TECHNICAL FOCUS-------------.I~Structural Analysis ofAsymmetrical Teeth:

Reduction of Size and WeightAbstract

The present article con-tains a preliminary descrip-tion of studies carried out bythe authors with a viewtoward developing asymmet-rical gear teeth. Then a com-parison between numeroussymmetrical and asymmetri-cal tooth stress fields underthe same modular conditionsfollows. This leads to the for-mulation of a rule for similarmodules governing variationsof stress fields, depending onthe pressure angle of the non-active side. Finally a proce-dure allowing for calculationsof percentage reductions ofasyrrimetrical tooth moduleswith respect to correspondingsymmetrical teeth, maximumideal stress being equal, isproposed. Then the conse-quent reductions in size andweight of asymmetrical teethare assessed.

IntroductionIn a paper read at the 24th

Annual Italian Association forStress Analysis (AlAS) Con-vention (Ref. 1), a geometricalformula for a new type oftooth was presented; this for-mula is characterised by thefact that the two sides of thetooth have different forms.

The teeth proposed havean asymmetrical form. Thetwo sides of each tooth arecharacterized by profileswith different pressure angles.

These teeth prove useful inmechanisms where the forcesemployed during rotation inone direction are greater thanthose engaged in rotation inthe opposite one.

The difference of form isobtained by adopting differ-ent reference pressure angles(aO) *- aQ2) for the two sidesof a tooth.

At a later stage (Ref. 2), inorder to compare this newform with the traditional one,a structural analysis usingfinite elements was carriedout; in this study, given thecomplexity of asymmetricalprofiles, it became necessaryto devise a specific and totallyautomatic finite element meshprogram. This permittednumerous structural checkscovering a broad spectrum ofcases and a heterogeneousselection of tooth contours.

All other factors beingequal and comparing theresults for teeth having differ-ent a02 values, it appears evi-dent that structural strengthincreases with the increase ofthe aforementioned angles.

This implies that in thepresence of equal maximumideal stresses at the root ofthe teeth, it is possible to cre-ate asymmetrically toothedwheels of smaller dimen-sions with a decrease inweight and bulk, not only ofthe gear wheels themselves,

SYMBOLSThe following symbols, based on those recommended by

ISO/R 701 (UNI 6773l for notations pertaining to gears, shallbe used in the present paper:

reference modulereference module in asymmetrical toothweight of asymmetrical toothweight of symmetrical toothreference pressure angle of the grip side,during rotation in preferential and non-preferential directions.maximum ideal stress in symmetrical toothmaximum ideal stress in asymmetrical tootha01 - a02

but also of their boxes andhousings.

In Ref. 2 variations ofstress fields regarding a02were calculated. The maxi-mum ideal stress, (Jimax' cor-respondi ng to the mesh'suppermost element, wastaken into account; then thevalue for (Jimax under thesame load and using thesame module was compared.

The Law of StressField Variation forSimilar Modules

Keeping in mind the con-clusions of the aforemen-tioned studies, the presentwork aimed at carrying outautomated structural analysesof numerous asymmetricaltooth profiles. Then a com-parison between these andcorresponding symmetricalteeth, having similar mod-ules, was carried out.

The outcome of this thor-ough investigation confirmedthe hypothesis that increasesin a02 led to a diminution in

(Jimax val ues; in versely,diminishing a02 led to in-creases in (Jimax values.

The diminution of (Jimax'

with a02 > aO] is essentiallydue to an increase in the sec-tion close to the internal rootcontact area. On the basis ofthis, it is interesting to seek agenerally applicable mathe-matical relationship between.6.(Jimax % and .6.ao, where.6.(Jimax%= «Jimaxa-{)~l00'(Jimax'

From the results obtainedby carrying out numerouscomparisons, it emerged thathighly accurate variations of.6.(Jimax % related to .6.ao areobtained by applying the fol-lowing formula:

where \jf and S are two coeffi-cients obtained by interpolatingthe numerous results obtainedfor external spur gears.

The relationship des-cribed in Eq. 1 is general; itpermits us, without carrying

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____ TECHNIICAL FO'CUS _

llG% 30IIII

<lilt ,,20" I I' .1"

,

hhmD = cost iLl - I, llG%_

Rk - , -

I- II "'1~ t-I

-., .....,il

, lll'"")%- I--

20

'0

o-10

-20

40'0 '2 '4 '6 18 20 22 24 26 28 30Fig. 1

out structural analyses, toestablish readily stress fieldvariations for every kind ofasymmetrical tooth in rela-tion to a corresponding sym-metrical one.!Establishment of ReducedModules in the Presenceof Identical Maximum

Ideal StressOn the basis of the results

thus obtained, it has beennoted that, in the presence ofidentical modules, the diminu-tion of maximum ideal stressin a tooth having 0;02 > 0;01 isgreater in terms of percentthan the correspondingincrease in weight

Fig. 1 illustrates trends inthe presence of identicalmodules and of 0;01 forLl.0imax% and for Ll.G% =(Ga-Go)/GOJ100 where t:Xo2 =Ll.o;o+nolo

When 0;02 ::::::30° (that isfor Ll.ao= 0;02 - 0;01 = 10°),Ll.O'imax% = -15.2 over anincrease in Ll.G%weight =2_54; therefore only a slightincrease in weight is re-quired to obtain a consider-able reduction of the stressfield at the tooth root.

On the basis of Fig, 1 wecould see that maximumideal stress being equal forboth the symmetrical andasymmetrical teeth, it waspossible to create asymmetri-cal et02 > etoI teeth far lighterthan traditional ones.

It was therefore importantto study the possibility ofobtaining, at identical maxi-

!

CIm[OI ,

mum ideal stress rates, asym-metrical teeth smaller (smallermodule) and lighter than cor-responding symmetrical ones.

The possibility of quicklyand precisely calculatingsmaller mo. modules having0;02 > 0(1'01 tooth and maxi-mum ideal stress equal tothat of a. conventionally pro-filed tooth, was then testedunder the same load condi-tions, that is, applying thesame quantity of force.

This second state wasdevoted to seeking a calcula-tion system capable of'estab-lishing IDoo. modules having0.02 > 0;01 in the presence ofidentical maximum idealstress and identical load.

On the basis of thenumerous cases studied, itappeared possible to calcu-late with a high degree ofaccuracy asymmetrical toothrnoa modules. This was doneby departing from the con-ventional tooth mo modulevalue. This rno module valueis reduced according to' therelationship described inRef, I by a percentage valueequal to the reduction of theLl.0imax % value, It is, there-fore, possible to apply, with ahigh degree of accuracy. thefollowing relational formula:

&no % = .6.0imax %whereLl.mo% = (moa-Illo) 100/mo'

As C!.aimax % can beunequivocally established 0.11

the basis of a chosen Ll.o;ovalue, it is thus possible to

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____ TECH,NICAL FOCUS ...

calculate, according to therelaeionship obtai ned the

value for nloa' when the moreference modale is known.

The calculation methodolo-

gy developed required theintroduction of an index, defin-

able as the la"asymmetryindex," which can be calculatedon the basis of the following

simple relational equation:Ia = 'IIt:\ao - 't:\cxo 2 (2)Thela index allows for

ready calculation of reduced

maR asymmetrical CX02 > etOItooth modules. departingfrom the cnnventional tooth

lIIo module:

mOil = mo (1 +]u 1'100) (3)where Ia is derived from Eq..2.

Eq, 3 wa also te ted bycarrying out nurneroa COID-

pari ons between traditionaland asymmetrical teeth. Forevery pair of teeth-one sym-metrical and one asymmetri-

cal tooth based on Eq. 3-themaximum ideal stress values,

calculated using automatedstructural analysis, were com-pared. The differenee betweenthese maximum ideal tressvalues always stood at about1% and never topped 2%.

Summing up, at the pre-sent stage of development, in

order to calculate the size of

<lG%1050

-5-10-15-20-25-30

an asymmetrical tooth, onemay proceed as follows. ltis

po ible to begin from the

size of a traditional syrnmet-

ricaltooth. Using the usualcalculation procedures, the

mo module is estimated.

Then one must choose avalue for <Xro > CIoI compati-ble with the filial geometricalcharacreri tics de ired.

By applying Eq, 3, it ispo • ible to calculate the I110amodule value for 'the asym-metrical 'tooth.

The above applie . in thepresence of equal force loads

being brought to bear uponthe symmetrical and asym-metrical teeth.

At this point it may be of

interest 10 verify the degree

of weight reduction applica-

ble on the ba is of thereduced module obtained byapplying <Xo2 > <XcI'

In the graphs in Fig. 2 the

values for t:\G% related to 000.for identical aOl values, andfor maximum ideal . tress at aconstant within the rangesdescribed above (±2%), this in

the following two cases: 1101=]7°30' and '4>1 = 20°.

In the case of aOI =17°30', for <Xc2 = 30° (that ist:\,cxo= 1r30'), the weight

-3521} 22 24 26 2B 30 11m'"'

I . !lo1 c 201' .I CDnst~nlldeBI Stress -

r-, I I

~ i I

, -',,- !

i "'1 ....... I

I r--,I

I I

AG'II.1050

-5-10-15-2{1

-25-30

!lo1 = 17·30'I

Constant Ideal Stress -

r-, I

........ ! I I

I <, I

I I <,! I I ~ I

I I ........I I .~

Fig. 2-35

11.5 21} 225 25 27.5 30 11m'"]

• TEcHN~CAL.iFOCUS _

Fig.]

reduction obtained is over30%; in this particular casethe I!imax a value was :1 %lower than that for the sym-metrical tooth.

fu Fig. 3, a traditional and

torques. It is interesting tonote how the reduction of thetransmitted torque, which, asis known, is equal in terms ofpercent to the reduction ofthe tooth module, is alsolighter in weight, It is there-fore possible to create teethhaving CCa2 >CCaI and smallermodules, capable of trans-mitting the same torque astraditional teeth, althoughtheir size is smaller. Withthis in mind the authors are

an asymmetrical tooth in iden- carrying out. a specific studytical maximum ideal stresscondition are shown. For bothaOl = 17°30', the number ofteeth is 20; furthermore Illo =2 mm, IDoa = 1.63 mm.

It emerges from this figurethat, ill the presence of identi-cal maximumideal stress, itrs more economical to 'Useasymmetrical rather than tra-diuonal symmetrical teeth.

ConclusionsIn the light. of these consid-

erations it appears evident that,in cases where weight reduc-tion is the primary need, theapplication of .1Xoo; > 0.0\ teethleads to noteworthy results andshows the undeniable benefitsof adopting teeth. with nOD-

conventional profiles.At this stage the structural

analysis and the entire studymight stop here, were it notfor the fact that the reductionof the module (adopted toachieve the same state ofstress in the presence of Iden-tical load) involves a corre-sponding reduction of thediameter of the gear wheel. areduction which, under con-ditions of identical normalforce, leads to. a reduction ofthe transmitted torque, Thetwo gears, conventional andasymmetrical, while equiva-lent from a structural point ofview, are not from the pointof view of the transmitted!

for the calculation of mOa

modules having the sametransmitted torques, 0

References:L Di Francesco. G. and S.

Marini: "Ruote Dentate Caratteri-zzate da Denti a ProfiloAsimmetrico." XXIV ConvegnoAlAS, Parma, 1995.

2. G. Di France co, S. Marini:"Structural Analysis of Tooth Havingan Asymmetrical Winding Profile."International Conference on MaterialEngineering, Gallipoli, 1996.

3. Di Francesco, G. et al."Calculation of the Maximum BendingStress at the Tooth Root Through anAnalytic and Graphic Identification ofthe Resisting Sections, and Cornpari-son of Their Respective StressValues." ICED '90 (InternationalConference on Engineering Design),Dubrovnik, 1990.

4. Di Prancesco, G. "Analisidelle Solleeitazioni su Denti di Ruoteper Unitll.Idrostatiche ad lngranaggi.Espressione Analitica Delle Tensionial Pi ede de i Denti." II ProgettistaIndustriale, Tecniche Nuove ,Gennaio-Febbraio, 1985.

5. Castellani, G. iIIId V. ParentiCastelli. "Rating Gear Strength."Transactions, ASME. April, 1981.

6 .. Dudley. D. W. Gear Hand-book. McGraw·HiIl ..New York, 1962.

7. Henriot, G. Traite Theoriqueet Pratique des Engrenages. Dunod,Paris, 1975.

Dr. G. D'i Francescois all the faculty of the Dept. ofMechanical and Industrial Engineer-ing oIrlle U,,;versiry cf Rome.

Pirof. S. Mariniis also 011 the mechanical engineeringfaclI1Jy aJ the University of Rome.

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