DESIGN AND FINITE ELEMENT MODE ANALYSIS OF NONCIRCULAR GEAR

Transcription

1 DESIGN AND FINITE ELEMENT MODE ANALYSIS OF NONCIRCULAR GEAR Cho Li 1, Ki Cheg 2, Dtog Qi 1, Cicho Zhu 1, Hu Qiu 3, Xiohu R 1 1. The Stte Key Lbortory o Mechicl Trsmissio, Chogqig Uiversity, Chi 2. School o Egieerig d Desig, Bruel Uiversity, UK 3. Deprtmet o Mechicl Egieerig, Kyushu Sgyo Uiversity, Jp Abstrct The ocirculr ger trsmissio is importt brch o the ger trsmissio, it is chrcterized by its compct structure, good dymic equilibrtio d other dvtges, d c be used i the utomobile, egieerig mchie, ship, mchie tool, vitio d spcelight ield etc. Studyig o the dymics eture o ocirculr ger trsmissio c improve the bility to crry lods o, reduce the vibrtio d oise o, icrese the lie o the ocirculr ger trsmissio mchie, provides guidce or the desig o the ocirculr ger, d hs sigiict theories d prcticl meigs. I this pper, the ger trsmissio techique is used to studied the desig method o the ocirculr ger, which cotis distributio o teeth o the pitch curve, desigs o the tooth tip curve d the tooth root curve, desig o the tooth proile curve, the ger system dymics priciple is itroduced to estblish dymics model or the ocirculr ger; bsic theory o iite elemet d mode lysis method re pplied, iite elemet model or the ocirculr ger is estblished, turl vibrtio chrcteristic o the ocirculr ger is studied. Ad the ovl ger is tke s exmple, the mthemtics sotwre MthCAD, the 3D modelig sotwre UG d the iite elemet sotwre ABAQUS re used to relize precise 3D model o the ovl ger. The iite elemet method is used, the turl vibrtio chrcteristic o the ovl ger is studied, the mi vibrtio types d turl requecies o the ovl ger d tht o the equivlet cylidricl gers re lyzed d compred, the coclusios received relect the dymics perormce o the ovl ger, d solid oudtio is lid or dymics reserch d egieerig pplictio o the ovl ger trsmissio. Keywords: Nocirculr ger, Fiite elemet method, Nturl requecy, Nturl vibrtio shpe. 73

2 1. Itroductio The ocirculr ger is importt brch o ger trsmissio, c be used to trsmit movemet d power betwee two itersectt xes, is chrcterized by its compct structure, good dymic equilibrtio d other dvtges, d c be pplied i utomobile, egieerig mchie, ship, mchie tool, vitio d spce light ield etc. The curretly, the studyig work o ocirculr ger cocetrtes o geometry modelig, kiemtics, mchiig etc, while tht o dymics is much less. Studyig o the dymics eture o the ocirculr ger trsmissio c improve the bility to crry lods o reduce the vibrtio d oise, icrese the lie o the ocirculr ger trsmissio mchie, provides guidce or the desig o the ocirculr ger, d there re sigiict theories d prcticl meigs. 2. Desig o the Nocirculr Ger The pitch curve o the ocirculr ger is ocirculr, which mkes the desig o the ocirculr ger diicult. The keys o the ocirculr ger desig re to determie the positio o the pitch curve o ech tooth, the tooth tip curve, the tooth root curve d the tooth proile curve o the ocirculr ger First, it give poit o the pitch curve s begiig poit. The determie the positios or let d right tooth proile o ech tooth by clcultig rc legth ccordig to pitch distce d spirl thickess [1]. 2.1 The Tooth Tip Curve d Tooth Root Curve The tooth tip curve d the tooth root curve o the ocirculr ger re orml equl-distce curves o the pitch curve, the orml distces betwee them d the pitch curve re the tooth ddedum d the tooth root height respectively[1], the show i Fig. 1. From Fig. 1 the tooth tip curve ormul c be writte s. 2 2 r = r + h + 2rh si µ (1) Where: r Tooth tip curve rdius, r Pitch curve rdius, h Tooth ddedum, µ Agle betwee tgetil directio d rdil directio o poit (P) o pitch curve. r µ = rct (2) dr dϕ h cos µ θ = ϕ rcsi (3) r Where: θ Polr gle o tooth tip curve, ϕ Polr gle o pitch curve. The tooth root curve ormul c be writte s. 74

3 Where: r Tooth root curve rdius, Where: θ Polr gle o tooth root curve. 2.2 The Tooth Proile Curve 2 2 r = r + h + 2rh si µ (4) h Tooth dededum. h cos µ θ = ϕ + rcsi (5) r The tooth proile curve o the cylidricl ger is ivolute o the bsic circle, d c be settled ccordig to the bsic circle. The tooth proile curve o the ocirculr ger is computed rom evolute o tooth proile, d the proile curve o ech tooth is dieret[2]. The tooth proile curve o the ocirculr ger c be derived rom pitch curve ormul by lytic method s show i Fig Pitch curve, 2 Root curvem, 3 Tip curve. 1 Pitch curve, 2 Let tooth proile, 3 Right tooth proile. Fig. 1. Tip curve d root curve Fig. 2. Tooth proile curve From Fig. 2 the right tooth proile curve ormul o the ocirculr ger c be writte s x r cosϕ cos( ϕ + µ + α ) r yr r si ϕ si( ϕ + µ + α ) (6) Where: xr X coordites vlue o right tooth proile, yr Y coordites o right tooth proile, α Pressure gle o tool, Distce rom itersectio poit betwee pitch curve d orml o tooth proile to tooth proile log orml directio o tooth proile. Let tooth proile curve ormul o the ocirculr ger c be writte s. x r cosϕ ± cos( ϕ + µ α ) y l l r siϕ ± si( ϕ + µ α ) (7) Where: x X l coordites o let tooth proile, y Y l coordites o let tooth proile. From the ger meshig theory. = cosα (8) S Where: S Arc legth o the pitch rom poit () to itersectio poit ( ) betwee the pitch curve d the tooth proile curve. The tooth proile curve o the ocirculr ger c be relized by two methods: 1) The progrmmig, which is diicult to commo desiger. 2) The equivlet method, which use the ivolute o the equivlet cylidricl ger to substitute the tooth proile curve o the ocirculr ger, d mke the model imprecise. All these 75

4 mke the lysis o the ocirculr ger diicult. This pper ims t this problem, tkes the ovl ger s exmple, uses the tooth proile curve ormul, combies mthemtic sotwre MthCAD, sotwre UG d iite elemet sotwre ABAQUS, d relizes the precise model o the ovl ger. 2.3 Ovl Ger Modelig The pitch curve ormul o the ovl ger c be writte s. r (1 k ) /(1 k cos( ϕ)) 2 = (9) Where: = 2, k Eccetricity, Rdius o log xis. The pitch curve o the ovl ger is symmetricl with the X-xis d Y-xis o crtesi coordite. For the desig coveiece, the tooth umber Z = 4 C + 2 (C is positive iteger), d the sectios t log d short xes should be tooth d lveolus respectively. The desig steps o ovl ger modelig re show i Fig. 3. The prmeters o the ovl ger i this pper: The tooth umber Z = 22, modulus m = 5mm, eccetricity e =.1, tooth ddedum h = 5mm, tooth height h = mm, tooth width B = 25mm, rdius (legth hl xles) 54. mm =. The two ovl gers re sme. = 728, ier rdius r i 25mm Give desig prmeters: tooth umber z, modulus m, eccetricity k, d solve rdius. Determie the polr gles o the itersectio poits o the pitch curve d the tooth proile curves. Use MthCAD d the tooth proile curve ormul to product coordite dt o key poits o the tooth proile curves. Import the poit dt ito UG, obti the tooth proile curve by ittig with cube splie, obti the ple sketch o the ovl ger by rc legth, corer, trim, d mirror uctio, sve the ple sketch s IGS ile, import ito ABAQUS. The get the 3D model o the ovl ger by extrude uctio, show i Fig. 4. Fig. 3. Desig o the ovl ger modelig 3. Fiite Elemet Model o the Ovl Ger The ABAQUS is oe o the most dvced lrge-scle geerl iite elemet sotwre i the world, d hs powerul uctio i big stri, olier (geometry, mteril d boudry), viscoelstic, dymic stress, d cotct problem ields etc [3]. 76

5 11 2 I this pper, the mteril o the ovl ger is 45 steel, Youg s modulus E = 2. 1 N / mm, Poisso s 3 3 rtio µ =. 3, d Desity ρ = kg / m. Crete mteril steel uder mteril module, crete the ovl ger sectio, set mteril steel s property o the ovl ger sectio, d ppoit to sectio o the ovl ger. The boudry coditio o iite elemet model or the ovl ger c be set ccordig to the ctul workig coditio. I the meshig process o the ovl gers, itererece it is pplied betwee iside surce d xis with splie, the itererece it betwee xis d the ger c be cosidered s rigid coectio i the iite elemet model, d the iluece o splie is eglected. The tolerce o this simpliictio is smll to dymic study. I order to relect the ctul coditio o ger meshig correctly, the ier surce o the ovl ger is restricted, d displcemets log X xis, Y xis, Z xis d rottios roud with X d Y xis re restricted. I the ABAQUS, the 3D solid uit oly hs three displcemet reedoms. I order to restrict rottio reedom o the ovl ger s ier surce, couplig must be dded to couple the ier surce to poit o the ceter rottig xis o the ovl ger, d reedoms o the ovl ger s ier surce c be restricted by settig the reerece poit s reedoms. The boudry coditio o the ovl ger c be pplied to iitil step. The mode is determied by turl property o the ger system, d it is irrespective with outer lods, so it is eedless to set the lod boudry coditio or the ovl ger. I course o the meshig, distortio should be reduced rthest, s or the problem tht the grids distorts bdly, smll sized lier reduced itegrtio uit c be used, or the 3D problem, hexhedro uit should be pplied utmost, which c get better result with lower cst, the result received orm tetrhedro uit is imprecise, so lrge umbers o uits must be pplied to get better result, which mkes computig cost icrese gretly. Accordig to the priciple bove, swept meshig techique is pplied i the this model, C3D8R uit (8 odes hexhedro reduced itegrtio uit) is used, d 196 uits d odes re received. The iite elemet model o the ovl ger completed is show i Fig. 5. Fig. 4. The 3D model o the ovl ger Fig. 5. The iite elemet model o the ovl ger 4. Clcultig the Nturl Mode d Nturl The methods o clcultig the turl mode d turl requecy. Accordig to the mechicl system dymics theory d the iite elemet theory, the movemet dieretil equtio o the multi-reedom system c be writte s[4]. [ M ]{ u } + [ C ]{ u } + [ K ]{ u} = { p (t )} Where [ M ] Mss mtrix, { u } Accelertio mtrix, [ ] [ K Stiess ] mtrix, { u} Displcemet mtrix, { p(t) } (1) C Dmpig mtrix, { } Outer lod mtrix. u Velocity mtrix, 77

6 Whe the dmpig orce is eglected d the system is ree o lod, the movemet dieretil equtio o udmped ree vibrtio system c be writte s[4]. ([ ] ω 2 [ M ]){ φ} = K (11) Where ω Frequecies o system, { φ } Eigevector o system. The LANCZOS method d the subspce itertive method re provided to clculte eigevlue. Whe the system hs my reedoms d pletiul chrcteristic modes re requested, it is much quicker by pplyig the LANCZOS method, while ew chrcteristic modes (< 2) re requested, it is much quicker by usig subspce itertive method. I this pper, the LANCZOS method is pplied. 5. Alyticl Result The structure vibrtio c be expressed s lier combitio o ech order turl vibrtio shpe, while lower order vibrtio shpe hs big iluece o structure vibrtio, d ply decisive role i structure s dymic chrcter. First 5 to 1 orders re eeded oly whe mode lysis. I order to expli the dymic chrcter o the ovl ger cotrstively, the iite elemet models o equivlet cylidricl gers ( degree, 3 degree, 6 degree, 9 degree) re estblished, d the turl vibrtios d turl requecies re clculted d lyzed. The 1st, 2d, 3rd, 5th, 7th, 1th mode vibrtio shpes re show i Fig. 6 () ~ (), d the vibrtio shpes d turl requecies re show i Tble 1. The Tble 1 d Fig. 7 show tht the turl requecies o the ovl ger lie betwee the correspodig order s turl requecies o the big sectio ( degree) d tht o the smll sectio (9 degree), they re bigger th tht o the big sectio d smller th tht o the smll sectio. The turl requecy icreses with the order icreses. The vibrtio shpes o the ovl ger re sme with tht o equivlet cylidricl ger, but the orders rise re dieret. Compred with the cylidricl ger, the requecies to ech order o the ovl ger is dieret obviously, while the requecies to ech order o the cylidricl ger my be sme or similr. The reso is tht the cylidricl ger is symmetricl with the rottig ceter bsolutely, while the ovl ger is symmetricl with the rottig ceter icompletely. The Tble 1 d Fig. 6 show tht the 3rd, 4th d 6th mode vibrtio shpes re sme, the 5th d 9th mode vibrtio shpes re sme, d the 7th d 8th mode vibrtio shpes re sme. The mi diereces lie i tht the vibrtio directios o ech tooth re dieret. The Fig. 8 shows tht the turl requecies to ech order o the equivlet cylidricl ger icrese while the polr gle o the ovl ger icreses d the pitch curve rdius decrese. The compred with cylidricl ger, the distce mog the mplitudes o ech tooth to ech mode o the ovl ger is quite big. For exmple, or the 2d SZ mode, the mplitude o the tooth bout the big sectio ( degree) is quite big, while the tooth bout the smll sectio does t vibrte bsiclly. 78

7 The Fig. 6 d Tble 1 show tht the mi vibrtio shpes o the ovl ger is the DZ mode d YZ mode, while rdil vibrtio is quite smll. So the DZ mode d YZ mode is the vibrtio shpe which is most possible to rouse resoce o the ovl ger. I desig o the ovl ger trsmissio system, the turl vibrtio shpes d turl requecies should be cosidered dequtely, the workig requecy should keeps wy rom the turl requecies to void resoce. Tble 1: Nturl requecies d vibrtio shpes o the ovl ger d the equivlet gers Model Order Ovl ger Type DZ1 SZ DZ2 DZ2 YZ DZ2 DZ3 DZ3 YZ DZ Type DZ1 DZ1 SZ DZ2 DZ2 DZ3 DZ3 YZ YZ DZ Type DZ1 DZ1 SZ DZ2 DZ2 DZ3 DZ3 YZ YZ DZ Type DZ1 DZ1 SZ DZ2 DZ2 DZ3 DZ3 YZ YZ DZ Type SZ DZ1 DZ1 DZ2 DZ2 YZ YZ DZ3 DZ3 DZ3 DZ1 1st olio vibrtio,dz2 2d olio vibrtio,sz Bevel vibrtio,dz3 3rd olio vibrtio,dz4 4th olio vibrtio,yz Circle vibrtio () 1st mode vibrtio (b) 2d mode vibrtio (c) 3rd mode vibrtio shpe shpe shpe Fig. 6. (d) 5th mode vibrtio (e) 7th mode vibrtio () 1th mode vibrtio shpe shpe shpe Mode vibrtio shpes o the ovl ger 3 Ovl deg 9 deg 25 (F/Hz) Order Fig. 7. Reltio o requecies betwee the ovl ger d equivlet ger 79

8 st 3rd 4th 6th 8th 1th (F/Hz) Pitch curev rdis (R/cm) Fig. 8. Reltio betwee requecies d rdius 6. Coclusio I this pper, the desig method o the ocirculr ger is studied by usig ger trsmissio techique, the mthemtics sotwre MthCAD, the 3D solid modelig sotwre d the iite elemet sotwre re combied to relize precise model o the ovl ger, d solid oudtio is lid or lysis o the ovl ger. The ger system dymics priciple is itroduced to estblish dymics model or the ocirculr ger. The bsic theory o iite elemet d mode lysis method re pplied, the iite elemet model or the ocirculr ger is estblished, d turl vibrtio chrcteristic o the ocirculr ger is studied. The iite elemet method is used, the turl vibrtio chrcteristic o the ovl ger is studied, the mi vibrtio shpes d turl requecies o the ovl ger d tht o the equivlet cylidricl gers re lyzed d compred, the coclusios received relect the dymics perormce o the ovl ger, d solid oudtio is lid or dymics reserch d egieerig pplictio o the ovl ger. Ackowledgemet The uthors wish to ckowledge the ssistce d support o the Ntiol Sciece d Techology Plig Key Project o Chi (No. 26BAF1B7-1). Reereces [1] Wu Xutg, Nocirculr Ger d Vrible Rtio Trsmissio, Beijig: Mchiery Idustry Press, Chi, pp. 52~53, [2] Li Fusheg, Desig o Nocirculr Ger d Specil Ger, Beijig: Mchiery Idustry Press, Chi, pp. 57~59, [3] Zhug Qu, Accidece Guide or ABAQUS Fiite Elemet Sotwre, Beijig: Tsighu Uiversity Press, Chi, pp. 49~8, [4] Li Rug, Dymics o Ger System, Beijig: Sciece Press, Chi, pp. 69~96,