Th,e Ioumalot Gear' Manufacturin',gSEPTEMBER/O'CTOBER 198B

Accurate gear inspection depends on retlat8ddataevaluation. Profit depends on speed. Maag CNC ~1 •• '"

three, in the lab and on the shop floor.• Maag mechanical machines 'combine the speed and automatic checldngthe reliability of a base circle disc design for profile and lead gear inspection.• Onlly Maag multi-axis electronic machines measure internal and external spur, helicaland cluster gears with up to 15 sets of teeth, all in a single setup, with accuracy of.000040 in.• Maag's unique easy-to-use menu programming and software automatically bring thestylus to the moth surface.If you'd like to know how to spot your bad apples quicker, contact American Pfauter,92,5Estes Ave., Elk 'Grove v.mage, IL 60007. Phone (312) 640-7500.

11111I !I.f" A .• 11-_..1•• ..:.,:.,uou,n,· menea. r;nNllnnf'iii

MIKRON brings you hob speeds upto2,000 rpm and high rigidity with theA33/2 gear hObblng machine

The model A33/2 is a completelyautomatic gear hobbing machinewith programmable controller. Usersrealize the full advantages of modemTiN coated hobs while hobbing gearsof from 0.151" to 4" (4mm -100mm)outside diameter.The Mikron A33/2 is step-by-stepexpandable and is available in bothautomatic and manual load models.It can be fitted with an automaticdeburring device and is also suitedfor skiving with tungsten carbidehobs because of the high rigidityachieved by its horizontal design.

write or phone for complete details on this flexible,expandable system and other Mikron hobblng.machines for gears up to 8" diameter.

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EDI'I'OIUAL STAPPPUBUSHER •EDITOR·IN.QOEFMichael GoldsteinASSOCIATE PUBLISHER •MANAGING EDITORPeg ShortASSOCIATE mrroRNancy BartelsART DIRECTORKimberly ZarleyPROOUCDONIADVER11SINGPatricia FlamSALES co-oRDINATORJan LaBargemrroRJAL ASSISTANTSNancy ConnMary MichelsonCandace Rose

IIANDAU. PIJBLISIIING STAFFPRESIDENTMichael GoldsteinVICE PRESIDENTRichard GoldsteinVICE PRESlDENTIGEN. MeR.Peg ShortART CONSULTANTMarsha Goldstein

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OUR COVERGalileo's Escapement. This model, built

110m a drawirW of GaJileo 's. ~ constructedin FIorenre. Italy, in 1883 and is now ondisplay in the Science MusetII1I. London.Probably desiItned by GaJiIeo in the 1610's.the device is one of the earliest to me a pen-dtMum to ~ a docJnr;ork. It is an ap-plication of the principle. discovered byCaMeo. thai the period cf osdlation cfa pen-d&bn is the Si1I11e ~ cfthe size of thean: made by its ~ motion. It consistsrfan esGlJ:Ie MbeeJ, pinion. pallets with 01.tchand the perxhium and its.suspension. (Photocourtesy of Sdence Museum, London,EtvJIand.)

CONTENTS PAGE NO.

'CROWNED SPUR GEARS: OPTIMAL GEOMErRY AND GENERATION 9Faydor L. Litvin, Jiao Zhang, Wei-Shing Chaing,University of Illinois at Chicagp, Chicago, lLJohn 1. Coy, Robert F. Handschuh, NASA Lewis Research Center,Cleveland, OH

CALCULATION OF OPTIM~ TOOTH FLANK CORRECTIONSfOR HELICAL GEARS 1.6

Manfred Week. Georg Mauer, Laboratory for Machine Toolsand hldusttial Management, Technical University, Aachen,West Gennany

ENHANCEU PRODUCT PERFORMANCE THROUGH eBN 'GRINDING 23Glenn Johnson and Ernest Rattennan,GE Superabrasives, Worthington, OH

EDITORIAl

TECHNICAL CALENDAR

IMTS ADV.ERTISER INDEX 8

BAiCK TO BASICS •..1NVOI.UTOMETRY

Harlan W. Van Cerpenand C. Kent ReeceVan Cerpen-Reeee Engineering, Cedar Falls, LA

34

CLASSIFIED 46,

September/October. 1988

withfor one-

time entry of partp!int and tolerancedata. 'Mouse" per-mits use of CADtE!(;hniques.

Our Model 3000 ac Gear .Ana:ryzeris a third generation CNCgear inspection system in-corporating all of the comprehensive analytical tests and evaluation capabiilities ofprevious M & M systems, such as our Model 2000, but with these added capabilities:.' iDramat,ically improved speed and accuracy through new mechanical system design

and advanced CNC control technology.• Computer hardware and applications software are modular to allow the user to buy

onl,y the required, capability. This makes the 3000 ac adaptable to laboratory test,ingor producfien-llne inspection.

• l:ntegrated Statistical, Process Control with local data base capability is an optlonalfeature. .

• Networking with MAPS compatibililty is available.• IRobotic interfacing for totally automatic :Ioadltestlunload operation can be

incorporated.' .For more information or applications assistance, write or call: M & M Precision Systems,300 Progress IRd.,West Carrollton, OH 45449,513/859·8273, TWX 810/450·2,626,FAX 513/859·445·2.

GraphiCS.- rlntercopies ClfT.

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THE 500 LBGORILLA IN THE WINDOW OF OPPORTUNITY

rMTS ISback. in [own, From Sept. 7 through Sept J 5. thelargest Industrial exhibrtron In the Western Hemisphere will f,Uone of the largest exhIbItion centers in the world, A show of thismagnitude is a little like the 500 lb. gorilla In your dining room -hard to ignore.

But like the gonlla in the dining room. IMTS raises certainambrvrlenr feelings, True. I['S big and Important but what does Itmean for our industry 7 The rest of the machine [001 businessrends [0 dwarf the gearing Industry, and the ternpranon to Sit theshow out is strong. After all. there are other trade shows andconferences that relate more directly to our Interests, BeSidesItsbigness, what does IMTS have to offer the gear manufacturer,engineer or designer7

A lot. J .078 companies 11\1111 dIsplay the very latest machinetool products - including gear cutting and finishing equipment- from 27 nanons. There IS no better place to see the directionmanuFactlJnng will take In the coming decades. Running concur-rently with the show is the Technology Conference, wheresome 200 industry experts from around the world Will coverevery aspect of the lalest techniques In manufacturing in 48separate half-day sessions. One whole session is devoted tomodem gear making methods. and at least a half dozen othergear-relared topes Will be covered. That's a lot of informationand opportunity all in one place at one orne - information andopporturrty we should nor Ignore.

America's industry has been through some tough times over[he last decade. We have suffered from a slightly dulled com-petmve edge and. more critical In my opinion, a doJlarlforelgn-currency imbalance that no amount of cornpennve sharpnesscould overcome Now things have changed, Currency balancesare much more favorable than they have been in years. and theharsh weeding out process has left Amencan Industry leaner.meaner and capable of faCing foreign competition Suddenly ourproducts are In demand again. Exp0r!3 are growing and our rae-tones are busy. We have been able to come up for air In ourstruggle to survive the comperl[Jon.

Bur we cannot be lulled Into complacency. This change forthe better ISa window of opportunity. nor a permanent condi-tion. Exchange rates and economic Indicators are never fixedvalues. Politics. weather. scientific advances, an work to makethe situation fluid, We have to make the oesr use of [he goodtimes while we have them. Not only must we repair financialdamage, but also reinvest some of our higher profits In our com-panies, in machines. people. education, systems and Informa-tion, In the years ahead. we should expect to face larger, betterfinanced and even more stra[eglcally oriented competition. Wemust take advantage of trus present opportunity to makeourselves ready.

lMTS is a good place to begin. Even If you're nor In the marketfor machinery today, trns isme place to seed what's out there.what the competlrion IS manufactUring. buying and will be usingin the future ..A show like IMTS or the Gear Expo In Pittsburghnext year is the place to survey all the manufaaurjng options.

processes and rechnqoes available to lower your costs andimproveyour quality,

In a chess game. the winner ISthe player who can plan themost moves ahead, The same is true In the highly competitive.global manufacturrng world In which we operate todayEducation, Information and strategic planning are crucial toeconormc survival.

Don't let opportunities like IMTS slip by, Windows cI oppor-tunity can close as quickly as (hey open. and the companies thathave not used their open time Wisely Will be shut out In the cold

Sep'emberjOc'ober 1988 5

-- -- - -

TECHNIC.LU C.LU4END1\.RSEPTEMBER27-29. American Scxlecyfor Metals 11th Annual' H'eat l\rea:tlngConference, McCormick Place, Chicago,ll. Presentations on suqjects including heattreating, statistical process control. newenergy applications, quenching and cool-ing improvements. For furtherinformadon.contact: ASM International, Metals Park,OH44073.(216) 338-5151.

OCTOBER 10-12...AGMA Fall Technl-

cal Meeting'. Falfmon,t Ho,tel', N'ewOrl'eans, LA. Presentations from U.S.and around the world on gear materialsand heat treatment design techniques.applications technology and the latesttechnological developments. Contact:AGMA. 1500 King Street. Suite 201, Alex-andria, VA 22314. 1703J 684..Q211.

NOVEMBER 1'-3. SME,Gear .Processlngand Manufacturing ClinIc, Sheraton

AVAIILA'BILITY, AFFORDABIUTY, DEPENDABILITY,THREE REASONS WHYAiEROSPACE SHOU LD BEYOUR DEALER!We specialize in gear cutting tools and gear machine accessories 01 manytypes inc.luding involute form CBN grinding wheels-hobs-dresser arms forGleeson" grinders-cuNers for Gleason 104" -Gleason 114" and Revex"machines. Call us today with your gear tooling needs. CALL 312-323-0737

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AEROSPACE, A. C. INC.m-.. ~UI!II,ca..:.1'4n .... ItIIdInIla,IItQuaW::lr'mEM!!I~~"&:i!IIIiIII!!iIIII!!I~~ G!IIIoII~WiE;i;ii!i!l~""~n"""""~~·''''''--''.ielPL

CiRelli A-4 ON READER REPlV CARD

,6 Gear Technology

MeridJan,.lndlanapolls, IN. The technical'conference will include papers on "Designand Selection of Hobs.' "Selection of High-Speed Steel for Gear Cutting Tools","Special Design Deburring EqUipment"and other gear related subjects. Attendeeswill have an opportunity to meet withpresenters to discuss papers and ask ques-tions on a one-on-one basis. Tuesdayeven-ing will feature a reception and tabletop ex-hibits from major manufacturers. For furtherinformation, call Dominic Ahearn at SMEHeadquarters. (313) 271-l 5()1Ox384.

NOVEMBER 5-10'. International ,Con-ference on Gearing. Zhengzhou',China'. ASME-GRI and several interna-tional gear organizations are sponsoringthis meeting. For more information contactInter-Gear '88 Secretariat. ZhengzhouResearch Instittlte of Mechanical Engineer-ing, Zhongyuan ,Rd.Zhengzhou. Henan.China. Tel: 47102. Cable 3()()(). Telex46033 HSTEC CN.

INOVEMBER8:.10. American Societyfor Metals Near Net Shape Manufac-turlng'Conferenc,e,. Hyatt Regency, Col-umbus. OH. Program will cover precisioncasting, powder metallurgy, design of diesand molds. forging technology and inspec-tion of precision parts. For further informa-tion contact: Technical DepartmentMarketing, ASM International, Metals Park.OH 44073.

CALL FOR PAPERS - TennesseeTechnological University for its list Inter-nat'l Applied Mechanical Sy.stemsDesign Conference', Mallch -19'-22.1989, ,N'ashvllJe,TNI.Papers are invitedon general mechanical systems sub.jectsin-cluding strength, fatigue life. kinematics,vibration, robotics, CAD/CAM andtribology. Deadline for first drafts is Oct. 1.1988. For further information, contact: Dr.Cemil Bagci. Dept. of Mech. Eng.. nu, .Cookeville. TN 38505. [61 S) 372-3265.

CALLFOR .PAPERSfor ASME 5th An-nual P·ower Tran,smlsslon & 'GeatJlngConference. April 25-27, 1989',Chicago,. !Il.Papers are Invited on emerg-ing technologies for gears. couplings. belts,chains and other power transmissiondevices - gear geometry. noise, manufac-ture. inspection, scoring. lubrication,materials, applications, efficiency.dynamics. For more .information, contactDonald Borden. PO. Box 502. Elm Grove.WJ53122.

ADVERTISERS' INDEXBooth #6263

• AMERICAN PFAUTER925 E. .EstesAve.Elk Grove, IL 6CXXJ7(312) 640-7500

Booth #6316• HAMMOND MACHINERY

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Booth # 6691.ielMA U.S.A.50 r Southlake Blvd.Richmond, VA 23236(804) 794-9764

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Booth #6673• M & M PRECISION

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8 Gear technology

Crowned Spur 'Gears:Optimal Geometry and Generation

F-' d L Litvin-' J" Zh W· . S"L!-(h'" U" "ty . f· Olin" , Chi'ay .or" . , _laO ang,· .eJ- -,rung ,·amg, ,mvers! ,'.- 0'- ,_-.015,,-.- ca.goJohn J. Coy, Robert ,F. Handschuh, NASA Lev.ris Researdll Genter, Cleveland, OH

Abstra,ct:

The authors have developed a method to. synthesize the pinioncrowned surface that provides a localized bearing contact and afavorable type of transmission error for misaligned gears. A methodfor generation of the pinion crowned surface bya surface of revolu-tion (it slightlydeviates from a regular cone surface) is proposed.Tooth Contact Analysis (TCA) programs for simulation of meshingand bearing oontact for misaligned spur gears with the crowned pin-ion have been developed. A computer graphic program for· thedisplay of the pinion crowned tooth surface in 3·D space has alsobeen developed.

IntroductionInvolute spur gears are very sensitive to gear misalignment

Misalignment will cause the shiH of the bearing contacttoward the edge of the gear tooth surfaces and transmissionerrors that increase gear noise. Many efforts have been madet,o improve the bearing contact of misaligned spur gears bycrowning the pinion tooth surface, WiJdhaber(1) has pro-posed various methods of crowning that can be achieved inthe process of gear generation. Maag engineers have usedcrowning for making longitudinal corrections [Fig.1a); modi-fying involute tooth profile unifonnly across the face width(Fig. Ib); combining 'these two functions in fig. Ie and per-fonning topological modification (Fig. Id) that can prov:ideany deviation of the crowned tooth surface from a regularinvolute surface. (2)

The main purpose of these methods for cr-owningis to im-prove the bearing contact of the misaligned gears, which ad-

a) b)

c) d)

Fig. 1

dresses only half the problem. The transmission errors ofmisaligned spur gears are a main source of gear noise ..Ac-cording to the open literature, 'the influence of gear misalign-ment on transmission error has not been inv,estigated.Maag'smethod. of topological modification does not determine therelationship between the surface deviations and the transmis-sion errors, Also, the optimal geometry for the pi.njoncrowned surface has not been proposed.

The contents of this article cover the solutions to the

AUTHORS:

DR. FAYDOR L. lITVIN.is Professor of Mechanical Engineeringat the University of Illinois at Chicago. He is the author of severa.1books and many papers on gears and holds several patents ..Dr. Litvinhas served as a consultant to a number of industrial corpor,UioTl5.He is chairman of the ASME subcommittee on Gear Geomefry anda member of the ASME Pouer Transmission and Gearing Commit-tee. He has won a number of professioMI awards, including threeNASA Tech-Brief ,awards for published papers.

MO .ZHANC is a doctoral candidate in the Mechanical Engineer-ing Department at the University of Illinois ,at Chicago. He is cur-rently researching computer modelling and simulation for gearingtheorems and design.

WEI~SHINC CHAING is a doctoral cllndidate at the Universityof Illinois at Chicago in the Mechanical Engineering Department. Hiscurrent field of research is computer"aided simulation and design forgearing.

DR.. JOHN ,. COY is Branch Chie!, Rotosystems Technol~gy, Pro-pulsion Systems Division, NASA Lewis Research Center, C1eue.1Dnd,OH. He has been~esearching helicopt.er d.rive trains since 1974. Dr ..CO'lIearned.his undergradl.lllte and grad'lJQtedegT'ees from the' U'niver-sityof Cincinnati. He has served on the faculty of the Rose' HillmanInstitute of T,echnology and the Ul'liveTSity of Cincinnati. He haspublished over 65 technical papers and is ,a consultant to ,industryand other government ,agencies..Dr. Coy is a member of A SME and.a licensed professional engineer in the State of Ohio,.

MR. ROBERT F. HANDSCHUH works for the u.s. ArmyAVSCOM Propulsion Directorate, NASA Lewis Research Center,Clevela.nd, OH, in the area of gear geometry, gear manufacturingand drive train performance. He did ,undergraduate work inmechanl'cal engineering at Cleveland State University and enmed' 1m

MS in mechanical engineering from the University of Toledo. Mr.Handschuhis a member of ASME and the National Society of Pro-fessional Engineers.

September/October 19,88 9

_______ ~ J

followmg problems: opdmalgeornetry ofa pinion crownedtooth surface; new method of crowning that is based on ap-plication of a tool. provided with a surface of revolution (thetool surface is slightly deviated from a cone surface); thedevelopment of TCA {Tooth Contact Analysis) programs forthe determination of transmission errors .for misaligned gearsand their bearing contact and a computer graphic programfor the display of the crowned surface and bearing contactin 3-D spaoe ..The first method of generation that providesthe desired optimal pinion 'tooth geometry is basedon theapplication of a computer controlled machine with fivedegrees of freedom ..The second method for generation needsonly a new tool. shape and is based on application of the ex-isting equipment.

The development of the optimal geometry of a pinioncrowned gear tooth surface is based on the followingconsiderations.

Misaligned spur gears with a pini.on crowned tooth sur-face can provide transmission errors .6.412(411)of two types,shown in. Figs. 2a and 2b,. respectively. The transmissionerrors are determined with the equation as

aq,2 = ~(q,1) - Nl. rbt (1)N2

Here: Nl and Nz are the numbers of gear Iteeth; q,,1 and q,2aile the angles of gear rotation; q"z(q,t) is the function thatrelates the angles of rotation of gears Hthe pinion is crowned

1988Fall TechnicalMeeting

New Orleans • October 10 - 12

Jazz Up Your TechnologyTopics Include:

Practical Carburizing SpecificationsCase Hardening in Large Gear UnitsCalculation of \lVormgear LoadMiner's Rule .Applied to Industrial DrivesSpecial Problems in Gear Tooth CouplingsIDevelopment of a Noise Test FacilitylDynamic Behavior of Epicyclic GearsA New Approach for the 'Derivation of

lDynamic LoadLoad Evaluation of CBN Finished Gears.A New Process for Finishing BevelsReduction of Transmission Error

For Reglstratlen Informatlon Conteet:AGMA Headquarters

(703) 684-0211

CIRCLE A-24 ON READER REPlY CA!RD

10 Gear Technology

a)

aq;2

~ /1 /'~v V ~I

b)

¢I

Fig. 2

and the gears are misaligned; 41,0= N1q,1 is the theoreticsl2 N2

relation between the angles of rotation of the gears in the idealcase where the gears are not crowned, not misaligned andthe transmission errors do not exist. Type 1 transmissionerrors are not acceptable because the change of tooth meshingis accompanied by an interruption or interference of toothsurfaces ..Type 2 transmission error is preferable if the levelof 'error does not extend the prescribed limit.

At first glance crowning should be directed toward pro-viding an exact involute shape in 'the middle cross section (Fig.3). In realjty, this ,type of crowning is not acceptable becausethe misaligned gears will transform rotation with transmis-sion errors of type 1 (Fig. 2a), but not of type 2 {Fig. 2b).For this reason the authors have synthesized a specific pinioncrowned tooth surface .. Sucha pinion,. while in mesh wi.tna. gear that has a regular involute surface, is able to providetransfonnation of rotation with a parabolic 'type of transmis-sion error function. This 'type of function of errors is syn-thesized for ideal gears that do not have any misalignment.Then, the tendency to provide parabolic transmission errorscan be extended Ito misaligned gears and the discontinuanceof meshing can also be avoided. Note that the proposedmethod of synthesis provides a shape in the middlecross sec-tion. of the tooth that deviates in a certain way from the in-volute curve that is shown in Fig. 3..The longitudinal devia-tion from a straight line is not related to the transmission er-rors, but to the desired dimensions of the instantaneous con-tact ellipse for the gear tooth surfaces. The proposed piniontooth surfacecan be generated by a plane chosen as the toolsurface. The rnotionsof 'the plane with respect to the pinionmust be controlled by a comput.er,and the machine representsan automatic system with five degrees of freedom (firstmethod for generation).

The second method of pinion crowning is based onap-plication of a surface of revolution that slightly deviates froma regular tool conical surface (Fig. 4). Such a tool can be usedlas a grinding wheel. or asa shaver. The motions of the tool

Involute shape

Hg.3

Fig .. 4

1/

11

Fig. 5,

and the gear being generated are related sim~larly to the mo-tions of a rack-cutter and the gear .. (See Section 3.) A toolwitha regular conical surface can generate a pini.on crown-ed surface whose middle cross section represents an involutecurve (Fig. 3). However, this type of pinion crowned sur-face is not desirable because the misaligned gea.rs provideType 1 transmission errors (Fig. 2a). Therefore, a tool witha surface of revo1ution instead of a conical surface is used.

The evaluation of the bearing contact and transmissionerrors for the misaligned gears, as well as the investigationof the influence of errors of gear assembly, needs the applica-tion of TCA programs. The programs are based on thefollowing algorithm:

(a) The contacting gear tooth suriacesare represented ina fixed coordinate system, Sf, thai! is rigidly connected to, 'thegear housing (Fig.. 5).

(b) The continuous tangency of gear tooth surfaces is pro-vided if the position vectors and surface unit normals for thecontacting surfaces coincide at the contact point at anyinstant. Then we are able to determine the path. of contacton the gear tooth surfaces and the relations between. the anglesof rotation of the output and input gears. Knowing function4>2(4)1) we can determine the deviations of this fundion fromthe prescribed linear function; i.e, the transmission errors.

(c) Due to the elasticity of the gear tooth surfaces, the sur-face contact is spread over an elliptical area ..The dimensions

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a)

o

b)

Middle-crosssect.ion shape

a

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12 Sear Technology

and orientation of the instantaneous contact ellipse dependon the principal curvatures and the principal directions ofthe contacting tooth surfaces. The bearing contact is deter-mined by the developed TCA program as the set of the con-tact ellipses that move over the contacting surfaces in the pre-cess of motion.

Synthesis of Pinion Crowned Tooth SurfaceConsider that the gear is provided with a regular involute

surface. The pinion will be provided with a crowned surface.The shape of this surface in the middle cross section is syn-thesized on the basis of the foHowing considerations: Twoshapes - a regular involute curve of gear 2 and the to-be-determined shape of the pinion tooth surface - are in meshin the middle cross section. Although the gears are notmisaligned,their shapes, being in mesh, must transform rota-tion with the function (Fig. 6a).

rp2{,rpl) ;;;;;; rp1 ~l + .a.rp2(rpl) =N2

arp21

Here: (2)

(3)

.tlr.b2(rpl) is a parabolic function that must satisfy the follow~ingequation :.

11"

1 N1 (b - arp2)dr.bl = 0 (4)11" 1

-Nt

Here 211"is the angular distance between two pinionNl

neighboring teeth. Equation 4: states that the arithmeticaverage of the transmission error tlrp2(·tPl) over theinterval (- 11', _!J is equal to zero.

N1 NIAfter some transformation we obtain

tPZ(rpl) = rpl NN1 + d [113 - (Nl)2r.b~]'2 7T

The magnitude of d represents the level of transmission er-ror. Using methods of -synthesis of planar gears(3) we maydetermine the sought-for shape of the pinion middle crosssection.

The longitudinal shape of the pinion crowned tooth sur-face may be determined from the requirements of the con-tact ellipse. The authors propose representing the pinioncrowned tooth surface as a surface of revolution that can begenerated by rotation about its axis a-a (Fig. 6b).

The advantage of the proposed geometry of the crownedpinion tooth surface is that ~thegears, while misaligned, havea parabolic type of transmission error, and the discontinuanceof meshing can be avoided.

An example is given below to demonstrate the results that

(5)

can be obtained through use of the proposed crowningmethod.Example 1

Given: numbers of teeth: N) = 20, N2 = 40; diametralpitch, P = 10 1; pressure angle tc = 20°. The pinion

in

tooth surface has been designed as a crowned surface witha parabolic tr,ansmission error maximum d = 2 arc secondsin the aligned condition. The developed TeA program hasbeen applied for the evaluation of transmission errors for thefollowing misalignments:

(i) The change of the center distance is 4C = 1%. Thec

gear axes are not parallel, but crossed, and the screw angleis five arc minutes. The function of transmission errors causedby the misalignments mentioned above is of a parabolic type.and it is represented in. Table L The maximal value oftransmission errors is 1.2 arc seconds.

(iii) The gear axes are not parallel, but intersected. and formthe angJe (l = 5 arc minutes. The function of transmissionerrors is of a. parabolic type. and the maximal! value oftransmission errors is 2.0 arc seconds (Table 2).

Generation of Pinion Crowned. Tooth Surface by a ToolWith a Surfaoe of Revolution

Fig. 7 shows the installment of a tool with a regular conesurface. The cone surface is tangent toa plan.e whi.ch is thesurface of a rack cutter. We can imagine that the two tools - acone and a, rack cut ter - being .rigidlyconnected, generate a.crowned pinion surface and a regular gear involute surface,r,espectively. In the process of generation the rack cutter andthe cone perlorma translational motion, while the pinionand the gear rotate about their axes (Fig. 8). The rotationof the con about i1s axis, C-C, is not related to other mo-tions thae have to be provided for the tooth surface genera-tion. The angular velocity in the rotational motion of the conedepends on the desired velocity of cutting ..The tool for thecrowning of the pinion can be designed as a grinding wheel

Table 1function '0£ Transmission Errors

Table .2

or as a shaver. The opposite sides of t.he pinion tooth aregenerated separately.

The described process of the crnwning of the pinion bya regelarcone provides an involute shape for the pinion toothsurface in its middle section. Th crowned pinion and theinvolute gear. if they are not misaligned. can transform rota-tion without transmission errors, and their bearing c,ontactcan be localized. However. the misaligned gears willtransform rotation with Type 1 transmission errors (Fig. 231).To avoid the disoontinuance of tooth surfaces that occurs atthe change of teeth in meshing. a surface of revolution mustbe used instead of a. cone surface, This surface slightlydeviates tram a regular cone surface and its application forcrowning provides Type 2 transmission errors (Fig. 2b). Also.

Fig. 7

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Septem'ber/October 1988 113

knowing the topology of the pinion crowned surface that isgenerated by the surface of revolution, we can generate itbya plane. The conditions of gear meshing and their bear-ingcontact have been simulated by the TCA program thathas been developed by the authors. An example is shownbelow to demonstrate the concepts developed in this article.

Example 2The input is the same as in Example 1. The pinion tooth

surface is crowned by a surface of revolution with the follow-ing parameters (Fig. 9): e = 20°, R = 500 ", The misalign-ment of gears has been simulated and the transmission er-rors have been evaluated by the developed TCA program

Fig. 8

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with the following results. The gear axes are not parallel, butcrossed, and the screw angle is a = lOarc minutes. Thechange of the center distance is .8.c = 1%. The function

cof transmissionerror is of a parabolic type, and the max-imal value, 0.35 arc seconds (Table 3) ..

The gear axes are intersected and form an angle, a = 10arc minutes. The function of transmission error is of aparabolic type and its max ..imal value is 0.34 arc seconds(Table4).

Table 3

Table 4

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1'4 Gear Technology

CIRCLE A-25 ON READERREPLYCARD

Fig. 10

Display o.f Analytical. Resullts iby Computer GraphicsA computer graphics program was developed to represent

in 3-D space the pinion crowned tooth surface and the mo-tion of the contact ellipse in the process of meshing ..

The graphics program is based on the analytical. solutionsthat have been obtained from the TCA program. It producesthe intricate high resolution picture on a graphic terminal andlaser printer. Fig. 10 shows the results of the computer graphicprogram that represents the pinion crowned surface and thelocation and orientation of the contact ellipses.

ConclusionThe authors devdoped:" A method for the synthesis .of a pinion crownedsurface

,that provides a localized beariagcontact and a limitedlevel of transmission error ofa parabolic type. This ap-proach provides favorable conditions of meshing andcontact for misaligned Spill gears.

• A method for generation of a pinion crowned tooth sur-face by a surface of revolution that slightly deviates froma regular cone surface. This method can be applied fercrowning by grinding and shaving.

" TeA programs to simulate the meshing and bearing 'con-tact of misaligned spur gears with th~ crowned pinienand to investigate the influence .of misalignment on thetransmission errors ..A computergraph:ic program to display in 3-D space thepinion crowned tooth surface and the Jecation and orien-tation of the contact ellipses.

•

References1. WILDHABER,E. 1962, Method and Ma.chine for Pr,oducing

Crowned Teeth. USA Patent 3.046,844.2.. Maag Information 18, Topological MDdjfica~jon. Zurich.3.. lITVIN, F.L. 1968, Theory of Gearing, 2nd ed, Moscow (.in

Russian). The English version is in press, NASA Publications.

Reprinted with permission of tire American Gew Mmll~fQclUFflrs Association.The opinions, statements Qlul conclusiom presented in this article are thoSl'of the authors and in noway represent the position or opinion of theAMERICAN GEAR MANUFACTURERS ASSOClA'HON.

Drewco Holblbing IFIil:l(tur,esmaintain ther~quilre_C!..~clos,e .~elationshilP betw.eenthe looatllng surface ,andltihe falce ofthe 'g,ealr. Tlhle rUlgged precjstenconstruction rnatntalns uhe close holdto face retatlonshlp whether locatingon smooth diameters or spline teetli.The expanding a..rbee w.iII. ~ff,ectiyelyhotd a gear fllankfor flnlSh, turnmg,gear cuning. glearfinishiin,g, and giea:rInspection operations. -

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OIOCIJE A-14 ON IREADER' REPLY CARD

September/OCtober 1988 15,

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

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

Abstract:Strider reqWre.ments are being imposed on

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

tions on spur and helical gears.

Main Influences on the Bearing Capacityand Run_ning Behavior of Gears

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

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

16 Gear feohnology

types 01 rootnr lank correct ions

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

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

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

calculation at theengagement cona it Ions

preset geOletrl.Ctooth aevlaUoos

calculation of ttlegear Dllalll.11 ties

alstortlona,l Ile!iavlourof ttle llesh

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

DeBrJngS

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

cootact III steoces s fordiscrete [JOInts

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

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

alstr1butlon of10<10 af'll1 pressure

contact oattern

course ct "1tota 1 llesh st trrness

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'Quantity to QPhmlze :

tooltl Hank correction

dJeclgng ofrestrjgtipns "

.' contact rano

• rOiling conditions

• contac1 conditions and

correctional motions

at manulacMing stage

oPhmjljDQ, meloom !lank correctIOn;

• varlafional calculation

.' modification 01theoorrec1ion topography

• checking theconvergence '1:lel1aviour

decrees 01freedom"number 01describingvanabiesfOi theflank cerrecnon

, ~tqyanti1ies"• ,engagemefll conditions

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

100111contact analysisof loaded gears tly the

methodol6nite elements

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

face ,wIdth

cnlerions lorHfIOk corrections'

• bene[icialload characteristics

• low dlsplaoementsensivity

• low dynamICexcitation

• distritJution or load

and pressure to

the mating, area

• rota.tional deviations

of loaded gears

I ~ ©

Fig. 3-Optimization of tooth Hank corrections.

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

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

G

lengtl1 of pathof contactcorrec t I,on

value40kim20

o

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

gear data:

lin ; 7 IIIiII

Z, = 27l2 ; 28

x, -0,006

x2 -0.031

fa" '.55

Et3 = 1.00Il:n = 20·fj = li6·

b ; 78,8 !11111

a = 200 ..

'1 = 4820 NIl

brings about an unwanted impact speedand shockacnvation.V'

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

Geometry of Bank Correctionsfor Helical Gears

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

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

AUfHORS:

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

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

volved strain and noise,longitudinal. flank corrections in the

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

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

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

Optimizing Method Based m, FiniteElement Ca!letdatwons

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

\1lIO\~ ._

0<1 roiling nilS

_<iii ... ,

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

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

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

Mtrl1tll1!lft preuurnOf! tllo_1IrH

! -~II I IIi

J 111

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

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

b ••11,' !!!! ~ • 16'

., • 71· ", - -.ID

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

~a· '1.55

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c~~tld .,,-

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I

10

100

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100000!ti.KIUMll 'Md reUd's

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

September/Octo'ber1988 J 9

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

the mesh and analytical models of shaftsand bearings.

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

The con tad and deformation analysis,

CIRCLE A-35 ON READER REPLYCARD

20, Gear Technology

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

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

Calculated ExamplesThe following section contains the

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

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

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

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

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

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

mesh stiffness course.Clear improvements of the dynamic

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

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

with a smaller specific variationalrange,

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

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

course.The unmodified area. on the middle

flanks of the corrected gear is sufficient to

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CIRCLE A-37 ONI READER REPLY CARD

22 Gear Technology

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

ConclusionsA method is given for optimizing

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

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

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

References:

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

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

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

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

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

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

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

Enhanced Product Performance- -

Through CBN Grinding,Glenn Johnson and EmestRatterman,GE Superaibrasives, Wo.rthington, OH

Abstract:The' extraordinary physical and thermal

properties of CBN abrasives area primary fac-tor in !:hegeneration ofbenefidal residual com-pressi.ve stresses, These residual stresses have,iI favorable influenceOI\, the fatigue life of com-ponents ground wi th CBN abrasives, This art-icle suggests 'thaI, the effects of CBN grin.dingshould be factor d into the original design ofhighly stressed, fatigue-prone drive trainccmponents,

InlrodurtionModem manufacturing processes have

become an aUy of the product designer inpmducing higher quality, higher perferm-ins components in the transportation in-dustry. This is particularly true in grinding;systems where the physical properties ofCBN abrasives have been applied to irn-pr,oving cycle 'times, dimensional con-sistency, surface integrity and everalleosts, Of these four factors, surface in-tegrity offers the greatestpotential for in-f1uencing the actual design of highlystressed, hardened steel components.

The purpose of 'this article is to' reviewboth the empirical studies and theoretical,analyses which substantiate the surface in-tegritycharaeteristics inherent with CBNgrind:ing. In addition, some importantnew empirical, findings of direct interest tothis subject are presentedand discussed.

BaCkgfOundSignificant evidence empiricaJly sub-

stantiates the fact that grinding with CONabrasives can produce a significant level ofcompressive' residual stress In the surfaceof hardened steels. Most bibliographies in-clude the work of Navarro, (11 the first toestablish that both CBN and diamondabrasives produce residual compressivestresses,

Navarro's key results are shown in figs ..Ia, b and 'c. Figs. 131 and Ibcontrasttypical residual Str1SS distributions foundafter grinding with CBN. diamond andaluminum oxide. The positive Conse-quences of CBN grinding are illustrated in

SIIlUAl SUI¥Jt.CE sm£SS ,INAI$I_

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i5 ...

~ ... IlLj--W

..r~+_.II_ COtOTlOOIt

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• .;" "" l1li ... ...

'"

Fig. I. - Residual stresses in SAE4340 after grindingwith CBN and diamond.

I I ------I

, \ - - _. -. \ -- - - -11 .1 .-I

, \'tJiMooI .... La .... ..\ ~-'- ... ..... ""1 .... '" I''''\1 \'1 \

1 ~cooova..- \ ~..l-, \,I -, '\\

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.",

.. ... "" ,.. ....Fig:..lb - Typical residual' tresses In SAE <t340aftergrinding with alumina.

FATIGU CHARACTER1S,T1CS OF AISI 4340(OUENCI'tED AND TEMPERED. 50 Rci

METAL REI,4OYAL CONornoNS SURFIICE GRINDINGfo!ODE CANTILEVER iBENDING ZERO MEAN STRESS

TEI\IIPERA.TU RE '7S"f

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~!

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_D' IIoNII-.uON IIIOTII CONDITIONI

, .IMIZ·Dow" 1-1

TlE ,GRINDXWHER

VEHTIOIU. _IlI.Ol! WHEEl.

Fig. Ic - ~N data for SAE4340 ground with various abr lves,

1e, where the bending [atiguecharacteri~tics of AISI 4340 samplesground with three different abrasives arecompared. In recent years, several. otherstudies (2-5) have served to substantiatethe work Hirstreported by Navarro.

However, Httle work has been done toincrease our fundamental. understandingof the mechanism by wh:ichthese residual.

compressivestresses are generated. Pro-duction grinding of steel components inthe metalworking industries has beendominated by the use of aluminum oxideabrasives for over 80 y MS. Therefore, ,thetotal context of OUl' ,thinking about thegrinding process is oriented aroundaluminum. oxide a.brasive grainJ>and theirspecific properties. Fer example. "grind-

September/October 1988, 23

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ing" will generally leave a residual tensilestress in the surface ofa workpiece, suchas AJSI 4340, unless special precautionsare taken to avoid this -phenomenon. (6) Itis also generally agreed that these residualtensile stresses are created because the"grinding" process rapidly increases thetemperature of the ground surface, whichis then subjected to various rates of quen-ching, depending upon the specific natureof the operation ..This process can lead tothe generation of bothuntempered andover-tempered martensite as weU as SUf-

face cracking. Expensive and tedious pro-

cedures, such as low stress grinding (LSG),nital etching or shot peening, may have tobe employed in certain cases to overcomethese inherent problems.

That the crucial importance of thespecific thermal properties of aluminumoxide grain have been overlooked isunderstandable; but in view of the grow-ing importance of CBN grinding, com-parison with the extraordinary thermalproperties of CBN abrasives must now bemade.

Selected physical properties of CBNand aluminum oxide abrasives are shown

in Table I. The density (e), thermal con-ductivity (k), and specific heat (c) arelisted along with the calculated value ofthermal djffugjvity ..Thermal diffusivilty (athermophysical property, k/,,,c), is theratio of heat conducted versus the heat ab-sorbed in a body. (7) In transient situa-tions, such as the grinding process, <I highvalue means that much heat is transmittedthrough the abrasive relative to theheating of the abrasive itself. As reflectedin Table I, the thermal diffusivity of CBNis almost two orders of magnitude greaterthan that of aluminum oxide.

In order to investigate the signficance ofthese properties on temperatures gener-ated in the workpiece surface, Shaw andRamanath (8) have determined the frac-tion of grinding energy (R) going into theworkpi.ece to be

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R - 1 + (~::;~;~rThus, the fraction of heat generated in thegrinding process which flows down intothe work is governed by the ratio of theproducts, kec ofabrasive/kec of work.

Fora1uminum oxide abrasive, thecalculation shows R = 0.76 and for CBN,R= 0.37. These writers demonstrate thatthis difference in R is sufficient for thealuminum oxide grinding process to gener-ate temperatures well in excess of thesoftening temperature of-steel, while CBNgrinding will not reach such temperature.This analysis strongly suggests that thealuminum oxide grinding process subjectsthe workpiece surface toa severe thermaldisturbance in. addition to the normalmechanical process of chip formation.CBN grinding, on the other hand, mayonly subject the workpiece surface to thenormal mechancial disturbance with min-imal thermal disturbance.

Johnson (9) Illustrates the importance ofthe extraordinary differences in thermody-namic properties by use of a simple finiteelement analysis. In a typical grindingprocess, any single abrasive grain will be incontact with the work for onJy80 micro-seconds. The analysis examinesthe tem-perature distribution in both an abrasivegrain and a steel workpiece durin.g the 80microseconds aftera grain at room temper-ature is placed in contactw:itha. steelworkpiece at 648°e It must be noted thatJohnson has not in any way attempted tosimulate the complex heat generation and

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O.oSlmm

-23.3

23.3 ~A 23.3

1:41.2

638.3 585.6 638.3 I,

, 646.7 640.0 646.7

648.3 647.8 648.3

Aluminum OXtde

22.2

22 ..2 22 ..2 22 ..2 III 0051mm

»> 22.2AdiabaticBoundary 648.9 648.9 648.9Condilions

"ii <,I:!!tIl 648.9 648.9 648.9~

648.9 648.9 648.9 i

Temperatu,e "C II

Fig. 2a - Finiteelem.entanalysis model of heat transferbetween workpiece and abrasive.

Fig. 2h - Comparison of temperature distributionafter BOJ.LSeCof contact between workpiet.:e andabrasive for three types of abrasives.

,.............., - - .

121.7 12<l.6

121.7 123.9 121.711 12<l.6 124.4 120.6 1

134..4 140.0

621.7 486.7 62U 622.2 490.0 622.2

642.8 626.1 642.8 642.B 626.7 642.B

647.8 645.6 647.8 647.8 646.1 647.B

Diamond C8N

Fig. 211 Fig.,2b

TtI:i!lle'llSELECTED PHYSICAL PROPERlIES OF CBN AND AWMINA ABRASIVES

'TRADEM:ARK OF GENERAL ELECTRIC CO

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heattransfer dynamics of Itheactual grind-ing process. The initial conditions areshown in Fig,. 2a, while the temperaturedistribution after 80. microseconds is illus-trated in Fig,. lb.. This study graphicallyreinforces the concept developed byRamanath and Shaw that heat is forceddown into the work in the aluminum oxidecase. but can flow up into the abrasivegrain itself in CBN grinding.

Dodd and Kumar(lOf have also studiedthis problem. using yet a diUerent analysis.Their work suggests that 6,3% of the heatgenerated in aluminum oxid grinding goesdown into the work, while in CBN grind-ing, only 4% goes into the work. Theyhave concluded that chip fermadentakesplace ata much lower temperature in thecase of CBN grinding than in the case ofaluminum oxide grinding.

While none of these studies in and ofthemselves eoadusively prove that the ex-traordinary thermal properties of CBNabrasives are solely responsible for theresidual compressive stress phenomenon,they dearly establish that these propertiesare the predominant factor'S.

Experimental Investigation.Most investigators have conducted. em-

pirical residual stress studies using someform of plain surface grinding and flatworkpiece specimens. This in.vestigationwill utilize cylindncally shaped specimens.In addition, most studies concentrate onthe residual stresses obtained on only two'or, at the most. thr-ee sample surfaces foragiven combination of abrasive grain. andgrinding conditions. This study will. utilizea total of 93 individual workpiece samples,all ground with the same CBN wheel spec-ification ..This investigation is comprised ofthe following steps:

1. Sample preparation - Two sets ofsteel cylinders, 1" in.diameter by SII long,have been carburized, heat treated andquenched to produce a hardness of Rc62-64. A total of 40 cydinders of SAE 8620and 501cylinders of SAE 4620 have been ac-cordingly prepared. Each sample cylinderhas been cylindrically plunge ground witha one inch wide wheel containing CBNabrasive. The details of this procedure areshown in Table II. Fig.. 3 illustrates thedimensions and configuration of the fin-ished samples.

2. X-ray diffraction residual stressanalysis - The surfaces of the samples pre-pared in. Step 1 above have been analyzed

September/OOtober 1988 27

t=ql..........-- I!----f"--I--r---IL---='EJe.m GOO"ND 1007 01" I

n,",

fig. 3-Details of SAE 4620 and 8620 cylinders used to compare residual stresses - as-heat treated andCBNground.

~!=---"-IfI TYP TVP

-1 -1 -I 1I

0 ~~w~!a

0 0 6 0 0

~n.

~ Q -

;:! a ;::

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1 1~oJ

:-'-

...,

fig. 4.- Details of SAE 4620 and 8620 cylinders used to determine effect of CBN grind depth on residualstresses.

using established techniques of x-ray dif-fraction analysis. This analysis has beenconducted on both the unground and theground surfaces of each sample. Peripheraland longitudinal stresses have been selec-tively measured on the unground surfaces ..The peripheral stresses in the direction ofgrinding and longitudinal stresses at rightangles to the grinding direction have beenmeasured on the CBN ground surfaces.

3. Effect of grinding depth on residualstresses - The amount of material to beremoved in production grinding opera-tions cannot always be precisely con-trolled. The slight distortions incompo-nents which have been heat treated canlead to variations in grinding depths insuch cases; thus, detemrining how residualstress will vary as material is removed fromthe unground, heat treated surface is im-portant. Therefore, another set of cylin-drical samples have been plunge ground toa range of finished diameters in. order todetermine this eff·ect. The configuration ofsuch samples is illustrated in fig ..4. Thegrinding conditions used to produce thesesurfaces are !he same as shown in Table IIexcept for the use of a 0..5" wide wheel.

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.Fig. S - Residual stresses in SAE 4620 bef,ore and after grinding with CBN abrasives(S1 samples) .

•..~------------+------------+------------+------------11

Fig. '] - Residual stresses in SAE 8620 after CBN grinding to various depths below hardened surface .

..-'1.'5

~

·30

-4.5

~ -10

~ -~iilw -110'" -""

-1m

~ • PEJlll'HEML I• I.OItQ!TUOIIU!. I

Jr

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-1:15o

Fig. 6 - Residual stresses in SAE 8620 before and after grinding with CBN abrasives (40 samples).

2.

-..-30

~ -tS

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• III: 10 !2 14DEPTH BHOW OR'GINAl SURF'''CE~~N)I 1()1

..

30 Gear Technology

Fig. 8- Residual stresses in SAE 4620 after CBNgrinding 10 various depths from as-hardened surface.

ResultsTne results of this work are detailed as

follows:1. Residual stress analysis - The

unground and ground SAE 4620 resul.ts in±Z sigma limit bars illustrated in Fig. 5.Analysis of these results using the student'T' test for signiticanee reveals that CBNgrinding has had the following effects onthe residual stress levels of the groundsamples:

4620 - 95% confidence that thelongitudinal stress is increased from- 96250 PSI to -148600 +/ - 5900 PSI,and the peripheral stress from- 61800 PSIto - 88UO+ I -5200 PSI.

8620 - 95% confidence that thelongitudinal stress is increased from- 45200 PSI to -133300 +/ - 3600 PSI,and the peripheral stress from -43300 PSIto - 92700 +/ - 3900 PSI.. 2...Effect of grinding depth on residual

stress - The effect on both the SAE8620and SAE 4620 ground samples are showngraphically in Figs. 7 and 8. In the case ofboth workpiece types, the consequences ofCBN grinding dearly have been to furtherincrease the residual compressive stress asthetotal grinding depth is increased.

DiscussionThe general character of the residual

stresses which result from CBN grinding inthis work is in agreement with previouslyreported work. This investigation, how-ever, has shown dearly that the residualcompressive stresses created by CBNgrinding are additive to the inherentresidual. compressive stresses found on as-heat treated surfaces (Figs. 5,6,7,8). Thisisa completely new finding which shouldtrigger further investigation.

Recently we obtained samples of auto-motive transmission gears ground with anelectroplated CBN wheel. Five gears wereselected at random during the course ofgrinding a. run of approximately 250,000gears. The midflank residual stress in theradial direction has been analyzed, and theresults of these analyses are shown inFigs, 9a and 9b. The residual stress distrib-ution is developed to a depth of.'OOB"below the flank surface. The results showa remarkable consistency of compressiveresidual stress level. at the flank surface.The small subsurface profile variations inthe residual stress distributions will be dueto variations in the heat treated profiles.Overall, these results again confirm the in-

-75

of). ==P

r- -

11 ,

'Ifl

I

i

-100

-125

-150

-175

-2256

DEPTH (IN • '0..31

Fig. 9~ - Mid-flank radial residual stress analysis.

0

-25

-50

-15

~Ul -100Will...OJa: -125t;-'~c -150iiiwa:

-175

-200

-225

r-(( -----'i r::"'t

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I!i

,

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'8 10

OEPTH (IN " '0-31

Fig. 9b - Mid-flank radial residual stress analysis.

AUTHORS •GLENN JOHNSON is superoisor at GE

Superabras.ive5 Pacific AppUcation Develop-ment Center in Gotenba, Japan. He is respon-sible forprograms associated with the applica-tion of CBN and diamond in grinding of metallicand nonmetallic materials. Prior to joining GESuperabrasive5, he worked in process develop-ment at GE Mining Products Division. Mr.lohnson has a degree in mechan iealengineeringfrom Penn State University.

plication Techno.logy Communications at GESuperabrasive5, Worthington, on. He hasworked in the helicopter and nuclear industriesand has over twenty years'experience in the ap-plication development of superabmsives. He hasbeen manager of GE Superabrasives Applica-tion Technology Operations and is now respOI1-

sibl'e for .applying new and conventional 001'11-

munications technologies to superabrasiveeducational needs. Mr. Ra.ttemu111 holds Q

degree in aeronautical engineering from theUniversity of Cincinnati.ERNESTRAITrnMAN is Manager - Ap-

September/October 1988 31

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herent capacity of CBN to produce signifi-cant levels of favorable residual stresses ingear components ..

All of these findings tend to support theresults of four-square fatigue testing ofCBN ground gears reported by Kim-met. (11) This work found significant im-provement in fatigue life of gears withCBN ground flanks and fillets over con-ventionally hardened and lapped gear sets(Fig. 10). In drawing these conclusions,

Kimmet placed emphasis on the benefits ofgear-to-gear uniformity and precisionwhich result in more uniform load distribu-tion in power transmission. In some cases,this aspect of CBN grinding may be as im-portant or even more important than theresidual stress benefits.

Yokagawa(12) has also reported in-creased wear resistance of the CBN groundsurfaces of hardened steels over similar sur-faces ground with aluminum oxide.

Honda Motor Co. has reported that theuse of CBN grinding has made it possibleto design and manufacture final drive gearsof reduced weight and which operate atlower noise levels. These attributes derivefrom both the increased uniformity oftooth form and beneficial residual stresses.

ConclusionsThis investigation has established that:

• CBN grinding of carburizedand hard-ened parts can impart additional residualcompressive stresses to the part surface.These stresses may be from as littl as30 % greater to as much as 250'% greaterthan the heat treated surface stresses.

• CBN grinding develops an increase incompressive residual stresses as grindingprogresses from an as-heat treated sur-face into the case. The generation ofcompressive stresses has been shown inFigs. 7 and 8 to be independent of theamount of case depth removed atequivalent hardness. CBN grinding wiUalso remove oxides, carbides and bainitefrom the surfaces.The major thrust of these findings is that

the beneficial effects of CBN grindingshould be considered in the original designof drive train gear components. Coupledwith the ability to remove all inherentdistortion from the previous heat treatmentprocess, one can also consider reducingbacklash, root clearance and foot con-figuration foru1timate beam strength. Suchgrinding processes offer the design engineerthe confidence of knowing exactly whatdesign loads a gear can withstand and opti-mizing gear size and weight in overall de-sign to take advantage of this knowledge.

References

1. NAVARRO, N.P .. "The Technical andEconomic Aspects of Grinding Steel WithBORAZON Type II and Diamond",Society of Manufacturing Engineers,MR70-198, April, 1970.

2. CHUKWUDEBE, t.o. "GrindingSuperalloys With 130RAZON GIN",Proceedings DWMI, November, 1975,Chicago, IL.

3. TONSHOFF, H.K., "Influence of theAbrasive on Fatigue in Precision Grind-ing", Milton C. Shaw Grinding Sym-posium, American Society of MechanicalEngineers, Miami Beach, Fl, 1985.

4. ALTIiAUS, P.G., "Residual Stresses in In-temal Grinding", Industrial DiamondReview, 3/85.

5., MATSUO, T., H. SHlBAHARA, Y.OHBUCHl, "Curvature in Suriace Grind-

ing of Thin Workpieces With Superabra-sive 'lNheels", ClRP Annals. Vol. 36,111987.

6. HELD, M., J.F. KAmES. 'The Influenceof Untempered And OvertemperedMartensite Produced During Grinding onThe Surface Integrity of AISI 4340, HRCSteel", Milton C. Shaw Grinding Sym-posium, American Society of MechanicalEngineers, Miami Beach, FL. 1985.

7. lNCROPERA, F.P., D.P. DEWITT, Fun-damentals of Heat Transfer, John Wiley& Sons, 1981.

8. RAMANATH, S., M.e. SHAW, "Role ofAbrasive and Workpiece PropertiesRelative to Surface Temperature andResidual Stress", Proceedings 14th NorthAmerican Manufacturing Research Con-ference, May, 1986.

9. JOHNSON, G.A., "Beneficial Com-

pressive Residual Stress Resulting FromCBN Grinding", SME Second lntema-tional Grinding Conference, 1986.

10. DODD, H.P., K.V. KUMAR,'Technological Fundamentals of CBNBevel Gear Finish Grinding", Society ofManufacturing Engineers, Superabrasrves'85, Chicago, IL, April, 1985.

11. KIMMET, G,J., "CBNFinish Grincii!\gofHardened Spiral Bevel and HypoiciGears", American Gear Manwactur rsAssociation. 1984.

12. YOKOGAWA, K., "Grinding Perfor-mance of CBN 'lNheel - Report #22",Japanese Society of Precision Engineers.Spring Conference, 1987.

13. PRIVATE COMMUNICATION - Hon-da Motor Co. to General Electric Com-pany, Worthington, Ohio, USA, July 2,1987.

September/OCtober 1988 33

- -

BACK TO BASICS ...- - - --

Involutometry Involute Curve FundamentalsOver the years many different curves have been considered

for the profile of a gear tooth. Today nearly every gear toothuses an involute profile ..The involute curve may be describedas the curve generated by the end of a string that is unwrappedfrom a cylinder. (See Fig.1.) The circumference of the cylinderis called the base circle.

Following are specific features of the involute curve.• A perpendicular to the involute surface is always tangent to

the base circle.• The involute surface is a uniform rise cam (equal rise per in-

crement of rotation).• The radius of curvature of an involute surface is equal to the

length of the tangent to the base circle.• The same force applied perpendicular to the involute surface

of a gear tooth at different tooth radii results in the same tor-que on the gear.Because the involute surface is a uniform rise cam, any in-

volute surface imparts unifonn angular motion when operatingagainst any other involute surface. To illustrate further, con-sider two pulleys with a crossed-belt drive as shown in Fig. 2.If we place knots in the string, then the knots will generate in-volute curves as they leave one circle and approach the othercircle as shown in Fig ..3. If the knots are spaced close enough,there will always be knots on an involute surface when they arenot on the circle. These generated curves can then be consideredas teeth on the gear.

Harlan W. Van Gerpen, P.E.C. Kent Reece, P.E.

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Fig. 2-Crossed belt drive. Fig. 3 - Multiple involute curves.

34 Gear Technology

Of course, there are otherconsiderations. The distance be-tween knots must add up to an integer around the base circle,and the knots must be far enough apart to provide a structurefor the tooth. When these criteria are met, the distance betweenknots is called the base pitch ..Any two spur gears that have thesame base pitch can be meshed together. (See fig. 4.)

Just as the diameters of the two, pulleys in the crossed-beltdrive establish the ratio of the two shafts, likewise, thediameters of the base circles establish the ratio .of the two gears.

The contact between two involute surfaces of two gearsalways occurs on a line that is tangent to the two base circles.This tangent line is called the "line of action" ..(See Fig. 5.) Thisphenomenon results from the from tact that aperpendicular tothe involute surface is tangent to the base circle.

The line of action will cross the center line between the twogears. This intersection is called the pitch point. If we drawcircles 'on each gear using the gear centers and passing throughthe pitch point, we have pitch circles. These pitch circles maybe considered as cylinders which would roll with each other byfriction with no slippage. The distance between the center linesof gear teeth on either pitch circle is called the circular pitch. Ifwe divide the circumference of the pitch circle by the numberof teeth in the gear, we get the circular pitch. These circles areimportant in the calculation 'of tooth thicknesses because it ishere that the two tooth thicknesses, plus the backlash, must addup to the circular pitch.

Each gear has two distinct pitch drdes to be considered. Thegear win have one pitch drdewith its cutter and another pitchcircle with the mating gear. If it is the middle gear in a string .ofthree gears,it may have a different pitch circle with each matinggear.

The pitch circles of mating gears will change as the centerdistance is changed. ITthe center distance is increased, the pitchcircles will get larger, and the circular pitch wilJ increase withthe same tooth thicknesses as before. The backlash will also in-crease. A gear does not have an operating pitch diameter untilits mating gear and center distance are specified.

Since gear teeth must make contact along the line of action,and since the contacting Iorcesare parallel to this line, there isa force which acts to push the tWD gears' centers apart. The

AlITHORS:

HARlAN VAN GmPEN has 28 years of experience at tile 10hn DeereProduct Engineering Center i'l Waterloo, Iowa, where Ile served asmanager of tedmica[ services tmd a pn'ncipaJ engineering specialist. Heis a licensed professional engineer in the State of Iowa and a member ofthe ASAE where he is a mamber of the Research Committee and the In-strumentatioPl and Controls Committee. He is also a member of the SAEEducation and Human Factors Committees. Mr. Van Gerpetl holds tI

masters degree in electn'cal engineering from the University of Illinois.

CARROLL K. REECE worked for the John Deere Product Engineer-ing Center, Waterloo. Iowa, as an engineering design analyst and super-visor in the mechanical elements department. He has seroedior 25 yearson the AN51-B92 National Spline Standards Committee and for 18years011 both the.lSO TC 32 International Spline Standards Committee andthe ANSi-B6Nationa} Gear Standards Committee. He holds a mastersdegree in agricultural engineering from Kansas State University and isa licensed professional en.gineer in the State of Iowa.

angle the line of action makes with a perpendicular to the centerline of the gears is called the pressure angle. The pressure angleis defined as B diase ra IUS

Pressure angle 0: = Acos (1)Pitch radius

Just as the pitch radius with another gear is unique, so thepressure angle with another gear will be unique.

This may be confusing until one is reminded of the crossed-belt drive ..The cylinders would rotate in the same manner, andthe knots in the string would generate the same involutes in-dependent of the center distance between the two cylinders. Thecrossover point of the line of action across the center line willresult in different pitch circles, and the pressure angle willchange as the center distance is changed. The force along theline of action will remain the same; it is the torque divided bythe base circle radius. As we will see later, there are considera-tions of tooth thickness and contact ratio which restrict the

Fig. 4 - Base pitch of a.gear and a rack.

Fig. 5 - Line of action and pressure angle.

September/October 1988 3S

range that the center distance may be changed.The insensitivity of the involute action to the changing of the

center distance of the gears is a distinct advantage over othertooth profiles eonsidered in the past As shafts deflect due tochanges in. load, the involute action is unchanged, assumingsufficient contact ratio still exists.

The involute profile is the path of the end ofa string un-wound from a base circle. The design of the gear involvesestablishing the ends of this curve. At the beginning of the curvethere must be some sort of fillet blend to the root and the in-volute of the adjacent tooth. It is not good practice to plan onusing the involute near the base circle because its radius of cur-vature is very small and changing rapidl y. It is very difficult tomanufacture the tooth accurately near the base circle.However, this end of the involute profile must be defined ac-curately to avoid problems of interference with the tip of themating gear tooth. The "form diameter" is used to specify themot end of the involute profile.

The tooth tip end of the involute profile can either extend tothe outside diameter of the gear, or if a chamfer is provided, itends with the start. of the chamfer. A chamfer is used by manymanufacturers to prevent damage to the involute surface fromnicks due to handling.

Helical GearsMore and more new machines use helical gears. The prin-

cipal reason for this is the demand for quieter gears. Not onlydo we have an increased emphasis upon quiet machines, butalso most machines operate at higher speeds and higher loads,both of which increase the noise made by spur gears. A pitchline velocity of 2000 feet per minute is considered by many tobe the top limit Ior using spur gears innon-aircraft applications.The writers have observed spur gears operating quietly at pitchline velocities of 600() feet per minute, but these were probablyvery accurate gears. With the normal indust:rial productiontolerances, we can expect spur gears to be noisy above pitch linevelocities, of 2000 feet per minute. If this speed must be exceededand quietness is preferred, then the design should incorporatehelical gears ..

We will introduce several new terms to describe physicalcharacteristics of helical gears ..The first of these is the helixangle. The helix angle is the inclination of the tooth in alengthwise direction ..The helix angle results in the gear havingend thrust or an axial force during operation.and the supportbearings must be able to resist this force. There is no rule for thebest helix angle. It may be up to 45°.

The helix angle of the tooth depends upon the radius underconsideration ..Nonnally we quote a helix angle associated withthe generating helix angle. This relates to the pressure angle oJthe cutter and the generating pitch radius. It is more importantto work with the helix angle at the basecircle, For two gears tomesh properly and operate on parallel axes, they must have thesame normal base pitch and the same base helix angle.

To determine the different helix angles on a.gear requires afamiliarity with the term "lead". The lead of a gear may be de-fined as the width of a gear required for a tooth to wind onerevolution around the gear. A large helix angle will result in asmall lead. A spur gear may be considered as having an "in-finite" lead.

36 Gear Technology

In very large power applications common practice dictatesthe use of two helical gears, side by side, with opposite anglesof the helix (hand) so that the end thrust is neutralized. Usuallythere is a small space between the gears to facilitate manufac-ture and also to give the tooth flexibility for load sharing.

Helical gears have involute tooth profiles similar to those inspur gears. Each gear will have a base cirde and a line of actionlike a spur gear ..[f we observe the gear from an axial posit ion,the gear contacts will occur along the line of action. However,an entire tooth will not make contact with the mating gear in-stantaneously, as does a spur gear tooth. The contact may startat the tip of one end of the tooth and proceed along the lengthof the tooth, with the first point of contact moving down thetooth before the other end of the tooth makes 'contact. The in-stantaneous load contact will be a diagonal line across the toothface with a different radius as we follow the contact across thetooth. For very wide gears the pattern may be repeated severaltimes across the gear as several teeth, with their helix angles,contact on the line of action.

With helical gears we have a second contact ratio to consider.The contact ratio expressed by the length of the zone of actiondivided by the base pitch is still used, The second, called facecontact ratio, is a function of the helix angle and the width ofthe gear. For smooth action both contact ratios should exceedone.

The teeth on a.helical gear have a slight "twist" in the radialdirection ..lf a "pin" is placed in the tooth space, as is done whenmeasuring the distance across pins to determine tooththickness, the pin will sit on a high spot and "rock". A ball mustbe used for making this measurement with helical gears, whichhave an odd number of teeth.

To study the tooth geometry we have to consider two planes,the transverse plane and the normal plane. The gear teethoperate in the transverse plane, but the teeth are generated inthe normal plane. The tooth thickness and generating pressureangle usually refer to the norma] plane. The base circles, out-side radius, root diameter and contact ratio are determined inthe transverse plane.

To determine the tooth thickness at a second radius requiresthe conversion of the given tooth thickness in.the normal planeto the transverse plane. The tooth thickness at the desired radiusis calculated in the transverse plane. It can then be convertedto the normal plane again, taking into account the change ofhelix angle at the desired radius.

Helical gears may be cut with the same hob that is used. to cutspur gears. The hobbing machine has the ability to position thehob at an angle corresponding to the helix angle.

For gears that must be cut with a shaper because there is noclearance for hob runout, the shaper machine must be equippedwith a "guide". The guide provides rotation of the cutter as itgenerates the teeth. Each guide has a specific lead, and the helixangle generated depends upon the diameter of the shaper cut-ter. A large diameter cutter will generate a larger helix anglethan a small diameter. The shaper cutter must have nearly thesame lead; "nearly" because the shaper cutter may stiill be usableeven though its helix angle is up to one degree different than thehelix angle being generated. The cutter has enough side reliefto permit this. This should only be done for small lots ..

When selecting shapers for a helical gear, the availability of

a guide is mast important because the cost 'Of a guide is manytimes that of a shaper,

The mathematieal formulae for helical gears aile nearly thesame as for spur gears. except for the indusian of a function ofthe helix angle In the equation, The formulae are basically setup to. wark in.the transverse plane (plane of rotation); therefore.u the tooth thickness and pressure angle a:rein the normal plane.they must be converted to the transverse plane.

Gear Design MathematicsThe tooth geometry 'Of a spur or helical gear may be de-

scribed by seven pieces 'Of data. These are number 'Ofteeth, basediameter. 'Outside diameter. form diameter. root diameter,tooth thickness at base diameter and lead (and hand).

The design of a gear involves the use of mathematical equa-tions to 'Obtain the above seven numbers. The mathematics in-valves extensive use of trigonometry . The designer must befamiliar with the use 'Of angles in radians and the involutefunction.

As mentioned previously.the involute curve is the path ofthe end 'Ofa string unwound from a cylinder.The fan owing relationships hold.

Tan a = a + ()

() =Tan a -a = iavo (2)

The involute function is the angular displacement of a pointon the tooth from its pointaf intersection with the base circle.This relationship is useful for calculating the circular thickness'Of a tooth. When the thickness at a given diameter is desired.there is a corresponding angle or "pressure angle" of the toothat that radius. By determining the pressure angle at a point ona tooth and then taking the involute of this angle, the angularshift of the curve from where it left the base circle is available.Now by multiplying this angle by the new radius, we have thelength 'Of arc of the shift of the tooth pl'afi1e. By subtracting twotimes this angle from the angle on the base cirde for the entiretooth, and multiplying this resultant angle by the desiredramus. we get the circular tooth thickness at the desired radius.(See Equation 3.)

a -= Acos I~lTR = 2R I' !b... - inv,al2'~

(3)

Good practice suggests use of the tooth base thickness as a[undamerual dimension on the gea:r because finding thethickness at any other radius 'Onlyrequires finding 'Oneinvolute.If the known thickness is at some other diameter, then it isnecessary to have the involute at this radius, and then use thedifference in involutes to determine the ·diHerence in knowntooth thickness and the desired tooth thickness.

Eq.2

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September/October 1988 31

Eq ..4Eq.J

It Is customary to use the circular dimensions for describingtooth thickness. The difierence between the circular and chor-dal tooth thickness is very small, usually less than one-tenth ofthe backlash that will be provided. However, in somemathematical analyses, such as following the path of a cutter,chordal dimensions must be used to get the accuracy required.

Gear designers who work with "standard" gears usually havethe tooth thickness at the pitch circle and then have to use twoinvolutes to find the tooth thickness at another radius.

Given the arc tooth thickness and pressure angle in the planeof rotation of a helical gear at a given radius, to determine thetooth thickness at any other radius, use 'the following formulae.

transverse plane is

where

R1· Casal

R2

PD = ZDP'COS~

(5)

Z =Number of teethDP = Diametral pitch

(normal)i3 =Helix angle

(generating)

Converting the norma] tooth thickness to the transverseplane requires the following fonnula:

Note that the helix angle must correspond to the diameter onthe gear corresponding 10 the tooth thickness. The helix angleis different for different diameters of the tooth. The lead is con-stant for a gear and can be determined from a helix ..angle givenat a given radius by the formula:

(4)

where where

Rt. = Given. radiusat. = Pressure angle at RlTl = Arc tooeh thickness at RlR2= Radius of unknown. tooth thicknessct2; = Pressure angle at R2T2 = Tooth thickness at R2

The above formula assumes all the data is provided in thetransverse plane (plane of rotation), Most helical gear data isavailable in the normal plane because the cutter is dimensionedin this manner. Such data as tooth thickness and pressure angleare usually in the normal plane.

The formula for the pitch diameter of a helical gear in the

38 Gear TechnologV

T = TnI

Cos~(6)

'r, = Normal tooth thickness(3 -" Helix angle

T, = Transverse tooth thickness

2'1!"RL = .. -Tan (3

(7)

Ag. 6-Diagram 1,0 illustrate helix angle variation.

where

R .,. Radius of given helix anglef) ~ Given helix. anglel = Lead of the gear

Oncethe lead is known, the foll.owing formula may be usedto find the helix an.g.Jeat 'the radius corresponding to, the given,tooth thickness ..

(2' r' RJf) = Atan L. (8)

where

R - Radius of given tooth thicknessL - Lead of the gearf) - Helix angle for R

Fig. 6 is helpful in remembering the relationship between thelead and helix angle at different radii of the gear tooth. The leadis constant for a gear, but the ciecumlerence depends upon theradius being studied ..

The study of the beam stress oJ the tooth and the form.diameter is done in the normal plane. Doing these studies in thenormal plane r quires converting to a "virtual" gear. The nor-mal plane of a helical gear is really an ellipse. The point of in-terest is the largest radius of curvature of the ellipse ..This radiusof curvature is used Ito construct a virtual gear.

The virtual gear is a simulated spur gear. Mathematically,the virtual, gear is derived from the helical gear by the follow-ing relationships:

V-· ] .- h di (R) Pitch radius (helical gear)rrtua pttc .ra JUS ,,= ----------'~-Cos 2 (Helix angle)

Virtual number .of teeth (Z ) = "{;eethin helical gear (Z) (10)v cos3 (Helix angle)

(9)

Virtuall base radius (Rb,,) ==Virtual p.itch radius' Cos (PAnol1lliill)

(11)

To 0brain the ou tsid.ediameter and root dia..rnel:er, their dif-

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The given pressure angle will usualJy correspond to thepressure angle of the cutter. To convert this into the transverseplane the following formula is available.

. Z1(T1+ Tz) - 2' 'K' Rl + .mvaz = . mval2 . R1 (Z1 + Z2)

C1= R1 + R2

Tan anTan at =--

Cos {3g(12)

whereCosaz

an = Normal pressure angle(38 = Generating helix angleat = Transverse pressure angle

Rl = Given radius of first gearR2 = Given radius of second gearZ1 = Teeth in first gearZz = Teeth in second gearat = Pressure angle at R1 and r2a2 = Pressure angle at meshing positionTl == Ar,c tooth thickness at R)T2 = Arc tooth thickness at RZC1 = Center distance for al

C2 - Center distance for 112

Another calculation that is important in designing a set ofgears is determining the center distance for two gears to bemeshed 'together. Again, it isoruy necessary that the gears havethe same base pitch for the involute action to be proper;however, there may be interference with the tips of teeth goingbelow theform circle. The tips of the teeth may also bottom outin the root of the mating gear if the members ·of the pair werenot designed together. If the gears are helical, they must havethe same base helix angle. If they are external gears, the handof the helixes must be opposite, assuming they tum on parallelshafts.

Given tooth proportions in the plane of rotation for twohelical gears, to determine the center distance when they meshtightly, use the following formulae,

The previous equations are useful for determining the centerdistance when run with a master gear. This is a tight meshoperating condition. If two gears are to be operated together,then the desired backlash may be added to the tooth thicknessesin the equation, and the corresponding center distance derived.

Technical Education Semin,arsYour technology has to be the best. To help vou, AGMA isinitiating a series of seminars - each coneentratlnq on a de-tali1ledaspect of gear technologlY. Regist,ration is limit,ed topreserve classroom interaction, so reserve your seat nowl

Topics, L.ocaUons and Speaken for 1988 - 11989Include:

Nov. 9 Controlling the Carburiziing Process - Cincinnati, OH - by Roy Kern, Kern Engineering Co.Dec. 7 Rational Loose, Gear Quality Requirements (using AGMA 2000 properly) for Specifyer and

Purchaser - AGMA Headquarters - by Chuck Schultz, QuaKer Ci,ly Gear, Iinc.Jan. 11 Speci,fyingand Controlling the Quality of Shot Peening - Cincinnati, OH

by Jim Daley, Me,tal' improvement CompanyMarch 7/8 Source Inspection of Loose Gears from the Customer's Standpoint - Rochester, NY

by Robert E. Smith. R. E. Smith and CompanyMay 2 Gear Math at the Shop Level for the Gear Shop Foreman - Olnclnnati. OH

by Donald McVittie, Gear Engineers., Inc.June 6 Specifying and Verifying Material Quality per AGMA Material Grades - AGMA Headquarters

by Rocky Andreini, iEarle Jorgensen Company

For Further Inf'ormatlon C'o,ntact eill! Daniels, AGMA. Headquarten1500 King Street, S,uUe 201. Alexa,ndir,ia, VA 22314

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40 Gear Technology

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Ference from the pitch diameter is maintained in each plane.

Contact RatioIf two mating gears are to provide smooth rotation, it is

necessary that there be a continuing 'contact on the line ofac-tion. A second pair of teeth must mesh before the first setseparate. This relationship is described by the "contact ratio".The contact ratio is the ratio of the length of adiveengagementof the gears on the line of action to the base pitch ..For spur gearsthe contact ratio must exceed one and may exceed two for gearswhere quiet operation is needed.

The following figure and equations show how contact ratiois determined.

CR =YF:fJ1-R2bl + YR52-R2b2 - C'Sincx

p. Cos e(14)

where

Ral= Outside radius of gear 1Ro2 = Outside radius of gear 2

C = Center distanceeR = Contact ratioRbI = Base radius gear 1Rb2. = Base radius gear 2

0( = Pressure angleP = Circular pitch

With helical gears we have a second contact ratio to consider ..

The contact ratio expressed by the length of the zone 0.£ actiondivided by the base pitch is sHUused. The second, called facecontact ratio, is a function of the helix. angle and the width ofthe gear. For smooth action both contact ratios should exceedone. The formula for face contact ratio is

CR - F'Tan~If ----race p (15)

where

F = Gear face widthf3 = Helix angle at pitch pointP ,= Circular pitch (transverse)

C~ace = Face contact ratio

Generation of Gear TeethWhile we discussed the forming of the involute surface ofa

gear tooth as identical to the curve of a string being unwoundfrom a base cylinder, the following is descriptive of how a geartooth is actually generated. There are two main types of cut-ters used to generate gears, ho bs and shapers. First consider the"shaper", This cutter closely resembles a gear. One edge is sharpand, by gradually meshing with a blank as it moves back andforth axially , it will cut a mating gear. The blank should be theappropriate diameter, and the two centers must be externallygeared together with the right ratio.

A second type of cutter is the "hob". Before discussing thiscutter we must discuss the "basic rack". A basic rack is a seg-

(continued on page 45)

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INVOLUfOMETRY •••(continued from page 42)

r-rI

ment of a gear of infinite size, but with teeth the size of the gearwe want to cut. As a result, the "involate" sides of the teeth areessentially straight lines. To generate a tooth, the basic rackmoves in a. straight line like a. rack and pinion arrangement,with the cuttingaction being up and down axially to the gear.In reality, instead of having a very long rack, segments are ar-ranged around a cylinder and offset to create a helix. With theaxis of the cutter inclined, the teeth will contact the gear being

Eq.14

'cut parallel Itothe gear axis. By inclining the cutter at a dilferentangle it can cut a helical gear. The cutting activity closelyresembles a worm. gear drive. Hobbing is the most popular andfastest method for cutting gears.

In our calculations it is necessary to determlnethe positionof the basic rack with the gear being generated in order toestablish the form diameter and root radius and determine

(continued on page 48)

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GEAR CALCULATI'O,NPROGRAMS F'OIR T:HE

GEA.IR MAINUFACTUIRIE.:R

liNEXPIENSIVE,. EASY TO RUNON IBM COMPATIBLES

EXl'EAN'ALINTERNAL

liNCH ID:P)METRIC ,(Mod.)

S'PU'RIHIEUCAL

STRAIGHT a.eViEL-(104)SPIRAL BEVEL. 1#16& 27 GRD

BLANK AND SUMMAFUESFor Brochure and More Information

CAU OR WRIITEUNIVERSAL GEAR' CO. INC ..- P.O.Box A.G., Anza, CA. 92306 - .

(I14) 763-4616

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GEAR DESIGN/ANALYSISfor IBM PC scompatibles.GEARCALC($995): design op-timum spur & helical gearsets from appli·cation data. AGMA218 ($1495): calculatepitting and Ilendingfatiguelives per AGMAStd .: 218.01. SCORING +- ($495):analyzefplot scoring & wear pro-babilities. Fast, menu-driven pro-grams work as one system (eembln-ed price $.2495), do extensive error-checking. Comprehensive manuals(theory, operation, examp'les) in·eluded. Demo avail'abl'e. GEAR.TECHSoftware, Inc., 1017 Pomona A.ve.,.Albany, CA 94706 (415) 524·0668.

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1- SERVICE

AiPiPlICAT;I,ONTESTING:-iDEVELOPMENT

Total Gear TeChnologyfrom comput,er ,aided

engineering of design to final,application testing .

• Loaded Contact and Axle Deflection• Fatigue Strength and Durability ,• Constant Velocity Testing .• Instrumented Vehicle Sound Analysis I

• G-Age with Zeiss CMM• Single HankI. ,Finite Element Analysis

'The 'Gleaso:n Works1000 University Ave.

Rochester, New York 14692(716) 461-8026, Larry Palmateer

--'

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GEAR: TES,TIING ANDDIESI:G'N FACIUTIES

• GEAR DESIGN (NOiSe: - STRENGTH)• ROTATING GEAR (TORQUE - SPEED

CONTROL) TEST MACHINes..' SINGLE TOOTH BENDING IFATIGUE

TESUNG.• STATIS-';ICAl PlJ>iNNING - ANALYSIS.'. WROUGHT STEELS, SINTEREO

METALS, NON-METALLIC MAT'L.S.'. CAD FACILITIES FOR LOW COST

SET-UP.'. CUSTOM TEST MACHIINE IDESIGN• IEXPERIENCEDPEASONNEL

PACKER ENGINE:ERING312/355-5722, ext. 21'4

BOX 353, NAPERVILLE, IL 60566

Bring in new customers for your business by advertising inGEAR TECHNOLOGY, The Journal of Gear M,anufacturing.

Can (312)437-6604Rates: Classified ,Display-per inch (rnlnl-mum 3") 1.X·$120, lX·$110, 6X·$100. Typewill be set to advertiser's layout or GearTechnology will set type at no extra charge.Word Count: 35 characters per line, 7 linesper inch.

46 GearTechnology

Payment.: Full payment must accompanyclassified ads. Mail copy to Gear Technology,P.O. Box 1426, Elk Grove Village. It 60007.Agency Commission: No agency commissionon elassifiacs.

lMater:ia.ls Deadline: Ads must be recei.vedby the 25th of the month, two, months priorto publication. Acceptance: Publisherreserves the right to accept or rejectclassified advertisements at his discretion.

HELP WANTED

M~~UFACTURING MANAGE:RNIE,W AIRCRAFT' FACIIUTV

An aggressive, aircraft & precision partsmanufacturer (gearboxes" gears, shafts,housings) la seeking an experiencedManufacturing Mana-ger withlhe ability I

to direct the msnufactu ningoperations &growth of an ,ex:paJ1c1lng200 personfaclHty. The candidate we seek mustIhave working knowl'edge of aircraftpariS" q,uaJity requirements" tools, fix-tures" proceSsing & manufacturing. linreturn we ,off,eran attractive salary, e.x-cellent benefits & a.futur,e dir,ected en-viro.nment.IP,lease submit your resumewithsal'ary history to:

ACR INDUSTRIES, INC.,15375 Twenty Three Mile Ad.Mt. Clemens, M148044-9680

GEAR IPERS'ONNEiLENGINEERS

GEA'R GRINDER.SMACHINISTS,

MANAGEMENT/SUPERVISIONIPRODUCTION CONTROL

QUALlTV OONTROLIINSPECTION:SALIES

ACR has moved 10 a new 85,000 squarefoot fa.cility in Mt. CI'emens, Michigan. Asa result, the above and other new pOsitions,8J'e beinQl created. If you are in the the gearbusiness in any capacity, are ambitious 'and thinking of making a. move, send usyour resume.

ACR INDUSTRIES. INC.15375 Twenty-Three Mile Ad •.M•. Clemens, MI 48Q44-9680

SALES AND SERVICE

A FULL SERVICE COMPANY SUPPLYING THE AEROSPACEAND COMMERCIAL mDUSTRIES WITH PRECISION PRODUCTS.

V1NOQ®B-lRADIUS/ANGLE WHEEL DRESSERVlNCQ® SPLINEDICATORS AND WEAR GAGES

VINCQ® GEAR MEASURING INSTRUMENTSGEAR COMPONENTS AND PARTS.oPERATIONAL GEAR GRINDING

SPUNE GAGES AND ARBORSMASTER GEARS

TIFCO SPLINE, INC.VINOO® PRODUCTS

2900S,ANTHONY DRIVE WIXOM. MICHIGAN 48096(313) 624-790.2

FAX (313) 624-1260

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There's stillltime ....order dosing date for aolasshied ad in the!Nov.lDee. issue lisSeipt. 110th.

Tell our advertisersyou saw it in

GEAR TEICHNOtiOGY

SUBCONTRACT

GEAR rOOTH' GRlND.ING& HONING .oNLY

Produclliol1 Quantities314" P.O. to 27.S" P.lD.3.5 D.P'. and 11 .. Face

'Gear Tooth IF,inishingllis our ,OnJy Business

we have notumfng. hobbing orshaping capability

ALIlE.GHIENIY ,GEAR C'OR;P.23 Dick IRoad .......

loe:,e.~.'~ ¢.FAX (716) ~i711 f :'"lVJ)l

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REBUILDtNG

H:O'BB'E'R IREBUILDIINGSPECIALlST:S

Having! trouble meeting today's d&mand quality col1trol to'lerances?iLet our factory trainedl and ex-perienced staff return yourmachine to optimum operatingoondition.We spectattae iln repairingl,rebuilding aM mode:miztng aUmakes o. 'hob'bers. -

• Cleveland Rigjdhobbers• Gould & Ebe~ha.rdt• Barber Co'lmall

PRESSftlATION IINC.522 Cottage Grove RoadBloomfield, Conn. 06002

(203) 242-8525,

September/Octob9ll' 1988 47

aJSlNESS OPPORTUNITY

••• FOR G!EAR CONSULTANTSPut EXCIiTING NEW GEAR OIAG-NOSTIC TECHNOLOGY to worksolving your clients' most criticallgearproblems.,Fortune 500s are nowgelfing BREAK-THIROUGH iENHANCEMENTS INMONIITORING GiEAR PERFOR-MANCE AND CONDITION .... forend of Iline (QIA), predictivemaintenanoe& engineering andtesting applications. -·J'oin MTC jin applying this technology ..CaJl1EO PAGE for more information.

IMONITORlNG JEOKNOLOG,Y 'CORPORA110N2n9 Han\and Roa.d

Falls Church, VA 22043Call (703), 16198-5520

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It's true. our Consumer InformationCatalog is tilled with booklets thatcan answer the questions Americanconsumers ask most.

TOsatisfy every appetne, theConsumer Information Center puistogether this helpful Catalogquarterly containing more than 200federal pubhcancns you can order.trs free, and soare almost half of Ihebooklets it lists. Subjects likenutrition, money management,health and iederal benefits help youmake the right choices anddecisions. .

So get a slice or Americanopportunity. Write today for yourfree Catalog: .

Consumer Infor::-mation centerDepartment APPueblo. COlorado, 81009US General Sep'!h.l'~ AdmIMLSlr,atIOr\

_ As.Amencan

INVOLUTOMETRY. ,..(continued from page 45)

whether 'the cutter will interfere with the outside diameter ,of thegear .. Usually a gear tooth thickness is given, and thecorresponding position of the basic rack is computed.

Given the tooth dimensions in the plane of rotation of ahelical gear, to determine the position ofa mating rack of di!-fe.lentcircular pitch and pressure angle (same base pitch), usethe fonowing formulae.

Sin (jb = Sin {31 . Cos cxnl

S".·-t:l _ Sin (jb _ Sin {31. Cos anIm"'2 - -

COS CXn2 COS an2

'T Tan O'n2Ian 0'2 = .--------'=Cos 13'2

X = R2 - a. + _--",,,....1 _2 Tan 0'.2

R - R,2----Cos 0'2.

,anI = Normal pressure angle at RIal == Transverse pressure angle at RlRl = Given pitch radiusR, - Base radiusT1 = Given tooth thickness at RlX == Dist, gear center to hob tip

Pn2 = Normal pitch of rack

(17)

Base pitch must be equal (Pnl • Cos al = Pn2 .. C.oSa2)

The position of the rack formula results in. at figure for theroot radius of the gear. The following formula wi]] give theminimum root radius without undercut. By comparison, itcanbe determined if undel1cutting is taking place. Undercutting isundesirable. By 'taking the minumum root radius withoutundercut a new tooth thickness for the gear can be determined.By keeping one gear from undercutting, the mating gear shouldbe checked to determine if it may be undercut as a result ofshifting t.ooth thickness.

Giventhedimension.s ·ofa hob, the helix angle and numberof teeth in a helical gea.r, to determine the mlnlmum root radiusto avoid undercut, use the following formulae,

Tan ahb

Tan 0' = Cos fJI I I I'. T . '/!'·R.2 ..R2 '__ 1_ + inv a1 - inv a2 - ----=;

2 ·R1 Z

where

(16) ZR=----2· Dp· Cos 13

where

Run.: = R· Cos2a - r : (1 - Sin a)131 = Given helix angle at RlfJb= Base helix anglean2 = Pr-essure angle of rack(1'2 = Transverse pressure ang1e of rackR2.= Pitch radius with racka = Addendum of rackZ = Number of teeth

Pnl = Normal pitch at Rl132 == Helix angle for mating rack

48 Gear Technology

Z = Number of teeth in gearan" = Pressure ang1e of hob

0' = Pressure angle in plane of rotationfJ = Helix angle

DP = Diametral pitch of hobr = Radius of hob tooth tip

Rune = Minimum root radius wi 0 undercut .1

Cetans at IIMTS, '88Find out more

about how the mostdramatic improve-ment in ,gear produc-[ion in decades canwork into your plans.See us .at Booth#I 6301 or contact usnow. The GleasonWorks, Division ofGleason Corporation.P.O. Box 22970,Roches[e~NY 14692j7161473-1000.

In Seven Decades,the~Baste rrit:'lciple~ of GeC!rMaking

Havent Changed ...Untd Now

Introducing PHOENIXINO CRADI!.iE., NO' ECCENTRIC.NO SWIVEL NO CUT1iERTlI%

PHOENIX is a revolutionarynew gear machine thareumlnatesall of the mecnankal setup adjust-ments necessary with convenuonalmachines used for cutting or grind'-ing bevell and hypoid gears~ Theresult Both uptime and flexibHityare multiplied.

Only 6 axes, of,lmIchlne motion'create aU ,gear ,toot.hl geometry

Wrth PHO,ENIX 6 CNC~control-led axes, do it all. The secret isGleason's ability to control these 6axes so precisely that [he mechan-ical motions we aJi know are nolonger required.

Changeove,rIn mfnutes,

Random batchprocessing can be assimple as calling upa program and put-ting the tooling inpla-ce. Cuner lengthadjustments are ac-complished auto-matically Trainingrequirements areminimized.

High Illepeatablllty. ..•sec~p~to-letup and'imathlne--to"'machine

Tlhe variables disappear. Setup in-consistencies between batches, orbetween machines, are replacedwith CNC control of all setupparameters. rrial-and",errm adjust-ments are eliminated, making scrapa real rarit)~.

,Ceslgnedf:orautomatJon .• "now or later

The PHOENIX design, combinedwith its high repeatability and com-p~ehensjve CNC control make itsimple to automate part and toolhandling. eel/and system integra,-don has never been easier.

I

PHOENIIXSlX'Iludsl precision ellmlnat,es uadl'e,

eeeentrlc, Clutter tilt •.SWlvell ....

,ilndtradlUonaJ setllp.

GleasonThe! W'arlcllaf Giellr,lng

CIRCIJE ."·2.0 ON IR,EADER IREPlV CA_RD

e,IMTS:IWJIIIIIMIUIl'.aII1I11l"

BOOTH630'l

HURTH Rotary Deburrlng Machinesand TOols

fOr fast, reliable gear chamfering and deburrlng• Short cycle nmes• High ouallty• LOwcost per PieceGearsand cluster shafts of up to4 gears of different configurationsare chamfered and deburred simul-taneously on different versions ofthe HURTHZEAline.HURTHloading and unloading equip-ment can be fully integrated in allZEAdeburring automatics.Also, HURTHbrings you the right toolsystem for every application. Fromthe basic version, consisting of twodeburring tools and a meshing gear,right up to tools incorporatingmaster gears, side trimming discsand tools, burnishing wheels andpressure plates, HURTHhas the righttool for your application.We would be pleased to review yourgear deburring and chamferingrequirements and invite your lncurv.

• Automatic rotary deburring machine.

SEND FOR FREE BOOKLET -HURTH Standard HN 115 206 -a practical guide forselecting the appropriate chamfer.

• Rotary deburring tools on a four station unit machining gears forsecond and fourth. while two side trimming tools and two TiN sidetrimming discs remove secondary burrs.

//lJiiij'"7~ARCUT SALES, INC.

&/~~~61 Industrial Park DriveFarmington Hills, MI48024

313/474-8200 Telex 230-411STARCUT SUPPUED PRODUCTS AND SERVICESstar MachilDe Tools I star cutting Tools / Gol'dstar coatl'ng!s / Hurt,h Machine TOols IMikron IHobblng! Machines and' Tools /TiN Coating Systems /Stieber Cil,ampln,g TOOlS

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