Generation of Crowned Parabolic Novikov gears

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1 Engneerng Leers, 5:, EL_5 4 Generon o Crowned Prol Novkov gers Somer M. Ny, Memer, IAENG, Mohmmd Q. Adullh, nd Mohmmed N.Mohmmed Asr - The Wldher-Novkov ger s one o he rulr r gers, whh hs he lrge on re eween he onvex nd onve proled mng eeh. In (June 8, 999), new geomery o W-N ger wh prol prole n norml seon hs een developed. Ths pper sudes he generon o rk-uers or prol rowned proles wh s generon n order o sele he requremens o W- N gers. where u s vrle prmeer h deermnes he loon o he urren pon n he norml seon nd s he prol oeen. Index Terms- rowned prol prole, generon o gers, Novkov gers I. INTRODUCTION Crulr r hell gers were proposed y Wldher nd Novkov. However, here s sgnn derene eween he des proposed y he prevously menoned nvenors. Wldher s de [] s sed on generon o he pnon nd ger y he sme mgnry rk-uer h provdes onjuge ger ooh sures h re n lne on every nsn. Novkov [] proposed he pplon o wo msmhed mgnry rk-uers h provde onjuged ger ooh sures h re n pon on every nsn. Pon on o Novkov gers hs een heved y pplon o wo msmhed rk-uers or generon o he pnon nd ger, respevely. There re wo versons o Novkov gers (wh rulrr prole), he rs hvng one zone o meshng, nd he oher hvng wo zones o meshng. The desgn o gers wh wo zones o meshng ws n emp o redue hgh endng sresses used y pon on. The proposed new verson o hell gers s sed on he ollowng des [3]:. The erng on s lolzed nd he on sresses re redued euse o he ngeny o onveonvex ooh sures o he mng gers.. The norml seon o eh rk-uer s prol s shown n Fg.. A urren pon o he prol s deermned n n uxlry oordne sysem y he equons u x = u y = () Mnusrp reeved Jnury 7, 007. S. M. Ny s wh Al-Khwrzm College o Engneerng, Unversy o Bghdd, Bghdd, Irq. Phone e-ml: smny@elernngq.ne M. Q. Adullh s wh he College o Engneerng, Unversy o Bghdd, Bghdd, Irq. M. N. Mohmmed s wh Al-Khwrzm College o Engneerng, Unversy o Bghdd, Bghdd, Irq. S Fg. - Prol prole o rk-uer n norml seon. II. DERIVATION OF PINION TOOTH SURFACE A. Pnon Rk-Cuer Sure The dervon o rk-uer sure s sed on he ollowng proedure:. The norml prole o s prol nd s represened n oordne sysem equons h re smlr o (): [ u u 0 ] T S, Fg. -, y r ( u ) = () where s he prol oeen; u s he vrle prmeer.. The norml prole s represened n y mrx equon r u ) = M r ( u ) (3) ( M ndes he 4 4 mrx used or he oordne rnsormon rom oordne sysem S o S [4]: 3. Consder h rk-uer sure s ormed n S whle oordne sysem S wh he norml prole perorms rnslonl moon n he dreon - o he skew eeh o he rk-uer, Fg. 3. Sure s S (Advne onlne pulon: 5 Augus 007)

2 Engneerng Leers, 5:, EL_5 4 omponens; k s he un veor o xs z. The rnsverse seon o rk-uer s shown n Fgs. 4- nd. Fg. 4- Rk-uer rnsverse proles. () Mng proles. () Pnon rk-uer prole. () Ger rk-uer prole. Fg. 3- For dervon o pnon rk-uer sure deermned n oordne sysem n wo-prmeer orm y he ollowng mrx equon r u, θ ) = M ( θ ) r ( u ) ( (4) 4. The norml N o rk-uer sure s deermned y mrx equon, [4] N u ) = L L N ( u ) ( S r N ( u ) = k u (6) nd he un norml o he sure s N N ( u ) n ( u ) = = (7) N + 4 u where o M L (5) ndes he 3 3 mrx h s he su-mrx nd s used or he rnsormon o veor B. Deermnon O Pnon Tooh Sure The deermnon o s sed on he ollowng onsderons:. Movle oordne sysems nd S, Fg. 5, re S rgdly onneed o he pnon rk-uer nd he pnon, respevely. The xed oordne sysem s rgdly onneed o he ung mhne.. The rk-uer nd he pnon perorm reled moons, s shown n Fg. 5, where s = rp ψ s he dsplemen o he rk-uer n s rnslonl moon, nd ψ s he ngle o roon o he pnon. 3. A mly o rk-uer sures s genered n oordne sysem nd s deermned y he mrx equon r u, θ, ψ ) = M S ( ( ψ ) r ( u, θ ) (8) M ( ψ ) = Mm M m (9) The pnon ooh sure s genered s he envelope o he mly o sure r ( u, θ, ψ ).Sure s deermned y,, ψ ) 0 (0) p ( u θ = smulneous onsderon o veor unon r,, ) nd he so-lled equon o meshng. ( u θ ψ S m (Advne onlne pulon: 5 Augus 007)

3 Engneerng Leers, 5:, EL_ To derve he equon o meshng (0), pply he heorem o [5] nd [4] o on,. The norml prole o s prol represened Fg. 5- Generon o pnon y rk-uer O O = rp + rp j () ψ () () v = ω rp j where ω = k () () () = ω r + O O ω v (3) The relve veloy s v = v v = ω [( rp ψ y ) + x ] (4) Thus, he equon o meshng s N v = 0 (5) Th yelds u, θ, ψ ) ( rp ψ y ) N + x N 0 (6) ω ( = x y = where ( x, y, z ) re he oordnes o urren pon o ; ( N ) s he norml o he sure ; ω s he ngulr veloy; v nd v re he veloes o he rkuer nd pnon respevely; represen he relve veloy (sldng veloy) eween he rkuer nd pnon, Fg. 5. Equons (8) nd (6) represen he pnon ooh sure y hree reled prmeers. Tkng no oun h he equons ove re lner wh respe o θ, hene θ my e elmned nd represen he pnon ooh sure y veor unon r, ). v ( u ψ III. DERIVATION OF GEAR TOOTH SURFACE n e S e, reerrng o Fg. -, [ u u 0 ] T r ( u ) = (7) whh s smlr o (). Use oordne sysems, Fg. -, nd S h re smlr o S nd S, Fg. 3, o represen sure y mrx equon, S r u, θ ) = M ( θ ) M r ( u ) (8) ( k ke e The norml o he sure s deermned y equons smlr o (5) o (7). The derene n he represenon o s he hnge n he susrp o. B. Deermnon O Ger Tooh Sure The generon o y rk-uer sure s represened shemlly n Fg. 6. The rk-uer nd he ger perorm reled rnslonl nd roonl moons desgned s s = rpψ nd ψ. The ger ooh s represened r = r,, ) ( u θ ψ (9) ( u, θ, ψ ) = 0 (0) Equon (0) represens n S S k he mly o rk-uer sures deermned s, r u, θ, ψ ) = M ( ψ ) r ( u, θ ) () ( M ψ ) = M ( ψ ) M ( ) () ( m m ψ A. Ger Rk-Cuer Sure The dervon o rk-uer sure s sed on he proedure smlr o h ppled or dervon o Fg. 6- Generon o ger y rk-uer (Advne onlne pulon: 5 Augus 007)

4 Engneerng Leers, 5:, EL_5 4 The dervon o he equon o meshng (0) my e omplshed smlrly o h o (6), u, θ, ψ ) ( rp ψ y ) N + x N 0 (3) ( = x x = Equons () nd (3) represen he ger ooh sure y hree reled prmeers. The lner prmeer θ n e elmned nd he ger ooh sure represened n wo-prmeer orm y veor unon r, ). ( u ψ IV. MATHEMATICAL SIMULATION OF RACK FILLET A lle pr s rulr r whh hs oordnes x nd y, k nd h ener oordnes nd rdus r. Ths r les eween pons A nd B, where pon A represen he mng pon, whh ssy smoohng on, eween he rulr-r or prol urve nd he lle urve, pon B represen he meeng pon, whh ssy smoohng on, eween he lle urve nd he horzonl srgh lne, s shown n Fg. 7. Thereore, o nd he x nd y-oordnes o pons A nd B, nglesθ A nd θ B whh ssy smoohng on mus e ound. To ssy smoohng on pon B, ngleθ my o equl o 90 euse he rdus o lle my e perpendulr on he ngen, whh s he horzonl srgh lne hs pon, lso o ssy smoohng on pon A he lle rdus my e perpendulr on he ngen hs pon, or n oher words he slope o rulr-r or prol urve mus e equl o he slope o he lle urve hs pon, [6]. Thus, o nd he slope o prol urve ny pon, usng (3), o ge, dx dx du os( α n ) u sn( α n ) = * = (4) dy du dy sn( α ) + u os( α ) n n B Also o nd he slope o lle urve ny pon, he rle equon s:- ( y h) y + ( x k) h y + h + x = r k x + k = r (5) derenng (5) wh respe o y s ollows: dx dx dx dx y h + x k = 0 h = y + x k (6) dy dy dy dy y susung (3), (4) nd (6) n (5), hus onng non-lner equon whh s solved numerlly usng Sen Mehod o ge he oordnes o smoohng pon (A). For he rowned prol prole, he lle urve n e represened s ollows: x r sn ( θ ) + xo r = = y r os( θ ) + yo (7) z 0, r s he lle rdus; x o, y ) re he r ener oordnes; ( o θ s he vrle prmeer. Usng oordnes sysems smlr o S nd S, Fg. 3, o represen sure S, he susrp mens he lle sure, hus r θ, θ ) = M ( θ ) r ( θ ) (8) ( The un norml o he sure n e ound s r r N N = nd n = (9) θ θ N The represenon o he pnon ooh lle sure y equons smlr o (8) nd (6), nd lso he represenon o he ger ooh lle sure y equons smlr o () nd (3). Then he generon o pnon nd ger or prol proles wh lle rdus n e oned s shown n Fg. 8. Fg. 7- Flle pr o rk-uer. Fg. 8- Generon o prol ooh wh lle rdus. (Advne onlne pulon: 5 Augus 007)

5 Engneerng Leers, 5:, EL_5 4 M, Mres o oordne rnsormon rom V. CONCLUSIONS The developed pproh o desgn nd generon o he rowned prol Novkov ger drves hs suessully een ppled. The onjugon o ger ooh sures wh prole rownng s heved y pplyng wo rk-uers wh rowned prole n norml seon. REFERENCES [] Wldher, E., "Hell Gerng", U.S. Pen No.,60,750, 96. [] Novkov, M.L., U.S.S.R., Pen No. 09,750, 956. [3] Lvn, F.L., Feny, P., nd Serge A. L., "Compuerzed Generon nd Smulon o Meshng o New Type o Novkov-Wldher Hell Gers", NASA/CR , 000. [4] Lvn, F.L., "Ger Geomery nd Appled Theory", Prene-Hll, Englewood Cls, NJ, 994. [5] Lvn, F.L, "Theory o Gerng", NASA RP- (AVSCOM 88- C-035), 989. [6] Mohmmed Qsm Adullh, "Compuer Aded Grphs o Cylodl Ger Tooh Prole", Unversy o Bghdd, Fh Engneerng Conerene, 003. Nomenlure Prol oeens o proles o pnon rk uer (=) nd ger rk uer (=). j Equon o meshng eween ooh sure () nd rkuer (j). l Prmeer o loon o pon ngeny Q or pnon (=) or ger (=). j Lj S o oordne sysem S j. ( j) ( j) n, N Un norml nd norml o sure n oordne sysem S j. r Poson veor o pon n oordne sysem S. r Flle rdus. rp s (=). Rdus o ylnder o pnon (=) or or ger (=). Dsplemen o rk-uer or pnon (=) or or ger S (O,x,y,z ) Coordne sysem (=,,p,g,,,m,,,,s,,r,k,e) α n β Helx ngle. φ Pressure ngle n Norml seon. Sures (=,,p,g,,). Angle o roon o he pnon (=) or he ger (=) n he proess o generon or rulr-r prole. ψ Angle o roon o proled-rowned pnon (=), he doule rowned- prole (=p) or or ger (=) n he proess o generon or rulr-r prole. ( u, θ ) Prmeers o sure. θ A, θ B Angles whh ssy smoohng on urves. ρ Prole rd (=p,g,,,,). (Advne onlne pulon: 5 Augus 007)

Abhilasha Classes Class- XII Date: SOLUTION (Chap - 9,10,12) MM 50 Mob no

Abhilasha Classes Class- XII Date: SOLUTION (Chap - 9,10,12) MM 50 Mob no hlsh Clsses Clss- XII Dte: 0- - SOLUTION Chp - 9,0, MM 50 Mo no-996 If nd re poston vets of nd B respetvel, fnd the poston vet of pont C n B produed suh tht C B vet r C B = where = hs length nd dreton