Instantaneous Cutting Force Model in High-Speed Milling Process with Gyroscopic Effect

Transcription

1 Advancd Matrials sarch Onlin: -8-6 ISS: , Vols , pp doi:.48/ rans ch Publications, Switzrland Instantanous Cutting Forc Modl in High-Spd Milling Procss with Gyroscopic Effct ZHOU Hongliang a, AG Wixiao b, SOG Qinghua c and JU Huawi d Ky Laboratory of High Efficincy and Clan Mchanical Manufactur, School of Mchanical Enginring, Shandong Univrsity, Ji nan 56 P.. China a pigpi3@6.com, b Corrsponding author: tangwx@sdu.du.cn, c ssinghua@sdu.du.cn, d uhuawi856@63.com Kywords: High-spd milling; gyroscopic ffct; quivalnt profil; cutting forc Abstract. High-spd milling (HSM) has advantags in high productivity high prcision and low production cost. hus it can b widly usd in th manufactur industry. Howvr, whn th spd of spindl-tool rachs a highr spd rang, th gyroscopic ffct will bcom an important part of its stabl milling. In this papr, a dynamics modl of HSM systm was proposd considring th influnc of gyroscopic momnt du to high rotating spd of nd milling. Finit lmnt modl (FEM) is usd to modl th dynamics of a spindl-milling systm. It obtains th tractory of cntral point in fac milling with considring gyroscopic ffcts through th dynamics modl at high spds. hn th cutting forc modl will b corrctd by th tractory of fac milling. So it can provid a basis for stability prdiction of high spd milling. Introduction High spd milling has bcom on of th advancd manufacturing tchnologis. And widly usd in many diffrnt filds,.g. aronautics and automobil. h cutting procsss nd fficint and stabl cutting []. So th study of gyroscopic ffct on nd milling can hlps a lot about tool s stabilitis. h high spd rotating tool systm can b rgardd as a rotor systm. hrfor, th rotor dynamics can b applid to study th vibration bhavior of high-spd rotary tool. sarching th ffct of inrtia on th stability of th nd milling at high spd, it can b mor accuratly prdict th bhavior of th nd milling. An important part of inrtial ffcts is th gyroscopic ffcts causing by rotary nd milling. With th spd incrass, th rol of th gyroscopic torqu on th nd milling bcoms vry important. In this papr, hrough Considring th gyroscopic ffct of high-spd rotary nd milling in modal analysis study th dynamic charactr of th nd milling. hr ar also spcializd post procssing faturs can us to analyz spcific bhavior, and to procss simulation rsults to dtrmin critical paramtrs. Dynamic Modl of Spindl-ools Systm Finit lmnt mthod is usd to modl th dynamics of spindl-tools systm. h modl is basd on classical bam thory of imoshnko bam thory. h bam thory and influnc of gyroscopic momnt is basd on th systm s assumption and applicabl condition. h imoshnko bam thory is chosn to formulat th mass and stiffnss proprtis for spindl lmnts. h imoshnko bam thory includs th mass momnt of inrtia and th shar ffct. For an infinitsimal lmnt of bam with lngth dx (Fig. ), w can driv th following quilibrium quation, w E w w ρ I w EI ρi A f ( x, t) ρ + = 4 x κg x t t κ w t () All rights rsrvd. o part of contnts of this papr may b rproducd or transmittd in any form or by any mans without th writtn prmission of rans ch Publications, (ID: , Pnnsylvania Stat Univrsity, Univrsity Park, USA-9/5/6,:3:3)

2 39 Advancd Manufacturing chnology, ADME γ α Fig. An infinitsimal lmnt of th bam Fig. Shar angl of a dformd cross-sction whr M is th bnding momnt; V in th figur is th shar forc. In addition, th bnding momnt M and th shar forc V ar rlatd to α and w (Fig. ) by th following formulas M α = EI x () w V = κ AGγ = κ AG α x whr G is th shar modulus of lasticity of th matrial of bam; κ is th imoshnko s shar corrction factor. Lagrang s quations indicatd that quations of motion of a structur can b calculatd from that nrgy of that structur, for an lmnt with two nd nods, th Lagrang s quation would b d U + = Qi ( t) i=,,...,8 (4) dt u i ui ui whr Q i (t) is th gnral forc. It could b xtrnal forcs or momnts acting on th lmnt, dpnd on th natur of th loads. h kintic nrgy of th lmnt can b obtain by intgrating th squar of point p vlocity with rspct to th ara a = ρvpda (5) A h potntial of th lmnt can b writtn as { } ( [ ] [ U = u KB + KS ] ) { u } (6) whr (3) l Ψ Ψ (7) [ K B ] = EI dx X X l [ KS ] = κ AG[ Ξ] [ Ξ]dX (8) Substituting quations (5) and (6) into th Lagrang s quations. W can obtain th quations of motion of on lmnt as ([ M ] + [ M ]){ u } +Ω ( ([ M ] + [ M ]) [ G ]){ u } + ([ KB ] + [ KS ] +Ω ([ M ] + [ M ] [ G ])){ u} = { Q ( t)} +Ω ({ UYY } + { U ZZ }) (9) Fig. 3 Elmnt of th spindl-tools systm

3 Advancd Matrials sarch Vols Using th abov mthod. W can obtain th systm quations of motion of th whol spindl-tools systm (Fig. 3) by using th Lagrang s quation ( ) Ω( ) + Ω ( ) = () [ M ]{ q} [ M ] [ G] { q} [ K] [ M ] [ G] { q} { B} whr, {B} is th unbalanc forc vctor. h spindl-tools systm is rprsntd by a 3-lmnt finit modl and th dtaild information can b found in th abl []. Instantanous Cutting Forc Modl h quations of motion of spindl-tools can b obtaind by FEM. In this sction. A MALAB program is dvlopd to solving th dynamic quations. By solving th quations formd by th finit lmnt modl. W can obtain th nodal displacmnts of nd fac milling. hn using thos nodal displacmnts crat fit curv. h curv is th tractory of cntral point [5]. W can obtain th quivalnt profil by th tractory. abl Dimnsions of th lmnts (Unit: mm) Elmnt Lngth Diamtr Elmnt Lngth Diamtr Elmnt Lngth Diamtr Fig. 4 shows a tractory of cntral point of fac milling. Curv rprsnts th actual tractory, curv rprsnts th fitting tractory, through th Fig. 4. h paramtric quations xprssion of quivalnt profil of milling can b writtn as: Fig. 4 ractory of cntral point in nd fac milling (Unit µm) x= 4.5 cosθ +.5sinθ y = 4 cosθ +.8sinθ Aftr obtaining th tractory quation of cntral point, th cutting forc modal can b updatd. First, th total chip lad can b xprssd as g( φ ) = φst < φ < φx h( φ ) = st sin φ + ( v, v ) g( φ ) () g( φ ) = φ < φst or φ > φx whr th static componnt of th thicknss (s t sinφ ) is droppd. And (v, -v ) ar th dynamic displacmnt of th cuttr at prvious and prsnt tooth priods. φ st, φ x ar start and xit immrsion angls of cuttr and from th cut, rspctivly. So dynamic cutting thicknss can b xprssd as ()

4 39 Advancd Manufacturing chnology, ADME h( φ ) = x sin φ + ( 4.5cosΩ t+.5sinω t) + y cos φ + (4cosΩ t+.8sin Ωt) g( φ ) (3) whr x=x-x, y=y-y, (x, y) and (x, y ) rprsnt th dynamic displacmnts of th cuittr structur at th prsnt and prvious tooth priods, rspctivly [3]. Substituting th chip thicknss (3) into cutting forc modl, th rsulting xprssions in matrix form yilds as, Fx axx axy x+ 4.5cosΩ t+.5sinωt = apkt F y ayx a (4) yy y+ 4cosΩ t+.8sinωt whr, a p is th axial dpth of cut; cutting cofficints K t is constant. And tim vary dirctional dynamic milling forc cofficints ar givn as a = sin φ + K ( cos φ ) g a = (+ cos φ ) + K sin φ g xx r xy r = = a = ( cos φ ) + K sin φ g a = sin φ + K (+ cos φ ) g yx r yy r = = h quation (4) is th nw cutting forc formula. It s obvious charactristics is basd on radial cutting dpth variation. So th cutting forc modl is mor accurat than th old on. h main chang is radial cutting dpth. (5) Conclusions A dynamic milling modl is stablishd considring gyroscopic ffcts in HSM. ractory quation of cntral point in nd fac milling was obtaind at a high spd. h gyroscopic ffcts impactd th cutting thicknss mainly. So th cutting forc modl can b corrctd with Axial cutting dpth variation. h Instantanous Cutting Forc Modl is obtaind. So it can provid a basis for stability prdiction of high spd milling. Acknowldgmnts h authors ar gratful to th China Postdoctoral Scinc Foundation for supporting this work grant no. 4859, th Scinc Foundation of Ky Laboratory of High-fficincy and Clan Mchanical Manufactur at Shandong Univrsity, Ministry of Education, and th 973 ational Basic sarch Program of China for supporting this work undr grant no. 9CB7445, ational High chnology sarch and Dvlopmnt Program ("863"Program) of China, Grant o.: 9AA4433. frncs [] G.L. Xiong, J.M. Yi and t al.: Journal of Matrials Procssing chnology, Vol. 38 (3),pp [] Chi-Wilin. High Spd Effcts and Dynamic Analysis of Motorizd Spindls for High Spd End Milling. Wst Lafaytt: Purdu Univrsity.,pp.7-55 [3] Y.Altintas, E.Budak. Analytical Prdiction of Stability Lobs in Milling. Annals of th CIP. 995, 44(): [4] X Q Xu, sarch and softwar dvlopmnt on th stability prdiction of high spd milling. Jinan: Shandong Univrsity,9,pp.9-34 [5] H.L.Zhou, W.X.AG, Q.H.SOG. Equivalnt profil of high-spd mill considring gyroscopic dffct. Intrnational Confrnc on Futur Industrial Enginring & Application ( ICFIEA ),.:6-65

5 Advancd Manufacturing chnology, ADME.48/ Instantanous Cutting Forc Modl in High-Spd Milling Procss with Gyroscopic Effct.48/ DOI frncs [] G.L. Xiong, J.M. Yi and t al.: Journal of Matrials Procssing chnology, Vol. 38 (3), p /S94-36(3)-X [3] Y. Altintas, E. Budak. Analytical Prdiction of Stability Lobs in Milling. Annals of th CIP. 995, 44(): /S7-856(7)634-7