Modelling of tangential vibrations in cylindrical grinding contact with regenerative chatter

Transcription

1 Modellng of tangental vbatons n cylndcal gndng contact wth egeneatve chatte Vel-Matt ävenpää, Lhong Yuan, Hessam Kalbas Shavan and asal Mehmood ampee Unvesty of echnology epatment of Engneeng esgn P.O.Bo 589, ampee, nland {vel-matt.javenpaa, lhong.yuan, hessam.albasshavan, fasal.mehmood}@tut.f} Abstact he gndng contact of a cylndcal gndng pocess may have unstable vbaton behavou such as egeneatve chatte o suface patten geneaton. hs wll have stong effect on the suface qualty of the wo pece. he stablty analyss of gndng contact has been studed n lteatue wdely. ypcally, the nomal decton s consdeed. n ths pape the tangental vbatons of the contact ae consdeed by focusng to the otatonal movements at the contact. A numecal model of the otatonal vbaton system s deved whch ncludes the gndng wheel, shaft, belt tansmsson and moto subsystem and the wo pecemoto subsystem. he nomal and tangental cuttng and sldng foces ae coupled by the fcton and the gndng penetaton tem and ths leads to the descpton of the ln between the tangental vbatons and the nomal decton stablty behavou. n the esults the tme doman smulatons and the stablty condtons n two dffeent unnng speed cases ae pesented. ntoducton he cylndcal gndng s a manufactung pocess to poduce hgh qualty suface fnsh on a cylndcal wo pece suface. Gndng n geneal belongs to the mateal cuttng pocesses such as tunng and mllng n whch chps ae emoved by a machne tool. he vbaton dynamcs of these cuttng pocesses s studed n lteatue wdely. Some eamples ae fo eample: hompson ntoduced the classcal theoy of chatte gowth wth double delays []; Moon eploed the dynamcal phenomena of seveal manufactung pocesses []; Nayfeh consdeed the chatte contol and the stablty analyss unde egeneatve cuttng condtons []; Altntas descbed the chatte vbatons n cuttng wth delay []; Stepan analysed cuttng pocess stablty by ncludng double delay effects [5]; and Lu (et al) nvestgated double delay stablty of cylndcal gndng [6]. he cylndcal gndng dynamcs s chaactezed by the egeneatve chatte phenomenon whch s elated to the tme delay effects. he nstablty of the gndng pocess esults fom the contact nteacton of the gndstone-wo pece and may cause damage and falue such as unacceptable suface fnsh and patten geneaton. he fst chatte souce s the wo pece due to the ovelap of the gndng path on ts suface. A egeneatve ectaton souce s geneated to the system because the pevous suface hstoy s entoduced to the gndng contact. he second chatte souce s the gndstone tself because the suface of the gndstone also loses mateal slowly and these out-of oundness damages cay on by the otatonal movement nto the gndng contact nteacton. Both of these chatte souces one fom the wo pece and anothe fom the gndstone can be descbed as ndvdual delay tems n the system dynamcs modellng and equatons. he vbatons of the gndng system n the tangental decton ae not consdeed usually because the obvous vbaton behavou clealy s n the nomal decton, and the dynamcal modellng at ths decton povdes suffcent equatons to detemne the stablty condtons. he tangental decton, howeve, povdes some addtonal nfomaton about the chatte vbaton development n the gndng system. he

2 man motvaton to nvestgate the tangental decton s to ncease the ablty to obseve the unstable chatte vbaton phenomenon. n ths pape the tangental decton s consdeed by modellng the otatonal degees-of-feedom system fo the gndng vbatons. Modellng of gndng system dynamcs he dynamcs of cylndcal gndng system wll be descbed now. he system equatons wll be pesented at fst and the gndng foce togethe wth the tme delay tems wll be pesented n the net chapte. gue llustates the otatonal dynamcs system and ts membes. b wopece gndstone dve dve pulley pulley belt gndng contact gue : he otatonal gndng system wth the gndstone, the wo pece, the shafts, the belt tansmsson and the dves. he system equatons can be wtten n the mat fom as b b b b, () whee and ae the otatonal degees of feedom, and the netas, the tosonal stffness, b the belt stffness, the aduses and the tangental gndng foce (compae wth gue ). he belt tansmsson system can be dealzed by lettng t ed ()

3 whee t s the tansmsson ato of the belt system. hs leads to the moe educed system pesented n gue. ed wopece gndstone dve dve system gue : he educed otatonal gndng dynamc system. he educed system equatons ae ed. () Now, by ntoducng the vaable tansfomaton fo the degees of feedom of each shaft le R () the system equatons tae the foms ed (5) and ed ed ed ed. (6) nally, the vbatons at the nomal decton can be descbed by -degees-of-feedom model pesented n gue. he system equatons n the nomal decton ae N N c c m m (7)

4 gndstone wopece c m m c gue : he gndng dynamc system n the nomal decton. whee ae the tanslatonal degees of feedom, m the masses, c the dampng coeffcents, the stffness, and N the nomal gndng foce. he complete system dynamcs equatons conssts of (5), (6) and (7). Gndng contact foces he gndng foces wll be descbed now. he nomal and tangental gndng foces of the suface gndng case pesented n [7] and [8] ae used. t s assumed that t povdes elable accuacy and the equatons can be updated to coespond to the cylndcal geomety n futue contets f necessay.. Suface gndng foces n the nomal decton one can descbe the cuttng penetaton n the wo pece at the followng way (gue ) [9]. Let be the total penetaton and the paamete defnes the mateal emoval aton between the wo pece and the gndstone. hus, the total mateal emoved s and beng close value to unty. n the tangental decton the chp emoval egons can be dvded to cuttng (chp fomaton) and sldng zones. he foce equaton composton dffes n these two zones due to the natue of cuttng pocess at each zone and ths lead to two tems n the gndng foce equaton n the nomal decton. he zone between them s not consdeed. he tangental gndng foce s elated to the nomal foce by fcton. cuttng zone sldng zone gndstone wopece gue : he nomal decton penetaton n the gndng contact and the cuttng and sldng zones. hus, the nomal and tangental gndng foces ae

5 N (8) Nc c s Ns (9) whee Nc, Ns, c and epessed as s ae the nomal and tangental cuttng and sldng foce tems. hese ae v G vg ( ) { K K ln b v v w w () v G vg ( ) { K K ln b v v w w () ( ) pab b () Nc } c } Ns ( ) () s Ns whee K ae the epemental gndng foce coeffcents, v G and gndstone and the wo pece, b s the wdth of the gndstone. Paametes v w the contact veloctes of the p A ae elated to the sldng fcton tems and s the fcton coeffcent. Because of the sepaate foce tems n (8) and (9) one can nto account only desed ones and nvestgate the effects ndvdually n the smulatons.. me delay tems he gndstone s movng hozontally on the suface of the wo pece. he gndng path ovelaps tself at some degee because ths ensues that the whole suface wll be gound. hs hozontal movement specfed two zones n whch the gndng foce wth the delay effects ae descbed as llustated n gue 5. gndstone t g ) t w) wopece gue 5: he ovelap of the gndng path and the two dffeent foce zones. he paamete specfes the ovelap ato. stly, the fesh cuttng zone les on the wo pece sde and thee s no delay tem fom ths zone. Secondly, on the ovelap suface aea the delay effect s geneated by 5

6 the delay of the wo pece w. nally, fom the whole gndstone suface ae the delay effect s geneated by the delay of the gndstone g. hese thee effects compose the total gndng foces wth delays by substtutng the penetaton values to equatons () to () as N ( ) N ( nom ) N ( ( t w)) ( ) N ( ( t g )), () ( ) ( nom ) ( ( t w)) ( ) ( ( t g )), (5) whee (6) and nom s the nomnal cut depth fo the gndng. Methods and analyss he tme ntegaton method used n the smulatons n MALAB s descbed n []. he method belongs to the Newma tme ntegato famly and t s also nown as the aveage constant acceleaton method. he method s mplct, uses a pedcto-coecto appoach and t ncludes the Newton-Rhapson teaton pocedue. he teaton mat used s K C M, (7) h h S whee and ae the tme ntegaton coeffcents and h s the tme step. he model also has a small Raylegh dampng wth the dampng mat whee C M K, (8) R R R and R ae dampng coeffcents. A P-speed contolle s defned by K t ) K ( ), (9) P ( d d whee d s the desed otaton speed and K P and K the contol coeffcents. he tme ntegaton pocedue must be upgaded fo the tme delay equaton. he method of steps can be used fo the tme doman soluton of tme delay equatons []. o a delay dffeental equaton an ntal hstoy peod s equed as a stat up fo the delay effects to develop. he method of steps pocedue specfes an ntal functon on nteval of [t -, t ], whee s the delay tme (gue 6). hs so-called ntal peod of the delay equaton s solved wthout delay effects at the begnng. hen the delay dffeental equaton can be solved by tang the delay tem values fom the hstoy. n the case of otatonal vbatons the tme delays ae non-constant and they ae dectly elated to the otaton speeds of the gndstone and the wo pece. hs eques moe comple soluton pocedue because the eact values of the delayed vaables ae not dectly avalable fom the data of pevous tme ncements due to the dscete soluton. he values must be estmated by a two-step polynomal ntepolaton pocedue [8]. n the fst step, the unnown value of the tme delay s detemned fom the nvese cuve of the otaton angle by subtactng fom ts cuent value and then ntepolatng the delayed tme t. n the second step the unnown values of the delayed vaables ae ntepolated accodng to the values of the delay tmes. hs pocedue s mplemented n a code but t s possble to ntoduce t fo eample n the SMULNK envonment. 6

7 gue 6: he ntal hstoy peod (t) n the method of steps pocedue. he geneal objectve of the analyss n ths pape s to llustate the stablty chaactestcs of the s degees-of-feedom gndng system. t s well nown that the system has multple unstable egons dependng on the values of the ey system paametes. As dscussed n the ntoducton ths topc has been consdeed n the lteatue wdely. ypcal paametes defnng the stablty ae the unnng speeds (the tme delay values) and fo eample the stffness of the gndng contact. he pmay objectve s to nvestgate how the couplng of the nomal and tangental gndng foces nfluences to the vbatons n the otatonal decton. he unstable behavou wll develop at the nomal decton and the vbatons eflectng ths at the tangental decton ae at the nteest. 5 Numecal esults and dscusson he numecal eample s ceated based on the paametes fom a heavy ndustal gndng machne used fo the gndng of lage cylndcal objects made fom steel. he mass of the gndstone s 5 g and the wo peces can have masses up to g. he hozontal lowest natual fequency of the gndstone mount s about Hz and the wo pece s long cylnde wth natual fequency between to Hz. Raduses of the gndstone and the wo pece ae 5 mm and about mm. he wdth of the gndstone s 8 mm. he unnng speeds ae - Hz and less than Hz, espectvely. he aveage cuttng depth s m and the paametes K and K n () and () ae about 6 N/m. Wth these system paametes a stablty bounday can be found between the gndstone tangental contact veloctes of 5 m/s and m/s. hus, two smulaton cases wee chosen to llustate the vbaton behavou of the gndng system. At the unnng velocty m/s (about Hz) the system s stable and at 5 m/s (about Hz) t s unstable. Only the cuttng foce tems wee consdeed n the smulatons. gue 7 shows the nomal foce and the wo pece otatonal vbatons at the stable case. he vbatons due to the ntaton of the gndng contact at the begnnng ae dyng out when the gndng contnues. he delay effect caes on n the smulaton but t does not act n unstable fashon. he model has vey modest vscous dampng at the nomal decton n the smulatons, whch s not lage enough to cancel the delay ectaton souces. gue 9 llustates the wo pece moto dynamc behavou. he bgge scale wavness n the fgues s due to the P speed contol. gues 9 and show the unstable case. Now the chatte vbaton effect s developng n the nomal decton clealy. Also at the tangental decton the chatte becomes detectable even though the scale of the effect seems to be at a smalle ange. mpotant ema s that the P speed contol does not eact to the tangental chatte vbatons because the contol paametes wee chosen n such way that the contol behaves athe wealy. he am s to nvestgate the delay effect phenomenon nstead of contollng t. t s, howeve, obvously possble to use P o P contol fo the speed whch can compensate the chatte vbatons at the tangental decton, but even n ths case the chatte becomes detectable then fom the contol sgnals. he obsevaton of the otatonal degees-of-feedom seems to povde a feasble method to detect the chatte vbatons n the cylndcal gndng system. 7

8 -8 - N [N] [ad] me [sec] gue 7: he stable case. he nomal foce s above and the otatonal vbaton of the wo pece below. he gndstone s contact velocty s m/s and the wo pece s m/s d [Hz] [Nm] me [sec] gue 8: he stable case. he angula velocty of the wo pece s above and the toque of the wo pece moto below. he gndstone s contact velocty s m/s and the wo pece s m/s. 8

9 N [N] [ad] me [sec] gue 9: he unstable case. he nomal foce s above and the otatonal vbaton of the wo pece below. he gndstone s contact velocty s 5 m/s and the wo pece s m/s d [Hz] [Nm] me [sec] gue : he unstable case. he angula velocty of the wo pece s above and the toque of the wo pece moto below. he gndstone s contact velocty s m/s and the wo pece s m/s. 9

10 Conclusons he numecal soluton of the delay equaton seems to wo well. he fst man pupose of ths soluton pocedue s to povde a tool fo the stablty analyss of comple delay systems. An altenatve use could be fo measuement data vefcaton to detect the pesence of delays n the system measued. uthe wo s equed to vefy the modellng appoach of ths pape. Acnowledgments he authos would le to epess the gattude to the Academy of nland, whch has povded fundng fo ths eseach. Refeences [] R. A. hompson, On the oubly Regeneatve Stablty of a Gnde: the heoy of Chatte Gowth, ounal of Engneeng fo ndusty, May 986, Vol. 8/75. []. C. Moon, ynamcs and Chaos n Manufactung Pocesses, ohn Wley & Sons, nc., 998 [] A. H. Nayfeh, Poblems n Petubaton, 99 [] Y. Altntas, Manufactung Automaton, Cambdge Unvesty Pess,. [5] G. Stepan, Modellng Nonlnea Regeneatve Effects n Metal Cuttng, Phl ans R Soc Lond A () 59, he Royal Socety, [6] A. Lu, G. Paye, Stablty Analyss of oubly Regeneatve Cylndcal Gndng Pocess, ounal of Sound and Vbaton, (7) [7]. ang,. u, Y. Chen, Modelng and Epemental Study of Gndng oces n Suface Gndng, ounal of Mateals Pocessng echnology, 9 (9) [8] S. Maln, Gndng echnology heoy and Applcatons of Machnng wth Abasves, [9] L. Yuan, E. Kesnen, V. M. ävenpää, Stablty Analyss of Roll Gndng System wth ouble elay Effects, UAM Symposum on Vbaton Contol of Nonlnea Mechansms and Stuctues, Sold Mechancs and ts Applcatons Volume, 5, [] M. Géadn,. Ren, Mechancal Vbatons-heoy and Applcaton to Stuctual ynamcs, ohn Wley & Sons, nc., 997. [] R.. ve, Odnay and elay ffeental Equatons, Spnge-Velag, New Yo, 997