003 a Basis for Solid Modeling of Gear Teeth With

8/11/2019 003 a Basis for Solid Modeling of Gear Teeth With

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@ ergsmem Mech. Mach. TheoryVo l. 29. N o. 5 . pp. 713--723, 1994Copyright C 1994 ElsevierScience LtdPrinted in Gre at Britain. All rights reserved0094-114)(/94 $7.00 0,0o

A BASIS FOR SOLID MODELING OF GEAR TEETH WITH

APPLICATION IN DESIGN AND MANUFACTURE

R O N A L D L . H U S T O NUnive rsi ty of Cincinnati . Cincinnati . OH 45221-0072, U.S.A.

D I M I T R I O S M AV R I P L ISEDS, Cincinnati , OH 45246, U.S.A.

F R E D B . O S WA L DNational Aeronautics and Space Administrat ion,Lewis Research Center. Cleveland, OH 44135, U.S.A.

Y U N G S H E N G L IU

Feng-Chia Universi ty. Taichung, Taiwan, R.O.C.

(Receired 21 April 1 992: in revised orm 13 M ay1993: receit,ed or pub lication 16 July 1993)

A b s t r a c t - - T h i spaper discusses a new approa ch to mo deling gear tooth surfaces. A com puter graphicssolid modeling pro cedure is used to simulate the tooth fab ricat ion processes. This procedure is based onthe p rinciples of differential geometry that pertain to env elopes of curves and surfaces. The procedure isillustrated with the m ode ling of spur. helical, bevel, spiral bevel and hypo id gear teeth. App lications indesign and m anufacturing a re discussed. Extensions to nonstand ard tooth forms, to cams. and to rol l ingelement bearings are proposed.

I N T R O D U C T I O NA difficult task facing analysts and designers ofg ear ed pow er t ransmiss ion systems is unders tand ingand ut i liz ing the com plex geom etry of the gear teeth . Even for involute spur gears where the toothgeo m etry is general ly well unde rs tood, the deta i ls (e .g ., t rochoidal g eom etry and prof ile modif i -cations) are neither simple nor accessible to most designers. For helical bevel, spiral bevel andhypoid gears , the geometry is even more complex so that analyses and designs are of necess i tyapproximate and empir ical . Opt imal gear design is thus e lus ive . Even i f the opt imal gear toothgeo m etry is know n, i t i s d is tor ted u nder load. H ence, the geom etry of meshing gear teeth isgeneral ly less than opt imal . Indeed, the geo me try of m eshing gear teeth und er load is v ir tual lybeyond analyt ical descr ipt ion.

Whi le the de ta il s o f gea r too th geo met ry may no t be impor tan t in many app l ica t ions , fo r

precis ion gears the to oth geom etry is the s ingle most im portan t factor influencing t ransmiss ionkinemat ics , gear s t rength an d ge ar wear. Tha t i s , gear tooth geo me try has a greater effect uponthe performance, s t rength and l i fe of gear ing systems than any other factor. Accordingly,com prehen sive gear ing kinemat ic , s trength an d l i fe analyses cann ot be conducted withou t anaccura te represen ta t ion o f the gear too th geom et ry.

in recent years , a num ber o f analysts [ I -9] have used the proce dures of di fferentia l geom etry andclassical geom etr ical analyses to descr ibe gear to oth surfaces . These analyses have great ly extendedunders tand ing o f su r face geomet ry and con tac t k inemat ics . Some o f these p rocedures have beencom bined w ith num erical metho ds to o bta in tooth con tact analyses [10-14]. H owever, there is s ti lla need fo r m ore ex tens ive and mo re acc ura te represen ta tions o f gea r too th geomet ri es if ana lys tsand designers are to achieve improvements in transmission efficiency, reliabil i ty and l ife. Such

imp rovem ents are especia lly desi rable for precis ion gears meshing under load. B et ter represen-ta t ions o f gear too th geom etr ies are a lso essent ia l for the developm ent of specia lized, n onstan dardtoo th fo rms .

Recent ly, a n ew app roac h has been p ropose d to s imulate the fabr icat ion of gear teeth by usingcom pu ter graph ics [15, 16]. This app roac h has led to a proced ure for developing solid m odels of

713


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714 R. U Ht~sTos eta/.

gears which in turn can be used for s tudying m eshing kinemat ics , con tact s t resses , roo t s t resses andlubr ica tion. Herein we discuss the analyt ica l bas is for the pro cedu re u sed to develop sol id modelsof gears. Th e pro cedu re i s i l lus t ra ted by developing mo dels o f spur. he lica l, bevel , sp i ra l bevel andhypoid gears .

This repo r t i s d iv ided in to f ive sec t ions , the f irs t of which p rovides so me back grou nd mater ia ltha t wa s needed to develop the proced ure for model ing gear too th surfaces . The succeeding sect ionsexamine involute tooth genera t ion, provide some graphical resul ts , d iscuss the procedure andpresent conc lus ions .

P R E L I M I N A R Y A N A L Y S I S

Envelopes of curves and sur faces ; model ing of sur face genera t ion

Consider a p lane curve C def ined by the equat ion

y = f ( x , t) ( I )

where t is a param eter. S upp ose that C has a typica l form as in Fig . 1. Let C be such tha t i t canbe moved (or reposi t ioned) in the p lane wi thout being dis tor ted . Let t be a parameter determiningthis reposi t ioning. The n as t var ies , the locus of pos i t ions of C form the family of curves seenin Fig . 2. The curve E, represent ing the l imi t ing locat ion of the family, is the env elop e of thefamily.

Sup pose that a family of curves i s represented in analyt ica l form as

y ~ - f ( x , t ) o r F ( x , y, t ) - - - O , (2)

where F is cont inu ous w i th nonvanishing par t ia l der ivat ives wi th respect to x an d y. Then i t isknown that an envelope of the family may be determined by us ing the re la t ion [17, 18]

OF/Or = O. (3)

That i s , the e l iminat ion of the parameter t be tween equat ions (2) and (3) provides an analyt ica ldescription of E, if the envelope exists .To i l lus t rate the const ruct ion of an envelope, consider a family of l ines such that each l ine of

the fam ily is a f ixed distance r fro m a fixed point O, as depicted in Fig. 3. Let ~ define the inclinationof a typica l l ine L o f the family and le t 0 define the inc l inat ion of the l ine no rmal to L as show n.The equa t ion o f L may be wr i tt en in the s t andard fo rm

Y - Ye = m (x - x , ) , (4)

Fig. I . A plan e curve.


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Modeling gea r tooth surfaces 715

Y

XFig. 2. A family of plane cruves and its envelope.

w h e r e m is t h e sl o p e o f L a n d w h e r e ( xp , y p ) a r e t h e c o o r d i n a t e s o f a p o i n t P o f L . I f P i s t h e p o i n to f i n t e rs e c t io n o f L w i t h t h e n o r m a l l i n e w h i c h p a s s es t h r o u g h O , t h e n m , 0 a n d ~ a r e r e la t e d b yt h e e x p r e s s i o n

m = tan ~ = - c o t 0 . (5 )

F r o m F i g. 3 w e s ee th a t t h e c o o r d i n a t e s o f P a r e

x p = r c o s 0 a nd y p = r s i n O . (6)

H e n c e , f r o m e q u a t i o n ( 4 ) , t h e e q u a t i o n o f L m a y b e e x p r e s s e d a s

y - r s in 0 = ( - c o tO ) ( x - r cos 0) . (7 )

I n t h e f o r m o f e q u a t i o n ( 2 ) , t h is b e c o m e s

y sin 0 + x cos 0 - r =F ( x , y , O) -- 0. (8)

E q u a t i o n ( 8 ) d e f in e s a f a m i l y o f l in e s w i t h 0 b e i n g t h e f a m i l y p a r a m e t e r . T h e n , f r o m e q u a t i o n(3 ), i f we d i f f e r en t i a t e w i th r e spec t t o 0 , we hav e

(~F/c~O = y co s 0 - x s in 0 = 0. (9)

Y

I.

I

Fig. 3. A family of l ines equidistant from a point.

X


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716 R .L . HUSTONe t a l .

Hence , the equa t ion o f the enve lope may be ob ta ined by e limina ting 0 be tween equa t ions (8) and(9) . That i s , by solving equat ions (8) and (9) for x and y we obta in

x = r c o s O a n d y = r s in O. (10)

Final ly, by e l iminat ing 0 , we have

x 2 + y 2 = r 2. ( I I )

This i s the equat ion of the envelope of the family of l ines . As expected, i t i s a c i rc le wi th radiusr and center O.

The env e lope o f a cu t t ing too l m ay be ob ta ined in a s imi l a r manner. The cu t t ing too l enve lopeis determine d by the inf in ite family of surfaces forme d by the tool . He nce, us ing a proce dure basedupon equat ions (2) and (3) . we may s imula te surface genera t ion in a manufactur ing process . Morespecif ica lly, we can s imula te an d mo del gear to oth surfaces .

I n r o l u t e o f a c i r c l e

To es tabl ish the procedures for model ing gear tooth surfaces , i t i s he lpful to br ief ly review the

proper t ies o f involute curves . Recal l tha t the involute of a c i rc le m ay be character ized as the locusof pos i tions o f the end po in t o f a co rd be ing unw rapped fo rm the cir cle . F igure 4 dep ic t s an invo lu teI of a c i rc le C. Let P be a typica l point o n the inv olute and le t #b me asure the unw rapping. Thenthe posi t ion vector p locat ing P re la t ive to the c i rc le center O may be expressed as

p = x n , + y n , , = r n , + r ~ n ~ 1 2 )

where x and y are the hor izonta l and ver t ica l coordinates of P, r i s the c i rc le radius , and n , , n , . ,n , and n , are the h or izonta l , ver t ica l , radia l and tangent ia l uni t vectors as shown in Fig . 4 . Theuni t vectors are then re la ted by the express ions

n , = s i n q ) n , + c o s ~ n , , a nd n ~ = - c o s ~ n , + s i n ~ n y . (13)

When a subst i tu t ion for n , and n~ f rom equat ion (13) i s made in equat ion (12) , x and y are seento be re la ted to r and ~ by the express ions

x - - r s i n ~ - r ~ - c o s ~ a nd y - - r c o s ~ + r ~ s i n ( k . (14)

Eq uat ion s (14) are param etr ic eq uat io ns def in ing the involute . The s lope of the involute a t Pis then

dy/dx = d y / d ~ ) / d x / d c ~ ) = c o t q ). I 5 )

I r

n x

F i g . 4 . Involu te geomet ry.


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Modeling gea r tooth surfaces

Fig. 5. A plastic disk rolling ove r a rigid triangular protrusion.

717

Use o f com puter graphicsThe concep t o f pa ramet r i c equa t ions de f in ing the enve lope o f a f ami ly o f li nes can be used a s

a bas is for com pute r graphics m odel ing o f involute spur gear tee th . To i l lus t ra te , conside r aperfectly plast ic circular disk roll ing in a straight l ine, as in Fig. 5. Let the disk encounter a r igidprot rus ion , such as a n i sosceles t r iangle , on the ro l l ing surface . A f ter the d isk ro l ls ove r theprot rus ion , the impress ion (o r footpr in t ) le f t in the p las t ic d isk represents the enve lope of the s idesof the t r iangle re la tive to a coo rdina te sy s tem f ixed in the d isk . This envelope is an involute of acircle wh ose rad ius is equ al to that o f the roll ing disk less the tr iangle height [15].

This s imula t ion can b e used to represent the man ufactu re o f an involute spur gear tooth: le t therol ling disk be replaced b y a gear b lank ro l ling on i ts base c i rc le as seen in Fig . 6 . Wh en the ro l l inggear b lank enc ounters a cut ter in the shape o f an involute rack tooth , the impress ion on the b lankis the gap be tween the t ee th o f an invo lu te spur gea r. Fur the rm ore ( a s dem ons t ra t ed in Refs[15, 16]) , the s im ula t ion a lso def ines the t roch oidal geom etry a t the roo t o f the gear tee th .

F igure 6 shows how th is p roce dure was used wi th com pute r g raph ics so f tware [19] to s imula tea gear b lank ro l l ing over a rack cut ter.

The ana lys is o f Ref . [15 ] and the image o f F ig. 6 show tha t com pute r g raph ics can be used to

s imula te invo lu te gea r too th manu fac tu re , a s w i th a r ack cu t t e r o r hob cu t t e r. H owev er, the successof th is s imula t ion ra ises severa l ques t ions : ( I ) Can the s imula t ion b e extend ed to the man ufactu reof o ther to oth forms, tha t i s , to noninv olute to oth forms? (2) Ca n the s imula t ion be extended tothe ma nufac ture of o ther gear form s, for example , to bevel , sp i ra l-bevel and hy poid gears? (3) Ca na procedure be developed for s imula t ing surface genera t ion in genera l? (4) i s there an analyt ica lbas is for such s imula t ions? We address these ques t ions in the fo l lowing sect ions .

E N V E L O P E O F A R O L L I N G I N V O L U T E C U T T E R

To develo p an an alyt ica l bas is for the graphical s imula t ion, cons ider f irs t two rol l ing disks W,a n d W2 wh ose radi i r~ and r2 def ine p i tch c i rc les of ma t ing spur gears (Fig . 7). L et a cut ter shap edl ike an invo lu te too th be p laced on W2 and le t Gt be a gear b lank placed on W, . The c ut ter w i llleave a gear tooth impress ion (or foo tpr in t ) on the gear b lank as WI rolls wi th W:.

To o bta in an analyt ica l representa t ion o f the impress ion, it i s he lpful to in t roduce coo rdina teaxes f ixed in W, an d IV., , as sho w n in Fig. 8. Let ~- ~ a n d X - Y be Car tes ian axes sys tem s f ixedin Wt and W2 wi th or ig ins O~ and 02. Let r, be the angle betwe en ) ( and X as WI rol ls on W2.

Fig . 6 A gear b lank ro l l ing over a r ack cu t t e r.


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Modeling gear tooth surfaces 719

a n d

a n d

y = (r, + r . )c os 02 - :~ s in a + ~ co s a

.~ = (r, + r2)s in( ' , - 02) + x c os - y sin a

(16)

.~ = - ( r t + rz)COS(~t - 02) + x sin ;t + y co s ,t. (17 )

Ob se rve t ha t . s i nce W t r o l l s on W2 , ; t and 0 : a r e no t i nd epen den t . I nd eed , f r om F ig . 8 we s eet h a t

= = 0 , + 0 2 . ( 1 8 )

T h e r o i l in g c o n d i t i o n r e q u i r e s th a t

rt O, = r20 :. (19)

H e n c e , b y s o l v in g f o r 02 a n d f o r a - 0 2 , w e h a v e

02 = r l= and = - 0 2 = R2___~ (20 )r l + rz r i + rz

Suppose that the graph of a funct ion y = f ( x ) descr ibes the cut ter prof ile in W2. Thenfrom equations (16), the cutter profile may be represented in W, (and, hence, also in gear blankG , ) as

y : c , y , = ) = A x . L : , = ) ] o r F ( . L . ~ , = ) = 0 . ( 2 1)

T h e r e p re s e nt a t i on f t h e c u t t e r p ro f i l e n G , t h u s d e p e n d s u p o n t h e r o l l n g l e =. H e n c e , f r o me q u a t i o n ( 3 ) , t h e e n v e l o p e o f t h e c u tt e r p r o fi l e n G , i s d e t e r m i n e d f r o m e q u a t i o n ( 2 1 ) a n d t h ee x p r e s s i o n

OF : c , . a ) I O = = 0 . ( 2 2 )

B y s u bs t i tu t i ng o r x a n d y f r o m e q u a t i o n s ( 1 6 ) i n t o e q u a t i o n ( 2 1 ) a n d b y u s i n g e q u a t i o n ( 20 ) ,w e h a v e

( r, + r 2 ) c o s ( i . ~ a '~ - . ~ s i n = + f c o s a= J r , . r , ) s , n _ _ _ _ + . ~ c o s a + . ~ s i n = . ( 23 )\ r , + r \ r , r21

T h e n b y p e r f o r m i n g t h e d i f f e r e n ti a t io n o f e q u a t i o n ( 2 2 ) , w e o b t a i n t h e ex p r e s s i o n

- r t s i n - . ~ c o s a - ~ s i n a = d -~ x - ~ r , c o s - . ~ s i n = + : c o s = . ( 2 4 )

B y u s i n g e q u a t i o n s ( 1 6 ) a n d ( 2 0 ) , e q u a t i o n ( 2 4 ) m a y b e e x p r e s s e d i n t e r m s o f x a n d y a s

d.f_ . + , . . J , . ., . + , , , , , L , ( 2 5 )

S u p p o s e t h a t t h e c u t t e r p r o f i l e is a n i n v o l u t e [a s i n e q u a t i o n ( 14 )] w i t h p a r a m e t e r ~ . T h e n f r o me q u a t i o n ( 1 5 ) ,d f / d x e q u a l s c o t 0 a n d e q u a t i o n ( 2 5) b e c o m e s

- x + r 2 s i n ( r , , '~ = = ( c o t ) [ y - r 2 c o t ( r , . , ' ~I . ( 2 6)\ r , + r z ] \ r l + r , . / I

M ult ip lyin g by cos ~ and rearrang ing terms resul ts in

x s i n O + y c o s O = r 2 co s ~ r ~ r 2 / ( 2 7 )

O b s e r v e , h o w e v e r , f r o m e q u a t i o n ( 1 4 ) t h a t

x s i n ~ + y c o s ~ = r . ( 2 8 )


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720 R. L Hcs-roNt a l .

Thus , we have

r ,~ ~ r , , 2 9 )cos 4 r ~ + - r , / = lo r 4 ' = r , + r 2

Final ly, consider the express ion of the involute cut ter prof ile in Ga: by subst i tu ting for x a nd yfrom equat ions (14) in to (17) and by using equat ion (20) , we obtain

. / ' ria '~. = ( r, r , ) s l n ~ , ~ ) r : s i n( 4 ' - - a ) - r , 4 ' c o s( 4 ' - - a ) ,

- (r ,+ r : ) c o s l ~ ) + r z c o s ( 4 - a ) + r 2in(4' - a) . (30)=

Observe f rom equat ion (24) that

\r~ + r2/ \ r d

where the parameter / / i s def ined by the f inal equal i ty. Hence, by subst i tu t ing into equat ion (30) ,we have

.? - - r, s i n / / - r,p cosp and .P - - - r l cos/~ - r,p s in// . (32)

by comp aring equat ions (32) wi th equat io ns (14) . we see that they have the same form. Therefo re ,the envelope (or footp r int ) o f the involute cut te r in G, is i tse l f an involute .

C O M P U T E R G R A P H I C S R E S U LT S

A co m pu ter graphics sof twa re system [19] was used to s imulate the gea r cut t ing process forseveral types of gears. Fi rs t, to s imulate an involute rack cut ter, a p las tic disk was rol led over a

ser ies of r ig id s t ra ight-s ided protrusions (Fig . 6). The disk w as exam ined and foun d to have involuteteeth cut in to i ts surface . Final ly, a s imilar pr oce dure w as used to gen erate images of hel ical, bevel,spira l bevel , and hypoid gears . Figures 9-12 show the s imulat ions .

D I S C U S S I O N

The discussion presented ear l ier on the envelop e of a family of curves es tabl ishes the basis fora compute r g raph ics p rocedure fo r s imula t ing gear too th genera t ion . The ana lys i s shows , fo rexam ple , that the envelope of an invo lute protrus io n is an involute . While th is i s not a new orunexpected resul t ( indeed, i t i s the basis of spur gear fabr icat ion by hobbing) , the analysises tablishes that the resul t i s based on the pr inciples of d i fferentia l geom etry.

The com pute r g raph ics p rocedu re desc ribed in th i s r epor t p rov ides a real is t ic s imula t ion o f gea r

generat ion processes. The o nly s impl i fying assum ptions involved are th e assumptio ns of a perfect lyr igid cut t ing tool and a perfect ly plas t ic workpiece (gear blank) .

The s ignif icance of th is resul t is that to oth form com plexi ty need no longer be a hindra nce tocom prehensiv e analysis and design; that i s, the com pu ter graphic p roce dure is unaffected by thecomplex i ty o f the too th fo rm whereas ana ly ti ca l p rocedures qu ick ly become in t rac tab le as thecom plexi ty increases. Hence , the proc edu re can be appl ied w ithout m odif icat ion for the analysiso f beve l gea r, hypo id gear, and non invo lu te , nons tandard gear too th fo rms .

The com pute r g raph ics p rocedure can a l so be used to de f ine the geo met ry fo r a f ini te -e lementmo del of a tooth form. Th e com bine d graphical and f ini te-e lement analysis i s thus a design toolfor s tudying st resses and d eform at ion s in too th forms, par t icular ly in the less well unders tood too thforms o f spi ra l beve l and hypo id gears. N oninvo lu te and nons tandard too th fo rms a re a l so read ily

accom mo dated . Indeed , the com pute r g raph ics p roc edure can be used to modi fy, deve lop andexam ine new form s of conjugate teeth . I t can a lso be used to s tudy f i llet geome try, t ip rel ief, dress ingopera t ions and con tac t pa tch geomet ry.

The c om pute r g raph ics p rocedure p rov ides accura te s imula t ions in a shor t t ime . F or example ,the s imulat ions shown in Figs 9-12, a l though they have only two or three teeth , are general ly


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Mo del ing g~ar too th sur faces 721

Fig. 9 . Hel ical gear s imulat ion.

Fig. I0. Bevel gear s imula t ion.


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Fig I I Spiral bevel gear simulalion


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Mod eling gea r tooth surfaces 723

s u f fi c ie n t f o r s t r es s a n a l y s i s ; n o n c o n t a c t i n g t e e t h h a v e l i tt le e f fe c t u p o n t h e s t r e ss e s o f c o n t a c t i n gt e e th . T h e r u n t i m e fo r a s p i r a l b e v e l g e a r s i m u l a t i o n i s a p p r o x . I h o n a n H P m o d e l 3 5 0 c o m p u t e r.

T h e c o m p u t e r g r a p h i c s p r o c e d u r e c a n a l s o b e a p p l i e d t o t h e m a n u f a c t u r e o f t o o t h s u r fa c e s.To o t h f o r m s f r o m a r b i t r a r y c u t t e r s h a p e s c a n b e p re d i c te d . C o n v e r s e l y, c u tt e r s h a p e s p r o d u c i n ga d e s i r e d t o o t h f o r m c a n b e d e t e r m i n e d . S p e c if i ca l ly, t h e c u t t e r p r o f il e f o r a n y g i v e n t o o t h f o r mi s t h e c o n j u g a t e o f t h a t t o o t h f o r m . T h i s c o n j u g a t e c a n b e g e n e r a t e d u s i n g c o m p u t e r g r a p h i c s b y

r o l l in g a b l a n k d i s k o v e r t h e d e s i r e d t o o t h f o r m . T h a t i s, t h e s h a p e o f th e c u t t e r p r o f i l e i s t h e s a m ea s th e i m p r e s s i o n l e ft o n t h e b l a n k b y t h e d e s i r e d t o o t h f o r m .

F i n a l l y, t h e c o m p u t e r g r a p h i c s p r o c e d u r e c a n b e e x t e n d e d t o s t u d y s u r fa c e g e n e r a t i o n i n g e n e r a l,a n d a p p l i c a t i o n s c a n b e m a d e t o c a m d e s i g n , b e a r i n g d e s i g n a n d s u r f a c e d e s i g n o f a r b i t r a r y r o l li n ge l e m e n ts . A n a l o g o u s p r o c e d u r e s m a y b e d e v e l o p e d f o r s i m u l a t in g t h e m a n u f a c t u r e o f s ki n s u r fa c e ss u c h a s a u t o m o b i l e f e n d e r s , a i r fo i l s a n d s h i p h u ll s . I n d e e d , t h e p r o c e d u r e s m a y b e d e v e l o p e d t od i r e c t ly g o v e r n t h e m a n u f a c t u r e o f t h es e s u r fa c e s .

C O N C L U S I O N S

A n a n a l y s i s o f t h e e n v e l o p e s o f f a m i l i e s o f c u r v e s , b a s e d o n d i f fe r e n ti a l g e o m e t r y, w a s p r e s e n t e da s th e f o u n d a t i o n f o r a s o li d m o d e l i n g p r o c e d u r e . C o m m e r c i a l c o m p u t e r g r a p h i c s s o f t w a r e w a su s e d t o s i m u l a t e g e a r t o o t h g e n e r a t i o n . T h e c o n c l u s i o n s a r e

I . A n a n a l y ti c a l b a s is f o r n u m e r i c a l c o m p u t e r g r a p h i c m o d e l i n g o f g e a r t o o t hs u r f a c e s h a s b e e n e s t a b l i s h e d .

2 . T h e c o m p u t e r g r a p h i c p r o c e d u r e c a n b e u s e d f o r t h e m o d e l i n g a n d a n a l y s e s o fa w i d e v a r i e t y o f t o o t h f o r m s , i n c l u d i n g s p u r g e a r s , h el ic a l g e a rs , b e v e l g e a rs ,s p i r a l b e v e l g e a r s , h y p o i d g e a r s a n d n o n s t a n d a r d g e a r s .

3 . T h e p r o c e d u r e m a y b e e x t e n d ed t o s t u d y t h e de s ig n a n d m a n u f a c t u r e o f ro l li n gs u r f a c e s i n g e n e r a l a n d , u l t i m a t e l y, s k i n s u r f a c e s .

Acknowledgement - -Researchsumm arized in this paper was partially supported b y the National A eronautics and SpaceAdm inistration und er Grant NSG -3188 to the University of C incinnati and is gratefully acknowledged.

R E F E R E N C E SI. M. L . Baxter Jr,J. Mech. Des.tO0, 41-44 (1978).2. M. L. Bax ter Jr,Ind. Math. II , 19-43 (1961).3. R. L. Huston and J. J. Coy.J. Mech. Des.103 , 127-133 (1981).4. R. L. Huston, V. Lin and J. J. Coy,J. Mech. Transm. Automn Des.105, 132-137 (1983).5. F. L. L itvin, NA SA TM-7513 0 (1977).6. F. L. Litvin, N. N. K rylov and M. L. Erikhov,Mech. Mach. Theory10, 365-373 (1975).7. F. L. Litvinet al., J. Mech. Transm. Auto mn D es.109, 163-170.8. F. L. Litvin, P. Rah ma n and R. N. G oldrich, NA SA CR-3553 0982).9. Y. C. Tsai and P. C. Chin,J. Mech. Transm. Automn Des.109, 443 -449 (1987).

10. F. L. Litvin and J. Z hang , NASA CR-4224 0989).II. F. L. L itvin W .-J. Tsung and J. J. Coy, N ASA TM-87273 0987).12. F. L. Litvinet al., SAE Paper 851573 0985).13. F. L. Litvin, W..J. Tsun g and H.-T . Le e, NA SA CR.4088 0987).14. J. A. R emscn and D. R. Carlson, Gleason Publication No. AD 1906, Gleason W orks (1968).15. S. H. Chang , R. L. H uston and J. J. Coy, ASM E Paper 84-DET-184 0984).16. R. L. H usto n, D. M avriplis and F. B. Oswald.Int. Power Transmission and Gearing Conf. 5th A S M E,pp. 539 -545

1989).17. W. C. Graustein,Differential Geometry Macmillanp. 64 (1935).18. D. J. Struik,Lectures on Classical Differential Geometrypp. 73, 167. Addison-Wesley, Reading , Mass.19. I. D E A S (Integrated Design and Engineering Analysis Software).Structural Dynam ics Research Corporation (SDRC ),

Milford, Oh io 0985).

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