JJMIE Jordan Journal of Mechanical and Industrial Engineering

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1 JJIE Joda Joual o echaical ad Idustial Egieeig Volume 8 Numbe 4, August 4 ISSN Pages 7 - Dyamic Aalysis ad Desig o Steel-Ball Gidig achies Based o No-Slip Cases Jigju Zhag *, Guoguag Li, Ruizhe Gao, Bi Ya, Huimi Xue Hebei Uivesity o Egieeig, Hada, Chia 568 Received 4 Ja 4 Accepted 8 Aug 4 Abstact Based o o-slip cases betwee steel balls ad lappig discs while a hoizotal gidig machie is wokig, the motio equatios o the ball lappig ae established ad the law o motio is evealed by the ATLAB sotwae. This pape shows that the dieece o the tace spacig is educed by iceasig the disc diamete, which ca make the tace distibutio o the steel ball suace moe uiom. I ode that balls do ot slip whe otay disc dives them to otate, the ages o the lappig pessue ad lappig disc speed is deduced by usig the dyamics aalysis cosideig om the vetical tech suace ad alog the goove suace, espectively. Fially, this pape takes the selected mateial as a example to aalyze the pocess paametes that aect the lappig cuves, ad explais the selectio piciples o alpha ad beta whe the tech has some bias, which ca povide a eeece to impove the quality o the steel balls suace. 4 Joda Joual o echaical ad Idustial Egieeig. All ights eseved Keywods: : Gidig achie, Lappig, Steel Ball, Dyamics, Desig, ATLAB Sotwae.. Itoductio Beaigs ae oe kid o vital pats o mode machiey ad equipmet. They ae widely used i may idusties. Thei mai uctio is to suppot the mechaical otatig body ad educe the coeiciet o load ictio i the tasmissio pocess. As show i Figue, steel balls ae impotat compoet o ball beaigs, which pocessig techology has uique specialized chaacteistics ad the pocessig quality will lagely aect the lie ad eliability o the ball beaigs [- ]. Figue. Ball beaig Lappig is the last step i steel balls pocessig techology. I the meatime, steel ball-billets ae squeezed, scaped by lappig discs ad abasive, which ca emove the machiig allowace o the ball suace ad impove steel balls spheicity as well as educe the oughess o the suace [4, 5]. The spheicity is a mai techical idex o steel balls ad it has a geat impact o beaig peomace (such as accuacy, oise, vibatio, etc.) [6-8]. About lappig, the covetioal view is that otay disc dives steel balls evolutio aoud the disc axis, ad the liea velocity dieece o the ac betwee the spheical suace ad goove makes balls otatio. The evolutio ad otatio complete the lappig o steel balls togethe [9]. This itepetatio is elatively simple because the evolutio speed is much geate tha the speed o the otatio, which will make balls become ellipsoid i the lappig pocess. Zhu [] poposed that the lappig disc gids alog the thee cicles o the steel ball suace epetitively though the cotact poits ude the lappig method o two discs alog a coaxial. It has bee poved that the lappig tajectoy is a tace cicle oudig a axis o the ball cete whe a steel ball is i motio. This pape pesets the motio equatios o the lappig model without slippig, ad the agula velocity o otatio, agula velocity o evolutio as well as agle o delectio ae calculated by usig the ATLAB sotwae. Based o the aalysis o the tace distibutio o the steel ball suace, it is show that educig the dieece o two tace cicles ca make the tace distibutio o the steel ball suace moe uiom. I ode o the balls ot to slip whe the otay disc dives them to otate, the ages o the lappig pessue ad lappig disc speed ae deduced by usig the dyamics aalysis cosideig om the vetical * Coespodig autho. satt88@6.com.

2 8 4 Joda Joual o echaical ad Idustial Egieeig. All ights eseved - Volume 8, Numbe 4 (ISSN ) tech suace ad alog the goove suace, espectively. Fially, this pape takes the lappig disc mateial HT, steel balls mateial GC5 as a example to aalyze the act that pocess paametes aect the appig cuves, ad explais the selectio piciples o alpha ad beta whe the tech has some bias, which ca povide a eeece to impove the quality o the steel balls suace.. Dyamic Aalysis o Ball Lappig.. otio Equatio Figue. The motio aalysis o a steel ball igid bodies, the equatios o lappig motio without slippig at thee cotact poits is expessed as ollows: ω R + ω ΩR ω R ω si( α θ ) () ω - R -ω si( β +θ ) whee Ω deotes the agula speed o otay disc, θ is the agle o delectio, ad is the adius o steel ball. Lappig paametes ae calculated, espectively, as: R si β + R siα taθ R cos β + R cosα R si( α + β ) Ω ω R si( α + β ) + R cos β + R ω () cosα RRΩ si( [ R β + θ ) + R ] I which R R -cosα, R R +cosβ. The above omulae povide the o slippig laws o lappig motio. Whe the positio o steel balls i the goove ad the speed o otay disk ae detemied, θ, ω adω will have a uique solutio, which meas the motio ca be uiquely idetiied. It also shows that ω, ω ae popotioal to Ω. Sice θ is geeally ot equal to, ω will have a omal compoet ad a tagetial compoet. The coespodig movemet o omal compoet ad tagetial compoet take a pat maily i gidig steel balls ad ollig steel balls, espectively. The lappig o steel balls is implemeted by the movemets togethe... Stuctue Optimizatio o Lappig Disc Figue. The tace distibutio o lappig Balls ad lappig discs ae egaded as igid bodies whe a hoizotal gidig machie is wokig ad the cotacts betwee them ae idealized as poit cotacts. As show i Figue, A, A, A ae thee cotact poits betwee balls ad lappig discs, the evolutio adius o the balls ae R, R, R. The motio o balls i the goove ca be divided ito ω that evolves aoud the cete o discs ad ω that otates aoud its ow sphee cete. Geeally, the otatio ω ca also be decomposed ito pivot motio ad oll motio []. Without cosideig the cotact deomatio ad applyig the geeal piciple o Thee exists the agle o delectio θ emaiig uchaged whe the tech tucate αβ45 is take ito cosideatio. I the same ciculatio with dieet otatioal loops, the lappig taces ae thee cicles aoud the axis o otatio. The ial machiig is completed by usig the epeated lappig. As show i Figue, the distace amog thee taces ae a ad b, which ca be expessed as: a cos( 45 θ ) + siθ b cos(45 + θ ) siθ () R + R taθ R + R Fom the above omulae, we ca get a b ( + ( + ) R + ) R + Thee ae thee tace cicles o the suace o steel balls, whee A is i the middle o A ad A. It ca be see om Eq. (4) that the iteval atio o tace cicles ae oly elevat to /R, ad the elatio cuve o which is show i Figue 4. It is showed by tests that the value o a/b ilueces the lappig balls quality ad eiciecy, ad (4)

3 4 Joda Joual o echaical ad Idustial Egieeig. All ights eseved - Volume 8, Numbe 4 (ISSN ) 9 the accuacy o balls suace will be impoved whe the value is close to []. G G cosγ -G si γ G -G cosγ G si γ,, G G cosγ G si γ G -G cosγ -G si γ (5) Figue 4. The cuve o /R -a/b.. Dyamic Equatio o the Vetical Tech Suace Without takig ito accout the ball iteactio ad the impact o lappig luid, a sigle steel ball without slippig is chose to be the eseach object. I ode o the balls ot to slip i all diectios, the vetical tech suace ad alog the goove suace ae aalyzed, espectively, as show i Figue 6 ad Figue 7. Figue 5. The gavity decompositio o steel balls Whe the odiate is equal to, the abscissa will be close to iiitely, which meas that the adius o the evolutio is much lage tha the ball adius. The abscissa o outside goove ca be educed by iceasig the disc diamete, ad the quality o balls suace ca be impoved. Howeve, thee ae some poblems whe iceasig the diamete. The liea velocities iside ad outside the goove will ievitably lead to a geat dieece, which meas that it caot make all balls get i ad out o the goove smoothly at the same time, ad the the steel ball suace will be destoyed. Thus, it is essetial to coside the above two aspects whe the value o /R is selected. Figue 6. The stess aalysis o the vetical tech suace. Dyamic Aalysis o Lappig Ball Figue 5 pesets the ball distibutio i the same tech o hoizotal lappig machie. Supposig the weight o a sigle ball is G, the compoets o ou quadats ae calculated as: N N si 45 N ( F + F + F ) N si 45 N si 45 + R si 45 + F cos 45 F F G + F cos 45 + F Figue 7. The stess aalysis alog the goove suace I case that steel balls ae ot allowed to slip aoud the diectio o z-axis i the V-shaped goove, the coditios that the dyamic equatio has to satisy ae as ollows: F x, F y, z ( F) (6) It ca be also witte i the om as ollows: cos 45 cos 45 (7)

4 4 Joda Joual o echaical ad Idustial Egieeig. All ights eseved - Volume 8, Numbe 4 (ISSN ) Whee N, N, N epeset the pessues at thee cotact poits, F, F, F deote the oces o the slidig ictio at thee cotact poits, R * ad * ae the ietia oce ad couple o a ball, espectively. Suppose that the citical pessues o thee cotact poits ae [N ], [N ], [N ], ad the citical oces o the slidig ictio ae [F ], [F ], [F ], espectively, thei elatio ca be expessed as ollows: [F ] [N ], [F ] [N ], [F ] [N ] (8) Combiig Eqs.(7) ad (8), it ca be obtaied as + ( + ) R G [ N ] + + ( + ) ( + + ) R ( + + ) G [ N ] + + ( ) ( + ) R ( + ) G [ N ] Each o citical pessues cotais R *, G ad *. Geeally, R *, G, */ have the same ode o magitude, ad the slidig ictio coeiciets ae elatively small. Thus, Eq. (9) ca be expessed appoximately as: [ N ] ( ) ( ),[ N ],[ N ] (9) ( ) () To the motio balls, the otatioal ietia J ad ietia couple * ae calculated as: J m, Jω ω 5 () The citical pessues eed to less tha the actual lappig pessues at thee cotact poits i ode ot to slip, which ca be witte as: [N ]<N, [N ]<N, [N ]<N () Usig Eqs.(), () ad (), the lowe limit iequality o N ca be deived as N ( ) mωω 5 > ().. Dyamic Equatio alog the Goove Suace Aalogously, to pevet steel balls om slippig aoud the diectio o x-axis ad y-axis, the coditios have to be satisied as ollows: z F, ( F), ( F) x y (4) Which ca be witte i the ollowig om: F + F4 + F5 + Gt ( F 4 cos45 + F5 cos45 ) + mx (5) ( F F4 si 45 F5 si 45 ) + my whee F, F 4, F 5 epeset the oces o the slidig ictio at thee cotact poits,,, deote the pivot ictioal momets at thee cotact poits, m, m, m ae the ollig ictioal momets at thee cotact poits, m x, m y ee to the algebaic sums o pojectio that pivot ictioal ad ollig ictioal momets wok o x-axis ad y-axis, espectively, ad i which mx + ( )cos 45 + ( m m)cos45 m y ( + )si 45 + m + ( m + m)si 45 Because the ollig ictioal momet is a less tha pivot ictioal momet at the cotact poits, oe ca wish to omit the ollig ictioal momet, which meas oly to coside the pivot ictioal momet. The the above expessios ca be simpliied as: m + ( ) cos, x 45 m y ( + )si 45 (6) By Eq. (5) ad (6), it ca be obtaied as: ( )( + Gt ) F ( ) F4 [ ( + ) + ( + ) G t ] (7) F 5 [ + ( ) + ( ) G t ] Supposig that the elastic modulus ad poisso's atio o steel balls ae E g, υ g, the elastic modulus ad poisso's atio o lappig discs ae E y, υ y. Accodig to the elastic cotact theoy [], the pivot ictioal momets ca be calculated as ollows: π N υ g υ y N ( + ) 4 Eg Ey π N υ y g υ N ( + ) (8) 4 Eg Ey π N υ y g υ N ( + ) 4 Eg Ey Aalyzig the statics o steel balls i the goove, it ca be oud that the atio o lappig pessue is appoximately equal to the atio o citical pessue. Thus, it is easoable that the elative elatio o actual lappig pessue is: N : N : N :: (9) Substitutig Eq.(8) ad Eq.(9) ito Eq.(7), it is obtaied as ollows ( ) F ( ) G t F F 4 5 ( + )( ) G t G t () I the lappig pocess, the otay disk has to ovecome the pivot ictioal momet ad ollig ictioal momet, ad the dives steel balls to otate. To stop balls om slippig whe they ae otated, the coditios that have to be satisied ae as ollows

5 4 Joda Joual o echaical ad Idustial Egieeig. All ights eseved - Volume 8, Numbe 4 (ISSN ) F <N, F 4 <N, F 5 <N () Fom Eq. (), it ca be see that F 4 is geate tha the othe two oces o the slidig ictio. Selectig F 4 as the epeseted omula ad substitutig Eq. () ito Eq. (), oe obtais [ 6 + ( ) + ( ) Gt ] < N ().. Selectio o the Pessue ad Speed It ca be daw om Eq. (8) that is popotioal to N 4/, ad it is obtaied that N is less tha a speciied value, eeed to as B by substitutig Eq. (8) ito Eq. (). Similaly, to solve the ist ad thid iequalities o Eq. (), it ca deduce that N is less tha A ad C, espectively. Theeoe, the ollowig iequality has to be satisied i lappig balls do ot slip i all diectios: ( ) mω ω < N < mi 5 { A, B, C} () Substitutig Eq. () ito Eq. () gives that the expessio o the lowe limit is popotioal to Ω. Sice A, B, C ae costat ad idepedet o the otay disc speed, it ca deduce Ω<D, which meas D is the maximum allowable speed o the otay disc. 4. Iluecig Paamete 4.. Pocess Paamete Whe the steel balls otate without slippig i the goove, the adius o the evolutio aects the lappig pessue ad the maximum allowable speed. Hee select HT as the mateial o lappig discs, GC5 as the mateial o steel balls, ad 5mm as the evolutio adius o R. Figue 8. Coeiciets o ictio that wok o lappig cuves Figue 9. Sizes o balls that wok o lappig cuves As show i Figue 8, it is the elatio cuve betwee the lappig pessue ad maximum allowable speed. The cuve shows that with the icease o the lappig pessue, the maximum allowable speed ises. The maximum allowable speed o the otay disc ca achieve a highe value whe the coeiciet o the ictio iceases, which geeates that steel balls ae less likely to slip i the lappig pocess. I Figue 9, it ca be see that the maximum allowable speed o the otay disc deceases while the sizes o the steel balls ae iceasig, which illustates that the steel balls will slip easily. The lappig pessue will be iceasig ad the speed o the otay disc will be educig at this momet. 4.. Tech Bias Thee ae may actos that aect the suace quality o the ball i the lappig pocess, such as the goove shape, machie accuacy, mechaical popeties o lappig disc, etc. The goove shape is elatively easy to cotol ad chage, which ca be achieved by chagig the paametes α ad β. As show i Figue, the omal agula velocity ad agula tagetial velocity at thee poits ae as ollows: t ω ω siθ, ω ω t ω ω cos( α θ ), ω ω si( α θ ) (4) t ω ω cos( β + θ ), ω ω si( β + θ ) Fom the above omulae, it ca be see that the omal compoet ad tagetial compoet ae the egative elatios, ad thee exist dieet lappig coditios at the cotact poits. I Figue, it is the elatio cuve betwee α ad taθ, which shows that taθwill be equal to whe α is ea to the value o π/4, ad at this time the dieece o the lappig coditio is elatively smalle. Thus, αad βshould comply with the cetai piciple whe the tech is bias. They should be uequal, the dieece betwee them is vey small, ad thei sum is aoud π/. It ca make the tace cicles have a good distibutio ad impove the eiciecy o the ball lappig simultaeously.

6 4 Joda Joual o echaical ad Idustial Egieeig. All ights eseved - Volume 8, Numbe 4 (ISSN ) () The possibility o slip cases ca be educed by iceasig the lappig pessue o the coeiciet o the ictio. Lappig big steel balls ae easie to slip tha the small oes, ad the ways o iceasig the lappig pessue ad deceasig the otay speed ca solve this poblem eectively. (4) A satisactoy distibutio o tace cicles ca be achieved i the dieece betwee αad β is ot big as well as thei sum is appoximatively equal to 9. It will povide a bette oudatio o the uthe eseach. Ackowledgmets This wok is suppoted by the sciece ad techology eseach ad developmet poject o Hebei Povice ude gat No.D. Reeeces Figue. The elative otatio o thee cotact poits. Figue. The cuve o α-taθ 5. Coclusios This pape eveals the motio law o gidig balls though establishig the motio equatios. Based o the vetical tech suace ad alog the goove suace, espectively, the dyamics aalysis o a sigle steel ball is achieved ad the iluece o the quality o the steel ball suace by dieet paametes is compaed. It ca be daw as ollows: () The motio law ca be uiquely idetiied whe steel balls do ot slip i the goove, ad the agula velocities o the evolutio ad otatio ae both popotioal to the otatioal speed o the otay disc. () Whe steel balls ae goud by a hoizotal gidig machie, i ode to pevet them om slippig, the lappig pessue has to satisy a cetai age, ad the otay speed o the otay disc also has to be less tha the maximum allowable speed. [] Y.Q.Zhag, Z.Y.Wu, The status quo ad developmet ted o Chia steel balls pocessig equipmet. Beaig, ()No. 4, 59-6 (i Chiese). [] J.L.Zhou, J.C.Che, G.Q.Wu, X.Y.Che, Eect o test ig stuctue paametes o cotact stess o specime. Egieeig echaics, Vol. 8()No. 6, 6-(i Chiese). [] T.H.C.Childs, S.ahmood, H.J.Yoo, agetic luid lappig o ceamic balls. Tibology Iteatioal, Vol. 8(995)No. 6, [4] X.L. Liu, Supe lappig o steel ball ad whet liquid. Joual o Habi Beaig, Vol. 7(6)No., -(i Chiese). [5] H. Liag, X.J. Zhao, Y.J. Qiu, Aalysis o the lakig o beaig steel balls i the poductio pocess, Beaig, (8)No. 8, 6-7(i Chiese). [6] H.Ohta, K.Kobayashi, Vibatios o hybid ceamic ball beaigs. Joual o Soud ad Vibatio, Vol. 9(996)No., [7] B.W.Huag, H.K.Kug, Vaiatios o istability i a otatig spidle system with vaious beaigs. Iteatioal Joual o echaical Scieces, ()No. 45, [8] T.. Bada, Soud ad Vibatio i Rollig Beaigs. otio & Cotol, (999)No. 6, 9-7. [9] L.F. Nie, X.J. Zhao, The coditios that pocess ito oud ball ad exploe the iluecig actos, Beaig. ()No., 6-8(i Chiese). [] C. Zhu, echaical aalysis o the two kids o ball lappig method. Beaig, ()No. 9, -(i Chiese). [] Z. Xue, C.A. Fu, T. Zhag, Reseach o mechaism o lappig balls, Joual o Jiaga Uivesity. Vol. 8(9)No. 4, (i Chiese). [] Xia XT, a W, Tacheg Xie, auactuig techology o ollig beaig. Beijig: Chia achie Pess; 7(i Chiese). [] Zhu C, The mechaics piciple o lappig ball. Zhegzhou: Hea Sciece ad Techology Pess; 995 (i Chiese).