the journal of gear manufacturing · high production cutting of identical gear shapes requires both...
TRANSCRIPT
The Journal of Gear ManufacturingMAY/JUNE 1988
High production cutting of identical gear shapes requires both speed and accuracy.But some fast, accurate machines incur maintenance and operator costs that cat up profits.With Kapp CBN form grinding, you don't give up productivity to maintain profitability.• Kapp CBN machines are faster than other grinding processes, with no heat checking orburning. Kapp CBN accuracy has proven itself on spur & helical involute fOnDS,radial formst
near-net forgings, internal & external profiles, vane pump slots and constant velocity joints.• Kapp CBN machines are operator friendly, for fast recall of setups and accuracy time aftc.r dmc:.• Kapp CBN machines use CBN wheels that incorporate the desired form right in the wheel withno need for dressing.Ifyou'd like to find out how Kapp CBN technology can improve your productivity. contactAmerican Pfauter, 925 Estes Ave., Elk Grove Village,IL 60007. Phone (312)640-7500.
ON brings you hob speeds up to2,000 rpm and high rigidity with theAS /2 gear hobblng machine
The model A33/21s a completelyautomatic gear hobblng machinewith programmable controller. Usersrealize the full advantages of modemTIN coated hobs while hobblng gearsof from 0.157" 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 anc other Mlikron hobbingmachines for gears up to 8" diameter,
/!LJioj'"7/liARCUT SAl'ES, INC..&,!~~~6~ Indust~lal Park Drive
Farminqton Hills, MI 48024313/474-8200 Telex 230-411
STARCUT SUPPLIED PRODUCTS AND SERVICESstar 'Machine TOols I 'star Cuttlnig Tool's I Go,ld Star Co,atlngs I Hurth, MaChine, Tools IMllkron IHob:bing Machllnes and Tools /TIN ,coating 'systems /stelber Clamping TOOlS
CIRCLE A~, ,ON READER REPLYCARD
EDITORIAl. STAFPPUBUSHER&:EDITOR·IN-ClDEFMichael GoldsteinASSOCIATE PUBLISHER &:MANAGING EDITORpeg ShortPRODUcnON/ADVERTlSINGPatricia FlamASSISTANT EDITORNancy BartelsEDITORIAL SECRETARYPamela NolanFREELANCE ARTISTCatherine Murphy
RANDALL .PUBI·JSUlNG STAFFPRESIDENTMichael GoldsteinVICE PRESIDENTRichard GoldsteinVICE PRESIDENT/GEN. MGR.Peg ShortART CONSULTANTMarsha Goldstein
RANDALL PUBLISHING1401 Lunt Avenue
P;O, Box 1426ELk Grove, II... 60007
(312) 437-6604
COVER"South Pointing Chariot" An ,ex;unpJe of
"an ancient Chinese vehide Mritha woodenfigure always pointing to the south~ " simi/arfigures .are found in archeological digs inChina Theireocact puIpOSe is open to debate,but One common theory is that they wereused as a kind of compass on long wildernessand desertjoumeys. Although they are notIIisib/e on this model,. curviJineargears are thelikelydriving mechanism on the original de-vces. The figure on our cover is a contem-porazy working model ,using a system of dif.ferential.gears. It was produced by Mr. DonFrantz of Bwbank. CA.
CONTENTS P,\(~E vo.- -----
CURVILIN-EAR. CYLINDRICAL GEARSLiu Shue- TsengShanghai MetailuIgical Machinery FactoryShanghai, People's Republic of China
8
TOOTH ROOT STRESSES OF SPIRAL .BEVEL GEARSHans Winter, Technical University of Munich,Munich, West GennanyMkhael Paul, Zahnradlabrik Passau GmbH,Passau, West Gennany
13
OOMPUTER.-.AlDED DESIGN OF THE STRESS ANALYSISOF AN INTERNAL SPUR IGEAR
§eng-Fong Hwang, Dennis A. Guenther,The Ohio State University; Columbus, OHJohn F. Wiechel, S. E. A., lnc., Columbus, OH
22
DEPART~IENTS
EDITORIAL 5
BACK TO BASICS ••.•HEUCAt GEAR MATHEMATICS FORMULAS & EXAMP.LES
Eliot K. Buckingham,Buckingham Associates, Inc., Springfield, VT
35
CLASSIFIED 46
T~CHNICAL CALENDAR 48
Mayi§une 1988 Vol. 5, No ..3
·GEAR TIOCKNOLOGY. The J""rn':l of Gea •. l'>Ianu(aClttl'Iq (lSSN 0743-6858) I. published bimonthly by R;o:ndoll 'PubliohLnQCo .. [nc .• 1425 Lunt Avenue .. P. O . Box 1426, Ell<. Grove VlUage. IL 60007. GEAR TECHNOLOGY, The Journal of Gear M..nu{",,-b.!ri~ is distributed free of charge In qUolI.fiedfndlvlduels and finns in the .1Iea!:' manuf"c~1"l1lllInd\!Stry. S<!bscrip~on ",teo for non-qualified. individual. and fi""" are: 535.00 ,n the 'United SIlII£>, $55.00 for foreign countries. Seoond-C~ postllQ!: !)IIld al. ArllnQIonHelllhts. IL and at additional mailing office.Po.tmaster: Send address changes 10 GEAR TcECHNOLOGY. The Journal of Gear Manuf~turing, 1425 Lunt A""m re ·, P. O. Box1426. Elk Crove Vill~e, :IL 60007.@Contents cop)'rillhted 'by RANDALL PUBLISHING 00 .• INC. 1987. Articles ill>lJearins in GEAR TECHNOLOGY :may not bereproduced in whole' or in port withoul the exp' .... permtssion of the pybli$ner o. the altU!or.MANUSCRlP1"S, We are requ.estinil technical papers with an educational emphasis for anyone h".lns anl!thini!'w do with ·thedesign. manufacture. I£>~IlIIor processins of gears. Subject.. ooU!lht are solutions to. specific :problems. explana.tiOl15of new: 'b!clInoICIIIY.'techniques, designs, processes, andalternative m;mufadu:ri11ll methods. These an .range from. the "How 10 .• ,"' of 11l= ruttiiia{BACK 1'0 BASICS) loth.· most advaru;ed ~chhOlogy ..All :manuscript.. .submitted wlll be carefully considered. Ho~, the Publisherass umes no 11lSJlQnsibility for the safelY or return of m;muscripts. Manuscripts ml.l$tbe accempanled by a ""lf~. selt-sramped'envelope, and be ..,nl 10 CEAR TECHNOLOGY, The Joumal of 'CeaJ" .Manyf~lurillll, P.o.. Sol!; 1426, Elk 'Grove, m. 60007, (312)437-6604.
Befo~e'you buy a CNC Gear-,..,.-~Inspection svstem, ask
these four qu,estions
1. Can we get complete inspection ofboth gears and cutUng tools?With tihe M &. M 2000 QC System, the answer is~. The system will test lead and. ,involute profilecharacteristics of internal or external
'gears and splines.
It is the only universal gear tester available which providestrue index testing: without expensive attachments ..In addition,we have fully-developed software, for checking' hobs, shavercutters, shaper cutters, and other cylindricall parts such asworms and cams ..
Hob Pressure Angle Check Shaver Culler Lead Check
2..Can you customize software to' meet our quanty-inspect'ion specifications?At M & M Precision, we write and develop our own inspectionsoftware. Our technicat team can and has implemented in-spection specifications into specific software for individual re-quirements. Our current library includes line/curve fitting aswell as modified K-chart analysis routines.
UnefCutve Fitting Modilied K·Chart Analysis
3. 'OK, you can inspect gearsand cutting tools. What else is available to,
aid us in quality controll of the manufacturingprocess? - -
At M & M Precision, we have fully integrated such advancedsoftware packages as Statistical Process Control and ToothTopography into our standard testing software. Our SPC pro-gram 'can lidentify non-random variations and provide earlywarning of variations which are approaching tolerance limits.Our Tooth Topography software features automatic testing of,lead and involute at multip'le locations and provides two- andthree-dimensional graphics.
SPC Run Chart Topological Map
4. Do you have the technical support team and in-staUation experience to back up the hardware andsoftware provided?At M& M Precision, we have a technical team with over 45man-years ofexper.ience in developing CNC gear inspectionhardware and software. All software for our QC 2000 Systemhas been devel'oped in-house. ln addition, we have working'lnstallafions at these leading companies: -
• General Motors • Cincinnati, Milacron• TAW • Chrysler, Ford Motor • Pratt & Whitney• Warner Gear • Rocketdyne
For details on our advanced QC 2000 System and availablesoftware, contact M &. M Precision Systems,. 300 ProgressRoad, West Carrollton, Ohio 45449, 513/859-8273. -
=-.MIii"tl..- PA. Se'SIDN".SYSTEMS
AN ACME·CLEVELAND COMP.£J.NY
CIRCLE A-6 ON READER REPLYCARD
"We have met the enemy and he IS us," says Pogo,the cartoon ch.3raaer. The enemy IS the cnss in oureducational .sysrem, and "cnss" is the only term that ac-curately describes the sillJation. It is every bit as serious, ifnot more so, than the crisis that followed the Sovietlaunctling of Sputl1lk in 1957 - and for many of thesame reasons. Our failing public education systemthreatens our position In the global political and businessarenas: and this time, it's not just the Soviets or theJapanese who need robe taken seriously as competitors.Every country in the world that graduates better preparedstudents [han we do - and'there are a great many ofthem - has us at a competitive disadvant.3ge.
My personal concem. and, frankly, sense of frustrationhave been raised by a number of Items which havecrossed my desk in the last few months. Late last year,Secretary of Educatlon, William Bennett, recommended abasic core cumculum for American high schools emphasiz-ing writing. reading. language. history, science. math. artmusic and physical education; and then he pointed outthat only 15% of the high schools in this country nowoffer such a progam to their students. At the same time.according [0 Business Week magazine. federal spendingfor education has dropped 17%. In my own state. Illinois,a school pioneering in math and science for giftedstudents is in d.3nger of closing because of lack of funds,and both the illinOIS Sr."3teChamber of Commerce and theIllinois Manufacturers Association oppose any tax increasesfor education - out of fear mal higher taxes will causebusinesses [0 move elsewhere. It is disheartening to realizethat neither government officials nor businesspeople seemto see the connection between poor education and apoor quality work force.
While your child or mine may not be victims of thiscnsis, all too many American children are barely literateafter twelve years in our school system. Far too manyyoung adults cannot read and write well enough to fiJIout a job application, and they cannot do math wellenough to make change if the electronic cash register fails.They don't know an atorn from an aardvark. much lesshave any ideas about or familiarity With "soner" su~ectslike err, literallJre or any music wntten before 1980. Whatis worse, they have been so turned off by their educa-tional experience that they have no interest in any kind' offormalized leaming. The system as it works now doesn'tprovide basics for the average student. suffiCient specialhelp for the culturally and economically disadvantaged oracademic challenge for me grfied.
Of course. not every school is a fallur;e. Not every childis :Iost. We have some marvelous school systems that pro-vide for the needs of most of their students and tum outfully prepared graduates. Unfortunately, there are not
nearly enough of such systems. Too many lose .SlUdentlto boredom. Indifference or simple incompetence.
We can already see some of the results of trus neglectAccording to a study by the U.S. Science Foundaoon.which compared the scientific education of students fromseveral countries, American students ranked below themiddle of the scale. U.S. high school seniors were nearthe bottom. Last year, 55.4 % of the engineering doctoraldegrees gramed in the U.S. went to overseas students. AtPenn Sr."3te.the figure was 74%.
The problem Isn't that we're training roo many foreignstudents, but that we're not training enough of our own.While our best universities are still providing advancededucation [hat is the envy of the world. many of ourown citizens are not well enough prepared to take advan-tage of it. Corporate recruiters for some of the country'sleading technological firms compJainthat they cannor finda single, qualified American to hire.
Easy answers to this problem oon'r exist. Wringing ourhands, pointing 'fingers of blame or eSt.3blishlng quotas tolimit the number of foreign students training in the U.Sare nor really helpful solutions. Neither is simply throwingmoney at the problem.
We must seriously reassess the place that educationholds in our own minds. Do we really care about ex-cellence in education, and are we willing to demand It ofour own children 7 Since the improvement of the sysremstarts at home, will we insist mat the t.v. beturned offand the homework done 7 When the option is offered.are we willing to insist our Children take a challengingcourse instead of one that is "run", and then are we will-ing to take the time [0 work with them and theirteachers, or go to school meetings and conferences? To a
May/June 1988 5
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~-~------------------------~,
~. "'I!!!!!§. - I.-.r • ., =~I -. 31g£OIIM~RC~~RIVE _ _ 0.11: '··.00· .•2'·217. .FAX: '.2U.512-fJ3J5 .1
.- FORT WAS.HINGfON" PA. 11034 '-tl' Pin" •• , 1-215-542·8000' =
large degree. our children will take their cues about theimportance of the If education from us.
More than this sacrifice of time and effort we are goingto have to be prepared to sacrifice some money. and.perhaps more painful, some of our most cherished presup-positions. This country has some enormous economic andfiscal problems to solve, and the educational crisis is partof this larger picture. We are facing some tough choicesand some favorite "untouchable" spending programs mayhave to be touched after all. Everyone's campaign pro-mises to the contrary, we all may have to pay highertaxes.
At the same time. we have to give up the Idea thatdealing with the educational crisis - either fiscally orphilosophically - is the exclusive responsibility of the stateor federal government. Education has always been a localissue, a place where individual citizens were deeplyinvolved. People like us are the ones who run for theschool board and the PTA. make the phone calls andwrite the letters tnat motivate change. A hassle perhaps.but historically, that is the way the job gets done.
Businesses can respond directly to the crisis in educationtoo, An entrepreneur and philanthropist in New Yorkadopted an entire sixth grade class in the South Bronx. Hepromised a complete scholarship to each child that com-pleted high school with good grades. Better than three-fourths of them accepted his offer. Few of us have thewealth to do something on that scale, but we can in-volve ourselves in creative ways like internship programs.joint ventures and corporate or trade association scholar-Ships. This is not the time to r~ect plans just because theyare Innovative or non-tradirional.
Certainly such ventures take time and money, but weeither become more involved in the business of improvingour schools now. or we pay later when we have to im-plement programs to retrain workers to tasks they shouldhave learned in school in the first place,
We need a change in our national attitude and resolu-tion, just as we did at the beginning of the space pro-
r------I-I---I-I-IHOW '1'0 CORRECTLYIDENTIFY YOUR GE.tR·S: sp,c.'!Jilll01'ol'd'l'.illr lit, "'1'0"1 siz·, I,a,. CIlII .61-.I:o.tly,. tim.-colluml", tlll4 Ctl"Cd"'.'dblutroll.S .d,'ay, III IOU' "odll.c,lolI.
0'1'011. Olun' Oag" /J'"'' Y·OII,. ·Gu/J·r;/Jll/n/0, .tcc.ur/Jc,J
gram. We must tell our elected officials that we are seriousabout improving the educational system and are willing tomake the necessary sacrifices to do it. We can't expectpoliticians to commit political suicide. If we routinely res-pond to the candidate who offers us the easy. short termsolution at the price of a mortgaged future farther downthe road, then we cannot be surprised if that is what weget.
Secretary of Education William Bennett demands thatwe "raise the expectations" for our children, demandingthe best from them. At the same time. we should "raisethe expectations" for our politicians and ourselves as well.We can demand more than the cheap. easy. popular orfadle solution to our problems, and by demanding it wewill eventually get it. A democratic country gets thegovernment it deserves - and the government it asks for.
This is a preSidential election year. It is the perfect oppor·tunity to get the message across [01 those we are sendingto Washington. to our state capitals. and to our localgovernment bUIldings. The message is that we need abetter educational system and that we are willing towork, sacrifice and support officials who will provide uswith the necessary leadership to get the job done.
What is at stake here is nothing less than our children'sand grandchildren's futures - futures conceivablypopulated with more and more under-educated peoplepushed out into society where they can funaion onlyminimally. The longer we wait, the higher the pricebecomes, If we don't give American children the besttraining possible, we deprive them of the tools necessaryfor their own welfare and for the welfare of the counrryas a whole.
The crisis in American education is real. It is as real athreat as any disease, invasion or high tech weapon, Ifwe ignore it it will go away - along with any oppor-tunity for our country to maintain its standard of livingand its position ofleader.ship in the world.
Michael Goldstein, Edltor/PubJisher
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6 GearTechnology CIRCLE A~3 01'111READER REPl V CARD,
time, after time, afteYou need Nonnac's Fonnaster CNC Grinding Wheel Promer
The Fonnaster gives you inspection guage accuracyNormae guarantees an accuracy of :t .0001" (.002 mm)from programed dimensions and provides less than ,0001"(.002 mm) positioning error throughout total slide travel.
Easily adaptable to meet YOUT need ..The Forrnaster is designed to mount easiIy on nearly any type ofgrinding machine. It iscompa.ct and lightweight and just amounting bracket and only minor machine modifications arerequired! for insta'llation on most grinders. Installation usuallytakes one da,y or less.
The unique features of NO.nnac'sFonnaster.• The unit is totally sealed and air- purged fortroublefree operation and protection fromcoolant contamination.
.' Double nut pre-loaded roller screwstransmit servo motor to slide movement
• Optional rotary diamond dressing wheelattachment provides greater accuracy andeliminates error caused by single pointdiamond wear.
Formaster users have reportedinc-reased production and greatera.ccuracy.'. Increase in wheel life of up to ,a·w timeslonger than with other units.
.' Production increases up to .20% in the flrsttwo, weeks of use.
.' Set-up-time is reduced to 80% of what itused to be with improved surface integrity I
oHheparts.
• Eliminates need for skilled operators as the unit isautomatic. Unit can be retrofit for use with automatic infeedl.
Now Nonnac offers you software for the Fonnasterfor automatic prt)Qmm generation.Normae has developed optional software that runs on apersonal computer in a question and answer format It allows,the user to enter part print specification!! (including root andprofile modifications) in standard gear nomenclature. Theprogram outputs N.C. code loaded into the Formaster controlfor wheel profiling, and: also detailed printouts suitable for
gear profile analysis. This software contains logic for all- types of root zone grinding, true involute and
I exponential flank modifications as well as "Proudness"of the enure profile as indicated in the 'K·diagram above.
Let us show you what the Fonnaster 'cando for your grinding operations
If you would like, we can arrange a demonstrationofthe Formaster for you, Call or write us today,
NORMAC, INC.P.O. Bo)( 69'
AJ'dell.N.C. 28704,(704) 684·1. 002
IP.O. Box 201Northville. Ml48167
(313) 34ig·2644
Liu Shue« Tseng,Shanghai MetaBurgica1 Machinery Factory
Shanghai, People's Republic of China
Curvilinear Cylindrical Gears
8 GearTecnno'iogy
(Photo on lem Wooclen model of 'South Pointing Chariot" ..Althoughthe curvilinear gear is not visible an this mod ], such gears wereused! in the manufacture of these devices, which. may have servedas compasses in ancient times.
The curved tooth ,cylindrical gear is one of ancient design.Samples which date &om the period of the Waning Sta.te(475-221 BC) have been excavated from archeological sitesin. China. One such sample is now on display in the Xi'anCia,y Figures of Warriors and Horses Exhjbition Hall. Thisexample is about 314'" in diameter and made ,of bronze. Itwas used in, 'the famous model, "Ancient Chinese Vehicle Witha. Wooden Figure Always P,ointing to the South." Al.thoughthis early gear is handmade and somewhat crude, it is a viablemodel.
Chinese ,technologists have been interested in thisancientdesign, and, after overcoming some technical difficulties, wehave suooessfully designed the special machine tools andmanufacturing processes necessary to bring the curvilinearcylindral gear into pradicali production, The first marketableapplications appeared in 1980. Photographs of some of thesegears are shown in fig. 1.
Basic: Curvilinear Cylindrical Gear ManufactureGear manufacturers and consumers are always searching
for more compact or higher capacity gear sets. This designof gearing offers a. w.ay to achievethese characteristics ..Thebasic manufacturing process is described below.
First imagine a tooth generating rack made of hard metal.The rack is kept stationery, and the form of its longitudinal.tooth line iscircular rather than the usual straight line. Aplastically compliant gear blank i.s used .and is run along therack witha. pure roning motien, that is.. with no slippage;thus. the blank is squeezed and extended to produce a cir-cular toothed. gear conjugal.e to the rack. (See Fig. 2.) [f thetooth line of the rack were not circular, but of another form,the teeth produced on the blank would be of another curvedfonn.conjugate to that rack. All such eusved tooth forms arecalled "curvilinear toothcylindr:ical gears."
To, produce such gears, the combined motion of the gearblank and rack must obey the following rule:
(1)
Va - linear translational velocity of Ithe gear blank axis inmeters per minute (m/min).
Vp - Circular velocity of the blank on time pitch diameterin m/min.
D'p - Pitch diameter in mete:rs.... - Pin Rotational speed of the gear blank. in rpms.
In practice we use a. rotating face mill cutter to producethe gear teeth. The cutter spindle rotates on a .fixed or sta-tionary axis while the gear blank is rotated and translatedacross the turning cutler teeth according to Equation 1. Bythls rolling gen.erating. process the cutter produces one spaceon the gear. The blank is then indexed one tooth and the
Fig. 1- Some samples of curvilinear tooth gears.
Ag. 2- The generatil1lg principle of ,a curvilinear tooth cylindriG31 gear.
AUTHOR:
UU SHlJL.l'Sl'l"JGgmduated {rom the Hong Chaw University withBacheror of Science ,ill Mechanical engineerillg. After his, universitywork. he received additl'onal trailling with! National AgriculwmlEngineering Corporation. For the last 38 years he has beetl· employedby the Shanghai; Metallurgical Machinery Flretory. working in foun'-dry technology, IZS chief of the maintenance md design deptfrtmentand, since .1980, as senior chief engineer in dUrrge of f1Uichine ,designQ1u1 reseaTch work. Mr. l.iu has beell working 011 t.11ecuroilinem toothcylind'rical gear project since 1978. -
IMcrt'iJ.lne 11988 9
Fig..3 - Cutting a circular tooth gear with a disc tool.
t - TIn
Fig. 4 - The supplementary contact ratio, ofa circular tooth gear.
generating cycle repeated. This sequence is continued untilall the spaces and teeth are formed. (See Fig. 3.)
Since the generating motions are relative,an alternatemachine construction would have the gear axis fixed and thecutter translated past the gear blank. The results win be 'thesame.
for ease in manufacturing, we made the profile on the'cutter blades in the form of an isoceles trapezoid similar tothe tool used in Acme thread turning. The generating actionofthis tool configuration develops a gear with an. involuteprofile of pressure angle a in the central section and a hyper-bolic evoluted profile in. theoff-center sections .. But at anysection across the gear face,. its concave and convex flanksare conjugate. so any pair of gears can be properly meshedtogether.
Obviously, the radius of curvature on both flanks of thegear must be alike to achieve proper engagement. The radiusof curvature on the outer form of the cutter teeth is largerthan that on the inner form. So after the gear is formed bythe first cutter, a second cutter is utilized to form the convexfaces to the same radius as the concave flank. (See Fig. 3.)
We have now developed a continuous cutting processwhich provides greater ~anufacturing efficiency.
Strength Analysis of a Curvilinear '[:oothThe Contact Ratio. According to the theory of mechanism,
we know that the contact ratio of a spur gear is
Eo ... El + E2 - EA
Eo = 4(Z1 + 2)2 - (Zl C050:)2 +2 1I:C050:
(2)
",(Z2 + 2)2 - {Z2 .cosa)2
2'/1'cosa
(2a)
where
ZI = number of teeth on the pinionZ2 = number of teeth on the gear.O! = the pressur'e angle ..
A curvilinear gear tooth possesses the same contact ratioas a spur gear, plus a supplementary contact ratio dueto faceoverlap, (See Fig. 4.)
The - supplementary contact ratio of a curvilinear toothgear is
h hEg='-=-- t IDW
VVhere
"'" Tooth pitch distance
m= Module of the gear
h = R, - JR! - (~)2
R, = Radius of tooth curvature
B = Gear face width
the total contact ratio of a curvilinear tooth gear
is
(3)
Evidently, EE is greater than Eo. That means that a pairof curvilinear tooth gears has a greater number of gear teethsimultaneously in mesh; thus the force exerted on every in-dividnal tooth may be reduced materially.
For instance, if Eo ofa spur gear equals 1.7. that meansat a certain instant, only one tooth pair is meshing. But fora pair of curvilinear toothed gears, I:E increased to 2 ...2, atleast two tooth pairs are in mesh at all times.
The Moment of Inertia a'nd Section Modulus of the ToothRoot of.a Curvilinear Gear. First, we observe the cross sec-tion of the tooth root of a spur gear. (See Fig .. S.)
The moment of inertia is
(4)
The section modulus is
1W - - BSlIp1I' 6-0 (5)
Then let us observe the cross section of 'the tooth J100t ofa circul!ar tooth gear. (F~g..6.)
By computation. we found l!:hat its modulus We is bigger'than Wspw.We tabulate the numerical value of a real exam-ple iin Table 1.
For instance, if we design a pair of curvilinear gears whoseB/2Ro· 0.6, from the table we find Wc/WIfN.J -, 1.20 andI:E --' 2.76 (tE of the spur gear is o~y i.7). Then thestrength of such gear is 1.20 X 2.76 = 1.95 times strongerthan the spur gear. 1.7
Tab~e 1. Moment ,of Iner:lia. and '5ecti.onModulus of Circ:u1a:r Gear Tooth
B/lRo VI.- Wc/WipIII EE
O.SO 2.52 1.059 2.55
0.52 2.64 1.076 2.60
0.54 2.79 1.105 2.64
I
0.56 2.96 1.139 2.67
0.58 3.14 1.173 2.72
0.60 3.31 1.20 2.76
0.62 3.53 1.236 2.81I
0.64 3.74 1.275 2.85
0.66 3.93 1.304 2.90
0.68 4.18 1.344 2.95
0.70 4.48 1.395 3.0
Ie - Moment of inertia of a circular gear tooth
~ - Moment of inertia of a spur gear toothWe - Section modulus of a circular gear toothW IpW' - Section modulus of a spur gear tooth
EE is calculated corresponding to a special case ifZl - 1.9. and Zl '""'60; whereas Eo of spur gear isonly 1.7.
Table 2, The Comparison of T,ooth LengthBetween Spur Gear and CiI'cuIar Tooth Gear
B/lRo LIB B/lRo LIB B0.50 1.047 0.60 1.072
~
0.52 1.052 0.62 1.079
0.54 1.056 0.64 1.085
0.56 1.060 0.66 1.092
0.58 1.066 0.68 1.10 I
looth foot thickness
I -.. JFig. 5-Cross section of the tooth root of iii 5Pw-geu.
B
.c
R1-R1-R"R"...nom:iN1 rdius
of ItoOth lineon pitch.cIlA.
Fig. 6-Cross section of the tooth root of a. circular t.oot" ear.
The Length of Tooth Contact Line. The length of the toothconeact lin· of a spur gear equals its width, but the Iengthof tooth contact line of a circular 'tooth gear is somewhatlonger; We may compare the two .in Table 2.
The Stress Induced On! a Curvilinear Gear Tooth. FromUris discussion, we can conclude the foUowing:
• The bending stress of circular gear teeth is reduced by Ithehigher contact ratio and the greater section modulus.
• The contacting stress of 'the circular gear tooth is alsoredueed by its higher contact ratioand longer ~ength oftooth contact
" If an alternate tooth curve is used and is .appmximateIyan ar,e,then it has features similar to a circular gear.
The Characteristics of Curvilinear Toala. Gears1. Curvilinear ItOOth gears possess a higher bending and ,con-
tacting stnmgth; Ithus, Ithe center distance of gear boxesmay be reduced, while maintaining power transmissions.
2. Since there are more '~eeth simultaneously engaged, these'gears run smoothly with low noise.
3. Oil is retained within the ccncave tooth surfaoe, so' there'is always an oil film between the two, engaging surf.ace5,.
May/June 1988 1111
resulting in good lubrication qualities.4. No axial foree is produced during operation, so axial bear-
ing load is negligible.5.. Gears with similar, but not identical modules can be cut
with a single tool set, saving tooling costs.6. Because of the simplicity of the cutting tool, it is easy to
implement carbide cutting, resulting not only in an in-crease inrnanufacturingeffidency, butalso in the abilityto cut gears w:ith much higher hardnesses, producing im-proved quali.ty.
At present, these gears are in use in. steel plants, aluminumrolling mills, 'cement equipment plants and other places. Theirsuperiority has been proven in practice. The service life ofmachinery has been improved by anywhere from two to fortytimes or more.
Editors' Note:Gearing of this type was avaiIabie in the U.S. some years
ago under .the name "Spuricals": The herringbone .type gear,in use worldwide, also has the same attributes and wouldbe the first choice by a gear applications engineer today. Somereasons are currently .available machine and cutting tools andthe availabliIity of some competitive gear vendors. Some ofthe variations in the longitudinal form of gear teeth that havebeen used are shown in Fig 7.
']:0. produce these get:lrs with proper contact, some formof profile and longitudinal adjustments may be necessary. Theprofile of the blades can. be modified to produce t.ip relief orroot relief on the gear tooth, and the radius of curvatur:e of
EQU~L TO THE' TASK
12 Gear lechnol'ogy CIRCLE A-4 ONI READER REPLYCARD
FE
Fig. 7 - Diagrammatical views of diffe.rent types of herringbone gears,
the cutters can be ,altered to provide a central bearing CO/'l-
tact, avo;iding tooth edg;e contact.
Acknowledgement; The edi.torswish to thank Mr. Willi4rn L /wminck forhis help with the technical editing.of this article.
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To,oth Root Stressesof Spiral Bevel Gears
Hans Winter,Technieal Umversity of Munidt
Munich, West GermanyMichael Paul,
Zahnradfabrik Passau GmbHPassau, West Germany
GEAR 5PEORCATION5
Nomenclature Pinion Gearnumber of teeth 11 36
normal pressure angle 20°
normal module inmidface width Imm I 12
outer transverse module Imml 17.50
outer pitch diameter Imm I 192.50 630.00
mean spiral angle 34.597°
face width Imm I 110.00
cutter radius Imml 210.00
total contact ratio (theoretical) 2.8addendum modification 0.34 -0.34
thickness modification 0.025 -0.025
material 17CrNiMo6(through-hardened)
Variation of lengthwise crowning:
Cutter Eccentricity ImmlPinion GearGear Set
AContact Pattern
IntroductionService performance and load carry-
ing capacity of bevel gears stron-81ydepend on the size and position of thecontact pattern. To provide an optimalcontact pattern even under load, the geardesign has to consider the relative dis-placements caused by deflections or ther-ma1 expansions expected under serviceconditions, That means that more or lesslengthwise and heightwise crowning hasto be applied on the bevel gear teeth.
In order to gain reliable informationon the interrelationship between stresses,tooth crowning and relative displace-ments between pinion and mating gear,extension investigations were carried outby the authors. The aim of these in-vestigations was to determine the quan-titative influences of different displace-ments on the tooth root stresses and, byevaluating the results, to give recommen-dations for choosing the optimal. amountof crowning.
±7 0 "small"
Test Gears and Investigation MethodTo measure the tooth root stresses,
strain gauges were applied on the testgears. They determined the stressdistribution over the root filletand overthe Face width ..A detailed description ofthe strain gauge application and measur-ing method is given in Reference 1. To,provide enough space for the straingauges, test gear sets having a largemodule had to be chosen. (Main Data.See Fig. 1.) Three sets of test gears wereavailable, differing only inthe amountof lengthwise crowning. (See Fig. 2,) Thetooth profiles were kept exactly thesamefor an pinions and gears.
First we shall discuss the case of op-timal mounting positions for the pinionand gear. By use of special features onthe test rig, these positions could bemaintained, even under heavy loads.Then the additional effects caused bydisplacements shall be described. Notethat throughout the discussions, "crown-ing" shan always be understood aslengthwise crowning.
Influence 'of Lengthwise Crowningon Tooth Root 'Str,esses
Fig. 3 shows the maximum tooth rootstresses during one load cyde in the caseof multiple and single tooth contact. The
Mcry/June 1988 13
B
C
"large"±1.2 o±2.S 0 "mean"
~ig. I-Main data.of test gear sets.
t 0,)0Ccin
mmO,20
I1,\ _ ~eor set A
'\ I /\ /, gear set C I~~
II V-:""<, '-. /1geYSj' B
"/ ......V,
~'"~~ 1/ ~ --~. ~b12------l
b = 110
0,10
o
heel toe
Fig. 2 - Lengthwise crowning of the test gears.
stresses measured with single tooth contact (no load distribu-tion between pairs of teeth meshing simultaneously) slightlyincrease with targer crowning" This obviously is an effect ofthe more and more concentrated load application when thesize of the contact pattern is reduced bya higher amount ofcrowning. This can be described as a problem of load dis-trlbutlonover the lines of contact.
When looking at the curves for multiple tooth contact inFig. 3, one can see that the stresses now are considerably
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1'4 Gear feohnology
0---0 sln~le tooth cDntactx-x l1'Iulhple tooth cantad
.--_--.-......:.gea.;,.;...t -,---,.400
fJ50"'E
!zE
1--*--+-~:loo-"=-....=..:.tJOO0-b
.. I
250+'--+----+---1
150+----+--+----1
"-Jt~~~tr-;;~·100+ b= 2 kNm
SO+----+--~---~ r----+-----l-----+50I B C gear set Ao -- B C gear 511 A
o2 3
ceria.! ".tOOO-o 1 2 3 0
cella./b.fOOD--
Fig. 3 -lnfIuence of lengthwise crowning on maximum tooth root stresseswith single and multiple tooth contact (optimal mounting condition).
lower than in the case of single tooth contact, especially forgear set B with smaUcrowning and when high loads are ap-plied. For gear set A with large crowning, there is almost nodifference in stresses between multiple and single tooth con-tact even with high loads. Gear set C also gives a dear resultiThere is quite a strong influence of crowning on load distribu-tion between neighboring pairs of teeth. In gear set A onepair of teeth has to carry almost the total load; on the con-trary, in. gear set B there is a considerable sharing of loadbetween two or three pairs of teeth,
Therefore, two different problems relating to theeHects
Nomenclature
b - face width, mmba tooth length, mmb-rB width of contact pattern, mmCc lengthwise crowning, mmdm mean pitch diameter, mmf displacement (in general), mmfa offset displacement, mmfv axial displacement, mmfr; shaft angle deviation, degreesg". length of path of contact, mmh tooth heighthw active tooth heightIB length of line of contact, mmIS' projected length of line of contact, mmw influence coefficient for load distribution
gear set0--.0(] A0----0 Bb--t!J. C
Fig. 4 -lnflumoe of lengthwise ao1.'lI1IDgon the load distribution over the lines of contact (single tooth conract).
ofcrowrUng have to' be discussed: Load distribution over linesof contact and load distribution between gear pairs meshingsimultaneously. We Can Solve Your
I Distortionand WearProblems ..
Load O.istributi.on 'OY,er Lines ,of Contact.By using the superposition principle the load distribution
over the lines of contact could be determined from root stressmeasurements with pointwise load .application on the teeth.lFig.4 shows the results for the test gears valid for the desired
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mesh positionforce, Ntangential force in mean face width, N
- optimization factor for lengthwise crowning- lengthwise crowning factor- displacement factor- measure point- position of point load
outer cone distance, mminner cone distance, mmtorque (on pinion) carried by one pair of teeth,Nmtotal pinion torque, Nm
- load sharing factor- base spiral angle, degrees- inclination angle of line of contact, degree- stress in depthwise direction, N/mm2
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gear set A
gear set C
pitchcone
pitchcone
FIg. S-Zones of action wifh paths of contact and load maximum on linesof contact.
position of contact pattern. In this case, with equal surfaceintegrals for each position, the load distribution is very un-symmetrical in the beginning (E2, E3) and the end (E7) ofthe contact. Only on the particular line of contact runningapproximately through the center of the tooth (E5) is therean almost symmetrical distribution. The gradient of thisdistribution depends on the amount .of crowning.
This ,effect can be explained when looking at the corre-sponding zones of action shown in Fig. S. The positions ofmaximum loads on the lines of contact obtained bymeasurements are compared with the theoretical paths of con-tact. There is a quite dose agreement that corresponds withthe theoretical idea of the Hertzian contact, This figure alsoexplains why unbalanced load distributions must appear inthe beginning and t:he end of the contact, while symmetricaldistributions can be expected on the center erossinglines,Since bevel. gears normally are designed so that the paths ofcontacts run through thecenter of the zone of ad ion, thisresult may be generalized.
It has to be pointed out that the results of Figs. 4 and 5are valid in the case of single tooth contact. In Reference 1it was mown tha.t the maximum stresses appear in mean meshpositions; for example, E5 in .Fig..4. So in order to find theinfluence of crowning on the root stresses caused by differentloaddistnbutions over t.ne lines of contact, only those meanmesh positions are of practical interest; therefore, a sym-
16 Gear Technolog,y
metrical load can be assumed over the corresponding linesof contact.
Based on the measurements, the influence of more or lesssteep load gradients was further investigated by theoreticalmeans ..Calculations using the plate theory(2.3J and the finiteelement method were performed with a three dimensional.tooth model. In Fig. 6a the fa.d.QI KHtI = wmD/wm describesthe nonun:ifonnity of the load application. Factor ~6-cdescribes the increase of tooth root stresses due to d_ilff,erentload distribution~.
KFt/-c "'" Up .rrw:(KHtI>l)/up tl'lOIXI(KHg = 1)
The results of Fig ..6b, which in some points are confirmedby experiment, give an overview of the influence of themaximum line load (KH,/l) on the maximum root stresses(Kp~-c). It can be seen that this influence strongly dependson the portion lB' of tooth length ba that is covered by theline of contact. For a low ratio of IB'/ba there is already a
Fig. 6a - Definitions on the tooth model for theoretical calculations.
I
0 plate Ih<!Of)'.1\0fl-zonlol lone 01 con -
~ 1.0 lociI, /b
-I---t---t---:;;""""-+----j B • plolelhl'Ol)'. in «
bzw. cLm-ed line 0' con-'s/ba loci
I 0,8 ,BYba .0,52+---~~~--~~~--~C FE"', Incl,ned
lone 01 contact
lB'/bo .0,/,6
;0 pinionA gear
!:..-O_-j--+--+--jO.2 lalba .0.52
Fig. 6b -Int'enelationship between KIiI' and KF/I-c· depending on the ratioIs-lb.; ,comparison between calculation and measurement results.
strong stress c-oncentration below the short lin.e of contact'even with a unifonn load distribution. In this casea moreor less steep load gradient has only a minor effect.. On theother hand, a large ratio 10'/b, is quite sensitive to loaddistribution.
In the case of spiTaJ bevel gears, usually one will have a ratio,of lo/ha "'" 10.5 under design load. Assuming approx-imatelythis value, the factors (KH,8) and (KF~-c)can be plot-ted versus the amount of lengthwise crowning. (See fig. 7.)This figureal[ows a general estimation of this ·effect for prac-tical applications.
lnflluence of Crowning on. Load SharingWith MUltiple 'Iooth Contact
Fig. 3 already showed that the crowningeffeds con-siderably the maximum portion of load carried by one pairof teeth during one load cycle. Fig. 8 shows this effectmeasured with the test gear sets under two different loads.
2.00.,.---r---..----,
te1.75 -1----1----1-----1
:.:
Flrltl b In NImm_
o 227 454 680 907 1131.3.0"'1---+---1---1----1----1
21.0+---+---+----1---+-----1o 4 6 8 10
TG In kNm_
i1.0
....I.:J 0.8-........
Fls. 7 - Influence of lengthwise crownlng 01'1 Ktip andKF~' (KHP. see fig. 4; KKP-c' see Fig. 6b.)
0.6
Il !1)\"\I
I
'/"
I \ _
'/ .~I 1\ \ \r 1\:\I V/ I \ \~,~11/ 11,,/ \1\\
J.t /'I
i~
I .
!I l" .,i,
0.4
0.2
mesh position
Fis. 9 - Influence of load on the effective total contact ratio measured £'01the test gear sets,
looking at the low load (TG = 4: kNm), one cansee thatfor gear sets A and C (large and mean crowning) there is anarea of eHective single tooth contact, (One pair of teeth car-ries 100% load, T E/TG = 1.) for gear set B (small crown-ing) the maximum portion of load is about 95%.
\l\lhen higher torque ITG =8 kNm) is .applied only in gearset A,. one pair of teeth still has to carry the 'to'taJ load fora short time: i.e., with this torque on gear set. A just thesignificant value of E')'W = 2 for the effective total contactratio is reached. So from these results, the in't,errelationshipbetween load, crowning and effective contact ratio 'can bederived. (See Fig .. 9.)
It is of interest to compare the effective (,ontact ratio deter-mined by measurement with calculated contact ratios aecerd-ing to AGMA(4) or DIN(S) standards ..Fig. lOa. compares the
values for the example ofgear set A. The resultsshow that the con tadratios aceording to DIN3991 are too high over thewhole torque range; andthat the AGMA values fitquite well at low torques ..Until higher torques arereached, the measuredcontact ratio. increasesmore rapidly than thecalculated contact ratio ..
The differences can beexplained by ,consideringthe zones of action as-sumed by the calculationmethods. (See Fig. lOb.)The real. zone of action (1)determined by experimentis smaller than the rec-
mesh positioll --
Fig. 8 - InfIutcnc:eof crowning onportion of load carried. by one pair of teeth during one lead cyd .
oEO E1 E2 EJ E4 E5 E6 E7 E8EO El E2 EJ E4 E5 E6 E7
gear set0---0 A0---.0 BA-lJ. C
r---,---r-..-..=.......,.....~"E"'''"""'I'--r-"Tl.0
f~
1--11--r~fLI7'-;'-+--I--'llMr+---l 0.6
May IJUne 1988 l' l'
3.0..---------r1-""T""1-...,,----------,georse! A
f 2.5+---+--+----+---1---1cclculaled 1___
cec, 10 DIN 3,991\~, ___o ~ -.~
2.+-~-..-..-+~-~~~-,,-.9-'~-m-.~-.~~s~ur~e~d-l.c'.....•..'/ _--1'-1.5+---.,J.,.,..__.c:._~_ ....II..,,-_ ...."~,- ~--+----I
~I , eclcutoted;;,.-' ace. to AGMA
I I1.0+---.......r--+--+----+---io t. 6 B 10TG In kNm_
2
fig. lOa - Comparison between calculated and measured totalcontact ratios. Example: Cearset A.
tangular zone of actionaccording to DIN (2) and larger thanthe elliptical zone of action asumed by AGMA (3). Cor-respondingly, the actual effective total contact ratio liesbetween the values of AGMAand DIN. Nevertheless, theshape of the real zone ofaction seems to be nearer to theAGMA than to the DIN assumption.
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FIg. lOb-Comparison between the actual effective zone of ac-tion and the zones, ·01action assumed by the calculation methods.Example: Gear set A. 1 - actual zone of acl:ion;2 - zone 'of ac-tion for virtual cylindrical gears per DIN 3991(4) ; 3,- ellipticalzone of action per AGMA 2003(51;4- theoretical zone of actlon~o.rteeth without crowning.
The greater increase of the measured total contact ratioin comparison with the calculated ratios .results from the factthat both the DIN and A:GMA calculations only consider theinfluence of load on the face contact ratio, but not on theprofile contact ratio', Nevertheless, due to tooth. deflectionsand the heightwise crowning, the profile contact ratioalsoincreases with higher loads. So with regard to this effect, thecurves of Fig, lOa seem plausible. A rough estimation of theinfluences of load and crowning on the effective total con-tad ratio of spiral bevel gears can betaken from Fig. 1l.
With regard to the tooth root stresses, the interrelation-ship between contact ratio and maximum amount of load car-ried by one pair of teeth is of interest. Fig ..12 shows the max-imum values of T EfT G, called load sharing factor Y'I'"ver-sus the effediv·e total contact ratio .. One can see that the
1,2 I
t I, O+""~:r-~b:-~_~';;;::;:::~--<:loJ
tee
1000 -%800 ,~
600 .a-'00 EII.
200
)oiO
1.1--!a ,S +-----1'""':::----1-""""-==..-1w
1 2eel b . 1000
3
Fig..ll-lnflu neeof load and crowning on the effective total contact ratio,l" - ../Eal + EI. Efj calculated with the lotal face width.
r 1.0 i
"o!..0.9+---+~-'j~---1-----l~-- I~O.8+----+---I ""-I----l"
0.6o 2.0 2.25 2.50
El'ol-2.75
Fig. 12 - Load sharing factor Y~. Comparative values accord-ing 1.0 Coleman. (6)
measuring results conespondquite well 'to an earlier formula-tion of Coieman(6) if the effective values of contact ratio areused here as well. From this investigation, we may ,concludethat the influenci! ,of lengthwise crowning on the tooth rootstresses resulting froma differing load distribution over thelines of contact (factor KF,B-c), and from a differing loadsharing (factor Y'I') can be estimated quantitatively.
Influence of Relative DisplacementsThe results discussed before were valid for optimal posi-
tions of the contact patterns even under heavy loads; i.e.,an ideal stiff ,configw..a.tion was simulated on the test rig.Nevertheless, under practical conditions some displacementof pinion and gear cannot be avoided. Therefore, this in-vestigation also covers the influence 'of those relativedisplacements on the root stresses. By performing similarmeasurements on the threet:est gearsets, 'the ,effect of crown-ing could be considered too .. F~g..13 gives a definition of allrelative displacements that can. occur between pinion andgear,
To describe the difJierence in maximum root stresses when
Fig. lJ- Definition ofrelative displacements be-tween pinion and gear. (71
fa
+1--.
IINr nl(].-._.(] AO~---<l B
t.JI...-_-.-_t::Pi;;;"' ...on......__ llo-,- F4.;;C;...-.,-_!l~~Fr~-,-_--.rI. J
luI'" ,"; r-,";1.1 '-.e I -...... r~, I
:II: '~I-~1.0 Kt-:~-':::i.
0,9~--+---+---+----":::--~
O.II+---I---t---+--I ~-+--+--~-1Q'-1,52 -0.76 0 0.76 1,52-1.52 -0.76 0 Q7S 1.521.,+- 'f'+
certain ,displaoements are adjusted (f ~ 0), the factor KF~was introduced .. It is defined as
.Kr:,B-f - O'rmax (f ~O)/aTmax (f -0).
Fig. 14 shows this factor (Kp,9-fvl) in case of axialdisplacements of the pinion (fvl).' The corresponding shapesand positions of the contact patterns on the gear teeth aregiven in Fig. 15, As expected, the shift of the contact patternis stronger with a smaUamount of crowning (gear set B) andless with the high erownedgear set A. Of course, the variationof the root stresses is graded correspondingly.
In all gear sets for a certain value of displacement, thereis a counteracting influence on the pinion and gear stresses.This is caused mainly by the opposite shift of 'the contact pat~tern in the heightwise direction on the flanks of pinion andgear. As a result, the increase of the stresses for contact pat-terns near the tip of the teeth is stronger than the equivalentstress decrease when thepattems are positioned bar-ely in theroot.
So far 'the discussion has been theoretical and does not havedirect practical. application. Under service condifionsa singleItype of displac-ement will not appear alon e, The total relative
AUTHORS:
PROF. DR,~ING. HANSWINT[R is the head of the laboratory ofGear Research and Gef.l1' Desigrl al flu? Tec-hnimJ Unfuer3ity of Munich.He has wor:ked in the GemUm ge;Qrindustry sinoe 1956 irt manufac-turing. devel'apmtmt, design and rlese.arch. He .receiued his doctorAtefrom the Technicw University of Munich and studied as an associateof Prof. Dr, -Ing h.c. Gustav Niemann.,
DRANG. MlCHA..ELPAUL is head of the department for technicalcalculatian$ af Zahnmdfabrik Passau GmbH, Passau, West G TI'!1any,Prior to taking .this post he worked as Chief Engil1eeT at the GearResearch Laboratory, ,and he has been extensively involved in theresearch 'of tooth root stresses of :;piral bevel gears. Dr. Paul receivedhis graduate degree from Technical University of Munich.
displacement between pinion and gear wiU be acombination of all the portions showed in Fig. 13.From the investigation 'of combined displacementsthe following conclusions could be derived:
• Axial displacements of the gear (fv2) anddeviations of the shaft angle (fd are of minoreffect on the root stresses. Within the range ofdisplacements that have to be expected in usualgear and housing designs this effect may beneglected!. This statement is valid for gear ratioslarger thanS,
• Deviations in pinion mounfing distance (fvl)
and offset (fa) do have a strong .influenoe. Pig.16 shows the root stresses measured. on the pi-nions for different combinations of fvl and fa.
lit becomes very clear how the gradients dependon the amount of crowning (pinions B -> C-> A).
1\ gear B gear Cgear A
~~-----~-.-.---~loll ~ I , S I " it
Ilheel toe heel toe heelloe (mill]
Fig. IS-Influence of axial displacements of the pinion (fvil on the contact pattern,
rvl
-0.B
-3 •.4
a.B
• In general, determination of the factor Kf6-f for a certaincombination of displacements (tv1• fa) and for a certainamount of crowning (Cc) is allowed by Fig. 17. By using,this factor the influence of the discussed parameters canbe estimated for a given practi.cal case.
Optimization 0.£ Lengthwise CrOwningFrom. the knowledge of the influence ,of crowning and
displacements discussed earlier. a criterion for the ,optimiza-tion of the crowning can be derived .. For a given design en-vironment with certain relative displacements (measured or
(continued on page 45)
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CIRCLE A,..8 .oNI READER IREPlY CARD
IntroductionAlthough there is plenty of informa-
tion and data on the determination ofgeometry factors and bending strength ofexternal gear teeth, the computationmethods regarding internal gear designare less accessible. Most of today'sdesigns adopt the formulas for externalgears and incorporate some kind of cor-rection factors for internal gears.However, this design method is only anapproximation because of the differencesbetween internal gears and externalgears. Indeed, the tooth shape of inter-nal gears is different from that of exter-nal gears. One has a concave curve,while the other has a convex curve.Usually, internal gear teeth are thick attheir base, and tooth height is short.These two characteristics make the useof the Lewis beam model for the calcula-tion of the stresses improper, because thebeam model assumes the gear tooth is along cantilever beam, and the bendingstress at the root is the dominant factor.However, for internal gears the stressdistribution around the root fillets is verycomplex. Factors such as rim thickness,rim support conditions, root fillet radiusand loading positions, must all be takeninto account when accurate stress estima-
22 Gear Technology
Fig. 1- FEM model of the internal gear.
tion of internal gears is required. There-fore, in order to meet the demand of to-day's high performance gear design,more information on the effects of thoseparameters and a more useful designcon-cept with better numerical results shouldbe investigated.
The work presented here providesmore information about the behavior ofinternal spur gears under specific situa-tions. Those areas examined were:
a. The effects of rim thickness on theroot fillet stress and the whole stressdistribution,
b, The effects of fmet radius on rootfmet stresses,
c .. The effects of loading position,
AUTHORS:
MR. }ENG-FONG HWANG .15a graduateof the National Taiwan University ofMechanical Engineering and received hisMasters of Engin.eering degree from the OhioState University in both Welding Engineeringand Mechanical En.gineering. His major areaof emphasis has been in computer aided design(CAP/CAM) and stress analysis using thefinite element method.
DR. DENNIS A. GUENTHER is anAssociate Professor of Mechanical Engineer-ing at The Ohio .state University where he hasserved since rec.eiving his Ph.D. in 1974. Hehas done extensive work in the a1let:! of failureanalysis and identi/ic:ation of failure mode,having inv.estigated and analyzed virtually all
:FIg.2-FEM model of the internal gear.
ment with the uniform thickness of 2.54em (1"". Basically, the models of variouscases have about 800 nodes and 250elements. In all, a total of 79 cases weretested with 162 to 261 nodes and 559 to869 elements for the different cases.Additional detail. is given by Hwang.(6J
The finite element analysis was carriedout through the computer facility at theAdvanced Design Method Laboratory(ADML), in the Department of Mech-anical Engineering of The Ohio StateUniversity. The establishment of theFBv1I model and finite elementcalculationof this research were conducted ,throughthe software package, I-OEAS, runningon DEC VAX 111750 computers.
phases of machinery failures. His research andteachin,g efforts in recent years haoecenteredon system dynamics, design of machineelements, ,comput,er ,aided design, vehicledynamics, and accident reconstruction. Hisinterest in gear design has grown out of .in-volvement with teaching machine design andevaluation of gear failures.
DR. JOHN F. WIECHEl is a projectengineer with S.E.A., inc. in Columbus, Ohio.He received his B.S. and M.S. degrees inmechanical ,engineering from Purdue Univer-sity and a Ph.D., from The Ohio Staw Univer-sity. Reconstruction' of vehicular andmachinery accidents COl'I5titutes the majorityof his curr:ent work with a' focus on locating,identifying, and analyzing failed machine com-ponents and /!Valuation of the potentiJlJ .of thefailure to produce injury.
the PEM mel .....Aida. (7) Aida', JIBM1251 nodes and 212Z ... _upon a thin shell ..., * .ment. He aJso conducted _ cIoty experiments for the verIftcatkm of idamodel. The finite element models of thIIstudy were then established and baedupon the same gear dimensions specifiedby Aida.
It should be noted that since the loadis applied at the tooth tip and the tip isfar away from the tooth root, the effectof the point load singularity upon theroot fillets is minimal and can beneglected. Therefore, in the FEM model,the mesh surrounding the point loadregion was not modified .. At the rootfillet areas, the element density hassignificant influenoe on the accuracy ofthe results. Type, size and aspect ratio ofthe elements in these areas should becarefully examined. Within this study,the FEM model. was establi hed to reflectthe general trends of the stress responsescoroesponding to different parameters ofinternal spur gears. The mesh and theelements used are shown in Fig. 2. Thismodel derived the fillet stress data in theaverage sense over the root regionssimilar to the data based upon strain gagemeasurements. The boundary conditionimposed upon this model is that the sec-tion is fixed at both ends.
The results of two FEM analyses andexperimental dataare presented in Fig.3. They indicate that the results of thetensile root fillet were better than Aida'smodel. and the error is about 13%.However, the compressive root filletgave worse results than that of Aida'smodel, but the average error was still
(continued on page 26)
May/June 1,'988 2'3
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periect gear profile. That's why we de-signed the 1160 with our patented thermalcompensation system. The system keepsthe relationship between the part andhob constant So, unlike other machines.the~ 60 gives you accuracy to the nthdegree. Instead of in varying degrees.
~ding The Limits Of OurImagination AlllThe Way To The USA..With CIMA as your gear hobbingmachine company, you're not limnedto Italian manufacturing standards. Now,all of CIMA USA'sequipmenl is buinin our American factory to yoUr Americanstandards. Giving you acoessio sparepartsand service as well as the mostimaginatively designed hobbing equip-ment on the continent.
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It isn't hard to imagine why the legacy ofLeonardo Da Vinci is arive and weir atCIMA USA Leonardo's fascinatibn withgea,rs, motion, and economy of designis part of our Italian heritage. II's theinspirational force behind every pieceof hobblng equipment we build. AndIT keeps our imagination poised totransform ideas that look good onpaper Into machines that performbrilliantly on the HOO1.
ClMA limagination lisGearedToward AccuracY.For accuracy, productivity, and highspeed, CIMA's160 is on the"cuttingedg~. W~h 5DP capability and 6"a«es ofadjustment. rt was designed specificallyto cut gears up to 6.5" in diameter forthe automotive Industry. The extraprecision that smaller gears requirecalled for extra stability in our design.So we bui" the 160 with enough heftto eliminate vibrations, for stabler cutsand longer tool life. Plus, the motor isconnected directly to the work area,effectively cancelling torsion stressors.
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Imagination To the Nth Degree.Fluctuations in temperature by as littleas (.e)1) degree will r,esult in a less than
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~IMTSThe ~ of mallmacUing fIdnJlog,I •
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COMPUTER~AIDED DESIGN ...(continued from page 23)
13% ..Therefore, the overaD performanceof the model can be regarded as satisfac-tory, and the same mesh design as wellas the element types can be applied toother cases. This comparison has laid thefoundation of the FE-M model employedfor the parametric study presented here.
Effects of Rim Thickness. Usually in-ternal gears have their rim thicknessrelatively thinner than that of externalgears and, unfortunately, the rim thick-ness does affect the total stress distribu-Han. Therefore, it is worthwhile to studyits influences. Fig. 4 presents the FEMresults of the minimum and maximumprinciple stresses at root fillets of the in-ternal spur gear under the influence ofdifferent rim thickness for a pressureangle of 20° .. As the rim thicknessbecomes thinner, the stresses at the ten-sile root stay nearly the same or go upa little; however, the compressive rootincreases its stress significantly. This ef-feet is especially apparent when the rimthickness is less than 4/ P (P=5 in thecurrent study). If the rim is thick, thestresses at the root fillet converge tospecific values.
The explanation is that when the rimis thin, the whole rim is not rigid enough.Therefore, the rim deforms as shown inFig. 5., and this creates high tensilestresses at the outside fiber of the rim andhigh compressive stress at the inner fiber·of the rim (which is around the root filletareas of teeth). At the tensile root, thetensile stress due to tooth bending iscancelled by the compressive stressgenerated by rim bending. However, ifthe tin! is thick, it is very hard to bendbecause of its high .rigidity. The wholemodel can be described as similar toacantilever beam, and the root filletstresses approach constant values ..fig. 6shows the corresponding Von Misesstresses of the case presented in Fig. 4.figs. 7 through 9 are the maximum prin-ciple stresss distribution within gears ..Figs. 10 through 12 shows the minimumprinciple stress distribution. Little dif-ference was found in stress distributionsfor rim thicknesses of SIP and 6/P. Thelocation of maximum principle stress atroot filletswilJ tend to move to the bot-tom of the tooth space apart from theloaded tooth as the rim becomes thinner.
.26 Gear Technolog¥
Rim Thickness, (in)0.50 0..75 100 1.25
• i~ 5! 0.
• PhaloelOsficily Ex~imenl [7]0
§til --FEM CoIcula1ion [ 7]~ .,
- ... -FEM ~lculOtion ~ontre '"en Model reefed in this eseach -5 ~~
iii-\0 :!lIi:!: ~----~-- I i:!:J::. -!CO -- -15$ -- .t::~... I
~.-:..,i-20,,,
I -25r
Fig. 3 - Comparison of the results based on FEM: calculations of dif-ferent mesh designs and experiment.
R'im Thickness(mm)
~u::J::.
.~r15 0:c!il -25(\?
ii'i -508..~ -75
d: I,,! -100~ :
-g -125 II'c: :/~ -150'
Rim Thickness (in)
KJ8.g- 0,6 O.a 1.0 1.2 1.4 1.615
.~10. §IS ~u::
J::.
0 8~'0
Fig. 4-Root fillet principal stress of internal gears (tJ:> - 20) with dif-ferent rim thickness and fillet radii.
--:-:':::"'l".=-=-~_._. __ .__
- Sharp Roo! F'llel Rochus--- Roo! F'IIe1 Ro:hus 0.0605,,,_.- Rool Fillet Rochus 00926 in
15 20 25 so 35
Rim Thicknesslmm)
=~
Fig. 5 - Schematic illustration of rim bending model.
Rim Thickness. (in)
10.4_10.6 10.8 1.0 1.2 1.4 1.625
.;;;c.
§15 ZI
!ci);:on
I ~----..110 ~
'•. F II~t Under' TenslO!'I
... F ,Ilel lk'dI!I: Convl!$'SIO!l
IRim Thiekness(1I'YT11
Fig. ,6- Root fillet Von Mises tresses of inleJTIal gears (41, - 20) withdifkl'flll rim thickness arid fillet radii.
Fill. '1-Maxlmum principal stR!SSdistribution of the internal gear(41 - 20, rim thickness - 6/P).
fig. 8- Maximum principal SlR!SSdistribution of tile internal geilT(4t - 20, rim thidness - 4/P).
Fig ..9 - Maximum principal SIres distribution of '!he inttrN.! geM(41 - 20), rim thickness - ZIP).
Fis. IO-Minimum principal stress distribution of the inttrN.! gear(1/1 - 20, rim thickness - 6/1'1.
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CI.RCIJE A~21 ON IREADER IR.!PLV CARD
May/June 1988 21
Fig. 11 (Left) - Minimumprincipal stress distribu-tion of the internal gear (rb-, 20, rim thickness -4/P).
Fig.. 12 (Right) - Mini-mum principal stress dis-tribution of the internalgear (4) - 20. rimthickness ... 2IP).
CIRCLE A-13 'ON READER REPLV CA~RD
.2B: Gear Teehnolog;y
As the rim becomes thinner, the neigh~boring teeth are also subject to higherstresses, Inother words, due to 'the severebending of the rim, high stresses are builtup within the rim near the loaded tooth.Under such situations, the maximumstress or the critical area may be locatedin. the rim. The defonnation of theloaded tooth and the whole rim is shownin Figs. 13 and 14 (rim thickness ., .sIP),and !Figs. IS and 16 (rim thickness2/P). The model should contain moreteeth in order to get better results for thecase of a very thin rim.
E.ffects of Root Fillet Radius. Root 61Ietradius is the major factor that detenn:inesthe extent of local stress concentrationand is the deciding factor for the localstress magnitude. For external gears, ala~ger fillet radius has, a. local stress levelsmaller than that of a small fillet radius.The effect is very significant. However,in this research, the effect of fmet radiusshowed the same trend, but was not aspronounced because of 'the mesh designat the fillet areas. More accurate anddetailed infonnaUon should be gathelledbased upon comprehensive FEv1I modell-ing together with experimental verifica-tion, Aspect ratio and element. densitywill be the key factors in this eff.ort
Effects of .Loading Positions. Whentwo gears mesh, the contact point winfollow the path of the lineeE action andfor each tooth in mesh, the force trans-mUted win be normalto the t.ooth sur-fa.ce and subsequently move along the in-volute profile. A simulation .of the Itoothstress variation can be made if subse-quent loads at diffeJle.nt positions on thetooth prome are applied to the model.and the fillet stress is calculated by FENll.This study has selected five positions.along the profile to carry out this simula-tion as indicated in Fig. 17. At all posi-tions,the tangential component of the
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CIRCLE A·2 ON READER REPLYCARD
Fig. 13-Defonnation of teeth (4)=2(1, rim thickness = 51?).
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CIRCLE A-M ON READER'IREPlYCARD
30 Gear Technology
Fig. 14-Deformation of rim (.p ••" 20, rim, thickness - 8/1').
Fig. 15-Defonnalion. of teeth (4)-20, rim thickness ~. 211').
Fig. ]6-Defonnation of rim (1/1 .,.. 20, rim thickness - UP).
P, ' Tooth TipP2, O.325/P" Away from the Tooth TipP3' O.902/P" Awoy from the,Too1h TipP4' 1.384/P" Awoy from the ToolhTipP5' 1.5931 p" Awoy from the Tooth Tip
Fig. 1.7- Different loading position on. a tooth.
load remains constant. fig. 18 demon-strates the effect of different loadingpositions. As those curves show, thevariation of the stresses at both tensileand compressive fillets is essentiallylinear unless the rim thickness is verythin. The explanation follows ..
The stress due to tooth bending islinearly proportional to the bendingmoment. In this study, the tangentialforce component, which is the main fac-tor determining the bending moment,stays the same; therefore, the bendingstress is direcdy proportional to themoment arm. The moment arm is thedistance between the tooth root and thelocation where load is applied. A linearrelationship is thus expected.
Rim bending also makes a contribu-tion to resultant stresses, but unless therim thickness is thin, the ·effect is verylimited. for thin rim cases, the deforma-tion of rim and the rim bending stressesare very sensitive to the applied load.Some nonlinear variation due to thecomplicated rim deformation occurs, andit also nonlinearizes the variation of theresultant stresses. Figs. 19 and 20 showthe fillet Von M:ises stress for the casesin Fig. 18. Figs ..21 and 22 show the max-imum principle stress distribution of thegear loaded at Qifferent positions. Thestress patterns in the rim and neighbor-ing teeth are the same except for the dif-ference in magnitude. However, thestress distribution within the loadedtooth changes acording to the loadingpositions. The location of the maximumstress behaves the same way as the rimthickness effect does; namely ,as theloading position approaches the rootfmet, the point of maximum stress shiftsits location to the bottom 'of the toothspace apart from the loaded tooth. Theloading position effect also shows its in-fluence the same way for those cases withdifferent support conditions.
Effects of Support Condition. The sup-port condition for a section of an inter-nal gear will be dependent upon howmany bolts are used in mounting thegear. The model used within this researchis a section spanning two neighboringbolts cut out from the whole gear .. Atboth ends, the boundary condition isassumed totally fixed. Four sectionsspanned by 60°, 80", 100°, and 120°were analyzed. The variation of the sup-port condition showed no. obvious effect
on the fillet stress. However, the varia-tion of the support condition does havesignificant influence on thin rim gears.
If the span is small, the rim is rigidand, consequently, the fillet stress will besmall. As the span increases, the rimbecomes more flexible and the fillet stresswiU then increase accordingly. Especiallyfor the compressive fillet as shown inFigs. 23 and 24, the stress magnitude mayincrease by 30% ..The reason is that if therim is thick enough, the whole rim isvery rigid and the loaded tooth is very
similar to a cantilever beam, Eventhough the support condition changes,the stress does not increase appreciablyunless the total span between two, endsis very large. For thin rim cases, the rimdeformation is important and the changeof the support condition win easily af-fect the rigidity of the rimas well as thefillet stresses.
Effects of Pressure Angle. Three dif-ferent pressure angles were in.vestigated inthis study, specifically 14 ..5°, 20°, and
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CIRCLE A-12 ON READER IREPLYCARD
May/June 1988, 31
L.OOding Position (in. Away fromlhe Tlp)0__ 0.1 0.2 0.3 0.4-20
if 125 - - Rim ThocknIM 8/P wrlh Root F,iIe1 . 'jS,fbi .., 0 0926 II1CheI I §.
~ - Rm Tlw:IuwM 8IP with Sharp Agot Filial 15---- Rin Thocknell 4IP w,th Sharp Agot fillet I -
_. - RimTluckneu UP WIth Slvp Roo! Fille1 10 ~I i.L
.J:;
~'5'"Q)en,'"e?
Vi
25"6en·Q)<II<II
l\! -25Vi°
Loading Position (in. AWf:1oJfrorntheTip)1759. 0 I 0 2 0 :3 04
I --Run Thi(;knHI 8/P with RootFiile1I RacIiuI 0.0926 inches
-- Rin TlIcknns SIP willi Sharp Root fillet-- -- RimThc:"- 41P with Sharp Root Fillet-- Rim Thick,.. UP With Slvp Root Fillet
'--'--.-.." .-......<,
"-10 "8.'i;j
-15 ;t'"c:::?!
-20 'Eoc
L..-+-~-........,J'----!:--+--l-25 .~o 2 4 6 8 10
--'-.-.--.....-.---
Loading Position (mm Away from the Tip)
Ag. IS-Root fillet principal stresses of ~temal.~ (cP ... 2O)with different rim thickness and loading POSItiOns.
loadingl Position (mm Away from the TIp)
Lood ing, Position ( in. Away frern the Tip)o 0.1 0.2__ Q;3 04 20
fig. 20 - Compressive root fillet Von Mise>stresses of in-ternal gears (cP = 2O)wilh different rim thickness andloading positions.
-.~
1
15 ~<II
'"2!Vi
10 .~:::?!c~
5 ~u,.J:;
~
- - Rim ThicI<onI !,lIP with Root FilletRadius 00926 ,nchII
-- Rim TIIcIawu SIP with Shcrp Root Fillet-- Rim T1w;tu_ 41P with Sharp /bit Fillet_.- RIm Thick"", 21P with Sharp /bit Fillet
ILoading Position (mm Away from the Tip)
fig. 19 - Tensile root fillet Von Mi.ses stresses of internalgears (q, ... 20) with different rim thickness and loadingpositions.
Fig. 21- Maximum principal stress distribution of internal gear(.p ... 20, rim thickness - SIP) loaded at O.325/P from the tip.
25° angles, Usually, the large pressureangle gear can. sustain a larger loadtransmitted because a larger cross sec-tionalarea exists at the root section.Therefor-e, for the same loading, smallerstresses are expected. Figs, 25 and 26substantiate this phenomenon. When therim thi.ckness is larger than 4/P, thelarger the pressure angle, the smaller theroot .fillet stresses. However, if the rimthickness is less than 4/P, then, as the
32 Gear fechnology
pressure angle increases, the stresses alsoincrease. Rim bending is the probablecause of this phenomonen. For a rigidrim, the loaded tooth is similar to a can-tilever beam, and the stress is then deter-mined by the root cross sectional area,
As discussed previously, there aremany parameters that affect the stressdistribution of internal. gears. Althoughthese ,effects vary, each parameter has thesame influence on internal gears regard-
less of pressure angle.
ConclusionsOther Approaches To Static Stress
Analysis of Gears. Although the finiteelement method has frequently beenap-plied to gear stress analysis, it usually isexpensive in terms of CPU time andman-hours required to establish or editthe model, Besides,FEM can only giveapproximate results rather than exact
It". 22-Maximum principal stress distribution of internal gear(41'... 20', rim thickness, .. SIP) loaded at 1.593/P from 'the tip.
500 I
75~-------------50 • Filet Under TensIOn.. F diet Lh:ief ~O'l -25 - Rm ThICkness SIP Inches ,
o f- - - - RIm Thickness 2/P Inches -I
251-
~501-
75 -
I- J25- ..._---~---- ........ -50 -----.
5
oIn
-5 ~Iii)
-10 1-15 :g
e,K
-20 .~
l-25 .40 50 60 i
Support Condition (8Degrees) -
Fig. 2J - Root fillet principal stresses of Internalgears (41 - 2-0) with different rim thickness undervarious supports.
answers. There is a relatively new ap-proach to this pr-oblem. This approachis called the Boundary Integral Method,which is based on two dimensionaltheory of elasticity and appropriatetransfonn functions. The solution is ex-act for the boundary curve chosen andas dose as possible ,to the given onewhich represents the shape of the stressedpiece. Severa] researchers(8,9] have doneinvestigations on external spur gears, but
1251-
__ - 20~-----~ ...-rP---
~, 75 ~---- - ... _
~~
en_____ "-1 0 .~
:!:
50-- X: ~~ • Fillet Under Tensxn .....~ 5 ~.E ...OF 25 I- <It. F lIIe! Under CompressIOn .t:
~ - Rim ThICknessBtP Inches ~OI---RlmThickness2/PInches 0
30 40 50 60Support Condilions (9 Degrees)
Fig. 24 - Root fillet Von Misesst:resse5 of intemalgeMs. (1/1 - 20) with dif£erenl rim thickness undervarious upports.
Rim Thickness (in)0_6_09 10 12 14 1615
---------• Ccrnpressive Root Fillet• Tensile Root Fillet
- Presslre AnQIe 20 Dec7ees-- - Presue AnQIe 14.5 Dec7ees- - Pr8UlSe Angle 25 Degrees-----~ - ._----
Rim Thickness (rnrn)
Fig .. 2S - Root fillet prill.dpa1 stre;ses of internal~ars with different p!'eSSUfI! angles.
no resear-ch has been done on the inter-nal spur gears. It is believed that theapplication of this method to the inter-nal gear stress problems can yieldsatisfactory results, especially in the in-vestigation of the stress concentrationaround Lite root fillets. This method ismore efficient as far as the CPU time isconcerned, and the stress values havebetter ac,curacy ...
OpnmizRtiol'l ,of Intemal Gear Design.
The traditional approach togea:r designis to utilize design formulasand gearratmgs found in standardsand codes inorder to meet the requirements of the ee-pected operating conditions. This, ap-proach does give a feasible solution fora. given des~gn; however, it does notguarantee this design is the best possiblesolution. Nowadays, it is possible to ltake.advantage of high speed digital com-puters to determine the optima] solution
IMay/ .kmeli9S9 3:3
hom among many feasible answers.Optimization of the tooth profilegeometry is an important area within thisfield. (10) It is possible to synthesize anoninvolute spur gear pair with optimalload carrying ability based on the bend-ing and contact strength. All these toothprofiles can be synthesized by usiIlg man-computer interactive design procedureson CAD/CA1v1I. systems.
Summarizing the results and discus-
sion presented here, the foUowing con-clusions have been reached:
1. Rim thickness has a dramatic effecton the fillet stresses and stress dis-tribution. VV'hen the thickness isless than 4/P, the stress at the ten-sile fillet changes, and the stress atthe compressive tinet increasessharply,
2. As the rim thickness decreases, the
po, Box 125, Sussex, WI 53089, (414) 246-4994
CIRCIJE .A-2A ONI READER REPLY C.A~RD
34 Gear Technology
position of the maximum stresswilJ tend to move to the bottom ofthe tooth space apart from theloaded tooth.
3. Rim thickness has a global effecton the stress distribution of thewhole gear, and it is recommendedthat the-thickness should be at least4/P .. If the rim is too thin, rimbending may be a severe problem,and the most critical area may shiftto the rim.
4. Root .fillet radius has a local ,effectonly" and, in order to accur.atelydetennine its influences, differentelement types and meshes shouldbe used. Experimental methods arealso suggested.
5. The root fillet should be careful!lymodeHed in order to derive accur-ate results. Element densiltyandtheaspect ratio of elements should bechosen based upon comprehensivestudy. EXperimental. data must beused to justify the validity of themodels.
6. The fiUet stresses vary linearly asthe loading position changes, butfor dun rim. cases, nonlinear rela-tionships are present due to the in-fluence of the deformation of theflexible rim ..Usually, if rim thick-ness is less than 31P', the rimbending effect will appear.
7. When the loading position ap-proaches the tooth root,the loca-
(continued 011 page 48)
Rim Thickness (in)0.4 0,6 0,8 10 12 1.4 1~5
• Corrvnsive!blt Fillet• Tensile Root Filla.
150 - Preswe An;t 2ODec;rees_ \ ---PreanAnl;lllI45~ 20'!8. \ -- Pr-.n Arw;t 25 DIc7ees .~.:E 1125 \ .C!.
~ " iQI \ 15 !!?'~ 00 \ m..JI~::E.~ ....~.Ii£: 50s::.
'8II-
2.5
----+---~--
°-15 20-25-30~35-4i5'Rim TIliCkneSSh,."j
Fig. 26 - Root Fillet Von Mises stresses of internalgears with different pressure angles.
Bl~CIi TO BASICS ...- -------------
Helical Gear MathematicsFormulas and Examples
Earle Buc:kin,ghamfliot K. Buckingham
Buckingham Associates,. Inc.S~eld, VT
The foUowing excerpt is from the Revised Manual of GearDesign, Section 1Il, covering helical and spiral gears. This see-tion on helical gear mathematics shows the detailed solutionsto many general helical gearing problems. In 'each case, a.definite example has been worked' out to illustrate the solution.Al!lequations are arranged in their most effec:ti ve form for useon a computer or calculating machine.
AUTHOR:
ElIOT K. BUCKINGHAM is presidmJ of Budringhmn' Associales.Inc .. a consulting finn worbng in the ar"lla5,ofdes(g7I, application andmanufacture of gears for any type of drive. Mr. Buckingham eamed hisB. S. from Massiuhusetts Institute of Technology and his M.S. from theUniv.ersity of New Mexico. He is the /luthor ofT ables tOT RalessActionGears tlnQ t1Umerous technical papers, as well as the revised edition ,ofThe Manuall of Gear Design by flute Buckingham. He is " member ofASME QJ1da Registered Professional.&lgineer in the State of Vemr.ont.
'Given tha pitch radius and lead of a helical gear, to determine the helix al'lgle:
When. R '" Pitch Radius of Gear
!l - !lead ot Tooth
"" == Helix Angle
Then, 2 ... FITA_N!It .. -Il-
Exampl'e: PI ",,3.000 l - 2U)OO
2 x 3.1416 X 3.0001TAN"" "" 21.000 '" .89760
The iinvolute ,of a cir'cle ls the curve that is described by 1he end of a line which is unwound fromthe circumference ota circ~'eaS shown in ,Fig'. 1.
When, Fib = Base Radius
I(} '" Vectorial Angler = Length of Ra.dius Vector
8 ..
'Given the arc tooth thickness and pressure angle in the plane of rotation of ahslieal gear at a givenradius, 10' determine its tooth thickness at any other radius:
When,
TIlen,
I!xample:
rl - Given Radius
tPt Pressure· Angle at rl
T1, - ARC Tooth Thickness, at rl
r2 - Radius Where Tooth, Tflickness Is To Be Determined:
1/12 "" Pressure Angle' at r2
T2 "" ARC Tooth, Thickness at '2
rl COS !PICOS tP.2 ..
r2 '" 2.600
!/l1 '" '4.500" Cos 411 '" .9Sa1S ..00554
,2.500' x .96815C~ 1/12 '" 2.600 :; .9309,11
---f-.- --- . ---
IINV 1/12 '" .01,845
(.2618 .'. 005"'''' 0,11845') - 2'051T - 2 x. 2.600 I~. +. . ..... - '. ..' -. "
2 .. _. . . 5.000Fig. 2
When"
Given the herix angle, normal diameUal pitch and numbers of teeth, to determine the center distance:
Then, 1'\11 + N2C ..
2 Pn COS I/!
I lEx.ample:
I/! = IHelix Angle
Nil = Number of Teeth in Pinion
N2 .. Number of Teeth in Gear
C... Center Distancs'
Pn "" Normal Diametral Pitch
I/! = 30° Nl '" 24 COS I/! ... 86603
24 +48C '" 2 x 8 x .86603 .. 5.1'9611
36 'Gear Technology
Gl,ven the arc tooth thickness in the plane of rotation at a given radius, to find the n.ormal chordalthickness and the nonnal chordal addendum:
Given the ,circular pilCh and pressure al1lgle in the pl'ane of rotation and the helix angle of a helicall gear, to,detennine the normalcircular IpilCh and the normal pressure 8n9[9':
When,
Then,
Example:
When.
Then,
Example:
T ,." ARC Tooth Thickness at R in Plane of Rotation
Tn "" Normall Chordal Thickness at IR
a'n '.. Normall Chordal Addendum
IRo roo Outside' Radius
R '''' IP,itch Radius
!/t - !Helix Angle at R
T COS2 'fARC B '" 2R
Tn .. 2 R SIN B
COS '"
0'11 "" Ro - COS B
T •. 2267 Ro = 11.8570 R "", 1.7320
!/t- 30" COS '" - .86603 COs2'" :: .75000
.2267 x .7500ARC B '" == .'049082.x 1.7320
SIN B ... 04906 COS B = .99880
2' x. 1.7320 )( .04906Tn - '" .1962.86600
Fig. 3
'Q\, - 1.8570 - (1.732,0)( .99880) .. .1271
!/t ., Helix Angle
.1/> - P,ressure Angle in Pl'ane ,of Rotaliion
IP .. Circular P,itch in Plane of Rotalion
I/ln .. Normal Pressure, Angle
Pn - Normal Circular Pitch
II' .. P COS '" TAN 'I/ln= TAN tP· COS '"
p II! .3927 TAN <p II•• 36397COS", ... 92050'" = 23"
Pn "" .3927 x .92,050 ... 361148 TAN <Pn "" .36397 x .92050 "" .33503
I/>n - 18.522°
May l.k.Lne 19,8837
I Given the arc tooth thickness and pressure angle in the plane of rotation at a given radius, to deter-mine the radius where the tooth becomes pointed:
Given the normal circular pitch, the normal pressure angle and the helix angle 01a he'licalgear, to determine the circul'ar pitchand the pressure angle in the plane ot retanen:
Wh.9n,
Then,
Example:
Then,
I Example:
rl: - Given Fladius
r2 - Radius where Tooth iBecomes P,oirlted
Tt "" ARC Tooth Thickness at TIl
!/it "'" Pressure, Angle at Ir1
t:P2 "" Pressure Angle at r2
rl COS 1/11r-""
2 COS 4>,2
r1 ;2.500 Tt - .2618 411 - 14.500°
INV tPl .. .00554
.2618INV t:P2 .. 2 x .2.500 + .00554 ... 05790 Radians Fig. 4
IP2 .. 30.6930 COS tP2 <= .85991 COS 411 ... 96815,
2.500 x .968115"" 2.8147.B5991
1/1 "" Helix Angle
¢n == Normal Pressure Angl'e
PM Normal 'Circular IPHoh
tP, .. Pressure Angle in PI'a_neo~Rotation
P .. Circular Pitch in ,P,lane of BOlation
prJp ... cos 1/1
TAN ¢n
TAN I/l' .. COS 1/1
COS ..J.o .. .90631 TAN ¢In .. .36397 Pn ... 5236
.5236P ,= .90631 ... 57772
.3639'7TAN I/l ". 90631 ... 40159 t/i .. 21.8800
38 'Gear Technology
Given the tooth proportions, Ilnlhe plane 'of rotaliion of a pair of !helical gears (par.al1el shafts), to deter-m]ne,lhe, center distance ,aJ whIch Ihey wUl mesh l'i9'htly:
f11 Given Radi'us of 151 'Gear
f2 Given Radius of 2ndl Gear
N1 Number of Teelh In 115tGear
N2: : Number of Teeth lin 2nd Gear
411 .. Pressure Angle at r, and If2
412' - Pressure .Angle aJ Meshing Position
T1 .... ARC TOOlh Thickness al f,
T2 - ARC l'OOIhTihlclmess al ~2
01 .. Center Distance for Pressu~e Angle 4>,C2 .. 'Center Distance for Pressur,e Angle 412
Then.Nl (Tl + T 2l -2wrl
IINV 4>2 - + INV 4112' fl (Nl + N.2l
ell cos 411C:!... COStb2 Ag.5
EXample: Irl "., :2.500 Tl ~ .2800 Nl ... 3()
r2 '''' 4 ..000. T2 = .2750 C,= 9.500
30 (.2800 + .2750) - 2r x 2~SOOINV!/>2 - 2 x 2.500 (30 + 48) + ..,00545= ..007955
oos ¢2 ,. .95973
6.500 x .968,15C2 - .95973 - 6.5570
Given the IPitch, radius and heli.x angle of a helical 'gear, to determine the lead ,o1ttls, tooth.
When, A PHch Radius
L "'~ Lead ,of Tooth
1/1 - llielix Angle
Then, 2T:RL ,. TAN T
TAN t/I '" .41421'!Example: R - 2.5001
2 )(3.1'416)( 2.500L - ... 37.9228.41'421
May/JllJne 1988, 39
Gillen the number of teeth, ,helix8ngle and proportions of the normal basic rack of a. helical gear,, to determine the pitch radius and the !base radius:
When, N = Number of Teeth
t/t = Helix Angle at R
Pn Normal'
:Dlametral Pitch
R- P,itch Radius
q,r! = Normal Pressure Angle
q,. - Pressure Angle in Plane' of Rotation
Rb ::: Base Radius
Then, N TAN cPr!TAN cP ""COS t/tR =2 Pn COS '"
N COS cPFib = FI COS cP =2 P
nCOS t/t
N = 30 COS y,. '" .90631 TAN cPn '" .25862Example:
30R == 275842 x 6 x .90631 =_.,' .
.25862TAN q, '" .90631 = .28535, cos cP '" .96162
30 x .96162Rb - ... = 2,65256" - 2 x 6 x .90631
Gillen the normaJ diametraJ pitch, numbers of teeth and center distance, to determine, the, lead and helix angle:
When, Number of Teeth in IPinion
N2 "" Number of Teeth In Gear
Pn Normal :Diamatral Pitch
C = Center :Distance
t/t = Helix Angle
Lli = Lead of P,in,ion
L2 = Lead of Gear
Then, Nl + N2COS "'= 2Pn C
~ N2l2 '" Pn SIN I/;
Example: C = 4.,500
18 + 30COS", = 2 x 6 x 4.500 - .88889 SIN y,. '" .45812
3,1416 x 18Ll = 6 x ..45812 == 20.5728
3.1416 x 30L2 "" 6 x .45812 "" 34.2880
40 Gear Technology
When. "'1" Given He'llx Angle at Rl
'1/12 .. Helix Angle for Mating Rack
l/tr, .. !Base IHelix .Angle
¢nl ., INa mallPressure Angle at R,
¢n2 .. IPressure Angle of Mating Rack
¢, .. Pf\8SSure Angle at lA, in Plane of Flotation
Rb 1- !Base, RadiLls
iii ,. Addendum of IFlack
T, - AAC Too~h TI'llckness at A,
N .. INumber of Teeth
.x .. Distane,e from Center of Gear to Tip of Rack Tooth
Pnl .. Normal Circular Pitch at A,
Given the tooth pro,port.ions in the Iplane of rotation of a hetica:i gear, to determine the' position ofamating rack of ,different ,ciroular pitch and pressure angle:
¢2 ., Pressure Angle ot lMaJing Rack in P,lane of Rotation Pn2" Normall Circular Pitch of IFiackIAl '. Given IPitch Radius Note: (Pnl COS¢nl Must IBe IEqual To (PI12COS ¢1121'IR2 - IPitch Radius with Mating Rack
TAN ¢n2TAN ¢2 .. COS ..1.
'1':2.
SIN ifb ., SINII/I, 'COS ¢nl
SINI "'b SIN !It, COS ¢l'I1
SIN, 1/12 .. ' 'COS ¢n2 .. 'COS ¢n2
Fig ..sExample: "'1 • ,250 ¢, .. 15.926° R, .. 2.7584 A,b .. ,2.65258
¢,n2 ... 20° a - .185 T, -.2888
SIN !ltl oz. .42262 COS "'1 ... .90631 TANtbn1 ... 25862
Po, .., .5236 COS IPnl "" .96815 COSIPn2 .... 93969
IIPn1 'COS IPnl .. .50692] .. [Pn2 COS IPn2 '" .50692:)
..42262 x .96815SIN Vt.2 ,.., .9,3969 - .43542 !It2 .. 25.812° 00SI/I2 ... 90023
.3639'7TAN ¢2 ... -.-. --, . - .40431
.90023, IP2 - 22.014° COS tP2, ... 92709'
INiV 4>2 "" .020093 INV tb1 .007387
2.65256,R2" - --c c '"' 2.86117..92709
,X '" 2.86,117 _ .185 + .. 1 .' [5,.72234( ... 2_888_ + .007387 _ .020093). _ .31416 ~~.86117]1 • 2.5729,2 )( .404311, . 5.51168 .
Given the center distance, number of teeth, and basic rack proportions (.hob proportions) of a pairof helicall gears, to determine the hobbing data:
When. <fine ,= Pressur,e Angle of Ho'b
Pnc = Diametral Pitch of Hob
ac = Addendum of Hob
C1 Center DiS1.ancewith Pressure Angle of <fIl
C2 = Given Center Distance of Operat,ion
Nl == Number of Teeth in Pinion
ROl = OIJ~side Radius of Pinion
Rrli Root Radius of Pinion
ILl - Lead of Pinion
Rll == P,itch Radius 'of Pinion
b1 == Oedel'ldl.lm of Pinion
1/11 == J-Ie'lixAngle of Generation
Then, Ma:ke trial calculation for lead as follows:
Nl "'" N2COS ""1 .. 2 Pnc C2
N.2 Number ·of Teeth in Gear
Ro2 = Outside Radius of Gear
Rr2 - Root Radius of Gear
l2 '" Lead of Gear
R2 - Pitch Radius ,of Gear
Oed'endurn of Gear
Helix Angle' of Operation
Pressure Angl'e of Generation in Plane of Rotation
Pressure Angl'e of Operation in Plane of IRotation
P1 = Diametral IPitch of Generation In ,P,laneof Rotation
hI = Total Tooth Depth of Gears
b2
1/12'
rPl<fI2 -
Then.Select values for Ll and: L2 which can be readily obtained on the hobbing machine:
TAN rP1TAN rPnc
- COS 1/11
:1'1 - Pne COS 'fllC1 COS <fIl
COS <fI2=
1:11 ..
C2 - (Rr1 "'" Rr21INote: When smallest N is 30 or more, then, b1 = b2= 2
Nl O2
Nl + N2N2 'C2
R2= Nt "'" N2
ht ,.'932 102 - (Rn "'" Rr2l I
2 7r R12 11" IR2TAN 1/12 '" -- = -----
- L1 IL2
42 Gear lechnolog;v
(Continued On next page)
,25862TAN 411 - ,88969 = ,29069' COS 411 ... ,96025 IINV 41, .... oonse,
Exam.pfe: Pnc = 5 Ac = .2314
tbnc ... 14.500 TAN tbnc .. .25862
Triall Calculation:
20 + 60COS "', - 2 x 5 )(9.00 .. .88889 SIN '" - .45812
60rIlll ... 5, x .45812 ". 2'7.4303 5 _",:0 2-' .. ,82.2909x.. """",,1 .
We, willi select the following values for Il, and L2:
Il, ... 27.500
20.".SIN", ... 5 x 27.500 :: .45696
p, - 5, x .88949 .. 4.4414520 + 160
C, "" 2' x 4.44745 - 8.'99392
8,,'9913·92 )( .96025COS th ..
9.959601 INV 4>2 - .007994
,8.99392.(Rft + A,v ::: B.99392 - 2 x .2314 + .29069 f·OO7994 - .0077961 =8.5372
9.00 -8,5372b, .. 1 ... ~ "" .169GB
9'.00 -8.5372'b2 "'" 1 + fo6O ...29340
20.x '9,00R, - -- - - = 2'.250
201"" 60
R,r1 '"' 2.250 - .16938 2,08062' 1=11'2 "" 6.750 - ,.29340 .. 6.45660
hi ., .932 [9.00 - 8.5372] ... .43133
Flo, .. 2'.oaos.2 "" .43133 - 2.51195 Ae:2 ... 6.45660 ... ,.431133 I ' 6.88793
May/June 1988 3
2 r2.25OTANlif2 ., .... 514019
21.5 "'2 .. 27.207
The specifications for this pair of gears are as ronows:
IN, ,.., 20Flo, .. 2.51195A, .. 2.250Il, ,..,27.500
1=1,1 ... 2.08062N.2 - 60Ao2 ... 6.88793R2 .. '6.750
L2 = 82.500R'r:2 .. '6.45660~ .. 9',00
Helix angle for hobbing .. 27.1910
Given the prcperticnaot an internal helical gear drive, to determine the contact ratio:
Pitch Radius oI Henes.1Gear FI.2 - Pitch IRadius of Internal Gear
Ri == Internal Radius of Internal Gea'r
Rb2 Base Fladius of Internal GeeJ
Then, ..JR012 - Ab12 + C SIN.cP - ..JFI? - Flb2
mp :: p COS tP
Glvenlhe preponlens of a. pair of helical gears, (external! or internal). to determine the face contact ratio:
When"
:Example:
I When.
Then,
Exarnp'le:
Rl
R1)1 '" Outside Radius of Helical Gear
RbI :: Base Ra.dius of Helical 'Gear
tP :: Pressure Angle in Pl'ane ,of Rotation
p - Circular Pitch in IP,laneof' Rotation
C -Center IDistanoe
mp ,., Contact Ratio
Rl "'" 1.250
R2",3.500
SIN tP - .34202
,R01 '" 11.4375
IAJ,== 3.4375
p "" .3927Abl '" 1.11746
Rb2 ,= 3.2888
cP'" 200
C = 2.250
COS tP, = .93969'
..j(1.4375)2 - (1.1746)2+ (2.250 x .3420.2) - ..j(3.4375)2 - (3.2888)2mp"" .3927 X .93.969' = 1.'62
F = Face Width
p .. Circular Pitch In PI!aneof Rotation
y, .. Henx Angle
m, '" Face Contact Ratio
F TAN y,m,oo
p
F = 1.50.0. p = .3927 y, = .300 TAN y,' == .57735
m, =l' .500 x .57735
'" 2.20..3927
I Given tile proportions of a pa.irof helical gears (external ,or internal)., to determine tile total contact ratio:
! When.,
I Then ..
Example:
mp '"" Contact Batlo,
mt = Face Contact Ratio
m, == Total Contact Ratio
mp = 1.59 ml' '" .2.20
ml = 11.59 *' 2.20 ;: 3 ..79'
44 Gear Technology
TOOTH ROOT 'S~RESSES,..••(continued !rom page 20)
estimated), the amount of crowning should be chosen in sucha w.ay 'that when applying the service load, the lowest rootstresses will be the result. This criterion is satisfied when theproduct
reaches a minimum.As an example this optimization is performed for the test
gears in Fig. 18. One can see that the curve for ICc has a fIatminimum in the area of small crowning values (near gear setB). This result seems to, be plausible because of the very stifftest rig.
It should be noted that the optimization method introducedhere is only based on the tooth root. stresses and should onlybe used if tooth breakage is the c:ritka~ failure criterion. Anoptimization for contact stresses may be quite different andusually provides a guide to higher amounts of crowning.
SummazyBy strain gauge measurements ,of spir.al bevel gears, the
influence of lengthwise crowning and relative displac1!mentsbetween pinion and gear on tooth root stresses was in-vestigated. It was found that the crowning effects the loaddistribution over the lines of 'contact and the load sharingbetween pairs of teeth meshingsimul,t.aneously, For both in-fluenc:,es.a. quantitative description. could be derived.
(c·ontinued on page 47)
SET-UPSTAKE
SECO,NDS* I,NTERNAlc-EXTERN'ALSPUR & HELICAL GEARS
ro 20 INCHES DIA_METER'11707 Mc8ean Drive, EI Monte, CA 9173.2
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pinion A
if>'I> ·0.5,.0.•... '" ~ 0 I .0.,.0,5·
500
..~~oo I
~Ii J50
p1nlon C
Fig. 16-Influence of combined displacements on the maximum root stresses at mu at the pinions. (Amounl of crowning, see Fig. 2.)
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46 Gear technology
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TOOTH ROOT STRESSES ...(continued from page 45)
-1.0 -0.5 o 0.5 1.0tlt_vl
-1.5
I 1,3
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~~ I, '-R:::::~9<:::-1--1,""u,~~ ,.O-l<,:......-:....:.:...-r----I---.:CJ"'iI
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In the case of relative displacements, deviations in pinionmounting distance and in offset have the strongest influenceon the root stresses. A method was introduced to determinethe increase or decrease of maximum stresses that have tobe expected tor a combination of certain values of theseparameters ..Further, a optimization criterion was derived thatallows finding the amount of lengthwise crowning produc-ing the lowest root stresses for a certain service condition.
References1. WINTER,. H., PAUl, M. "Influence of Relative Displacements
Between Pinion and Gear on Tooth Root Stresses of Spiral BevelGears." Gear Technology, July/August, 1985.
2. JARAMILl:O, T.J. "Deflections and Moments due toa Con-
F.[g. HI- Optimization of lengthwise crowning.
centrated Load on a Cantilever Plate of Infinite length." Jour-nal of Applied Mechanics, VoL ]7, Trans., ASME, Vol. 72 1950,S. 67-72, 342-343.
3.WELLAUER, E.J., SEIREG, A. "Bending Strength of Gear Teethby Cantilever Plate Theory." Journal of Engineering for In-dustry. Trans. AS ME,. Aug, 1960, S ...213-222.
4. AGMA.2003 - A86: Standard tor Rating ,the Pitting Resistanoeand Bending Strength of Cenerated Straight Bevel, Zerol Beveland Spiral Bevel Gear Teeth, 1986.
S. DIN 3991: Tragfahigkeitsberechnung von Kegelradem oboeAchsversetzung: Normentwurf, 1986.
6, COLEJ:vlAN, W. "Improved Method for Estimating Faligue Lifeof Bevel and Hypoid Gears." SAE Quarterly Transactions, Vol.6, No.2, 1952.-
'7, COLEMAN, W. "Effect of Mounting Displacements 001'1 Beveland Hypoid Gear Tooth Strength." SAE-Paper 75 01 51. 1975.
May/June 1988 47
CALL FOR PAPERS- The Society ofanufacturing Engineers for its 1988Gear Processing & .Manutactur.lngClinic, Nov. J 5-17• Indianapolis. IN.Some suggested topics include gearbasics, shaper cutter applications,automotive applications, aerospacegears, heat treating and workholdingdevices. For further information, con-tact Joseph A. Franchini, SME. OneSIv1EDrive, P.O. Box 930. Dearbom. Mr48121. (313) 271-1500 x394.
-
1~ECI'1NI(_;l\L (~lU-,I~NI) l~ltJUNE 22-24. University of Wls-consln-Mllwaukee, MicrocomputerApplications In Worm Gear Design& Analysis. A workshop enablingstudents to develop customizedcomputer-aided gear design systems.For further information. contact: JohnM. Leaman, Center for ContinuingEngineering Education, UW-M, 929 N.6th Sr., Milwaukee, WI 5.3203. (414)227-3 J 10.
AUGUST UH2. Ohio State Univer-sity. Gear Noise Course. Materialcovered includes noise measurementand analysis, causes, reduction techni-ques. modeling and modal analysis ofgear boxes. For further information.contact: Mr. Richard D. Frasher, Collegeof Engineering. OSU, 2070 Neil Ave ..Columbus. OH43210. (614) 292-8143.
SEPTEMBER 27-29. American Soci-ety for Metal·s 11th .Annual HeatTreating· Conference, McCormickPlace, Chicago. IL. Presentations onsutijects including heat treating. sta-
COMPUTER-AIDED DESIGN ...(continued from page 34)
tion of the maximum stress willtend to move to the bottom of thetooth space apart from the loadedtooth.
8. Support conditions do not changethe stress distribution or filletstresses very much for a rigid rim.For thin rim gears, support condi-tions become an important issue.It may affect the rigidity of the rimand thus increase the stressesconsiderably.
9. For a rigid rim, the larger thepressure angle, the smaller the rootfillet stress. But for thin rims, thelarger the pressure angle, the largerthe fillet stress.
10.. Internal spur gears with differentpressure angles respond the sameway to the influences of rimthickness, support conditions,loading positions, etc.
48 Gear Technology
nstical process control. new energyapplications, quenching and cooling im-provements. For further information.contact: ASM International. Metals Park.OH 44073. (216) 338-5151.
NOVEMBER 5-10.. InternationalConference on Gearing. Zhengzhou,China. ASME-GRI and several interna-tional gear organizations are sponsoringthis meeting. For more information con-tact: Inter-Gear '88 Secretariat.Zhengzhou Research Institute of Mech-anical Engineering, Zhongyuan Rd.Zhengzhou, Henan, China. Tel: 47102.Cable 3000. Telex 46033 HSTEC CN.
NOVEMBER 8-10. American Soci-ety for Metals Near Net Shape Man-ufacturing Co,nference, HyattRegency, Columbus, OH. Program willcover precision casting, powder metal-lurgy, design of dies and molds. forgingtechnology and inspection of precisionparts. For further information contact:Technical Department Marketing. ASMInternational. Metals Park. OH 44073.
CALL FOR PAPERS - TennesseeTechnological University for its 1st In-temat'l Applied Mechanical Sys-tems Design Conference, March19-22, 1989. Nashville, TN. Papersare invited on general mechanicalsystems subjects including strength,fatigue life. kinematics, vibration,robotics, CAD/CAM. and tribology.Deadline for first drafts is Oct. I, 1988.For further information, contact: Dr.Cemil Bagci. Dept. of Mech. Eng .. TTU.Cookeville. TN 38505. (615) 372-3265.
of Internal Spur Gear". Bulletin of ]SME,Volume 25, No ..202, April, 1982.
8. BARONET, C.N. TORDION, G.V."Exact Stress Distribution in StandardGear Teeth and Geometry Factors" [our-nal for Industry, ASME: Transaction,Volume 95, November, 1973.
9. RUBENCHlK,. V. "Boundary IntegralEquation Method Applied to GearStrength Rating". Transaction of AMSE,[oumal of Mechanisms, Transmissionand Automation. in Design. Volume lOS,March, 1983. -
10. SEIREG, ALI, "State-of-the-Art Reviewof Computer Optimization of GearDesign", Journal of Mechanical Design,Transactions of ASME. Volume 103,January, 1981.
Reprinted with permission of the Amen'can GearManufactw:ers Association. The opiniol15, statemenl5and conclusions presented {n this paper are those afthe authors and in no way represent the position oropinion of the AMERICAN GEAR MANUFAC-TURERS ASSOCIATION.
References1. DRAGO, RAYMOND J.. PIZZIGATI.
Gary A. "Some Progress in the AccurateEvaluation of Tooth Root and FilletStresses in Lightweight, Thin-RimmedGears", AGI'vIA Paper 229.21, FallTechnical Meeting, Washington, D.C.,October, 1980,
2. AGMA Publication 225 ..01.3. AGI'vIAGear Rating Coordinating Com-
mittees. "Coordinated Rating for theStrength of Gear Teeth", Publication229.03, June, 1956.
4. AGI'vIA. "AGI'vIAManual for MachineTool Gearing", AGMA 360.02,.December, 1971.
5. AGMA. "USA Standard Tooth Propor-tions for Coarse-Pitch Involute SpurGears", AGMA 201.02, August, 1968.
6.. HWANG, JENG-FONG. "AdvancedComputer-Aided Design Method on theStress Analysis of Internal Spur Gears".Masters Thesis, The Ohio State Uni-versity, 1986.
7. AIDA, TOSHIO, et aI. "Bending Stresses
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