_------------TECHNICAILFOCUS------------II

tooth (aol .. a02) and theasymmetrical one (,aOI ~ am)examined had the same haO

asthe tool. the same r, thesame n1(}' the same x and 'tileame LThe calculation methods

applied to lite state of tre swere also 'Used 10 verifywhether and to what extentactual calculatien modesapplied to asymmetricaltooth profiles may prove use-ful when designing asym-metrical teeth, and whetherand .how said modes ought tobe modified, or whether theyneed to be replaced by newones devised ill order [0

obtain procedures useful tothe task at hand.

Mesb Generation: Studyand Oper:ative PropesalIn this work, reference i

made to external, cylindrica]wheels having straight teeth.

A study of a system for cal-culations capable of tracingthe tooth profile was carriedout and involved the rootdiameter, tooth root fillets,sides and major diameter.

The particularity of theasymmetrical profije formand the need 'to concentrate alarge number of elements(discrete variable speedmesh) regarding the morecritical areas of the structure,with a view to carrying alit a

Structural Analysis of TeethWith Asymmetrical Profiles

G. DilFrancesco & S. Mari:ni

AbstractThis article illustrates a

structural analysl of asyrn-metrical teeth. This studywas carried out because ofthe impossibility of applyingtraditional calculations toprocedures involved in the:specific case. In particular,software for the automaticgeneration of meshes wasdevised because existingsoftware does not produceresults suitable for the new

Having carried out the struc-tural calculations, a compara-tive study ofthe stress fieldsof symmetrical and asym-metrical teeth wa carriedout. The structural advan-tages of the Jailer type ofteeth emerged.

IntroductionIn a previou tudy (Ref. 1),

the po sibility of creating agear wheel. with teeth havingasymmetrical profiles capa-ble of conferring a more effi-

geometrical model required, cient structure upon the teeth

Symbols

lb, following symbols. Ibased on those recommendedby ISO/R 701 fUNI 67731 for notations penaining I,Dgears, shall be used:

h.,- reterence tool addendumk-Iowering ot head coefficient

m,l- reference module1'- reference pitch radius't - bale circle radius11- addendum modification coefficient

atw aaz- reference pressure angle during rotation inpreferential and non-preferential directions.

lliig. 1 - Gear Wheels wilh aSYllIJl1l1tricalllootbprofiles.

16; GEAR TECHNOLOGY

was discussed. This holds forall those numerous u eswhere the forces employedduring rotation in one direc-tion are greater than thoseengaged in rotation in theopposite one.

The teeth proposed have anasymmetrical fonn. The twosides of each tooth are, infact, characterized by loadedprofiles with different pres-. liTe angles.

As pointed out in theabove-mentioned study,asymmetry i achievable byadopting different aOI and,a02 values for the profile ofthe two sides of the rack. Itis, therefore, not necessaryto further modify the cuttingtools. The two differenttooth profiles thus obtainedhavetwo base circles, eachwith a different radius: FbJ ;;;;;

F cosaOJ' FbI" r easaw'Fig. I is anenlargement of

gear wheels having teethwith asymmetrical a()l ~ a02

profiles.In order to estimate abe

quantitative advantages ob-tainable using the kind oftooth proposed, the authorscompared the stress field of atraditional tooth and that ofan asymmetrical one, alloilier conditions being equal.The study was carried outapplying the finil.e elementsmethod, The symmetrical rapid comparison between

... TECHN'ICAL FDCOS, _. tre fields. urged theauthors to devise some spe-cific sefrware for the tudyof tbe problem. Thi waselaborated and supported by

Here is a run-down. of 'Ihechief features in the program:

• The geometric parame-ter of the teeth and of tbecutting tool. III w.eU as the

o.ftware capable of precise exact number of the sectorsand accurate analysis of Ule into which the profile is. toloolh areas under consider-.able degrees of stress ..

To thi end, th - tooth pro-file was di.vi:dedinto a serieof variable sectors based ondegrees of accuracy withwhich the various areas. needto be studied!. On the basi ofthe software drawn up totrace profiles. it. i pos ible'10 find the coordinates forthe points of each sector intowhich one decide to· dividethe profile itself.The structuralanalysis was

carried out on the section ofthe tooth contained on theaxial median of the gearwheel. bt this mesh genera-tion, the four-juncture.plane stre -element fermu-la wa .applied.

hl the construction of theme h. we havetsied to main-tain as regul!U'a fannas pes-ible, above all where the

need for deeper structuralinvestigation existed.

Automatic Tracio,gof the Mesb

The me h generation. pro-gram was studied and per-fected in uch a manner as '10

allow the u er to interact.through a "dialogue win-dow" devised pecifka]Jyfor that purpose.

The program was eon-eeived in uch a waya tocreate .31 basic me hautnmat-ic lIy with a profile dividedinto a number ofpre-chosensectors. It i therefore pas i-ble '10 depart from this meshby varying the mesh den ilywherever and whenever theexamination and experiencemay require it.

be divided. are included .• The program elaborates

inpu! dataand carries. our a.basic division of the variablepace profile,

.•If required, it is possibleto distribute (depending onthe nomber of sectors on theprofile) the diversified den-sity in a manner differentfrom that offered by theba ic program; this isachieved simply by usingthe dialogue window.

• The program, on tbebasis of the two tooth pro-files, recognize point uponthe same radial quota andtraces circumferencearcsbetween them. These arcsintercept the profiles per-pendicularly with a highdegree of preci ion in boththe tandard and the asym-metrical teeth.

• The program, havingcarried outthi _ubdivisiena~ong the radial direction,then doe likewi e along thecircumferencial direction. .orthis latter divi ion. it is uJfi-cient to input the appropriatenumber of sectors into whicheach of Ih arc is to be divid-ed ..The div:i ion along the cir-eumtesencial direction. too, isestablished in such a way asto obtain Ithe highe t concen-ttation in those areas (oear thetooth' lateral urfaces) wherethe tension field i al its peak.

The program has been stud-ied so as to match radial andcircumference concentrations.rhus obtaining element assquare as possible in thoseareas considered most impor-tant to structural analy i .

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••• _TECH,NICAL FOCUS _The program allow for di -

location of the areas of great-e t concenrrauon at anypoint whatsoever along thetooth. In the case of SlJ'\IctUJ-81 analysis of symmetricaland asymmetrical teeth, it isconsidered convenient to dis-locate the concentrations (asin the case of symmetricalteeth) at the fillet at~he rootof the tooth (Fig. 2).

It. is important to under-line the fact that programsdevised especially for thestudy of gears are based ongeneral characteristics toallow 'them to be used forteeth with different geomet-rical. patterns: high or lownumber of teeih (even in thecase of considerable under-cut, as .in gear pumps), toolshaving harp or rounded COf-

ners, symmetrical forms orasymmetrical. contours, as inthe present study.

As mesh generation pro-grams were devised to beapplied to, a broad range ofgeometric solutions, the fol-lowing must be kept in mind:

• In asymmetrical teeUl,the tangents to the profilesat points dong tile sameradiu may be (If the samesign Ubi occurs when 'Oneof the profiles has consider-able undercut).

• In the vicinity ofangularpointsof the tooth profile(teeth with considerableundercut). the arcs may curveinlhe opposite directions.

.' The consecutive sidesof an element may be farfrom acceptable perpendicu-larity values,

'. The ratio between thelengths of the greater andlesser sides 'Of an elementmay not fall within accept-able parameters.

Thi oftware was de igned

ings •. automatically modify-ing the mesh parameter . Inparticular, one or more eir-cumference arcs may beadded or eliminated or auto-matically substituted bycurves based 011 the coordi-nates of the nearby arcpoints. In any case, the pro-gram automatically optimizethe me haccording to criteriaestablished for a specific kindof structural analysis.

It is evident that in areas oflesser interest from a struc-

Angular transmissionor tooth coupling?With Cylkroefa,,~ gears thechoke is up to you. Paired With acylindrical pinion, the Cylkro® gear is ther·evolytionary .al~em",tive to bevel gears. The'combination of two spur Cylkro® gears. sunds for azero-backlash tooth coupling. Intereued?

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CIRCLE 153 to compensate for these fail-

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____ ITECHINIICAL FOCUS ..

fore. of minor concentrationof elements, a kind of meshquality closer to admissibleLimitsis acceptable.

The calculation programgenerate two output files:an editable .dxf file using acommon CAD program andan infonnation summary filefor the user.

Calculati.on of Stressand Results

The program drawn up.perfected and described inthe previous paragraph per-mits many easy and rapid

tructural analyses ofteethhaving considerably differ-ent parameters.

In all the cases mentioned,the binding system consid-ered is that with the joints ofthe line external to the cir-cumference of the internalcircle ..Numerous trials havepermitted ILlS, in fact, to limitthe area SUbjected to struc-rural analysis 10 those 0111-

side of which the stress andstrains are practically of noimportance even in the pres-enee of considerable loading.

The analysis was carriedout by charging the struc-ture with a normal. concen-tration of force along theside of the tooth. Among thevaricuspossihilities, partic-ular emphasis was placed allthe extreme case wherepres ure was placed on thetop of the tooth.

The comparison betweenthe stress fields of symmetri-cal and asymmetrical teethobtained by varying the aozvalue only while keeping thesame module, was carriedout. applying the same forcevalue to the top of the tooth.

Every series of geometricalcalculations for teeth differ-ent from each other (eachhaving a pecific a01 andvariable aoz) and the objectof the study. was rendereddiscrete by imposing thesame mesh generation para-meters for all geometricaltypes (for example, the samenumber of ector in whichto divide the profile. erc.) onthe calculation program.Each serie in the geornetri-

caI cslculations involved fix-ing a constant for aO] andvariablevaIue for a02•

The structural analy is wascarried. out by comparing thestress fields obtained byFEM a.naIy is.

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CIRCUE 154IIfIg. 3 -Ihree seb 01 ,ideal een-stant slrass.'curv8S.

______ T1EC:HNICAL IfUCUS _

An overall comparisonbetween the stress fieldsobtained using variou types.of geometry was carried out.Fig. 3 illustrates the idealconstant stres curves for thefollowing three cases: ,(101 =a02 = 20D; aOi = 20", ..1102 =14°; aOJ= 20D

, a02;:;;; 26".Furthermore, because the

areas that undergo the higheststress are those closest to thetooth root fillet. a qualitativeand quanticati ve comparisonof these areas. was provided.In. particular, it is possible tocompare significantly themaximum ideal sues valuescalculated both for symmetri-cal andasymmetricali teeth.

Fig. 4 shows the trends for..1.0';% of the function of thea02 angle, where ..1.Qj% is thepercentage variation of thevariation of the ,auz angle, ofthe idea] maximum stress incase of aOI = 1lo2'

Two geometrical serieswere examined. The first hadvalues of ,(X01 :: 20" and anaV2 varying from ]00 to 30".The second eries had a01 =17"30' and an a02 that variedfrom between 10° and 30° ..

The structural. analy is car-ded out u ing a calculationcode of proven trustwortlu-nessvpermitted the calcula-tion of element by elementideal stre s values; the maxi-mum ideal stress value pre-sent ill the correspondingtooth root fillet area ituatedon the same side .8S that fromwhich the solicitation camewas calculated for each toothgeometry type.

The outcome shows thatmodules and active side a.alpressure angles being equal,the maximum ideal: stress atthe root of the tooth diminish-es whenlhe Uo2 pressure angleam the inactive side increases.

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CIRCLE 135

____ ITECHN!ICAL. FOCUS _The results from the struc-

tural analy e emphasizedthe noteworthy convenienceof using ashigh a02 pressureangle values as possible(Ref, 1). aOI pressure anglesbeing equal.

In fact, for aO! - a02= 12°30', the diminution of themaximum ideal stress value.10'; = - 18.5%.

ConeluslonThe "me h generation" pro-

gram for asymmetrical teeth,

characterized by aOI =F a02

values projected and de-signed within the presentlimits, permits the swift and

easy structural analysis of abroad range of geometricaltypes for teeth.

Comparing the results

obtained, all else beingequal, for teeth whose a02

angle value differ, it appearsevident that structuralstrength increa e with an

increase in the aforemen-tioned angle,

This implies, when maxi-mum ideal tress at the toothroot is equal, that it is possi-ble to create an a ymraetri-cally toothed wheel having

dimensions less than those of

a symmetrically toothed one;Uris allows for a convenientreduction in the weight andsize, not only of gear wheelsbut also of the box and hous-ing containing them.

011 the basis of the resultsof the present tudy, 'Iheauthors are now examiningthe possibility of elaboratinggeneral criteria so thatsmaller (lighter) asyrnmetri-cally toothed wheels withthe same strength as syrn-metrically 'loathed ones maybe created. 0

References:1. Di Francesco, G. and S.

Marini: "RUOle Demate Car-atterizzate da Denti a Profile

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Asimmetrico." XXIV COIl-

vegno AlAS, Panna, 1995.2. Di France co, G. et al,

"Calculation of the MaximumBending Stress at the ToothRoo! Through an AIlaJ1ylk andGraphic Idemification of theResisting Sections. and Com-pari on of Their RespectiveStress Values." ICED '90(International Conference onEngineering Design), Dubroy-nile. 1.990.

3. Di Francesco. G."A.11aIi idelle Sollecitazioni suDenti di Ruote per UnitaIdro tatiche ad Ingranaggi,Espressione Analitica DefleTensioni al Piede dei Denti." nPrcgettlsta Industriale, Tee-niche Nuove, Gennaio-Febbraio, 1985.

4. Castellani. G. and V.Parenti Castelli. "Rating GearStrength." Transactions.ASME, April. 198].

5. Dudley., D. W. GearHandbook. McGlI'3w-Hili. NewYork,1962.

6. Henriol, O. TraiteThtorique et Pratique desEngrenages. Dunod, Paris,.1975.

Thanks to William Margolin ofSouthfield, MI. for help' withthe technical editi'lg of thisartide.

Prot. IG,. IDil1r,ancesco,is on the facllit)' of the Dept. ofMechanical ana Industrial EII-gineeriPlg of the University ofROrM.

Prof. S.lMa,riniis also OIl the mechanical engi-neering/acuity ar the University ofRome.

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