direct gear design for spur and helical involute gears ... gear design for spur andhelical involute...
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Direct Gear Design for Spurand Helical Involute Gears
Alexander L Kapelevich and Roderick E Kleiss
Introductionodem gear de ign i generally based on
tandard tool Thi make gear de ign quite im-pie (almost like electing fa tener ) economicaland available for e eryone reducing toolingexpense and inventory At the arne time it iwell known that univer al tandard tool providegear with le than optimum performance anel-in orne cases--do nOI allow for finding accept-able gear elutions Application pecific includ-ing low noi e and vibration high den ity ofpower transmi ion (lighter weight mailer ize)and others require gears with nonstandard para-meters That why for example aviation geartran mi ion usc tool profile with cu torn pro-portion uch a pre ure angle addendum andwhole depth The following con iderations makeapplication of non tandard gears uitable andco t-effi ientbull C C cutting machine and CMM gear in pec-tion equipment make production of non tandardgear as easy a production of tandard onesbull Co t of the cu tom cutting tool i not muchhigher than that of the cutting tool for standardgears and can be amortized if production quantityi large enoughbull The cu tom gear performance advantage makesa product more competitive and ju tifies largertooling inventory e pecially in rna productionbull Gear grinding i adaptable to cu tom toothhapes
bull Metal and pIa tic gear molding cost largely doesnot depend on tooth hape
Thi article presents the direct gear designmethod which parates gear geometry definitionfrom tool election to a hievc the be lpo ible per-formance for a particular product and application professional engineer is
The direct de ign approach that i commonly owner and president of
u ed for most part of mechani m and machine Kleiss Gear I Hi com-
(f I Imiddot k pany engineer and mam-or examp e earn mage compre or or tur- o 10 X I factures high precisionbine blade etc) determine their profile accord- plastic molded gear u illg
ing to the operating conditions and de ired per- Figure i-The zone allowed by the tandard 20deg the direct gear de ignformance Ancient engineer u ed the arne rackor a gear pair zJ = 14 Zz = 28 approach
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This paper present an aJtemative method of Ianalysi and de ign of pur and helical involute
Igears i
IiIiiI
N n tbw face width in the meshd outside circle diameter mm (in)d base circle diameter mm [in]dj tip circle diameter mm (in)m proportional top land tooth thicknessmb proportional base tooth thicknessPb base pitch mm [in]S top land tooth thickness mm (in)Sb base tooth thickness mm [in)u gear ratio
number of teethoutside circle profile angle degreesprofile angle in the bottom contact pointdegreesoperating pressure angle degreesbase circle helix angle degreescontact ratioaxial contact ratiotip circle profile angle degrees
z
Subscriptpiniongear
15
10
5
a
-5
Dr Alexander LKapelevichi all on1r of the consult-ins firm AKGlars ofShoreview M and prin i-pal engineer for KleissG ar Inc of CentervilleM H has more than 20year of experience indevelopment of aviationarid commercial gear Irans-mi ions irl Russia WId theUnited Slates
Roderick E Kleiss
ting tool (generating ra k) were tandardized Thi ha made modern involute gear de ign indi-rect because the gear tooth profile depend on apre elected usually tandard et [or parameters fthe generating rack (diametral pitch or modulepre sure angle addendum and dedendum propor-tion tip radii etc) and it I cation (addendummodification or x- hift) relative to a tandardpitch diameter of the gear
Table 1 how a typical helical gear pe ifica-tion where gear parameter and the generatingproces (rack and its location) parameters arc sep-arated The gear a a part does not ha e a pre urangle pitch diameter diarnetral pitch or moduleheli angle addendum or addendum modific -tion All the e parameters are related to the tooland generating proce The involute gear ha anumber of teeth base diameter out id diameterhelical lead and bas tooth thickne
he generating rack method of gear de igndoes not guarantee sufficient gear de ign Theminimum number of pinion teeth i limited toavoid undercut The addendum modification or -shift of the generating rack i introduced t bal-ance bending fatigue tre e and pecific lidingfor pinion and gear and to redu e undercut forpinions with mall numbers of teeth
Why must to th profiles be modified or cor-rected at the vcry earliest stages of the geardesign The modification must occur 0 earlybecau e the traditional approach i limited by itown arbitrary election of generating rack par -meter The zone depicted (Ref I) in -shift coef-ficient coordinate XI and x2 for a pair of purgear ZI = 14 ~ = 28 formed by a tandard gen-erating rack with 20deg pre ure angle i hown inFigure I The zone hown contain all gear com-bination that can be produced u ing thi particu-lar generating ra k It area i limited by the min-imum contact ratio for pur gears Eo = 10 (i 0-
gram A) the harp tip of the pinion (i ogram B)and the tip-fillet interference (i ograms C and D)The undercut isograms E and F put additionallimitations on the zone area Oth r available gearcombination exist outside the zone border butin order to r alize them the generating rack para-meters would have to be changed Tn other worda range of po ible gear combination i limitedby the cutting tool (generating rack) parameterand the machine tool etup ( -shift)
Direct gear de ign i the way to obtain all po -sible gear cornbinati n by analyzing their prop-erties without using any of the generating proceparameter Tho e parameter can be defined
30 SEPTEMBEROCTOBER 2002 bull GEAR TECHNOLOGY wwwgaarrechnologycom bull wwwpowerrransmissioncom
approach for gear de ign developing the tooth hape fir t and then figuring out a way to get it
During the technological revolution in the 19thcentury the highly productive gear generatingproce wa developed New machine toolsrequired complicated and expen ive tool hobor gear shaper Common parameter f the cut-
Table 1Drawing Specification Generating Process Parameter Gear ParameterNumber of Teeth XStandard Normal Pitch XNormal Pressure Angle XStandard Pitch Diameter XHelixAngle XHand of Helix XHelixlead XBase Diameter XFormDiameter XRoot Diameter XOutside Diameter XToothThickness onStandard Pitch Diameter XAddendum XWhole Depth X
Table 2Parameter Symbol Equation I ValueNumber of teeth Igivenl 11 14
z 28Proportional top land thicknesses (givenl mal 0075
m 0075Center distance inlgiven) 8w 3000Proportional base tooth thicknesses mb1 0755
(chosen from area of existence) mb2 0645Profile angle in the involute intersection point deg VI 14a) 4214
v 3285Profile angles on outside diameters deg as1 (5al 4123
aa2 3183Operating pressure anole dell Clw liD) 2498Transverse contact ratio E 12 160
Profile angles in the bottom contact points deg Clp (13) 887ClD2 (14) 1461
Base diameters in dbl (l1al 1813db2 dlf2 = db1bull U 3626
Base pitch in Pb (3) 0407Operating pitch in Pw (7l 0449Operating pitch diameters in d bullbullbullbullbull (8) 2000
dwz 4000Operating tooth thicknesses in Sbullbullbullbull 19) 0279
Swz 0170Outside diameters in dabull (2a) 2410
da2 4267Outside diameter tooth thicknesses in SSl 15b) 0030
5s2 0030
after the gear de iga i completely ani hedTher were attempt to u th bar circle as a
f undati n for the in lute gear th ry parat-ing the gear analy i from th g ar g n ratingpr e Pr fe r EB ulgako d I peel th
ailed the ry of generalized param teinvolute ge Ref 2 1R C I urn (Ref )d ribed an alt mati e definiti n f th in lutwith ut using the generating ra k Th s lf-g n-crating m thod gear f ems g ar pr sedfor pia tic m Ided gear (Refs 4 and 5)
c rding to thi method the top land f tht th of on of the gears form th fillet f thmating gear and vice er a I a gl n e it I kimilar 10 a gear haping r ge r lJing p
but th fa t that both ge are de ribed with utth generating ra k param te mak a dill r-n e in their go m try and hara teri ti
Involute Tooth Param ten involute tooth i f nn d tw in lute
un ound from the ba e circle db out ide circldiameter da and fillel (Ref 2) ( ig2) nle
therwi rated the foll win equ ti n are r-rect for spur gears and f r heli al gears in thtransver e e lion (the ecti n perpendicular I
th axi of the gear) Equati n numbers withalphabetic modi ier are gi en f r u in lhnumeri example Ii ted in Table 23 and 4
e profile angle in the interse ti n point fthe two involute (tip angle) i
v=aco dJdA (I)where dA i th harp tip irele diam t c
Th profile angle on th ut id diamet r da iaa = a 0 dJda) (2)da = dJ 0 (aa) (2a)
Th base pitch iPb = rt 0 dJz 3)
where z i the number f teethe proportional ba t th thickn
mb = SJpb = z 0 in (v)ht (4)invtv) = rt 0 InJz (4a)
wh r Sb is the bas thi kne The pr portional t p land thi kne s is
rna = Sjpb = Z 0 (invtv) - inv(aa))(lt 0 c et) (5)
co (a) +in (a)(rt 0 rna = mJma (5a)Sa = r 0 rna 5b)
inv(v = (n 0 maco aa + zinv araquo)z 5c)where Sa i the I p land thickne Th recom-mended alue of Ina h uld be I n 006 and01210 avoid a harp t th tip and pro ide u 1-
ient onta t ratio in th hIn olut Param t
Figure 3 ho th L n f I th Ii n f thpinion and the gear in cl me h ( a klash i
Figurlaquo 2-The involute tooth parameter definl-tion
Figurtion
da
zero cl e m h condition i
pbullbull=S bullbullbull+ bullbull2 (6)where
PM = rt 0 d bullbullz = rt 0 d bullbulliz2 7)is perating circular pitch
dbullbull = dblcos(CtbullbullJ dM2 = dbtco a ()are the pinion and the gear perating pitch diame-t and
bullbull = (inv(v ) - inv(a bullbullraquo 0 dbCO middot(a bullbullJ 9)bullbullbull2 = (inv(v2 - inv( ) 0 (dbtc (a bullbull)
are th pini nand th gear peratin 1 th thick-n ses
The perating pressure angle can be und bub tituuon of uati n with 78 and 9
in ( )=(inv(v)+Ioin V2)-ltIz1)1 +u (10wh re u i th gear rati I = 1
Th operating pre UTe angle i a gear m hparameter and it cant be de 1000 f r ne sepa-rate gear
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The center distance isa = dbl deg (l + u)(2 deg cos(awraquo (11)dbl = a deg (2 bull cos(a)(l + u) (11a)
The contact ratio (for spur gears and for helicalgears in the transverse section) is
en = z deg (tan(aa) + u deg tan(aa2)- (l + u) bull tan(a)(2 deg It)
The profile angle in the bottom contact pointmust be larger than or equal to zero to avoid invo-lute undercutfor the pinionap = atanlaquo(l + u) deg tan(a) - U deg tan(aa2raquo ~ 0 (13)for the gearap2 = atanlaquo(l + u) deg tan(a)u - tan(aa)u) ~ O (14)
The axial contact ratio for helical gear isE~= zolgt(2olt) (15)
where lgt (in radians) i the angular hift betweenthe oppo ite transverse section in the helicalmesh (see Fig 4) and
lgt = (2 deg b) deg tan(~b)db (16)where bw i the width of the helical mesh and ~b isthe helix angle on the base circle
The fillet profile must provide a gear mesh withsufficient radial clearance to avoid tip-fillet interfer-ence The fillet also must provide necessary toothbending fatigue resistance and me h tiffnes Thedirect gear design approach allow selection of any
I fillet profile (parabola ellipsis cubic pline etc)that would be t ati fy tho e condition Thi profileis not necessarily the trochoid fanned by the rack orhaper generating proce s
Tool geometry definition is the next step indirect gear design This will depend on the actualmanufacturing method For plastic and metal gearmolding gear extrusion and powder metal gearprocessing the entire gear geometry-includingcorrection for shrinkage-will be directly appliedto the tool cavity For cutting tools (hobs shapercutters) the reverse generating approach gearforms tool can be applied In thi case the tool-ing pitch and profile (pre sure) angle are selectedto provide the best cutting condition
Area of Existence of Involute GearsFigure 5 shows an area of existence for a pinion
and gear with certain numbers of teeth zl ~ andproportional top land thicknes e mal ma2 (Ref 2)Unlike the zone shown in Figure 1 the area of exi -tence in Figure 5 contains all possible gear combina-tions and is not limited to restrictions irnpo ed by agenerating rack This area can be shown in propor-tional base tooth thicknesses mb - mb2 coordinateor other parameters de cribing the angular di tancebetween two involute flanks of the pinion and gear
I teeth like aal - aa2 or v I - v2A sample of the area32 SEPTEMBEROCTOBER 2002 bull GEAR TECHNOLOGY wwwgearechnologycom bull wwwpowerransmissloncom
Figure 4-The angular shift of the transverse sec-tions for a helical gear
IDb2
20
15
10
5
o I5 10
Figure 5-The area of existence for the gear pairzJ = 14 Z2 = 28
a c
b d
Figure 6-lnvolute gear meshes 6a at point A of Figure 5 (aMmax = 395deg Eu =10) 6b at point B of Figure 5 (abullbull= 167deg ecaliu = 201) 6c at point C ofFigure 5 (amnx = 296deg Ea = 10) 6d at point D of Figure 5 (aM = 159deg EamllX
= 164)
(12)
of exi tence f r a pair of gears ZI = 14 = 2 mol
= ma2 = 0075 i shown in Figure 5 e area f exis-tence include a number of isogram reflecting con-tant value of different gear paramete u h
operating pre ure angle CX contact ratios Ea etcThe area of existence of pur gears (thi k lin I) ilimited by isogram Eo= 10 and undercut i gramCX 1 = 00
CX = 00bull HeLi al gears can have a tran -p p-
ve contact ratio Ie than I becau the ialcontact ratio can provide p per me h Th area fexi tence of helical gears i th refore mu h greater
h point on the area of e i len e refl IS a pair fgears with dimen ionle propertie that can fit a par-ti ular appli arion Th propelti are pres ureangle c ntact ratio pitting re istance g merryfactorl pecifi liding ratio et
The absolute area f e i ten e in Iud pur garcombinatio with any alue f proportional t p Iland thickne ber een mal = ma2 = 0 t TIol = mbl
and m 2 = mb (phantom lin 2) Thi area i ub- a _ I
tantially larger than the area ith gi en alue f iproporri nal t p land thi knes The zone f r a tandard generating rn 1ltwith 200 pre sure angle ( hown in ig I) i only a fractional part of the a ail-
able area of exi tence as h wn by hidden lin 3 in iFigure 5 An appli ati n of a traditi nal gear gener-
Iating app a h for gear pairs out ide th z ne ut- ilined by hidden line 3 require Ie Lionof a gen r-ating ra k with differentparam ters The generationof orne gear combinations (t p left and bottom rightcomers of the area of xi t n e hown in Fig 5) willrequire different generating ra ks for the pinion andfor the gear
Analy i of the area of exi tcnce haws how imany gear olutions could be left out of c nsid ra- tion if a traditional approach based n a predet r-mined t of ra k dimension i applied F r e am-ple pur gears with a high operating pressur angl(point A on the Figure 5 where th rating pres-ure angle cx= 95deg conta t ratio Eex= 10) r with
a high contact ratio (point B on the Figure 5 whereonta t ratio Eex = 20 I operating pre ure angl cx
= 1670) could not be produced with tandard ra k
dimen ion Fi ure 6a and 6b h w th geFigures 6c and 6d are the gears that are achie ableusing a tandard g nerating ra k that arc pre nted I tandard designed gear pair h a conta I ratio ofbypoinlSC(a bullbull=296degE
Q= 10)andD(a= 15cr i only a= 116in lose m hlnana tual application
En = 164) in igure 5 E en gears with the same with real manufacturing ~ leran and operatingoperating pre ure angle (point in ig 5) 1 k c nditi ns the ntact ratio of a tandard de ignedquite different (Fig 7) Th tandard d igned gear i gear pair uld be reduced t an unacceptable Ie elpair (Fig 7b) has alrn t sharp-pointed pini n t th Enlt 10and hort and tubby gear teeth with excessi e top ynthe i of Garing umerical xampJland tooth thi kne The direct de igned gear pair Th r arc everal w y to define gear parame-hownin Figure 7a has a contact ratio Ea = 147The i I ruing the direct gear de ign approach Thi
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a
b
Figure 8- helical gear pair with higl operatingpressure angle I = 9 2 = 12 a = 688~ Ea =0504 3 = 30~ EfJ = 055
Figur 9-A Jpur gear with a Itigll contact ratio I= 55 2 = 55 a = J29~ Ea= 40 lIIal = lIIal = 0075
article considers some of themArea of existence is known The initial data for
the synthesi of a pair of gear (zl Z2 01 f1l2)could be taken from the area of exi tence at omeparticular point The coordinate of thi point andcenter di tance abullbull describe all operating gearparameter This calculation procedure andnumerical example are presented in the able 2
Area of existence is 1101 known pical pr b-lern could be finding the maximum pre ureangle if the tran verse contact ratio i ch en orfinding the maximum tran verse contact ratio ifthe pre sure angle i cho en Both of the e ca erequire finding the point of area of exi tencewhere i ogram ltxwand Eo have the arne tangentThis condition is de cribed (Ref 2) as
cos(0)2 (l + 1t bull mal sin(ltXI)Iz)=CO (02)2 (I + 1t ma2bull in(oa2)1z2) (17)
and allows solution of the e problem withoutknowing the area of existence The calculationprocedures and numerical example are pre entedin Table 3 and 4
The fillet between teeth is not involved in gearmesh operation but it shape greatly affect gearperformance and durability In traditional gearde ign the fillet profile is a function of the cutlershape and the machine tool setup It typically haexce ive radial clearance resulting in high bend-ing stresses Direct gear de ign doe n t limit fil-let hape definition One po ibility i to de cribethe fillet profile as a trace of the top part of themating gear tooth (with corre ponding minimumradial clearance) (Ref 4 and 5) Application offinite element analysis allows for forming the fil-let profiles to balance and minimize bendingstresses
Extreme Parameters of Involute GearsPoint A (tangent point of i ogram Eo = 10 and
0bullbull= max) of the area of exi tence de cribe gearwith the maximum achievable operating pre ureangle There is no such limit for helical gearbecause a lack of the transverse contact ratio (Eo lt10) i compen ated by the axial contact ratio Ep
A ample of a helical gear with high operatingpressure angle (Ref 6) i shown in Figure InFigure 5 the point B (inter ection point of inter-feren e i ogram up I = 0deg and op2 = 0deg) of thearea of existence de cribe the gears with themaximum achievable transver e contact ratioTable 5 pre ents maximum value f r operatingpressure angle owA (Point A of the area of exi -abc renee) and tran ver e contact ratio Eo B (Point B of
Figure lo-Spur gears with minimum number of teeth a) ZJ = 5 Z2 = 5 a =33 1deg E 104 b) Z 4 Z 6 a 326deg 10) 3 11 the area of existence) for gear pairs with different oa = J = 2 = = ea = ~ C zJ = Z2 = a= 243 ea = 101 numbers of teeth and the proportional top land34 SEPTEMBEROCTOBER 2002 bull GEAR TECHNOLOGY wwwgesrtschnologycom bull wwwpowerlrsnsmlssioncom
Table 3Parameter Symbol Equation ValueNumber of Teeth (givenl Z 14
z 28Proportional top land thicknesses 19iven) ml 0075
m 0075Center distance in (given) 8w 3000Operating pressure angle deg [chosen] (lw 33Profile angles on outside diameters deg (ll (l0) amp 117) 4002
(lll 3884Transverse contact ratio (maximum 1 famagt 121 1246Profile angle in the involute intersection point deg VI (5cl 4103
v2 4036Proportional base tooth thicknesses mbl (41 0687
mb2 1296Use Table 2 to identify remaining equations
Table 4Parameter Symbol Equation ValueNumber of teeth (givenl Zl 14
lz 28Proportional top land thicknesses (given) ml 0075
m2 0075Center distance in (given) a 3000Transverse contact ratio (chosen) ElY
105Profile angles on outside diameters deg (lal (l0J(l21 4329
(laz amp (17) 4312Operating pressure angle deg [rnaximum] (lwmu 10J 021 amp (17) 3797Profile angle in the involute intersection point deg v (5c) 4406
v2 4352Proportional base tooth thicknesses mb1 (4) 0886
mb2 1693Use Table 2 to identify remaining equations
Table 5
owAIE8I Pinion ZI
5 10 20 30 40 505 315deg10 350deg114 378deg121 39deg123 396deg124 40deg124
~ 10 350deg1 14 368deg143 385deg166 394deg172 399deg175 408deg176bull 20 378deg121 385deg166 394deg1214 399deg237 402deg248 404deg254Ggt(l 30 39deg123 394deg172 399deg237 402deg274 404deg297 406deg31
40 396deg124 399deg175 402deg248 404deg297 406deg328 407deg34950 40deg124 408deg176 404deg254 406deg31 407349 408deg378
thickn s mal 11102 0075 An example of apur gear me h with a high coma I ratio is sh wn
in Figure 9pur gear (contact ratio En 10) with a min-
imum pos ible number of t th (Ref 2) areshown in Figure 10The minimum po ibl num-ber of teeth f r helical gears is n t limit d bytran erse contact ratio and could be as few aone (Ref 6) n e ample of a helical gear with thenumber of t eth - = ~ = I is shown in Figure IIIn elute Gea with mmetric ToothProlile
Oppo ite nan (profile ) of the gear I th arefunctionally different for m t gear Th w rk-I ad on one profile i ignificantl higher and ri applied for longer period of lime than n thoppo ite one The a ymm tric tooth shape a com-modates thi fun tional diff ren e
Th design intent of a ymmeuic teeth i Iimpro e performance of main nta tin p filby degrading oppo ite profile Th ppo ire pro-fil are unload d or lightl I aded and usuallwork for a relatively h rt period The impro edperformance c uld mean in rea ing load capacityor reducing weight noise vibration tc
Degre of asymmetry and drive pr file elec-tion for these gears depend on the applicati n
ymmetric profiles make it possible to managetooth stiffnes and 1 ad haring while k ping ade irable pre ure angle and conta t rati n thedrive profile
Dire t design of gears with asymmetri teeth iscon idered in detail in ther articles (Refs 7 and8) covering topic u h a analy i and synthe isof asymmetric gearing area e i t n e andapplication ample of gea ith asymmetritooth profile are hown in Figure 12 Gears witha yrnrnetric teeth h uld be c n sidcred f r geary terns that require extreme perf rmance like
aero pace drive Th Y are al a applicable f rmas production lransmi ion where the hare ofthe tooling co I per one gear is relatively in ignif-icant The mo t promising application for asym-metric profile i with m Ided gears and wdermetal gea Molded gear tooling u ually requirea custom shape 0 ih as mmetric profile dnot ignificanl1y affect c t
umm ryDirect gear de ign i an alternati e appr ach to
traditional gear design It allow analysi of awide range of parameters for all po ible gearcombination in order to find th m st uitablesolution for a parti ular appli arion This opti-mum gear oluli n can exceed the limits of tradi-tional rack generating methods of gear design
Figure ll-Helical gears with one tooth zL I Z2= I abull= 688~t = 056 b = 349~EIJ012
Figure 12- pur gear with a ymmetric teeth 12u)a generator gear drive with a = 41 ~ Ea = 12 fordrive flanks ~lfIda = 18~ Eq= l64for coastflanks12b) a pia tICgear pump WIt a = 457~E 101for drive flanks and a JO3~Ea = I09ltr coastflanks
Dire t gear d sign for asymmetric tooth profilespen additi nal re erve f r improvement f
gear dri es with unidirecti nal load cycles thatarc typical for many mechanical transrnissi n 0
cknowl dgm ntsThe authors expre deep gratitude 10 Gear Technologytechnical editors Robert Errichello of Geartech 1 ated inTownsend MT and Dan Thurman for their help in prepar-ing thi article
ReferencesJ Gr man MB The Zone f Involute Me h Ve tnikfa hiw troeniva 1962 [ ue 12 pp 12-17 (in Ru ian)
2 ulgakov EB Theory of Involute GearsMa hino troenie Mo 0 1995 (in Ru ian)
olbourne lR The Ceometrv of Involute Gearspringer- erlag ew York 1987
4 IAGM 1006-A97 Tooth Proportion for Pia ticG ppendix F -o nerating Gear Geometr WilhoutRa I GMA Ale andria A 19975 Kupelevich L 1 Klei s and RE Kleis ewOpportunities with Molded Gear AG Fall Techni alMecling Detroit October -52001 (01 FTM9)6 Kapelevi h AL and pound8 ulgakov Expanding therange f involute helical gearing Yestnl]MasIJlllomiddottroeniya 19 2 I sue 3 pp 12-14 (in Russian)
ranslated [0 nglish 0 iet Engin ering Research Vol 2I lIC 3 1982 pp 8-97 Kapelevi h AL Gc rnetry and design of in oluiepur gear with a ymmctric teeth Mechanism and
Machine Theory 2000 I sue 35 pp 117-1308 Kapclevich AL Q Lian and FL uvmA ymmetric rnodif d g ar drive reduct jon or noi elocalizati n or contact imulauon of meshing and stresanalysis Computer Mehods in Applied Mechanics andEngineering 2000 Issue 18 pp 63-390
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ting tool (generating ra k) were tandardized Thi ha made modern involute gear de ign indi-rect because the gear tooth profile depend on apre elected usually tandard et [or parameters fthe generating rack (diametral pitch or modulepre sure angle addendum and dedendum propor-tion tip radii etc) and it I cation (addendummodification or x- hift) relative to a tandardpitch diameter of the gear
Table 1 how a typical helical gear pe ifica-tion where gear parameter and the generatingproces (rack and its location) parameters arc sep-arated The gear a a part does not ha e a pre urangle pitch diameter diarnetral pitch or moduleheli angle addendum or addendum modific -tion All the e parameters are related to the tooland generating proce The involute gear ha anumber of teeth base diameter out id diameterhelical lead and bas tooth thickne
he generating rack method of gear de igndoes not guarantee sufficient gear de ign Theminimum number of pinion teeth i limited toavoid undercut The addendum modification or -shift of the generating rack i introduced t bal-ance bending fatigue tre e and pecific lidingfor pinion and gear and to redu e undercut forpinions with mall numbers of teeth
Why must to th profiles be modified or cor-rected at the vcry earliest stages of the geardesign The modification must occur 0 earlybecau e the traditional approach i limited by itown arbitrary election of generating rack par -meter The zone depicted (Ref I) in -shift coef-ficient coordinate XI and x2 for a pair of purgear ZI = 14 ~ = 28 formed by a tandard gen-erating rack with 20deg pre ure angle i hown inFigure I The zone hown contain all gear com-bination that can be produced u ing thi particu-lar generating ra k It area i limited by the min-imum contact ratio for pur gears Eo = 10 (i 0-
gram A) the harp tip of the pinion (i ogram B)and the tip-fillet interference (i ograms C and D)The undercut isograms E and F put additionallimitations on the zone area Oth r available gearcombination exist outside the zone border butin order to r alize them the generating rack para-meters would have to be changed Tn other worda range of po ible gear combination i limitedby the cutting tool (generating rack) parameterand the machine tool etup ( -shift)
Direct gear de ign i the way to obtain all po -sible gear cornbinati n by analyzing their prop-erties without using any of the generating proceparameter Tho e parameter can be defined
30 SEPTEMBEROCTOBER 2002 bull GEAR TECHNOLOGY wwwgaarrechnologycom bull wwwpowerrransmissioncom
approach for gear de ign developing the tooth hape fir t and then figuring out a way to get it
During the technological revolution in the 19thcentury the highly productive gear generatingproce wa developed New machine toolsrequired complicated and expen ive tool hobor gear shaper Common parameter f the cut-
Table 1Drawing Specification Generating Process Parameter Gear ParameterNumber of Teeth XStandard Normal Pitch XNormal Pressure Angle XStandard Pitch Diameter XHelixAngle XHand of Helix XHelixlead XBase Diameter XFormDiameter XRoot Diameter XOutside Diameter XToothThickness onStandard Pitch Diameter XAddendum XWhole Depth X
Table 2Parameter Symbol Equation I ValueNumber of teeth Igivenl 11 14
z 28Proportional top land thicknesses (givenl mal 0075
m 0075Center distance inlgiven) 8w 3000Proportional base tooth thicknesses mb1 0755
(chosen from area of existence) mb2 0645Profile angle in the involute intersection point deg VI 14a) 4214
v 3285Profile angles on outside diameters deg as1 (5al 4123
aa2 3183Operating pressure anole dell Clw liD) 2498Transverse contact ratio E 12 160
Profile angles in the bottom contact points deg Clp (13) 887ClD2 (14) 1461
Base diameters in dbl (l1al 1813db2 dlf2 = db1bull U 3626
Base pitch in Pb (3) 0407Operating pitch in Pw (7l 0449Operating pitch diameters in d bullbullbullbullbull (8) 2000
dwz 4000Operating tooth thicknesses in Sbullbullbullbull 19) 0279
Swz 0170Outside diameters in dabull (2a) 2410
da2 4267Outside diameter tooth thicknesses in SSl 15b) 0030
5s2 0030
after the gear de iga i completely ani hedTher were attempt to u th bar circle as a
f undati n for the in lute gear th ry parat-ing the gear analy i from th g ar g n ratingpr e Pr fe r EB ulgako d I peel th
ailed the ry of generalized param teinvolute ge Ref 2 1R C I urn (Ref )d ribed an alt mati e definiti n f th in lutwith ut using the generating ra k Th s lf-g n-crating m thod gear f ems g ar pr sedfor pia tic m Ided gear (Refs 4 and 5)
c rding to thi method the top land f tht th of on of the gears form th fillet f thmating gear and vice er a I a gl n e it I kimilar 10 a gear haping r ge r lJing p
but th fa t that both ge are de ribed with utth generating ra k param te mak a dill r-n e in their go m try and hara teri ti
Involute Tooth Param ten involute tooth i f nn d tw in lute
un ound from the ba e circle db out ide circldiameter da and fillel (Ref 2) ( ig2) nle
therwi rated the foll win equ ti n are r-rect for spur gears and f r heli al gears in thtransver e e lion (the ecti n perpendicular I
th axi of the gear) Equati n numbers withalphabetic modi ier are gi en f r u in lhnumeri example Ii ted in Table 23 and 4
e profile angle in the interse ti n point fthe two involute (tip angle) i
v=aco dJdA (I)where dA i th harp tip irele diam t c
Th profile angle on th ut id diamet r da iaa = a 0 dJda) (2)da = dJ 0 (aa) (2a)
Th base pitch iPb = rt 0 dJz 3)
where z i the number f teethe proportional ba t th thickn
mb = SJpb = z 0 in (v)ht (4)invtv) = rt 0 InJz (4a)
wh r Sb is the bas thi kne The pr portional t p land thi kne s is
rna = Sjpb = Z 0 (invtv) - inv(aa))(lt 0 c et) (5)
co (a) +in (a)(rt 0 rna = mJma (5a)Sa = r 0 rna 5b)
inv(v = (n 0 maco aa + zinv araquo)z 5c)where Sa i the I p land thickne Th recom-mended alue of Ina h uld be I n 006 and01210 avoid a harp t th tip and pro ide u 1-
ient onta t ratio in th hIn olut Param t
Figure 3 ho th L n f I th Ii n f thpinion and the gear in cl me h ( a klash i
Figurlaquo 2-The involute tooth parameter definl-tion
Figurtion
da
zero cl e m h condition i
pbullbull=S bullbullbull+ bullbull2 (6)where
PM = rt 0 d bullbullz = rt 0 d bullbulliz2 7)is perating circular pitch
dbullbull = dblcos(CtbullbullJ dM2 = dbtco a ()are the pinion and the gear perating pitch diame-t and
bullbull = (inv(v ) - inv(a bullbullraquo 0 dbCO middot(a bullbullJ 9)bullbullbull2 = (inv(v2 - inv( ) 0 (dbtc (a bullbull)
are th pini nand th gear peratin 1 th thick-n ses
The perating pressure angle can be und bub tituuon of uati n with 78 and 9
in ( )=(inv(v)+Ioin V2)-ltIz1)1 +u (10wh re u i th gear rati I = 1
Th operating pre UTe angle i a gear m hparameter and it cant be de 1000 f r ne sepa-rate gear
wwwpowrlnmiuQncam bull Wwwgurrchnolagycam bull GEAR TECHNOLOGY SEPTEMBEROCTOBER 2002 31
The center distance isa = dbl deg (l + u)(2 deg cos(awraquo (11)dbl = a deg (2 bull cos(a)(l + u) (11a)
The contact ratio (for spur gears and for helicalgears in the transverse section) is
en = z deg (tan(aa) + u deg tan(aa2)- (l + u) bull tan(a)(2 deg It)
The profile angle in the bottom contact pointmust be larger than or equal to zero to avoid invo-lute undercutfor the pinionap = atanlaquo(l + u) deg tan(a) - U deg tan(aa2raquo ~ 0 (13)for the gearap2 = atanlaquo(l + u) deg tan(a)u - tan(aa)u) ~ O (14)
The axial contact ratio for helical gear isE~= zolgt(2olt) (15)
where lgt (in radians) i the angular hift betweenthe oppo ite transverse section in the helicalmesh (see Fig 4) and
lgt = (2 deg b) deg tan(~b)db (16)where bw i the width of the helical mesh and ~b isthe helix angle on the base circle
The fillet profile must provide a gear mesh withsufficient radial clearance to avoid tip-fillet interfer-ence The fillet also must provide necessary toothbending fatigue resistance and me h tiffnes Thedirect gear design approach allow selection of any
I fillet profile (parabola ellipsis cubic pline etc)that would be t ati fy tho e condition Thi profileis not necessarily the trochoid fanned by the rack orhaper generating proce s
Tool geometry definition is the next step indirect gear design This will depend on the actualmanufacturing method For plastic and metal gearmolding gear extrusion and powder metal gearprocessing the entire gear geometry-includingcorrection for shrinkage-will be directly appliedto the tool cavity For cutting tools (hobs shapercutters) the reverse generating approach gearforms tool can be applied In thi case the tool-ing pitch and profile (pre sure) angle are selectedto provide the best cutting condition
Area of Existence of Involute GearsFigure 5 shows an area of existence for a pinion
and gear with certain numbers of teeth zl ~ andproportional top land thicknes e mal ma2 (Ref 2)Unlike the zone shown in Figure 1 the area of exi -tence in Figure 5 contains all possible gear combina-tions and is not limited to restrictions irnpo ed by agenerating rack This area can be shown in propor-tional base tooth thicknesses mb - mb2 coordinateor other parameters de cribing the angular di tancebetween two involute flanks of the pinion and gear
I teeth like aal - aa2 or v I - v2A sample of the area32 SEPTEMBEROCTOBER 2002 bull GEAR TECHNOLOGY wwwgearechnologycom bull wwwpowerransmissloncom
Figure 4-The angular shift of the transverse sec-tions for a helical gear
IDb2
20
15
10
5
o I5 10
Figure 5-The area of existence for the gear pairzJ = 14 Z2 = 28
a c
b d
Figure 6-lnvolute gear meshes 6a at point A of Figure 5 (aMmax = 395deg Eu =10) 6b at point B of Figure 5 (abullbull= 167deg ecaliu = 201) 6c at point C ofFigure 5 (amnx = 296deg Ea = 10) 6d at point D of Figure 5 (aM = 159deg EamllX
= 164)
(12)
of exi tence f r a pair of gears ZI = 14 = 2 mol
= ma2 = 0075 i shown in Figure 5 e area f exis-tence include a number of isogram reflecting con-tant value of different gear paramete u h
operating pre ure angle CX contact ratios Ea etcThe area of existence of pur gears (thi k lin I) ilimited by isogram Eo= 10 and undercut i gramCX 1 = 00
CX = 00bull HeLi al gears can have a tran -p p-
ve contact ratio Ie than I becau the ialcontact ratio can provide p per me h Th area fexi tence of helical gears i th refore mu h greater
h point on the area of e i len e refl IS a pair fgears with dimen ionle propertie that can fit a par-ti ular appli arion Th propelti are pres ureangle c ntact ratio pitting re istance g merryfactorl pecifi liding ratio et
The absolute area f e i ten e in Iud pur garcombinatio with any alue f proportional t p Iland thickne ber een mal = ma2 = 0 t TIol = mbl
and m 2 = mb (phantom lin 2) Thi area i ub- a _ I
tantially larger than the area ith gi en alue f iproporri nal t p land thi knes The zone f r a tandard generating rn 1ltwith 200 pre sure angle ( hown in ig I) i only a fractional part of the a ail-
able area of exi tence as h wn by hidden lin 3 in iFigure 5 An appli ati n of a traditi nal gear gener-
Iating app a h for gear pairs out ide th z ne ut- ilined by hidden line 3 require Ie Lionof a gen r-ating ra k with differentparam ters The generationof orne gear combinations (t p left and bottom rightcomers of the area of xi t n e hown in Fig 5) willrequire different generating ra ks for the pinion andfor the gear
Analy i of the area of exi tcnce haws how imany gear olutions could be left out of c nsid ra- tion if a traditional approach based n a predet r-mined t of ra k dimension i applied F r e am-ple pur gears with a high operating pressur angl(point A on the Figure 5 where th rating pres-ure angle cx= 95deg conta t ratio Eex= 10) r with
a high contact ratio (point B on the Figure 5 whereonta t ratio Eex = 20 I operating pre ure angl cx
= 1670) could not be produced with tandard ra k
dimen ion Fi ure 6a and 6b h w th geFigures 6c and 6d are the gears that are achie ableusing a tandard g nerating ra k that arc pre nted I tandard designed gear pair h a conta I ratio ofbypoinlSC(a bullbull=296degE
Q= 10)andD(a= 15cr i only a= 116in lose m hlnana tual application
En = 164) in igure 5 E en gears with the same with real manufacturing ~ leran and operatingoperating pre ure angle (point in ig 5) 1 k c nditi ns the ntact ratio of a tandard de ignedquite different (Fig 7) Th tandard d igned gear i gear pair uld be reduced t an unacceptable Ie elpair (Fig 7b) has alrn t sharp-pointed pini n t th Enlt 10and hort and tubby gear teeth with excessi e top ynthe i of Garing umerical xampJland tooth thi kne The direct de igned gear pair Th r arc everal w y to define gear parame-hownin Figure 7a has a contact ratio Ea = 147The i I ruing the direct gear de ign approach Thi
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a
b
Figure 8- helical gear pair with higl operatingpressure angle I = 9 2 = 12 a = 688~ Ea =0504 3 = 30~ EfJ = 055
Figur 9-A Jpur gear with a Itigll contact ratio I= 55 2 = 55 a = J29~ Ea= 40 lIIal = lIIal = 0075
article considers some of themArea of existence is known The initial data for
the synthesi of a pair of gear (zl Z2 01 f1l2)could be taken from the area of exi tence at omeparticular point The coordinate of thi point andcenter di tance abullbull describe all operating gearparameter This calculation procedure andnumerical example are presented in the able 2
Area of existence is 1101 known pical pr b-lern could be finding the maximum pre ureangle if the tran verse contact ratio i ch en orfinding the maximum tran verse contact ratio ifthe pre sure angle i cho en Both of the e ca erequire finding the point of area of exi tencewhere i ogram ltxwand Eo have the arne tangentThis condition is de cribed (Ref 2) as
cos(0)2 (l + 1t bull mal sin(ltXI)Iz)=CO (02)2 (I + 1t ma2bull in(oa2)1z2) (17)
and allows solution of the e problem withoutknowing the area of existence The calculationprocedures and numerical example are pre entedin Table 3 and 4
The fillet between teeth is not involved in gearmesh operation but it shape greatly affect gearperformance and durability In traditional gearde ign the fillet profile is a function of the cutlershape and the machine tool setup It typically haexce ive radial clearance resulting in high bend-ing stresses Direct gear de ign doe n t limit fil-let hape definition One po ibility i to de cribethe fillet profile as a trace of the top part of themating gear tooth (with corre ponding minimumradial clearance) (Ref 4 and 5) Application offinite element analysis allows for forming the fil-let profiles to balance and minimize bendingstresses
Extreme Parameters of Involute GearsPoint A (tangent point of i ogram Eo = 10 and
0bullbull= max) of the area of exi tence de cribe gearwith the maximum achievable operating pre ureangle There is no such limit for helical gearbecause a lack of the transverse contact ratio (Eo lt10) i compen ated by the axial contact ratio Ep
A ample of a helical gear with high operatingpressure angle (Ref 6) i shown in Figure InFigure 5 the point B (inter ection point of inter-feren e i ogram up I = 0deg and op2 = 0deg) of thearea of existence de cribe the gears with themaximum achievable transver e contact ratioTable 5 pre ents maximum value f r operatingpressure angle owA (Point A of the area of exi -abc renee) and tran ver e contact ratio Eo B (Point B of
Figure lo-Spur gears with minimum number of teeth a) ZJ = 5 Z2 = 5 a =33 1deg E 104 b) Z 4 Z 6 a 326deg 10) 3 11 the area of existence) for gear pairs with different oa = J = 2 = = ea = ~ C zJ = Z2 = a= 243 ea = 101 numbers of teeth and the proportional top land34 SEPTEMBEROCTOBER 2002 bull GEAR TECHNOLOGY wwwgesrtschnologycom bull wwwpowerlrsnsmlssioncom
Table 3Parameter Symbol Equation ValueNumber of Teeth (givenl Z 14
z 28Proportional top land thicknesses 19iven) ml 0075
m 0075Center distance in (given) 8w 3000Operating pressure angle deg [chosen] (lw 33Profile angles on outside diameters deg (ll (l0) amp 117) 4002
(lll 3884Transverse contact ratio (maximum 1 famagt 121 1246Profile angle in the involute intersection point deg VI (5cl 4103
v2 4036Proportional base tooth thicknesses mbl (41 0687
mb2 1296Use Table 2 to identify remaining equations
Table 4Parameter Symbol Equation ValueNumber of teeth (givenl Zl 14
lz 28Proportional top land thicknesses (given) ml 0075
m2 0075Center distance in (given) a 3000Transverse contact ratio (chosen) ElY
105Profile angles on outside diameters deg (lal (l0J(l21 4329
(laz amp (17) 4312Operating pressure angle deg [rnaximum] (lwmu 10J 021 amp (17) 3797Profile angle in the involute intersection point deg v (5c) 4406
v2 4352Proportional base tooth thicknesses mb1 (4) 0886
mb2 1693Use Table 2 to identify remaining equations
Table 5
owAIE8I Pinion ZI
5 10 20 30 40 505 315deg10 350deg114 378deg121 39deg123 396deg124 40deg124
~ 10 350deg1 14 368deg143 385deg166 394deg172 399deg175 408deg176bull 20 378deg121 385deg166 394deg1214 399deg237 402deg248 404deg254Ggt(l 30 39deg123 394deg172 399deg237 402deg274 404deg297 406deg31
40 396deg124 399deg175 402deg248 404deg297 406deg328 407deg34950 40deg124 408deg176 404deg254 406deg31 407349 408deg378
thickn s mal 11102 0075 An example of apur gear me h with a high coma I ratio is sh wn
in Figure 9pur gear (contact ratio En 10) with a min-
imum pos ible number of t th (Ref 2) areshown in Figure 10The minimum po ibl num-ber of teeth f r helical gears is n t limit d bytran erse contact ratio and could be as few aone (Ref 6) n e ample of a helical gear with thenumber of t eth - = ~ = I is shown in Figure IIIn elute Gea with mmetric ToothProlile
Oppo ite nan (profile ) of the gear I th arefunctionally different for m t gear Th w rk-I ad on one profile i ignificantl higher and ri applied for longer period of lime than n thoppo ite one The a ymm tric tooth shape a com-modates thi fun tional diff ren e
Th design intent of a ymmeuic teeth i Iimpro e performance of main nta tin p filby degrading oppo ite profile Th ppo ire pro-fil are unload d or lightl I aded and usuallwork for a relatively h rt period The impro edperformance c uld mean in rea ing load capacityor reducing weight noise vibration tc
Degre of asymmetry and drive pr file elec-tion for these gears depend on the applicati n
ymmetric profiles make it possible to managetooth stiffnes and 1 ad haring while k ping ade irable pre ure angle and conta t rati n thedrive profile
Dire t design of gears with asymmetri teeth iscon idered in detail in ther articles (Refs 7 and8) covering topic u h a analy i and synthe isof asymmetric gearing area e i t n e andapplication ample of gea ith asymmetritooth profile are hown in Figure 12 Gears witha yrnrnetric teeth h uld be c n sidcred f r geary terns that require extreme perf rmance like
aero pace drive Th Y are al a applicable f rmas production lransmi ion where the hare ofthe tooling co I per one gear is relatively in ignif-icant The mo t promising application for asym-metric profile i with m Ided gears and wdermetal gea Molded gear tooling u ually requirea custom shape 0 ih as mmetric profile dnot ignificanl1y affect c t
umm ryDirect gear de ign i an alternati e appr ach to
traditional gear design It allow analysi of awide range of parameters for all po ible gearcombination in order to find th m st uitablesolution for a parti ular appli arion This opti-mum gear oluli n can exceed the limits of tradi-tional rack generating methods of gear design
Figure ll-Helical gears with one tooth zL I Z2= I abull= 688~t = 056 b = 349~EIJ012
Figure 12- pur gear with a ymmetric teeth 12u)a generator gear drive with a = 41 ~ Ea = 12 fordrive flanks ~lfIda = 18~ Eq= l64for coastflanks12b) a pia tICgear pump WIt a = 457~E 101for drive flanks and a JO3~Ea = I09ltr coastflanks
Dire t gear d sign for asymmetric tooth profilespen additi nal re erve f r improvement f
gear dri es with unidirecti nal load cycles thatarc typical for many mechanical transrnissi n 0
cknowl dgm ntsThe authors expre deep gratitude 10 Gear Technologytechnical editors Robert Errichello of Geartech 1 ated inTownsend MT and Dan Thurman for their help in prepar-ing thi article
ReferencesJ Gr man MB The Zone f Involute Me h Ve tnikfa hiw troeniva 1962 [ ue 12 pp 12-17 (in Ru ian)
2 ulgakov EB Theory of Involute GearsMa hino troenie Mo 0 1995 (in Ru ian)
olbourne lR The Ceometrv of Involute Gearspringer- erlag ew York 1987
4 IAGM 1006-A97 Tooth Proportion for Pia ticG ppendix F -o nerating Gear Geometr WilhoutRa I GMA Ale andria A 19975 Kupelevich L 1 Klei s and RE Kleis ewOpportunities with Molded Gear AG Fall Techni alMecling Detroit October -52001 (01 FTM9)6 Kapelevi h AL and pound8 ulgakov Expanding therange f involute helical gearing Yestnl]MasIJlllomiddottroeniya 19 2 I sue 3 pp 12-14 (in Russian)
ranslated [0 nglish 0 iet Engin ering Research Vol 2I lIC 3 1982 pp 8-97 Kapelevi h AL Gc rnetry and design of in oluiepur gear with a ymmctric teeth Mechanism and
Machine Theory 2000 I sue 35 pp 117-1308 Kapclevich AL Q Lian and FL uvmA ymmetric rnodif d g ar drive reduct jon or noi elocalizati n or contact imulauon of meshing and stresanalysis Computer Mehods in Applied Mechanics andEngineering 2000 Issue 18 pp 63-390
For more informationabout AKGearsor tocontact the companyvisit its website twwwkgll bullbullcOtfl
For more infonutionabout KIi Glrs orto contact the compa-ny visit its website Itwwwklibullbullbullbullbull lIcom
Ten Us WhitYouThink VisitwwwpIfffCIMoIfyctobull RIte this artielebull Request mora
infonnltiOlbull ContlCt the lutltOrs
or cOlRpani n-tioned
bull Make I IIIIglltionOr cIII (8471437-6amp04 totalk to one of our di-tors
www poweransmtssioncom bull www geartechnologycom bull GEAR TECHNOLOGYmiddot SEPTEMBEROCTOBER 2002 35
after the gear de iga i completely ani hedTher were attempt to u th bar circle as a
f undati n for the in lute gear th ry parat-ing the gear analy i from th g ar g n ratingpr e Pr fe r EB ulgako d I peel th
ailed the ry of generalized param teinvolute ge Ref 2 1R C I urn (Ref )d ribed an alt mati e definiti n f th in lutwith ut using the generating ra k Th s lf-g n-crating m thod gear f ems g ar pr sedfor pia tic m Ided gear (Refs 4 and 5)
c rding to thi method the top land f tht th of on of the gears form th fillet f thmating gear and vice er a I a gl n e it I kimilar 10 a gear haping r ge r lJing p
but th fa t that both ge are de ribed with utth generating ra k param te mak a dill r-n e in their go m try and hara teri ti
Involute Tooth Param ten involute tooth i f nn d tw in lute
un ound from the ba e circle db out ide circldiameter da and fillel (Ref 2) ( ig2) nle
therwi rated the foll win equ ti n are r-rect for spur gears and f r heli al gears in thtransver e e lion (the ecti n perpendicular I
th axi of the gear) Equati n numbers withalphabetic modi ier are gi en f r u in lhnumeri example Ii ted in Table 23 and 4
e profile angle in the interse ti n point fthe two involute (tip angle) i
v=aco dJdA (I)where dA i th harp tip irele diam t c
Th profile angle on th ut id diamet r da iaa = a 0 dJda) (2)da = dJ 0 (aa) (2a)
Th base pitch iPb = rt 0 dJz 3)
where z i the number f teethe proportional ba t th thickn
mb = SJpb = z 0 in (v)ht (4)invtv) = rt 0 InJz (4a)
wh r Sb is the bas thi kne The pr portional t p land thi kne s is
rna = Sjpb = Z 0 (invtv) - inv(aa))(lt 0 c et) (5)
co (a) +in (a)(rt 0 rna = mJma (5a)Sa = r 0 rna 5b)
inv(v = (n 0 maco aa + zinv araquo)z 5c)where Sa i the I p land thickne Th recom-mended alue of Ina h uld be I n 006 and01210 avoid a harp t th tip and pro ide u 1-
ient onta t ratio in th hIn olut Param t
Figure 3 ho th L n f I th Ii n f thpinion and the gear in cl me h ( a klash i
Figurlaquo 2-The involute tooth parameter definl-tion
Figurtion
da
zero cl e m h condition i
pbullbull=S bullbullbull+ bullbull2 (6)where
PM = rt 0 d bullbullz = rt 0 d bullbulliz2 7)is perating circular pitch
dbullbull = dblcos(CtbullbullJ dM2 = dbtco a ()are the pinion and the gear perating pitch diame-t and
bullbull = (inv(v ) - inv(a bullbullraquo 0 dbCO middot(a bullbullJ 9)bullbullbull2 = (inv(v2 - inv( ) 0 (dbtc (a bullbull)
are th pini nand th gear peratin 1 th thick-n ses
The perating pressure angle can be und bub tituuon of uati n with 78 and 9
in ( )=(inv(v)+Ioin V2)-ltIz1)1 +u (10wh re u i th gear rati I = 1
Th operating pre UTe angle i a gear m hparameter and it cant be de 1000 f r ne sepa-rate gear
wwwpowrlnmiuQncam bull Wwwgurrchnolagycam bull GEAR TECHNOLOGY SEPTEMBEROCTOBER 2002 31
The center distance isa = dbl deg (l + u)(2 deg cos(awraquo (11)dbl = a deg (2 bull cos(a)(l + u) (11a)
The contact ratio (for spur gears and for helicalgears in the transverse section) is
en = z deg (tan(aa) + u deg tan(aa2)- (l + u) bull tan(a)(2 deg It)
The profile angle in the bottom contact pointmust be larger than or equal to zero to avoid invo-lute undercutfor the pinionap = atanlaquo(l + u) deg tan(a) - U deg tan(aa2raquo ~ 0 (13)for the gearap2 = atanlaquo(l + u) deg tan(a)u - tan(aa)u) ~ O (14)
The axial contact ratio for helical gear isE~= zolgt(2olt) (15)
where lgt (in radians) i the angular hift betweenthe oppo ite transverse section in the helicalmesh (see Fig 4) and
lgt = (2 deg b) deg tan(~b)db (16)where bw i the width of the helical mesh and ~b isthe helix angle on the base circle
The fillet profile must provide a gear mesh withsufficient radial clearance to avoid tip-fillet interfer-ence The fillet also must provide necessary toothbending fatigue resistance and me h tiffnes Thedirect gear design approach allow selection of any
I fillet profile (parabola ellipsis cubic pline etc)that would be t ati fy tho e condition Thi profileis not necessarily the trochoid fanned by the rack orhaper generating proce s
Tool geometry definition is the next step indirect gear design This will depend on the actualmanufacturing method For plastic and metal gearmolding gear extrusion and powder metal gearprocessing the entire gear geometry-includingcorrection for shrinkage-will be directly appliedto the tool cavity For cutting tools (hobs shapercutters) the reverse generating approach gearforms tool can be applied In thi case the tool-ing pitch and profile (pre sure) angle are selectedto provide the best cutting condition
Area of Existence of Involute GearsFigure 5 shows an area of existence for a pinion
and gear with certain numbers of teeth zl ~ andproportional top land thicknes e mal ma2 (Ref 2)Unlike the zone shown in Figure 1 the area of exi -tence in Figure 5 contains all possible gear combina-tions and is not limited to restrictions irnpo ed by agenerating rack This area can be shown in propor-tional base tooth thicknesses mb - mb2 coordinateor other parameters de cribing the angular di tancebetween two involute flanks of the pinion and gear
I teeth like aal - aa2 or v I - v2A sample of the area32 SEPTEMBEROCTOBER 2002 bull GEAR TECHNOLOGY wwwgearechnologycom bull wwwpowerransmissloncom
Figure 4-The angular shift of the transverse sec-tions for a helical gear
IDb2
20
15
10
5
o I5 10
Figure 5-The area of existence for the gear pairzJ = 14 Z2 = 28
a c
b d
Figure 6-lnvolute gear meshes 6a at point A of Figure 5 (aMmax = 395deg Eu =10) 6b at point B of Figure 5 (abullbull= 167deg ecaliu = 201) 6c at point C ofFigure 5 (amnx = 296deg Ea = 10) 6d at point D of Figure 5 (aM = 159deg EamllX
= 164)
(12)
of exi tence f r a pair of gears ZI = 14 = 2 mol
= ma2 = 0075 i shown in Figure 5 e area f exis-tence include a number of isogram reflecting con-tant value of different gear paramete u h
operating pre ure angle CX contact ratios Ea etcThe area of existence of pur gears (thi k lin I) ilimited by isogram Eo= 10 and undercut i gramCX 1 = 00
CX = 00bull HeLi al gears can have a tran -p p-
ve contact ratio Ie than I becau the ialcontact ratio can provide p per me h Th area fexi tence of helical gears i th refore mu h greater
h point on the area of e i len e refl IS a pair fgears with dimen ionle propertie that can fit a par-ti ular appli arion Th propelti are pres ureangle c ntact ratio pitting re istance g merryfactorl pecifi liding ratio et
The absolute area f e i ten e in Iud pur garcombinatio with any alue f proportional t p Iland thickne ber een mal = ma2 = 0 t TIol = mbl
and m 2 = mb (phantom lin 2) Thi area i ub- a _ I
tantially larger than the area ith gi en alue f iproporri nal t p land thi knes The zone f r a tandard generating rn 1ltwith 200 pre sure angle ( hown in ig I) i only a fractional part of the a ail-
able area of exi tence as h wn by hidden lin 3 in iFigure 5 An appli ati n of a traditi nal gear gener-
Iating app a h for gear pairs out ide th z ne ut- ilined by hidden line 3 require Ie Lionof a gen r-ating ra k with differentparam ters The generationof orne gear combinations (t p left and bottom rightcomers of the area of xi t n e hown in Fig 5) willrequire different generating ra ks for the pinion andfor the gear
Analy i of the area of exi tcnce haws how imany gear olutions could be left out of c nsid ra- tion if a traditional approach based n a predet r-mined t of ra k dimension i applied F r e am-ple pur gears with a high operating pressur angl(point A on the Figure 5 where th rating pres-ure angle cx= 95deg conta t ratio Eex= 10) r with
a high contact ratio (point B on the Figure 5 whereonta t ratio Eex = 20 I operating pre ure angl cx
= 1670) could not be produced with tandard ra k
dimen ion Fi ure 6a and 6b h w th geFigures 6c and 6d are the gears that are achie ableusing a tandard g nerating ra k that arc pre nted I tandard designed gear pair h a conta I ratio ofbypoinlSC(a bullbull=296degE
Q= 10)andD(a= 15cr i only a= 116in lose m hlnana tual application
En = 164) in igure 5 E en gears with the same with real manufacturing ~ leran and operatingoperating pre ure angle (point in ig 5) 1 k c nditi ns the ntact ratio of a tandard de ignedquite different (Fig 7) Th tandard d igned gear i gear pair uld be reduced t an unacceptable Ie elpair (Fig 7b) has alrn t sharp-pointed pini n t th Enlt 10and hort and tubby gear teeth with excessi e top ynthe i of Garing umerical xampJland tooth thi kne The direct de igned gear pair Th r arc everal w y to define gear parame-hownin Figure 7a has a contact ratio Ea = 147The i I ruing the direct gear de ign approach Thi
poeensmlssloncom bull g6llechnologycom bull GEAR TECHNOLOGY SEPTEMBEROCTOBER 2002 33
a
b
Figure 8- helical gear pair with higl operatingpressure angle I = 9 2 = 12 a = 688~ Ea =0504 3 = 30~ EfJ = 055
Figur 9-A Jpur gear with a Itigll contact ratio I= 55 2 = 55 a = J29~ Ea= 40 lIIal = lIIal = 0075
article considers some of themArea of existence is known The initial data for
the synthesi of a pair of gear (zl Z2 01 f1l2)could be taken from the area of exi tence at omeparticular point The coordinate of thi point andcenter di tance abullbull describe all operating gearparameter This calculation procedure andnumerical example are presented in the able 2
Area of existence is 1101 known pical pr b-lern could be finding the maximum pre ureangle if the tran verse contact ratio i ch en orfinding the maximum tran verse contact ratio ifthe pre sure angle i cho en Both of the e ca erequire finding the point of area of exi tencewhere i ogram ltxwand Eo have the arne tangentThis condition is de cribed (Ref 2) as
cos(0)2 (l + 1t bull mal sin(ltXI)Iz)=CO (02)2 (I + 1t ma2bull in(oa2)1z2) (17)
and allows solution of the e problem withoutknowing the area of existence The calculationprocedures and numerical example are pre entedin Table 3 and 4
The fillet between teeth is not involved in gearmesh operation but it shape greatly affect gearperformance and durability In traditional gearde ign the fillet profile is a function of the cutlershape and the machine tool setup It typically haexce ive radial clearance resulting in high bend-ing stresses Direct gear de ign doe n t limit fil-let hape definition One po ibility i to de cribethe fillet profile as a trace of the top part of themating gear tooth (with corre ponding minimumradial clearance) (Ref 4 and 5) Application offinite element analysis allows for forming the fil-let profiles to balance and minimize bendingstresses
Extreme Parameters of Involute GearsPoint A (tangent point of i ogram Eo = 10 and
0bullbull= max) of the area of exi tence de cribe gearwith the maximum achievable operating pre ureangle There is no such limit for helical gearbecause a lack of the transverse contact ratio (Eo lt10) i compen ated by the axial contact ratio Ep
A ample of a helical gear with high operatingpressure angle (Ref 6) i shown in Figure InFigure 5 the point B (inter ection point of inter-feren e i ogram up I = 0deg and op2 = 0deg) of thearea of existence de cribe the gears with themaximum achievable transver e contact ratioTable 5 pre ents maximum value f r operatingpressure angle owA (Point A of the area of exi -abc renee) and tran ver e contact ratio Eo B (Point B of
Figure lo-Spur gears with minimum number of teeth a) ZJ = 5 Z2 = 5 a =33 1deg E 104 b) Z 4 Z 6 a 326deg 10) 3 11 the area of existence) for gear pairs with different oa = J = 2 = = ea = ~ C zJ = Z2 = a= 243 ea = 101 numbers of teeth and the proportional top land34 SEPTEMBEROCTOBER 2002 bull GEAR TECHNOLOGY wwwgesrtschnologycom bull wwwpowerlrsnsmlssioncom
Table 3Parameter Symbol Equation ValueNumber of Teeth (givenl Z 14
z 28Proportional top land thicknesses 19iven) ml 0075
m 0075Center distance in (given) 8w 3000Operating pressure angle deg [chosen] (lw 33Profile angles on outside diameters deg (ll (l0) amp 117) 4002
(lll 3884Transverse contact ratio (maximum 1 famagt 121 1246Profile angle in the involute intersection point deg VI (5cl 4103
v2 4036Proportional base tooth thicknesses mbl (41 0687
mb2 1296Use Table 2 to identify remaining equations
Table 4Parameter Symbol Equation ValueNumber of teeth (givenl Zl 14
lz 28Proportional top land thicknesses (given) ml 0075
m2 0075Center distance in (given) a 3000Transverse contact ratio (chosen) ElY
105Profile angles on outside diameters deg (lal (l0J(l21 4329
(laz amp (17) 4312Operating pressure angle deg [rnaximum] (lwmu 10J 021 amp (17) 3797Profile angle in the involute intersection point deg v (5c) 4406
v2 4352Proportional base tooth thicknesses mb1 (4) 0886
mb2 1693Use Table 2 to identify remaining equations
Table 5
owAIE8I Pinion ZI
5 10 20 30 40 505 315deg10 350deg114 378deg121 39deg123 396deg124 40deg124
~ 10 350deg1 14 368deg143 385deg166 394deg172 399deg175 408deg176bull 20 378deg121 385deg166 394deg1214 399deg237 402deg248 404deg254Ggt(l 30 39deg123 394deg172 399deg237 402deg274 404deg297 406deg31
40 396deg124 399deg175 402deg248 404deg297 406deg328 407deg34950 40deg124 408deg176 404deg254 406deg31 407349 408deg378
thickn s mal 11102 0075 An example of apur gear me h with a high coma I ratio is sh wn
in Figure 9pur gear (contact ratio En 10) with a min-
imum pos ible number of t th (Ref 2) areshown in Figure 10The minimum po ibl num-ber of teeth f r helical gears is n t limit d bytran erse contact ratio and could be as few aone (Ref 6) n e ample of a helical gear with thenumber of t eth - = ~ = I is shown in Figure IIIn elute Gea with mmetric ToothProlile
Oppo ite nan (profile ) of the gear I th arefunctionally different for m t gear Th w rk-I ad on one profile i ignificantl higher and ri applied for longer period of lime than n thoppo ite one The a ymm tric tooth shape a com-modates thi fun tional diff ren e
Th design intent of a ymmeuic teeth i Iimpro e performance of main nta tin p filby degrading oppo ite profile Th ppo ire pro-fil are unload d or lightl I aded and usuallwork for a relatively h rt period The impro edperformance c uld mean in rea ing load capacityor reducing weight noise vibration tc
Degre of asymmetry and drive pr file elec-tion for these gears depend on the applicati n
ymmetric profiles make it possible to managetooth stiffnes and 1 ad haring while k ping ade irable pre ure angle and conta t rati n thedrive profile
Dire t design of gears with asymmetri teeth iscon idered in detail in ther articles (Refs 7 and8) covering topic u h a analy i and synthe isof asymmetric gearing area e i t n e andapplication ample of gea ith asymmetritooth profile are hown in Figure 12 Gears witha yrnrnetric teeth h uld be c n sidcred f r geary terns that require extreme perf rmance like
aero pace drive Th Y are al a applicable f rmas production lransmi ion where the hare ofthe tooling co I per one gear is relatively in ignif-icant The mo t promising application for asym-metric profile i with m Ided gears and wdermetal gea Molded gear tooling u ually requirea custom shape 0 ih as mmetric profile dnot ignificanl1y affect c t
umm ryDirect gear de ign i an alternati e appr ach to
traditional gear design It allow analysi of awide range of parameters for all po ible gearcombination in order to find th m st uitablesolution for a parti ular appli arion This opti-mum gear oluli n can exceed the limits of tradi-tional rack generating methods of gear design
Figure ll-Helical gears with one tooth zL I Z2= I abull= 688~t = 056 b = 349~EIJ012
Figure 12- pur gear with a ymmetric teeth 12u)a generator gear drive with a = 41 ~ Ea = 12 fordrive flanks ~lfIda = 18~ Eq= l64for coastflanks12b) a pia tICgear pump WIt a = 457~E 101for drive flanks and a JO3~Ea = I09ltr coastflanks
Dire t gear d sign for asymmetric tooth profilespen additi nal re erve f r improvement f
gear dri es with unidirecti nal load cycles thatarc typical for many mechanical transrnissi n 0
cknowl dgm ntsThe authors expre deep gratitude 10 Gear Technologytechnical editors Robert Errichello of Geartech 1 ated inTownsend MT and Dan Thurman for their help in prepar-ing thi article
ReferencesJ Gr man MB The Zone f Involute Me h Ve tnikfa hiw troeniva 1962 [ ue 12 pp 12-17 (in Ru ian)
2 ulgakov EB Theory of Involute GearsMa hino troenie Mo 0 1995 (in Ru ian)
olbourne lR The Ceometrv of Involute Gearspringer- erlag ew York 1987
4 IAGM 1006-A97 Tooth Proportion for Pia ticG ppendix F -o nerating Gear Geometr WilhoutRa I GMA Ale andria A 19975 Kupelevich L 1 Klei s and RE Kleis ewOpportunities with Molded Gear AG Fall Techni alMecling Detroit October -52001 (01 FTM9)6 Kapelevi h AL and pound8 ulgakov Expanding therange f involute helical gearing Yestnl]MasIJlllomiddottroeniya 19 2 I sue 3 pp 12-14 (in Russian)
ranslated [0 nglish 0 iet Engin ering Research Vol 2I lIC 3 1982 pp 8-97 Kapelevi h AL Gc rnetry and design of in oluiepur gear with a ymmctric teeth Mechanism and
Machine Theory 2000 I sue 35 pp 117-1308 Kapclevich AL Q Lian and FL uvmA ymmetric rnodif d g ar drive reduct jon or noi elocalizati n or contact imulauon of meshing and stresanalysis Computer Mehods in Applied Mechanics andEngineering 2000 Issue 18 pp 63-390
For more informationabout AKGearsor tocontact the companyvisit its website twwwkgll bullbullcOtfl
For more infonutionabout KIi Glrs orto contact the compa-ny visit its website Itwwwklibullbullbullbullbull lIcom
Ten Us WhitYouThink VisitwwwpIfffCIMoIfyctobull RIte this artielebull Request mora
infonnltiOlbull ContlCt the lutltOrs
or cOlRpani n-tioned
bull Make I IIIIglltionOr cIII (8471437-6amp04 totalk to one of our di-tors
www poweransmtssioncom bull www geartechnologycom bull GEAR TECHNOLOGYmiddot SEPTEMBEROCTOBER 2002 35
The center distance isa = dbl deg (l + u)(2 deg cos(awraquo (11)dbl = a deg (2 bull cos(a)(l + u) (11a)
The contact ratio (for spur gears and for helicalgears in the transverse section) is
en = z deg (tan(aa) + u deg tan(aa2)- (l + u) bull tan(a)(2 deg It)
The profile angle in the bottom contact pointmust be larger than or equal to zero to avoid invo-lute undercutfor the pinionap = atanlaquo(l + u) deg tan(a) - U deg tan(aa2raquo ~ 0 (13)for the gearap2 = atanlaquo(l + u) deg tan(a)u - tan(aa)u) ~ O (14)
The axial contact ratio for helical gear isE~= zolgt(2olt) (15)
where lgt (in radians) i the angular hift betweenthe oppo ite transverse section in the helicalmesh (see Fig 4) and
lgt = (2 deg b) deg tan(~b)db (16)where bw i the width of the helical mesh and ~b isthe helix angle on the base circle
The fillet profile must provide a gear mesh withsufficient radial clearance to avoid tip-fillet interfer-ence The fillet also must provide necessary toothbending fatigue resistance and me h tiffnes Thedirect gear design approach allow selection of any
I fillet profile (parabola ellipsis cubic pline etc)that would be t ati fy tho e condition Thi profileis not necessarily the trochoid fanned by the rack orhaper generating proce s
Tool geometry definition is the next step indirect gear design This will depend on the actualmanufacturing method For plastic and metal gearmolding gear extrusion and powder metal gearprocessing the entire gear geometry-includingcorrection for shrinkage-will be directly appliedto the tool cavity For cutting tools (hobs shapercutters) the reverse generating approach gearforms tool can be applied In thi case the tool-ing pitch and profile (pre sure) angle are selectedto provide the best cutting condition
Area of Existence of Involute GearsFigure 5 shows an area of existence for a pinion
and gear with certain numbers of teeth zl ~ andproportional top land thicknes e mal ma2 (Ref 2)Unlike the zone shown in Figure 1 the area of exi -tence in Figure 5 contains all possible gear combina-tions and is not limited to restrictions irnpo ed by agenerating rack This area can be shown in propor-tional base tooth thicknesses mb - mb2 coordinateor other parameters de cribing the angular di tancebetween two involute flanks of the pinion and gear
I teeth like aal - aa2 or v I - v2A sample of the area32 SEPTEMBEROCTOBER 2002 bull GEAR TECHNOLOGY wwwgearechnologycom bull wwwpowerransmissloncom
Figure 4-The angular shift of the transverse sec-tions for a helical gear
IDb2
20
15
10
5
o I5 10
Figure 5-The area of existence for the gear pairzJ = 14 Z2 = 28
a c
b d
Figure 6-lnvolute gear meshes 6a at point A of Figure 5 (aMmax = 395deg Eu =10) 6b at point B of Figure 5 (abullbull= 167deg ecaliu = 201) 6c at point C ofFigure 5 (amnx = 296deg Ea = 10) 6d at point D of Figure 5 (aM = 159deg EamllX
= 164)
(12)
of exi tence f r a pair of gears ZI = 14 = 2 mol
= ma2 = 0075 i shown in Figure 5 e area f exis-tence include a number of isogram reflecting con-tant value of different gear paramete u h
operating pre ure angle CX contact ratios Ea etcThe area of existence of pur gears (thi k lin I) ilimited by isogram Eo= 10 and undercut i gramCX 1 = 00
CX = 00bull HeLi al gears can have a tran -p p-
ve contact ratio Ie than I becau the ialcontact ratio can provide p per me h Th area fexi tence of helical gears i th refore mu h greater
h point on the area of e i len e refl IS a pair fgears with dimen ionle propertie that can fit a par-ti ular appli arion Th propelti are pres ureangle c ntact ratio pitting re istance g merryfactorl pecifi liding ratio et
The absolute area f e i ten e in Iud pur garcombinatio with any alue f proportional t p Iland thickne ber een mal = ma2 = 0 t TIol = mbl
and m 2 = mb (phantom lin 2) Thi area i ub- a _ I
tantially larger than the area ith gi en alue f iproporri nal t p land thi knes The zone f r a tandard generating rn 1ltwith 200 pre sure angle ( hown in ig I) i only a fractional part of the a ail-
able area of exi tence as h wn by hidden lin 3 in iFigure 5 An appli ati n of a traditi nal gear gener-
Iating app a h for gear pairs out ide th z ne ut- ilined by hidden line 3 require Ie Lionof a gen r-ating ra k with differentparam ters The generationof orne gear combinations (t p left and bottom rightcomers of the area of xi t n e hown in Fig 5) willrequire different generating ra ks for the pinion andfor the gear
Analy i of the area of exi tcnce haws how imany gear olutions could be left out of c nsid ra- tion if a traditional approach based n a predet r-mined t of ra k dimension i applied F r e am-ple pur gears with a high operating pressur angl(point A on the Figure 5 where th rating pres-ure angle cx= 95deg conta t ratio Eex= 10) r with
a high contact ratio (point B on the Figure 5 whereonta t ratio Eex = 20 I operating pre ure angl cx
= 1670) could not be produced with tandard ra k
dimen ion Fi ure 6a and 6b h w th geFigures 6c and 6d are the gears that are achie ableusing a tandard g nerating ra k that arc pre nted I tandard designed gear pair h a conta I ratio ofbypoinlSC(a bullbull=296degE
Q= 10)andD(a= 15cr i only a= 116in lose m hlnana tual application
En = 164) in igure 5 E en gears with the same with real manufacturing ~ leran and operatingoperating pre ure angle (point in ig 5) 1 k c nditi ns the ntact ratio of a tandard de ignedquite different (Fig 7) Th tandard d igned gear i gear pair uld be reduced t an unacceptable Ie elpair (Fig 7b) has alrn t sharp-pointed pini n t th Enlt 10and hort and tubby gear teeth with excessi e top ynthe i of Garing umerical xampJland tooth thi kne The direct de igned gear pair Th r arc everal w y to define gear parame-hownin Figure 7a has a contact ratio Ea = 147The i I ruing the direct gear de ign approach Thi
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a
b
Figure 8- helical gear pair with higl operatingpressure angle I = 9 2 = 12 a = 688~ Ea =0504 3 = 30~ EfJ = 055
Figur 9-A Jpur gear with a Itigll contact ratio I= 55 2 = 55 a = J29~ Ea= 40 lIIal = lIIal = 0075
article considers some of themArea of existence is known The initial data for
the synthesi of a pair of gear (zl Z2 01 f1l2)could be taken from the area of exi tence at omeparticular point The coordinate of thi point andcenter di tance abullbull describe all operating gearparameter This calculation procedure andnumerical example are presented in the able 2
Area of existence is 1101 known pical pr b-lern could be finding the maximum pre ureangle if the tran verse contact ratio i ch en orfinding the maximum tran verse contact ratio ifthe pre sure angle i cho en Both of the e ca erequire finding the point of area of exi tencewhere i ogram ltxwand Eo have the arne tangentThis condition is de cribed (Ref 2) as
cos(0)2 (l + 1t bull mal sin(ltXI)Iz)=CO (02)2 (I + 1t ma2bull in(oa2)1z2) (17)
and allows solution of the e problem withoutknowing the area of existence The calculationprocedures and numerical example are pre entedin Table 3 and 4
The fillet between teeth is not involved in gearmesh operation but it shape greatly affect gearperformance and durability In traditional gearde ign the fillet profile is a function of the cutlershape and the machine tool setup It typically haexce ive radial clearance resulting in high bend-ing stresses Direct gear de ign doe n t limit fil-let hape definition One po ibility i to de cribethe fillet profile as a trace of the top part of themating gear tooth (with corre ponding minimumradial clearance) (Ref 4 and 5) Application offinite element analysis allows for forming the fil-let profiles to balance and minimize bendingstresses
Extreme Parameters of Involute GearsPoint A (tangent point of i ogram Eo = 10 and
0bullbull= max) of the area of exi tence de cribe gearwith the maximum achievable operating pre ureangle There is no such limit for helical gearbecause a lack of the transverse contact ratio (Eo lt10) i compen ated by the axial contact ratio Ep
A ample of a helical gear with high operatingpressure angle (Ref 6) i shown in Figure InFigure 5 the point B (inter ection point of inter-feren e i ogram up I = 0deg and op2 = 0deg) of thearea of existence de cribe the gears with themaximum achievable transver e contact ratioTable 5 pre ents maximum value f r operatingpressure angle owA (Point A of the area of exi -abc renee) and tran ver e contact ratio Eo B (Point B of
Figure lo-Spur gears with minimum number of teeth a) ZJ = 5 Z2 = 5 a =33 1deg E 104 b) Z 4 Z 6 a 326deg 10) 3 11 the area of existence) for gear pairs with different oa = J = 2 = = ea = ~ C zJ = Z2 = a= 243 ea = 101 numbers of teeth and the proportional top land34 SEPTEMBEROCTOBER 2002 bull GEAR TECHNOLOGY wwwgesrtschnologycom bull wwwpowerlrsnsmlssioncom
Table 3Parameter Symbol Equation ValueNumber of Teeth (givenl Z 14
z 28Proportional top land thicknesses 19iven) ml 0075
m 0075Center distance in (given) 8w 3000Operating pressure angle deg [chosen] (lw 33Profile angles on outside diameters deg (ll (l0) amp 117) 4002
(lll 3884Transverse contact ratio (maximum 1 famagt 121 1246Profile angle in the involute intersection point deg VI (5cl 4103
v2 4036Proportional base tooth thicknesses mbl (41 0687
mb2 1296Use Table 2 to identify remaining equations
Table 4Parameter Symbol Equation ValueNumber of teeth (givenl Zl 14
lz 28Proportional top land thicknesses (given) ml 0075
m2 0075Center distance in (given) a 3000Transverse contact ratio (chosen) ElY
105Profile angles on outside diameters deg (lal (l0J(l21 4329
(laz amp (17) 4312Operating pressure angle deg [rnaximum] (lwmu 10J 021 amp (17) 3797Profile angle in the involute intersection point deg v (5c) 4406
v2 4352Proportional base tooth thicknesses mb1 (4) 0886
mb2 1693Use Table 2 to identify remaining equations
Table 5
owAIE8I Pinion ZI
5 10 20 30 40 505 315deg10 350deg114 378deg121 39deg123 396deg124 40deg124
~ 10 350deg1 14 368deg143 385deg166 394deg172 399deg175 408deg176bull 20 378deg121 385deg166 394deg1214 399deg237 402deg248 404deg254Ggt(l 30 39deg123 394deg172 399deg237 402deg274 404deg297 406deg31
40 396deg124 399deg175 402deg248 404deg297 406deg328 407deg34950 40deg124 408deg176 404deg254 406deg31 407349 408deg378
thickn s mal 11102 0075 An example of apur gear me h with a high coma I ratio is sh wn
in Figure 9pur gear (contact ratio En 10) with a min-
imum pos ible number of t th (Ref 2) areshown in Figure 10The minimum po ibl num-ber of teeth f r helical gears is n t limit d bytran erse contact ratio and could be as few aone (Ref 6) n e ample of a helical gear with thenumber of t eth - = ~ = I is shown in Figure IIIn elute Gea with mmetric ToothProlile
Oppo ite nan (profile ) of the gear I th arefunctionally different for m t gear Th w rk-I ad on one profile i ignificantl higher and ri applied for longer period of lime than n thoppo ite one The a ymm tric tooth shape a com-modates thi fun tional diff ren e
Th design intent of a ymmeuic teeth i Iimpro e performance of main nta tin p filby degrading oppo ite profile Th ppo ire pro-fil are unload d or lightl I aded and usuallwork for a relatively h rt period The impro edperformance c uld mean in rea ing load capacityor reducing weight noise vibration tc
Degre of asymmetry and drive pr file elec-tion for these gears depend on the applicati n
ymmetric profiles make it possible to managetooth stiffnes and 1 ad haring while k ping ade irable pre ure angle and conta t rati n thedrive profile
Dire t design of gears with asymmetri teeth iscon idered in detail in ther articles (Refs 7 and8) covering topic u h a analy i and synthe isof asymmetric gearing area e i t n e andapplication ample of gea ith asymmetritooth profile are hown in Figure 12 Gears witha yrnrnetric teeth h uld be c n sidcred f r geary terns that require extreme perf rmance like
aero pace drive Th Y are al a applicable f rmas production lransmi ion where the hare ofthe tooling co I per one gear is relatively in ignif-icant The mo t promising application for asym-metric profile i with m Ided gears and wdermetal gea Molded gear tooling u ually requirea custom shape 0 ih as mmetric profile dnot ignificanl1y affect c t
umm ryDirect gear de ign i an alternati e appr ach to
traditional gear design It allow analysi of awide range of parameters for all po ible gearcombination in order to find th m st uitablesolution for a parti ular appli arion This opti-mum gear oluli n can exceed the limits of tradi-tional rack generating methods of gear design
Figure ll-Helical gears with one tooth zL I Z2= I abull= 688~t = 056 b = 349~EIJ012
Figure 12- pur gear with a ymmetric teeth 12u)a generator gear drive with a = 41 ~ Ea = 12 fordrive flanks ~lfIda = 18~ Eq= l64for coastflanks12b) a pia tICgear pump WIt a = 457~E 101for drive flanks and a JO3~Ea = I09ltr coastflanks
Dire t gear d sign for asymmetric tooth profilespen additi nal re erve f r improvement f
gear dri es with unidirecti nal load cycles thatarc typical for many mechanical transrnissi n 0
cknowl dgm ntsThe authors expre deep gratitude 10 Gear Technologytechnical editors Robert Errichello of Geartech 1 ated inTownsend MT and Dan Thurman for their help in prepar-ing thi article
ReferencesJ Gr man MB The Zone f Involute Me h Ve tnikfa hiw troeniva 1962 [ ue 12 pp 12-17 (in Ru ian)
2 ulgakov EB Theory of Involute GearsMa hino troenie Mo 0 1995 (in Ru ian)
olbourne lR The Ceometrv of Involute Gearspringer- erlag ew York 1987
4 IAGM 1006-A97 Tooth Proportion for Pia ticG ppendix F -o nerating Gear Geometr WilhoutRa I GMA Ale andria A 19975 Kupelevich L 1 Klei s and RE Kleis ewOpportunities with Molded Gear AG Fall Techni alMecling Detroit October -52001 (01 FTM9)6 Kapelevi h AL and pound8 ulgakov Expanding therange f involute helical gearing Yestnl]MasIJlllomiddottroeniya 19 2 I sue 3 pp 12-14 (in Russian)
ranslated [0 nglish 0 iet Engin ering Research Vol 2I lIC 3 1982 pp 8-97 Kapelevi h AL Gc rnetry and design of in oluiepur gear with a ymmctric teeth Mechanism and
Machine Theory 2000 I sue 35 pp 117-1308 Kapclevich AL Q Lian and FL uvmA ymmetric rnodif d g ar drive reduct jon or noi elocalizati n or contact imulauon of meshing and stresanalysis Computer Mehods in Applied Mechanics andEngineering 2000 Issue 18 pp 63-390
For more informationabout AKGearsor tocontact the companyvisit its website twwwkgll bullbullcOtfl
For more infonutionabout KIi Glrs orto contact the compa-ny visit its website Itwwwklibullbullbullbullbull lIcom
Ten Us WhitYouThink VisitwwwpIfffCIMoIfyctobull RIte this artielebull Request mora
infonnltiOlbull ContlCt the lutltOrs
or cOlRpani n-tioned
bull Make I IIIIglltionOr cIII (8471437-6amp04 totalk to one of our di-tors
www poweransmtssioncom bull www geartechnologycom bull GEAR TECHNOLOGYmiddot SEPTEMBEROCTOBER 2002 35
of exi tence f r a pair of gears ZI = 14 = 2 mol
= ma2 = 0075 i shown in Figure 5 e area f exis-tence include a number of isogram reflecting con-tant value of different gear paramete u h
operating pre ure angle CX contact ratios Ea etcThe area of existence of pur gears (thi k lin I) ilimited by isogram Eo= 10 and undercut i gramCX 1 = 00
CX = 00bull HeLi al gears can have a tran -p p-
ve contact ratio Ie than I becau the ialcontact ratio can provide p per me h Th area fexi tence of helical gears i th refore mu h greater
h point on the area of e i len e refl IS a pair fgears with dimen ionle propertie that can fit a par-ti ular appli arion Th propelti are pres ureangle c ntact ratio pitting re istance g merryfactorl pecifi liding ratio et
The absolute area f e i ten e in Iud pur garcombinatio with any alue f proportional t p Iland thickne ber een mal = ma2 = 0 t TIol = mbl
and m 2 = mb (phantom lin 2) Thi area i ub- a _ I
tantially larger than the area ith gi en alue f iproporri nal t p land thi knes The zone f r a tandard generating rn 1ltwith 200 pre sure angle ( hown in ig I) i only a fractional part of the a ail-
able area of exi tence as h wn by hidden lin 3 in iFigure 5 An appli ati n of a traditi nal gear gener-
Iating app a h for gear pairs out ide th z ne ut- ilined by hidden line 3 require Ie Lionof a gen r-ating ra k with differentparam ters The generationof orne gear combinations (t p left and bottom rightcomers of the area of xi t n e hown in Fig 5) willrequire different generating ra ks for the pinion andfor the gear
Analy i of the area of exi tcnce haws how imany gear olutions could be left out of c nsid ra- tion if a traditional approach based n a predet r-mined t of ra k dimension i applied F r e am-ple pur gears with a high operating pressur angl(point A on the Figure 5 where th rating pres-ure angle cx= 95deg conta t ratio Eex= 10) r with
a high contact ratio (point B on the Figure 5 whereonta t ratio Eex = 20 I operating pre ure angl cx
= 1670) could not be produced with tandard ra k
dimen ion Fi ure 6a and 6b h w th geFigures 6c and 6d are the gears that are achie ableusing a tandard g nerating ra k that arc pre nted I tandard designed gear pair h a conta I ratio ofbypoinlSC(a bullbull=296degE
Q= 10)andD(a= 15cr i only a= 116in lose m hlnana tual application
En = 164) in igure 5 E en gears with the same with real manufacturing ~ leran and operatingoperating pre ure angle (point in ig 5) 1 k c nditi ns the ntact ratio of a tandard de ignedquite different (Fig 7) Th tandard d igned gear i gear pair uld be reduced t an unacceptable Ie elpair (Fig 7b) has alrn t sharp-pointed pini n t th Enlt 10and hort and tubby gear teeth with excessi e top ynthe i of Garing umerical xampJland tooth thi kne The direct de igned gear pair Th r arc everal w y to define gear parame-hownin Figure 7a has a contact ratio Ea = 147The i I ruing the direct gear de ign approach Thi
poeensmlssloncom bull g6llechnologycom bull GEAR TECHNOLOGY SEPTEMBEROCTOBER 2002 33
a
b
Figure 8- helical gear pair with higl operatingpressure angle I = 9 2 = 12 a = 688~ Ea =0504 3 = 30~ EfJ = 055
Figur 9-A Jpur gear with a Itigll contact ratio I= 55 2 = 55 a = J29~ Ea= 40 lIIal = lIIal = 0075
article considers some of themArea of existence is known The initial data for
the synthesi of a pair of gear (zl Z2 01 f1l2)could be taken from the area of exi tence at omeparticular point The coordinate of thi point andcenter di tance abullbull describe all operating gearparameter This calculation procedure andnumerical example are presented in the able 2
Area of existence is 1101 known pical pr b-lern could be finding the maximum pre ureangle if the tran verse contact ratio i ch en orfinding the maximum tran verse contact ratio ifthe pre sure angle i cho en Both of the e ca erequire finding the point of area of exi tencewhere i ogram ltxwand Eo have the arne tangentThis condition is de cribed (Ref 2) as
cos(0)2 (l + 1t bull mal sin(ltXI)Iz)=CO (02)2 (I + 1t ma2bull in(oa2)1z2) (17)
and allows solution of the e problem withoutknowing the area of existence The calculationprocedures and numerical example are pre entedin Table 3 and 4
The fillet between teeth is not involved in gearmesh operation but it shape greatly affect gearperformance and durability In traditional gearde ign the fillet profile is a function of the cutlershape and the machine tool setup It typically haexce ive radial clearance resulting in high bend-ing stresses Direct gear de ign doe n t limit fil-let hape definition One po ibility i to de cribethe fillet profile as a trace of the top part of themating gear tooth (with corre ponding minimumradial clearance) (Ref 4 and 5) Application offinite element analysis allows for forming the fil-let profiles to balance and minimize bendingstresses
Extreme Parameters of Involute GearsPoint A (tangent point of i ogram Eo = 10 and
0bullbull= max) of the area of exi tence de cribe gearwith the maximum achievable operating pre ureangle There is no such limit for helical gearbecause a lack of the transverse contact ratio (Eo lt10) i compen ated by the axial contact ratio Ep
A ample of a helical gear with high operatingpressure angle (Ref 6) i shown in Figure InFigure 5 the point B (inter ection point of inter-feren e i ogram up I = 0deg and op2 = 0deg) of thearea of existence de cribe the gears with themaximum achievable transver e contact ratioTable 5 pre ents maximum value f r operatingpressure angle owA (Point A of the area of exi -abc renee) and tran ver e contact ratio Eo B (Point B of
Figure lo-Spur gears with minimum number of teeth a) ZJ = 5 Z2 = 5 a =33 1deg E 104 b) Z 4 Z 6 a 326deg 10) 3 11 the area of existence) for gear pairs with different oa = J = 2 = = ea = ~ C zJ = Z2 = a= 243 ea = 101 numbers of teeth and the proportional top land34 SEPTEMBEROCTOBER 2002 bull GEAR TECHNOLOGY wwwgesrtschnologycom bull wwwpowerlrsnsmlssioncom
Table 3Parameter Symbol Equation ValueNumber of Teeth (givenl Z 14
z 28Proportional top land thicknesses 19iven) ml 0075
m 0075Center distance in (given) 8w 3000Operating pressure angle deg [chosen] (lw 33Profile angles on outside diameters deg (ll (l0) amp 117) 4002
(lll 3884Transverse contact ratio (maximum 1 famagt 121 1246Profile angle in the involute intersection point deg VI (5cl 4103
v2 4036Proportional base tooth thicknesses mbl (41 0687
mb2 1296Use Table 2 to identify remaining equations
Table 4Parameter Symbol Equation ValueNumber of teeth (givenl Zl 14
lz 28Proportional top land thicknesses (given) ml 0075
m2 0075Center distance in (given) a 3000Transverse contact ratio (chosen) ElY
105Profile angles on outside diameters deg (lal (l0J(l21 4329
(laz amp (17) 4312Operating pressure angle deg [rnaximum] (lwmu 10J 021 amp (17) 3797Profile angle in the involute intersection point deg v (5c) 4406
v2 4352Proportional base tooth thicknesses mb1 (4) 0886
mb2 1693Use Table 2 to identify remaining equations
Table 5
owAIE8I Pinion ZI
5 10 20 30 40 505 315deg10 350deg114 378deg121 39deg123 396deg124 40deg124
~ 10 350deg1 14 368deg143 385deg166 394deg172 399deg175 408deg176bull 20 378deg121 385deg166 394deg1214 399deg237 402deg248 404deg254Ggt(l 30 39deg123 394deg172 399deg237 402deg274 404deg297 406deg31
40 396deg124 399deg175 402deg248 404deg297 406deg328 407deg34950 40deg124 408deg176 404deg254 406deg31 407349 408deg378
thickn s mal 11102 0075 An example of apur gear me h with a high coma I ratio is sh wn
in Figure 9pur gear (contact ratio En 10) with a min-
imum pos ible number of t th (Ref 2) areshown in Figure 10The minimum po ibl num-ber of teeth f r helical gears is n t limit d bytran erse contact ratio and could be as few aone (Ref 6) n e ample of a helical gear with thenumber of t eth - = ~ = I is shown in Figure IIIn elute Gea with mmetric ToothProlile
Oppo ite nan (profile ) of the gear I th arefunctionally different for m t gear Th w rk-I ad on one profile i ignificantl higher and ri applied for longer period of lime than n thoppo ite one The a ymm tric tooth shape a com-modates thi fun tional diff ren e
Th design intent of a ymmeuic teeth i Iimpro e performance of main nta tin p filby degrading oppo ite profile Th ppo ire pro-fil are unload d or lightl I aded and usuallwork for a relatively h rt period The impro edperformance c uld mean in rea ing load capacityor reducing weight noise vibration tc
Degre of asymmetry and drive pr file elec-tion for these gears depend on the applicati n
ymmetric profiles make it possible to managetooth stiffnes and 1 ad haring while k ping ade irable pre ure angle and conta t rati n thedrive profile
Dire t design of gears with asymmetri teeth iscon idered in detail in ther articles (Refs 7 and8) covering topic u h a analy i and synthe isof asymmetric gearing area e i t n e andapplication ample of gea ith asymmetritooth profile are hown in Figure 12 Gears witha yrnrnetric teeth h uld be c n sidcred f r geary terns that require extreme perf rmance like
aero pace drive Th Y are al a applicable f rmas production lransmi ion where the hare ofthe tooling co I per one gear is relatively in ignif-icant The mo t promising application for asym-metric profile i with m Ided gears and wdermetal gea Molded gear tooling u ually requirea custom shape 0 ih as mmetric profile dnot ignificanl1y affect c t
umm ryDirect gear de ign i an alternati e appr ach to
traditional gear design It allow analysi of awide range of parameters for all po ible gearcombination in order to find th m st uitablesolution for a parti ular appli arion This opti-mum gear oluli n can exceed the limits of tradi-tional rack generating methods of gear design
Figure ll-Helical gears with one tooth zL I Z2= I abull= 688~t = 056 b = 349~EIJ012
Figure 12- pur gear with a ymmetric teeth 12u)a generator gear drive with a = 41 ~ Ea = 12 fordrive flanks ~lfIda = 18~ Eq= l64for coastflanks12b) a pia tICgear pump WIt a = 457~E 101for drive flanks and a JO3~Ea = I09ltr coastflanks
Dire t gear d sign for asymmetric tooth profilespen additi nal re erve f r improvement f
gear dri es with unidirecti nal load cycles thatarc typical for many mechanical transrnissi n 0
cknowl dgm ntsThe authors expre deep gratitude 10 Gear Technologytechnical editors Robert Errichello of Geartech 1 ated inTownsend MT and Dan Thurman for their help in prepar-ing thi article
ReferencesJ Gr man MB The Zone f Involute Me h Ve tnikfa hiw troeniva 1962 [ ue 12 pp 12-17 (in Ru ian)
2 ulgakov EB Theory of Involute GearsMa hino troenie Mo 0 1995 (in Ru ian)
olbourne lR The Ceometrv of Involute Gearspringer- erlag ew York 1987
4 IAGM 1006-A97 Tooth Proportion for Pia ticG ppendix F -o nerating Gear Geometr WilhoutRa I GMA Ale andria A 19975 Kupelevich L 1 Klei s and RE Kleis ewOpportunities with Molded Gear AG Fall Techni alMecling Detroit October -52001 (01 FTM9)6 Kapelevi h AL and pound8 ulgakov Expanding therange f involute helical gearing Yestnl]MasIJlllomiddottroeniya 19 2 I sue 3 pp 12-14 (in Russian)
ranslated [0 nglish 0 iet Engin ering Research Vol 2I lIC 3 1982 pp 8-97 Kapelevi h AL Gc rnetry and design of in oluiepur gear with a ymmctric teeth Mechanism and
Machine Theory 2000 I sue 35 pp 117-1308 Kapclevich AL Q Lian and FL uvmA ymmetric rnodif d g ar drive reduct jon or noi elocalizati n or contact imulauon of meshing and stresanalysis Computer Mehods in Applied Mechanics andEngineering 2000 Issue 18 pp 63-390
For more informationabout AKGearsor tocontact the companyvisit its website twwwkgll bullbullcOtfl
For more infonutionabout KIi Glrs orto contact the compa-ny visit its website Itwwwklibullbullbullbullbull lIcom
Ten Us WhitYouThink VisitwwwpIfffCIMoIfyctobull RIte this artielebull Request mora
infonnltiOlbull ContlCt the lutltOrs
or cOlRpani n-tioned
bull Make I IIIIglltionOr cIII (8471437-6amp04 totalk to one of our di-tors
www poweransmtssioncom bull www geartechnologycom bull GEAR TECHNOLOGYmiddot SEPTEMBEROCTOBER 2002 35
article considers some of themArea of existence is known The initial data for
the synthesi of a pair of gear (zl Z2 01 f1l2)could be taken from the area of exi tence at omeparticular point The coordinate of thi point andcenter di tance abullbull describe all operating gearparameter This calculation procedure andnumerical example are presented in the able 2
Area of existence is 1101 known pical pr b-lern could be finding the maximum pre ureangle if the tran verse contact ratio i ch en orfinding the maximum tran verse contact ratio ifthe pre sure angle i cho en Both of the e ca erequire finding the point of area of exi tencewhere i ogram ltxwand Eo have the arne tangentThis condition is de cribed (Ref 2) as
cos(0)2 (l + 1t bull mal sin(ltXI)Iz)=CO (02)2 (I + 1t ma2bull in(oa2)1z2) (17)
and allows solution of the e problem withoutknowing the area of existence The calculationprocedures and numerical example are pre entedin Table 3 and 4
The fillet between teeth is not involved in gearmesh operation but it shape greatly affect gearperformance and durability In traditional gearde ign the fillet profile is a function of the cutlershape and the machine tool setup It typically haexce ive radial clearance resulting in high bend-ing stresses Direct gear de ign doe n t limit fil-let hape definition One po ibility i to de cribethe fillet profile as a trace of the top part of themating gear tooth (with corre ponding minimumradial clearance) (Ref 4 and 5) Application offinite element analysis allows for forming the fil-let profiles to balance and minimize bendingstresses
Extreme Parameters of Involute GearsPoint A (tangent point of i ogram Eo = 10 and
0bullbull= max) of the area of exi tence de cribe gearwith the maximum achievable operating pre ureangle There is no such limit for helical gearbecause a lack of the transverse contact ratio (Eo lt10) i compen ated by the axial contact ratio Ep
A ample of a helical gear with high operatingpressure angle (Ref 6) i shown in Figure InFigure 5 the point B (inter ection point of inter-feren e i ogram up I = 0deg and op2 = 0deg) of thearea of existence de cribe the gears with themaximum achievable transver e contact ratioTable 5 pre ents maximum value f r operatingpressure angle owA (Point A of the area of exi -abc renee) and tran ver e contact ratio Eo B (Point B of
Figure lo-Spur gears with minimum number of teeth a) ZJ = 5 Z2 = 5 a =33 1deg E 104 b) Z 4 Z 6 a 326deg 10) 3 11 the area of existence) for gear pairs with different oa = J = 2 = = ea = ~ C zJ = Z2 = a= 243 ea = 101 numbers of teeth and the proportional top land34 SEPTEMBEROCTOBER 2002 bull GEAR TECHNOLOGY wwwgesrtschnologycom bull wwwpowerlrsnsmlssioncom
Table 3Parameter Symbol Equation ValueNumber of Teeth (givenl Z 14
z 28Proportional top land thicknesses 19iven) ml 0075
m 0075Center distance in (given) 8w 3000Operating pressure angle deg [chosen] (lw 33Profile angles on outside diameters deg (ll (l0) amp 117) 4002
(lll 3884Transverse contact ratio (maximum 1 famagt 121 1246Profile angle in the involute intersection point deg VI (5cl 4103
v2 4036Proportional base tooth thicknesses mbl (41 0687
mb2 1296Use Table 2 to identify remaining equations
Table 4Parameter Symbol Equation ValueNumber of teeth (givenl Zl 14
lz 28Proportional top land thicknesses (given) ml 0075
m2 0075Center distance in (given) a 3000Transverse contact ratio (chosen) ElY
105Profile angles on outside diameters deg (lal (l0J(l21 4329
(laz amp (17) 4312Operating pressure angle deg [rnaximum] (lwmu 10J 021 amp (17) 3797Profile angle in the involute intersection point deg v (5c) 4406
v2 4352Proportional base tooth thicknesses mb1 (4) 0886
mb2 1693Use Table 2 to identify remaining equations
Table 5
owAIE8I Pinion ZI
5 10 20 30 40 505 315deg10 350deg114 378deg121 39deg123 396deg124 40deg124
~ 10 350deg1 14 368deg143 385deg166 394deg172 399deg175 408deg176bull 20 378deg121 385deg166 394deg1214 399deg237 402deg248 404deg254Ggt(l 30 39deg123 394deg172 399deg237 402deg274 404deg297 406deg31
40 396deg124 399deg175 402deg248 404deg297 406deg328 407deg34950 40deg124 408deg176 404deg254 406deg31 407349 408deg378
thickn s mal 11102 0075 An example of apur gear me h with a high coma I ratio is sh wn
in Figure 9pur gear (contact ratio En 10) with a min-
imum pos ible number of t th (Ref 2) areshown in Figure 10The minimum po ibl num-ber of teeth f r helical gears is n t limit d bytran erse contact ratio and could be as few aone (Ref 6) n e ample of a helical gear with thenumber of t eth - = ~ = I is shown in Figure IIIn elute Gea with mmetric ToothProlile
Oppo ite nan (profile ) of the gear I th arefunctionally different for m t gear Th w rk-I ad on one profile i ignificantl higher and ri applied for longer period of lime than n thoppo ite one The a ymm tric tooth shape a com-modates thi fun tional diff ren e
Th design intent of a ymmeuic teeth i Iimpro e performance of main nta tin p filby degrading oppo ite profile Th ppo ire pro-fil are unload d or lightl I aded and usuallwork for a relatively h rt period The impro edperformance c uld mean in rea ing load capacityor reducing weight noise vibration tc
Degre of asymmetry and drive pr file elec-tion for these gears depend on the applicati n
ymmetric profiles make it possible to managetooth stiffnes and 1 ad haring while k ping ade irable pre ure angle and conta t rati n thedrive profile
Dire t design of gears with asymmetri teeth iscon idered in detail in ther articles (Refs 7 and8) covering topic u h a analy i and synthe isof asymmetric gearing area e i t n e andapplication ample of gea ith asymmetritooth profile are hown in Figure 12 Gears witha yrnrnetric teeth h uld be c n sidcred f r geary terns that require extreme perf rmance like
aero pace drive Th Y are al a applicable f rmas production lransmi ion where the hare ofthe tooling co I per one gear is relatively in ignif-icant The mo t promising application for asym-metric profile i with m Ided gears and wdermetal gea Molded gear tooling u ually requirea custom shape 0 ih as mmetric profile dnot ignificanl1y affect c t
umm ryDirect gear de ign i an alternati e appr ach to
traditional gear design It allow analysi of awide range of parameters for all po ible gearcombination in order to find th m st uitablesolution for a parti ular appli arion This opti-mum gear oluli n can exceed the limits of tradi-tional rack generating methods of gear design
Figure ll-Helical gears with one tooth zL I Z2= I abull= 688~t = 056 b = 349~EIJ012
Figure 12- pur gear with a ymmetric teeth 12u)a generator gear drive with a = 41 ~ Ea = 12 fordrive flanks ~lfIda = 18~ Eq= l64for coastflanks12b) a pia tICgear pump WIt a = 457~E 101for drive flanks and a JO3~Ea = I09ltr coastflanks
Dire t gear d sign for asymmetric tooth profilespen additi nal re erve f r improvement f
gear dri es with unidirecti nal load cycles thatarc typical for many mechanical transrnissi n 0
cknowl dgm ntsThe authors expre deep gratitude 10 Gear Technologytechnical editors Robert Errichello of Geartech 1 ated inTownsend MT and Dan Thurman for their help in prepar-ing thi article
ReferencesJ Gr man MB The Zone f Involute Me h Ve tnikfa hiw troeniva 1962 [ ue 12 pp 12-17 (in Ru ian)
2 ulgakov EB Theory of Involute GearsMa hino troenie Mo 0 1995 (in Ru ian)
olbourne lR The Ceometrv of Involute Gearspringer- erlag ew York 1987
4 IAGM 1006-A97 Tooth Proportion for Pia ticG ppendix F -o nerating Gear Geometr WilhoutRa I GMA Ale andria A 19975 Kupelevich L 1 Klei s and RE Kleis ewOpportunities with Molded Gear AG Fall Techni alMecling Detroit October -52001 (01 FTM9)6 Kapelevi h AL and pound8 ulgakov Expanding therange f involute helical gearing Yestnl]MasIJlllomiddottroeniya 19 2 I sue 3 pp 12-14 (in Russian)
ranslated [0 nglish 0 iet Engin ering Research Vol 2I lIC 3 1982 pp 8-97 Kapelevi h AL Gc rnetry and design of in oluiepur gear with a ymmctric teeth Mechanism and
Machine Theory 2000 I sue 35 pp 117-1308 Kapclevich AL Q Lian and FL uvmA ymmetric rnodif d g ar drive reduct jon or noi elocalizati n or contact imulauon of meshing and stresanalysis Computer Mehods in Applied Mechanics andEngineering 2000 Issue 18 pp 63-390
For more informationabout AKGearsor tocontact the companyvisit its website twwwkgll bullbullcOtfl
For more infonutionabout KIi Glrs orto contact the compa-ny visit its website Itwwwklibullbullbullbullbull lIcom
Ten Us WhitYouThink VisitwwwpIfffCIMoIfyctobull RIte this artielebull Request mora
infonnltiOlbull ContlCt the lutltOrs
or cOlRpani n-tioned
bull Make I IIIIglltionOr cIII (8471437-6amp04 totalk to one of our di-tors
www poweransmtssioncom bull www geartechnologycom bull GEAR TECHNOLOGYmiddot SEPTEMBEROCTOBER 2002 35
thickn s mal 11102 0075 An example of apur gear me h with a high coma I ratio is sh wn
in Figure 9pur gear (contact ratio En 10) with a min-
imum pos ible number of t th (Ref 2) areshown in Figure 10The minimum po ibl num-ber of teeth f r helical gears is n t limit d bytran erse contact ratio and could be as few aone (Ref 6) n e ample of a helical gear with thenumber of t eth - = ~ = I is shown in Figure IIIn elute Gea with mmetric ToothProlile
Oppo ite nan (profile ) of the gear I th arefunctionally different for m t gear Th w rk-I ad on one profile i ignificantl higher and ri applied for longer period of lime than n thoppo ite one The a ymm tric tooth shape a com-modates thi fun tional diff ren e
Th design intent of a ymmeuic teeth i Iimpro e performance of main nta tin p filby degrading oppo ite profile Th ppo ire pro-fil are unload d or lightl I aded and usuallwork for a relatively h rt period The impro edperformance c uld mean in rea ing load capacityor reducing weight noise vibration tc
Degre of asymmetry and drive pr file elec-tion for these gears depend on the applicati n
ymmetric profiles make it possible to managetooth stiffnes and 1 ad haring while k ping ade irable pre ure angle and conta t rati n thedrive profile
Dire t design of gears with asymmetri teeth iscon idered in detail in ther articles (Refs 7 and8) covering topic u h a analy i and synthe isof asymmetric gearing area e i t n e andapplication ample of gea ith asymmetritooth profile are hown in Figure 12 Gears witha yrnrnetric teeth h uld be c n sidcred f r geary terns that require extreme perf rmance like
aero pace drive Th Y are al a applicable f rmas production lransmi ion where the hare ofthe tooling co I per one gear is relatively in ignif-icant The mo t promising application for asym-metric profile i with m Ided gears and wdermetal gea Molded gear tooling u ually requirea custom shape 0 ih as mmetric profile dnot ignificanl1y affect c t
umm ryDirect gear de ign i an alternati e appr ach to
traditional gear design It allow analysi of awide range of parameters for all po ible gearcombination in order to find th m st uitablesolution for a parti ular appli arion This opti-mum gear oluli n can exceed the limits of tradi-tional rack generating methods of gear design
Figure ll-Helical gears with one tooth zL I Z2= I abull= 688~t = 056 b = 349~EIJ012
Figure 12- pur gear with a ymmetric teeth 12u)a generator gear drive with a = 41 ~ Ea = 12 fordrive flanks ~lfIda = 18~ Eq= l64for coastflanks12b) a pia tICgear pump WIt a = 457~E 101for drive flanks and a JO3~Ea = I09ltr coastflanks
Dire t gear d sign for asymmetric tooth profilespen additi nal re erve f r improvement f
gear dri es with unidirecti nal load cycles thatarc typical for many mechanical transrnissi n 0
cknowl dgm ntsThe authors expre deep gratitude 10 Gear Technologytechnical editors Robert Errichello of Geartech 1 ated inTownsend MT and Dan Thurman for their help in prepar-ing thi article
ReferencesJ Gr man MB The Zone f Involute Me h Ve tnikfa hiw troeniva 1962 [ ue 12 pp 12-17 (in Ru ian)
2 ulgakov EB Theory of Involute GearsMa hino troenie Mo 0 1995 (in Ru ian)
olbourne lR The Ceometrv of Involute Gearspringer- erlag ew York 1987
4 IAGM 1006-A97 Tooth Proportion for Pia ticG ppendix F -o nerating Gear Geometr WilhoutRa I GMA Ale andria A 19975 Kupelevich L 1 Klei s and RE Kleis ewOpportunities with Molded Gear AG Fall Techni alMecling Detroit October -52001 (01 FTM9)6 Kapelevi h AL and pound8 ulgakov Expanding therange f involute helical gearing Yestnl]MasIJlllomiddottroeniya 19 2 I sue 3 pp 12-14 (in Russian)
ranslated [0 nglish 0 iet Engin ering Research Vol 2I lIC 3 1982 pp 8-97 Kapelevi h AL Gc rnetry and design of in oluiepur gear with a ymmctric teeth Mechanism and
Machine Theory 2000 I sue 35 pp 117-1308 Kapclevich AL Q Lian and FL uvmA ymmetric rnodif d g ar drive reduct jon or noi elocalizati n or contact imulauon of meshing and stresanalysis Computer Mehods in Applied Mechanics andEngineering 2000 Issue 18 pp 63-390
For more informationabout AKGearsor tocontact the companyvisit its website twwwkgll bullbullcOtfl
For more infonutionabout KIi Glrs orto contact the compa-ny visit its website Itwwwklibullbullbullbullbull lIcom
Ten Us WhitYouThink VisitwwwpIfffCIMoIfyctobull RIte this artielebull Request mora
infonnltiOlbull ContlCt the lutltOrs
or cOlRpani n-tioned
bull Make I IIIIglltionOr cIII (8471437-6amp04 totalk to one of our di-tors
www poweransmtssioncom bull www geartechnologycom bull GEAR TECHNOLOGYmiddot SEPTEMBEROCTOBER 2002 35