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1 한국정밀공학회지제 32 권 2 호 pp J. Koean Soc. Pec. Eng., Vol. 32, No. 2, pp ISSN Pnt, ISSN Onlne Febuay 2015 / 특집 회전유니트모델링기술 복합베어링으로지지된스핀들의동적해석 Dynamc Analy of Spndle Suppoted by Multple Beang of Dffeent ype 통반칸 1, 배규현 1, 홍성욱 2, Van-Canh ong 1, Gyu-Hyun Bae 1, and Seong-Wook Hong 2, 1 금오공과대학교기전공학과대학원 Gaduate School, Depatment of Mechatonc, Kumoh Natonal Inttute of echnology 2 금오공과대학교기전공학과 Depatment of Mechatonc, Kumoh Natonal Inttute of echnology Coepondng autho: whong@kumoh.ac.k, el: Manucpt eceved: / Reved: / Accepted: h pape peent a dynamc modelng method fo the ndetemnate pndle-beang ytem uppoted by multple beang of dffeent type. A pndle-beang ytem uppoted by ball and cylndcal olle beang condeed. he de Mul beang model extended fo calculatng ball and cylndcal olle beang tffne matce wth ncluon of centfugal foce and gyocopc moment. he dependence between pndle haft eacton foce and beang tffne effectvely eolved ung an teatve appoach. he pndle oto dynamc etablhed wth the mohenko beam theoy baed fnte element. he pndle eacton foce, beang tffne and pndle natual fequence ae obtaned wth takng nto account pndle adal load, ball beang axal peload and otatonal peed effect. he developed method vefed by compang the mulaton eult wth thoe fom a commecal pogam. Key Wod: Spndle-beang ytem 스핀들베어링계, Angula contact ball beang 각접촉볼베어링, Cylndcal olle beang 원통롤러베어링, Stffne matx 강성행렬, Natual fequency 고유진동수 1. Intoducton Spndle an eental pat n machne tool becaue t oto dynamc chaactetc have a geat nfluence on the oveall pefomance of machne tool uch a pecon machnng, evce lfe of toolng and acceoe, poductvty, etc. Nowaday, combnaton of dffeent beang type n a pndle ytem pevalent to mpove the pefomance of machne tool. he combned ue of ball and cylndcal olle beang n a pndle haft beleved to gve an enhancement n load cayng capacty and themal tablty. 1 he fnte element model commonly ued n oto dynamc analy of pndle-beang ytem. 2-4 It ha been acknowledged that the fnte element technque can povde an accuate modelng of oto-beang ytem and be applcable to a wde ange of complex poblem n pactcal engneeng degn. 5 In the pat, howeve, the fnte element baed oto dynamc model nomally adopted the tffne coeffcent of ollng element Copyght C he Koean Socety fo Pecon Engneeng h an Open-Acce atcle dtbuted unde the tem of the Ceatve Common Attbuton Non-Commecal Lcene whch pemt unetcted non-commecal ue, dtbuton, and epoducton n any medum, povded the ognal wok popely cted.

2 한국정밀공학회지제 32 권 2 호 pp Febuay 2015 / 118 beang a gven paamete that wee aumed to be contant dung opeaton of pndle ytem. Recently, many eeache have hown that the beang coeffcent tongly depend on the otatonal peed and loadng condton. 6-8 hu, t neceay to adde the couplng between the beang and pndle n ode to obtan accuate beang tffne element along wth the pndle dynamc chaactetc. Seveal eeach eult have been publhed egadng the couplng of pndle and beang ytem. Jogenen and Shn 9 analyzed the angula contact beang tffne and natual fequence of the pndle wth the adal load appled. In the tudy, the pndle haft wa dcetzed nto lumped element wth whch the nfluence coeffcent method wa appled to fnd the load deflecton elatonhp. h method could be appled only fo pndle uppoted by two beang o two et of beang n whch each et of beang modeled a a tffne matx at a ngle node. hu, the accuacy of the model wth multple beang wa mpope, a well a the model wa confned to pndle uppoted by two et of beang. In addton, the detemnaton of beang nduced moment wa not mentoned. Hong et al. 10 mpoved the Jogenen and Shn model 9 o a to apply fo ndetemnate pndle-beang ytem uppoted by moe than 2 beang. A new teatve poce wa popoed to deve eacton foce and beang tffne baed on the tatc fnte element pndle model. Cao and Altnta 11 outlned a geneal method fo modelng of pndle-angula contact beang ytem. he combnaton of beang dynamc chaactetc and pndle haft ung fnte element method wa pefomed to deve the dynamc equaton fo pndle-beang ytem, whch wee then teatvely olved by the Newton Raphon technque. Although the method wa vefed to be accuate, much computatonal effot wa equed to olve non-lnea equaton multaneouly. Mot of the pevou tude attempted to couple beang tffne n the pndle haft baed on fnte element model. hen the unknown of the ytem wee obtaned ung teatve method. Howeve, few eeache ae avalable on the oto dynamc of pndle wth combned multple beang of dffeent type unde geneal loadng condton. h due to the lack of an effectve method to model the couplng between haft and beang, epecally n ndetemnate pndle-beang ytem. h pape extend the eale wok popoed by Hong et al. 10 to develop a new cheme fo calculaton of beang tffne and pndle haft eacton foce. he mot mpotant contbuton of th wok modfyng the pndle load and deflecton fomulaton ung the ocalled modfed tanfe matx method. he oveall computatonal pocedue n the ame manne a that n Hong et al. 10 he developed technque then appled fo detemnng natual fequence of a pndle aembly uppoted by angula contact ball and cylndcal olle beang unde adal, axal loadng and otatonal peed effect. Fnally, the peented model vefed wth a commecal pogam Ball and olle beang model he ball and olle beang model popoed by de Mul et al. 13 ae extended n th tudy. he ball beang model take nto account the effect of gyocopc moment, whch neglected n the de Mul model. Regadng cylndcal olle beang, a mplfed model developed by applyng a few appopate modfcaton fom the tapeed olle beang model. 8 In th pape, all the beang fcton and cage foce, themal expanon of component and lubcaton flm ae neglected. Only the defomaton at the contact locaton between olle and ace condeed, whle the beang ace ae aumed to eman ccula unde loadng. 2.1 Ball beang model A fve-degee-of-feedom DOF ball beang model adopted a hown n Fg. 1a. he beang loaded by extenal load vecto {F} = {F x, F y, F z, M y, M z } and dplace by a dplacement vecto {δ} = {δ x, δ y, δ z, γ y, γ z }. Dplacement of nne ng co-ecton and contact load ae defned, epectvely, by {u} = {u, u x, θ}; {Q} = {Q, Q x, } 1 whee {u} depend on global dplacement by: {u} = [RΦ]{δ} 2 whee the tanfomaton matx [RΦ] gven a

3 한국정밀공학회지제 32 권 2 호 pp Febuay 2015 / 119 a b Fg. 1 Ball beang dagam a Coodnate ytem and loadng b Ball fee-body dagam coφ nφ 0 x nφ x coφ P P Rφ = nφ coφ P P nφ coφ [ ] he ball cente dplacement ndcated by {v} = {v, v x } 3 he ball loadng ncludng the centfugal foce and gyocopc moment hown n Fg. 1b. F c and M g ae the centfugal foce and gyocopc moment of the ball ee Ha 14. he contact foce ae calculated ung the Hetzan theoy, a Q 3/2 = Kδ ; Q = K δ 4 3/2 e e e he contact defomaton can be calculated fom the geometc elatonhp hown n Fg. 2, a δ = l l ; l l e e 0e δ = 5 Havng obtaned all the load actng on the ball, one can obtan equlbum equaton at each ball a: M g Q coα Q coα + F nα nα = 0 e e c e D 6 M g Q nα Q nα + coα coα = 0 e e e D Fg. 2 Ball cente, nne and oute ace cuvatue cente befoe and afte loadng he Newton-Raphon method ued to olve th ytem of non-lnea equaton fo the 2 unknown {v, v x }. Afte the ball equaton ae olved, one can obtan the contact load of nne ace a Q co α + M / Dnα g Q = Q n α M / Dcoα g 0.5 M / D g { } Summaton of extenal load and all contact load actng on the nne ng gve the global equlbum of beang a { F} + [ RΦ ] { Q} = { 0} 7 n j 8 j = 1 he teatve Newton-Raphon method alo employed fo olvng the above global equaton to obtan the unknown {δ x, δ y, δ z, γ y, γ z }. he beang tffne matx can be calculated by efeng to de Mul et al. 13 a { F} { } n { Q} j [ ] [ ] 9 j 1 {} u k = = RΦ RΦ δ = j 2.2 Rolle beang model he cylndcal olle beang can feely move n the axal decton. hu, the extenal load and dplacement vecto need only 4 DOF, whch can be expeed a, Fg. 3a.

4 한국정밀공학회지제 32 권 2 호 pp Febuay 2015 / 120 y z F z,δz M y, γ y O F y,δ y M z, γ z {F} = {F y, F z, M y, M z } 10 {δ} = {δ y, δ z, γ y, γ z } 11 Due to the fact that the numbe of DOF educed to 4, modelng pocedue fo the cylndcal olle beang a mplfed veon of afoementoned ball beang model. he dplacement of nne ng co-ecton and contact load can be deduced fom Eq. 1 a {u} = {u, θ}; {Q} = {Q, } 12 Fom the olle fee-body dagam hown n Fg. 3b, one can obtan the olle equlbum equaton a Q Q + F = 0 e c M M M = 0 e g 13 he olle-ace contact load can be calculated ung the lcng technque. 13 he olle contact length dvded nto n numbe of lce, and the contact foce then evaluated fo each lce a 10 δ 9 q = cδ Δ l ; > 0 14 k k k k he total contact foce and moment become n P k k = 1 u,v, Q a b y η x θ,ψ, Τ n Q = q ; M = q l 15 k k k = 1 M Q e M e a b Fg. 3 Cylndcal olle beang dagam a Coodnate ytem and loadng b Rolle fee-body dagam M g F c Q η z O 1 2 l -1 x l he nne ng contact load found a { Q} Fg. 4 Spndle haft wth n poton Q = M 16 he global equlbum equaton fo the cylndcal olle beang and the tffne matx expeon ae mla to Eq. 8 and 9. It noted that both olle and global equlbum equaton ae non-lnea and hould be olved ung an teatve method uch a the Newton- Raphon method. 3. Spndle haft load-deflecton calculaton ung modfed tanfe matx method A new cheme fo calculatng pndle haft deflecton condeed hee. It noted that the Eule-Benoull beam theoy doe not gve accuate deflecton eult, patculaly n the cae of thck beam, nce the otay neta and hea defomaton ae not taken nto account. heefoe, th ecton am to mpove the computatonal accuacy by ung the mohenko beam theoy. Fg. 4 how the pndle haft, whch dvded nto n poton coepondng to the haft tep. Condeng the elatc deflecton v x at the co ecton poton x l -1 x l of poton whoe total length L = l l -1. v x can be expeed n the followng fom: x l 1 v x = v + v x l + v 1 2! x l x l + v + + 3! n! 3 n 3 1 n 1... v 17 whee v denote the elatc deflecton at the left end of poton. v k the k th devatve of v wth epect to x k = 1 n. n x

5 한국정밀공학회지제 32 권 2 호 pp Febuay 2015 / 121 Conde a functon a below, k x l 1 ; Φ x l k 1 = k! < x l x l 0; Hee, functon Φ k poee the followng featue 18 In the cae of a haft unde concentated adal load, t found that v Q κ GA 1 = ϕ ; 0 k k v v v x 0; 0 M EI 2 = ; 0 v = = 3 Q = 23 EI 3 k > 24 d dx Φ k =Φ k 1 19 Subttutng Eq. 18 nto Eq. 17 and then takng devatve the eultant equaton wth epect to x gve n v x = v Φ + v Φ + v Φ + v Φ v Φ n v x = 0 + v Φ + v Φ + v Φ v Φ n n v x = v Φ + v Φ v Φ 0 1 n n v x = v Φ v Φ n 0 n 3 Eq. 20 can be e-expeed n a matx fom a v 1 v x Φ Φ Φ Φ... Φ n v 1 2 v x 0... Φ Φ Φ Φ n v 1 2 = 3 v x Φ Φ Φ 0 1 n 2 v Φ... Φ 0 n 3... v x n v Fom the mohenko beam theoy, the followng elatonhp can be appled 1 dv x Q x v x = = ϕ x dx κ G A 2 dϕ x M x v x = = dx E I 3 1 dm x Q x v x = = EI dx EI 22 whee ϕ x, M x and Q x ae the haft angle of deflecton, bendng moment and hea foce at the coecton x of poton, epectvely. E and G ndcate the elatc and hea modul of the haft membe. he geometc paamete A, I and κ denote the coecton aea, moment of neta and hea coeffcent, epectvely. whee v, ϕ, M and Q epeent the elatc deflecton, deflecton angle, bendng moment and hea foce at the left end of poton, epectvely. hee value ae detemned by v v Δv 1 ϕ ϕ ϕ 1 Δ = + M M M 1 Δ Q Q Q Δ 1 25 whee v -1, ϕ -1, M -1, Q -1 ae the coepondng value at the co ecton x = l -1, whch have been detemned fom the pevou tep ung a mla poce. Δv -1, Δϕ -1, ΔM -1, ΔQ -1 ndcate the added value fo the left end of poton. Subttutng Eq nto Eq. 21, the loaddeflecton equaton fo an abtay poton of the haft obtaned a { S x } [ B ]{ S 1} { 0 } { 0 } { 0 } S H H whee = + Δ Δ + Δ 26 { S x } { v x ϕ x M x Q x } = 27 { S } { v ϕ M Q } = { S } { v ϕ M Q } Δ = Δ Δ Δ Δ ΔQ Δ = κ GA { H } [ B ] Φ Φ Φ 0 1 EI Φ 0 Φ EI 0 0 Φ Φ EI Φ EI Φ Φ =

6 한국정밀공학회지제 32 권 2 호 pp Febuay 2015 / 122 Ung the fomula hown n Eq. 26, the govenng equaton fo the ft and econd poton can be deved. Fo the ft poton, one can get { S x } [ B ]{ S 1 1 0} { S01} { H01} { H01} = + Δ Δ + Δ 33 Becaue {S -1 } = {S 0 } = 0, Eq. 33 can be futhe mplfed to { S x } [ B ] 1 1 { S01} { H0 } { H0 } = Δ Δ + Δ 34 Smlaly, the govenng equaton fo the econd poton can be wtten a * { } { } S x = [ B ] S +ΔS Δ H + Δ H It hould be noted that * { S1} { S x l 1 1 } = = 36 he above cheme ued fo fomulatng the elatonhp of the pndle haft deflecton and eacton foce. It obvou that the developed cheme could be applcable to a vaety of co-ecton aea, wth dffeent mateal and alo wth geneal loadng. 4. Natual fequency pedcton ung fnte element model In th tudy, the pndle model ung mohenko beam element a peented n Hong et al. 4 ued fo haft dynamc modelng. he model cont of a pndle haft and beang located at node. he ntenal dampng gnoed. he equaton of moton fo a haft element decbed a m 0 y 0 g y +Ω 0 m z g 0 z k 0 y f y + = 0 k z f z 37 In Eq. 37, m and g denote the 4x4 ma and gyocopc matce of haft element, epectvely. {y} and {z} ndcate 4x1 dplacement vecto n x-y and x- z plane. Wth neglectng axal dplacement of the haft, the beang epeented a b b b b k k xx xy y fx b b b = b k k f zy yy z y 38 whee {f y b f z b } the beang foce vecto. he matx k j b,,j=y,z epeent 2x2 beang tffne matx obtaned fom the beang model. Combnng the haft element and beang equaton gve whee Mq +Ω Gq + Kq = f t 39 he tate pace fom of Eq. 39 can be wtten a Ah + Bh= P 40 M 0 0 M q 0 A = ; 0 M B = ; K ΩG h = ; q P = f 41 he egenvalue poblem n aocaton wth Eq. 41 can be wtten a [ A] [ B] { h} { 0} α + = 42 whee α and h denote the egenvalue and coepondng egenvecto, epectvely. 5. Calculaton pocedue Becaue the beang tffne depend on haft eacton foce, and vce-vea, the beang tffne and eacton foce hould be detemned multaneouly. Fg. 5 how the block dagam of the ente calculaton poce. he pocedue tat wth aumng the adal dplacement of beang. At the ft teaton, the nduced moment at beang ae aumed to be zeo. In the next tep, the haft eacton foce and deflecton ae detemned ung the modfed tanfe matx method. he output fom th tep ued to calculate the beang behavo, e.g., adal dplacement, tffne, nduced moment. hen, the teaton poce epeated untl the haft eacton foce dffeence between the cuent and pevou tep ae mall enough. Afte the convegence attaned, the pndle natual fequence ae etmated ung the dynamc fnte element model.

7 y y θ z z θ 한국정밀공학회지제 32 권 2 호 pp Febuay 2015 / 123 Spndle-beang data, peed, load, = 0 F B B Fa B Intal gue of beang adal dplacement, aume no nduced moment at beang Ø70 = + 1 Calculate: - Shaft eacton foce; - Shaft deflecton; Ung modfed tanfe matx method >1 Fg. 5 Calculaton block dagam 6. Computatonal eult =1 Calculate: - Beang adal dplacement; - Beang tffne; - Induced moment; Ung beang model Conveged? No Ye Natual fequence ung dynamc FEM model In th ecton, mulaton pefomed ung Matlab to demontate the popoed method. wo knd of beang uch a angula contact beang B7014 and cylndcal beang NU1014, ae condeed thoughout the mulaton wok. he computatonal eult ae vefed by compang wth thoe fom a commecal pogam. 6.1 Model vefcaton he haft beang ytem loaded wth a adal load of F = 10,000 N. An axal peload at the mddle beang F a = 250 N a hown n Fg. 6. he otatonal peed of pndle n = 5,000 pm. able 1 compae the haft eacton foce n adal decton fom the mulaton eult ung the developed model and the efeence pogam. able 2 ummaze the nduced beang moment fom the developed pogam and the efeence pogam. Both able 1 and 2 exhbt only mno eo between the developed pogam and the efeence pogam eult. he beang tffne coeffcent ae alo compaed n Fg. 7. A can be een n Fg. 7, the maxmum dffeence between the mulaton and efeence 1.17% n tem of angula tffne coeffcent k θyθy Fg. 6 Smple ndetemnate pndle angula contact beang ytem able 1 Radal eacton foce at beang locaton N Beang Refeence Smulaton 1.57E E3 Eo % able 2 Induced beang moment load Nmm Beang Refeence Smulaton 1.68E5 5.99E3-7.54E4 Eo % /m N 200 M k y y d 200 a / 150 m N K100 k θ Beang # Beang # Refeence % o E % o E Smulaton /m N 200 M k z z d200 a / 150 m N K100 k θ 50 0 Eo Beang # Beang # Fg. 7 Beang tffne compaon 6.2 Applcaton to an actual pndle ytem Fg. 8 how the nvetgated pndle-beang ytem. he adal load F appled at the left end of the pndle to emulate the cuttng load dung opeaton. he load F elected fom 1,500 to 2,000N, whle the otatonal peed of ytem eman contant at n = 5,000pm. Fg demontate the adal tffne coeffcent of cylndcal olle and ball beang, natual fequence of pndle-beang ytem a a functon of appled adal load. he tffne and fequency % o E % o E

8 한국정밀공학회지제 32 권 2 호 pp Febuay 2015 / 124 F Ø97 Fg. 8 Invetgated pndle-beang ytem 7. Concluon A modelng pogam fo pndle-beang ytem ha been developed fo pndle uppoted by multple beang of dffeent type. An ndetemnate pndlebeang ytem wth angula contact ball beang and cylndcal olle beang nvetgated to demontate the pogam. he mulaton eult how that the developed pogam can well pedct the beang tffne and pndle dynamc behavo unde geneal loadng condton. ACKNOWLEDGEMEN h eeach wa fnancally uppoted by the Koea Inttute of Machney and Mateal. Fg. 9 Radal tffne of cylndcal olle beang Fg. 10 Radal tffne of angula contact beang Fg. 11 Spndle-beang ytem natual fequence dffeence between the developed pogam and efeence pogam ae vey mall. REFERENCES 1. SKF Goup, Beang Aangement, kf.com/goup/poduct/beang-unt-houng/upepecon-beang/pncple/degn-condeaton/ beang-aangement/ndex.html Acceed DEC. 7, Nelon, H. D. and McVaugh, J. M., he Dynamc of Roto-Beang Sytem ung Fnte Element, J. Manuf. Sc. and Eng., an. ASME, Vol. 98, No. 2, pp , Hong, S. W. and Pak, J. H., An Effcent Method fo the Unbalance Repone Analy of Roto- Beang Sytem, J. Sound & Vbaton, Vol. 200, No. 4, pp , Bae, G. H., Lee, C. H., Hwang, J., and Hong, S. W., Etmaton of Axal Dplacement n Hgh-Speed Spndle due to Rotatonal Speed, J. Koean Soc. Pec. Eng., Vol. 29, No. 6, pp , Chen, L. K. and Ku, D. M., Fnte Element Analy of Natual Whl Speed of Rotatng Shaft, Compute and Stuctue, Vol. 40, No. 3, pp , Hong, S. W., Cho, C. S., and Lee, C. H., Effect of Beang Aangement on the Dynamc Chaactetc of Hgh-Speed Spndle, J. Koean Soc. Pec. Eng., Vol. 30, No. 8, pp , Noel, D., Rtou, M., Fuet, B., and Loch, S. L.,

9 한국정밀공학회지제 32 권 2 호 pp Febuay 2015 / 125 Complete Analytcal Expeon of the Stffne Matx of Angula Contact Ball Beang, J. bol. an. ASME, Vol. 135, No. 4, Pape No , ong, V. C. and Hong, S. W., Chaactetc of apeed Rolle Beang Subjected to Combned Radal and Moment Load, Int. J. Pec. Eng. Manuf.-Geen ech., Vol. 1, No. 4, pp , Jogenen, B. R. and Shn, Y. C., Dynamc of Spndle-Beang Sytem at Hgh Speed ncludng Cuttng Load Effect, J. Manuf. Sc. Eng., an. ASME, Vol. 120, Vol. 2, pp , Hong, S. W., Kang, J. O., and Shn, Y. C., Dynamc Chaactetc of Indetemnate Roto Sytem wth Angula Contact Ball Beang Subject to Axal and Radal Load, Int. J. Pec. Eng. Manuf., Vol. 3, No. 2, pp , Cao, Y. and Altnta, Y., A Geneal Method fo the Modelng of Spndle-Beang Sytem, J. Mechancal Degn, an. ASME, Vol. 126, No. 6, pp , Schaeffle echnologe, BEARINX -onlne Shaft Calculaton, chaeffle.de/en/poduct_evce/nafagpoduct/cal culatng/beanxonlne/beanx_onlne.jp/ Acceed DEC de Mul, J. M., Vee, J. M., and Maa, D. A., Equlbum and Aocated Load Dtbuton n Ball and Rolle Beang Loaded n Fve Degee of Feedom Whle Neglectng Fcton - Pat I-II, J. bol., an. ASME, Vol. 111, pp , Ha,. A. and Kotzala, M. N., Advanced Concept of Beang echnology, New Yok, aylo & Fanc, 5 th Ed., pp , 2007.

10 한국정밀공학회지제 32 권 2 호 pp Febuay 2015 / 126

Contents 3 2 Intoducton Etenal and Intenal oces ncple of Tansmssblt: Equvalent oces Vecto oducts of Two Vectos oment of a oce bout a ont Vagon s Theoe

3 1 Rgd odes: Equvalent Sstems of oces 기계공학부최해진 School of echancal Engneeng Contents 3 2 Intoducton Etenal and Intenal oces ncple of Tansmssblt: Equvalent oces Vecto oducts of Two Vectos oment of a oce