Optimal Selection of Addendum Modification Coefficients of Involute Cylindrical Gears

Transcription

1 56 JOURNAL OF COMPUTERS, VOL. 8, NO. 8, AUGUST 03 Opimal Selecio of Addedum Modificaio Coefficies of Ivolue Cylidrical Gears Xueyi Li College of Mechaical ad Elecroic Egieerig, Shadog Uiversiy of Sciece ad Techology, Qigdao, Chia Shoubo Jiag ad Qigliag Zeg College of Mechaical ad Elecroic Egieerig, Shadog Uiversiy of Sciece ad Techology, Qigdao, Chia Absrac Based o he priciples of equalied slidig coefficies ad equalied eeh bedig sregh, he opimiaio mahemaical model for calculaig ad allocaig he addedum modificaio coefficies of ivolue cylidrical gear is firsly esablished. The he measures o calculae he ooh parameers i real-ime durig he opimiaio seps are achieved. Fially, he opimiaio program for calculaig ad allocaig he modificaio coefficies is developed usig he Opimiaio Toolbox i MATLAB combied wih VC++. Compared wih he radiioal mehods such as eclosed char ad graph mehod, he proposed opimiaio mehod i his paper ca updae he cosrais durig he opimiaio process ad obai he accurae modificaio coefficies. The examples show ha he proposed opimiaio mehod is more raioal ad accurae. Idex Terms opimiaio desig, addedum modificaio coefficies, cylidrical gears, slidig coefficies, ooh bedig sregh I. INTRODUCTION Gears wih addedum modificaios have he advaages of improvig he meshig performaces ad loadig capaciies. A raioal selecio of addedum modificaio coefficies ca efficiely help improve he faigue sregh of gears, reduce vibraios, suppress he oises ad exed service life [-]. Hece, gears wih addedum modificaios are widely used i he fields of machiery idusries, ad he selecio of addedum modificaio coefficies is oe of he mos impora research orieaios i he gear desig area. Amog he umerous selecio mehods, he eclosed char mehod iroduced by Niema ad Wier [3] is mos frequely used by he desigers. As boh he limiaios of selecios ad priciples of allocaios are well balaced i his mehod, he desig process is usually direcive ad fas. Bu his mehod cosiss of hudreds of graphs wih differe eeh umbers ad Copyrigh credi, correspodig auhor: Xueyi Li. *The mai fiacial suppor of Shadog Provicial Naural Sciece Foudaio of Chia (Gra No.ZR00EM03). pressure agles, which resuls i a icoveie usage. The graph mehod proposed by Wag [4] is widely applied i he civil gear maufacures due o he brief procedure ad well-balaced limiaios. Bu whe i comes o he allocaio of he modificaio coefficies, oly oe priciple based o equalied slidig coefficies is provided i his mehod wihou oher priciples, e.g., equalied bedig sregh. Therefore, he limiaio of he graph mehod is obvious. Wih he developmes of compuig echology, umerical mehods ad opimiaio heories have bee applied i he desig of egieerig areas [5-7]. Based o he allocaig priciples of equalied slidig coefficies, GA is used by Aal i he desig of helical gears [8]. Wih he improved cosrai codiios ad odimesioal gear ooh modelig, he Complex opimiaio algorihm is applied i he process of opimiig ivolue gear desig by Spias ec. [9]. Ad four differe allocaig mehods of modificaio coefficies are compared by Baglioi e al. [0], ad based o he compariso resul he ifluece of he addedum modificaio coefficies o gear efficiecy is researched. Ad GA is also used by Zhag e al. o carry ou a opimiaio for bevel gear drive []. Differe opimiaio algorihms for he opimal desig of he modificaio coefficies are applied i he above pracices. Bu i he opimiaio seps, he cosrais fail o updae wih he desig variables due o he complex ad ime-cosumig calculaio process of gear parameers []. I order o simplify he calculaio, approximaed curves are usually used o simulae he cosrai parameer, which resul i he iaccurae opimiaio resuls. I his paper, he dyamic opimiaio mehod is used o obai he opimal selecio of modificaio coefficies i order o overcome he limiaios i he above mehods. Firsly, he mahemaical model is esablished based o he priciples of equalied slidig coefficies ad equalied bedig sregh. The, he opimiaio process is achieved by he Opimiaio Toolbox i MATLAB ad he opimiaio program for gear desig is implemeed by VC++. I he ed, wo opimiaio examples usig differe mehods are 03 ACADEMY PUBLISHER doi:0.4304/jcp

2 JOURNAL OF COMPUTERS, VOL. 8, NO. 8, AUGUST carried ou for a pair of gears. The opimiaio resuls show ha combied wih he radiioal mehods, he proposed mehod i his paper has a faser soluio ad a more accurae allocaig resul, which ca improve he desig efficiecy. II. METHODOLOGY Wih he esablishme of he mahemaical model ad he dyamic cosrais, he opimiaio process is achieved usig he maure opimiaio program. A. Limiaios o Selecig Addedum Modificaio Coefficies Before he opimiaio process, he followig cosrais eed o be cosidered [3]: () Make sure here is o udercu ad excessivelyhied ooh; () Make sure he sufficie ooh hickess i order o guaraee he bedig sregh; (3) Make sure here is o ierferece bewee he addedum ad he correspodig dededum fille curve; (4) Make sure he coac raio is greaer ha.0 i order o saisfy he gears coiuous rasmissio codiio; (5) Uder some circumsaces, he rasmissio of o flak clearace is demaded; (6) Make sure he sregh codiios afer he modificaio. B. Mahemaical Model The opimiaio desig of modificaio coefficies is egaged uder may cosrais ad based o he specified allocaig priciples. Ad is aure is a oliear cosrai programmig problem. Accordig o he hree elemes of opimiaio mehod, he desig variables [4], opimiaio goal ad cosrai codiios are respecively deermied. ) Desig Variables Before he opimiaio, we assume ha he gear srucure parameers such eeh umbers, module, rasmissio raio, ec. are desiged ad obaied. So he desig variables are acually he modificaio coefficies x,x of wo gears [5], as show i (). a) Objecive fucio based o he priciple of equalied bedig sregh Accordig o ISO sadard, he calculaio equaio of bedig sress is defied as i (). F σ F YFYSYβ A v Fβ Fα () bm v N Where, he facors ca be referred by he sadard ISO For a pair of gears, all he facors excep for he ooh form facor Y F ad sress correcio facor Y S are equal. So he objecive fucio ca be esablished as i (3). ( X ) Y Y Y Y mi f (3) v v v F S F S b) Objecive fucio based o he priciple of equalied slidig coefficies Amog may geomery parameers ha ifluece he meshig performace, he relaive slidig velociy is he mos impora facor [3]. The larges slidig coefficies occur a he posiio whe he ooh addedum is meshig wih he correspodig ooh. Uder he circumsaces of high-speed ad heavy-load, he slidig coefficie will severely ifluece he meshig performace. Hece, i order o allocae he modificaio coefficies based o he priciple of equalied slidig coefficies, he equaio of slidig coefficie should firsly be derived. ω α α B α α O P α α N B T [ x, x ] [ x x ] T X, () ) Opimiaio Goal I order o overcome he limiaio of he eclosed char mehod ad graph mehod, boh he priciple of equalied slidig coefficies ad he priciple of equalied of bedig sregh are ake io cosideraios whe allocaig he sum of modificaio coefficie. The priciple of equalied slidig coefficies ca make sure he wo gears have equal slidig, which ca reduce he abrasio ad exed he service life. Ad he priciple of equalied bedig sregh ca guaraee boh he piio ad he wheel have equal bedig sregh i order o avoid he broke failure. So wo ypes of opimiaio goals are provided i his paper ad differe priciple leads o differe objecive fucio. Figure. Skech for he calculaio of slidig coefficies. Fig. shows he meshig skech of a pair of ivolue cylidrical gears, ad he piio ad wheel are meshig a poi. The liear velociies of piio ad wheel are correspodigly v ad v, which are o equal. I order o achieve he coiuous rasmissio, he velociies of wo gears should be equal alog he direcio of commo ormal lie a poi. Sice he velociies alog he ageial direcio v ad v of wo gears are o equal, slidig pheomeo happes a he meshig pois alog he meshig lie N N excep for he pich poi P, ad ω O 03 ACADEMY PUBLISHER

3 58 JOURNAL OF COMPUTERS, VOL. 8, NO. 8, AUGUST 03 he relaive slidig velociy v ca be expressed as i (4) show. ν ν ν (4) The slidig coefficie ca be used o express he relaive slidig degree of wo gears, ad i s defied as he raio of he relaive slidig velociy ad he ageial velociy. Therefore, he slidig coefficies ad of piio ad wheel ca be obaied as i (5) ad (6) show [3]. v v (5) v v v (6) v From he geomeric relaioship, he slidig coefficies ca be furher derived as i (7) ad (8) show. v v P u + (7) v N u v v P (8) ( ) u + v N Where, ω ad ω sad for he agular velociies of piio ad wheel. u deoes he rasmissio raio of wo gears. Hece, he slidig coefficie is he fucio of he posiio of meshig poi. A poi N,, ad a he pich poi P, 0, while a poi N,,. I fac, he wo gears ca oly mesh alog he acual meshig lie B B. A poi B, he slidig coefficie of piio ges is maximum max ad for B is max. The calculaio equaios of max ad max ca be derived as (9) ad (0) show. max a α a α + a α a αa u + u a a α a α u ( ) a max + + a α a αa (9) (0) I order o reduce he wear of wo gears ad exed lifeime, he maximum of slidig coefficies should be equal as much as possible. Ad he objecive fucio based o he priciple of equalied slidig coefficies ca be defied as () show. ( X ) max max mi f () 3) Cosrai Codiios Accordig o he limiaios o selecig he addedum modificaio coefficies, he equaliy ad iequaliy cosrais ca be formed [0]. a) Udercu cosrai The cosrai codiios o avoid udercu for piio ad wheel ca be expressed as i () ad (3). mi * x h a () x mi mi * h a (3) mi Where, Zmi is he miimum eeh umber o avoid udercu ad h a * sads for he ormal addedum coefficie. b) Esseial ooh hickess cosrai I order o maiai he ecessary ad expeced coac ad bedig sregh afer he modificaio, he ooh addedum should be greaer ha esseial hickess, as (4) ad (5) show. π + 4x a α d a π + 4x a α d a + ivα ivα Sa + ivα ivα S a 0 (4) 0 (5) Where, S a ad S a respecively sad for he miimum addedum hickess. For gears wih sof surface, S a 0.5m, ad for gears wih hard surface, S a 0.4m (m deoes he module). c) Ierferece cosrai Durig he meshig process, o avoid he addedum ierferes wih he fille curve of he respecive dededum, he ierferece cosrai codiio should be defied, as (6) ad (7) show. a α a α * si α ( ) ( 4 ha x a α a a α a α ) ( ) 4( ha x a α a a α a α ) * si α (6) (7) d) Coac raio cosrai Accordig o he codiios of coiuous rasmissio, he coac raio ε α should be greaer ha.0. I acual siuaio, i order o obai a sable rasmissio ad reduce vibraio, he coac raio eed o be greaer ha., as i (8). π [ ( a α a α ) + ( a α a α )] a a. (8) e) No flak meshig cosrai To mee he codiio of o flak meshig, he sum of modificaio coefficies should saisfy he equaliy cosrai as (9) show. + x x ( ivα + ivα ) (9) a α 03 ACADEMY PUBLISHER

4 JOURNAL OF COMPUTERS, VOL. 8, NO. 8, AUGUST f) Sregh cosrais Afer he modificaio, he calculaed ooh surface coac ad bedig sressed of piio ad wheel should be less ha he allowed coac ad bedig sresses. These cosrais ca be expressed as (0) ad () show. σ Hi [ σ Hi ] (0) σ Fi [ σ Fi ] () Where, σ Hi, σ Fi (i,) sad for he calculaed coac ad bedig sresses ad [σ Hi ], [σ Fi ] (i,) deoe he allowed coac ad bedig sresses. C. Implemeaio of he Opimiaio Program Durig he ieraive process, he modificaio coefficies may chage a each ieraive sep. Ad he chages of he coefficies will ifluece some ooh parameers such as he pressure agle a addedum circle ad he diameer of addedum circle. Sice hese ooh parameers ca direcly deermie he cosrais, he cosrais eed sychroism wih he desig variables a each ieraive sep. Based o he programmig echology, he opimiaio based o dyamic cosrais is achieved. I his mehod, afer each ieraive sep, he desig variables are saved o local files ad he used o re-calculae he ooh parameers. Ad he ew cosrais are formed usig hese parameers i order o carry ou a secod ime ieraive sep. These processes will loop uil he accurae resuls are obaied. Sice MATLAB has he fucios icludig solvig umerous equaios ad compuig mahemaical expressios [6], o impleme his process, he fmico() fucio i MATLAB Opimiaio Toolbox is used ad opimiaio program for modificaio coefficies opimiaio is developed by VC++. Fig. shows he ierface of his program. Usig he program, he modificaio coefficies ca be calculaed fas ad direcly. Figure. Ierface of he opimiaio program. III. CASE STUDY AND ANALYSIS A pair of cylidrical spur gears i he gearbox i a cerai machie ool is used o carry ou a opimiaio for modificaio coefficies respecively based o he priciples of equalied bedig sregh ad slidig coefficies. The iiial parameers are kow as: eeh umber of piio, eeh umber of wheel 33, module m.5mm, he acual ceer disace a 70mm, pressure agle α0 ad addedum coefficie h a *.0. A. Based o he Priciple of Equalied Bedig Sregh Based o he priciple of equalied bedig sregh, he radiioal eclosed char mehod (TECM) ad dyamic cosrai mehod (DCM) are correspodigly used o calculae ad allocae he modificaio coefficies. Ad he modificaio coefficies x, x ad he re-calculaed bedig sregh σ F, σ F are abulaed as Table show. TABLE I. COMPARISON BETWEEN TM AND DCM Mehod x x σ F (MPa) σ F (MPa) TECM DCM From he daa i Table, whe he eclosed char mehod is used, here is combiaio for eeh umber /33. So he eclosed char for eeh umber 0/33 is isead for allocaig he coefficies, which resuls i 03 ACADEMY PUBLISHER

5 60 JOURNAL OF COMPUTERS, VOL. 8, NO. 8, AUGUST 03 he iaccurae resuls. Ad afer he allocaio, bedig sregh of wo gears are o equal due o he allocaio error. Usig he mehod i his paper, he accurae modificaio coefficies are obaied afer he opimiaio ad equalied bedig sregh is guaraeed. B. Based o he Priciple of Equalied Slidig Coefficies Based o he priciple of equalied slidig coefficies, he allocaio for modificaio coefficies usig graph mehod (GM) ad DCM are respecively carried ou. Ad he coefficies x,x ad slidig coefficies afer he opimiaio are show i Table. TABLE II. COMPARISON BETWEEN GM AND DCM Mehod x x GM DCM From Table, i s obvious ha resuls obaied by wo mehods are differe o some exe. The slidig coefficies ad are o acually equal usig GM because he resuls are approximaed i he diagram, which leads o cerai error. Bu usig DCM, wo modificaio coefficies are obaied afer he opimiaio, which ca make sure he wo gears have he same slidig coefficies. Hece, DCM is more accurae ha he radiioal mehods. I ca be see ha he modificaio coefficies obaied by he radiioal mehods chage grealy wih hose obaied by he mehod i his paper. This is because a maual selecio is used o he diagram which causes grea error, ad cao guaraee he esseial equalied slidig coefficies or bedig sregh. While usig he opimiaio mehod combied wih compuer echology, he coefficies ca be allocaed fas ad accuraely. IV CONCLUSION I his paper, usig dyamic cosrai opimiaio, he mahemaical model for he allocaio of modificaio coefficies is esablished based o he priciples of equalied slidig coefficies ad bedig sregh. Ad he process is achieved usig he Opimiaio Toolbox i MATLAB combied wih VC++. Compared wih radiioal maual mehods such as eclosed char mehod or graph mehod, he dyamic opimiaio mehod ca allocae he coefficies more accuraely, also reduce he calculaio ime. This mehod ca improve he desig efficiecy ad has a impora pracical value. ACNOWLEDGMENT This work was suppored i par by gras from Shadog Provicial Naural Sciece Foudaio of Chia (Gra No.ZR00EM03), Sciece & Techology Research Guidace Projecs of Chia Naioal Coal Associaio (Gra No.MTJ0-059, Gra No.MTJ0-348), Projec for Scieific developme pla of Shadog Provice (No. 0GGX030) ad Docoral Fud of Miisry of Educaio of PRC(No: ). REFERENCES [] H.H. Mabie, E.J. ad V.I. Baema, Deermiaio of hob offse required o geerae osadard spur gears wih eeh of equal sregh, Mech. Mach. Theory, vol.8(3), pp. 8-9, 983. [] E. Rober, Raioal procedure for desigig miimumweigh gears, Gear Techol., vol.8(6), pp. 0-4, 99. [3] G. Niema ad H. Wier, Maschieelemee, d ed, Berli: Spriger-Verlag, 003. [4] Z. Wag, The ew mehod for he choice of profile shif coefficie of ivolue gear, Joural of Harbi Isiue of Techology (Harbi Gogye Daxue Xuebao), vol.z, pp. 9-47, 978. [5] H. Dog, Z. Zhu, W. Zhou ad Z. Che, Dyamic simulaio of harmoic gear drives cosiderig ooh profiles parameers opimiaio, J. Compu., vol.7(6), pp , 0, doi: /jcp [6] S. Wag, Y. Ji ad S. Yag, A sochasic combiaorial opimiaio model for es sequece opimiaio, J. Compu., vol.5(9), pp , 00, doi: /jcp [7] Q. Che, X. Che ad Y. Wu, Opimiaio algorihm wih erel PCA o suppor vecor machies for ime series predicio, vol. 5(3), pp , 00, doi: /jcp [8] T.A. Aal, A New Algorihm for Helical Gear Desig wih Addedum Modificaio. Mechaika, Mechaika, vol.77(3), pp , 009. [9] V. Spias ad C. Spias, Opimiig ivolue gear desig for maximum bedig sregh ad equivale piig resisace, Proc. Is. Mech. Eg. Par C J. Mech. Eg. Sci., vol.(4), pp , 007, doi: 0.43/ JMES34. [0] S. Baglioi, F. Ciaei ad L. Ladi, Ifluece of he addedum modificaio o spur gear efficiecy, Mech. Mach. Theory, vol. 49, pp. 6-33, 0, doi: 0.06/j.mechmachheory [] X. Zhag, Y. Rog, J. Yu, L. Zhag ad L. Cui, Developme of opimiaio desig sofware for bevel gear based o ieger serial umber ecodig geeic algorihm, J. Sofw., vol.6(5), pp. 95-9, 0, doi: /jsw [] M. Ciavarella ad G. Demelio, Numerical mehods for he opimiaio of specific slidig, sress coceraio ad faigue life of gears, I. J. Faigue, vol. (5), pp , 000, doi: 0.06/S04-3(98) [3] M.A. Sahir Arika, Deermiaio of addedum modificaio cofficies for spur gears operaig a osadard ceer disaces, i Proc. ASME Des. Eg. Tech. Cof., vol. 4A, pp , 003. [4] B. Peg, L. Zhag, R. Zhao, H. Zhag ad Y. Zheg, Srucure desig of wi-spirals scroll compressor based o 3C, J. Sofw., vol.7(9), pp , 0, doi: /jsw [5] X. Zhag ad S. We, Opimiaio mehod of modificaio coefficies of cylidrical gear pair havig hard ooh flaks i closed drive based o geeic algorihm, i Proc. World Cogr. Iellige Corol 03 ACADEMY PUBLISHER

6 JOURNAL OF COMPUTERS, VOL. 8, NO. 8, AUGUST 03 6 Auom. WCICA, vol. 3, Shaghai, 00, pp , doi: 0.09/WCICA [6] F. Gao, Applicaios of malab i mahemaical aalysis, J. Sofw, vol.6(7), pp. 5-9, 0, doi: /jsw Xueyi Li was bor i Hubei Provice, Chia i 97. He received his PhD. i Mechaical Egieerig from Xi a Jiaoog Uiversiy (XJTU), Chia, i 004. He currely serves as Associae Professor i Shadog Uiversiy of Sciece ad Techology (SDUST), Chia. From , he worked as a pos-docor i he Sae ey Laboraory of Auomoive Safey ad Eergy, Tsighua Uiversiy, Beijig. His research ieress iclude CAD/CAE, mechaical rasmissio ad virual prooype echology. Shoubo Jiag was bor i Shadog Provice, Chia i 988. He received his B.S. i Mechaical Egieerig from Shadog Uiversiy of Sciece ad Techology (SDUST), Chia, i 0. He is currely pursuig his Maser of Egieerig Sciece i SDUST. His research ieress are i area of fiie eleme aalysis, dyamics, ad applicaio program desig. Qigliag Zeg received his PhD. i Mechaical Egieerig from Chia Uiversiy of Miig ad Techology (CUMT), Chia, i 000. He currely serves as Professor i Shadog Uiversiy of Sciece ad Techology (SDUST), Chia. Ad his major researches cover he areas of hydraulic drive ad corol, mecharoics, CIMS, CE ad virual prooype. 03 ACADEMY PUBLISHER