New Pfauter 3-,4- and 5-axis CNC gear shaping machines put greatly improvied perlonnancein a compact, self-contained and economical package .•. the perlect fit for today'sgem-manufacturing operations.,

Consider these outstandulgfeatures:I!fGE Fi\NUC CNC controls and

digital servo drive systems,completely eliminating change gears

I!fSmall-footprint design:Only 4.8 sq. meters of floor space required

!!1I'Fivemachine sizes and configurationsto choose from:3-,4- and 5-axis models; up to 20" pitchdiameter; models available with CNCcontrolled cutter slide

I!f Quick change toolingf!1i' High continuous strold-'-ngrates, with

hydrotatic guide and spindle bushing designI!fAU maintenance points accessible

without interference

-------.t ~~OJQD'IT~rn

PS180

All told, you get:r!fHigb machine efficiency and utilization ratesI!f'Vast improvements inthrougbputI!f"Greatl)' increased cutter life 'if Superior gear quality, surface finishes I!f Fast ROI

Then factor in American Pfauter's unsurpassed support capability ...design teams ...application expertise ...the industry's most advanced and productive shaper cutters ...a11available from a single-sourceconveniently located in Rockford, Illinois,

But why not see for yourself!Call (815), 282-3000 for moreinformation or to scheduJea visit to our facility ..

~AmERIU!n .,! - UIIIILimited Partnership

PMos: 815-282-3000Telefax: 815·282,3075

1351 Windsor RoadLoves Park, IL 61132-2698 U.S.A.CIRCLE A.-1 '01'1 READER REPLY CARD

More CllStol1leTS aroundthe world are tunling toPMCfto meet their IOOstdelnmuJiJlg Slwper Cutterneeds. Here's why:U't?'re innovators. PMCf hasn'tstood pat on conventional shaper cut-ter technology. New approaches.designs. and materials m-cin constantdevelopment. And our uniqueIsoform grinding process gives ow'shapercuners the longest effectivetool length available. Our cutlers

average 25% lower cost per th0u-sandth of usable tool life as a resultFirst ill customer suppon. PMCf israted No.1 in 'customer support' byleading automotive, DUcking, andother industry OEMs. They've foundthat only PMCf has the design teams,applications expertise. and modemmanufacturing technology to ensureon-time deli very of a qualified product.Product quality is assured throughoutmanufacturing with SPC. andadvanced inspection equipment.Here 10 stay. PMCT is in for the long

haul. Just look at the investmentwe've made in modemfacilities, newequip-ment and pe:ople-all committed to pr0-ducing the best geartools in the world.But don't take ourword tor it. Just call(81S) 877-8900 andask for the sales andservice support teamin your areaThen put us to work!

1351 Windsor Road, P.O. Box 2950, Loves Parle,IL 61132-2950 USATelephone 815-877-8900 • FAX 815-877-0264CIRCLE A·2 on READER REPLY CARD

FORMASTER. .. BUIILT FOR LASTIINGIACCURACY AND EASY CONTROLIN ,GEAR GRINDING APPLICATIONS

Norrnac's FQ,AMASTER CNC Grinding Wheel Prof Her has been rigidlydesigned and constructed to resist deflection and maintain an accuracyof ±.0001" (O.002mm) indefinitely.

Normae's proprietary offline software makes data input quick andeasy. This application software. (a collection of programs running on anIBM-PC compatible), completely removes the burden 01 complicatedmath calculations and NC programming. You can grind what youdesign.

An added plus is the FORMASTER'S compact design enabling it tobe mounted quickly and easily on old or new gear grinding machines.

Call or write today toflrr41lge a demonstration

INCORPORATED

P.O. IBox691 AirpoM !Road Industrial Par\(. Arden.NC28704 USATel: (704)684·1002 J Tele~~57-7437 Normae Hevl

Fax: (704) 684·1384P.O. Box 207/720' E Baseline Road 1 Northville,. MI48167 USA

Tel: (313) 349-2644/ Fax: (319) 349-1440'

Cover photo courtesy ofThe Falk Corporation,Milwaukee, WI.

CONTENTS7'\OVE:\lBER/DEC E~IBER 1492

FEATURESNon-!Invol'ute H10bbed IFormsThomas D, Ware

Starcut Sales, Farmington Hills, ML" """ ,, "."." " U

Tooth Contact Shift in Lo'sded Spiral Beve~G,earsDr. Michael Savageet a l .

University of Akron, Akron OH" ..2:4,

SPECIAL FEATURESLewis, Be~ding. Strength E,~uations CentennialInvestigatIOn of the Strengtbl of IGear TeethWi/Iud M. LewisWilliam Sellers & C.o. Philadelphia, PA"." .."""."."'''" .. ,,'''''''', ,, ,, ..18

Gear FundamentalsCla~ssifi;calion ot 'Types, ot IGear Tooth Wea,r - Part ILouis F'a u r e

CMD Transmissions, Carnbrai, France "..,.., , , ,..,.." ,..,.., , 32---- - -

DEPARTMENTS- - - - - - - - - -- ---

IPubHsher's PageThe Sum of OUf Fears 9

Sho,P R'oorICamshaft GearsWill i a 111 L. Jan n inc k 111

Adv,ertis'er IndexFind the products and services you need".",", "',, J·11

Management IMattersManagement By W:alkiing Around _Ric h a r d G, Ens 111, an, Jr " .." , 143

Cilassiti:edsProducts, services, and information you can use, "." "." " " 46

'CalendarEvents of Interest., , ,.. "." .. ,"".", .."."." .."." "." .. ,.. "., , ,." " ,.." 48

HIGH NOON

(310)' 202·901iO by appoim_ment(310)202·1340 (fax)

,:-WESTERN COLLECTIBLES ,~

BougJht & Sold:

"* Bridles'* Buckl'es'* Oolthing'* Bits'* Books

"* Belts... Bolos*Chaps* Spurs* Saddles

CIRCLE. A·6 on READER REPLY CARD'

1I1./1lsr4t:~.SubSldlarv of Star Cut1er Company23461 Industrial Park Drive. Farmington '·Wls, MI48335313/474-8200 FAX 313/474-9518CIRCI.E A-7 on IR~ADER REPLY CARD

,JI_II,.'t G'EAR TeCHNOLOGY

GEAR TECHNOLOGY

EDITOR! \L

Publishers; Edi.tor-in-CbiefMicbael (;Qldstein

Associate Publisher & Managing EditorPeg Short

Senior Editor Nancy Bartels

Technical EditorsRobert Errichelle

William L. JanntnckDon McVittie

Robert E. Smith

:\RT

Art Director Jean SykesArt Director jenn:iler Goland

,\ [)\'ERTISI:\:(;

Advertising Sales Manager PatriciaFl.sm

Sales Coordinator Don_na-Mllrie Weir

CIRClllATIO:\

Administra:tive CoordinatorDeborah Donigiao

Circulation AssistantsJanlee JacksonMary Sauerland

R,\:'\\),\l.l.l'lTBI.ISIII:\'(; ST.\FF

President Michael 'Goldstein

Vice President Richard Goldstein

Vice President/General Manager Peg Sllort

Controller Patrick. Nash

Accounting Laura .Ki:onane

R:\~n.\I,I, PllBLlSl-ll7"C, 17"l'.

1425 Lunt AvenueP.O. Box 1426

Elk Grove Village, IL 60007(708) 437-6604

- -

vor 'I, :\'0_ (,I

G.EA8 TECllNOUIGY. 11>< Journal of G...- M'''' __ ''J!& IISSNU143=M5K) is publu.hed blfftOfllh1)"by Randall Publish"wg.li'I::' .. 14:2:SlulUA,yc:~P.O. BoA 14l6.E1-.kGroveVil1qe.1L 6!)))'r. Subs:riptiQIII"MeJIIiR':540.00;" "'" u.s,; $3Q,OOin CIIIOd" 555.00 in aU ocII

THE8 e c a use

WORLDIISYou

NOW AH ear Our

QUIETERG ear s

or many years, gearnoise has continued to

Are you p'lague the industry.concerned Today there is aabout noisy sotunon .. .me I

ADVERTORJAL

Line of Action Testing:THE DEFINITIVE TEST FOR HOSS

AND THE GEAR'S THEY GENERATE

A variety of tests are used by tooling experts todocument hob accuracy, The most common testemployed to meet AOMA, British and MCn stan-dards focuses 01'] checking the lead ill 3 revolutions(the maximum number of revolutions necessary togenerate an involute profile), Although effective. thisparticular test does not incorporate hob profile errorsin the resul r,

Another test that meets thestandards of DIN 3968 and[SO is the Line of Actiontest. In concept, Line ofAction documents asuccession of points ofcontact between the hobflankand the gear flankduring the generatingprocess, This line isperpendicular, or normal.to the hob flank profile andform the pressure anglevalue with the line of generation, The generationsegment (length of action) is the necessary generatinglength to generate a complete involute f1aILK.

Wh,y Une 'of .A,etiontestingi is importan,tto' t,he purchaser ,of hobs.

The Line of Action test is an indication of the finalresult you can expect in the generated gear as cut bythe hob you've purchased, The gear' profile can bedirectly related to the Line of Action test you receivewith GMI-FHUSA hobs,

While Line of Action testing is not universallyacknowledged, it is specifically recognized by theDIN 3968 and IS0' standards, It is an effective testbecause it's a. documented measurement of thefinal result the end user experiences from the hobhe purchased. In effect it is an absolute verificationthat the hob. when properly mounted in a rigid

hobbing machine and runningwith a properly mounted piecepan, win generate a toothprofile in accordance withthe desired accuracy,

This test is absolute assurance of the quality of thehob and its re-sultant impact on the generated gearprofile, The Line of Action test is directly related tothe total hob error-e-not including mounting errors.machine errors, etc. Once again. the involute profileon the generated piece part will be identical to thehob Line of Action. Shown above is the typical hobLine of Action test as well as the gear profile obtainedwith this hob.

6· G~AR TECH~OLOGV

We at GM[-FHUSA supplythi test from both mechamcaland CNC in pectionequipment.As an added documentation,GMI-FHUSA will fax Line ofAction Test results to customer,before the ordered hob leaves thefactory .. .another assurance oftotal commitment ~O hob quality.

'GMII-FHUSAFor further information, contact:

P.O. Box 31038Independence. OB 44] 31Phone: 216-642-0230Fax: 216-642-0231

CIRCLE A-SS em READER REPLY CARD

,51 - eaks loude:rthan wordls.

"Line' of Action testing insures that each cutting edge of tile hob is in proper position relative 10lead. flank form and rotational accuracy of Ihe hob.

Meet1ing A or AA qualiity requirements is easiersaid than done'. That's why each IFHUSA hob isguaranteed - every tooth is in proper positionto generate the desired gear profile,.When Igear quality lis cfliticailly important,ILine of A.,ct,ion can assure success.

Insist on FHUSA hobs that are inspected andverified to meet your requirements. ALLIFHUSA hobs are' ... and can,Once you've checked our Line 0,1 Actionyou've taken the' first step ..,..and best steptowards a better bottom line ....tool

GMI PO lBox 31038Independence, OH 44131Phone (216) 642-0230FAX (21'6)642-0231'CIRCLEA-36 ,on R,EA'DER REPLY CARD'

M&M Precision svst-ems-I~NC'G'ear In5De.C~/onlSvstems~Wo~/dwldeSolutions.Quality-from s.ad10 Dn s'h. Our total processinspection capabilities ean assist you in monitoring'gears, part blanks, and gear ,cutting tocls such ashobs, shapers. shaver ,cutters, and broaches. We,also ,offer Interfacing ,capabilities to transfer testresults to standard SPC modules for evalua-tionlprocess monitoring.IReUablllty-true, generatlvel Index" lead" ,andIlnvDute testing. Our generative metrology tech-niques offer superior accuracy; efficiency and sim-plicity ,over touch probe methods. Plus, U,9setechniq,ues are enhanced by ,computerized automa-tion and analysis.

SoluUan '-we ofte stancial'd or customenglneerad Ipackages. Our technlcal suppon teamwUl work with you on your speclfi~c requirements.With eur years of experience we willi provide you withsuperior training and after-the-sale technlcall support.Shown above, ,our 3012-4 QC G.ear ,Analyzer ts loneof a family of gear ,and ,gear cuning too'il analyzers.Other ,opt'ional inspection ca,pabili1!ies include,unknown gear, gear surfa.ce finish" spirall bevel andhypold gears, worm inspection, Involute scrolls,and mal'alfemal'e helical rotor vanes. IFor a full co'iorbrochure write or calli M&Mi Precision Systems Corp' .•300 Progress lAd."West Carrollton, OH 45449-Phone:513-'8S~273, Fax: 51'3-.6594452.

M&M PRECISiaN,avs,EMS ,CDRRAN ACMEClEV£L4NO COU.9lNY

CIR.CLEA-5 on READER REPLY CARD'

The Sum of Our Fears"Prospel"i'fy' :isnot ,,,,ithout loa.ny f- -,-s---d d"'st- t-··_~__ _ _ __ _ _____ __', ea an, I as es."

dv-rSI"t' y I""Snot witbout' comf t- -- dC'-h - - --~ c' - --~-'-- '-- ' Dr sail" ope

Sir Francis BacuD.

and

a t month I artended a meeting inMexico City sponsored by CLATEQ,a quasi-governmental organization inMexico, which has as one of its aimsthe encouragement of the growth of

the gear industry in Mexico. The purpose of themeeting was to provide a catalyst among theattendee '10 form iii Mexican equival,em ofAGMA and to encourage:m alliance withAGMA. Joe Franklin, the Executive Directorof AGMA, BiU Bogge . the President, Vice-Pre ident R~IYHaley. and ~were among the fewAmerican. at the meeting.

ot surprisingly, one of the major topics ofdiscussion at the meeting was the NAFT A - theNorth American Free Trade Agreement, whichwin probably be passed by the U.S. Senate.jfnot thi year. then in 1993. While the detail ofthe treaty are still vague ,in many respect ,'thegeneral thrust ofthe arrangement \s quite clear,H NAFf A is approved, trade barrier betweenthe United State. Canada. and Mexico wiUgraduajly be all but eliminated. And the ques-tion on the minds of all of us at the meeting wasthe same, what will this mean for our busine s?

The interesting thing to me was that regard-le of which ide ofthe border we resided nn,QUI' hope and, particularly our fears, werealmo tidentical. Ll.Sibusines es areconcerned,rightly enough, that free trade with Mexico andCanada will mean 3. los of jobs in the U,S, Weee Mexico a a giaO[ to the oulb. with vast

quantities of cheap labor that will be able tomanufacture goods at far lower prices than wecan. How win we be ableto compete with acountry that pays its workers less than half ofwhat we do?

On the other hand. our Mexican counter-parts feel dwarfed by what they imagine as an

industrial colossus on their northern border,filled with state-ot-the-art factories capableof 24-hour shifts staffed by robots, NCmachines, and the late I. in high-tech won-ders they cannot hope to afford to purchasefor years. They ask themselves. how can wehope to compete against that?

The reality on both sides of the border, ofcourse, falls somewhere in between these fWO"worst case" visions, If and when tile trade

PUBLISHER'S PAIGE

Attending the CIATEQ meeting i" Mexico Cily were kf/ to right: GTpubJis/J"" Miclra~1Goldstein. AGMA. President, Bifl.Boggess. Dr. Iuventino Balderas MorellO oflhe M'exicandelegation to the NA.FTA talks, and AGMA Executive Director, Joe Frankli», Jr.

barrier between our countrie come down.there will be winner and loser on both ides ofthe Rio. 'Grande. The ell tornary, comfortableways of doing things both here and in Mexicowill of necessity have to change. We'll all haveto learn some tough new lessons,

At times like these, when a multitude ofconflicting interests all demand to be heard,and a variety of conflicting needs have to be

N,OVE'~BER/DECE~BER' i a s a '9

met.It is always tempting £0' act out of our fearsrather than out of our hopes. But is doing so thebest way to meet the challenge of an industri-ally growing Mexico?

A look at the positive side is in order. Since1988, U.S. direct investment in Mexico hasgrown from $5.5 billion to $9.4 billion. Two-way trade between the two countries was $64.5billion last year. That's .1 lot of dollars andpesos changing hands, and people all both sideof the border are benefitting - even before theNAFTA is in place. And access to' a share ofthose dollars and pesos is not limited to afortunate few. It i open to any businesspersonwining to do what it takes to become involved.

Successful jrading with Mexico is governedby the same rules a succe sful exporting any-where. Besides the necessary capital and will-ingne s to take a risk. the, uccessful exporterneeds time, imaginarion, flexibility, and open-ness to new ideas: Time to learn the country" thelanguage, the business culture, theneeds of thelocal markets. and to let investments grow and

PUBLISHER'S PAGE

"No nadon was ever I-llinellby tra.de."

Belljamill FI-ankUIi.

develop at a reasonable pace, which may beslower than the accustomed one; imaginationto see how a particular business can fit into thelocal environment; flexibility enough to alterplans that don 't work or don't fit the changingcircumstances across the border; openness toways of thinking and doing business that aren't"the way we've always done it back horne .."

It is also important to remember that ex-porting does not have to be an all or nothingpropo ition. By means of local representa-tives, joint venture • and other kinds of coop-erative arrangement. busines e on both idesof the border can get their feet wet in theexport market without having to go in at thedeep end of tile pool. In cooperative ventures,each member can play to its own. trengths,whether those be extremelycompetitive labor,tate-of-the art equipment, valuable local

knowledge, or a well-developedcustorner base.

1'0 GEAR TECHNOLOGV

Each partner can continue to service hiS or herown customer base, do business in his nativetongue, and follow local customs ..At the samelime, each partner has vastly expanded thekind of products and services he can offer.

Finding the perfect match for a tradingpartner is hard work. It has been suggestedthat finding the right partner for overseas jointventures may be the management challenge ofthel990s. That may be overstating the case,but finding the right partner is key to successin joint ventures. To find the right partner withwhom to do business across the border, youhave 10 travel, meet with many of your coun-terparts in the other country. research the po-tential partners, learn as much as you canaboutthe local economy; in short, you have todo a lot of homework and be open to newpossibilities .. Perhaps this is a place whereAGMA and CIA TEQ can help by arrangingmeetings where potential investment partnerscan meet and get to know one another and thekinds of opportunities available on both sidesof the border.

Success in the new global economy de-mands that we operate out of our hopes ratherthan our fears and see the promise offered bytrade with our neighbors rather than only thethreat. Living next to a strong, economicallyhealthy neighbor with industries that. competedirectly with ours is without a doubt more of achallenge than living next to one with no suchindustries. But such strong neighbors also pro-vide better markets for our products and awealth of opportunities for the shrewd andimaginative businessperson.

To respond to the new global economicrealities out of the sum of our fears is [0 envl-ion a nightmare world which doesn't really

exist. Making business plans to cope only withthis nightmare in the end does us more harmthan good. To respond our.of our hopes gives usthe chance to evaluate our fears in the clearlight of day and deal with them - and with theopportunities that exist side by side with them- in a way that can benefit us all.

~ Michael GoldsteinPublisher/Editor-in-Chief

Camshaft Gears

mne of our reader in England haasked for our help in locatingpublis~ed technical d.3taand in-formation on the design, manu-

fact 1.1re, and inspection of camshaftgear. Althnugh millions ofthese gearshave been made and are :inconstant lise,we are not aware ofany formal materialhaving been published. We would bepleased to hear from anyone who hasknowledge of such information,

The camshaft gear gels its namefrom the factthm it is leeated 011 an

I Figl.11

15 Teeth141NDP14.5 NP.A56° Helix

PlungeCuI Form

Fig. 2

,- -0,0.-

'I I1--r- ~~-1,....,,....-, I-- I,.001 ',(P.D, .....f.ljlt-l+----f+l+---++V-+~__

I

jR.D, I

.EndI F,ig.3

Cenler - EndProfile Check

WiUiam L J:anninck

engine camshaft where it i usuallynested lightly between the cam lobes orbearing journals, The gear is of a highhelix angle and drives a much smallerdiameter gear which turn a distributoror oi:1pump, Both gears havethe samenumber of teeth, rotating on a '1-10-1ratio, Since these gear operate at a'9Uo-ax.is angle, the distributor gear hasII complementary helix angle to thecamshaft gear helix .. Fig. I illustratethe arrangement of these gears.

Since the distributor gear is a con-ventional helical gear. il is usuallymanufactured by the finish hobbingprocess. The cam haft gear, however,being confined to available space onthe shaft, poses a special problem,There is usually insufficient space fortooltravel to completely generate a truehelical gear, andthe compromise is toplunge cut the gear with a large a hobas pas ible without damaging the adja-cent cam or joumal surfaces. This leavesa non-involute gear which is totallyfunctional in use, but present anotherproblem. that of gear in pection,

Fig. 2 shows a comparison of thetrue involute helicoid surface with thatactually cut by infeed hobbing. Thedimensions shown on the sketch showthe material that is left on the gearrelative to the involute helicoid form,Along the diagonal line of COIl 'lac I.swept out.the form is correct. and theset functions properly. Fig. 3 showsthe involute trace on the gear flank

Non -InvoluteHobbed Forms

Thomas !D.W·alreSta.rcut Sales, Inc., F,armi1ngtolil Hills. MI

Fig. I - Major diameter fit.

Fig. 2 - Minor diameter fit.

Fig. 3 - Side fit.

Round Bottom

Straight-sided splines have maximum strength when a full-fillet design is utilized. Suchfull fillets (round bouoms) can be simply reproduced by the hobbing tool, The fillet onthe tool differs from that on the produced fillet on the spline.

Fig. 4 • SUi'.aight side round bettom.

12 GEAR TECHNOLOGY

Altbeughhobbing is usually associated withthe manufacturing ef'involute gears and spline ,many other tooth forms can be produced by agenerating hob. A generating hob is defined asone whose tooth form :isnot reproduced on thepart. These tooth forms must beequally spacedabout an axis. but do not have to be symmetri-cal, The types of parts this article will coverare parallel key splines, serration. sprockets,ratchets, and special tooth forms.

Parallel Key SplinesThis istbemosecommon non-involute tooth

form. A parallel key spline can be made formajor diameter fit, minor diameter fi.t, or sidefit. (See Figs. 1-3 ..) These fits are divided intothree classes: permanent. sliding without load,and sliding with load. The different fits andclasses require different clearances betweenthe spline tooth andthe mating part

The hobs -can be designed to producedifferent features in the spline root or themajor diameter. These features are full fillet(round-bottom), flat root, and clearancesgrooves. (See Figs. 4-6.) Clearance groovesare produced on the workpiece by lugs onthe tip of the hob teeth. Non-topping. semi-topping, full topping, and shoulder clear-ance are hob features that affect the majordiameter of the spl ..ine.

AU parallel key spline hobs are single-purpose tools designed for a specific numberof'teeth, key width, and diameter. The tools aresupplied in three quality levels: A, C, and D,The quality level is controlled by the piece partdata. A sample is cut with each hob and in-

spected to assure tool quality. Normally thiscoupon is shipped with each hob.

ANSI Standard B94.7·1980 ha toler-ances for hob manufacturers' 'lest cut. PartA is for spline tooth thickness held ..Part B isfor minor diameter held. The tandard cov-ers 6-, ]0., and 16-toolh splines. Part Ccontrols the maximum undercut (per ide).Hob are al 0 designed and built to pecificcu turner part tolerances.

B94.7-1966 ala established standard hobsize for ingle-thread tools baed on the num-ber of splines, depth of cut, and part diameter.These sizes are recommended minimums basedon all acceptable number of generating flatsand regrindable length. This diameter call bere meted by sweep-out radius limitation. Mul-tiple-thread hobs can also be used to cut paral-lel key splines, but as the number of generat-ing teeth are reduced, the surface finish rough-nes . and Chip load will increa e.

. wo of tile specsal features, need orneexplanatron. Lugs are used only as alast re artto obtain the required straightand paralleldepth and [lot completely remove the minordiameter. Because of their shape and location,these lugs wear very rapidly and must be, harpened often. The space width must belarge enough to leave a sufficient minor diam-eter bearing. (Fig. 6)

Shoulder clearance type. hob are used whena full-depth pline is required up to the back- Fig. 7 - Shoulder clearane 'hob.

ing shoulder. The hob tooth i of sufficient

A semi-topping hobcan be used on straight-sidedsplines to produce .0 chamfer on theoutside comers of the spline teeth.

Fig, 5 • 'semii.topplng Rat root.

Proper fit of straight-sided spline members isachieved by hobbing a clearance groove inthe area where the straight side meet' the ml nor diamelerof!he p:1ine, Small projeclionscan be provided on the comers of the hob leel1llo provide a blended dcarnnce groove,

Fig. 6 • Straight s de clearance j:lroove.

depth to dear the shoulder diameter. In thehoulder portion of the shaft. the key profile

will can ist of a traight side 'to the out idediameter of the spline. topped by an involutecurve generated by the straight or s.lightlyangled clearance groove. (Fig. 7)

The hob tooth form is a curve. Its hape is

determined by the generating diameter se- l.25 GC!lJ" Dill.

Hob Tooth

Spline Tooth

lee ted by the hob manufacturer. Normally it i Fig. 8 _Large genu.' ling diameter.atthe O.D. of the' pline to allow as muchcutting clearance in the hob a pos ible. (SeeFigs. 8-9.) This dimension must be balancedwith the height of the start of rraight-sideddiameter. Approximately one quarter of thewhole depth will be fillet radius. Once tile hobdesign is established for a specific tooth thick-ne s and root diameter, the tool will not accu-rarely cut a different. tooth thickness or rootdiameter. If the hob is sunk-in deeper, as in

Thomas D..Ware'Spline Tooth is 111l! Chief Engineer,

Gear Tools Dep art-mellI, at Slar('1II Sales,Inc .• Farmington Hills.MI. He is the (III/hoI' of

........ --'1 ....O...8...G~e""!L~r"",D=."ia. several papers all gear.Fig. '9 - Small generating diameter. ;fl8 subjects.

NOVS MB ERIOECEMBE R 1 U2 113

Spl ine 1'001.11

Fig. 10· "Sunk in" hob.

Hob Tooth

Spline Tooth

Fig. 11 - "PUlled out" hob.

Fig. n -Sharpenlngprnblems: Ilfustrations showing three types ofrunout on a spline hob and their effect on splines.

Resulting SplineTooth FormWhen SpacingIs Too Great.Spacing Is1'00 Small

Fig. U • Sharpening error: cutting faces of spline hob unevenly spaced.

'14, GEAR TECHI\IOLOGY

Fig. 10, it will produce a tapered tooth minusat the root. If it is pulled out, as in Fig .. 11..the spline win be tapered with plus tooththickness at the outside diameter.

Sharpening errors and mounting runoutcan also adversely affect the qual ity of thespline. Parallel key splines are very sensi-tive to errors due to the nature of the gener-ating action of the hob. Uniform or Singledirection runout will cause the keys to be-come undercut on both sides or tapered onboth sides. With one-end runout or wobble.the key will be undercut on one side andtapered on the other side. (Fig. 12)

Excessive flute spacing error win gener-ate either a bump or a hollow on the sides ofthe teeth. (See Fig. 13.) The root di.ameterwill also have high and low flats and, if thespacing error i progressive. it may show upas an eccentric root diameter. Negative rakeerror produces a tapered tooth and a plus rootdiameter. (See Fig ..14.) Positive rake or hookcuts a key with undercut and an undersizedroot diameter. (See Fig. 15.) Adjusting thebob to cut the proper root diameter onlymagnifies the taper or undercut problems.

It is also important to remember that asthe hob tip wears, the fillet radius becomeslarger and can cause interference.

SerrationsStraight-sided serrations or straight-sided

splines are similar to parallel key splines inthat they have a flat tooth surface that isgenerated with a curve-sided, single-purposehob. (See Fig. 16.) The obvious difference isthe sides of the teeth ace alan angle to oneanother rather than parallel.

oThe SAE has a standard for 90 included

angle straight-sided serrations, (See Fig. 17.)The standard is based on nominal outsidediameters for 36 and 48 teeth only. Side fit isthe only type of fit specified. Allowable er-rors and machining tolerancesare the same asan equivalent pitch involute serration. Toinsure that the bobs will produce the partwithin tolerances, a sample is cut and in-pected. For finer pitch hobs, an involute

serration hob can be used in place of thecurved form hob.

As with the parallel key spline, when thestraight-sided serration hob is "sunk-in" orpulled out, the angle and form will change, as

· hewn in Fig. 18. It is possible by u inganintermittent hob de ign to produce a sharppoint at the 0.0 .. but a practical limit of.003R is obtainable in the root.

SprocketsThere are many different types of prock-

et . They are roller chain, silent (invertedtooth)" synchronous belt drive. hinge type ..block chain, and film procket. All have anen-lnvclute tooth form. Most are coveredby an ANSI tandard or a special applica-'lion design,

Roller Chain Sprockets. Most roller chainprockets are manufactured to the ANSI

18.29.1 "Power Tran mission Roller Chains"tandard, (Figs. 19-20) These sprockets are

specified in two qualitylevel . commercialand preci, len. Precl iOI1 sprockets are usedfor high speed and high load • critical timing,or accurate regi ter. The sprocket can be aType [ (with pitch line clearance) or Type II(without pitch tine clearance). Another com-men tandard i the British Standard :OS228.The main difference from the ANSI tandardisthe roller ize, A numberof special rollerchain sprocket tooth forms are in use.

Standard sprockets are classified by chainnumber. Thi .number specifies both the pitchand roller diameter. Minimum hob size areabo baed on the chain number for bothsingle and multiple thread hobs and are foundin ANSI B94,7-1.966.

Sprockets can be cut witheither a fulltopping or non-topping hob. Only one nOI1-toppnsg hob i needed to cut all number ofteeth of a ' pecific chain number. A toppinghob can cut only a small range of teethwithout exceeding the standard tolerance .As the number of teeth increase. the wholedepth must be increa ed, These ranges are 7-8 teeth, 9-11 teeth, 12-17 teeth, 18-34 teeth,and 35 and over.

The most critical features of roller chainsprockets are tooth spacing, seating curve ra-dius and root diameter. When single-threadhob are used, 100lh spacing is primarily de-pendent upon the hobbing machine. and theroot diameter ami 'eating curve radiu wi'll beuniform, ince all the teeth are finished by thesame group of hob teeth. Multi-thread hobswith a prime thread-to-tooth ratio will pro-duce accurate tooth spacing and a uniform Fig. 17 ..Sem Uon tooth.

Correct Cutting EdgeSharpenedWilh "-Negative -.Rake &~OOF7Ili;i~IiiiF====;t'''

Sharpening Hob With Nega-live Rake Will. Result In AChange Of Tooth ProfileAlong The CUlling Edge.

Correct SplineTooth Form

Fig. 14· Shaflileniing ,error: euttfng face of pUne hob i!ll'.ound with negalh,crake \ hen i.t should be radial,

Sharpening Hob WiU] HookWill Rc~uh In A Change orTooth Profile Along The

utting Edge,

Correct: Plillme .Toolh Form .1

• esuhing pllne -, -. i. .'..Tooth form . . '

Fig. 15· Shol"pening error: Cutting face ,olispUnc hebgreund wUh bookwhen it should be radial.

Fig..16 • Serratton tooth and hob,

·Na v EMil E RID E C EM B E A I; 12 1!5

Small End

Fig..19· Roller chain prock,et tooth and bob.

Fig. 20· Roller chain procket tooth.

Sprocket looth form

Fig. 21 • Silent chain procket. teeth.

16 ClEAR TECHNOLOGY

root diameter, but the seating curve radiumay vary. An even thread-to-tooth ratio willproduce an accurate seating curve radius butthe tooth pacing may not be accurate, and theroot diameter may 110t be uniform. Sharpeningerrors will affect the tooth form, but they arenot as ensitive as gear or spline bobs.

Silent Chain Sprockets. Silent chain or in-verted tooth sprockets are specified by ASA829.2 "Inverted Tooth (Silent) Chains andSprocket Teeth" or special tooth forms. (Fig.21) The chain links have no sliding action onor off the teeth. This re ults in a smooth.practically noiseless action. Standard prccketteeth have straight sides, requiring hobs withcurved teeth. Theoretical Iy , each number of teethrequiresa di fferent hob. but tolerances are broadenough 'to allow ranging. The range of teeth forthe hob are 1.7-23, 24-32, 33-43, 44-58, 59-79,80-BO. and ui-rso.

Each standard pitch i specified by a chainnumber, designated Sen,. The number is thepitch in eighths of an inch; i.e., SC5 = 5/8 "pitch.

Tooth spacing and tooth profile are themost crirical features of a silent chainsprocket. Accurate tooth spacing is neededbecause of the number of teeth engaged atone time. Tooth form must be held (with noconcavity) to eliminate chain whip. Of thesprocket type hobs, the inverted tooth hob imost sensltive to sharpening errors.

Synchrorlous Belt Sprockets. Synchronoubelt or timing belt sprocket come in manydifferent tooth forms. (See Fig. 22.) A sam-pliag of the various types are Trapezoidal(ANSVRMA IPZ4) shown in Fig. 23 andSynchroflex (mN 7721) shown in Fig. 24.These sprockets are specified in public stan-dards. The balan e were developed by dif-ferent manufacturers and are patented. Someof the tooth form' are Gilmer, Goodyear,Gates Rubber (HTD and Poly Chain GT),Dcdge-Re liance Ele ctr ie (Dynasink),Dayco, and Pire ll i.

The tooth forms vary from a traight-sideto mooth radiu to complex. mathematicalcurve . Each come. in various pitche •most ofwhich are metric. One common factor is thatthe pitch diameter i' above the outside diam-eter of the procket. This matchesthe solidsection of rhe belt

Hobs used to generate timing belt sprockets

are toppingand can cut only a small range ofteeth ..A sample cut is included as part of thehobin pection proce s. The bobs used to manu-facture the belt molds have a different toothform than the sprocket hob.

RatchetsThe ratchet-rype tooth form can be gener-Fig ..22 - Timing bell toothond hob.

ated using a convendonal multi-poshion hob.These ratchets must have a large fillet radius.(See Figs. 2S~26.) The straight portion of thetootb face is approximately two-thirds of thewhole depth. Ai 0 the tooth face may beradial. negative, or slightly positive. A sharpoutside diameter can be produced with anintermiuent or staggered-tooth hob. Ratchetswith hooked faces, small fillet radii, or nearly Fiig..23 - Timing belt rack Iteeth

parallel sides require a special single posi- r----------------------------.tion hob .. (Figs. 27·28)

Special Tootb FormsNOR-circular shapes and unHke tooth

~OITl1s. can also be generated using a 'con-ventional multi-po ition bob. Examplesof Ibis type of part Me squares and otherpoly-sided part • ovals. semi-circular teeth.and slots and grooves with different toothforms ..As with aU these types of bobs, theyare single-purpose for the exact part andhave no flexibility.

In conclusion, if the user is willing tusacrlfiee some Flexibility, a generating hobcan be used to cut. much more than invo-lute tooth forms. With proper care and'the correct machine adjustments, such aobscan also cut parallel key splines, erra-tions, roller chain. silent chain, and syn-chronous bell sprockets, ratchets, and pe-cia! tnoth forms. •

Acknow~edgement: From the Society of Meehani-ca'/ Engineers Fundamentals of Gear Manufae.tur-ing Clinic. 1992. Repritued with permission .

.ReferencesAC ;SI B29.l-197S. Precision Power Transm:issions- Roller Chains, Attachments & Sprockets.ANSr IP-24-1983. Synchronous Belts.ASA 829.2-1957 .Inverted Tooth (Silent) Chains &Sprocket Teeth.BARB R-COLMAN CO . .Hoh Handbook .. Est ed,Rockford. IL. [954.lLUNOIS TOOLS WORKS. Catalog E. Chicago,IL,1944.SOCIETY Of AUTOMOTIVE ENGINEERS. SAEHandbook. Warrendale. PA. 1'950.

r:t

HobToolh

Fig. 24 - TLming belt sprocket tooth.

Fig. 25 - Typical ra.tchet t.ooUt.

../'--t7(...

Fig. 26 - Large radius I"atf:het toaLb.

Fig. 27 • SmaU ra.dlus ratcbet tooth.

Fig. 28 - ParaUe] tooth ratcbet.

N QV!; M·B E RID E CE. M 8 E:R IR 11.2 1'1

'99;JVLarL Jmyorta»t

Investigation ofthe Strength of

GearTeethWUfredi Lewis, Member, Engineer:s' ,Club 'of IPhUadelp,hia

Read October 15, 1892

To mechanical engineers, the trength of gearteeth is a question of con tant recurrence. andalthough the problem to be solved is quite el-ementary in character, probably 1110 other ques-tion could be raised upon which such a diversityof opinion exists, and i:1l upport of which suchan array of'rule and authoritie might be quoted.In 1879, Mr. John H. Cooper, the author of awell-known work: on "Belting," made an exami-nation of the subject and found there were thenin existence about forty-eight well-establishedrules forhorsepower and working trength, sane-tioned by orne twenty-four authorities, and diJ-fering from each other in extreme cases of500%.Since then, a number of new rules have beenadded, but as no rules have been given whichtake account of the actual tooth form in com-mon use, and as no attempt has been made toinclude .inany formula the working stress on thematerial o that the engineer may see al onceupon what as umption a given result is based, Itrust I may be pardoned for suggesting that afurther investigation is necessary or desirable.

• .11 summing up his examination, Mr. Cooperelected the following formula from an English

rule publi hed at Newca tle-under-Lyme in 1868,and. a an expression of good genera] averages,it may be con idered pa ably correct.

X = 2,rxxYJ p /in which

x = breaking load of tooth in poundp ... pitch of teeth in inches/= face ofteeth in inche

In conclusion he makes this pertinentob ervation: "It must be admitted that theshape of the tooth has something to do withits ~trength, and yet no allowance appears tohave been made by the rules tabulated abovefor uch distribution of meta]. the breakingstrength being based upon the pitch orthickness of the teeth at the pitch line orcircle, as if the thickness at the root of thetnoth were the same in all cases as it is atpitch line." Notwithstanding the fact thatthe necessity for considering the form ofthe tooth, as wen as it pitch and face, wasthus clearly set forth over ·Ihirleen yearsago, I am not aware that anyone else hastaken the trouble to do it, and.as the re ults10 be presented have been well-te ted byexperience in thecompany with which Iam connected, I believe they will be ofintere t and value to others.

In estimating the strength of teeth, thefirst question to be considered is the pointor line at which the load may be appl.ied toproduce the greatest bending stress. In theniles referred to, the load is sometimesas umed to be applied at the pitch line ..ometimes at the end of'the tooth, and

sometimes at one corner; but in good mod-ern machinery. the agricultural type ex-cepted .. there is seldom any occa ion toassume that the load is not fairly distrib-uted across the teeth. Of course, il may beco.ncentrated at one comer as the result of

; " rs '"7""U,J"'f"J", roo ,''''' '"',"" i~I1rO{~ittcllfw·i S'11F

tooth. it ·15 rsib(t: tomrljltt: ~ str(Ss f II {i-JimrscJ OfitO il rm(,J/J I

--dIS obstnratwil 15 tk ~15Jor ~ LewiS ~IJ.~~tn;~~~Jortllllfn.

NOV E. M S E. RIO E C E M B E R 1 9 9 2 119

20 G~AR TE,C!H!OL.OQ,'Y

careless alignment or defective design. inwhich the shafts are too light or improperlysupported. and. for a rough class of work, al-lowance should certainly be made for this con-tingency, but" in an 'cases where a reasonableamount of care is exercised in fitting, a fullbearing across theteeth win soon be attained inservice, It must be admitted. however, that onaccount oftheinevitable yielding of shaftsandbearings, even of the stiffest construction, thedistribution of pressure may not be uniform

teeth. Having thus concluded that gear teethmay fairly be considered as cantilevers loadedat the end, the influence oftheir form upon theirstrength remains to be disposed of.

In interchangeable gearing. the oycloidal isprobably the most common form in general use,but a strong reaction in favor of the involutesystem is now in progress, and I believe aninvolute tooth of 22- ]12° obliquity win finallysupplant all other forms. There are many goodreasons why such a system should be generally

under variable loads, and that. the assumption of adopted. but it is not my purpose at present touniform pressure across the teeth. is not alwaysrealized in the best practice. To what extent it isrealized I shall not attempt to estimate, but ingeneral practice. where the width of the teeth isnot more than two or three times the pitch. thedeparture can 110tbe regarded as serious ..Theconclusion is therefore reached that in first-class machinery, for which the present investi-galion is intended, the load can be more prop-erly taken as well-distributed across the tooththan as concentrated at one comer. Having thusdisposed of the first question, the second is, atwhat part of the face should the load be as-sumed to be carried in estimating the strength.ofa tooth?

Evidently the load may be carried at anypoint. within the arc of action, and it might beargued that when a tooth is loaded at its end,thereare always two teeth in gear, and that theload should be divided between them. This istheoretically true of all teeth properly formedand spaced •.but it must be admitted that me-chanical perfection in forming and spacing basnot yet. been reached, and that the slightestdeviation in either respect is sufficient to con-centrate the whole load at the end of a singletooth. In time this concentration may be re-lieved by wear, but it is not so easily correctedas unequal distribution across the teeth, and. asthe present practice of cutting gears with alimited number of equidistant cutters makes italmost impossible to obtain teeth of propershape. it is evident that the load cannot safelybe assumed as concentrated at a maximumdistance less than the extreme end of the tooth.In some cases. of course, the teeth will not be soseverely tested. and the error in this assumptioncompensates in a measure for the error in thefirst assumption of equal distribution across the

discuss the merits and defects of different sys-tems of interchangeable gearing, and I nowproposetoexplain how the factors given in thetable herewith were determined by graphicalcon struction.

A number of figures were carefully drawnon a large scale, to represent the teeth cut byacomplete set of equidistant cutters, making thefillets at the root as large as possible to clear anengaging rack. See Plate L

The addendum was .3p and root ..35p, asshown in the ilfustrations, and the cl'ear-ance was .02p.

Wben the load is applied at the end of atooth and normal to its face in the direction ab,it may be resolved into two forces, one tendingto crush the tooth. and the other to break itacross. The radial component •.which tends tocrush the tooth directly, has but a slight effectupon its strength. In material which is stron-ger ill.compression than ill tension, the trans-verse stress due to the other component W ispartially counteracted on the tension side,and the teeth are stronger by reason of theirobliquity of action; but i.n material which isweaker in compression the reverse is thecase, and, in general, it may be said that thestrength of teeth will not be affected more (han10% either way by the consideration of thisradial. component.

It should therefore be understood that forthe sake of simplicity, the factors given havebeen determined only with reference to thetransverse stress induced by the force W•.which may be regarded as the working loadtransmitted by the teeth. This load is appliedat the point b. but it does not .at once appearwhere the tooth is weakest. and, to determinethat point. advantage is taken of the fact that

any parabola in the axis be and tangent to bWat the point b encloses a beam of uniformstrength .. Of all the parabolas that may thusbe drawn, one wiU be tangent to the toothform, and it is evident that the point of tan-gency win indicate the weakest section ofthetooth. ..III the rack Woth of 200 ob~.iql[lity, thisis found at. once by prolonging ca. to its inter-section g with the center line jb, and layingoff bf= bg; and in other cases, the weakestsectioned may be found tentatively to a nicedegree of accuracy in two or three trials.Having found the weakest section, the strengthat that point is also determined graphicallyby drawing be and erecting the perpendicularce to intersect the center line in e. Then ef orx is taken to measure the strength of thevarious forms of teeth.

To understand the reason for this construc-tion and the actual relation which the distancex bears to the strength of the tooth, it will beobserved that the bending moment Wi on thesectioned is resisted by the fiber stress s intoone-sixth of the facej'times the square of thethickness t, or, by the wen-known formula forbeams. we have

WI:::: sfr or6'

sft2W = -6771--

But by similar triangles

rx=----. 41

and substituting this value in.Equation 1 we have

2xW=sf-,3

or we may write

W=spf~3p

2xThe factor 3p or y:isdetermined by graphical

construction and is given in Table 1 for conve-aientrefereace, This is multiplied by the pitchface and fiber stress allowable in any case whenthe working load W is 'to be determined. Whatfiber stress is allowable under different cir-cum stances aad conditions cannot. be defi-nitely settled at present, nor is it probable that

(1)

(2)

,,' .

Racks

any conclusions will be acceptable to engi-neers unless based upon carefully madeexperiments, In the article referred to, cer-tain factors are given as applicable to cer-tain speeds, and in the absence of any lateror better light upon the subject, Table 2 hasbeen constructed to embody in convenientform the values recommended. It cannot bedoubted that slow speeds admit of higherworking stresses than high speeds, but itmay be questioned whether teeth running at100 feet a minute are twice as strong as at600 feet a minute. or four times as strong as

'N a v E Po! e E RID E C E 1MB.E RI8 92 211

(4)

Plate I

r-----------------------------~~~~~~~~~~~~~~~~-------.

TABLE 1

Factor for Strength, y

Number of Involute 200 Involute ISO RadialTeeth Obliquity and Cycloidal FlanKs

12 .078 .067 .052

13 .083 .070 .053

14 .088 .072 .054

15 .092 ..075 .055

16 .094 .077 .056

17 .096 .080 .057

18 .098 .083 .058

19 .100 .087 .059

20 .102 .090 .060

21 .104 .092 .(l61

23 .106 .094 .062

25 .108 .097 .063

27 .HI ..100 .06430 .114 ..102 .065

34 .118 .104 ..066

38 ,[22 .107 .067

43 .1.26 .110 .068

50 .130 .[12 .069

60 .1.34 .U4 .07075 .138 .116 .071

100 .142 .118 .072

ISO' .146 .[20 .073

300 .150 .122 .074

Rack .154 .124 .075

TABLE 2Safe' Working Stress, s, for Differene Speeds

II II~

IISpeed ofTeeth in I IFLper WOo. I I IMinute less 200 300 600 900 [200 J800 2JOO

;

Cast lron 8,000 6,000 4,800 4,000 3,.000 2,100 2,000 1,700

Steel 20,000 .15,000 12,(lOO 10,()00 7,500 6,()OO 5,000 1,300!

22 GEAR TECHNOLOGY

the same teeth at 1,800 feet a minute. Forteeth which are perfectly formed and spaced,it is difficult to see how there can be a greaterdifference in strength than the well-knowndifference occasioned by a live load or a deadload, or two to one in extreme cases. But forteeth as they actually exist, a greater differ-ence than two to one may easily be imaginedfrom the noise sometimes produced in run-ning, and it should be said that this table issubmitted for criticism rather than for gen-eral adoption .. It is one which has given goodresults for a number of years in machinedesign. and its faults, such as they may be,are believed to be in the right direction fromanother point of view; for when the durabil-ity of a train of gearing is considered, itwould seem that all gears in the train shouldhave the same pitch and face, because alltransmit the same power and are thereforesubject to the same wear. But this argumentis modified by the further consideration thatequal. wear does not mean equal. life in a trainof gearing, and a compromise between theconsiderations, of life and strength must re-sult in the adoption of different values fordifferent speeds, somewhat similar to thosegiven in Table 2.

To illustrate the use of Tables 1 and 2, leti.t be required to find the working strength ofa 12-too~hed pinion of 1'" pitch, 2 [/2" face,driving a wheel of60 teeth at 100 feet or lessper minute, and let the teeth be of the 200

involute form. In the formula W = spfy, wehave for a cast-iron pinion,

s = 8,000, pf= 2.5, and y = .078,and multiplying these values together wehave W = 1,560 pounds .. For the wheel wehave y = .134, and W = 2,680 pounds.

The cast-iron pinion is, therefore, themeasure of strength, but if a steel pinion besubstituted, we have s = 20,000 and W =3,900 pounds, in which combination thewheel is the weaker, and it therefore be-comes the measure of strength. In teeth ofthe involute and cycloidal forms, there is amarked difference between racks and pin-ions in working strength, while in radialflanked teeth, which are used more espe-orally on bevel gears, the difference is notso pronounced. These teeth are to be found

in the great majority of all cut bevels, be-cause they can be more cheaply produced onmilling machines and gear cutters. but the15° involute bevel tooth, a made by theBilgram process, is superior in accuracy ofform and finish, and is often preferred forpatterns and fine machinery. There are, there-fore, two well-defined forms of bevel gearsto be considered: and to bring the strength ofbevel gearing within the cope of the pre entinvestigation, it will be necessary to under-stand how the variation in their pitch andradius of action is allowed for. and withoutgoing into a demonstration of the formula.Itssimple statement will probably be ufflciem,

Referring to Plate II,D' = large diameter of bevel.d = small diameter of bevel.p = pitch at large diameter.TI = actual number of teeth.

N = formative number of teeth = 71secant a, or the number corresponding toradius R..

f = face of bevely = factor depending upon shape of

teeth and formative number N.W = working load on teeth referred

to in Diagram D.Then it can be hewn that

w - s t, IY - ell (5)- P Y 3 D' (D . d) .

To illustrate the use of this formula, letit be required to find the working strengthof a pair of cast-iron miter gears of 50teeth. 2" inch pitch, 5" face, at 120 revolu-tions p -r minute.

In this case, a = 45°, D == 31.8, d"" 24.8,and secant a = 104. N = 50 X l .4 = 70, forwhich y ;:::.071. The speed of the teeth is1,000 feet per minute, for which, by interpo-lation in the table" s = 2,.800, and the formulabecomes, by substituting these values ..

w= 2,800 X2 X SX. Or'll X 31.83- 24.8

3X3L82 (31.8 - 24.8)= 1.988 X .795 ;;;;;;.I ,580 Ib5 ..

This result is attained by some laborwhich is practically unnecessary. because dshould never be made less than 213 D, and

I

Plate II

comparison on which all authorities mayultimately unite. I.!

NOV~MBER/DECEMBER 1992 23

al-~

when this rule is observed, the approximateformula,

dW;;;;;;spfy-· ---D'

(6)

give. almost identical results. Tile reason for

fixing a limit to the ratio -~ is found in theD

fact that a further increase in face adds veryslowly to the strength and increases veryrapidly the difficulty of properly distribut-ing the pressure transmitted. Where long-faced bevels are used, the teeth near theshaft are generally broken by improper fit-ting, or, when properly fitted, by the springof the shaft or the yielding of its bearings,and, as the limit imposed gives about 70%of the strength attainable, by extending theface to the center, it is thought to be asliberal as experience can justify.

111 presenting certain forms and propor-tions for teeth, on which Table 1 is founded,I am aware that other forms and propor-tions aTe in common lise, which have someclaims to recognition, but my chief objectat present is to show that the strength ofgearing can be reduced to a rational basis of

Tooth Contact Shiftin Loaded Spiral

Bevel GearsMichael Savage!, University ·01 Akron'. Akron. IOH

P.C. Altidis.'Clark Materia,l Handliing Co.• Lexin.gtonl,IKYD,.G. Lewicki & J..,J. Co,y. N!ASA Ilew;is IReseal\ch ICenter, Cleveland. OH

IF.L Litvin. University of Ulino'jsa.t Chicago. Chicagol, IlL

Nomenclaturea Tooth mid-cone plane addendum, m

A Distance from gear to fir I bearing, mAo Outer-cone distance, m

B Distance from gear to second bearing, mn Tooth contact shift, mD,o Mid-cone distance, m

E1 Elastic support stiffness, N-m2EIo Elastic support stiffness of base design, N-m2

r Tooth face widlh,. mN Number of teeth

o Center of curvaturePb Base pitch equivalent spur gear, m

r Mid-plane phch radius, mR Mid-cone pil.ch radius, m.Re Effective pitch radius, m

R' e Loaded effective pitch radiusT Torque, N-m

U Pitch point tangential motion, mW Forcecomponent, N

X Bearing radial deflection, mY Pitch point deflection, mZ Gear center deflection, m

~ Orthogonal coordinate framer Cone angle, rad

'9 Pitch point slope, rad:p Radius of eurvanne, m

.:E Shaft angle. rad'$0 Normal pressure angle, rad

$:'n Loaded normal pressure angle, rad\II Spiral angle, rad

S~b5criptsa Axial first bearing

b Second bearingbi, i '"I, 2, 3 motion from bearing in first coordinate frame directionei, i == I, 2, 3 motion, from shaft in firs] coordinate frame direction

di, i. = 1, 2, 3 first coordinate frame directionei, i= I" 2, 3 second coordinate frame directionn, I = I, 2, 3, third coordinate frame direction

gi" I..I,. 2, 3 fourth coordinate frame directiong Gear (as last subscript)

p Pinion (as last subscript)r Radial

t Tangential

2.4 GEAR TECHNQ~QGY

Abstra.ct: An analytical method is presentedto predict the shifts of the contact ellipses onspiral bevel gear teeth under load. The contactellipse shift is the motion of the tooth contactposition from the ideal pitch point to its locationunder load. The shifts are due to the elasticmotions ofthe gear and pinion supporting shaftsand bearings. The calculations include the elas-tic deflections of the gear shafts and the deflec-tions of the four shaft bearings. The methodassumes that the surface curvature of each toothis constant near the unloaded pitch point. Resultsfrom these calculations will help designers re-ducetransmission weight without seriously re-dueing transmissionperformance ..

IntroductionSpiral bevel gears are important elements for

transmitting power. In designing spiral bevelgeartransmissions, the designer meets a tradeoffbetween a transmission's weight and its lifeand reliabihty, By removing weight from. trans-mission components. one increases the overallflexibility of the transmission. This flexibilityaffects the deflections of the loaded compo-nents in the transmission ..

The design of an efficient spiral bevel gearbox includes the design of gear and support geom-etry, bearing and shaft sizes, ami material proper-ties. The gear tooth interaction is morecompJexthan that of spur or helical gears. The loadedregion of the gear mesh. shifts due to the deflec-tions of the gear and the pinion. A primary causeof these motions is the flexibility of the gearsupport shaft and bearings ..Although the teethalso deflect, tooth beam deflections are small incomparison to lootll gear support deflections, If

the shift of the loaded region is large, the life ofthe gear mesh may reduce significantly.

The classic work of Wildhaberl-2 de cribesthe kinematic operation and the generation ofspiral bevel gears. More recently, Baxter andColeman" expanded on this fundamental. theory.They described a tooth. contacranaly i:sprogramwhichanalyzes the kinematic action of twospiral bevel gears in mesh. The program. consid-ers effects of tooth manofacnmng parameterson the gear mesh kinematics.

Litvin and CoyS and Litvin. Rahamn andGoldrich'' also presented the theory of spiralbevel gear generation and design. These worksdescribe kinematicerrors induced in the trans-mission by errors Ln gear manufacturing andassembly. The articles also suggest tooth profilemodifications to improve kinematic transmis-sion. They give direct relation bip for the gen-erated curvature and direction on the terms ofthe principal curvatures and direction on thebevel. gear teetlt. surface . The e relation hipsare in terms of the principal curvatures anddirections ofthe tool generating' urfaces.

Coleman 7 described the experimental mea-surement of existing bevel and hypoid geardeflections under load. The article cites theimportance of these deflections for the capacityand noise level of the gear set. The test resultsalso suggest gear mounting and tooth manufac-turing changes which can improve the capacityand life of the gear set.

Taha, Ettles. and Ma.cP.herson8 presentedthe in~.eraction of the structural rigidity andperformance from a helicopter tail rotor gearbox. They u ed a finite element model for mehousing. The article pre eats effects of deflec-tions on lbearing roller Ioad distribution. bearingfatigue life, and the gear motions at the un loadedpitch point Tbeirworkdemoastretes the impor-tance of rigidity to minimize deflections andmaximize bearing tife in a tran mission.

Winter and Paul9 presented work on theinfluence of spiral. bevel gear deflection ontooth root stresses.

This article pre ents an analytical. method topredict the shift of the loaded region on thetooth. The method treats the sbjft as a result ofthe elastic deflectiens ofthe gear support shaft-ing and bearings. The analysis is sequential.

The first stage define. the gear geometry andloading. The econd stage determinesthe elastic

!Fig. l - Spiral, bevel. :gear mesh Dnd support bearmg :geometry,deflections of the bearings and shafts and theslopesof the shafts at the gears. The third stagefinds me motions of me gear teethcau ed byeach elastic deflection. The total deflections ofthe gear teeth are the algebraic sum of thesemotions. The fourth stage determines the geom-etry and curvatures of the gear and pinion teeth.These directions and curvatures combine withthe separate motions ofthe gear teeth to producethe contact shift. The contact shift is the motionof the contact ellipses on the two tooth surfacesunder load from the unloaded pitch point. Theanalysis assumesthat the.curvatures are constantover the affected surfaces of the teeth.

A gear box model similar to 'the main rotorbevel gear reduction of the U.S. Army OH-58light duty helicopter serve as an example for[be method. The gear box includes a singlespiral bevel gear drive with a pinion input andthe supporting shafts and bearings. The analy-sis includes a paramet:rk study. The articlepresents radial and tangential shifts of the con-tact ellipses on the pinion and gear teeth asfunctions of shaft stiffness.

Geometry and LoadingFig. 1 shows the geometry of a spiral bevel

gear mesh. The mid-cone distance, Do' isthedistance from the apex of the bevel cones to themid-place pitch point. This distance is along thepitch line .of the two pitch cones. his equall to theoater-cone di stance, A0' rninus one-half the toothface width, f. The mid-cone distance and the'tooth face width describe the basic gear blank

!M!icha.el Savageis Professor of Mechani-cal Engineering at theUniversity of Airon, Ak-ren.OH.

p, C.,AI~idisis a Senior Design Engi-neer at Clark MalerialHandling Company, Lex-ington, KY.

0" G. Lewi.ckiisa Research Bngineerandat NASA Lewis ResearchCenter, Cleve/tmd, OH.

Dr. J. JI. Coyis rhe Chief of the Me·chanica! Syslems BranchOf NASA Lewis ResearchCenter, Cleveland, OH.

B'r. F. L ILitvinis on lhe facullY of IheMechanical EnginetringDept. at the Universlty o]Illinois at Chicago.sizes. The cone pitch angle. Tg and! rp' forme

NOVEMBER/DECEMBER ·"92 25

gear and pinion also contribute to the gear sizes.The shaft angle, I, is the sum of the cone pitchangles. The shaft angle defines the relative orien-tation of the gear and pinion shafts. The pitchradius of the gear in its mid-line, 'g is:

r = D *si11' rg . 0 - gThe cone pitch angle, rg, ill terms of the

numbers of teeth on the gear and pinion, Ng andNp' and the shaft angle, I, .IS:

r sin, Ltall = ---~...;...::=------ - - g (N,fNg) + cos I

Simihrr equations define the pitch radius, rp' andcone pitch angle, Fp' of the pinion with thesubscripts p and g interch!Ulged.

Fig. 2 shows the spiral angle, .fII, and norma]pressure angle, 4>,,. The spiral angle, 11', Is theangle of inclination of the teeth to the pitch ray.It is in the place tangent to the two pitch cones.The gear of Fig. 2 has a right hand spiral Thenormal pressure angle, tPn, is the angle between anormal to the tooth. and the tangent to the pitchcone surface sn the tooth normal plane.

Fig. 2 includes four orthogonal right handedcoordinate frames. All four coordinate framesare atthe mid-plane pitch po intof the spiral beveltooth. The first of these. {3di' has coordinates inthe tangential, axial. and radial directions ofthegear body. The second coordinate frame, 13ei, isthe first coordinate frame rotated through thecone angle, rg• in a negative direction about thef3~11 vector. The unit vectors of thi frame are intile tangential, pitch ray,and mid-cone radialdirections, respectively. The third coordinate

CA) ConeTangentPlane Pflpg.\

J3e3J3f3

J3g3(B) Tooth Normal Plane

-..p...~.,>!1-- J3e2

131'2J3g2

(C) Axial Section Plane

Fig. 2 - Spiral bevel gear tooth coordinate frames.

,26 G EAR TECHNOLOGY

(1)

frame, 13ji' is the second coordinate frame rotatedthrough the spiral angle, V'. in a positive direc-lion about the Pel vector fora right handed spiralangle. The unit vectors of this frame are tangentto the pitch cone in the tooth normal place,tangent to the tooth in the cone tangent plane,and in the mid-cone radial direction respec-tively. The fourth coordinate frame. 13gi, is thethird coordinate frame rotated through the nor-mal. pressure angle, ifill' in a negative directionabout the .f3j2 vector. The unit vectors of thisframe are in the tooth normal direction, tangentto the tooth in the cone tangent plane. and tan-gent to the tooth in the tooth normal planerespecti vely.

Fig. 2 shows the mid-plane pitch radius oftbe gear, rg, in the axial section plane. The mid-cone radiu of the gear, Rg, is in the ame view.

This equivalent spur pitch radius is:

(2)

rgRg = ----!!....--

cos Tg

The gear assembly includes the support bear-ings and their locations. The position of thebearings relative to the gear or pinion describesthe support system geometry. The two mostcommon bearing configurations are straddle andoverhung mountings. Fig. I shows both. Thepinion has a straddle mounting, while the gearhas an overhung mounting. III both cases, A isthe distance from the gear mid-plane to thebearing close t to the pitch cone apex . .B is thedistance from the gear mid-plane to the bearingfurthest from the apex. Distance A is positive foran overhung mounting and negati ve for a straddlemounting. B is always positive ..

The normal force acts on the tooth at the mid-

(3)

plane pitch point. The method assumes that theforce is a concentrated force. The force has threeorlho gonal components in the directions of the f3'dicoordinate frame relative to the tooth. The tan-gential component, WI' producesthe torque on I.hegear. Itacts in the f3d I direction. The axial compo-nent. Wa' acts in thef3d2 direction. And the radialcomponent. WI" acts in the f3d3 direction.

Fig. 3 shows these forces for both the gearand thepinion with their respective coordinateframes. For the gear, the components are:

(4)

Wa '" J wll [ tan ¢n * sin. rg + sin. IjI * cos r:~J(5)cos'll

Wr = IwJ [tan. 41/1 * cos ~ - sin VI * sin r,J (6)cos 'I'

where Tg is the torque on the gear, Replacing thesub cript g with the subscript pin Equations 4through 6 gives the equations for the pinionforce components. InEquations 5 and 6. the signof the last term depends on the direction ofrotation, the hand of the spiral, and whether thegear is driving or being driven. These equationsare valid for a right-handed piral driving gearrotating counterclockwise about the f3d2 direc-tion. Each change in the spiral hand. powerdirection. or rotation direction reverses the signs.

The sign of the tangential load. W"~ alsodepends on the directions of the rotation andpower flow. The tangential cornponent is posi-tive for a driving gear or pinion which is rottingcoanterclockwi e. It i also positive for a drivengear of pinion which is rotating clockwise. Thetangential load is negative otherwise, However,Equations 5 and 6 use the absolute value of WI'

Com.pon.ent DeflectionsUnder load. the motion of the ideal pitch

point of a spiral bevel gear is mainly thesuperposition of three elastic deflections. Thesedeflections result from shaft deflections at thegear center .ahaft slope at the gear centers. andthe deflections of the support bearings,

Table I lists the results of a strength of mate-rials shaft analysis for the deflections, Ycj' andslopes (rctations), ¢cj' of the gear at the pitchpoint in and about thef3di directions, In theanalysis, both the axial. and radial forces contrib-ute to both rhe radial deflection and the tangen-tial rotation. The axial deflection is due to thetangential rotation and the radial location of thepitch point. In this instance. the straddle andoverhung ca es require different formulae due tothe different ela tic configurations of the twocases ..As before, interchanging gear and pinionsubscripts yields valid equations for the pinion.

The bearings which upport the shaft alsodeflect, Each bearing has a nonlinear stiffnessand its own contact angle. HarrislO andHoughton" present methods for determiningthe rolling element load hating and resultingdeflection, The method of this article COID-bine the radial and axial. loads on each bear-

Pinion II

Apex

Fig. 3 • Gear and pinion tooth force eempeaents,

DEFLECTIONSStraddle Overhung

WtA2B

3EI

W A2B2Yc];::: __ l _

-W. r A (2B + A)

6El

W A2B+ '

3EI

3EI (B-A)

r*ecl

Y _ -_W_o_r_A_B_(B_+_,_A_) ,L W, AZ

132

28 GEAR T:ECHNOLOQIY

lug to find the bearing deflections. The pro-gram then resolves each bearing deflectioninto its radial and axial components in the(3dicoordinate directions.

The deflections at the bearings are X . and(II

Xbi. Here the subscript a identifies the bear-ing located the distance A from the gear. Andthe subscript b identifies the bearing locatedthe distance B away. The last subscript, i,denotes the deflection direction in thef3dicoordinate frame ..

For i= 2, both bearings have the same axialdeflection, Xz' Table IIlists the deflections, Ybi,and the rotations, f3di of the gear pitch pointcaused by these bearing deflections. The mo-tions are due to the rigid body motion of tlIe gearand support shaft in the bearings.

The total motions of the gear pitch point inand about the I(}di coordinate directions, ne-glecting .any elastic motion of the transmissionhousing are:

Similar equations describe the motion of thepinion pitch point. The motions of the gear andpinion pitch points combine in the analysis forthe shift ofthe contact ellipse under load.

Contact AnalysisThe principal. curvatures of the teeth are

needed in addition to the teeth deflections in thecontact ellipse shift model. Unfortunately, nodirect equations exist for the spiral bevel toothsurface due to the complexity of the surfacegeometryS.6. However, an indirect approach candetermine the principal curvatures and direc-tions of curvature. Litvin5-6 has developedequa-tions to determine these principal curvaturesandtheir directions at the pitch point.

Toe

Gear or Pinion Tooth_/

Fig. 4 - Tooth contaet point motion.

The analysis assumes that direct conjugaterelationships exist between the gear cuttingtool surface curvatures and motion and thesought quantities. These are the principal cur-vaturesand their directions on the gear toothsurface. The analysis uses the gear and pilliongeometry along wi.th the cutter machine set-tings for both the gear and pinion. The cuttermachine's manufacturer provides the gear cut-ter settings. With this method, one can deter-mine the curvatures of the gear tooth. surfacewithout equations for the surface.

The separate pitch point motioas of the twogear teeth effect the shift of the contact ellipseunder load. In addition, the motions of the cen-ters of transverse curvature of the teeth surfaceseffect the tangential contact ellipse shift. Themotions of the mid-plane base circles of the teethalso effect the radial contact ellipse shift. Inaddition, the standard Hertzian contact stressformulae12 yield the s:ize and orientation of thecontact ellipse on the gears. All these caleula-

(7) tionsassume that the principal. curvatures re-main constant over the contacted portions of the

(8) spiral bevel gear teeth.Relative Motions

Tnecontact shift motion occurs in. the planetangent to the two teeth. Relative motions inlhedirection of the common normal to the teethproduce kinematic errors in the motion trans-mission. Kinematic errorS is a forward or back-ward rotation of the output gear from its idealposition relative to tile input gear. This motiondoes not produce a shift of the contact ellipse.Only the radial. shift on the teeth in the tooth.normal plane and the tangential relative motionin the cone tangent plane are of interest, Theradial shift of the teeth produces a radial shift inthe contact ellipse locations on both teeth. Thetangential motion produces a tangential shift inthe contact ellipse locations as shown in Fig. 4.

In the motion analysis, the small pitch pointdeflections and rotations are vectors. One canobtain the vector components in arty of the fourcoordi.nate frames by direct matrix rotations.

Radial Shift - The radial shift ofthe two teethappears in the mid-cone plane and in the toothnormal plane. Fig. 5 shows two equivalent spurgears in the tooth normal plane before and afterapplication of the separating load, Equivalentspur gears have the tooth geometry of the sp.im-albevel tooth ill its mid-cone plane. The involute

action of these spur gears is the same as that ofspiral. bevels in the mid-cone plane.I-3 The ra-dial shift of the two gears is due to an increase ofthe center distance in this rotated mid-coneplane. The separating motion of each gear centeri the pitch point motion in the Pel directionminu a small relative motion. The relative ra-dial mot:ion ofthe gear center j .towardjhe pitchpoint It is dueto Ute gear rotation about theoriginal tangential direction Pel'

The displ acement of the pitch point in the l1e3direction is:

.y y *. r-· . Y * ....T.. 3· =. -'2 sm .. + d·' COSt .8

point. The relati ve motion is due to the gear bodyrotation perpend icular to this plane about the f3'g3direction. For the gear, this deflection is:

Z1l2. II = Y g2, 8 + P8 * sin. 8 83.8where

YgZ, g"" -Y,lJ, II *' sifltp'+ Ytl2.g *' cos rg cos "¥ (16)-Yd3.8 * sin rg cos lJf

RadialShift(Dr'ag) .041

Fig. 7 - Tangential motion of spiral bevel pitchpoint.

,020---Gear-Pinion

,0015

Radial

Shifl .0010(Dr'Ph)

00 1.00.6 .8Stiffnes __Ratio (EI/EIoO_J_

Fig ..8,- Tuoth cootactpositlon shift in radial direction.

00I TangentialI Shift

I-.00005

,I (2Dt/f)I -

- ..000010.

-.0001.5

.0001

, Tangential

Shifl -.00'1(DtlPb) ____ Gear

-- Pinion-.OOQ

30 GEAR TECHNOLOGYFig. 9 - Tooth contact po itlon shin in tangential direction.

and:

(J= (J' - * cos 1jI sill n'" cos Tg cos l/J,,)

The motion on the tooth surface from theunloaded pitch point to the loaded pitch point. inthis plane is Ug2' This is the deflection necessaryto keep the centers of curvature on a commontooth normal

(18)

This locates the new loaded pitch point rela-tive to the unloaded pitch point. The relativeshift of the gear pitch point in the tangentialdirection on the tooth surface is D,g' This is thedifference between the original pitch point'smotion and the pitch point shift, Ugi

(19)

A similar analysis determines the tangentialpitch point shift on the pinion, DIP'

Tooth Con.tact ShiftThe radial and tangentiaJ components of

tooth contact shift are valuable measures ofthe rigidity of a bevel gear me h. Knowledgeof the shift components can help a designerevaluate important properties of the gear meshand support bearings. An interactive inputFortran 77 computer program 13 calculatesthese tooth contact shifts for both the gear andpinion. The program, SLIDE. FOR, can run ona personal computer.

fig. 4 shows the shift of the nominal. pitchpoint contact ellipse on a gear tooth. The princi-pal radii of curvature of the teeth and the normaltooth load determine the size and orientation ofthe contact ellipse. One C3_n compare the tangen-tial. shift to the difference between the half toothwidth and the tangential radius of the ellipse. Thiscompari on demonstrates whether edge loadingcan occur in the design. By subtracting the radialhifts from the teeth addenda, one aI 0can evalu-

ate the reduction in the ideal contact ratio ..This model. can evaluate performance

tradeoffs where transmission weight and trans-mission life are competing objectives. Figs. 8and 9' are dimensionless plots of contact shift asfunction of support shaft stiffness for a singlestage transmission. The plots vary both the gear

and pinion shaft stiffnes es by the same percent-age. The parametric study reduce the. ti ffnessesfrom the design values to show the effect of shafttiffne on performance.

Fig. 8 is a plot of the radial contact hift onboth the gear and the pinion. Fig. 9 is a plot ofthe tangential contact shift for the two gears ..Bethplot show the contact shift a. a. ratio 'tothe mid-cone ba e pitch of the teeth. Thisgives a dimensionless comparison of the shiftmagnitudes. A second vertical axis on theradial plot, haws the shift as a percent .of thetooth addendum. This compares the shift ofthe contact point to the actual tooth size. In thetangential shift plot, the second vertical axisshows the tangential shift as a percent of thehalf tooth width. The horizcntal axis of bothplot i the ratio of the support shaft stiffnes ,EI, to the full support shaft. stiffness, EI,]"

The figures demonstrate a definite hyper-belie relatiensbip between the riffne es andthe re ulti ng pitch poi nt contact shift, A val uableconclusion i that one may reduce the stiffnessesin the example to about 20% of the nominaldesign values. At thi point. there is a significantincrease in the contact position shift. With thisinformation available at the design stage of atransmi sian's development, important weightsaving may be possible. The weight savingswin not sacri flee the transmi sion' slife or poweroverload performance seriously.

Summary and ConclusionThi artidepre ents a method to predict the

shift ofthe loaded region on the gear teeth in aspiral bevel reduction. This shift is a result of theeta tic deflections of the gear support shaftingand bearings under load. The reduction is asingle spiral bevel gear drive with a pinion inputand upportmg shafts and bearings. The a-Slimed deflection were caused by the shaft de-flection. , the shaft slopes, and the bearing de-flection . The analysis assumed that the curva-tures of fhe teeth near the pitch point. wereconstant, It determined tile curvature using theenvelope of cutting tool positions.

The analysis was equential. The first [agedefined the gear geometry and loading. The ec-ond stage determined the elastic deflectams oflhebearings and shaft and the slopes of the hafts atthe gear .. The third tage determined the motionsof'the gear teeth caused by each elastic deflection,Superposition of the e motion produced the total

deflections of the gear teeth. The fourth. tageanaJyzedthe interaction between the tooth geom-etry and motion to predict the tooth contact el-lipse motion underload ..A Fortran 77 computerprogram determined 'Ihe radial and tangentialtooth contact shifts of the gear and pinion.

An example representative of the main rotorbevel gear reduction found in the U.S. ArmyOH-58 helicopter was modelled. A parametricstudy illustrated the determination of the radialand tangential tooth contact ell ipse shifts. Graphsshowed the variation in contact hin with reduc-tion in upport shaft stiffness. An importantconclusion was that, for the example, one couldreduce the shaft stiffnesses to about 20% of thepresent stiffne ses, At this level, significantchanges in the shifts of the tooth contact p051.-lions occurred. Significant weighl reductionsm.ay bepossible without .. eriously affecting thegear action or the gear life .•

cknowledgem nts: .originally published as NASATechnical Memorandum 10/438 and ill the Proceed-ings of tile /iSM E Fifth Imernatiolla/ Power Trans-mission and Gearing Conference 1989.References:1. WILDHABER. E. "Basic Relation.ships of BevelGears." American Machinist. 1945. Vol. 89, No.20.,2.1.22; pp, 99-102. 118-21, 122-25.2. WILDHABER, E. "Basic Relationships of HypoidGears." American MacirifJi.fl.1946, Vol. 90. No. 4,5,12. 13, 15.16, 17; pp. 108·11, 131-4, 132-5.110-4,150-2,106-10,104-6,122·8.3. BAXTER, M.L,. JR. "Basic Geometry and ToothContact of Hypoid Gear." Industrial Mathemat-ics. Vol. II, PI .. 2. pp.I!9-42.4., COLEMAN. W. "Guide 10 Bevel Gears." ProductEngineering, 1963, pp, 87-95.5. UTVIN, F.L. & COY. J.J. "Spiral Bevel GearGeometry and Gear Train Precision." ASA CP 2210,1981. pp. 335-344.6. LITVIN, R.L .. RAHMAN, P. & GOLDRI.CH. R.N."Mathematical Models for the Synthesis and Optimi.-zation 01 Spiral Bevel Gear Tooth Surface ." 1982.NASA CR 3553.7, COLEMAN. W, "Analysis of Mounting Deflectionson Bevel and.H ypoid Gear .. " 1975. SAE Paper 750 152.8. TAHA, M.M.A., ETTLES,. C.M., &MACPH ERSON, P. B. "The R igi di ty and Performanc eof a Helicopter Gearbox with a Cantilevered Hou ingand Two Taper Roller Bearings." Journal of Me-chanica/Design. 1978. Vol!. 100, No.6. pp, 696-702.9. WINTER. H. & PAUL, M. "Influence of RelativeDisplacements Between Pinion and Gear on ToothRoot Sire ses of Spiral Bevel Gears." Journal ofMecha<"ism. Transmission. and Automalion in Design, 1985.Vol. 107. No.3, pp. 43-48,10. HARRIS, T.A. Rolling Bearing AIIQlysis. 2nd. ed.John. Wiley & 00s,1984.II. HOUGHTO •P.S. Ball and Roller Bearings. Ap-plied Science Publishers. 1976..12. ROARK. R.I. & YOUNG. W.C. Formulas forStress and Strain, 5th ed. McGraw-Hili. 19'75.13. ALTIDIS, P.' . & SAVAGE, M, "Flexibility Ef-fects on Tooth Contact Locntion in Spiral Bevel GearTransmissions." 1987, NASA CR-4055.

32'~~, G~AR T.CHNOLOOY

Classification of Types ofGear Tooth Wear - Part I.

Louis IF,a:ur,eCM'D Transmissiions,

Camlbrai. France'

IntroductionThe phenomena of deterioration of surfaces

are generally very complexend depend!on nu-merous conditions which include the operatingconditions, the type of load applied. the relativespeeds of surfaces in contact, the temperature,Iubrication, surface hardnessand roughness,and the compatibility and nature of materials.

Wear isa general term covering the localphenomena describing the removal of somematerial and occurring when two surfaces slideonto one another, This term also applies to theremoval of material resulting from the pres-ence of impurities in the lubricant. Other typesof gear failures, such as surface fatigue, corro-sion, plastic flow, and breakage are not cov-ered by this article.

Wear may be divided into two distinct clas-sifications: Qualitative, which is based on theaction modes of different wear phenomena ongear teeth and characterized by surface appear-ance; and quantitative, which takes into ac-count. the intensity of the wear phenomena.

The qualitative wear phenomena we willdiscuss include abrasive wear with two bodies,streaks and scoring, polishing, hot and coldscuffing, abrasive wear with three bodies,scratches or grooves, and interference wear. Inthe quantitative analysis we win define normalwear, moderate wear, and excessive wear. Asynthesis of these two classifications has beenmade under a chart found in Part H.

eurrent Wear (c:aued Two Body Abrasion)Current wear is revealed very early in the

life of the gear or gear train and is evidenced bythe removal of micro-particles of metal onthegear teeth surfaces. This phenomenon is causedby contact and metal-to-metal sliding, which

occurs through the oil film. The distinctivemachining marks of the cutting procedureor finish (for example, the facets resultingfrom hobbing, the streaks left by a gearcutter or by ,3 rack cutter. the surface patternfrom certain grinding operations) are di-m:in:ishedor erased by wear.

This wear brings about a progressive reduc-tion of gear tooth thickness, along with a moreor less marked distortion of the profile in theheavy sliding zones, but without noticeabledegradation of foe surface roughness ..

As indicated in Fig, Lthe wear is almost nil

'Wear on Tooth Tipl

_. iPitchCircl:eTooth Root Wear

Fig. 1 . Distinctive aspect at distortion of geartooth profile under wear action

in the pitch zone where the liding peed are[ow or nil, and becomes more and more pro-flounced a we move away from thi zone. Thiszone i maximum at the tipcircle and at theactl ve dedendum circle, wherethe sliding speedsare the highest

The wear zone i generally gray colored andlightly dun, with ometimes lu trou areas and

the presence of scoring. (See Fig. 2). Note thepresence of a dull zone below the pitch, wherethe wear i greater, and of a lu trnu area oneach 'ide of the pitch and slightly higher. Alsonote the transver e coring Ughtl.y markednear the tooth tipand the machine streaks tillvisible near the pitch.

The appearance of thi wear, as well as itsdeveloping peed, variesgreatly according tothe pres ure level between the contact surface ,the hardne of the material . the roughnes ofsurface. and the thickness of lubricant film. Forexample, in 'Ihe ca e of lightly loaded gearsoperating with an oil of relatively high vi co -ity at medium peeds, we will have an oil filmsufficiently thick to avoid metal-to-metal con-tact. Thi will not generate wear except duringstarting or stopping. The original machiningtrace will still be intact en the gear teeth afterlung periods of operation ..

In practice it is not always possible to havea continuous lubricant film between the gearteeth according to the load transmitted. Thencontact occurs between the top 'of the asperitiesmade during machining, and there i a tendencyto polish or score the surface in 'contact

In the ca e of urface-hardened gears ..be-cause of the high hardness of flank in con-tact, we eacounter only slight wear which isoften difficult to see.

Scoring - Str,eaksThis type of wear appears in the form of

fine groove or lines in rhe liding direction.The e streak or core are formed progres-ively in tile zone characterized by a highIiding peed. at the tooth tip and near the root

of the gear teeth. (Fi.g. 3)Tiley are etten developed 011 pinion teeth

and wheel teeth that are in contact at the verybeginning of meshing of mating profiles. Thebottom of the striae are smooth compared tothose faund in traces of scuffing.

Duses. The pre ence of these scores orstreaks reveals the existence of relatively

high pressure locally affectingthe gear teethflank . Under high load action. the asperitiecaused by the roughnes of the mating flank •a well a . foreign particles of small dimen-ions that imprint intothe gear teeth flank •

along with the. lippage effect, cause cavitieswhich appear as streak. .

The formation of coring can be con id-ered as a preliminary stage preceding scuff-ing. These treak generally lead to more e-vere local wear of gear teeth in the zonewhere there is a higher pressure. Frequentlywe note in time a. stabilization of these treakwhen the wear level of the cored urface habeen ufficient '10 generate a beuer distribu-tion of the load. which a a con equenee di-mini hes the maximum pre ure on the gearteeth .. In this case only the black. color base ofstreak, or coring will remain. This tabiliza-tionphenomenon can be accelerated by in-creasing the oil vi cosity and by improvingfiltration. This type of wear is oftenencoun-teredin worm gear " (Fig. 4)

Adheslve WearAdhesive wear appears on 'two surfaces

Iiding aero one another when tile pres urebetween the asperities in contact is sufficienttocau e localized pia tic deformations. rni-era-welding, or local, adhesion . When thereis generation or plastic deformation, there isenergy ab orptlon which leads to 0 erhearingdue to friction.

is 111(' Technical Mmlallt'r01 Ml) Transmissions 0/CamlJrlil. France. HI! isall active member of ISOamJAFNOR (!.ssac/aliallFrancois de Normalis-ation {Jlld the Unltm dtNormulisation lit' laM canique. He Jill also/(Jljglll gear Ihf'Uf.l' oud

inten e, and when the gl'C1r failure anoiysis.NOVEMBER/DECEMII R 1882 33

The mildest form of adhesive wear i theformarion of t'polishing' on the active flanks ofthe gear teeth, When the load condition andfriction become more

temperature at contact level increases. we maywitness the appearance of localized metal ad-hesions 0[1 the teeth. hot cuffing.or cold cuff-ing (for low speeds and heavy loads).

Polishlng. Polishing is a type of very slowprogressive wear in which the a peri tie of themating flanks are distorted and laminated andthen appear on the gear teeth a very moothand polished surfaces, which take on a mirroraspect. Such wear conditions are caused bymetal-to-metal contact during operation. Gen-erally the polishing occurs in applications atlow speeds «20 mlsec) when the lubricant oilfilm ill elastohydrodynarnic yield rate is notsufficiently thick and is near the limit of thelubricant performance ..Generally, this type of

34 GeAR TECHNOLOGY

wear does not cause large variations in theshape and in the dimensioru of gea:r teeth.However, when examined under a microscope,the structure just below the contact surfacereveals the presence of plastic flows in themateria] under slippage. Sometimes very local-ized overheating traces near the surface are alsoencountered. The polishing encourages estab-lishing gear teeth contact pattern in serviceand allows obtaining a good conformity ofsurfaces in contact.

This type of wear is not "damage" and canbe tolerated in service unless the initial mate-rial specifications forbid it. If the load increases,or if the lubrication conditions become insuffi-cient. this type of gear tooth wear can developbecause the temperature between surfaces incontact will increase and render them superfi-ciaIJy less hard and more sensitive to localizedmicro-welding formations ..The appearance ofadhesion or scuffing traces on the gear teethflanks can then be seen. To prevent such traces,we can, after the appearance of polishing. ill-crease slightly the oil viscosity to obtain innormal working conditions a thicker hydrody-namic film. We can also, when possible. reduceslightly the load to be