hanaw is the reversal ofthe curvature, as shown in Fig. 11..Onecan look at hollow as a variation of tileform error. This characteristic is wide-ly monitored for evaluation in theautomotive Indu try.
The K'Cbart Method ofProfile Evaluation
The K-chart is probably the mostwidely used technique for qualifying ordisqualifying agear, TIle K-chart is asimple appraisal for deciding whetheror not the gearprofile is within thespeclflcatiou However, it is importantto note that the K-chart is .110t a goodtool for analyzing the source of a prob-lern. It is only a go/no-go gage whichtells whether the profile is good. or bad.W11ena gear does not fit the K-chart, amore detailed analysis must be conduct-ed in order to find and eliminate theource of the problem; e.g., total errormust be broken down into 'the slope andform components.
Despite its seeming simplicity, the K-chart can become a matter of controversy.Many companies-and ometimes differ-ent people within the same company-differ about how to interpret a K-chart.For example, is the tracing in Fig. 12
~g. 13- AKchan with more than, three points.. inside or outside the K-chart tolerance?
52."
O'EAA T'ECH'NQ~OOV
EvalnationRange
filg. 111-"Mollow" is the reversallof curvllturB.
of the figure or a "minus" as shown on.the right, Sometime . Kschart band aredefined with more than three points.This opens up a further proliferation ofevaluations, Some people may ju lifythe high point located anywherebetween SAP and EAP, as shown on theright of Fig. 13. But some may use aspecific range of ron angle for ju lify-ing the high point. An example isshown in Fig .. 13 on the left, where amiddle portion of the tracing is used fDrjustifying a high point of the iuvollute.ASEl result, the same tracing could beconsidered as outside (left) or inside(right) a K-chart.
ConcillSioDBasic principles of gear inspection
have not changed during the last 30 to40 years. But there has been a dramaticproliferation of gear inspection tan-
-
lead evaluation are by no meaas com-prehel'lSlve and conclusive. For exam-ple, AGMA does not classify fonn andslope error components. Perhap that isone of the reasons why many Americangear manufacturers have created theirown, more detailed, but frequently con-tradicting standards and techniques. Ex-amples of thee contradiction are theaforementioned K-charl. and crown vari-eties of evaluations. These varieties arewide open for different interpretationsandre ulting disagreements.
To avoid the e di agreements, it ishelpful torecognize the e varieties and.if necessary, develop or adapt, documentand communicatethe company's policyregardiog profile evaluation methods andinterpretations of the K-chan: or crown,
In contrast. the European gearinspection _tandards are more compre-hensive and adopted widely. As a result,these standards are much more effectivein helping gear companies fwd commonground when dealing with one another.These standard can also be more effec-rive for debugging gear manufscturiagproce es, for example, by applyingslope and form error component .
Regardless of where one stands onthe merit of home-grown standard andevaluation proliferation, understandiagthe basics can help one navigate in thissea of mspectionta:ndards and evalua-tion teclmiques.O
Acknowledgement: The article is basedon paper presented ar the SME Gearin.gConference. Feb. 3, 1997. Chicago, IL
'tefim K'D'd,aris Ihe Gear TecflllOiogy and Processing Managerat Bodine Electric Company, Chicago, IL He isalso the author of a ",umber of aniell's 0" gearrelaltd ubjtCI.
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