Stability of Serrated Milling Cutters

Transcription

1 Machnng Dynamcs Stablty of Serrated Mllng Cutters Z. Dombovar 1, 3*, Y. Altntas 2, G. Stepan 1 1 Budapest Unversty of Technology and Economcs, Department of Appled Mechancs, Budapest 1521, Hungary 2 Unversty of Brtsh Columba, Manufacturng Automaton Laboratory, 2324 Man Mall, Vancouver, BC, Canada, V6T 1Z4 3 Ideko, Department of Mechancal Engneerng, Elgobar, Span * dombo@mm.bme.hu Abstract Hgh performance machnng of super-alloys such as Ttanum and ckel alloys are dffcult due to ther thermal resstance and hgh strength, and they are most wdely used n aerospace and bomedcal ndustry. The low thermal conductvty of such alloys prevents the applcaton of hgh cuttng speeds to avod premature tool falures and excessve tool wear. The performance of machnng can manly be mproved by ncreasng the depth of cut at low speeds. However, hgh strength of the materals leads to large cuttng force coeffcents, whch decrease the chatter free stable depth of cuts. Serrated cutters are most commonly used to mprove the chatter free depths of cuts n roughng operatons. However, except few tme-doman numercal smulaton models, analytcal treatment of chatter stablty of serrated cutters s not studed. Ths paper presents the stablty model of serrated cutters n analytcal sem-dscrete tme-doman. The cutter s dvded nto dscrete axal elements where the engagement and chp load condtons vary. The phase shft between the waves ground on each flute and varyng ptch angle among the successve flutes are ncorporated to the sem-dscrete tme doman model of the cutter. The stablty of the overall cutter s nvestgated by checkng the egenvalues of the lneared sem-dscrete dynamc model of the system. Snce the stablty s dependent on the serraton waves on the flutes, the cutter ptch angle and the ampltude of the chp along the flute, the cutter geometry and cuttng condtons are optmed to acheve maxmum materal removal rate. Sample smulaton examples are shown to demonstrate the vablty of the proposed stablty soluton and optmaton model. 1. ITRODUCTIO The essental am of manufacturng s to acheve the hghest possble materal removal rate wthout volatng the cuttng tool breakage and machne tool torque, power, chatter stablty lmts. The chatter vbraton lmts depth of cut, radal mmerson and speed. In the mddle of the last century, Tlusty and Tobas showed [15, 16] that the so-called regeneratve effect plays an mportant role n the stablty of machnng operatons. Generally the dynamc machnng systems lead to delayed dfferental equatons (DDE), whch lead to mathematcal challenges [11, 17]. Relable stablty predctons of the regeneratve dynamc machnng models (e.g., turnng, mllng, drllng, etc.) are needed by many researchers n the feld. One can fnd technques related to frequency doman (ero order and mult frequency soluton [1, 4, 14]) and to tme doman, too (sem dscretsaton, tme fnte elements and collocaton methods [5, 6, 12, 18]). The current lterature ndcates that the present chatter stablty theorems are most successfully used n mllng operatons where the rato of natural frequences of the machne tool structure to tooth passng frequency of the spndle can be set to nteger number. Whle such nteger numbers from one to fve often correspond to deep stable pockets (.e., lobes), wth decreasng ampltude, the lobes become narrow and shallow at low speed. Whle the theoretcal lnear stablty lmts predct the behavor of the machnng operatons well at hgh speeds at the frst few lobes, the low speed machnng has added complexty due to process dampng whch has yet to be modeled wth satsfactory accuracy [3]. The machnng speeds are restrcted for the dffcult-to-cut thermal resstant materals [7, 8], thus, the hgh speed machnng technques to penetrate nto the stable lobes are not applcable. However, the low frequency machne tool modes wth hgh nerta lead to severe chatter 873

2 12 th Crp Conference on Modellng of Machnng Operatons. Donosta-San Sebastán, Span, May 7-8, 2009 wth large magntude of forces and vbratons n roughng operatons. These are the cases where the serrated cutters are wdely used to reduce the regeneraton mechansm of chatter. Ths paper presents a mathematcal model of mllng wth serrated cutters. The tme-doman model s converted nto sem-dscrete form to evaluate the stablty of the process. Examples show the effectveness of the appled analytcal method to predct stablty and to evaluate the effcency of dfferent types of serrated cutters. 2. MODELIG OF MILLIG WITH SERRATED CUTTERS An end mll wth serraton waves descrbed by cubc polynomals s consdered for 2+½ axs mllng operatons [10, 13]. Snce the frst harmonc of the serraton wave has the relevant effect on stablty, only ths harmonc component s used n the model. Ths way, the number of parameters n the model s also reduced. The dervaton of the mechancal model s presented for a general 3-axs mllng case, but the constant feed f s always set n the x drecton as shown n the fgures, and the examples are presented for a machne tool structure that s stff n the drecton. 2.1 Serraton The serraton s generated by varyng the local radus R of flute defned by, R ( ) R R ( ). (1) where R s the radus of tool shank envelope and the cylndrcal tool, and s the axal coordnate measured from tool tp. The frst harmonc of the varaton R () can be wrtten as a s ( ) R ( ) sn2 1. (2) 2 L where a s the peak-to-peak serraton ampltude and L s the wavelength of the serraton (Fgure 1). s() s the local coordnate along the flutes, whch can be expressed explctly wth the axal coordnate for a smple cylndrcal geometry as s ( ), cos where s the helx angle of the tool (Fgure 1). ndcates the phase shft of the varaton (2) on the th flute whch s, n practce, a unform dvson of the revoluton expressed as, 1 p, k k 1, wth the ptch angles p,k between the flutes. 2.2 Regeneratve delays between the serrated flutes The serrated waves are ground wth relatve phase shft from one flute to the next, whch leads rregular regeneratve phase delays durng mllng. When the ampltude of the chp load s less than the ampltude of the serrated waves on the cuttng edges, some flutes may not cut any chp, whch leads to doublng or trplng the effectve ptch angle at the partcular locaton. Instead of havng a sngle phase delay as n regular end mlls,.e. one tooth passng perod, the system experences multple tme delays. The angular postons of flutes and +l become equal at a certan segment : (, t ) l (, t,l ), where the angular poston of the th flute at has the form 1 (, t ) t p, k ( ). (3) k 1 Fgure 1. Geometry of serrated cutter Ths angle s measured clockwse from the y axs (Fgure 1) wth three ones: the revoluton part, the ptch shft part and the lag angle 874

3 Machnng Dynamcs ( ) tan. R The tme delay between the actual th flute and the flute at the (+l) th poston can be expressed as l 1 1 1, l, l p, k. k 1 The delay s generated from the angular dstance,l between the th and (+l) th teeth at constant spndle speed (rad/s) (Fgure 2). Maxmum of dfferent delays can occur n case of a unform ptch serrated cutter, whereas 2 -(-1) amount of dfferent constant delays are produced by serrated cutters wth nonunfom ptch angles. where r (,t) and r +l (,t,l ) are the vectors, n the absolute coordnate system, pontng to the tp of the edges at, whle f,l (,t) s the correspondng feed moton durng,l.the regeneraton between the flutes and the (+l) s r, l ( t ) r( t ) r( t, l ). ote that, the relatve moton of the tool center s gven by r(t)=col(x(t), y(t), (t)). The normal vector of the th flute at for uneven rad s gven by sn ( ) sn (, t ) n ( ) sn ( ) cos (, t ). cos ( ) The axal mmerson (lead) angle can be expressed as the spatal dervatve of the radus, dr ( ) cot( ( )). d The effectve geometrc chp thckness can be calculated as a mnmum of the geometrc chp thcknesses by takng all the flutes nto account backwards [19], hg, e (, t ) : hg,, e (, t ; e ) mnhg,, l (, t ). (5) l 1 Fgure 2. The possble angular dstances between the flutes n the rotatonal drecton (R +1 ()< R ()< R +l ()) 2.3 Chp thckness The chp thckness s defned approxmately as the local dstance between the already cut and the just cut surface n the drecton of the local normal vector n () of the flute. However, t s not clear whch flutes are n connecton through the chp thckness or, n other words, whch delay causes the regeneraton n the case of serrated flutes. A geometrc chp thckness s ntroduced here to model the chp, whch s not restrcted to be postve, as the real chp thckness s. The chp between the th flute and the (+l) th flute has the followng form hg,, l (, t ) r, l (, t ) n ( ). (4) The local movement of flute compared to flute (+l) at the same angular poston s (Fgure 3), r, l (, t ) r (, t ) r l (, t, l ) ( R ( ) ( )) sn (, ) (, ) ( ) R l t f, l t x, l t ( R ( ) ( )) cos (, ), ( ), R l t y l t, ( t ) l Fgure 3. The possble mssed-cut effect of the serrated tool Snce the number of the mssed-cuts can change along and may be dfferent on each flute, the effectve ndex and the effectve delay have a smplfed notaton and e :, e (, t ). e : e (, t ) The real chp thckness of the th flute at level can be determned as 875

4 12 th Crp Conference on Modellng of Machnng Operatons. Donosta-San Sebastán, Span, May 7-8, h (, t ) : h, e (, t ) g (, t ) hg, e (, t ), where the swtchng functon g, t ) g (, t ) g (, t ) ( r, h, s a multplcaton of two dfferent functons related to radal mmerson 1, gr, (, t ) 0, and to the mssed-cut effect 1, gh, (, t ) 0, en (, t ) mod 2 ex, otherwse; hg, e (, t ) 0, otherwse. The entry angle en and the ext angle ex are measured from the y axs, too, as n (3). 2.4 Cuttng Force Model The cuttng force for unt depth of cut at a partcular edge locaton s expressed (Fgure 4) n edge (tra) coordnate system as, ftra, (, t ) : ftra, ( h (, t )) ( K e K ch (, t )), where the tangental (t), radal (r) and axal (a) edge force coeffcent vectors are T e [ K t e K r e K a e] K (/m) and T c [ K t c K r c K a c] K (/m 2 ) Fgure 4. Dfferental forces on the cuttng edge The dfferental cuttng forces are projected n feed (x), normal (y) and axal () drectons as f (, t ) g (, t ) T (, t ) ftra, ( h (, t )), where the transformaton matrx between the (tra) and (xy) coordnate systems has the form cos T (, t ) sn 0 sn cos cos cos sn sn sn cos sn cos wth : (, t ) and : ( ). The resultant force actng on the cuttng tool s evaluated by ntegratng the dfferental force along the flute and summng the contrbutons of all flutes n cut, F ( t, rt ( )) f ( ( ), t ) d, 1 where s the arc-length coordnate along the cuttng edge of the correspondng flute (Fgure 4), and r t ( )=r(t + ) s the so-called shft functon [17] whch emphases that force F contans regeneraton of the poston of the tool wth multple constant delays, hence, [ max, 0]. Consequently, the force can be calculated as ap f (, t ), rt ( )) cos sn 1 ( 0 F ( t d. (6) ) In numercal calculatons, the ntegraton n (6) s approxmated by a sum of forces at dscrete ponts along the axs, of course. 2.5 Dynamcs For the smplcty of presentaton, the mllng cutter s consdered to have one essental vbraton mode just n each correspondng drecton x, y and of the Cartesan coordnate system. Accordngly, the governng tme perodc DDE wth fnte number of constant delays has the form M r ( t ) Cr ( t ) K r( t ) F( t, rt ( )), (7) where M (kg), C (s/m) and K (/m) are the mass, dampng and stffness modal matrces and all of them are dagonal n ths specal case. Snce the force s lnear, t can be separated to statonary and vbratory part n closed form as F ( t, rt ( )) Fp ( t ) H j ( t ) ( r( t j ) r( t )). (8) j 1 F p (t) and H j (t) are T-perodc (T=2/) and are derved from the rearrangement of (6) wth many () but fnte delays j. The soluton of the lnear DDE (7) and (8) has the followng form r( t ) rp ( t ) u( t ), (9) where the statonary (or partcular) soluton r p (t)= r p (t+t) s a perodc functon, and ts perturbaton u(t) s the homogeneous soluton that

5 Machnng Dynamcs mght be assocated to chatter. Wth the substtuton of (9) nto (7), the perturbed equaton of moton can be obtaned as follows 2 1 u ( t ) [2 n ] u ( t ) ([ n ] M H( t )) u( t ) 1 M H j ( t ) u( t j ), j 1 where H ( t ) H j ( t ), j 1 (10) [2 n ] and [ 2 n ] are dagonal and contan the modal dampng ratos and the natural frequences n (rad/s) of the th modes, respectvely. the orgnal form of (5) s not sutable to determne the possble mssed-cuts due to the serratons. Clearly, nstead of the nstantaneous geometrcal chp thckness h g,i,l (,t), one can use ts statc part only, whch has the form (see (4)): hgst,, l (, t ) ( R ( ) R l ( )) sn ( ) x, l (, t ) sn ( ) sn (, t ) and the effectve ndex e can be determned from hgst,, e (, t ; e ) mn hgst,, l (, t ), l 1 whch s mportant to rearrange the dfferent parts of the resultant force (8) accordng to the delays j occurrng n the dynamc model of the serrated tools. 3. CHATTER STABILITY MODEL The frequency doman solutons for lnear stablty are not adequate n ths case, snce the depth of cut of the serrated tool s not well defned whch s used n the fnal soluton of the constructed egenvalue problems [1, 4, 14]. The sem dscretaton technque can be used to determne the lnear stablty of the serrated cutters, but at the expense of hgh computatonal load. 3.1 Sem-dscretaton The man dea of that method s to dscrete the phase space ut ( ) yt ( ), (wth u u t ( ) u( t ) ) t ( ) of the governng equaton (10), and approxmate the tme perodc coeffcents (H(t) and H j (t)) for a certan perod of tme where analytcal soluton can be obtaned [12]. The orgnal nfnte dmensonal tme perodc DDE can be approxmated wth large but fnte number of ordnary dfferental equatons (ODE). Accordng to the Floquet theorem [9] a lnear map can be constructed. Ths s based on the transton matrx, whch projects the dscrete representaton of the phase space y t () to the next perod [12]. The stablty of (10) can be determned through the egenvalues of, namely the characterstc multplers k. The statonary perodc soluton r p (t) of (7) s orbtally asymptotcally stable f all of the characterstc multplers have modulus less than one, otherwse, t s unstable that leads to chatter [9]. Durng the calculaton of the multplers, the vbraton tself cannot be nvestgated, therefore Fgure 5. The stablty chart of the serrated cutter performed wth the followng parameters: =3, a=1[mm], L=3[mm], K t =900 [MPa], K r =270 [MPa], =30 [deg], D=20[mm], x =0.02, x =0.03, k x =20.16 [/m], k y =25.5 [/m], f x =510 [H] and f y =700 [H]. Dashed lne, =1 non-serrated; thn sold lne, =3 non-serrated; thck sold lne, =3 serrated tool. 3.2 Stablty chart An example for the stablty of the serrated cutter s gven n Fgure 5, where the stablty chart s presented for a certan feed. The tool has =3 serrated helcal flutes, and the structure s assumed to have two modes n the x and y drectons, respectvely. The stablty chart s presented wth an equvalent non-serrated one and three-fluted tools (dashed and thn lnes). The thck lnes n the fgure s the lnear stablty lmt of the serrated cutter related to the dmensonless feed as defned by f f t, a a 877

6 12 th Crp Conference on Modellng of Machnng Operatons. Donosta-San Sebastán, Span, May 7-8, 2009 where f t (mm/flute) and a(mm) are the feed per tooth and the peak-to-peak value of the serraton, respectvely. 4. COCLUSIO Dynamcs and stablty of serrated cutters are modeled. The analyss showed that the stablty of these cutters vares between the stabltes of the one-fluted and the equvalent non serrated counterparts of the orgnal tool. A dmensonless feed was defned whch can charactere the behavor. The stablty results provde an essental tool for any optmaton method n the applcaton of serrated tools for mproved effcency. 5. ACKOWLEDGEMET Ths research was partally supported by the Hungaran Scentfc Research Foundaton OTKA Grant o. K68910, the Spansh- Hungaran Scence and Technology Program Grant o. 8/07, SERC Pratt & Whtney Canada and Unversty of Brtsh Columba Industral Research Char Grant. 6. REFERECES [1] Altntas, Y. and Budak, E., 1995, Analytcal Predcton of Stablty Lobes n Mllng, CIRP Annals Manufacturng Technology 44, [2] Altntas, Y., 2000, Manufacturng automaton: metal cuttng mechancs, machne tool vbratons and CC desgn, Cambrdge Unversty Press, Cambrdge. [3] Altntas Y., Eynan, M. and Onouka, H., 2008, Identfcaton of dynamc cuttng force coeffcents and chatter stablty wth process dampng, CIRP Annals- Manufacturng Technology, 57, [4] Budak, E. and Altntas, Y., 1998, Analytcal Predcton of Chatter Stablty n Mllng Part 1: General Formulaton, J. Dyn. Sys., Meas., Control, 120, [5] Bayly, P.V., Halley, J.E., Mann, B.P. and Daves, M.A., 2003, Stablty of Interrupted Cuttng by Temporal Fnte Element Analyss, Journal of Manufacturng Scence and Engneerng, 125, [6] Engelborghs, K., Luyanna, T. and Roose, D., 2002, umercal bfurcaton analyss of delay dfferental equatons usng DDE BIFTOOL, ACM Trans. on Math. Software, 28, [7] Eugwu E.O. and Wang, Z. M., 1995, Ttanum alloys and ther machnablty a recew, Journal of Materals Processng Technology, 68, [8] Eugwu, E.O., Wang, Z.M. and Machado, A.R., 1999, The machnablty of nckelbased alloys: a revew, Journal of Materal Processng Technology, 88, [9] Farkas, M., 1994, Perodc Motons, Sprnger-Verlag, Berln and ew York. [10] Ferry W.B. and Altntas, Y., 2008, Vrtual Fve-Axs Flank Mllng of Jet Engne Impellers-Part I: Mechanncs of Fve-Axs Flank Mllng, ASME Journal of Manufacturng Scence and Engneerng, 130, (11p). [11] Hale, J. K., 1977, Theory of functonal dfferental equatons, Sprnger, ew York. [12] Insperger, T. and Stepan, G., 2002, Semdscretaton method for delayed systems, Int. J. umer. Meth. Engng, 55, [13] Merdol S.D. and Altntas, Y., 2004, Mechancs and Dynamcs of Serrated Cylndrcal and Tapered End Mlls, ASME Journal of Manufacturng Scence and Engneerng, 126, [14] Merdol, S.D. and Altntas, Y., 2004, Mult Frequency Soluton of Chatter Stablty for Low Immerson Mllng, J. Manuf. Sc. Eng, 126, [15] Tlusty, J. and Spacek, L., 1954, Selfexcted vbratons on machne tools (n Cech), Prague: akl CSAV. [16] Tobas, J. A., 1965, Machne tool vbratons, Blacke, London. [17] Stepan, G., 1989, Retarded Dynamcal Systems, Longman, London. [18] Sala R, Stepan, G. and Hogan, S.J., 2007, Contnuaton of Bfurcatons n Perodc Delay-Dfferental Equatons Usng Characterstc Matrces, SIAM J. on Scen. Comp., 28, [19] Wang, J.J. and Lang, S.Y., 1996, Chp Load Knematcs n Mllng Wth Radal Cutter Runout, ASME J. Eng. Ind., 118, [20] Zataran, M., Muñoa, J., Pegné, G. and Insperger, T., 2006, Analyss of the Influence of Mll Helx Angle on Chatter Stablty, CIRP Annals-Manufacturng Technology, 55,