DEMONSTRATION AND METHOD FOR CALCULATING THE EFFICIENCY OF DIFFERENTIAL MECHANISM

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1 DEMONSTRATION AND METHOD FOR CALCULATING THE EFFICIENCY OF DIFFERENTIAL MECHANISM Assoc. Prof. Barbu PLOSCEANU, PhD, POLITEHNICA Uversty of Bucharest, Assst. Prof. Ovdu VASILE, PhD, POLITEHNICA Uversty of Bucharest, Abstract - To solve the problem of calculatg the power crculato ad effcecy, the mechacal trasmsso cycles are reform the oto of effcecy. Usg the prcple of vrtual veloctes preseted a demostrato to calculate power flux dcator. Fally, examples of the method for calculato of dfferetal type mechasms. Keywords: kematc, forces, trasfer rato. Itroducto I the case of trasmssos formed by mechasms coected seres the problem of power flux trasmsso aturally results from the put to the output, whch meas from the motor elemet (drver) to the resstat elemet (drve). For trasmssos wth coectos seres ad parallel, whch form cycles (see fg..a) the power flux trasmsso does t result a aturally way. I that case the trasmsso of power flux ca take place, by example, accordg to oe of varats outled fgure.b. I fgure s oted: MM - car eges; ML work mache; R reducer; D- dfferetal mechasm. As a cosequece such stuato oe asks whch s the power flux through trasmsso braches, how much of exteral motor power pass through each brach, whch s the torsor forces whch operates o trasmsso elemets, how oe calculates the effcecy, ad so o. For aswerg at such questos, we must make frst some reformulatos of otos we are talkg about. Sgfcat resolvg the relatoshp for calculatg the coeffcet of dstrbuto of power trasmsso dustres. Based o data kow lterature [2], [] [4] [5] [6], gve a demostrato ut for ths deducto factor ad exemplfes Fg. 2.Kematc trasmsso rato, forces trasfer rato, effcecy [], [2], [] Let s cosder the trasmsso wth gears wth fxed axs (see fgure o. 2) at wtch the elemet s the motor elemet, ad the M ω 0 ad the elemet 2 s drve elemet ad the M 2 ω2 0. As a cosequece, the trasmsso effcecy (gearg) s: Pu M 2ω2 η2 = =, () P M ω m Fabltate s Durabltate - Fablty & Durablty o 2(6)/ 200 Edtura Academca Brâcuş, Târgu Ju, ISSN X

2 M ω η 2 η 2 M ω 2 2 Fg. 2 Fg. From were: M 2 ω = η2 M ω2 (2) Oe otes: - the forces trasfer rato (of torsor forces), wth υ 2 = M 2 M () -the rato of trasmsso veloctes (of torsor veloctes) wth 2 = ω ω2 (4) Wth ths otato the relato (2) ca be wrtte: υ 2 = η2 2 (5) The η 2 = υ2 2 (6) If oe eglects the effcecy ( 2 = ) υ 2 = 2 (7) Let s cosder ow, the trasmsso composed by mechasms coected seres from fgure. The kematcal trasmsso rato s: = 2 2 (8) The trasfer forces rato, f s the motor elemet ad s the drve elemet, wll be: υ = η = (9) η2 = 2 the But the relato (8) ca more be wrtte: 2 2 υ = 2 η = 2 2η 2 From relato (0) results that f kematcal trasmsso rato s take the sese of power flux, that the forces trasfer rato ca be obtaed by multplyg the kematcal trasmsso rato wth the effcecy at power +, ad f the kematcal trasmsso rato s take opposte sese of power flux, tha the forces trasfer rato s obtaed by multplyg the kematcal trasmsso rato wth the effcecy at power. The, for ay trasmsso, f oe otes wth x = ±, the expoet of effcecy, tha the force s trasfer rato s a fucto lke: x x x 2 x ( η ; 2η 2 ; η η ) υ 2 = f ;...; () Therefore the force s trasfer rato s a fucto whch depeds o, 2,,...,. - the kematcs trasmsso ratos of compoet mechasm; o ther effceces η, η 2, η,...,. η ad also depeds o sese of power crculato trough the expoets x x, x,...,., 2 x. The trasmsso veloctes rato s a fucto termedate trasmsso rato, that meas: = f (, 2,,...,. ) (2) Tha, geeralzg the relato (6) the effcecy oe calculates wth the expresso: (0) 2 Fabltate s Durabltate - Fablty & Durablty o 2(6)/ 200 Edtura Academca Brâcuş, Târgu Ju, ISSN X

3 x x x 2 x ( η ; 2 2 ; ;...; ) η η η υ f η = = () f (, 2,,...,. ) I the relato (), the expoets x, x2, x,...,. x take the value + f the power flux cocdes (has the same sese) wth the kematcal oe, ad take the value, to the cotrary. As a result, for kowg whch s the put ad whch s the output (whch s the motor elemet ad whch s the drve elemet) for a compoet mechasm of a mechac trasmsso as that from fgure.b, t must be vestgated (establshed) the expoets x. Also oe asks how much of exteral power crculates through each brach, ad fal whch s the effcecy of trasmsso.. Coeffcet of power repartto. Power flux dcator Let s cosder the trasmsso from fgure 4, where s the motor elemet ad the M ω 0, ad s the drve elemet ad the M ω m 0, ad R, =,..., p, are the compoet elemets. For the establshg how much of motor power crculates through each brach we ca compare the motor power P = M ω 0 wth the power applcat to the elemet k whch s a rotato moto, Pk = M k ωk, whch belogs to compoet mechasm R m. The rato of two powers show how much of exteral motor power pass through that mechasm. Ths rato s called the coeffcet of power repartto [], [4], [5] ad oe ote: λ k = Pk Pm (4) Wth ths, for kowg f the k elemet s a motor oe or a resstat oe, we wll calculate the sg of ths rato, whch meas the dcator of power flux: xk = sgλk (5) Wth ths, t meas that we ca say whch s the sese of power flux through each mechasm, respectve through each brach of trasmsso; how much of motor power (the exteral) crculates through each brach; ad takg to accout the relato () whch s the effcecy of trasmsso. The soluto of problem the coeffcet of power repartto oe obtas applcato the prcple of vrtual veloctes (vrtual powers) to trasmsso whch the coecto let be ths a gearg couplg from elemets k ad h s chaged wth forces F ad F, ad these (the elemets) take the vrtual values ω k ad respectve ω h (fg. 4.b). For the cosdered trasmsso, the motor power s P = M ω. As a result of coecto chagg betwee elemets k ad h, the elemet h gets the agular velocty ω h, wth ukow sze ad sese. Let be tha ω wth the same sese of resstat momet M, the real sese followg to result from calculus. Therefore P = M ω. The power accordg to elemet k s Pk = Fr k ω k, ad the power accordg to elemet h s Ph = Fr h ω h. Tha, accordg to prcple of vrtual powers: P = M ω + M ω Fr ω Fr ω (6) k k + h h Fabltate s Durabltate - Fablty & Durablty o 2(6)/ 200 Edtura Academca Brâcuş, Târgu Ju, ISSN X

4 Fg. 4 Ad because at the applcato of prcple of vrtual powers, oe eglects the frctos, accordg to (7) υ 2 = 2 = M 2 M. As a result relato (6) oe wrtes: r = h ωh P M ω M ω Fr kωk r (7) k ωk Oe otes: ω = ω the ketc trasmsso rato accordg to vrtual veloctes mpressed, M k = Fr k the momet applcat to elemet k, kh = rh rk the ketc trasmsso rato accordg to real veloctes, kh = ω k ωh the ketc trasmsso rato accordg to vrtual veloctes, Wth ths otato the relato (7) ca be wrtte: = kh P M ω M kωk = 0 kh or: kh kh P = M ω M kωk = 0 kh From where: M kωk kh = M (8) ω kh kh But s a fucto of kh ad s a fxed value of fucto for kh fxed ( kh = kh ). Tha the coeffcet of power repartto o elemet k s: M k k kh ω kh kh k M = = k k kh λ = lm = lm lm lm ω ω kh ω kh kh kh Therefore: kh λ k = (9) kh From the relato (9) results that both the power flux sese, by meas (5) ad the power repartto coeffcet ca be determed o kematcs codtos. 4 Fabltate s Durabltate - Fablty & Durablty o 2(6)/ 200 Edtura Academca Brâcuş, Târgu Ju, ISSN X

5 4. Cocluso I cocluso for calculus of the effcecy of a mechacal trasmsso wth cycles we suggest to pass the followg steps: a) the calculus of trasmttg rato veloctes, that meas fucto (2); b) the settg up of fucto wtch gves the trasfer rato of torsor forces, that x υ kh = khη kh, ad the replacg () kh wth υ kh ; c) oe calculates all coeffcets of power repartto λ k, wth relato (9); d) oe calculates the power flux dcator, that meas all x k, wth relato (5); e) oe chages the fluctuato set up at step b takg to accout that η kh = ηkh f η kh s approxmate equal to hk = f ηkh dffers very much of η hk ; η ad ηkh ηhk f) oe calculates the trasmsso effcecy wth relato (). A very smple example of mechacal trasmsso wtch geerates the pheomea of power crculato s that of a dfferetal mechasm ad prvate the plaetary mechasm[8]. Thus, the dfferetal mechasm smple aggregate relatoshp (9) wrte: 0 λd = (20) 0 where 0 s the basc gear rato, the rato of kematcal trasmsso mechasm caused dfferetal arm fxed plaet carrer. Whether the trasmsso show Fgure 5 s a key factor drvg states, led elemet, worm gear Z = 20, Z 4 = 40 ad Z 4 = 0, Z 2 = 0 ad dfferetal gear mechasm Z = 20 Z 2 = 60 ad Z 5 = 20. The mechacal effcecy of the gear torque s equal to 0.98 ad the basc mechacal effcecy (the dfferetal arm brought fxed plaet carrer) s η 2 = η0 = 0, 96. Is requred to calculate trasmsso effcecy η. Kematcal trasmsso s descrbed by the equatos: ω4 = 4ω ; ω2 = 24ω4 ; ω = 2ω2 + ( 2 ) ω wth 2 = 0 - base gear rato. Fg. 5 5 Fabltate s Durabltate - Fablty & Durablty o 2(6)/ 200 Edtura Academca Brâcuş, Târgu Ju, ISSN X

6 From relatos (20) by substtuto yelds: ω 0 = = ω 4 0 For dfferetal mechasm 0 0 ( ) λd = = = 0,45 xd 0 ( 0 )( 4 0) = sgλ D = For gear wheels wth Z ad Z 4, λ4 = = = 0, 6 x 4 = For gear wheels wth Z 4 ad Z 2, λ24 = = = 0, 6 x 24 = Results = η ν,57 ν =, 57 ad η 0, 98 = = =.,6 Refereces [] Krasekov, V.I., Vaşet, A.D., Proectrovae plaetarâh mehazmov trasportâh maş, Maşostroee, Moskova, 986 [2] Krees, M.A., Rozovsky, M.S., Zubcatâe mehazmâ, Nauka, Moskova, 972 [] Krdaşev,Iu.A., Mogopatocâe peredac dffereţaloe tpa., Maşostroee, 98 [4] Mlou, Gh., Dudta, Fl., Dacoescu, D.V., Trasms mecace modere, Ed. Tehca, Bucureşt, 980 [5] Muller, H.W., De Umlaufgetrebe, Bd.28, Sprger-Verlag, Berl, 97 [6] Plosceau, B., Cotrbuţ la studul damc mecasmelor dfereţale, Teza de doctorat, 988, I.F.T.M. [7] Plosceau, B., Vasle, O., Plaetary safety clutch - qualtatve aspects. Proceedgs Twelfth Word Cogress Mechasm ad Mache Scece, Besaco, Frace, 2007, p [8] Plosceau, B., The Workg Lmts ad Effcecy of Dfferetal Mechasm the Aggregate. Nth World Cogress o the T.M.M., Italy 995, Vol 2. 6 Fabltate s Durabltate - Fablty & Durablty o 2(6)/ 200 Edtura Academca Brâcuş, Târgu Ju, ISSN X