Prof. Dr. -Ing. Manfred WeekDipl, -Ing. Georg Mauer

laboratory for Machine Tools and Industrial Management (WZL)Technical University, Aachen,West Germany.

Abstract:Strider reqWre.ments are being imposed on

heavy duty gears in terms of running behaviorand load carrying capacity.Purthermere, theexpanding teclmiquesin the field of gearmjU\ufacturing are demanding 'the develop-ment of suitable designing methods for toothflank corrections. In order to calrulate thesecorrections, the spatial stress and deformationstate in the mesh has to be determined. Thisarticle reports on the further development of afinite element calculation method into an op-timizing and designing system for flank OOITec-

tions on spur and helical gears.

Main Influences on the Bearing Capacityand Run_ning Behavior of Gears

The load carrying behavior of gears isstrongly influenced by local stress concen-trations in the tooth root and by Hertzianpressure peaks in the tooth flanks pro-duced by geometric deviations associatedwith manufacturing, assembly and defor-mation processes, The dynam:iceffectswithin. the mesh areessentially determinedby the engagement shock, the parametricexcitation and also by the deviant toothgeometry, (1)

The engagement shock results from adisplaced starting point ofengagementdue to rotational deviations or pitch errorswithin the gear system. This transferredstart of engagement is located outside theplane of action. Here deviations occur inthe value and direction of the normalvelocity 'components of the contactingtooth flanks; thus, vectorial difference

16 Gear feohnology

types 01 rootnr lank correct ions

II p!o'fLJe: cor"Ffc.UDf'I ., lmal tytllnti c:arrCUIM

ji§ ~c, tit Df.fj] till

() ~ _aJlel 1Q JI.. !; 11'1 the" illlC:MlWI 1:oruc.upn

ro···c.- tlJ:."

F'lg. 1- (upper left) Objectives oftooth flank corrections:• reduction of pressure peaks• decrease ofdisplacement senstvity• diminishing of engagement shocks• lessening of parametric excitation

Fig. 2- (below) Basic principles ofcontact analysis of loaded gears ..

calculation at theengagement cona it Ions

preset geOletrl.Ctooth aevlaUoos

calculation of ttlegear Dllalll.11 ties

alstortlona,l Ile!iavlourof ttle llesh

diS tor tIona I. .tlehav.IOOI'ofllheel. si1aft !J1d

DeBrJngS

.'alscrete cootact oorms of • utrh ,tt. _Uh defOr.lllltlon IndicesO!lffelent roiling [JOsltloos ~ for discrete [JOInts

cootact III steoces s fordiscrete [JOInts

soiu; Ion of the contllCt-Drool.eII In !)ear eng&gell!ntsfor dtscrete oolnts of different roll.1119 pas I tloos

n' F + 5 • ,Srj91!1' marolna·1 cOOClltlons : I:F I • ftot ' Sn91a ' I." censt,

alstr1butlon of10<10 af'll1 pressure

contact oattern

course ct "1tota 1 llesh st trrness

These precision Bryant CNC grinders, the finestavailable anywhere, are ideally suited for eithershort-run or high production performance. OurTeachable® III CNC unit permits us to precisiongrind parts so varied in size and shape (evenout-at-round intentionally) that you wouldn'tbelieve the same type of machine could do it all.They perform contour grinding - both rotary andlinear, vector grinding, conjugate grinding, andsimultaneously or sequentially grind multiplesurfaces in a single set-up to provide ultimateprecision and concentricity.Our time-tested Commonponeni® constructionpermits the use of identical slides, drives, ballscrews, controllers, dressers, etc. on eithermachine. That's one reason why they are soflexible, precise - and affordable. Commonalityof parts - bet your bottom line will love it! Weoffer slide resolutions as fine as 0.000001" andwhen it comes 10 sizing capability, we've got it.The Lectraline machines offer the world'swidest capacity range- 0.040" to 88" indiameter with strokes up to 7 feet. They comein over a half-dozen standard configurationsto suit your exact requirements.

The Lectraform grinders can handle partsfrom 0.050" to 24· 00 and are available intwo standard slide arrangements. Unlike othermanufacturers a compound slide is out of thequestion. The closer that big grinding wheel is tothe bearing support, and the Lectraform can uselarger wheels than other makes, the greater theprecision - and that's the name of the game.Tell us what you have to grind and we'll show youhow to do it - precisely. Bryant Grinder Corporation,Springfield, Vermont 05156, (802) 885-5161.

BRYANT GRINDER CO'R,PORATIOiN

CIRCLE A~7 ON IREADER IREPlY CARD

'Quantity to QPhmlze :

tooltl Hank correction

dJeclgng ofrestrjgtipns "

.' contact rano

• rOiling conditions

• contac1 conditions and

correctional motions

at manulacMing stage

oPhmjljDQ, meloom !lank correctIOn;

• varlafional calculation

.' modification 01theoorrec1ion topography

• checking theconvergence '1:lel1aviour

decrees 01freedom"number 01describingvanabiesfOi theflank cerrecnon

, ~tqyanti1ies"• ,engagemefll conditions

In 'the mesh• ccmpliarces• outer torque.' deviations

100111contact analysisof loaded gears tly the

methodol6nite elements

:::=::::"0-- 0---- 1~~OfPlnlon~5 _ o~1 I - ,~ 1 1 ---r---~-o 10 20 30 40 50 mm 70

face ,wIdth

cnlerions lorHfIOk corrections'

• bene[icialload characteristics

• low dlsplaoementsensivity

• low dynamICexcitation

• distritJution or load

and pressure to

the mating, area

• rota.tional deviations

of loaded gears

I ~ ©

Fig. 3-Optimization of tooth Hank corrections.

... ~o)6...<: mm0-u.... 20Q

s 15~C1...0 11).c...... 5<:QJ

G

lengtl1 of pathof contactcorrec t I,on

value40kim20

o

Fig. 4-0ptimized three-dimensional correction. with basic gear data.

gear data:

lin ; 7 IIIiII

Z, = 27l2 ; 28

x, -0,006

x2 -0.031

fa" '.55

Et3 = 1.00Il:n = 20·fj = li6·

b ; 78,8 !11111

a = 200 ..

'1 = 4820 NIl

brings about an unwanted impact speedand shockacnvation.V'

The parametric excitation is a conse-quence of t.hechanging number of matingteeth and the contact line movement overthe tooth flank during engagement. As aresult, a periodic modulation of the meshstiffness follows and induces rotationaldeviations and activates unwanted vibra-tions, even when the outer load f.or thegear is constant. (3)

Geometry of Bank Correctionsfor Helical Gears

The flank geometry of helical gears canbe modified in various ways with respectto manufacturing requirements and thedifferential improvements on the bearingcapacity and dynamic behavior, Fig, 1gives an overview of the principle correc-tionforms.

Hanl. corrections in the direction oftooth depth as shown inFig. la are simplycarried out by involute tip or root reliefs.In this case, the part of the flank to betaken back consists of a corrected involuteprofile, which is defined by a. modifiedbase circle diameter and by the intersec-tion point with the uncorrected involutecurve, This is achieved by tools having abasic rack system with altered profileangles or by using a modified workingpitch diameter at generation. Anotherform of profile correction which has asmooth transition to the original, uncor-rected involute curve is manufactured bytools with crowned basic rack profiles .Profile corrections are mostly used todecrease the engagement shock and the in-

AUfHORS:

PROF. DR.-INC .. MANFRED WECKholds the university chairoi Machine Tools atthe Technical University Aachen. West Ger-many. He is also director of the Laboratory forMachine Tools and lndustrial Management(WZL) and manager of the machine tool de-partment of the Fraunhoier 1I15titutefor Pro-duction Techno.logy ([PT) in Aachen. Prof.Wecks professional interests are in the areas ofmachine testing and evaluation, machinedesign and calcula.tion, ,development of auto-matic control and monitoring systems, auto-mation of production and ha.ndling ,and gearcutti.ng machines and gears.

OWL. -ING. GEORG MAUER received hisdegree in mechanical engineering at ,the Tech-nical University of Aachen and is curren.tlywo rking as a scieniiiic assistant in the gearresearch group of WZL.

volved strain and noise,longitudinal. flank corrections in the

direction of the fac,e width (Fig, tb) areused to attain. a, low displacement sen-sitivityand to avoidstl'ain peaks ocruringdue to displaced positions of the gear axis('twisting, tilt) or helix deviations of theHanks,

For hehcsl gears, a correction runningparallel to the lines ofeontact is advan-tageous. (See Fig. Ie.) The intersection lineof the corrected and uncorrected area isidentical to a line of contact between themating teeth, The maximum correctionval!ue is found at the end or starting point,of the engagement. The corrections arecarried out by helical invol.ute areas whichare mathematically defined by a modifiedbase cirdeand an additional incrementalchange made in the helix angle, This f.onnof flank correction has the advantage 'ofproducing a smaller loss in. the contactratio, so that the geometric correctionmay have a greater r,oning length andmore beneficial normal vector conditions.

The correction fonns described aboveare to a eertain extent funited in optimiz-ing the rumnngbehavior and load carry-ir!g capacity. because the flank correctionsdo not meet the gear geometry and matingconditions adequately, These conditionscan be met by using three dimensional(oHen called topologjc(4l) flank correc-Hon~ (See F.ig. ld.), which are character-ized by variable correction forms inbothdirections of tooth depth and facewidth,(Sl There are, however, variousmarginaiconditions which have to betaken into, consideration. these being thecontinuity of dilierentials.minimumcon-tact ratio, rolling conditions and manufac-turing processes. (l)

Optimizing Method Based m, FiniteElement Ca!letdatwons

To obtain. the required starting positionto optimize flankcorrections. one has toreturn to the basic principles of contactanalysis ,of loaded gears illustrated infig. 2. The continuous, spatial load defor-mation problem is replaced by a system ofdiscrete contact points. defined by amatrix with deformation indicies !a-I, aload vector {F}and a vector containingcontact distances [s], which describes thegeometric flank deviations or corrections.The equation system is then set up for dif-ferent rolling positions and is solved byconsidering Ithe marginal conditions con-c,eming the 'totalload and the rigid body

\1lIO\~ ._

0<1 roiling nilS

_<iii ... ,

•• II.I!!!!I ., ••• , "2 • ·,03' ~. '5"

II ·200.. ta· VB 'IJ· 1.001 1'. !I!! W201"

Fxg, 5 - Calailated tooth loads for a full cyde of meshing contact.

Mtrl1tll1!lft preuurnOf! tllo_1IrH

! -~II I IIi

J 111

o • • WII.·ID' IIrltYDLuUDftlID.llfItIlt "1

11.10

1.05

1 2 , • ·S 6 1 , 9' ID

,,~ ."tl' "'D1.00 ml:;:;...~=::;:::;=i

o • I U/IIln'IO" n'_luU .... 0••1...11.'"

~.., Ii '1 _ iln ~ 2Oi!

b ••11,' !!!! ~ • 16'

., • 71· ", - -.ID

'1 • ,.OJI

11"18. ·6 - Periodic courses of the total mesh stiffness.

~a· '1.55

'II' 1.00 " • 1iI2O: ..

'1"2 • l'S," ,

11/12 • O.5OQI1l~26I J e • J«IJ I!I!!!

~l/a02 • 599._ .... 1~1'.~· 011', '·20'

1~s:e~jfi

~"~

iIi

110 III~i

~ 7iI_M_::: .,'U.

~~ ,.......... -~r .. t ..

5 ffc,'''_ •. ....- .. --- ......Cattltt .. I

I IL I~

c~~tld .,,-

1\..:)1' I '"-0 .iI'./"" I ~

r..1nttoa--~,-" ,/ ,.~--

-,"'lCtld \

I

10

100

1500

.'UJOO

750500

2SO

199th'l - COUKtll7n 'ot '1 • .a UII :

100000!ti.KIUMll 'Md reUd's

.Fis. 7 - Smiples of geometrically simple corm:tion forms.

September/Octo'ber1988 J 9

displacement. (6) The results provide in-formation on the load and pressuredistribution on the field of action and theshape of the rotational deviation curves,which is influenced by the alternatingmesh stiffness. It becomes clear, therefore,f.haf Ioad distribution and mesh stiffnessare direct functions of the correction to-pography. The pliability indicies tal arecalculated by finite element structures of

the mesh and analytical models of shaftsand bearings.

This contact analysis is the central start-ing point to optimize tooth flank correc-tions ..(See Fig. 3.) By varying thecorrec-tion topography, a beneficial load andpressure distribution on the flanks, a I'owdisplacement sensitivity and a lowerdynamic activation can be achieved.

The con tad and deformation analysis,

CIRCLE A-35 ON READER REPLYCARD

20, Gear Technology

as described above, presents all the criterianecessary to assess a particular correctionwith regard to the various objectives. Themathematical variation provides informa-tion regarding the trends and dependentvariables of the topography and the targetobjectives, so that the correction can beimproved in a stepwise fashion. Thisnumeric process has to be repeated severaltimes after checking that all the restrictingmarginal conditions are met.

Other research developments at WZlconcern the simulation of grinding proc-esses for corrected tooth flanks, wherebycorrection movements and contact condi-tions are analyzed ..Control data for NCgrinding machines are also generatedPJ

Calculated ExamplesThe following section contains the

calculation results of two flank correctionsfor typical helical gears in industrial. use ..Fig. 4 shows an optimized three-dimen-sional correction with the basic gear data ..The correction values are drawn up on theplane of action with a maximum relief ofapproximately40p.m. In the middle of theengagement area an uncoerected regionensuring a contact ratio greater than 1.0can be seen. f1) The correction topographyis observed to run parallel to the originalinvolute flank at the approach contact,which is favorable fQr lowerexcuationproduced by engagement shocks,

The calculated tooth loads for a fullcycle of meshing contact are illustrated inFig. 5. A lower force level at the start ofengagement and a shallower Ioadreeep-tion gradient for the corrected gear ean beseen dearly. The pressure distribution onthe plane of action depicts lower strainsfor the corrected gear, and the pressurepeaks have been successfully reducedwhere the contact lines end at the root ·0£one of the mating teeth.

InFig. 6 the periodic courses of the totalmesh stiffness are drawn up on one basepitch. The specific range of variationwhich is characteristic for the intensity ofthe pararnetricexcitation could be reducedby the three-dimensional flank correctionfrom 3.3% to 0'.9%. The simulation of thedynamic behavior of the gear at variousrotational speeds shows that the max-imum dynamic load factor Kv could bedecreased from 1.17 to 1.06., This isachieved by depressing the maximumresonance peaks as well as the first andsecond order Fourier coefficients of the

• Eliminates need for skilled operators as th unit isautomatic. Unit can be retro·fit for use with automatic infeed.

You need Nonnac's Fonnaster CNC Grinding Wheel Promer

The Fonnaster gives you inspection guageacQlracyNormae guarantees an accuracy of! .0001" (.002 mm)from programed dimensions and provides less than .000 l"(.002 mm) positioning error throughout total slide travel.

Easily adaptable to meet your need.The Formaster is designed to mount easily on nearly any type ofgrinding machine. It is compact and lightweight and just amounting bracket and only minor machine modiflcations arerequired for installation on most grinders. Inslallation usuallytakes one day or less.

The unique features of Normae'sFonnaster ...' The unit is totally seal'ed and air- purged fortrouble-free operation and protection fromcoolant oontamination.

• Double nut pre-loaded roller screwstransmit servo motor to slide movement.

• Optional rotary diamond dressing wheelattachment provides greater accuracy ande'liminates error caused by single pointdiamond wear.

Fonnaster users have reportedlncreased produdiionand greateraoourag-.• Increase in wheel: li~eof up to 8·10 times'longer than with other units.

'. Production increases up to 20% in the firsttwo weeks of use.

• Set-uptime is reduced to 80% of what itused to be with improved surface integrityof the parts.

Now Nonnac offers you software Jar the FonnasterfOIi automatic pliogramgeneration.Normac has develop d optional software 'that runs on apersonal computer in a question and answer format It aHowsthe user to enter part print specifications (including root andprofile modifications) in standard gear nomenclature. Theprogram outputs :N.C code iloaded into the Formaster controlfor wheel profiling, and also detailed printouts suitable for

gear profile analysis. This software contains logic for alltypes of root zone grinding, true involute and

i exponential Rank modifications as weJlas "Proudn ss"of the entire profile as indicated! in the K·diagram above.

Let us show you what the Fonnaster cando for your grinding operaUons

If you would like. we can arrange a demonstrationof the Formaster for you. CaU or write us today.

NORMAC, INC.P.O. Box 69

Arden, N.C. ,28704(704) ,684·1002

P.O. Box 207Nol1hvllle, MI 48167

(31,3) 349·2644

CIRCLE A-19 ON READER REPLYCAJ~,I:)

mesh stiffness course.Clear improvements of the dynamic

and bearing behavior can be achieved alsoby geometrically simple correction forms.(See Fig. 7.) Here the results of the toothcontact analysis ofa loaded helical gearfor the corrected and the uncorrected caseare illustrated. The tooth flank correctionconsists of longitudinal end reliefs and twoinvolute profile corrections, The relieflengths and values were optimized in themethod described previously.

The corrected gear shows several op-timized characteristics. These are• a much smoother mesh stiffness curve

with a smaller specific variationalrange,

• a reduction of the force levels at thebeginning and end of engagement,

• a lower load reception gradient,.' a.well-balanced and lower pressure

course.The unmodified area. on the middle

flanks of the corrected gear is sufficient to

IG.···,I _ I

Eliminate Carburizing P!roblems,

Commercial Processing• COmplete high tech lab• ln-house, high powered radio

frequency

• Full production quantmes- largeor small

• Development or heat treatpatterns and specifications

• Prototylles for test purposes

Induction AdY,antages'. Greatly reduces distortion'. Minimizes part finishing require'

ments• Facilitates cellular and in-line

processing• instant start-u p and sh ut down'. Individual part quality control• Exclusive process Signatu re

monitoring -

World Wide L,eadersInductoheat IS the leaderIrI inducuon hardening .technology.

Now we have developedadvanced the process of contourgear hardening. The experienceand resources of our internationalopera~ons have been combinedto :brmg you this new technology.Calltooay and we will show you theadvantages of contour gear

•

- hardening and how itwill benefit you.

I _NDUCTOHIIAT32251 North AVIS DriveMadison Heights, M14!W71(313) 585-9393 TWX 61 (1.232·52081'~4.6<'91 FAX(313)SS9-1062

Manuf.c1urlng PI.ntl In:.. Australia. Belg!um • Brazil., England• France ,. India 'II' Japan. Taiwan• U.S.A. 11.11,TX) .' West Germany

CIRCLE A-37 ONI READER REPLY CARD

22 Gear Technology

ensure that a minimum contact ratio of1,.10 is achieved; thereby guaranteeing a:smooth operation, free from kinematicand dynamic disturbances.

ConclusionsA method is given for optimizing

various forms of tooth flank correctionswith regard to the load carrying capacityand the running behavior of spur orhelical gears. Two examples for correctedgears 'confirm that it ispossible to optimizeflank corrections at the designing stageand to make empirical attempts un-necessary,

Even when flanks are corrected by sim-ple geometric modifications which do notrequire advanced manufacturing meth-ods, the gear characteristics can be im-proved so that an increased transferablepower and a reduction in noise emission isachieved. A more complex correctiontopography allows for an optimum con-formation with the gear geometry and theengagement conditions, while, at the sametime, keeping the correction topographywithin manufacturing restrictions.

It becomes dear that this new calcula-tion makes it possible to influenceallessential effects which determine thedynamic and load carrying behavior in aspecific way.

References:

1. SALlE. H. 'Optimizing the Running Behaviorof Cylindrical Involute Heavy Duty Gears."Dissertation, RVVTH Aachen, 1987. (Printed inCerman.l

2. TESCH, F, "The Defective Tooth Engagementand its Effect on. Noise Emission," Dissertation.RWfH Aachen, 1965. (Printed in.German.)

3. MOLLERS, W. "Parametric Excitation ofVibrations in Cylindrical Gear Pairs." Disser-tation. RVVTH Aachen. 1982. (Printed inGerman.)

4. MAAG 'Topological Modification - A fur-ther Step in. Gear Technology." Technical PaperES-42.2. Maag-Zahnrader AG, Zurich, 1985.(Printed in Gennan.)

5. WECK, M., MAUER, G., SALrE, H. "J-D·Tooth flank Corrections - Improving theBearing and Running Behavior of Gears."Jlldustrie-Anzeiger, Leinfeldell·Echl'r!rdingllll,Vol. 109, No. n. 1987, pp. 3()'3J.. (Printed inGennan.)

6, NEUPERT, B. "Calculation of Tooth Forces.Pressures and Stresses of Cylindrical and BevelGears." Dissertation, RWIH Aachen. 1983.(Printed in Cerman.)

7. WECK. M.& REUTER, W. "Simulation ofTooth Plank Grinding Processes for FinishingSpur and Helical Gears." Paper for th 28thConference - Gear and Gear Cutting MachineResearch, WZL RWTH, Aachen, 1987,(Printed in Cerman.)