Trajectory Research about the Rolling-Pin Belt Transmission

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1 he Ope Mechaica Egieeig Joua, 2012, 6, (Supp 1-M ajecto Reseach about the Roig-Pi Bet asmissio Ope Access Huiog Zhao * ad Qigog Zhag Hubei Uivesit of Automotive echoog, Shia, , Chia Abstact: A ew bet tasmissio mechaism, havig the fuctio of o-sip divig, is descibed i the pape. It is composed of pues with caved ais ad bet with oed pis spaced out o both sides. Fist, the tajecto cuve is deduced aog which pis embedded i the pue. Ad the, paametic eatioships ae estabished, which ae about cete offsets of two pues, wokig adius of each pue, the umbe of meshig cuves o eve pue ad the umbe of pis o the bet. A of those povide a theoetica basis fo the desig of this tpe of o-sip bet tasmissio mechaisms. Kewods: Roig pi, sepaate cuve, bidig cuve, mechaism. INRODUCION paticua defied, coodiate sstem S =[O ; x,] is the eft pue coodiate sstem, S =[O ; x,] is that of the ight oe [8]. Coodiate sstem S=[O; x,] is a fixed oe whose oigi supeimposes with the eft pue s cete, ad S =[O ; x,] is aothe fixed coodiate sstem whose oigi ocates at the ight pue s cete. α ad α ae the ages which descibe x-axis to the ies coectig the iitia itegatio poit with the oigi of pue. So the foowig eatio fomuas ca be obtaied immediate. ( " = acsi (1 a Fig. (1. Achitectue diagam of bet tasmissio. With the pioities of high efficiec ad compact aout, mechaica bet tasmissio pas a ve impotat oe i mechaica egieeig, ad is wide used i automobie egies, geaboxes ad othe powe tasmissios [1-3]. As the taditioa bet dive beogs to the fictio dives, thee ae ma we-kow pobems such as ow tasmissio powe, high sippig ate duig tasmissio, easi faiig ad so o. Schoize gea-bet is a geea used ateative wa to sove it [4-7]. Howeve, the stegth of the schoize gea-bet is imited to a ve ow eve. heefoe, this pape poposes a ew fexua tasmissio mechaism to impove the efficiec ad tasmitted powe, ad aso to avoid eastic sip b embeddig the oig pis i the bet. 1. INRODUCION O HE NEW BEL DRIVE SRUCURE As show i Fig. (1, the bet is assembed with two pues ad the bet is embedded with eve distibuted oig pis. he cete coodiate sstem o each pue is *Addess coespodece to this autho at the Depatmet of Automotive Egieeig, Hubei Uivesit of Automotive echoog, Shia, , Chia; e: (office, (Mobie; Fax: ; E-mais: zh9823@ahoo.com.c, @qq.com X/12 whee, α deotes the age betwee bet ad x-axis i coodiate sstem S, ad epesets wokig adius of eft pue ad the ight oe espective, a idicates distace betwee the cete of eft pue ad that of the ight pue [9, 10]. Poit P (x 0p, 0p is the iitia sepaatio poit of the eft pue, P (x 0p, 0p is the simia poit of the ight oe. Fom the simpe geometic eatioship, coodiates of poit P ad P ae descibed i the foowig fomuas. x0 p = " si( (2 0 p = (3 x0 p = " si( ( Betham Ope

2 74 he Ope Mechaica Egieeig Joua, 2012, Voume 6 Zhao ad Zhag 0 p = " (5 whee, x 0p ad 0p is x-coodiate ad -coodiate of poit P i coodiate sstem S, x 0p ad 0p ae x-coodiate ad - coodiate of poit P i coodiate sstem S, espective. I ode to fid the eatioship betwee α ad α, geometic eatio betwee poit G (x 0g, 0g ad coodiate sstem S is show i Fig. (2 whee α is epeseted b fomua (6. Simia, eatioship betwee α ad α is specified with fomua (7. whee, x 0g ad 0g ae the coespodig x-coodiate ad - coodiate of poit G i coodiate sstem S, x 0g ad 0g ae the coespodig x-coodiate ad -coodiate of poit G i coodiate sstem S. 2. LEF PULLEY CURVE EQUAION 2.1. Equatio of the Left Pue Sepaatio Cuve I Fig. (3, ω is the agua veocit of the eft pue, poit P 1 (x 1p, 1p ad poit P ae ew paces of poit P o bet ad pue i the fix coodiates sstem S at time t, ad θ epesets the vaue of " P O P 1. Whie the oig pi us fom the sepaatio poit P to the poit P 1, poit P 1 o the pue otates to the poit P 1 which supeimposes with poit P 1 i coodiate sstem S, ad poit P 1 o the bet evoves to the poit P i coodiate sstem S. So the eatios about ω ad θ ae epeseted i the foowig fomuas. 2 2 ( " = (12 t 0 Fig. (2. Geometic eatio diagam. = acsi + (6 = acsi " (7 whee, is the adius of each pue. G (x 0g, 0g is the iitia itegatio poit of the eft pue, G (x 0g, 0g is the coespodig poit of the ight oe. Immediate, coodiates of poit G ad poit G ca be obtaied i the foowig fomuas. x0 g = (8 0g = " si ( (9 x0 g = " (10 0g = si ( (11 t 180 = " (13 # whee, t 0 is the tota sepaatio time, t deotes uig time duig sepaatio pocess. Fom kiematics eatioship, the speed of oig pi is. Accodig to the geometic eatioship, coodiate cacuatig eatioships of poit P 1 ae descibed as: x1 p = x0 p + t " (14 = " t si( (15 1 p 0 p + whee, x 1g ad 1g ae the coespodig x-coodiate ad - coodiate of poit P i coodiate sstem S. At this time, " P O P 1 is the age that coodiate sstem S otates eative to coodiate sstem S, which is equa to the vaue of " P O P So it is deduced that the coodiate of P is the positio that P 1 1 otates θ coutecockwise aoud the coodiate oigi. Fig. (3. Poits eatio diagam of eft puet.

3 ajecto Reseach about the Roig-Pi Bet asmissio he Ope Mechaica Egieeig Joua, 2012, Voume 6 75 Defiig the coodiate of poit P as (x 1 p, p i coodiate sstem S ad coodiate tasfomatio matix as A, the coodiate (x p, p is descibed as: ( x p p = A x 1p p1 ( (16 whee, A is the coodiate tasfomatio matix fom coodiate sstem S to S [11, 12], the matix is descibed as, # cos A = % % si $ "si cos & ( ( ' 2.2. Equatio the Left Pue Itegatig Cuve (17 As descibed i the Fig. (3, poit G (x 0g, 0g is the iitia itegatio poit betwee the eft pue ad bet, G 1 (x 1g, 1g is some poit befoe the ed of itegatio. Because poit G is statig itegatio whie poit P stats sepaatio, otatig speed ad the time ae equa that two poits sped o espective itegatig ad sepaatig, which ae amed as t 0 ad ω above. Accodig to the geometic eatioship, the coodiate of Poit G 1 is descibed as: x = x " t (18 1 g 0 g # = " t si( ( g 0 g whee, x 1g ad 1g ae coespodig x-coodiate ad - coodiate of poit G 1 i coodiate sstem S. Whie oig pi us fom the iitia itegatio poit G to the poit G 1, poit G 1 o the pue otates to the poit G 1 which supeimposes with poit G 1 i the fixed coodiate sstem S, ad the pue coodiate sstem S aso otates age θ eative to the oigia fixed coodiate sstem S. So it is deduced that the coodiate of G 1 is the positio that G 1 otates θ coutecockwise aoud the poit O. Accodig to the geometic eatioship, the coodiate of poit G is descibed as: 1 ( x g g = A x 1g 1g ( (20 whee, x g ad g ae coespodig x-coodiate ad - coodiate of poit G i coodiate sstem S EQUAION OF HE RIGH PULLEY CURVE 3.1. Sepaatio Cuve Equatio he deivatio of the ight pue sepaatio cuve equatio is simia to that of the eft oe. As show i Fig. (4, ω is the agua veocit of the ight pue ad t 0 is the time the sepaatio pocess speds, θ is the age offset of coodiate sstem S duig the sepaatio pocess. So cacuatig eatios about ω ad θ ae obtaied i the foowig fomuas. ( 2 2 " = (21 t 0 # = 180 " t (22 Poit P (x 0p, 0p is the statig sepaatig poit of bet ad pue, P 1 (x 1p, 1p is a poit befoe the ed of the sepaatio. Based o kiematics eatioship, the speed of oig pi ca be descibed as. Accodig to the geometic eatioship, the coodiate of Poit P 1 is descibed as: x = x " t (23 # 1 p 0 p = " t si( (24 1 p 0 p + whee, x 1p ad 1p ae the coespodig x-coodiate ad - coodiate of poit P 1 i coodiate sstem S. As eft pue s deivatio, defiig the coodiate of poit P 1 as (x p, p i coodiate S ad the coodiate tasfomatio matix as A, the coodiate (x p, p is descibed as: ( x p p = A x 1p p1 ( (25 whee, A is the coodiate tasfomatio matix fom coodiate sstem S to S, the matix is descibed as, Fig. (4. Poits eatio diagam of ight puet.

4 76 he Ope Mechaica Egieeig Joua, 2012, Voume 6 Zhao ad Zhag A & # " (26 Whe descibig eatios i the fixed coodiate sstem S, the offset fom the ight pue coodiate oigi to the eft is a. So the coodiate equatio is descibed as: ( x p p = A x 1p 1p ( + a Itegatio Cuve Equatio ( (27 Simia to the deductio of the eft pue, ight pue s otatio speed, time cost i the itegatio pocess ad the age ight pue otatig duig itegatig peiod, ae the same as those of ight pue duig sepaatio pocess, whose paametes ae t 0, ω ad θ. As show i Fig. (4, poit G (x 0g, 0g is the iitia itegatio poit betwee the eft pue ad bet, poit G 1 (x 1g, 1g is some poit befoe the ed of itegatio. Accodig to the geometic eatioship, the coodiate of poit G 1 is descibed as: x = x0 g " t (28 = " t si( (29 1 g g 0 g whee, x 1p ad 1p ae the coespodig x-coodiate ad - coodiate of poit G 1 i coodiate sstem S. I a simia wa, defiig the coodiate of poit G 1 as (x g, g i coodiate S, the coodiate of poit G 1 is descibed as: ( x g g = A x 1g 1g ( (30 Whe descibig eatios i the fixed coodiate sstem S, the coodiate equatio is descibed as: ( x g g = A x 1g 1g ( + a 1 0 ( (31 4. MESHING CURVE AND ROLLING PIN NUMBER Duig divig, sidig shoud t occu betwee pues ad bet. I othe wods, oig pi shoud t side o the pue. So, it is deduced that iitia sepaatio poit of the sepaatio cuve must supeimposes with the statig poit of the itegatio cuve ad the whoe meshig cuve is the combiatio of above two cuves. Fig. (5 shows sevea meshig cuves paced eve o the eft pue ad the thickeed cuve is oe sige meshig cuve. he iitia sepaatio poit ad the edig itegatio poit of each pue ae i the same espective wokig cce, ad offset ages i thei espective coodiates ae expessed as θ z ad θ z, as show i Fig. (2 ad Fig. (3. With affie tasfomatio, edig itegatio poit coud supeimpose with iitia sepaatio poit o each pue b meas of coutecockwise otatio. So thee is the foowig eatioship betwee (x 0p, 0p ad (x 1g, 1g ( x 0 p 0 p = A z A x 1g 1g ( t =t 0 (32 whee, A z is affie tasfomatio matix of eft pue fom the edig itegatio poit to the iitia sepaatio poit. It is descibed as: A z & z z z # z " (33 he eatioship betwee (x 0p, 0p ad (x 1g, 1g is simia. ( x 0 p 0 p = A z A x 1g 1g ( t =t 0 (34 whee, A z is affie tasfomatio matix of ight pue fom the edig itegatio poit to the iitia sepaatio poit. It is descibed as: A z & z z ( x z # z " he compete meshig equatio is show as foows: A ( x1 p 1 p ( 1 1 ( " " " # " " Az A x p p + a 1 0 $ = # " " " " # " Az A x p p + a 1 0 $"$ A ( x1 p 1 p ( 1 1 ( Left pue sepaatio cuve Left pue itegatig cuve Right pue sepaatio cuve Right pue itegatig cuve (35 (36 Accodig to the geometic eatioship, the bet egth is cacuated as: L = (1" # a # + (1+ # 90 (37 Suppose the umbe of oig pis is, ad meshig cuves ae eve distibuted i the pue, ad the umbe of meshig cuves is o the eft pue ad o the ight, the eatios ae as foows: L = (38 = 2 (39 = 2 (40 Fig. (5. Eve paced meshig cuves o the eft pue. he, the paamete ca be soved ad expessed as foows.

5 ajecto Reseach about the Roig-Pi Bet asmissio he Ope Mechaica Egieeig Joua, 2012, Voume 6 77 " (1 # + 2a + " (1 + = " a " (1 # " ( = 2" (41 B settig pioities ad adjustig vaues of thee paamete, ad a, a seies of discete soutios ae cacuated to meet itege paametes:, ad. CONCLUSIONS Impovig divig toque ad educig sip ate have impotat ad pactica sigificace i powe tasmissio idusties. his pape pesets a ew wa that is eve paced oig pis o the bet ad egaved cuve tajectoies o pue. Fist, sepaatio ad itegatio cuves ae deduced i the pape, ad the, meshig cuve equatios ae estabished. Fia, the method to detemie the umbe of tajectoies ad oig pis is obtaied. A those ca be used fo mechaism desig ad futhe eseach. CONFLIC OF INERES Noe decaed. ACKNOWLEDGEMEN Noe decaed. REFERENCES [1] A.H. Mei, ad J.H. Wag, "Reseach fo the Damic desigig of the asmissio of Schoize Gea-bet", J. Mach. Des. Res., vo. 15, pp , [2] Z. Wag, "he cacuatio ad the effect of eastic side o efficiec i bet tasmissio", J. Mech. asm., vo. 33, pp , [3] G.L. Pu, ad M.G. Ji, Mechaic Desig. Beijig: Highe Educatio Pess, Chia, [4] Y. Zhag, "Damic eseach o sepetie bet dive sstems", M.S. thesis, Nothweste Potechica Uivesit, XiA, Chia, [5] M. Jiag, "Capabiit eseach o automotive schoous bet", M.S. thesis, Chagchu Uivesit of echoog, ChagChu, Chia, [6] P. Zhou. "Automotive schoous bet dives desig method ad dives capabiit eseach", M.S. thesis, Chagchu Uivesit of echoog, ChagChu, Chia, [7] P.L. Cog. "Desig of ew odia v-bet CV", M.S. thesis, Daia Jiao og Uivesit, Daia, Chia, [8] Y.K. Wag, Z.J. Yag, L.N. Li, ad X.C. Zhag, "he equatio of meshig of spia beve geas maufactued b geeatig-ie method", Ope Mech. Eg. J., vo. 5, pp. 51-5, [Oie] Avaiabe: [Accessed Sept. 1, 2011]. [9] J.S. Zhao, Z.J. Feg, ad F.L. Chu. Aatica theo of degee of feedom fo obot mechaisms. Beijig: Sciece Pess, Chia [10] J. Opea. Diffeetia geomet ad its appicatios. Beijig: Machie Idust Pess, Chia, [11] Y.P. Cheg, K.Y. Zhag, ad Z. Xu, Matix heo Xi. A: Nothweste Potechica Uivesit Pess, Chia, [12] C. Stege, M. Uich, ad C. Wiedema. Machie Visio Agoithms ad Appicatios. Beijig: sighua Uivesit Pess, Chia, Received: Septembe 09, 2011 Revised: Octobe 16, 2011 Accepted: Novembe 18, 2011 Zhao ad Zhag; Licesee Betham Ope. his is a ope access atice icesed ude the tems of the Ceative Commos Attibutio No-Commecia Licese ( which pemits uesticted, o-commecia use, distibutio ad epoductio i a medium, povided the wok is pope cited.