IDENTIFICATION OF DYNAMIC RIGIDITY FOR HIGH SPEED SPINDLES SUPPORTED ON BALL BEARINGS

Transcription

1 IJRET: Internatonal Journal of Research n Engneerng and Technology eissn: pissn: IDENTIFICATION OF DYNAMIC RIGIDITY FOR HIGH SPEED SPINDLES SUPPORTED ON BALL BEARINGS Jaeer Hussan Sha, Srnvas J Research Scholar, Assocate Professor, Department of Mechancal Engneerng, Natonal Insttute of Technology, Rourela, Odsha, Inda, jaeersha786@yahoo.co.n Abstract The wdespread use of hgh-speed machnng n recent decades has led to a sgnfcant area of research on ssues that lmt ts productvty. Regeneratve chatter s a well nown machnng problem that results n unstable cuttng process, leads to the poor surface qualty, reduced materal removal rate and damage on the machne tool tself. The man requrement for the stablty of system dynamcs s the nformaton of tool tp frequency response functons (FRF s).the present wor consdered a coupled model of spndlebearng system by usng the angular contact ball bearng forces on stablty of machnng. Usng Tmosheno beam element formulaton, the spndle unt s analyzed by ncludng the gyroscopc and centrfugal terms and the bearng contact forces are arrved from Hertzan contact theory. Then, the model s used for studyng the effects of vscous dampng to obtan the tool pont FRF for the dynamc spndle. Index Terms: Spndle dynamcs, Stablty, Hgh speed effects, Bearng contact forces, Hertzan contact theory, Fnte element modelng *** INTRODUCTION In recent years there has been een nterest by manufacturers to ncrease Productvty and Surface qualty of part by usng Hgh speed mllng machnes. Ths has drect mpact on cost of machnng and Qualty. Dymamc rgdty s one of the most crtcal characterstcs of machne tools, especally for hgh precson and hgh performance machnng applcatons. Ths technology s manly lmted by the performance of the spndle whch has a sgnfcant nfluence on the machnng accuracy. Self-excted vbratons of the tool result n unstable cuttng process whch leads to the chatter on the wor surface and t reduces the productvty. At large rates of materal removal, tool chatter creates the unavodable flexblty between the cuttng tool and the wor pece; f t s uncontrolled the chatter causes a rough surface fnsh and dmensonal naccuracy of the wor pece along wth unacceptable loud nose levels and accelerated tool wear. The chatter stablty of the tool s dependent on the dynamc behavor of the spndle system, whch s often expressed as the frequency response functon (FRF) at the tool tp. In other words, tool-tp FRF s a ey varable n determnaton of stablty lmt []. The objectve of the desgn engneer s to predct the cuttng performances of the spndle durng the desgn stage by relyng on engneerng model of the process and system dynamcs. Early spndle research has focused manly on statc analyss, where as current research s extended to the dynamcs for optmal desgn. Over the last two decades, numerous approaches have been addressed to elaborate the tool-tp FRF through modelng and experments. Wth ntroducton of receptance couplng substructure analyss by Schmtz n early 000, several authors [-7] have llustrated the approach to predct the tool pont FRFs. Ertur et al.[8] presented analytcal models to obtan the tool-tp FRF. Faassen et al. [9] predcted chatter stablty lobes of hgh speed mllng on the bass of expermental FRFs at dfferent spndle speeds. Abele and Fedler [0] ntroduced a sub-space-state-space-dentfcaton method to measure and calculate the dynamc behavor of spndle-tool systems durng hgh speed machnng, consequently, stablty lobe dagrams were determned based on the dentfed FRFs. Zaghban and Songmene [] used operatonal modal analyss to estmate the machne tool dynamc parameters durng hgh machnng operatons, and the dynamc stablty lobes were calculated usng the extracted modal parameters. The expermental method s used to dentfy speed-varyng dynamcs of spndles n the above wors, whch s drect and feasble. However, t s tme-consumng to repeat the modal tests for every spndle speed and prone to errors. Wth nown detals of the spndle geometry, drawbar force, bearng parameters and preload, etc., an alternatve method to obtan the dynamc behavor of spndle system s the fnte element (FE) method. Wth the accurate spndle model, the FRFs at the tool tp are smulated and then chatter stablty lobes can be predcted. Chen and Wang [] modeled the ntegrated spndle-bearng system and they also found that sgnfcant errors occurred n predctng stablty lobes f the load and speed effects on the Volume: 0 Issue: 07 Jul-03, 46

2 IJRET: Internatonal Journal of Research n Engneerng and Technology eissn: pissn: shaft/bearng dynamcs were neglected. Tan and Hutton [3] consdered the gyroscopc effects of the rotatng spndle and proposed a chatter model n mllng systems. They found that gyroscopc effects reduced the crtcal axal depth of cut. Smlarly, Movahhedy and Mosaddegh [4] predcted the chatter n hgh speed mllng ncludng gyroscopc effects on the bass of FE spndle model. Gagnol et al. [5] developed a dynamc model of a hgh-speed spndle system and then proposed a dynamc stablty lobe dagram by ntegratng the speed-dependent FRFs of spndles nto analytcal approach of Altntas and Buda [6]. Angular contact ball bearngs are commonly used n hgh-speed spndles due to ther lowfrcton propertes and ablty to wthstand external loads n both radal and axal drectons. Although the ball bearngs appear to be a smple mechansm, ther nternal geometry s qute complex and often dsplay very complcated nonlnear dynamc behavor. It s obvous that the spndle machnng system supported by ball bearngs present nterestng stablty characterstcs and nonlnear responses. Several recent studes [7-] llustrated the dynamcs of hgh speed spndles under dfferent preload mechansms and operatng speeds. Present paper descrbes a coupled model of spndle-bearng system by consderng the effects of ball bearng Hertz contact forces on the spndle dynamcs. The governng dfferental equatons of moton of spndle system are employed and numercal smulatons are carred-out usng Tmosheno beam element model.. MODELING OF SPINDLE-BEARING SYSTEM The spndle system generally conssts of spndle housng carryng spndle shaft over the front and rear bearngs, a tool holder and tool as shown n Fg.. angular contact bearngs. The x and y drectonal cuttng forces act on the tool tp. Tool s assumed to be rgdly connected to the tool holder whch s fxed to the spndle shaft. Fg-: Equvalent analyss model The equatons of moton for the spndle shaft due to rotaton are gven by [-3]: d v v v ρa sag θz P Ω ρav = F dt x x () d w w w ρa sag + θy P Ω ρaw = F dt x x () () θy Rear bearngs Front bearngs θy d dθ z w ρi + ΩρJ EI + sag + θy = M dt dt x (3) x x y d θ dθ z y θz v ρi ΩρJ EI sag θz = M dt dt x (4) y Fg-: Schematc of end-mll spndle unt The dynamc behavor of the spndle-tool unt can be well establshed through the modelng of the restrcted spndle rotatng system as shown n Fg.. The spndle has sx elements, seven nodes and total 8 degrees of freedom. Both the ffth and seventh nodes of the shaft are supported by two Here, v and w are the bendng deformaton n two perpendcular drectons of the spndle unt. ρa s mass per unt length of spndle shaft, s s shear deformaton factor, EI s flexural rgdty, AG s shear rgdty of the shaft materal, ρj s polar modulus, Fx and Fy are the components of external forces, Mx and My are the components of transverse moments and Ω s the speed of rotaton of spndle shaft. A rotatng shaft wth eght degrees of freedom per node fnte beam element wth gyroscopc terms based on Tmosheno beam theory formulated by Nelson [4] s employed to model the rotatng spndle. The element translatonal and rotatonal mass matrces, the stffness matrx and gyroscopc matrx are Volume: 0 Issue: 07 Jul-03, 47

3 IJRET: Internatonal Journal of Research n Engneerng and Technology eissn: pissn: obtaned by applyng Hamlton s prncple. The governng equaton n matrx form obtaned usng FE method s: [ M]{q} && + [C]{q} & + ([K] Ω [Mc ]{q} ) = F (5) Where the system element matrces are gven by [8]: [M] = [MT] + [MR] (6) [MT] = [M0] + φ[m] +φ[m] (7) [MR]= [N0] + φ[n] +φ[n] (8) [K]= [K0] + φ[k] (9) [C] =-Ω[G] + α[m] + β[k] (0) EI φ= s AGl () The front and rear postons of the spndle shaft are supported by the smlar angular contact ball bearngs. Wthout consderng the axal forces and moment loadngs of the spndle system, the angular contact ball bearngs can be assumed as deep-groove ball bearngs and the dynamc problem of the angular contact ball bearng s made as one of the two degrees of freedom. The local Hertz contact forces and deflecton relatonshps for a ball wth the nner and outer races may be wrtten as a followng set of restorng forces: Fx= Fy= Fx= x& b y& x& b = = b = b (4) sn ϕ δ) () sn ϕ δ) (3) b.5.5 sn ϕ δ) snϕ.5 Ωr π φ = t + ( ) ( R r) + N (where =,, Nb) angle b s the angular locaton of the th ball. Here the term: xbcosφ+ ybsnφ-δ s accounted only when t s postve, otherwse t s taen as zero and the bearng races are at loss of contact wth bearng balls. An dealzed condton s consdered n present wor wth δ=-0.05 µm (consderng practcal condtons, the nterference ft generally adopted to chatter problems where negatve clearance s justfed) and neglgble value of bearng dampng, nner radus r=40 mm, outer radus R=60mm, number of ball Nb=8, =b= N/m3/. The spndle FRF consstng of real and magnary parts can be gven expressed as: [H(jω)]=[Re(ω)]+j[Im(ω)]=[-[M]ω +jω[c]+([k]-ω[mc])]- (6) Here, Re and Im are, respectvely, the real and magnary part of the transfer functon of the spndle tool tp. 3. RESULTS The parameters of the fnte element model of the spndle are llustrated n Table. Except the element- (Slcon carbde tool) and all other elements are taen as steel. Denstes and shear modulus are consdered for the problem s obtaned from tables. Table -: Parameters for fnte element model of the spndle [8]. Parameter Elements of the spndle Length(mm) Outer da.(mm) Inner da.(mm) E (GPa) Parameter Fy= b y& b b = b (5) b sn ϕ δ).5 sn ϕ Here δ refers to the ntal clearance of the bearngs, Nb s number of balls, x and y, xb and yb are the dsplacements of mass elements dstrbuted at the front and rear bearng nodes along x and y drectons respectvely, and c, b and cb are the stffness and dampng of front and rear bearngs respectvely, and the Computer program s developed n MATLAB to analyze the spndle system. The assembly matrces and statc condensaton approach to elmnate the rotatonal degrees of freedom are ncluded n the program. Intally, by assumng the bearngs to be perfectly rgd, Campbell dagram s obtaned as shown n Fg.3. Usng undamped natural frequences, the coeffcents of Raylegh s dampng α and β are obtaned as 7.3 and 3.87e-6 respectvely for % dampng rato. Volume: 0 Issue: 07 Jul-03, 48

4 IJRET: Internatonal Journal of Research n Engneerng and Technology eissn: pissn: natural frequency (Hz) CAMPBELL DIAGRAM rotatonal speed (rpm) BW FW BW FW BW 3 FW 3 Fg-3: Frst three natural frequences of spndle rotor Here the vscous dampng forces and centrfugal stffenng are not accounted. The forward and bacward whrl modes are obvous due to the gyroscopc effect at three frst natural frequences Hz, Hz and Hz respectvely. In order llustrate the centrfugal stffenng effect and bearng dynamcs, drect frequency response functon at the tool tp hxx(jω) s llustrated at dfferent spndle speeds (Ω). Fg.4. depcts the FRF wthout consderng Raylegh s dampng for the spndle mounted on rgd bearngs at two dfferent speeds. The effect of centrfugal stffenng s observed at hgher speeds only as seen n Fg.5. Bacward Forward Fg-6: Cross frequency response at 5000 rpm The effect of Raylegh s (vscous) dampng on the tool tp FRF s shown at 000 rpm as shown n Fg.7. It s seen that dampng affects consderably the whrl modes. Fg-7: Effect of vscous dampng at a speed 000 rpm For studyng the bearng effect, the soluton s obtaned as a transent analyss problem frst n tme doman and then the frequency spectrum s obtaned from fast Fourer transformaton (FFT) algorthm. The reduced coupled dfferental equatons (4 n number) were solved by explct Runge Kutta solver tme ntegraton schmes. Fg.8. and Fg.9. shows the tme hstores and FFT dagram at the tool-tp node. Fg-4: Frequency response at 7000 rpm Fg-8: Tme hstores Fg-5: Frequency response at 5000 rpm Unle drect FRF, the cross FRF s a smooth curve wthout showng bacward (BW) and forward (FW) whrl modes as seen from the Fg.6. obtaned at 5000 rpm. Fg-9: Frequency responses Volume: 0 Issue: 07 Jul-03, 49

5 IJRET: Internatonal Journal of Research n Engneerng and Technology eissn: pissn: It s has been observed clearly from the frequency responses that the natural frequences of the spndle have been affected by the bearng contact forces. CONCLUSIONS In ths wor a coupled model of spndle bearng system s consdered for llustraton and t was analyzed usng fnte element analyss by consderng the effect of shear deformaton and rotary nerta of spndle shaft. It was observed that the Centrfugal and gyroscopc forces on the spndle rotor affect respectvely the stffness and dampng of the system. The vscous dampng was also accounted and the bearngs were treated frst as a rgd supports. Bearng forces expressed as functons of correspondng nodal dsplacements usng Hertz contact theory have also been ncluded n the fnal model n order to nvestgate effect of the angular contact bearngs. The resultng FRF obtaned by the spndle bearng model can be used n the exstng analytcal and numercal models for constructng the accurate stablty lobe dagrams for the hgh speed machnng process. REFERENCES [0]. E. Abele and U. Fedler, 004: Creatng stablty lobe dagrams durng mllng. Annals of the CIRP. 53, []. Zaghban and V. Songmene, 009: Estmaton of machne-tool dynamc parameters durng machnng operaton through operatonal modal analyss. Int. J. Machne Tools & Manuf. 49, []. C.H. Chen and K.W. Wang, 994: An ntegrated approach toward the dynamc analyss of hgh-speed spndles.. Dynamcs under movng end load. J. Vbraton and Acous., Trans.ASME. 6, [3]. J.F. Tan and S.G. Hutton, 00: Chatter nstablty n mllng systems wth flexble rotatng spndles a new theoretcal approach. J.Manuf. Sc. and Engg. Trans.ASME. 3, 9. [4]. M.R. Movahhedy and P. Mosaddegh, 006: Predcton of chatter n hgh speed mllng ncludng gyroscopc effects. Int. J. Machne Tools & Manuf. 46, [5]. V.Gagnol, B.C. Bougarrou, P. Ray and C. Barra, 007: Stablty based spndle desgn optmzaton. J. Man. Sc. Eng. Trans. ASME. 9, [6]. Y. Altntas and E. Buda, 995: Analyss predcton of stablty lobes n mllng. Annals of CIRP. 44, [7]. S. Jang and S.Zheng, 00: A modelng approach for analyss and mprovement of spndle-drawbar-bearng assembly dynamcs. Int. J.Mach. Tools & Manuf. 50, 3 4. [8]. S.H.Gao, G.Meng and X.H.Long, 00: Stablty predcton n hgh-speed mllng ncludng the thermal preload effects of bearng. J.Process Mech.Engg., Proc.IMechE. 4, -. [9]. H.Cao, T.Holup and Y.Altntas, 0: A comparatve study on the dynamcs of hgh speed spndles wth respect to dfferent preload mechansms. Int J Adv Manuf Technol. 57, [0]. V.Gagnol, T.P. Le and P. Ray, 0: Modal dentfcaton of spndle-tool unt n hgh-speed machnng. Mech. Sys.Sg.Proc. 5, []. H.Cao, B.L and Z.He, 0: Chatter stablty of mllng wth speed-varyng dynamcs of spndles. Int.J.Mach.Tools and Manuf. 5, []. Y.Cao and Y.Altntas, 004: A general method for modelng of spndle bearng system. J.Mech.Desgn, Trans.ASME. 6, [3]. M.Rantatalo, J.O.Adanpaa, B.Goransson and P.Norman, 007: Mllng machne spndle analyss usng FEM and non-contact spndle exctaton and response measurement. Int. J. Machne Tools & Manuf. 47, [4]. H.D Nelson, 980: A fnte rotatng shaft element usng Tmosheno beam theory. J.of machne desgn.0: []. Altntas and E. Buda, 995: Analytcal predcton of stablty lobes n mllng. Annals of the CIRP. 44, []. T.L. Schmtz, M.A. Daves, K. Medcus and J. Snyder, 00: Improvng hgh-speed machnng materal removal rates by rapd dynamc analyss. Annals of the CIRP. 50, [3]. T.L. Schmtz, J.C. Zegert and C. Stanslaus, 004: A method for predctng chatter stablty for systems wth speed-dependent spndle dynamcs. Trans. North Amer. Manuf. Res. Insttuton of SME. 3, 7 4. [4]. T.L.Schmtz and G.S.Duncan, 005: Three-component receptance couplng substructure analyss for tool pont dynamcs predcton. J Manuf Sc Eng. 7, [5]. C.H. Cheng, T.L. Schmtz and G.S. Duncan, 007: Rotatng tool pont frequency response predcton usng RCSA. Machnng Scence and Technology., [6]. Z.Jun, S.Tony, Z.Wanhua and L.U.Bngheng, 0: Receptance couplng for tool pont dynamcs predcton on machne tools. Chnese J.Mech.Engg. 4, -6. [7]. U.V.Kumar and T.L.Schmtz, 0: Spndle dynamcs dentfcaton for Receptance Couplng Substructure Analyss. Precson Engneerng. 36, [8]. A.Ertur, E.Buda and H.N.Ozguven, 007: Selecton of desgn and operatonal parameters n spndle-holdertool assembles for maxmum chatter stablty by usng a new analytcal model. Int. J. Machne Tools & Manf. 47, [9]. R.P.H. Faassen, N. V.Wouw, J.A.J. Oosterlng and H. Njmejer, 003: Predcton of regeneratve chatter by modellng and analyss of hgh-speed mllng. Int. J. Machne Tools & Manf. 43, Volume: 0 Issue: 07 Jul-03, 50