ON THE COMPUTATIONAL DESIGN METHODS FOR IMPROOVING THE GEAR TRANSMISSION PERFORMANCES

Transcription

1 ON THE COMPUTATIONAL DESIGN METHODS FOR IMPROOVING THE GEAR TRANSMISSION PERFORMANCES Flavia Chira 1, Mihai Banica 1, Dinu Sticvici 1 1 Assc.Prf., PhD. Eng., Nrth University f Baia Mare, Flavia.Chira@ubm.r Abstract: The cmputatinal design permit t have the pssibility f chsing the gemetrical gears parameters nt in relatin with the standard gear rack but in relatin with the request f the beneficiary. The parametrical design permit als t study what are the better values f the designing parameters fr btain the ptimal slutin fr the functinal parameters f the transmissin. The paper present sme aspects abut the facilities ffered by sme riginal cmputer applicatin, results f the authr s research, t ptimise the nnstandard gears, cnsidered as a slutin fr imprving the gears transmissin quality. It became very easy t design, fr the same initial date, symmetric gears, direct r inverted asymmetric gears, final find the better fr the bjective functin. Key wrds: cmputatinal, direct, gear, design, asymmetric 1. INTRODUCTION Invlutes gears are the mst frequent because f the already knwn advantages. Asymmetric gears characterized by asymmetric invlutes prfiles f the tth are used nly in the last few years. Because the great numbers f gemetrical parameters, in relatin with the classical gears, designing and manufacture thse special gears it is mre difficult. By using the cmputatinal methd this incnvenient can be slved withut significant supplementary csts [5], [6]. Asymmetric gears, frmed f gear wheels with asymmetric invlutes prfile teeth, add further advantages like an increase in the lad capacity, a decrease in weight, a reductin in vibratins, an increase in efficiency and durability [4]. The rati between the diameter f the base circle f the invlutes inactive prfile and the diameter f the base circle f the invlutes active prfile represents the asymmetry cefficient f the tth [3]. In relatin with thse mst imprtant parameter can be defined: the direct asymmetric gear, fr which the direct" prfile is the active ne and the asymmetry cefficient is higher than 1; the inverted asymmetric gear, fr which the inverted prfile is the active ne and the asymmetry cefficient f is lwer than 1 and the symmetrical gear with the asymmetry cefficient equals ne. It has been called direct prfile the prfile with a high 7

2 gearing angle, and the prfile with a lw gearing angle it have been called inverted prfile. There are als ther designing parameters, the angles and the number f the generatin gear racks, which influence the gemetrical and functinal parameters f the designed transmissin. 2. APPLICATION AREA In the fllwing are presented part f results btained with riginal applicatins in Matlab and AutLisp develped fr designing, mdelling and studying gemetrical and functinal parameters f the nn-standard spur gears with invlutes asymmetrical prfiles f the teeth and als fr direct design f the symmetrical gears, that have been cnsidered nly a particular case f thse asymmetrical. All the designed gears that can be btained represent nn-standard gears because the applicatins use the direct design f the spur gears. That ffers the pssibility f determining first the parameters f the gears, fllwing by the determinatin f the asymmetric gear rack s parameters n the base f thse f the gears. The vectr f the designing parameters r variables is: ( 1 2 3, 4 5 ) ( wd wi ) X= x,x,x x,x = α, α,f,cr,var, (1) where: α, α are the pressure angle n the direct respectively inverted prfile f the gear; wd wi f called cefficient f mdificatin the gear rack angle determine the difference between and the gear rack direct prfile angle cr = 1,2 establish design with single r tw αwd α ; { } dc generatin gear rack and the last var = { 1, 2} is in relatin with the using f the gear as direct r inverted gear. A set f variables generate an asymmetric gear fr which can be determined the gemetrical parameters. Intrducing these parameters in the equatin f the invlutes and filet prfiles result the teeth prfiles (figure 1 - with Matlab applicatins), can be btained the 2D and 3D mdels f the gears (figure 2 with Autlisp applicatins), sme functinal parameters mentined befre and n the base f thse parameters can be evaluate the perfrmances f the transmissin and easy cmpared with anther pssibly slutins. Fig. 1. Pinin and gear teeth prfiles examples 8

3 Fig. 2. Examples f the 2D gearing representatin the meshing zne The applicatins develped in AUTOLISP fr AUTOCAD permit the rapid representatin f the asymmetric gears and the finite element methd, being verified in many applicatins, can be cnsidered as a methd fr determine but mre fr verify and cnfirm the results btained with the MATLAB applicatin, that is mre efficient and easily t use. 3. MATLAB APLICATIONS FACILITIES The behaviur f the designed asymmetric gear under the lad ne can rapid evaluate by the MATLAB applicatins, many rutines that permit t apprach the fllwing aspects [2]: mathematical mdelling f the teeth and f the meshing f the asymmetric gears; the elasticity, implicitly the rigidity, f the pairs f teeth in cntact and the variatin fr these parameters during the meshing cycle; n the base f the tth elasticity it can be slved the statically unknwn prblem f lad distributin between the tw pairs f teeth in meshing and s it was determined the diagram f variatin f the nrmal frce; the relative sliding speed and the variatin during the meshing perid; the instantaneus pwer lses, the variatin f the instantaneus efficiency and than the medium efficiency fr a meshing perid ; the bending stress at the bttm f the tth in relatin with the number f the cntact pint; the variatin during the meshing cycle f the bending stress it was s resulting; the cntact stress has been determined als fr every the cntact pint and represented the diagram f variatin fr a meshing cycle. Fr all the mentined parameters have been established the calculus algrithms, the mathematical relatin t btain the values crrespnding fr a number f cntact pint f the line f actin and frm these the significant values. Fr example the cntact stress has been determined as a functin depending n the pressure angle crrespnding t the cntact pint n the pinin active prfile: 0,3 E1 E2 Fnjd,i tgαwd,i ( 1+ u) ( ) ( ) σ Hjd,i = E + E b r tgα tgα 1+ u tgα where: E 1, E 2 are the elasticity mdules f the pinin and gear materials; 1 2 b1d,i j1d,i wd,i j1d,i 9, (2)

4 F njd,i (Fnjd, F nji ) cntact (fr direct gear, fr inverted gear); 2 1 is the nrmal frce n the tth prfile crrespnding t the j pint f u = z /z is the rati f the number f teeth f the gear and pinin; r b1d,i (rb1d, r b1i) direct gear, fr inverted gear); is the radius f the base circle f the active prfile f the pinin (fr α α α ) is the pressure angle crrespnding t the j cntact pint n the j1d,i ( j1d, j1i pinin active prfile. Anther imprtant facilities ffered by the develped applicatins is the rapid cmparisn between the significant functinal parameters crrespnding t a number f pssible slutins fr the same initial date. In the table 1 are given examples f such cmparisn diagrams [1] that indicate the cmbinatin f designing parameters that increase the transmissin perfrmances. z1= 15,z2 = 60,a = 120mm, n = 1000 rt / min, P = 18kW, α wd = [30, 32, 34, 36, 38 ], α wi = 20, f = [2, 4, 6, 18, 10] table 1. Functinal parameters cmparisn diagrams Variatin f the maximum bending stress t the pinin in relatin with gear rack direct prfile angle fr different values f the cefficient f asymmetry Variatin f the maximum bending stress t the gear in relatin with direct gear rack prfile angle fr different values f the cefficient f asymmetry 4. AUTOLISP APLICATIONS FACILITIES 10

5 The gemetric parameters which are necessary t intrduce in the first line f the AUTOLISP applicatin, are: the numbers f the teeth fr the pinin and fr the gear z,z ; the centre distance a ; the pressure angles n the direct and inverted prfiles prfile angles f the asymmetric gear rack dc, ic α wd, α wi 1 2 ; the α α ; the prfile angles n the addendum circles, n the direct teeth prfiles, fr the pinin and fr the gear αa1d, α a2d ; the tip radius f the gear rack R1 (r R 1,R 2 in the case f generatin with tw different racks) and the rack shift fr the pinin X 1. In the table 1 are given fr examples the results btained with finite element methd analysis fr the asymmetric gears with the numbers f teeth z = 16;z = 57, the centre distance a =120mm, the mesh angles, α = 40 ; α = 20, the angles f the gear rack prfiles equals with the mesh angles f wd wi 1 2 = 0, the generatin f pinin and gear with ne gear rack, fr P=18 kw, n=1000 rt/min, b=30 mm and in table 3 the cmparisn between gears with different cefficient f asymmetry, different mesh angles n the asymmetric prfiles [2]. fr pint Bd f the line f actin AdEd table 2. Stress distributin btained with FEA The stress distributin fr the direct asymmetric gear fr pint Dd f the line f actin AdEd The stress distributin fr the inverted asymmetric gear fr pint Bi f the line f actin AiEi fr pint Di f the line f actin AiEi table 3. Stress distributin btained with FEA Gear1: 40/20 Gear 2: 35/20 Gear 3: 30/20 Gear 4: 25/20 Gear 5: 20/20 11

6 5. CONCLUSION The cmputer applicatins permit t btain in very shrt time many pssible slutins fr designing a nn-standard gear, cmparing the perfrmances and establishing the request cmbinatin f designing parameters and implicitly the gemetrical parameters. Having the 2D mdel f the gear any design engineer can t btain the 3D mdel, as drawing file, and that can be change in initial graphics exchange file. S in the sft f crdinate manufacturing machine, the btained data will be transfrmed in machine language and thus all the data necessary fr manufacturing are prvided. 6. REFERENCES 1. Chira, F., Banica, M.& Lbntiu, M., 2009, Optimising Invlutes Asymmetrical Teeth Gears Sftware, Chapter 38 in DAAAM Internatinal Scientific Bk 2009, B. Katalinic (Ed.), Published by DAAAM Internatinal, ISBN , ISSN , pp , Vienna, Austria 2. Chira, F. & Banica, M., 2007, Studii asupra rtilr dintate cu dinti asimetrici/studies n the gears with asymmetric teeth, Risprint Cluj-Napca, ISBN , Rmania 3. Kapelevich, A.L., 2000, Gemetry and design f invlutes spur gears with asymmetric teeth, Mechanism and Machine Thery, Vl.35, Issue 1, , ISSN X 4. Karpat, F., Cavdar, K. & Babalic, C.F. (2006). An investigatin n analysis f invlutes spur gears with asymmetric teeth: Dynamic lad and Transmissin errrs, Prceedings f the 2 th Internatinal Cnference Pwer Transmissins 2006, pp , ISBN , Nvi-Sad, April 2006, Serbia and Mntenegr 5. Litvin, F.L., Lian, Q. & Kapelevich, A.L. (2000). Asymmetric Mdified Gear Drives: Reductin f Nise, Lcalizatin f Cntact, Simulatin f Meshing and Stress Analysis. Cmputer Methds in Applied Mechanics and Engineering, N.188, , ISSN Nvikv, A.S., Paikin, A.G., Drfeyev, V.L., Ananiev, V.M., & Kapelevich, A.L. (2008). Applicatin f Gears with Asymmetric Teeth in Turbprp Engine Gearbx, Gear Technlgy, January/February 2008, Marcinčin, J. N. - Kčišk, M.: Technlgical Prcess Plannig f Disassembly Technlgy. In: Prceedings f the Cnference Technlgical Prcess Planning Plitechnika Pznan, 2006, pp , ISBN