january/february 1989 gear technology · it is the spirit of leonardo we at gear technology wish to...

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(312} 437-6604

The Advanced Technologyof

Leonsrdo ds Vinci1452·1519

COVERThe only known documented likeness of

Leonardo. this self-portrait was done duringhis last}ears in ltalJ~He was around (j()~old at the time thed!:awing was done. Thedimensions of the original are 13 11112 }I

8 5112,and the medium is red chaUt. A t thebottom of the clrawing is an inscripfion addedby a later hand. "Leorurdo cia Vinci. portraitofhimseJ( as an old man ." The oripjnal IIDrkis in the Biblioteca Real. Thrin. italy.

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CONTENTS PAGE NO.

FEATURES

FACTORS INFLUENClNG FRACTURE TOUGHNESS OFHIGH· CARBON MARTENSrTlC STEELS

v. K. Sharma. Navistar lntemational, Ft. Wayne, ING. H. Walter, J. 1. Case, Hinsdale, ILD. H. Breen, ASME Gear Research Institute, Naperville, IL

7

.FRLET GEOMETRY OF GROUND, GEA.R TEE1:HG. Castellani, Studio Castellani, Modena, ItalyV. Zanotti, Rossi Motoriduttori S.P.A., Modena, Italy

19

-DEPARTMENTS

EDITORIAL 5

TECHNICAL CALENDAR

BACK TO BASICS ••.SPUR GEAR FUNDAMENTALS

Uffe Hindhede, Black Hawk College, Moline, IL36

CLASSIFIED 46

January/February, 1989 Vol. 6, 0.]

GEAR TE"CHNOLQG.v. The J""rn ..J of Cur I\hnufad u ",lnll (lSSN 0743-68581 ls pUblished bl!nonthly by Randall P.ublJ.tu11llCo.• Inc., 1425 Lum .... enue, P. O. Box 1426. Elk Grove Village. IL 60007. GEAR TECIiNOLOCY. Th .J""mal of Gear Manufac·'tunn!!. Subscripuon rate. are: $40.00 In the UlIlted States, 550.00 in Canada, $55.00 in all other (oreigneeuntries, Second-Class

pO~~~a~~rneAn~i~'t~~~~g.!\~nUUdf~'t'H}lo"t'JGI 't~~ejoumal of Gear Manufacturing. 1<125 Lun! Avenue, P. O.Bo~ 1426. Elk Grove Village. IL 60007.

e on tents COl'yri!lhted by RANDALL PUBL.ISHING CO" INC. 1969 .... rticles al'l"'Illfi"gln CEAR TECHNDJ:.OCY may not bereproduced in whole or in part without the: express permisslen of the publisher or the author.

MAN- SCRIPTS: We are requesting technical pape •• with an educational emphasis for ;>-"JIOn havil'\l! ..n~ln~ ro do ,"11ll th.desogn, rrarrufectare. tc;S1illllor processing "r.gears. Sub.Ie<;l5 ..,ugh! are ..,Iutions to ."""itk I'",ble ...... ""planat>!",. of...."., 'tedtooIOl!\'.l\!cl>nlques, desIgns. processes .•.and alternative trulnufactu''-I\~ methods. These can r;onll from the "Howto ... " or lie'" ~uttil\(BACK TO BASICS. to the most advanced Il!l:hJlOIOl!\', AIlnranwcnpl:s eubrraued wiU be carel'UJly co ns rdered, H"","" r. the Publisher"MU""" no respol'ISihihb'{or the safety or return "f manuscripts. Manwcnpts. must be aaompal'lled bf ,,<elf-add~. II-s tampedenvelope. and b. sent to GEAR TECHNOLOCY. The Journal of Cear Manufacturin!l. P.O. Box 1426. I:;lk ernv e. Il, 60009.(312) 437-6604.

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EDITORIAl.,LE'O.NARDO, THE EN'GINEER

.'Mechanical science is most noble and useful above allothers. for by means of It all animated txaes In nxxoo performmer operations ."

Leonardo de Vinci.

These lines. Interestingly enough, are from the notebooks ofan artist whose Images are part of the basic Iconography ofWestem culture. Even people who have never set foot In amuseum and wouldn't know a painting ,by Correglo from asculpture by Calder. recooruze the Mona Usa. But Leonardo daVinCI was much more than an arnsr. He was also a man ofscience who worked In anatomy. botany. cartography. geology.matnernaacs. aeronautics. Optics. mechanics. astronomy.hydraulics. SOniCS. civil englneenng. weaponry and CIty planning.There was little in nature that did not Interest Leonardo enoughto at least make a sketch of It. Much of It became a matter oflifelong study The breadth of hiSmterests. knowledge. foreslghr.Innovation and Imagination ISdifficult to grasp.

ChOOSing leonardo da vmo, the protmyprcal RenaissanceMan. as a role model for a highly specialized technical magazinemay seem a little peculiar What can Leonardo. the man whoseemed to know almost everything and seemed to do it all very.very well. say to us in our highly specialized world of gear deSignand manufacture?

It is the Spirit of leonardo we at GEAR TECHNOLOGY wishto honor and to emulate - the spirit of excellence and the splntof cunosty.

Leonardo was never satisfied With the way things were. Henever ceased to questron. to expenment, to Improve rus ownskillsand deSigns. The 5.0C0<x1d pages of tus notebooks that stiliexst are full of sketches of an Infinite variety of natural andmechanical objects. He drew hundreds of sketches of variousplans for bndges. weapons. clocks and hydraulic systems; forgeared machines for lifting, moving. cutting and drilling: for or-nithopters. which he hoped would enable humans to fly. andfor parachutes. "horseless carnages" and other devices far aheadof rus time. Leonardo's VISIon frequently outstrrpped [he scienceof hiSday, and many of hiS "mvennons" were never pur Inro useat the time simply because there was no practical means topower them.

ThiS capaoty for dreaming. questioning. expenmentlng and[inkerrng untrl the opnrnurn deSign IS actueved lies at the root of[he soence of enqmeennq We cannot all expect to have thebreadth of knowledge and [alent that Leonardo had. but we cancertainly emulate rns attitude - that the natural world is full ofwonders that can be known. and, once known. turned mtomachines mat make life better. easier and safer for everyone

Researching and reading about Leonardo is both an In-teresting and a fulfillIng expenence One ISconstantly astonishedby the scope of hiSskill; even rough sketches In hiSnotebooks arelittle works of art. His vision - of nature, soence and engineer-Ing. as well as art - ISawesome. Every staff member at GEARTECHNOLOGY that has worked on the Leonardo covers hasbeen enrrched by the expenence

The Leonardo covers have been one of our most. consistentlypopular features among both our readers and our staff You may

have noticed, however. that on several recent occasons we didnot feature one of tus drawmqs on [he cover The reason for tnlSISsimple: We are runnIng outof Leona roo's sketches that featuregeanng. and we have little desrnpnve Information for the onesthat are left

Part of the problem ISthat Leonardo's sketches were neversystematically organized. Some of both the charm and the diffi-culty of the sketchbooks is mer soontaruery Leonardo wrotedown or sketched Ideas as they came to him. and he rarely wentback to edit or categorrze hISwork. It was a project he always in-tended to undertake . Then. after hISdeath, the 5,000 unor-ganized pages were separated. scattered. In some cases.destroyed Those portions of the manuscrrpts rernammq arefound in a number of museums and libraries throughout theworld.

The vanety of teonaroo's mterests also complicate searchIngfor approprrate sketches. They are apt to show up In books onalmost any su~ect. A hunt for Leonardo's sketches can cover anennre library. The place to start ISany good biography of the art-ist. but from there. the trail can lead almost anywhere

So we ISSueyou a challenge. Go to your lIbrary and .spend anafternoon absorbing some of the spint and Wisdom of LeonardoIfyou come across In your reading any of hiS sketches featurrnggears or geared mecnarusms that have not been featured on ourcovers. send copies to US.and we Will be pleased to share themwith the rest of the GEAR TECHNOLOGY audience. We Will runthese sketches as cover art and credit you and your company astheir source.

Eventually. of course. we Will completely run out of Leo-nardo's gear-related sketches. The chances of gettlng the artistto produce more are slim However. sketches or no. the splrr£ofLeonardo Will continue to inspire us here at GEARTECHNOLOGY. and. we hope. Will continue to Inspire you aswell - for mvennveness. intellectu 15k taking and simplecuriosity are a(w<lYs the r t of s ess for <lny enqmeer.

--;7"

January/February 1989 5

TECHNICAL CALENDAR

CALL FOR PAPERS. AGlV1A Fail TechnicalMeeting, ovember 8-9, 1989. Papers. on geardesign, analysis, manufacture and applicationsof gear drives and related products 1NiII beao-cepted, Deadline for abstracts is JANUARY5, 1989. Contact AGMA headquarters formore information.

MARCH. 19-22., 1989. FIfSt bltemationil Ap-plied Mechanical Systems Design ConfuJmce.Convention Genter, Nashville, TN. Presenta-lion of papers, tutorials and panel sessions onvarious design topics. For further information,contact: Dr. Cemil Bagci, TennesseeTechnological University, Cookeville, 1N38S05. (615) 372-3265.

APRll. 25-27; 19S9: ASr\4E Slit Annual PowerTransmission & Gearing Conference,Chicago, Il., Presentations on em rgingl"echnologies for gears. couplings and otherpower transmission devices, gear geometry.noise, manufacturing and other gear-relatedsubjects. For more information, contactDonald Borden~ P.O. Box. 502, Elm Gmve.W] 53122.

SEPTEMBFR 12--20. 1989. European MachineTool Show, .Hannover;. West 'Germany. Ex-hibits from J6 countries 1NiII show rutting andforming equipment, machine tools.CAD/CAM, robotics, etc. For more infor-mation, contact: Hannover Fairs, USA, Inc.,103 Carnegie Center, Princeton, NJ. 08540.(609) 981~l202.

NOVEMBER CXI, 1989 .. AGMAGear Expo'89, David lawrenoe Convention Center,Pittsburgh, PA, Exhibiticn of gear machintools, supplies. accessories and gear products.For more information, contact: Wendy reid!,AGlV1A. 1501) King Street, Suite 201. Alex-andria, VA 22314, (703) 684..(12ll .

.AGM.A T[CHNKAL EDUCATIONSfMlNARS. AGNlA is offering a newsmesof technicaleducalion seminars, each onefocusing on a different aspect of gearmanufacturing and taught by industryexperts.

]an. ll, Cincinnati, OH. "Specifyingand Con-trolling the Quality of Shot Peening."March 7/8, Rochester. NY. 'Source Inspec-tion of Loose Gears from the Customer'sStandpoint."May 2, Cincinnati, OH."Gea3' Math at thShop level for the Gear Shop Foreman."tune 6, AGMA Headquartl!'IS. "Specifying andVerifying Material Quality per AGlV1AMaterial Grades."For more information. contad: Bill Daniels.AGMA, 1SOCIKing St., Suite 201, Alaandrla,VA 222]4 .. (703) 684-0211.

Factors Influencing Fracture 'Toughness of'High-Carbon Martensitic SteelsV. K. Sharma, Navistar International Transportation Corp.,. Ft. Wayne, IN

G. H .. Walter, J. I. Case, Hinsdale, ItD. H. Breen, ASME-Gear Research Institute, Nape,rviUe, IL

Abstract:

Plane strain fracture toughness of twelve high-carbon steels hasbeen evaluated to study th influence of alloying elements. carboncontent and retained austenite, The steels were especially designedto simulate the carburized case microstructure of commonly usedautomotive type gear steels, Results show that a small variation incarbon can infl.uence the Krcsignificantiy. The beneficial eHect ofretained austenite depends both on its amount and distribution, Thealloy effect, particularly nickel, becomes significant only after thealloy content exceeds a minimum amount, Small amounts of boronalso appear beneficial.

IntroductionThe issue of toughness of materials used for machine

elements remains controversial, although much is understoodabout the subject. In the past few decades. our quantitativeunderstanding of the subject has increased significantly.Concerned by brittle fractures in World War 11liberty shipsand, more recently, by rocket motor case failures, engineershave developed a comprehensive understanding of the coo-

cepts and test methods of fracture toughness and its measure-ment. Yet many myths concerning the use of toughness con-cepts in design persist It is certainly an acceptable principlethat designs must be strong and tough; however, applicationof strength and toughness concepts in the real world mustbe done through compromise, especially when fatigue is oneof the design criteria.

Increased strength is usually accompanied by decreasedtoughness. In designs for which steel strengths below approx-imately 1700 MPa (40 Rcl are acceptable, material systemsare available which offer appreciable toughness, However,in carburized applications, such as gears which are case-hardened to obtain a high-hardness, high-carbon martensiticcase with a relatively low-hardness, low-carbon core.toughness becomes a specialized area of knowledge, Consider-able controversy surrounds the problem of defining andevaluating the toughness required for heavy duty automotivegears. (1f

The present study was undertaken to. obtain data useful

Jcnuary/february1989 7

Fig. 1 As-rolled microstructure in 0.99% C PS-15 and 5195 steels. Similargrain boundary carbide network surrounding pearlitic grains was d velopedin 8697, lH-50 and ERCN-l.

in designing and selecting carburizing steels. The effects ofalloying elements, carbon content and retained austenite onplane strain fracture toughness properti s of high-carbonmartensitic steels especially designed to simulate carburizedcase microstructure of several commonly used automotivesteels were examined.

ProcedureTwelve 60 kg, high-carbon steel ingots were poured from

air induction melted heats made with fine grain practice us-ing aluminum and silicon. Chemical composition of the steelsinvestigated is given in Table 1. All steels have a high car-bon content to simulate the surface of a carburized compo-nent. The steels were designed to study the effect of alloyingelements, case carbon 'content and retain d austenite. Tostudy th effect of carbon content on KIC Mn-Cr-Mo PS-15steel was poured at 0.99%, 0.86% and 0.72% carbon, andNi-Mo 4800 steel was produced at a carbon content of 0.99%and 0.70%. At equivalent carbon levels, SAE PS-15 steel isa cost-effective replacement steel for Mn-Cr-Ni-Mo 8600 seriessteel. Mn-Cr-Mo type steels have been used successfully inaxles and power train components for over 15 years.

Comparison of the fracture toughness of 0.99% CPS-ISsteel (residual nickel) with 8697 (intermediat nickel levels)

Table 1 - Chemical Composition

C MN Cr Ni Mo 51 AI S P- - - - - - - - -P5-15 0.99 1.09 0.54 0.03 0.16 0.3.2 0.05 0.02 0.02'P5-15 0.86 1.19 0.54 '0.03 0.16 0.30 0.04 0.02 0.02PS-15 0.72 1.11 0.51 '0.03 0.17 0.28 0.04 0.02 0.028697 0.97 0.83 0.52 0.60 0.22 0.31 0.04 0.02 0.01IH-50 0.97 1.37 0.05 0.03 0.01 0.68 0.05 0.02 '0.024895 0.95 0.73 0.06 3.44 0.26 0.32 0.04 0.02 0.014870 '0.70 0.76 0..05 3.50 0.25 0.31 0.04 0.02 0.01ERCH-l 1.00 1,57 1.25 0.03 0.01 0.70 0.05 0.02 0.02ER-8 0.95 1.32 0.95 1.57 0..01 0.31 0.03 0.02 0.019399 0.99 0.65 1.16 3.22 0.13 0.34 0.04 0.02 0.015195 0.95 1.00 0.93 0.00 0.01 0.31 0.04 0.02 0.01PS1S+B· 1.00 1.11 0.56 0.00 0.16 0.35 0.04 0,03 0.02

~PS15+B had '0.0008% B.B content of all other steel < 0.0005%

provides information about the influence of low Ni on Krcof high-carbon steels. IH-SO is a high-manganese, high-siliconsteel. This type of steel is generally not used for carburizingapplications because it has a relatively low case harden ability.The 4800 series steels, ERCH-1 and ER-.S, were included tostudy the ,effect of nickel on fracture toughness. ERCH-l andER-8 are nickel-free and reduced nickel replacement steels(same case and base hardenability) for 4800 steels. The 4800steel was poured at. carbon contents of 0.95% and 0.70% todetermine the eHect of carbon content on KIC of high-nickel,high-carbon steels. Other steels include 9300 steel with 0.99%C (9399), 5100 steel with 0.95% C (5195) and PS-IS withboron. The 9399 is a 1.16 Cr-3.22 Ni. steel. The 5195 is a Mn-Cr ste I with ha.rdenability equivalent to high-carbon PS-15and 8697 steels. Boron additions in Mn-Cr-Mo (PS-IS withboron) steel were made using commercially available Battsalloy.

The ingots were furnace-cooled and appropriate sectionsfrom each ingot were wiled into two 15 cm wide, 1 em thickplates. Proper care was exercised to prevent cracking anddistortion during the processing. Four to six compact tensionspecimen with width (W) equal to Iour times thickness (B)were machined from each grade of steel per ASTM E-399.The specimens were prepared in the "L-T' crack plane orien-

AUTHORS:

V.K. SHARMA is Chief Materials Engineer at Nauistar Intema-tional Transportation Corp .. Fort Wayne, IN. Prior to taking thispost he worked at Navistar as research engi~leeri/lg manager in tileareas of slmclural metallurgy and me/al and lubricants technology.He did his undergraduate work at the Indian Institute of Technologvand earned a MS in metallurgiCQ/ engineering /rom the Universityof Michigall and Q PllD in rtIJ1.teria/sengineering from !lli/lOis Instituteof Tecllnology. He .also bolds an MBA from tire University of Chicago.Dr. Sharma is .a member of ASM. SAE, ASTM and AIME.

G.H. WA1Tm is Manager, Maten'als Technology - AgriculturaiEquipment and Component Engineen'frg at 1.1. Case Co, He Iras alsoworked for Intemational Harvester Co. with responsibilities ill thearea of materials specifications developmerlt and metallurgicalresearch. Mr. Walter Iw/ds a B.S. il'l metallurgjcal engineering /rom

8 Gear Technology

Illinois ltlsti!ute of Techf1ology. He is a member of SAE and currentchairman. of SAE Dioisio» 8011 Carbon and Alloy Steel Hardenabilityalld a Fellow of the Amen'can Society for Metals. He is also a memberof Tau Beta Pi engineering honorary /ratemity.

DA~E H. BREEN is Secretary arid Director of ASME-Gear ResearchInstitute. Prior to laking this post. he worked for sixteen Y.l!arsforInternational Harvester Co., managing tile corporate metallurgicalresearch laboraton·es. His technical interests include gear tecim%gy.fatigue alld {racture of metals, alloy steel and iron technology,tribology and mecha':lics. Mr. Breen did undergraduate work i'lmechanical engineering at Bradley University. Peoria. IL. He holdsa MS degree in metllJJurgy from the University of Midligan ana aMBA {rom tile University of Chicago. He is the author of IltJmerousbooks and papers on gears. metallurgy and fatigu and is a memberof ASM, ASME. SAE, ASLE and the Americall blstitute of Mit1ingand Metallurgical Erlgin.eers.

tation. Before final machining, 3.75 x 3.75 x 1 em pieces fromthe plates were ground to 0.6 em. A minimum of 0.1 emmaterial was ground from each surface to remove any decar-burized layer. Although an the data in this report were ob-tain d using Standard ASTM E-399 compact tensionspecimens, tapered double cantilever beam (DCB) and over-sized compact tension specimens (W -8B) were also used inthe preliminary studies. DCB and oversized CT specimensresulted in 10-20% higher KIC values. (S~ Table 2.)

Completely machined specimens were austenitized for twohours at 930 C. cooled to 845 C and vertically submergedinto an oil quench tank at room temperature and temperedat 190 C for one hour. The surfaces of the specimens wereprotected from decarburization with a protective "decarb"stop coating. Representative heat treated specimens werechecked for residual stresses, distortion and quench cracks.Quench cracks were not detected, and all specimens werewithin the dimensional tolerances permitted by ASTM E-399.Average residual stresses measured on the surfaces of twotapered DCB specimens were found to be +1.3 MPa. Con-sidering the size and geometry of the compact tensionspecimen, it is assumed that the KIC values reported hereinare independent of macro residual stresses.

The specimens were fatigue precracked, loaded to fracturein tension, and fracture 'toughness values were calculated fromthe autographic plot of load versus displacement in accor-dance with standard ASTM procedure. The KIC valuesreported herein are averages of at least three replicate tests.

Since the specimens were heavily textured, the amount ofretained austenite could not be determined by routine x-rayanalysis. The volume fraction of retained austenite wasmeasured by x-ray diffraction using a tilting and rotatingstage.(21 The use of a rotating and tilting stage averages thex-ray peak intensity for a set of planes from a maximumnumber oE possible crystallographic orientations.

Fractographic features on several selected compact tensionspecimens were studied using a scanning electron microscope.In addition. several compact tension specimens were sectionedand evaluated metallographically.

Results and DiscussionAs-Rolled Microstructure.

The cleanliness level of all steels met typical JKT cleanlinessrequirements for commercial quality steels, The sulphideswere pancake-shaped, having no abnormal size or distribu-tion. Although each steel experienced a similar heating, roll-ingand cooling cycle, as-rolled microstructure of the platesvaried significantly depending on the chemistry. (See Table3.) lower carbon, 0.72% C and 0.86% C, PS-IS steelsdeveloped a fully pearlitic microstructure. A proeutectoidgrain boundary carbid network formed in 0.99 % C PS-15,8697, [H-50, ERCH-1 and 5195 steels. Typical as-rolledmicrostructures for 0.99% C PS-15 and 5195 steels are shownin Fig. 1. The as-rolled grain size in 0.99% C PS-15 steel wassignificantly coarser (ASTM grain size 1-2 as compared to4-5) than all other steels showing proeutectoid carbides. TheER-8 steel had a primarily spheroid microstructure with smallamounts of grain boundary carbides. High carbon 4895 steel,developed a bainitic microstructure with patches of marten-

Fig. 2-As-rolled microstructures of 4895 and 9399 steels. Photomicrographfor 4895 steels shows essentiaily bainitic microstructure wi!h patch- ofmartensite. The 9399 steel developed proeutectoid carbides in acompletelymartensitic matrix.

site. (See Fig. 2. ) Fig. 2 also shows as-rolled microstructurefor 9399 steel. Because of its high hardenabiliry, 9399 st Ideveloped a completely rnartensitic matrix with proeutectoidcarbides in the grain boundaries.

As-rolled microstructure can influen e subsequent heattreated mechanical properties of high-carbon, martensiticsteelsYU41 Nakazawa and Krauss'f" evaluated the effect ofproeutectoid carbides on the fracture toughness of quenchedand tempered 52100 steel. Two series of specimens - one withgrain boundary proeutectoid carbides in pearlitic matrix(Series A), and the other with well-distributed, fine spherical

Tab'le 2 - Specimen Geometry EHect(iHigh Carbon IH·50 Steel)

Fracture Toughness MPa.mV2ASTM Compact

STD CT TensionW=4B W=8B

HardnessRc Tapered DCB

.24.223.323.3

59.560.0'60.059.5

Average

17.618.017.517.917.7

21.017.918.021.719.6 23.2

Table .3, - As-'Rolled Microstructures

SteelPS-1S(.72% C, .86% C)PS-1S (0.990;0 C)

Microstructure

Fully PearliticPearlilic With GB CarbidesASTM 1-2 GSPearlitic With GB CarbidesASTM 4-5 GSSpheroidized With SmallAmounts of GB CarbidesBainitic With Patches OfMartensiteMartensitic With ProeutectoidCarbides

8697, IH-SO.ERCH·" 5195

ER-8

4895

9399

January/February 1989 '9'

--0-- Se'''1 A

- -Lr - Serl •• 8

.k----~./.,a'

12 - - -800 850 900 950 1000 1050 1100

X30 MIN

Fig. 3-lnfluence of starling microstructure on KIC of 52100 steel. SeriesA had proeutectoid carbides. Series B contained only fine carbides .• '

carbides in a spheroidized matrix (Series B) -were produc-ed. and their fracture toughness was determined as a func-tion of the austenitizing temperature, The results arereproduced in Fig. 3. When austenitized below 950°(,martensitic specimens with proeutectoid carbides in the start-ing microstructure (Series A) are significantly tougher thanSeries B. which contained only fine spherical carbides. Above

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10 Gear Technology

Fig. 4 - M icrost rucrure of high carbon IH-50 steel. The m icrosrruct U Fe showsa uniform, lightly tempered, high-carbon martensite and retained austenitewith no carbides or microcracks.

950 C. where all carbides dissolve to form a homogenousaustenite, both series have about the same toughness,Nakazawa and Krauss explained the increased toughness ofSeries A on the basis of higher toughness of carbide-free.transgranular areas present between the proeutectoid car-bid s. Below 950 C. specimens containing only fine spheroidparticl s have low toughness because of th very finemicrovoid coalescence (small ligaments of tough carbide-freeareas) associated with the closely spaced carbides. In the pre-sent studies, all specimens were ausrenirized at 930 C whereessentially all proeutectoid carbides were dissolved.Therefore, variations in as-rolled microstructure are expectedto have little or no influence on the KIC values.

Heat Treated Microstructure.

Typical microstructures for various steels after the heat treat-ment (austenitized at 930 C for two hours, quenched in oilfrom 845 C and tempered at 190 C for one hour) are shownin Figs. 4-7. As shown in Fig. 4. IH-50 developed a uniformmicrostructure containing lightly tempered, high-carbonmartensite and retained austenite with no carbides ormicrocracks. This is the typ of surface microstructure whichis most desirable in carburized parts. However, as indicatedpreviously, IH-50 (high-manganese, high-silicon) is not asuitable steel. for most automotive gearing and shaft applica-tions because it does not have sulfident case hardenabiltty.

With the exception of rH-SO steel, all other steels exhibitednon-uniform, banded microstructures. The severity of car-bon and alloy segregation varied depending upon the type

Table 4- Undissolved CarbidesalA 720 MeasllIrements,

SteelPS-15 (.99% C)8697EACH-1EA-S93995195PS-15+B

% Carbides1..0%1.0°/09.0%1.5%8.5%1.0%1.0%

.72% C PS-15, .860/0 C PS-15, 4870 and 4895 containedno undissolved carbides.

I~

Fig. 5 - Microstru lure ot 0.72 % C PS-] 5 steel. The wI-ill Iching 'bands arehigher carbon areas which are relatively soft and contain higher amountsof retained austenite. No carbides or microcracks are visible.

and amount of alloying elements. PS-1S steels with 0.72%and 0.86% carbon, when examined at lOOX, showednumerous white etching bands (Fig. 5), The white etchingbands were relatively soft (HRC 62 vs 65) and contained ahigher volume fraction of retained austenite as compared tothe surrounding matrix. Carbides or microcracks were notobserved in the 0.72 % CPS-IS steel. The 0.86 % CPS-ISsteel also did not have carbides, but occasionally microcrackswere observed. Microstructures of high-nickel 4870 and 4895steels were similar to the 0.72 % PS-1S steel. High-nickel steelsalso showed non-uniform microstructure with no carbidesor microcracks. The frequency of white etching bands,however, was low. The rnartensitic plate size in the nickelsteels was somewhat coarser.

The 8697 and 0.99% C PS-15 steels had similar microstruc-tures. In addition to microcracks, the white etching bandscontained numerous undissolved carbides. A typical micro-structure for heat treated 8697 is shown in Fig. 6, The worstsituation with respect to carbides and banding was observedin ERCH-1, a nickel-free substitute steel for the 4800 gradeof steel (Fig. 7). The microstructure of 9399 steel was verysimilar to ERCH-l.

The amount of undissolved carbides was determined us-ing a quantitative image analyzing system. As illustrated inTable 4, 0.99% C PS-15, 8697, 5195 and PS-15+B steelshave less than 1 % carbides. ER-B, a substitute steel For 4800steel with reduced nickel content, had 1.5% undissolved car-bides. The highest amount of carbides, 8.5 and 9%, were pre-sent in 9399 and ERCH-1 steel, respectively.

The influence of carbides on K[C was investigated by deter-mining the fracture toughness of ERCH-l steel after austenitiz-ing it at 930 C for two, four and seven hours. Samples ofcarbide free 0.86% C PS-15 steel were also included in the

Table 5 - EHect of AlistelliUzll1Q Time at 9'30"C

K1c MPa.mV2

4 Hours23,8

2 Hours22'.4

SteelPS-1S (.87% C)

ERCH-l

7 Hours21.8

20.7 .21,1 20.0

2GJJ!--1

8697I

Fig.6 Microstructure at 8007 steel. PS-lS WIth 0.0<),"" also showed asimilar microstructure. In addition to the microcracks. the white etching bandscontained undissolved carbides,

study for comparison. Increasing the austenitization timefrom two to seven hours did not change fracture toughnessof either steel significantly (Table 5). The influence of massivecarbides on fracture toughness of austenitic and martensiticwhite cast iron has been studied by Gahr and Scholz.(SI Onthe basis of the results shown in Fig. 8 from that study, theauthors concluded that increasing the volume of carbidesfrom 7% to 30% had no significant influence on K[C Onlywhen the volume of carbide increases beyond 30%, did thefracture toughness start decreasing. The matrix of mart n-

sitic white cast iron is essentially the same as the high car-bon steel. None of the steels included in the present investiga-tion contained more than 9% undissolved carbides. In fact.

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Fig. 7 - Microstructure of ERCH·l steel. ERCH·l steel showed maximumbanding with great t amount of undissoved carbides.

as given in Table 4, the majority of steels have less than 2 %undissolved carbides. It is, therefore, reasonable to assumethat the fracture toughness values reported herein have not

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12 Gear Technology

~ 35!'J0.::;:

U) 50U)u.;Z

~:l 250I-

:..L::.: 20:lI-U<:CQ:;:t.1. 15

0

21~

10 20 30 40CARB IDE \'01.U~'E. ~

50

Fig. 8 - Influence of carbides all fracture roughness of martenstric andaustenitic white cast irons."

been influenced by the presence of carbides, Since more than90 % of the fracture propagates through a martensite-austeni-tic matrix, fracture toughness values should be dominatedby the matrix properties rather than the carbides,

Influence of Retained Austenite.The presence of retained austenite, particularly as a con-tinuous thin. film surrounding martensitic plates rather thanas a discreet blocky phase, is considered to. enhance fracturetoughness of martensitic steels. Thomas(6J has emphasizedthe importance of mechanical and thermal stability of retainedaustenite. Retained austenite has been proposed to improvecrack propagation resistance and, thereby, increase fracturetoughness by crack branching or blunting, m by strain orstress induced transformation resulting in the developmentof compressive stresses in front of the advancing cracK,i81bypreventing the formation of brittle boundary carbides, andby breaking the continuity of the cleavage planes acrossvarious rnartensitic plates.(9J

The influence of retained austenite on fracture toughnessof high carbon steels was studied by varying the amount ofretained austenite in 4895 steel by deep freezing at -84cCand -207"C. The results in Table 6 show that a decrease inthea ustenite from 40 % to 18.5%, caused by deep freezing,quenched and tempered 4895 steel at -84 C, decreased frac-tUI1etoughness from 24.5 MPa.m" to. 14.6 MPa.m"_ Fur-ther deep freezing at -207 C reduced retained austenite to15.5% causing an additional decrease in the toughness to U.5MPa.m I,. The amount of retained austenite, therefore, hasa significant effect on the KIC of high-carbon, martensiticsteels. The microstructure of 4895 steel after the subzerotreatments is shown in Fig. 9. Deep freezing transforms re-tained austenite to martensite. A careful examination of themicrostructure revealed that additional microcracks weregenerated as a result of the transformation. A 50% decreasein the KIC value, however, cannot be explained on the basisof a.n increase in the density of microcracks. This decreaseis primarily related to the reduction of retained. austenite. Sui-

Fig. 9- Microstructure of 4895 steel austenitized at 930° C. quenched in oil,tempered at 190° C for 1 hour and deep frozen at -84" C and -207° Cfor two hours.

ficient data were not generated to determine the minimumlevel of retained austenite necessery to increase toughnesssignificantly. The beneficial effect of retained austenitedepends on its nature and distribution. Retained austenite ismost effective when present as continuous films surroundingmartensite rather than a discontinuous blocky phase.(Q) Heattreatment and alloying elements could be selected to max-imize the influence of retained austenite at lowest levelsnecessary so that the toughness can be improved withoutdetrimentally affecting the hardness and other relatedproperties.

Effect of Carbon Content.,

Increased strengthening caused by carbide precipitation orinterstitial solid solution strengthening by carbon is usuallyaccompanied by decreased toughness. In designs for whichsteel strength below 1700 MPa (4:0 HRC) are acceptable,material systems are available offering appreciable toughness.In high-hardness, carburized applications, (2000 MPa (57HRC) and above) such as gears, toughness becomes aspecialized area of knowledge. Earlier work in this area wasdone by Schwartzbart and Sheehan.(1O) Results of some oftheir work, replotted, are shown in Fig. 10. This work wasdone with Charpy V-notch specimens. The same hardnessat different carbon levels was obtained by tempering.

In the present investigation, the effect of carbon on KICof high-carbon, martensitic steel was st udied by test ing PS-lSsteel at 0.99%,0.86% and 0.72% carbon. Also, the fracturetoughness of 4800 steel was determined at 0 ..95% and 0.70%carbon. The results are shown in Table 7. Lowering carbonenhances fracture toughness. The effect of carbon, however,

Table 6 - Effect of Retained Austeni,te4895 St,eel

RA40.0%18.5%15.5%

KICMPa.mV224.514.612.5

IHeat Treatment190°C TemperDeep Frozen ·84°CDeep Frozen ·207°C

Rc55.564.065.0

~30HRC

...o...i 10

o 0.10.2 0.30 •.40.50.6 0.70.80.9 1.0% Cllt'bon

Fig. 10 -lsohardness ~J~tion of carbon and toughness. Toughness determinedby Charpy V-notch.

must be considered in conjunction with retained austenite.In general, the amount of retained austenite decreases withdecreasing carbon. In the case of PS-lS steels, a change inthe 'carbon from 0.99% to 0.86% decreased retainedaustenite from 39% to 23% ..Discounting other effects, this

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Fig. n- Microstrudure of 4870 and 4895 steels.

"to'

2 J..) d 7 1'1 l .II "ii 7 1"'1 2 l ~

10' leTCyele.e To Fracture Or Initial, Pitting

S .'l1li11.Fig. 12-Comparuon of bending and con lac! fatigue life durability of car-buriaed Mn-Cr steel lYi!h CJ o. GSD. and C90 (50lid and doned Jines) bandsfor carburized 8622 steel.

reduction should have lowered the toughness. However, thetoughness increased, indicating the importance of carbon.KIC decreased slightly when the carbon content was reduc-ed from 0.86% to 0.72% C. Improved toughness resultingfrom lowering the carbon in this case is more than offset bya decrease in the toughness caused by a change in the retain-ed austenite from 23% to 16%.

The eHect of carbon on fracture toughness of high-carbonsteels becomes more evident in the case of high-nickel 4800series steels. A decrease in the carbon from 0.95% to 0.70%increased K1C from 24,5 MPa.mljz to 34.5 MPa.m1fz. Nickelpromotes the stablization of retained austenite. Therefore,even at 0.70% C. 4870 steel containing 3.50% nickel retained21 % austenite. The highest fracture toughness was recorded

Tab'le 7 - IEff,ect of Carbon Content on KIC

'RA39%23%16'lfo400/0210/0

K1CMPa.m1/216.6220421.724.S34.5

SteelPS-1SPS·1SPS-1S48954870

%e0,990,860.720.950.70

Rc60.060.560.555.557.0

14 Gear Technology

for this steel, Notice that the fracture toughness improvedin spite of an increase in the hardness by nearly 2 Rc.

Lower carbon martensites are Inherently more ductile. Itis well known that the morphology of martensite changesfrom dislocated laths to twinned plates with increasing solutecontent. Carbon has the strongest effect in promoting twin-ning. Alloy composition and martensitic transformationtemperature influence the transition of martensite from lathto plate martensite. The dislocated laths are composed ofbundles of slightly misoriented grains having high density ortangled dislocations. The fine structure of plate martensiteconsists of fine transformation twins.Ill' Lower carbondislocated lath martensites are relatively less brittle, becauseunder stress, dislocations can move, resulting in deformationby slip. Higher-carbon martensite, which consists primarilyof plate martensite with internal twins, is britt.le. Twinnedrnartensites have extremely low fracture toughness. A reduc-tion in the percentage of twinned martensite, resulting fromlowering carbon content, can be expected to improve K1Csignificantly. It has been shown that plane strain fracturetoughness of AISI 4130 (medium-carbon Iow-hardenabllitysteel) can be improved two-fold by a high temperature andstep-cooling austemtizing treatment, which virtuallyeliminates twinned martensit plates.112l

The microstructures of 4895 and 4870 steels are shown inFig. 11. Photomicrographs show a difference in the marten-sitic plate size and the amount of retained. austenite. The 4870has finer plate size and less retained austenite. It is also likelythat the 4870 steel contains lower amounts of plate marten-site, although the structure for both steels is mostly plate type.

Alloy Effects .Our past experience indicates that in automotive gear steels.small amounts of nickel and molybdenum can be replacedwith other alloying elements. A basic assumption in thedevelopment of cost-effective carburizing steel is thatengineering performance of a component made from lowalloy steel is dictated by the base and case carbon content,microstructure and residual stresses. As long as the steelselected meets these requirements, equivalent engineering per-formance can be expected. The microstructure and r sidualstresses are governed by the carbon content, hardcnabilityand martensite start and finish transformation temperatures.The major reason for alloying these steels is to influencehardenability and transformation temperatures. Sufficienthard nability, coupled with adequate and uniform quenching,provides microstructure optimization and control. Thedependence of performance/strength on microstructure isbasic.

Considerable laboratory test and field history data existto support these basic assumptions. SAE PS-16 (Mn-Cr-Mo)steel, which is a nickel-free, co t-effective replacement steelfor standard AJS[ 8622 (Mn-Cr-Mo~Nj) steel. has been usedsuccessfully in the production of heavy duty truck rear axlering gears and in tractor and combine power trains for ov r15 years. SAE PS-59, a cost-effective, Mn-Cr replacementsteel, has also been used to substitute for PS-16 and 8622 steelfor the last five years. Fig. 12 shows torque-life gear data fromPS-59 steel at two carbon levels. superimposed on th GIO,Coo and G90 bands for 8600 type steel. The bending and con-

tact fatigue durability data were obtained by testing carburiz-ed and hardened six-pitch test pinions in a power circulatingrig. The results show that the Mn-Cr steel and Mn-Cr-Mo-Ni (8622) steel have equivalent fatigue properties,

The data in Fig. 12 indicates that in low alloy gearing steels,small amounts of nickel and molybdenum can be replacedwith other more cost-effective alloying elements. Nonetheless.the effect of alloying elements on the performance of gearsis quite controversial, Using a modified Brugger test,Diesburg(l3) studi d the influence of carbon, alloy andresidual stress on .fracture stress and very Jow cycle fatiguelife, Carbon and residual stress were shown to be very im-portant. Alloy content was also found to be important. butthe amount of alloy appeared more important than thespecific alloy, DePaul,!I41 SheaCl51 and Love and Camp-bell, (10) among others, provide a technology base concern-i:ng the nect of major alloys on properties of carburized steel.Most of this relatively small amount of work on alloy ef-fects has been related only to gear tooth bending fatiguestrength, not gear tooth flank durability (pitting fatigue).Kraussl17lconduded, based on roller specimens, thai. thealloying dements alone had no effect on flank durability.Allsopp, Weare and love (181 also were not able to d teet analloy influence in test. using gears. None of the above pro-grams ..however, were designed specifically to systematicallystudy alloy effects.

Effect of Nickel at intermediate Levels.Fracture toughness K1C for 0.99% CPS-IS ste I, Mn-Cr-

Ni-Mo steel 8697 and 5195 steels are given in Table 8. Atequivalent carbon 1 vels, K]C for PS-15 and 8600 series steelsall exactly the same; i.e .. 16.6 MPa,m". The 8695 steel hasa slightly lower retained austenite. However, the amount ofaustenite is sufficient to provide its maximum contributionto toughness. Fracture toughness of 5195 Cr-Mn steel issuperior to both the high carbon PS-1S and 8699 steel. Thisimpr ved toughness is thought to be related to a lower matrixcarbon concentration in 5195 steel caused by the formationot chromium carbides. The data in Table 8 show that small

Table 8 - IFrac.ure Toughness ofHigh 'CaribOIl PS·15,860Ql and 5100 :Steels

-----

Steel C Mn Cr Ni Mo HRC %RA K,cMPa.m1/2- - - -pg·15 0.99 1.09 0.54 0.03 0.16 60 39 16.68697 0.97 0.83 0.52 0.60 0.22 60 33 16.65195 0.95 1.00 0.93 0.03 0.01 !60 34 20.8

amounts of nickel, molybdenum and oth r alloying elem ntsdo not influence th fracture 'toughness of high carb n st Isignificantly.

Effect of Nickel at Higher I.e·ve/s.Table 9 shows the cHct of nickel on fracture toughn s

of high-carbon rnartensitic steels. A comparison of K1C for4895, ER-8and ERCH-1 reveals that fr cture toughn ss in-creases w.ith increasing nickel content. ER-8 andER, B-1 arereduced nickel and nickel-free replacement steels for 4800steel. K1C of ER-8, in which half of the nickel in 4800 st elis replaced with other alloying elements. has approximately10% lower fracture toughness as compared to the 4895· steel.ERCH-l, which contains no nickel, has the lowest toughnes .The effect of nickel can also be seen by comparing tn K]Cof 0.72% C PS-15 steel with 4870 and 9399 teel with 5195.Although some of the improvement in toughness may beplained on the basis of higher amounts of retained austeniteand lower hardness of 4870 and 5195 steels, a nominal 3.5%nickel in these steels seems to promote improved toughness.

Effect of Boron.Limited data obtained on testing a steel containing boron

reveals that small additions of boron can enhance fracturetoughness of high-carbon martensitic steels significantly. (SeeTable 10.) The addition of 0.0008% boron to 0.99% CPS-ISsteel improved fracture toughness from 16.6 MPa.m" to21.2 MPa.m I,. Boron is known to be a potent contribut rto hardenability in the low and medium carbon range. buthas very little influence on the hardenability of high-carbonsteels. This non-hardenability related improvement in 'fhe

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Fig. 13- Photomicrographs showing fracture modes in fatigue precrack andsingle overload regions of 8697 steel. (A) Fatigue precrack area. (8) and (C)fatigue crack/single overload interface, and (0) transgranular area in singleoverload.

fig. 14 - Photomicrographs showing fracture modes, fatigue precrack andsingle overload regions of 4895 steel. (A) and (C) fatigue preerack/singleoverload interface, (B) transgranular area in fatigue precrack area, and (0)transgranular area in the overload region.

Tab~le9 - EHect of Nickel Content On K:lc

Steel %C %Ni Ac AA KlcMPa.m'h-- - -4895 .95 3.44 55.5 40% 24.5ER·B .95 1.57 60.5 47% 22.4ERCH-1 1.00 0.08 58.5 42% 20.7PS-15 .72. 0.03 60.5 16% 21.74870 .78 3.50 57.0 21% 34.59399 .99 3.22 53.0 49% 27.15195 .95 0.08 60.0 34% 20.8

Table 10 - EHect of Boron On KIC

Sleell %C Rc RA K1c'MPa.m'h-- - -PS-15 0.99 60 390Al 16.6PS-15+B· 1.00 60 31% 21.2

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16 Gear Technology

Table 11 - ICritical Crack Size andLoad Carryingl Capability of Various Steels

ac (MPa)(1)

16909\30900690840101514108609301125870885

R(2)

1.001.351.301.001.221.472.041.251.351.631.261.28

K1dMPa. m112)16.622.421.716.620.324.534.520.722.4.27.120.821 ..2

Ac(mm)0.3630.6600.6150.3630.54\30.7871.5200.5660.6600.9650.5710.589

STEEL (%C)PS-15 (.99)PS-15 (.86)PS-1S (.72)8697 (.97)IH-50 (.97)4895 (.95)4870 (.70)ERCH-1 (1.00)ER-8 (.95)9399 (.99)5195 (.95)PS-15 + B (1 .00)

(1) Critical applied stress for a. crack size of 0.363 mm.(2) R = uc/u for PS-15.

fracture toughness of high-carbon boron steel may dependon how boron additions are made. Unlike the domestic prac-tice of alloying with "protected" boron by simultaneouslyadding titanium, zirconium and aluminum, the Gennan prac-tice is to alloy with ferroboron. The German practice is aninefficient way to use boron for hardenabilityenhancement,but the practice is said to improve the distortion, toughnessand low and high cycle fatigue characteristics of carburizedsteel.(19)According to the German practice, when added asa ferroboron, boron acts as a nitrogen fixer during both thesteel making proce s and subsequent carburizing treatment.After combining with nitrogen during the steel making pro-cess, enough boron remains in the soluble form to combinewith nitrogen picked up in carburizing and, thus. high car-bon martensite-austenite surface is low in solubl nitrogenand has enhanced properties.f201 Additional research is re-quired to fully assess non-hardenability related beneficial ef-fects of boron-containing carburized steels.

Significance of Fracture Toughness in Design.

Small diff rences in the KlC resulting from carbon con-tent, retained austenite or alloy effects can have a significantinfluence on the critical size (Ac) and load carrying capability((1) of hi.gh-carbon steels. The data in Table II werecarculated using basic fracture mechanics concepts involvingrelationships between stress intensity factor. applied stress andcrack length. The critical size (Ac) was calculated for an ellip-tical surface crack with a.geometry A/2c of 0'.5, assumingapplied stress of 690 MPa and applied stress-to-yield strengthratio of 0'.33. The results show that a change in the KIeoEhigh-carbon steel from 16.6 to 22.4 MPa.mVz nearly doublesthe size of the crack that the material can sustain before anunstable crack growth occurs. Similarly, for a given cracksize, the load carrying ability of the steel improves by 35%.

Fractography.

The fatigue precrack and single overload regions of severalsamples were examined using SEM. Typical fractographsareshown in Figs. 13-15, Stable fatigue crack growth in theprecrack area was characterized by both transgranularandintergranular fracture modes. In all samples, fracturetopography at low stress-intensity range was smooth andprimarily transgranular, As the stress-intensity range in-

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Fig. IS - Fractographs showing diff rences in the 'topography 01 transgranularfractures in the single overload and fatigue precr ck regions.

creased, the fracture surface generally became coarser, andthe percentage of transgranular areas decreased, while theamount of intergranular fracture mode increased ..Further ex-amination of the transgranular area at higher magnificationrevealed topography typically associated with crack propaga-tion in high-hardness steels. The fracture surfaces were flatand mottled, but occasionally fatigue striation could beresolved.

The single overload region was primarily intergranular withsome tra:nsgranular areas present. These transgranular areaswere distinctively different from the topography observed inthe transgranular Iatigue precrack areas. (See Fig. 15.) Thesetransgranular areas were characterized by cleavage and flatdimples.

ConclusionThe results show that a complex interdependency existing

between the carbon content, retained austenite and variousalloying elements makes interpretation of the data difficult.Experimental steels must be designed carefully to isolate theeffect of each of thes variables.

A small variation in the carbon can influence the fracturetoughness of high-ca.rbon steels significantly. Lowering car-bon increases toughness. The effect of carbon, however, mustbe considered in conjunction with retained austenite, whichincreases with the increasing carbon content. Higher retain-ed austenite is usually beneficial, but the beneficial effect isdependent on its nature and distribution.

The alloy effect becomes significant only after the alloyexceeds a minimum amount. Results show that small amountsof nickel (-0.5%), molybdenum (-0.2%) and other alloy-ing elements do not have significant eHect on the plane strainfracture toughness properties of high-carbon steels. Highernickel promotes toughness. K1C of high-carbon steels con-taining a nominal 1.5% and 3.5% nickel improved signi-ficantly. Small amounts (0.0008%) of boron also appearbeneficial. This non-hardenability related improvement maydepend on how boron additions are made.

References:1. RAZIM, C. and STRENG, H. "Properties of Ca -Hardened

Components - Basic Considerations Concerning the Defini-

18 Gear Technology

tion and Evaluation of Toughness" Proceedings, Case-HardenedSteels: Microstructumt and Residual Stress Effects, D.E.Diesburgh, Ed. A[ME, 1983, p. 1-15.

2. SHIN, s.w. and SHARMA, V. K. "Application of a Tiltingand Rotating Specimen Stage to. X-Ray Retained AusteniteMeasurements in Textured and Coarse Grained Steels;' SAEPap r N800428, SAE Tmnsactions, 1980.

3. STICKELS, CA. "Plastic DeformaHon of Quenched andTempered 52100 Steels Under Compression:' Metallurgical Tran-sactions A. 1977, Vol. SA pp. 63-70.

4. NAKAZAWA K. and KRAUSS, G., 'Microstructure and Frac-ture of 52100 Steel" Metallurgical Transactions A. Vol. 9A,1978, pp. 681-689.

5. GAHR ZUM, K.H., and SCHOLZ, W.G. "Fracture Toughnessof Whit,e Cast Irons'; fournal of Metals. October. 1980, p. 38.

6. THOMAS, G. "Retained Austenite and Tempered MartensiteErnbrittlement,' Metallurgical TrallStlctions A Vol. 9A, March,1978, p. 439.

7. WEBSTER, D."Development ol a High Strength Stainless Steelwith Improved Toughness and Ductility; Metallurgical Tml'1-sactiolls Vol. 2, 1971, p. 2097.

8. GERBERICH. W.W., et .11. Fracture 1969, Proc. of Znd, Inter-national Conference on Fracture, Brighton, U.K. Chapman andHall, ua. 1969, p. 288.

9. xoo. J., RAO, B.Y.N., and THOMAS, G. "Designing HighPerformanace Steels with Dual Phase Structures" Metal Pro-gress, Septemb r, 1979, p. 67.

10. SCHWARTZBART, H. and SHEEHAN, J. "Impact Propertiesof Quenched and Tempered Alloy Steel'; Fi'lal Report ONRContract N6()NR244TO.1.. September, 1955, Research con-ducted al liT, Chicago.

11. KRAUSS, G. and MARDER, A.R. 'The Morphol.ogy ofMartensite in Iron Alloys: Metallurgical TrarrsQctions, Vol. 2,September. 1971, p. 2343.

12. ZACKAY, V. F., "Fundamental Consideration in the Designof Ferrous Alloys:' Alloy Design for Fatigue and FractureResistance, AGARD-CP-.I8S. NATO, January. 1976,

13. DIESBURG, D.E. and SMITH, YE. "Fracture Resistance in Car-borized Steels, PE II: Impact Fracture" Metal Progress. 115 (6),June, 1979, pp. 35-39.

14. DEPAUL, R.A Impact Fatigue Resistance of Commonly UsedGear Steels, SAE paper 710277, 1971.

IS. SHEA, M.M. Impact Properties of Selected Gear Steels, SAEPaper 780772, 1978.

16. LOVE, R.A. and CAMPBELL, J.C. "Bending Strength of GearTeeth;' Report 1925/5, Motor Industry Research Association0.£ U.K., December, 1952.

17. KRAUSS, G. "Influential Factors in the Fatigue Behavior of CaseHardened Steels Subjected to Rolli.ng-Sliding Loads; Lecture tothe Union of German Engineers, 3/22/72.

is, ALLSOPP, a.c.. WEARE, AT, and LOVE. R.J. "Resistanceto Pilling of G ar Teeth, Re" Report 1959/2, Motor IndustryResearch Association of U.K., June, 1957.

19. LLEWELLYN, D.T. and COOK, W.T. "Metallurgy of Boron-Treated Low Alloy Steels; Metr:!lsTechnology, December, 1974,p.517.

20. GERMAN PATENT 1608732, "Procedure for Producing ToughBoron Treated Steels" September, 1969.

Acknowledge~t: .Reprinted wilh pennission of the American Gear Manu/r.lc-hirers ~tion. The opinions. statements and conclusions presented in thispaper a~ those of the ,flUthoTSami in no way represent the position or opi-Plio" of the AMERICAN GEAR. MANUFACTURERS A55OCIATlO

Absl:raclThis article investigates fillet ~eatures consequent to tooth grind-

ing by generating m thods. Fillets resuJting from tooth cutting andtooth grinding at different pl'e5sure angles and with different posi-'tions of th grinding wheel aN compared. Wa.'j'5to improve the finalfillet of the ground teeth with regard to toolh strengthand noise,as well as the grinding conditions, are shown. "Undergrindlng" isdefined and special designs for noiseless gears are described.

IntroductionTooth fillets of involute gears oHen are more important

than the involute itself in determining manufacturing cost,precision and gear pair ,operating success. The purpose of thisstudy is to illustrate fea.tures of tooth Fillets as they aHoctmanufacturing conditions on grinding machines and as theypertain '10 gear strength and noise in gear operation.

Tooth generating methods are considered. The computa-tions extend the procedure in Reference 1 to helical. gears,using the criterion of Salamoun and SuchyW and are appliedto both tooth cutting and grinding. (See Appendix A.) Arigorous approach for helical fillets should consider the con-jugation of either tools or grinding. wheels with the generated

teeth in the space, but an approximation of the adopted pro-cedure is more than sufficient for the engineers investigations,as it does not concern the mating flanks of the gears ofapalr,but only Hllet forms.

AUTHORS:

DR. ENG.GIOVANNI CASTELlANI is a gear.consultant to privateindustry specializing in the relationship of gear design to manufac-turing problems. Prior to enterillg tke private .consulting field. Dr.Castella'ii was a profPS5ar of mechanics at the State Technical 111-5itute. He has published two books 011 gear design as well as numerousarticles and papers for technical ,conJeTf!P1ce5allover the world. Dr.Castellani is a memb· r of AIPI (Assaciaztione Italitmtl Progetti.sti 111-dustriali}, chief of the teclmical committee of ASSlOT (Associaziol1eItal'iana Cos/ruttori Orgarri di Trasmissian« e .lngnmnggiJ. a memberof the 150 and the IFTaMM Gearing Committee.

MR. vrrrORlANO .ZANorn is ,in charge of the Calculation andApplications Department of ROSSI MOTORlDUTTORI S,p.A. QI1

Italian manufacturer of sp ea reducers, geared motors and motor-variators. He is a member of tile company's working group for UNl(The Italian Standardization Board) and II mem!:ler of ASSIOT. Mr.Zanotti earned his bachelor's degree in mechanical engineering fromthe University of B%grla in 1974.

January/February 1989 19'

The computation is extended for helical gears to a "nor-mal coordinate," whose abscissa is the half-ellipse chord nor-mal for the local helix according to Castellani'!'. Thiscriterion is preferable to that of actual spur gears because thecomputed limit between involute and fillet remains exactlythe same as for the transverse section, and is also the samefor tooth undercutting or undergrinding. Nevertheless, wechecked that no appreciable difference exists between the p!ot-lings based on "normal coordinates" and plottings of actualteeth as far as the assessment of the fillet quality is concerned.

Plotted and manufactured examples of typical grindingsof hobbed or rack-cut industrial gears are given and ageneralization is made. The effects of different grindingmethods and criteria ale compared. Special attention is givento an uncommon grinding method adopting an increasedpressure angle, whose mathematical basis is the same as forthe contrary method used in older machines for 150 grindingof teeth cut by a 200 pressure angle.(4)

Regarding gear strength, neither AGMA 218.01(5. nor ISO6337(6) give any indications of the effect of grinding steps atthe tooth root of gears that are cut without a. protuberanceor with an insufficient one. Entwurf DIN 399011980(7) does,but, in our opinion. its experimental basis(8) should bewidened. On the other hand, the number of possibilities isinfinite. Computer analysis can serve not only to avoiddangerous "notches in the notch" in specific cases, but also

1<,,1 (Ksl)

to identify the grinding methods and parameters which aremore likely to improve strength. In the future, tests mightbe restricted to convent nt cases.

Fillet analysis has two effects on gear noise. It helps avoidraise contacts for industrial gears and enables special geardesigns to be adopted for special cases.

Grinding Tooth FiUets of Gears for a Speed Reducerlet us consider a. gear pair, AlB, designed for a nominal

gear ratio of 3.55 and a center distance of ill mm. The maindata are given in Fig. 1a for the pinion and in Fig.2 for thegear. The teeth are hobbed without protuberances and groundby a wheel that has no facility for rounding the tip edge, (1.lG

- 0, but permits any pressure angle.Pinion A. Fig. Ib is the complete drawing of a tooth and

Fig. Ic is the detail of the tooth fillet. both ground. by a 24grinding wheel. Figs. Id, e, and f show fillets obtained bya 20' grinding wheel in various radial positions. An arc, cf.indicates the limit of the contact with the mating gear. Thearcs, pfO and pfG, relate to the involute limits obtained byhob bing and by grinding respectively. The point IG is thelower limit of the ground fillet.

In Fig. Ld, we consider the same grinding limit r,pIC -25.36 as in Fig. 1c. In Fig. Le the limits of the involutes ob-tained by hobbing and by gr:inding coincide. In Fig. If the

radius of the fictitious root circle generated bythe grinding wheelradius at the inner end of grindingradius of the diameter dp10radius at the inner limit of the ground involuteellipse chord normal to tip helices; i.e.. normaltip thicknessprotuberance amountgrinding stocknominal coefficient of addendum modificationtooth numbernormal reference pressure anglenormal reference pressure angle at grindingreference helix anglereference helix angle at grindingradius of the tip edge rounding of the 100]

radius of the tip edge rounding of the grindingwheel

We shall call "grinding step" the variation in the slopeof the tooth profile where a ground zone connects with a

Nomenclature

(Note: symbols in parentheses relate to computer outputs.)

a' (A')da! {DaDdd (Dd)dr (Of)dpfQ (Dpf0)

h.u (Hal')hao (Ha0)

IbnO (lbn0)

m; (Mn)mnG (MnG)raJ (Ra!)fd (Ref)rf (Rf)rfG (RfG)

operating center distancetip diameter at the outer contact endroot diameter at the inner contact endroot diameterdiameter at the inner limit of the involutegenerated by cuttingaddendum of the reference rackaddendum of the generating rack, either hobor rack-cutterfinal reduction of normal base thickness of atooth with regard to the nominal one"reduction" of the normal base thickness aftercutting: It is usually negative if the tooth mustbe ground.tooth shortening coefficient for the outer con-tact endreference normal modulereference normal module at grindingradius of the diameter darradius of the diameter ddroot radiusradius of the fictitious root circle generated bythe center of the tip edge arc of the grindingwheel

20 Gear TechnolOOY

rf(,,o (RfG0)

fIG (RIG)rplO (Rpf0)rpfG (RpfG)s~t\ (S-aN)

llo (U0)US (Us)x (X)z (Z)an (an)allG (cmG)t3 ({3)t3G (t3G)(1••0 (roa0)QaG (roaG)

cut one.Further nomenclature is given in Appendix A.

CIRCLE A-lION READER REPLYCARD

DRAWING·HOBBIill G'ROUND GEAR TEETH

General Data

Har/Mn .. 1Mn .. 4Z .. 13HaO/Mn - 1.337lJO/Mn-OIbnO/Mn .. - .05Dal'" 64.7S- aN .. 2.019DptO - SO.fl73Oct - 51.17 !theoretical),

an - 20fJ - 16.2666667X - .348B3roaD/Mn- .2Us(mm) ... 12Ilbn/Mn - .'01'IKsl - .03239':Df = 46.848

aNORMAL roors COORDINATES

RaJ Rpro RI32.35025.337 23.424

PINION Aof a g,ear-lPai rA/B

Center distance:

a' = 1 25

Grinding Data

anG .. 20 roa.G/M n - 0RrG .. 24.162 RIGO .. 24.152A/1fG - 25.38 AI • 23.424Lower limit 01ground flUetRIG- 24.302Tip thickneas 01grinding wheel:<2.92

..nG- 24RpIG.25.36

rosG/Mn- 0RIGO - ,23.524

d'NORMAL TOOTH FILLET COORDINATES

c33

l

Grinding Oats

cmG .. 20 roaG/Mn - 0Rlg- 23.956 RrGO - 23.958Apm - 25.337 AI .. ,23.424Lower IIImil of groy ndflllet:RIG .. 24.025 -Tip thickness 01 grinding wl1eel:<2.778

NORMAL rOOTH FILLET COORDINATES

1IIIi

Fig. t - Data (a), drawing of a tooth (b) and various fillet grindings (c.d.e.f) of a pinion A.

22 Gear Technology

Grinding Da!a

"nG .. 24 roaGIMn II 0trG-.16.7457612 MnG .. 4.1144865RIG - 23.524 AIGO .. 23.524ApIG - ~5.36 _ Af .. 23.424lower limit 01 ground mle!:AIG ... 23.527np thickness 01grinding wheel:< 1.9'72

NORMAL TOOTH FILLET COORDINATES

Grinding Da.l~with Undergrinding(undergr. IImll: RIG ~ 23.68)

anG - 2.0 roaGIMn _ 0RIG·, 23.524 RrGO - 23.524RplG - 25.326 Flf - 23.424lower limit of ground flllet:RIG ~ 23.524-np thickness 01grinding Whlllll:<2.463eNOAMAL TOOTH FILLET COORDINATES

c

f

.Ii.

1

,-,

ORAWING·HOBBEO GROUND GEAR TEETH NORMAL TOOTHCOOAOINATES NORMAL TOOTH FI LLET COORDINATES

General Data

an - 20fJ ~ 16.2666667X -.2roaO/Mn = ..2Us{mm)m .12lbn/Mn= .018Ksl = .03412Of = 183.068

HarIMn,= 1'Mn = 4Z ~ 46HaO/Mn.= 1.337UO/Mn= 0IbnOIMn = - .042Oal = 201S-aN= 3.023OpfO =185.195Del = 188.38 (theoretical)

GEAH Bof a gear-pair A/8

Center distance:a';;;; 125 10mmL-- ---' lmm

anG- 24RpfG- 92.93

roaGIMnz 0RrGO ~ 91.634

\, \". \"", pIG~.--~----------~

\.\ pro .- - '"',,-',- - - , - - - - - - - - - -1\..""" ".....

\;"'.

\~~ ,"-",- 1

.."'--------l.'-------'

1

Fig. .2- Data. tooth and fillet of a Gear B. ground by 24" pressure angle.

radial position of the grinding wheel is the same as in Fig. lc.Its tip is 0.1 mm from the tooth root, and the fictitious rootcircle generated by grinding has a radius rfGo - 23.524,while the actual root radius obtained by hobbing is rf -23.424 mm.

Theoretical and pra:::tical false contact. The contact limitradius, rd, must be greater than the calculated involute limit,rpfG' to avoid theoretical false contacts. A good margin be-tween the two circles is needed, especially if the grindingmachine has no apparatus for sharpening the grinding wheelperiodically. In fact, the tip of the grinding wheel wears morerapidly than. the internal zone of its profile. Thus, a promi-nent error can arise in the generated involute in the extremeregion towards the tooth root, so that we would get a prac-tical false contact, even if not a theoretical one, as well asnoise and dynamic overloads.

(Tip reliefs of the mating gear may somehow compensatefor the effects of a fillet contact or of an involute error, butthis compensation is unpredictable, and it is not the intendedpurpose of the elastic deformation of the teeth already inmesh.)

In our case, the grinding machine is equipped withautomatic sharpening apparatus and the margin between rdand rpfGis sufficient. We observe that it varies very little ac-cording to the methods and the positions of the grindingwheel. (See Fig. 1.)

Observation.s on the ground fillets of the Pinion A. Grind-ing operators very often bring the grinding wheel very doseto the root of the tooth space for two main reasons. First,if they are conscious of the risk of false contacts, they wishto achieve the maximum margin against it. On the otherhand, if they are not, they prefer that no grinding step betoo visible. Such indiscriminate practice usually removes alarge amount of metal at the tooth root when grinding withthe same pressure angle as in cutting. (See Fig. 1f.) This has

two disadvantages: The carburized layer of case-hardenedgears is reduced, and the grinding wheel needs frequentsharpenings. Furthermore, the grind operation is heavy,which can badly af.fect the precision grade of the teeth. If thereis any vibration, the heavy operation will worsen it.

On the other hand, if we content ourselves with shorterground fillets like those in Figs. Id or Ie, then the grindingstep is not too bad. In our case, it is much worse For hightooth numbers, high helix angles and high addendummodifications. However, such a step seriously impairs gearstrength because it is not very far from the point of max-imum stress of the decisive cut profile. (8) In any case, theplottings enable a compromise choice.

But we have a better option. If we grind the tooth withan increased pressure angle, we obtain a gradual diminutionof the ground metal amount and a lighter operation. We canalso expect improved gear strength because the ground fillethas a smaller curvature and covers the zone of maximum localstress(9) where it is advantageous to have low surface rough-ness .. Both influences are taken into account by the generalISO and DIN ratings.(6,7) See the example of Fig. lc in thecase of Pinion A.

Plotting enables us to optimize the combination of pressureangle and grinding wheel position, as it is not always prac-tical to bring the grinding wheel close to the tooth root.

Gear B. In Fig. 2 we give data and plottings for Gear Bas ground with a 24° pressure angle. The contact limit is outof the field of the plotted fillet. This is usual for the biggergear of a pair. There isa good margin between contact andinvolute limits.

We performed two manu.facturing tests of this gear bygrinding it close to the tooth root with 24° and 20° pressureangles. (See Figs. 3a and 3b.) The fillets are plotted in thetransverse section in Figs. 4a and 4b to compare them withthe photos. Note that the operator brought the grinding wheel

Jonuory!iFebruory 1989 23

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Global Technology with a V.,S. BaseCIRCLE A-12 ON REA!DER REPLYCARD

Fig. 3a.- Teeth of Gear B ground by 24 pressure angle. Fig. 3b- Teeth of Gea:r B ground by 20° pressure angle.

TRANSVERSE TOOTH FILLET COORDINATES TI'IANSVERSE TOOTH FILLET COORDINATES

a\\\

\. \ pIG .. '

\\\ pro . !-\--------------1'. , '

\\ III

"\,~\.\~"'"".-,-.........~-----llmm

Grinding Data

"nO - .24 roaGJMn ~ .0~O ~ 16.7457612 MnG= 4.1144865RIG = 91.634 RIGO = 91.634RptG = 92.93 Rt =91.534LOwer limit 01 ground fillet:RIG= 91.652Tip thickness of grindmg wheel:< 1.736

b1

\ \\ \

\ \

\. \ ptO ,-~~--------------1\. \. pIG !\ \\ \, \

\ \\', .~~".

~ .......lmm -----~

Grinding Data

anG ~ 20 roaG/Mn = 0RfG =91.634 RIGO. 91.634RplG = 92.3.02 Rt - 91.534~Qwer Iimil of ground !illet:RIG= 91.634Tip thickness of grinding wheel:<2.463

.Fig.4- Tooth fillets of Fig. 3 plotted in the transverse section.

a little closer to the tooth root in these areas. Similar obser-vations can be made for Pinion A. (See above.)

Gear Hobbed with an Insulfictent ProtuberanceFig, 5 gives data and plottings of a gear specially designed

for bobbing with a non-standard hob with protuberance andincreased tooth height. The gear works as the driving ele-ment of a speed increaser, which motivates the relatively highaddendum modification.

The grinding was planned for the 0° metnod.f10J Thismethod gives no ground fillets out of the basis circle. The

26 Gear Technology

ground involute should connect with the hobbed fillet withoutany step or with a very slight one. But for the examined gear,case-hardening caused distortions and general size increases,so that the data given in Fig. 5 correspond to the actual situa-tion after neat treatment, and the grinding stock to be re-moved was too large for the protuberance. (See AppendixA) Therefore, it was impossible to avoid grinding steps suchas presented in Fig. 6.

Jn such cases, plotting of hobbed fil1ets helps. us to choosethe grinding limit and minimize steps, as in fig. 6a. Other-

NORMAL TOOTH IFILLET COORDINATEORAWING·HOBBE[)'GROUNO GEAR TEETH NORMAL TOOTH COORDINATES

General Oala

i-IarfMn Kl.2 an .. 20Mn .. 6 (j - 30Z .. ,88 X _ .68433HaOfMn - I .525, roaO/Mn - .35UOfMn - .026 Us(mml- .21IbnOJMn.. - .05 Ibn/Mn - .02Dill .. 618.4 Ksl ... 00312S - aN - 3.383 01 .. 586.615OplO - 591.533 (with protuberance)

'-I., '", '." ..\ I\ !

\ \\\ plO I

- ~,; ~ - - \ I, - - - - - - - - - - - - - -1r

Grinding Oala

anO z 24 roaGIMn .. 0tIG - 30.9514001 MnG- 6.1717298RIG ~ 293.657 RIGO - 293.657RplG = 295.228 Rf .. 293.307Lower umu 01ground fillel:Am .. 293, 736Tip thickness 01grinding wheel.< 1.724 10mm----~

,-,

lmm

Fig. s- Tooth involute and fillet grinding by 24 pressure angle of a gear hobbed with protuberance.

wise, worse steps carroccur, and, in fact, 'they are frequentin industrial practice, even dose to' the tooth root, (See Figs.eb and c.) Of course, this does not invalidate the 10"method;it just indicates that a greater protuberance is needed, butit also raisesthe problem of saving a costly gear.

Common 200 grinding (rigs. 7a and7b) makes the stepless sharp, apparently by creating some ground fillets, but'these are extremely short with a high curvature due to highertooth number, higher addendum modification and higherhelix angle than the teeth in Figs. 1 and 2. A completelysatisfactory connection between cut and ground fillets by 20"grinding can only be obtained if the grinding wheel can haveits tip edge rounded. (See Fig. 7c where a rounding radius,eaG = 0.24, has been adopted.)

A similar good connection and an even better curvatureradius of the ground fatet can be achieved by using a 24"grinding as in Fig, 5 without any rounding of 'the tip edgeof the grinding wheel. (Note that not all grinding machinespermit such pressure angles, but they do not enable tip edgerounding either. In general, in available machines the pressureangle is obtained by means of sharpening rather than by theinclination of the grinding wheel.) InaU cases, plotting isnecessary to optimizeeither the tip edge rounding or thepressure angle of the grinding wheel. as wen as its radialposition.

UndergrindingLet us compare "undergrinding" to "undercutting": a tooth

is underground when the generation of the nllet removes aportion of th generated involute, so that the radius at theinner limit of the ground involute is greater than the baseradius with a step. The computer signals undergrinding inFig. If because the tip of the grinding wheel should be at aradius of rfG - 23.68 mm instead of 23.524 to avoid theproblem.

Deliberate Undergr:inding ola Plnjon ToothLike undercuttin-8,[]lI undergrinding does not always

mean poor design criteria. On the contrary. it can improvereliability against false contacts to such an extent that thetheoretical involute limit can coincide with t.he contact limitwithout drawbacks. There are two reasons for this: The slightstep between the involute and the ground fillet, and the factthat the involute limit is ground, not by the tip edge of 'thegrinding wheel. but by a point a little away from the edge.

This possibility contributes to obtaining higher contactratios, together with low pressure angles and/ or greater toothheights. Theoretical investigations (121 show that SpW' gearswith contact ratios greater than 2 have less tendency to dynamicoverloads, and industrial experience with some of Castellani'sdesigns confirms that gears (both spur and helical) wirh suchcontact ratios can be satisfactorily noiseless. This conclusionis supported by some published industrial l'esults.C1JI

Undergrinding in itself does not diminish gear strength. Thstrength diminution due to a lower module and unlavorabltooth form 'C<1nbe compensated for in part by good curvaturradius and low roughness of the fillet zone. Such compensa-tion requirescareful determination of the fillet features so thatgrinding covers the zone of maximum stress, and 'the trendof the ground fillet.is correct. The choice of the pressure angleand of the grinding wheel position greatly affect the filletfeatur,es.

As an example, let us propose the problem of designinga spur gear pair for 3. low noise level, with the same 'Genterdistance and about the same gear ratio as the gear part A lB.(See Figs. 1 and 2.) We want to employ a readily availableEuropean hob which has a 150 pressure angle.

The main data for the pinion are reported in Fig. 8. Thegear has 98 teeth and a tip diameter of 197.7 mrn, whichrelates to the contact limit of the pinion cI." - 54.092.coin-ciding in turn with the base diameter. FoUowing the choioeof the grinding data with an 18 pressure angle, we obtaina.good ground fillet with a moderate undergrinding. We po i-tion the grinding wheel at a fictitious radius rfGO- 25.485;whereas, [!GO - Tre - 25.722 m:m would avoid undergrind-


",nG= 0RpIG= 294.8

roaGIMn- 0RI .. 293.307


\\\\

\ '.\ \\\\ \ oro

- - - - - -~T' of" ..... - • - - - - - - - ,- - - ...

\ I I\." !\\ IPfG\ j

\ ." I<,

"'---J

a

1mm

Fig.6-0' grinding of the gear in Fig. 5.

",no' = 0RpfG= 294

roaGIMn= 0RI = 293.307


I\\\ \ I. "

b \.'. I\ "-\ \-----\,\--------------i

\\ I,.;:,

\\\\ ~ I'--. "...

'~------i1mm

"nG~ 0RpfG - 293.363

roaGIMn= 0RI ,= .293.307

",nG = 20 roaG/Mn = 0RIG = 295.02 RIGO = 295.02ApfG = 295.1 Af = 293.307Lower limit 01 grQund fi lIet:RIG= 295.02np thickness 01 grlndingl wheel:<4.011

NOAMAL TOOTH FILLET COORDINATES

\\\ \\\

.., \\ \ pia !------\1--------------1

piG \\ .

\\ -, i

"'''--J

a

lmm

Fig. 7-20' grinding of the gear in Fig. 5.

",nG= 20 roaGIMn=ORIGI = 293.407 RIGa = 293.407RpfG = .293.603 Rf = 293.307Lower limit 01 ground fillet:RIG = 293.407Tip thickness 01grinding wheel:<2.837


\ \\ \\. \b ',',\ '\'\ ..., .

....\ I-- ----'\.,,-------------_..:\ J {

'.\ .\~ !'\\ I...:;:.. I\- -. ,

\

'.tmm

ing. The difference is small, but sufficient to ensure that nopractical false contact will arise. The involute limit, rpfG -27.048 mm, is just a little greater than the base radius anddoes not hinder the achievement of a 2.21 contact ratio, asrd and rpfGare practically equal.

SummaryIn the previous paragraphs we have seen typical examples

of fillets generated by both common and less usual pro-cedures. Now we want to classify fillets by some other well-known features.

Influencing parameters. The f.eatures of the fillet as wellas the trend and the amount of metal. to be removed dependon the following:

28 Gear Technology

NOAMAL TOOTH FILLET CooRDINAT'ES

c

- the grinding pressure angle Cl:nC;

- the radius of the tip edge rounding of the grinding wheel,eaC, which is often zero;

- the radial position of the grinding wheel, expressed bythe difference, rpGO - rf, between the root radiusgenerated by grinding, usually fictitious for industrialgears, and the radius generated by cutting, usually real.

As for the influence of the parameters of the gear itself,higher normal pressure angle, higher helix angle, higher toothnumber and higher addendum modification create shorterfillets, both cut and ground. (For OC grinding, see below.)

00 grinding. There is no ground fillet except for undergrind-ing, A proper choice of protuberance ensures a good con-nection between involute and fillet. Sharp grinding steps are

"nG, = 20 roaGIM n = .24RIG -294.605 ArGO = 293.658AprG~ 294.71 AI = 293.307Lower limit of ground lillet:RIG - 293.951Theorelicallip Ihickness or grinding wheel< 3.019'

NORMAL TOOTH FIL'LETCOOADINATES

c\

'. II

\ '\\". \

'. \

'-- \ ". \- . ---\'1- -------------I

\1 I

.\ I

\'. I\\ '

\.

lmm

v-, •

..~

Matrix CNC Thread Grinders ...

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Appropriate to a. multitude of applica-tions, Matrix CNC Thread Grinders areideal for ball screw production, internaland external A. P. I. grinding, worm grin-ding, micrometer and surg,ical screwproduction, g.rinding the tracks in recir-culating ball nuts, and many more.These Matrix machines can handlealmost any aspect of precision grinding.

16001 4150 5150 69001 7000Series Series Series lnlemal Lg. Capacity

Swing over table ways 8" 15"-75" 15"-75" Hi" 15.5"Distance between centers 18" 49" 90" - -

1~MaxImum ground length 8" 41" i- 78" 118" I 161"Maximum jjfound diameter 8" 12.5" 12.5" 10" I 10"Grinding wheel dameter 14 ' - I 20" 1" to 5" I 20"HP of wheelhead drive ! 5 and 7'L~ 10, 20 or 25 '10, 20 or ~ __ 5 10, 20 or 25Helix of wheel, LH & RH 200 1 45° 45D 100 -

24°I

Workhead thru bore 1.5" 3 97" J- 3 97" NlA .I 61.4". ,-'

Internal or external threads 01 virtuallyany form, single or multi-start, right orleft hand, parallel or tapered can beeasily ground with or without relief. 1\only lakes the right Matrix CNC ThreadGrinder for the Job.

Choices of wheel dressing units areavailable in conventional template ordresser arm style, diamond roll, crushroll plus a versatile CNC Dress un.t,

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IMATRIX-CHURCHILL I

CIRCLE .A..13 ON READER REPLY CARD

DRAWING HOBBE[)'GROU ND·GEAR TEETHITOOTH COORDINATES

General Data

HarlMn = l' an .. 15'Mn - 2 I'l .. 0Z .. 28 X~OHaO/Mn .. 1.40749 roaOIMn ... 2UOIMn .. 0 Us(mm) = .082IbnO/Mn .. - .062 Ibn/Mn = .02Dal - 60.4 Ksl = -.15 - aN - 1.557 Of .. 50.849'DpfO - 54.111 (with undercut)Del .. 54 092 (theoretical)

Grinding data with undergrinding(undergri nding limit: RIG .. 25.722)

anG- 18MnG= 2.031269RIG .. 25.485RpfG - 27.048Lower Iimit '01ground flllet;RIG .. 25.485-Tip thickness of grinding wheel:< 1.404

roaG/Mn=ORIGO =25.485RI = 25.425

TOOTH FILLET COORDINATES

\\\ "... ~.\\, ,

'\ \

\\\\ pfO

lmm~"~L.

-.-..:....--.~---!I

Fig. a-Pinion hobbed by 150 and ground by 180 pressure angles with undergrinding.

unavoidable for gears cut without protuberance or with in-sufficient protuberance. (See Fig. 6.)

a"G ,- an, eaG = O. With sufficient protuberance-noground fillet except for ground fillet stretches following in-correct positioning of the grinding wheel.

Without protuberance - ground fillet stretch of widely dif-ferent form and extent and various steps with the cut pro-file. (See Pigs, Id, e, F, 4b, 7a, b.)

O:"G - an,eaG > 0 or a"G > an' !laG -0. Both methodsare applicable for pressure angles obtained by dressing ratherthan by inclination of the grinding wheel. The first one re-quires special dressing facilities. Both methods enable smallersteps a_ndlower .fillet curvature, provided the grindingparameters are well chosen ..(See Fig. 7c for the first case andfigs. lc, 2, 4a, 5 and 8 for the second use.)

anG < an- Some grinding machines do not permit opera-tion beyond a given pressure angle. For instance, let us sup-pose that we cut at 250

, but must grind at 200• Then the op-

posite tendencies must be expected, because of an increaseof the pressure angle: Wider margins between contact andinvolute limits at pinion root, but worse fillet features forgears cut without protuberance.

Ilndergrinding. Undergrinding can occur when using the00 method following incorrect choice or grinding parameters,and in this case it should be considered an anomaly, as itcan accompany pronounced (even if not sharp) steps, badcurvature and excess metaJ removal, See, for instance, the"Pnilradvariante Nr. 2(Y' (8) that showed a loss of nearly70% in fatigue resistance as compared to similar teeth freefrom grinding steps.

If the teeth are ground by a proper choke of the pressureangle, then undergrinding can help achieve a high contactratio. It enables extension of the contact as far as thetheoretical limit of the involute toward the pinion rootwithout any risk of practical falsecontact. This may be essen-tial to achieve noiseless working of the gear pair.

Limitations. When deciding the grinding specifications, twolimitations cannot be disregarded: Theoretical false contact

30 Gear Technology

and inadequate tip thickness of the grinding wheel when thepressure angle is obtained by means of dressing.

It is always necessary to obtain an involute limit radius,rpfG, less than the contact limit radius, rd, to accomplish thefirst condition, A further margin is necessary to guard againstpractical false contacts due to involute errors. or due to thefrequency of grinding wheel dressings, except in undergrind-ing. For sharp and deep grinding steps, it may be necessaryto ascertain that the intersection point of the ground zonewith the cut fillet does not interfere with the tip edge of themating gear ,(10)

The second condition is bound up with the necessity ofgrinding one tooth flank. at a time and is related to thematerial of the grinding wheel and the amount ·of metal tobe removed.

Both conditions are more restrictive for grinding witharounded tip edge of the grinding wheel or by an. increasedpressure angle. On the other hand, tip thickness of the grind-ing wheel usually is not a problem for industrial grindings,that do not affect the root circle. As for the false contacts,this is never a problem for the greater gear of a gear pair.For the pinion, it just. means that proper investigation mustbe made.

Strength and Strength Rating. Fillet features greatly affectfatigue strength and, to lesser extent, static strength of the toothroot. Even for case-hardened gears, Winter and Wirth (8) statethat problems in reducing the case-hardened layer by grindingand in altering the residual compressive stresses are less impor-tant than the geometrical form of the fillet. Then fillet cur-vature, grinding extension and depth, and localization of theintersection point between grinding and cut fillet are decisive.

If a ground fillet covers the zone of maximum stress, thenthe general ISO or DIN rating of the stress correction fac-tors can be provisionally applied by taking into account bothIillet curvature and surface roughness.(6,7)

Of course, specific tests would be welcome in the future,and the comparison between gradually ground fillets andfillets free from grinding as cut by protuberance tools would

be interesting. But in any case, there is no doubt that eithersolution is far better than continuing to create fillets affectedby dangerous steps that are still so frcequent in industrial gear-ing. Such fillets are not likely to be justified in the futurebecause improved computation will enable us to avoid them.

Noise. Fillet features aHect gear noise in two ways:Negatively if false contacts, either theoretical or practical,arise, and positively by ensuring extension of the contact andincrease of the contact ratio when desired by means ofundergrinding.

Manufacturing Conditions Or! Grinding Machines. Forgears cut without protuberance, fiUetsground graduallywi.thout excessive metal removal make operation easier,reduce the risk of vibrations and improve tooth precision.If we compare teeth ground to near the root by differentmethods, the rounding of the tip of the grinding wheel im-proves fillets, but increases costs. But an increased grindingpressure angle (with a sharp tip edge of the grinding wheel)lowers costs of the machining in itself, as less frequentsharpening of the grinding wheel is needed. The cost of thefirst sharpening of the grinding wheel does not practicallyincrease general costs if a constant pI'eSSUlIeangle'C(nG, isadopted for a given kind of gear design, or if it is not impor-tant when proper grinding of a large gear is involved. Cut-ting with wide protuberances lowers grinding costs, but in-creases the cost of the hob cutters, 'especially if specific pro-tuberances are adopted for specific gears.

General Observations. The analyses of some of these ex-

amples may seem to imply that the problem of setting upthe grinding wheel is critical, especially for big case-hardenedgears, which can grow and distort in irregular ways becauseof heat treatment. However, computer analyses enable us toidentify the kind of procedure that affords the widest pro-tection against every possible drawback.

Let us consider the gear in Figs. 5, 6, and 7. The gear wasplanned for a grinding stock of 0.12 mm and was hobbedwith a hob addendum, hao - 1.5 m.; (The value reportedin Fig. 5 is fictitious according to Appendix. A, "Fillet AnalysisAfter Heat Treatment.") Let us suppose that in some toothzone it did not grow and distort at aU, whereas, in some otherzone it grew and distorted much worse than in Fig. 5; thatis, it reached a root diameter of dr - 586.95 mm and re-quired a grinding stock, Us -0.294 mm to be removed. Iiwe then set up the grinding wheel as in Fig. 7b, we wouldobtain the fillets of Fig. 9a, where there is an unground filletstretch anda bad notch at the tooth root, or 9b, where afull grinding of the tooth root is shown. This last conditionis unusual for industrial gears and dangerous in this case,because of the great variation of the grinding stock. But ifwe set up a 240 grinding wheel exactly as in Fig. 5, we ob-tain the fillets of Fig. 10, both more than acceptable. Thusthe plotting after heat treatment enables us to choose the grin-ding parameters more likely to avoid trouble. It must bestressed that the grinding problems of the cited gear were dueto a manufacturing error: a greater protuberance should havebeen adopted, On the other hand, it is well known that

Technical !Ed'u,cation SeminarsYour technology has to be the best. To help you, AGMA hasinitiated a series of seminars - each concentratiiing 'on onedetailed aspect of gear t'echnology" Registration is limited topreserve classroom interaction, so reserve your see: now!

March 7-8, 1989Source Insp,ection of Loose Ge,arsFrom the Customer's Standpoint

Rochester, New YorkBob Smith, R.E. Smith and Co., Inc.

Customers and Quality Assurance personnel areamong those who will benefit from Bob Smith'sseminar on inspection techniques for determin-ing the quality of unassembled gears. Inspectiontechniques will be examined and discussed in de-tail. Following the seminar, a hands-on demon-stratton at the Gleason Works will equip you todetermine if gear quality meets specifications.

May 2, 1989Gea~rMath at the Shop Level for the

Gear Shop lForemanCincinnati, Ohio

Don McViWe, Gear Engineers,. Inc.The math required to set up machine tools forcutting standard and non-standard gears can beeasily [earned by a foreman with shop levelbackground and experience. Some algebra andtrigonometry are used. but no college level mathis required. The participants will learn how toapply the precise calculations necessary to set upthe tools to cut non-standard gears with a stan-dard cutter.

Cost of each seminar is $29'5 for AGMA members and $395 for non-members.Can AGMA Headquarters at (7'03,) 684-0211 for more information on these seminars.

CIRCUE A-l!4 ON R,EADER REPLY CARD

January/,february 1,'989 31

.'\\,~\ a\\\', I\ I- - -- - . --It - - - - . - - - - - - --1

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\ I

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\\ '

\ \ I\\ '

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b

lmm lmm

Fig.9-20 grinding of a gear similar to Fig. 7b: (a) undeformed: (h) badlydeformed after heat treatment,

growth and distortions, especially of the largest case-hardenedgears, are so unpredictable that a costly preliminary heattreatment with the sole aim of ascertaining the trend of a par-ticular gear to deform has been suggested.(l4l If the plannedo grinding is to be maintained for the gear, either a veryample protuberance must be adopted or th risk of a situa-tion similar to that of Fig. 6 arises.

In the authors' opinion, grinding steps like those of Fig. 6should be avoided, although some gears, even with steps likethose in. Fig. 6c, have been known to work for years withoutfailures. But in other gears, breakages did occur. Of course,

\

a i\ !\ I-- ----\1-----· --------1

\\ II

" i~ .~lmm

F:ig. 10- 24 grinding of" gear similar 10 Fig. 5· (a) und formed; (h) badlydeformed after heat treatment.

higher safety factors would be needed to avoid such failures,but, even then, the strength is aleatory because, besides thebad notch, the grinding may burn the tooth root and causecrackings,

If grinding facilities and computer analyses do enable usto obtain a regular fillet, then even a full grinding of the toothroot may be obtained, like those usually adopted in specialfields to improve reliability, as in the case of certain gearsground for use in helicopters. Nevertheless, this increasescosts, and we do not think it feasible for common industrialpractice.

,8511 OHIO PIKE,. CINCINNATI. OHIIO 45245, .• 1(5131 752-'60001

32 Gear Tecrmology CIRCLE A-IS ON READER REPLY CARO

TRUE DIMENSI'ON GEAR .IN:S'PECTIO'N

Provides actual overball/pin measure-

ment of anyhelical or

spur gear orspline with-

out the needof costly

setting masters.

Provides vitalS.P.C. information ..

CAPACITY:9'" O.D.8" I.D.

ConclusionsMathematical methods are supplied in Appendix A to com-

pute cut or ground fillets of both spur and helical gears.Typical fillet features resulting from different methods ofgenerating grindings and from various choices of the influ-encing parameters are examined. A final classification ismade.

Special attention is given to an uncommon grindingmethod, namely, grinding at a pressure anglegreater thanthat of the cutting tool, which permits fillet features andlighter manufacturing without protuberance tools or specialsharpening apparatus for the grinding wheels.

False contacts of the operating gear pairs at the pinion toothroot are either "theoretical" or "practical" ones.

"Undergrinding" is defined and its contributions to themanufacture of noiseless gears by excluding false contacts andincreasing contact ratios is discussed,

A proper choice of grinding method and of grindingparameters enables one to obtain gradual fillets covering thezone of maximum stress with low fillet curvature and lowsurface roughness, thus improving the fatigue strength of thetooth root and making the general strength ratings applicableto ground teeth while awaiting specific tests.

References

r. CASTELLANI, G., and PARENTI CASTELU, V. "Rating GearStrength", I. of Mech. Design, Trans. ASME. Vol. 103, April, 1981.pp. 516-527.

2. SAlAMOUN, C" and SUCHY, M. "Computation of Helical and SpurFillets", Proc. of the Int. Symp. ASMEI AGMAIlFToMM, San Fran-cisco, Oct" 1972, Paper RM4.

3. CASTELLANI. G.. "Calcolo Della Spessore di Controllo per Den-tature Esteme a Evolvente", Rassegna lntemazionale di Meccrmica,n.S/6, Maggio-Giugno, 1968.

4. HENRIOT, G. Trait« Theoriqu.e et Pratique des Engrenages, TomeII, Paris, Dunod. 1961, and other edit.

S. "AGMA Standard for Rating The Pitting Resistance and BendingStrength of Spur and Helical Involute Gear Teeth". AGMA 218.ot.Dec., 1982.

6. "Deuxieme avant-projet ISO/DP 6336 I ill. Calcul des Tensions au Piedde Dent", ISO/TC60 424F, 1981,

7.. "Crundlagen fur die Tragfahigkeitsberechnung von Gerad- undSchragstlrnradern, Berechnung der Zahnfussrragfahigkelt". EntwurfDIN 3990, Teil 3, Marz, 1980.

8. WINTER. H .. and WIRTH. X. "Einfluss von Schleifkerben auf dieZahnfussdauertragfahigkelt oberflachengeharteter Zahnrader", An-lriebstechnik 17 (1978) NT. 1-2, pp. 37-41.

9. WINTER. H .• and HIRT. M. "The Measurement of Actual Strainsat Gear Teeth, Influence of Fillet Radius on Stresses and ToothStrength",lr. of Eng. for industry, Trans, ASME, Feb. 1974, 'pp. 33-40.

10. "Underlying Theory for the Maag O· Grindlng Method," MaagZahnrader A.G" Zurich. 956.801100.01.

11, COLBOURNE. J.R. "The Design of Undercut Involute Ge rs", Proc.of the Int'l, Power Transm, &; Gearing Cent .• San Francisco, 1980,ASME Paper No. 8O-CUDET -e.

12. KASUBA. R.. and EVANS, J.W. "An Extended Model for Determin-ing Dynamic Loads in Spur Gearing". Jr. of Mech. Design, Trans,ASME. Vol. 103. Apr, 1981. pp. 398-409.

13. MURRELL, P.W. "Developments in High Powered Marine EpicyclicGearing, 3{),00Q S.H.P. - Prototype Trials Results", Proc. of the Int.Symp. on Gearing and Power Transm .• Tokyo, 1981.

14. BlDCH, P., "Deformazloni da Trattamento Termico", Proc. of the3a Giornata dell'Ingranaggio, Milan, 1978·.

15, CASTELLANI, G., "IJ Calcolo Delle Ruote Dentate", E<lit. Rivistadi Meccanica, Milan, 1962.

Appen~ix A - Fillet Coordinates

NomenclatureThis nomenclature is an addition to the general one.

haG addendum of the grinding wheelactual gear centerreference radiusbase radiusradius at a tooth chord

reference arc thickness in the transverse sectiontransverse chordal thicknesstooth thickness normal to local helices (ellipsechord)coordinates of the center of the tip edge radiusof the generating rackcoordinates of a point of the tip edge arc of thegenerating rack

0r

rbry;;

StSty

SyN

u. w

, ,u, w

XgO

Xg

Or:t

{3b{;

actual coefficient of the addendum modificationat cuttingactual coefficient of the final addendummodification, here for ground gearstransverse reference pressure anglebase helix angleangle between a fillet tangent and the tooth axisin the transverse section, or in general 0 - VI

-11-rotation angleangle between the radius vector of a fillet pointand the generating line of tool or grinding wheel

SubscriptsG referring to grindingn normal to generating rackt transversey referring to a generic cylinder

Tooth data referred to gn'nding, A given pressure angle,ClnG *- Cln, can be adopted when grinding, provided that thebasic geometric parameters of the tooth flank, rb and {3b're-main the same and that we obtain the desired tooth thickness.The reference tooth parameters become fG, {3G,mne, O:tGa~dthe usual gear formulae apply. As in Reference 15

(1)coso, / cos{3 = cosa,/ cos{3b

We have similarly

cosatG/cos{3G = C050:nC/cosi3b (2)and we deduce that the normal module relating to grinding.depends solely on the given pressure angle:

As for the helix angle itself, from Reference (15) we deduce

(3)

January !iFebruarv 1989 33

nortlln~I_~_Involyll!'

Fig. n - Thickne s "reductions" in the transverse section.

for a generic cylinder:

sinl3b = sinl3y msany

Hence,I3G - arcsin (sinl3 cOSQ'n/c05andfG = Z mnG1(2 cos{3d

(4)

(5)(6)

Coefficients of the addendum modification. The so-called "ad-dendum modification" is the distance between reference andgenerating lines of the generating rackand is positive if thereference line is external with respect to the generating line.This may seem a,simple concept, but when enteringthedetai!sof cut and ground profiles we must distinguish not less thanfour values of the addendum modification (none of whichis the true modification of the addendum, except for particularcases).

The nominal addendum modifications x mn refer tonominal gear pairs without tooth backlash; i.e., to the dot-ted involute in Fig. 11.

(Note that it is just a convention to refer the addendummodification coefficient to the normal module; thus, we canmaintain it when considering the transverse gear section ..)

A reduction ibt of the transverse base thickness is usuallyadopted to contribute to the tooth backlash for the operatinggear pair. Then the final generating addendum modification is

Xg mn = x m, - ibl/{2 sino,} (7)

Let us assume for the moment that 0nG ... an' If the grind-ing wheel is inclined by f3, the tangents to the base cylinder,normal to the tooth flank, are inclined by f3b with respectto the gear axis. Then,

(8)

and

X-g - )( - (ibn/mn)/(2 sinOn)'

as in Reference (15).

(9)

34 'Gear Technology

sinol - sinOnIcos{3b (10)

Tooth cutti.ng must leave a grinding stock

1.1, - (ibn - ibnO)/2 (11)

which defines the value of ibnO' usually negative, becauseit is defined as a "reduction" of normal base thickness forpurposes of generalization. Thus a formula similar toEquation 9 applies for a coefficient relating to cutting:

)(gO - x - (ibnO/mn) 1{2 sinOn) (12)

If we grind the teeth by a pressure angle One '* an,the totaladdendum modification is )(gG mnG '* x mnas it refers tofe '* r. The condition that the transverse reference tooththickness remains the same can be applied to calculate xgG.Since Ste - 51'

(13)

andXgG- (s, cosl3G/mnG - 7i/2)/(2 tanord (14)

Thus, the formulae regarding addendum modifications alsoextend to the grinding conditions.

Note that the tip diameter remains dependent on thenominal x:

d~1 - m, z/cos{3 + 2 mn (har/mn + x - ksl) (15)

where da] is meant for the contact limit towards the toothtip that may be aHected by "sernitopping." and da.1 issomewhat arbitrary because it depends on our choice of atooth shortening coefficient, k.1'

Positioning of the grinding wheel. Unlike the tip radius, theroot radius depends on the actual generating conditions.Tooth roots usually are not completely ground in industrialgears so that -

(16)

whereas, the tip of the grinding wheel generates a "root cir-de" entirely or partly fictitious:

rrcc - mnG z/(2 cos{Jd - haG + XgG mnG (17)

The addendum of the grinding wheeL h.G, is defined simi-larly to the tool addendum, hao, if the grinding wheeloperates on both flanks at the same time. More often it grindsone flank ata time. Then we assume freo directly by posi-tioning the grinding wheel at a. distance, rrco - rr, from thetooth root, Then haG becomes, fictitious and is defined in-versely by Equation 17.

Generation of the tooth fillets. The coordinates of point Sin the normal section of a hob or of a rack cutter, Fig. 12,are given by Equation 31 and Equation 17 of Reference (1),

By using the present symbols, with m" '* 1:

l~m. '1:/2- a U(J

Un - __ ,_n+ hao tanan. + 0.0 tan n - --4 2 (OSCin

w -h..o - "so mn -0..0

(18)

(19)

Let F be a contact point of the fillet with the generating rack.Its position is defined in the transverse section of the generatedgear by the angle "t that we assume as an independentparameter, (Fig. 13.) The fillet field is defined by

"t lim ~ lit ~ 7r12 (20)

where 7[/2 means the limit point between root circle and fillet.and v, lim corresponds to the lim.it between fillet and in-volute. For non-undercut teeth, II! lim -au whereas forundercut ones, the intersecting between fillet and involutemust be determined by iteration. If the tool has protuberance,the tooth is always undercut. Otherwise undercut arises ifXgO is less than the limit value.

z sin2at

2 cos(3 (21)XgO lim

Such a condition assumes that the limit point of the straightprofile of the rack generates the involute, beginning exactlyfrom the base circle.

An angle such that

1lin - arctan (tanJ.lt rasp) ~ "I < 7r12 (22)lin - 7r/2 if lit -7f/2

corresponds to lit in the normal section of the rack inFig. 12, and defines the coordinates of point F in the axis setof the rack.

1u'n - Un -l)aO cos», (23)w' - w +liaO sinvn (24)

Now we come back to the transverse section of both gearand generating rack, inclined by (3 with respect to the nor-mal section of the rack, and we have

(25)

In Fig. 13, U'I - JK, Geo - JCo and we obtain the ro-tation angle of the tooth

p: - (u't + w' cotllt)1r

By assuming

o - "I - P:

(26)

(27)

the coordinates of the point. F, in the axis set of the geartooth,are

- r sinp: - ~ cos/)sim'l

w' . ~- r cosj.! - -.-- SInu. Sin!'t

(28)

(29)

In the case of a grinding wheel (Fig. 14) the coordinatesof the point S are

'1:12 - O!'nG~ UnG - ifIJInC/4. +hac tananG + eaG tan 2 (30)

~We - haG - Xgc moc - eaG (31)

(continued 011 page 47)

n:mn-,,-

w u~

Fig. 12-D<l13 of the generating point F on the rack in the normal section,

Fig. 13 - Calculation of the coordinates of a Point F of the tooth fille! ina transverse section.

[xIIG>Ol

gener. line , I h.aC)1

UnG

Fig. 14 - Data of a rack-like grinding wheel.


BACK TO BASICS ...'Spur Gear Fundamentals

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

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

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

36 Sear Technology

UHe HaidhedeBlack Hawk College

Moline, n

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

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

function and DesignProbably the earliest method of trans-

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

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

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

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

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

Law of GearingGears are provided with teeth. shaped so

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

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

AlITHOR:

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

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

$trl1.gh~ he! ical

fbJ IntorMC1.ng ""fts.

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

Fig. J -Important types of gears.

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

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

a = addendumb = dedendumB = backlash, linear measure

along pitch circleC = center distance

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

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

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

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

gears

NomenclatureM = measurement over pinsm = module

mf = face contact ratiomG = gear or speed ratio

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

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

N, = critical number of teeth for noundercutting

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

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

helical gearp.. = normal circular pitch of

helical gear

R = pitch radius gearr = pitch radius pinion

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

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

Se = endurance limittc = circular tooth thickness

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

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

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

~11 = pressure angle in normalplane

'" = helix angle

January jFebruary 1989 37'

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

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

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

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

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

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

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

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

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

Ihi Rolling beam met,",odHbeed base errelel

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

Fig, 5 - Generation of the involute.

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

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

38 Gear Technologv

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

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

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

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

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

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

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

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

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

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

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

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

rb Rbcos 0 = --;= Ii

where

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

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

Also,

db Dbcos~ = d = 0

where

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

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

(1)

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

Pitch circle diameter ~ d

P,lch Circle drameter lJ

CenterD,stance

(2)

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

Fig. 8-Curved involutes in contact.

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

70 to 50° (still large by normalstandards).

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

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

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

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

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

.Fig.10- Effect of changing center distance.

JanuaryjFebru.ary 1989 39

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

III\\\\,

\,"""

(u)

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

PilCh (nell!

'hi

tOOl" spacing onbulh PIU!; cv-ves

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

40 GearTechnol'ogy

19'

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

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

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

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

Gear TerminologyBasic Terminology. To make further

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

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

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

C = 0.5(D + d) (3)

where

D = pitch diameter of geard = pitch diameter of pinion

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

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

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

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

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

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

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

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

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

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

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

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

whe,,;e

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

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

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

BOl1om ItndTop ~lrld

FUIet nlCh'!A

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

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

.fig. 14- Tooth parts ofspur gears.

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

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

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

Fig. IS -Measurement over pins.

because they involve whole numbersonly.

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

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

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

Dedendum. Depth of tooth below the


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

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

Base pitch

j

I'-'-'rI

/1,,/ r

Pinion

"\,\,

I "I\\\

\II

II

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

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

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

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

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

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

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

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

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

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

Involute Gear TeethSo far we have considered only two

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

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

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

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

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

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

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

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

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

Standard Spur 'GearsSpur gears can be made with greater

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

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

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

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

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

p)J = 1fD

7lDPc = N

Module: D mmm = N tooth (definition)

Diametral pitch:N teeth

Pd = D· -.- (definition) (8)m.

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

Pb = Pc cos cp

Diametral pitch is related to the module asfollows.

mPd = 25.4 (10)

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

p~ = 1frn (ll)

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

gears must. have the same diametral pitch.

(12)

(5)

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

(6)

Standard Tooth Pmpod:iofl5of Spur Gears

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

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

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

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

(7)

(9)

.' AGMA has yet to complete its SIstandards.

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

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

Janucry/Februcry 1989' 43

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

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

~iJr;». T

I

Fig.1S-Contact ratio, mI"

Lccahzed ~COr!"9

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

44 'Gear techno'iogy

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

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

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

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

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

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

TABlEl National Pitch SystemCoarse Pitch

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

Fine Pitch

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

Stub

TABLE 2 Involute Gear Tooth Dimensions Based on Pitch-Coarse

Full Depth

Pressure angle, 0 degAddendum, CI

Dedendurn, bTooth thickness, te(theoret ical)

200.801P1.00/ P7r/2P

Full Depth Stub

20liP1.2SIP7rI2P

2S0.801P1.00IP7rI2P

2SliP1.251 P1I'12P

TABLE 3 Preferred Values for Module m (mrn)

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

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

Stub Full_Depth Stub Full Depth

Pressure angle, 0'"Addendum, aDedendum, b

20 20 25 25

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

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

1l'm 1rn1 7r17l 1rn1

I '

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

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

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

(Fig. 13-18).

Zmp = Pb

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

Pc cos cp(13)

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

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

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

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

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

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

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

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

Interference can be avoided if:

Ro ~ ~ Ri + C2 sin2 cp (15)

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

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

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

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

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

Modifications of Spur Gears(Nonstandard)

The teeth of a pinion will always be

(continued Or! page 48)

(14)

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

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


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fILLET GEOMETRY or GROUND ...(continued from page 35)

and the calculation proceeds based onformulae similar to E-quatjons 20 to 29except that"'t lim -atG for non-underground teeth, and the parametersfJ. O:t, haD' 12.0.O:n, mn, XgO' Un. wand rare replaced by fJG. O:IG. haC' ,QaG. ,00nC.

mnG' XgG, UnG, Wc and rc respectively.Fillet .analysls after heattICeatment. Let usconsider a gear which is deformedbecause of heat treatment after cutting,The x value of the addendum rnodifica-tion coefficient does not change, as it is"nominal," and the prescribed tina Ithickness must be attained, so the ibn

value of the normal base thickness reduc-tion has to be maintained,

The following procedure can beadopted:

'. Measure the actual rootdiameter df. hence Tf

• Assess the necessary grindingstock Us

.' Imagine that the gear has beencut by a fictitious tool: rate ibnOfrom Equation 11 inversely,and XgO from Equation 12 anddeduce a fictitious tool adden-dum, hao, from Equation 16.

The general computer program can beapplied to analyze fillets.

If the gear deformations have been ir-regular, the analyses must be repeated forvarious tooth zones by maintaining thesame positioning of the grinding wheel;i.e., the same value of rrco.

ACKNOWUDGfMENT: This article was originallyprinted by the American Society of MechanicalEngineers ,as their Paper No, MDET-181.

Janu.ory/FebruOlY 1989 47

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

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

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

\\\

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

Fig. 23- tong and short addenda.

48 Gear Technology

Fig, 24 - Backlash,

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

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

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

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

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

These modifications will improveprimarily the kinematics of spur gears.

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

IJ.

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