The Study of Gearing Line Limits of the Cycloid Gear With Roller Teeth

Transcription

1 ISSN , Volume 2013 No.XXVII The Study o Gearng Lne Lmts o the Cyclod Gear Wth Roller Teeth Dăscălescu, Anamara Abstract: From dynamc pont o vew, at the cyclod gear transmsson s not ratonal to utlze the entre length o the cyclod prole o the satellte gear. The avorable nterval o the cyclod prole o a tooth s lmted by the tooth addendum crcle wth r a radus and the tooth dedendum crcle wth r. radus. The dynamc avorable nterval represent 45% rom entre cyclod prole length o a satelte gear tooth. Keywords: gearng lne, pressure angle, reacton values. 1 INTRODUCTION The study o the gearng lne lmts o the cyclod gear wth roller teeth ncludes determnatons on two types o cyclod reducer wth roller teeth: M1C-23 wth gear rato r = z 1 = 17, number o roller teeth z 2 = 18, eccentrcty a = 2,4 [mm], drvng engne power P = 1,1 [kw], the drvng engne nomnal speed n m = 1500 [rot/mn] and the roller tooth dameter d bo = 10 [mm], respectvely RPR 150 wth gear rato r = z 1 = 25, number o roller teeth, z 2 = 26, eccentrcty a = 2 [mm], drvng engne power P = 1,5 [kw], the drvng engne nomnal speed n m = 1500 [rot/mn] and the roller tooth dameter d bo = 12 [mm]. The pressure angle between a roller tooth and the cyclod prole represents an essental ndcator on the estmaton process o the energetc lux transmsson (orces). The satellte gear cyclod prole optmzaton takes nto account the varaton o ths angle durng a cycle o gearng o a roller tooth. When the pressure angle s zero, or has small values, the energetc lux transmsson s carry out n good condton. Based on the transmsson angle γ(ϕ) varatons graphs, we conclude that the central nterval o the cyclod prole o a tooth, between , s avorable to the transmsson o the orces and the begnnng/endng part o the cyclod prole o a tooth corresponds to some transmsson angle unacceptable or the orces. We conclude also that the entre gear lne utlzaton, rom ϕ= , s not possble because we see the pressure angle γ tends to 90 0 on the extremty parts o the gearng lne, wtch proves that exsts blockng stuatons. The analyze o the R 12, R 32, R 42 reactons varaton graph shows us that, theoretcally, the reactons are nnte at the begnnng/endng o the gearng o the sun gear roller tooth wth the satelte gear cyclod prole The reactons decrease n the central part o the actve prole and, practcal, became constant on the nterval ϕ= By dynamcal pont o vew, on the orces transmsson s not ratonal to use the entre length o the cyclod prole. The cyclod prole avorable nterval to transmt the orces s delmtated by the tooth addendum crcle wth r a radus and the tooth dedendum crcle wth r radus. In the paper, we present the mathematcal expressons o the tooth addendum crcle radus, respectvely tooth dedendum crcle radus. These crcles lmt the optmal cyclod prole consderng the orces transmsson avorable angle. The dynamc avorable nterval represent 45% rom entre cyclod prole length o a satelte gear tooth. We obtan ths shorten length o the gearng lne by supplementary manuacturng process o the begnnng / endng part o the tooth actve prole. To use only the dynamc avorable doman o the cyclod prole s possble by elmnaton o the begnnng / endng part o the tooth actve prole through decreasng the tooth addendum crcle dameter, respectvely through deepenng the tooth dedendum prole. 2 THE PRESSURE ANGLE AND THE GEARING LINE At the cyclod gear, the pressure angle γ o the roller tooth and the cyclod prole s ormed between the normal drecton o the CA segment n the gearng pont o a roller tooth and the lne AD, perpendcular to the centre lne drecton (g.1), (1). When the pressure angle s zero, or has small values, the energetc lux transmsson s carry out n good condton. r3 cosϕ rg γ = arc tg (1) r sn In the case o the cyclod gear wth X-prole, the gearng lne don t pass through the gearng pole (extrapole gearng), (g. 1). We consder the begnnng o the gearng lne n the contact pont o the roller tooth wth the tooth dedendum o the satellte gear. Is stuated at ρ mn =r 3 -r g -r bo dstance rom the pole C. The endng o the gearng lne s n the contact pont o the roller tooth wth the tooth addendum o the satellte gear. Is stuated at ρ max = r 3 +r g -r bo dstance rom the pole C. The gearng lne s ponts are stuated on the normal lne o the cyclod prole, n the contact ponts o the roller tooth wth the cyclod prole, at r bo dstance rom the pont B o the theoretcal prole. In the coordnate system xated on the sun gear wth roller teeth, the parametrc equatons o the gearng lne ponts coordnate are: xa = rg + ( ρb rbo )snγ (2) y = ( ρ r )cosγ (3) 3 ϕ A B bo 18

2 ISSN , Volume 2013 No.XXVII Fg.1. The pressure angle and the gearng lne 3. THE ANALYZE OF THE VARIATION OF THE REACTIONS WHO ACTS ON THE SATELLITE GEAR. THE ANALYZE OF THE PRESSURE ANGLE Based on the determnaton relatons o the reactons values, we present the varaton o the R 12 (ϕ), R 32 (ϕ) ş R 42 (ϕ) reactons who act on the satellte gear, calculated n MathCAD utltaran program, or derent poston o the satellte gear expressed by the rotaton angle o the centrodes tangent pont, ϕ,. (g. 2). Fg. 2. The R 12 (ϕ), R 32 (ϕ) ş R 42 (ϕ) reactons graphs who act on the cyclod satellte gear - ccloyd reducer M1C-23 19

3 Fg. 3. The varaton o the pressure angle correspondng to M1C-23 Fg. 4. The R 12 (ϕ), R 32 (ϕ) ş R 42 (ϕ) reactons graph who act on the cyclod satellte gear - cyclod reducer RPR 150 Fg. 5. The varaton o the pressure angle correspondng to RPR

4 We present the graphs o the R 12, R 32, R 42 reactons varatons, gures 2 and 4, calculate n hypothess I (each satellte gear gears wth a roller tooth and a thumb -o the transversally couplng- on the entre length o the gearng lne). Analyzng the reactons graphs, we conclude that, n the begnnng/endng o the roller tooth gearng wth the cyclod prole, theoretcal the reactons are nnte and decrease on the central nterval o the actve prole, practcal beng constant on the nterval ϕ= The accepted values o the transmsson angle we consder to be untl 50 o, and these accepted values are stuated between the lmts 12,951 o and 112,951 o o the angle ϕ or the rst analyzed reducer, respectvely 16,768 o and 116,768 o o the angle ϕ or the second analyzed reducer. Analyzng the graphs o transmsson angle varatons γ(ϕ) (g. 3 and 5), we consder that the central nterval o the cyclod prole lmted by s avorable to the orces transmsson. The begnnng / endng o the cyclod prole correspond to some transmsson angles unacceptable or the orces. The usage o the entre length o the gearng lne, rom ϕ= , s not ratonal, because, at the endng part o the gearng lne, the pressure angle ncrease to 90 0 (g. 3 and 5) showng us the exstng possbltes o blockng. We conclude that, rom dynamc pont o vew, to transmt the orces s not ratonal to utlze the entre length o the cyclod prole From pont o vew o the transmsson o the orces, the avorable nterval o the cyclod prole s lmted by the tooth addendum crcle wth r a radus and the tooth dedendum crcle wth r. radus. We present the determnaton relaton o these crcles radus values (4.) Porţunle prolulu cclodal avantajoase dn punct de vedere al transmter orţelor sunt delmtate de cercul de cap de raza r a ş cercul de pcor r. r = x + y (4) a, Aa, Aa, where: x Aa and y Aa, respectvely x A and y A, are the coordnate o the gearng entrance pont, respectvely the coordnate o the gearng ext pont o a tooth. ( 2 cosϕ ) 3 3, x = r a + r + r r r r Aa, g g g a bo r cosϕ 3 a, r r + r 2r r cosϕ 3 g 3 g a, g (5) a, ( 2 cosϕ ) 3 3, y = r + r r r r A g g a bo r snϕ 3 a, r + r 2r r cosϕ 3 g 3 g a, (6) In gure 6 we present, wth thck lne, the eectvely length o the gearng lne between the lmts ϕ and ϕ a +ϕ, The actve nterval o the cyclod prole corresponds to a-b nterval to the drecton o rotaton, respectvely c-d to the nverse drecton o rotaton. The dynamc avorable nterval represent 45% rom entre cyclod prole length o a satelte gear tooth. We obtan ths shorten length o the gearng lne by supplementary manuacturng process o the begnnng / endng part o the tooth actve prole. To use only the dynamc avorable doman o the cyclod prole s possble by elmnaton o the begnnng / endng part o the tooth actve prole through decreasng the tooth addendum crcle dameter correspondng to b-c nterval, respectvely through deepenng the tooth dedendum prole correspondng to d-e nterval. The cyclod reducer has multple advantages that are not ound n any other type o reducer: the reducng report s untl 119:1 n one step, the mpute pour s between 30W and 200kW, the gear transmsson ecency on a reducng level s untl 98% because the rollng o the elements, a robust desgn, the shats algnment and perectly symmetrcal poston o all peces ensure a smooth and slence uncton wth very low level o vbraton, the centruge orce are compensate by usng two satelltes gear dspose at or three satellte gear dspose at The dsposal o the shats on the same axe allows a drect transmsson o the power and ast assembly. The most mportant dsadvantage o the cyclod gear reducer s the complex orm o the cyclod gear prole wtch nvolves the necessty o hgh precson manuacturng. A hgh precson manuacturng technology mples precson machnes tools and expensve tools. Also, an mportant dsadvantage s that the modcaton o axal dstance modes the nstantaneous transmsson report. All the advantages, plus the low gauge and mass to the same power and the same transmsson report as n a tradtonal gear reducers and the extended perod o workng hours wthout reparng n heavy work condtons, recommends ths type o gear reducer. 21

5 Fg. 6. The lmts o the gearng lne 4. RESULTS AND CONCLUSIONS Analyzng the graphs o R 12, R 32, R 42 reactons varatons, gures 2 and 4, calculate n hypothess I (each satellte gear gears wth a roller tooth and a thumb -o the transversally couplng- on the entre length o the gearng lne), we conclude that, n the begnnng/endng o the roller tooth gearng wth the cyclod prole, theoretcal the reactons are nnte and decrease on the central nterval o the actve prole, practcal beng constant on the nterval ϕ= Analyzng the graphs o transmsson angle varatons γ(ϕ), we consder that the central nterval o the cyclod prole lmted by s avorable to the orces transmsson. The begnnng / endng o the cyclod prole correspond to some transmsson angles unacceptable or the orces. The usage o the entre length o the gearng lne, rom ϕ= , s not ratonal, because, at the endng part o the gearng lne, the pressure angle ncrease to 90 0 showng us the exstng possbltes o blockng. We conclude that, rom dynamc pont o vew, to transmt the orces s not ratonal to utlze the entre length o the cyclod prole From pont o vew o the transmsson o the orces, the avorable nterval o the cyclod prole s lmted by the tooth addendum crcle wth r a radus and the tooth dedendum crcle wth r. radus. REFERENCES [1] Chmurawa, M.,John, A., (2000) Numercal analyss o stress and stran dstrbuton n planetary wheel o cyclodal gear usng FEM. In: Numercal Methods n Contnuum Mechancs, Lptovsky Jan, Slovak Republc. [2] Chmurawa, M., Lokec, A., (2001) Dstrbuton o loads n Cyclodal Planetary Gear (CYCLO) ncludng modcaton o equdstant. In 16 th European Mechancal Dynamcs mscsotware.com/support/lbrary/con. [3] Dăscălescu, A.: (2005) Contrbuton to the Knematcs and Dynamcs Studess o the Planetary Gears wth Cyclod Toothng and Roller Teeth, PhD Theses, Techncal Unversty o Cluj-Napoca, [4] Dăscălescu, A.: (1998) Research on the reactons varaton n knematcs pars o the cyclodal gear reducer, In Proceedngs McroCAD 98 Internatonal Computer Scence Conerence, February 26-27, Mskolc, vol. I, pp [5] Janjun, Z., Zchen, C.,: (2000) Contact stress analyss n planetary cyclodal ball reducers. The 4 th nternatonal Conerence on Fronters o Desgn and Manuacturng, pp , Hangzhou, Chna. Author address 1 Dăscălescu, Anamara Assoc. Pro. Ing. PhD, Techncal Unversty o Cluj-Napoca North Unversty Center o Baa Mare, RO Baa Mare, Dr. V. Babeş St. 62A, e-ma:l anamara.dascalescu@gmal.com 22