5th Internatonal Conference on Advanced Engneerng Materals and Technology (AEMT 05) Predcton of Chatter Stablty for Mllng Process Usng an Improved Sem-dscretzaton Method LI Zhongqun, a, ZHU Fan,b, XIA Le,b, PEG Yuerong,c School of Mechancal Engneerng, Hunan Unversty of Technology, Zhuzhou, 4007, Chna a zhqunl@63.com, b 5657039@qq.com, c 3437870@qq.com, d 87976898@qq.com Keywords: Mllng Process; Stablty Lobe Dagram; Anaytcal Soluton; Sem-dscretzaton Method; Abstract. On the bass of the classcal sem-dsretzaton method (SDM) proposed by Insperger, an mproved SDM and ts algorthm were mplemented n ths paper. Wth the developed module, the chatter stablty of mllng process can be predcted. As the dchotomy search rather than sequental search s appled, the smulaton tme s greatly reduced. As a result, t s pratcal to determne the optmzed chatter-free cuttng condtons of mllng process n shop floor applcatons. Introducton In mllng process, chatter has negatve effect on the machned surface qualty, the cuttng tool and even the machne tool. Chatter stablty predcton s an mportant and effectve way for optmal selecton of spndle speed and cuttng depth to avod chatter and mprove producton effcency. Based on the classcal dynamc model consderng the regeneratve effect descrbed by a Delay Dffental Equaton (DDE) wth tme varyng coeffcents [], the stablty lobe dagrams (SLDs) can be obtaned to fnd the relatons between the cuttng depths and the spndle speeds [,3]. It contans stablty boundares, below whch vbraton des down and above whch vbraton grows to fnte ampltudes eventually. Except expermental methods [4], numercal algorthms of predctng stablty have been developed. The basc way s to tranform DDEs from tme doman to frequency doman usng the Laplace Transform. Then to calculate the crtcal axal cuttng depth utlzng real and mage part of the characterstc equaton of the system. Altntas [5] presented an analytcal method for predctng mllng stablty based on the mean of the Fourer seres of the dynamc mllng coeffcents. ZOA (Zero-order Analytcal) method s effcent and fast, but t cannot predct the exstence of the addtonal stablty regons n low mmerson mllng. To overcome the problem, Mult-frequency Soluton (MFS), Sem-dscretzaton Method (SDM) and Full-dscretzaton Method (FDM) have been developed. The MFS was frst explored by Budak [6] and then extended by Merdol [7]. The SDM was developed by Insperger [8], and t s effcent n stablty analyss of lnear delayed systems whch can be appled to predct mllng stablty. Dng [9] presented the FDM to predct of mllng stablty whch has hgh computatonal effcency wthout loss of a numercal precson compared wth the SDM. As the effect of hgh harmoncs due to hghly nterrupted cuttng was consdered, the computaton effcency of SDM s not hgh. Moreover, to obtan a SLD wth SDM, the core program must be looped for both dscretzed spndle speeds and avalable cuttng depths, t s very tme consumng. Therefore, a dchotomy search based SDM s proposed n the paper to predct the chatter stablty for mllng operaton. Stablty analyss of DOF mllng process wth SDM The governng equaton of a -DOF mllng model wth a couple and delay dfferental equaton reads 05. The authors - Publshed by Atlants Press 39
ω 0 + ζω & x + ω kx axx t axy t ( ) = ak p t + ζω & ayx t ayy t y + ω ω ( ) && xt xt xt xt xt T && yt yt yt yt yt T 0 ky where ω, ζ x, k x, ω, ζ y, k y are the natural frequency, relatve dampng and stffness n x and y drectons; a p s the axal cuttng depth; K t s the tangental specfc cuttng force; T s tooth passng perod; x(t-t) and y(t-t) are the delayed term; the tme varyng drectonal coeffcents reads = snφ + cosφ a t g t t K t xx j j r j φ φ ( cos ) rsnφ a t = g t + t + K t xy j j j ( cos ) rsnφ a t = g t t + K t yx j j j = snφ + cosφ a t g t t K t yy j j r j () where s the number of the teeth, K r s the rato of radal specfc cuttng force wth respect of tangental specfc cuttng force, φ j (t) s angular poston of tooth j defned as = ( Ω ) + ( ) φj t π /60 t j π / where Ω s the spndle speed n r/mn. The functon g(φ(t)) s a screen functon, f the tooth j s n cut, t s equal ; otherwse, t s equal to 0 In the th sem-dscretzaton nterval, Equaton can be approxmated as (3) ω 0 k ζω & x ω x αxx αxy xt x τ, = ak p t αyx α + ζω & yy yt y y + ω ω τ, && xt + xt + xt && yt yt yt 0 ky where the average drecton coeffency reads α= t t t+ at dt By Cauchy transformaton, Equaton (5) can be wrtten n the followng form q& t = Aq t + Bq + + Bq where m m (5) (6) (7) 40
0 0 0 0 0 0 0 0 0 0 0 0 0 0 a pωhxx apωh xy apωhxx apωh xy A x 0, 0 0 = ω ζω B = (8) kx kx kx kx apωhyx a pωh yy apωhyx apωh yy ω 0 ζω y 0 0 ky k y ky k y (,,, ) and j ( j) ( j), ( j), ( j), ( j) u t = xt yt xt & yt & u = u t = xt yt xt & yt & (9) Then the soluton Equaton (9) for the ntal condton u where + = + m+ + m/ t = u reads u Pu R u u ( ( t) ( t) P= exp A R = exp A IA B As xt &( τ), yt ( τ) & does not appear n Equaton (5), u + depends on x, y, x&, y&, x -m+, y -m+, x -m,, x& m, y& m. Ths leads to the (m + 4)-dmensonal state y -m, but t does not depend on x& m +, y& m + vector ( x y x y x y x y ) z = col,, &, &,,, L,, ( m m The resultng dscrete map reads z = Dz + (3) 0) ) Where the (m+4)-dmensonal coeffcent matrx s R R R R P, P, P,3 P,4 0 L 0 R R R R P, P, P,3 P,4 0 L 0 R R R R P,3 P,3 P,33 P,34 0 0 L R R R R P,4 P,4 P,43 P,44 0 L 0 D = 0 0 0 0 L 0 0 0 0 0 0 0 0 0 L 0 0 0 0 0 0 0 0 0 L 0 0 0 0 0 M M M M M O M M M M M 0 0 0 0 0 L 0 0 0 0 0 0 0 0 0 L 0 0 0 0 0 0 0 0 0 L 0 0 0 0,,,,,,,,,3,3,3,3,4,4,4,4 ( 4) 4
Where P,hj and R,hj are the elements of matrces P and R n the hth row and jth column. The (m+4)-dmensonal transton matrx Φ s determned by couplng Equaton (5) for = 0,,, k-: Φ=D D L DD k k If the egenvalues of Φ are n modulus less than one, then the system s stable. Implementaton of stablty lobe dagram wth SDM Based on the proposed SDM algorthm for stablty analyss of -DOF mllng system, a Matlab-based smulaton module has been developed, whch reads the nput data such as the modal parameter, tool geometry, specfc cuttng forces and cuttng condtons. The flow chart s shown n Fg.. It can be found that two mprovements have been made wth regard to the orgnal program provded by Insperger []. Frstly, an teratve dchotomy search nstead of a sequental search was used to determne the stable borders whch results n a decreased calculaton amount by an order of magntude. Secondly, a one-dmensonal array nstead of a two-dmensonal array was used to store the SLD whch results n a decreased of memory space. Thus, the smulaton tme s greatly reduced. Smulaton and analyss SLD are determned for down mllng where the radal depth of cut a e s set as.0, 0.5, 0., 0.05 tmes tool dameter D wth a two-fluted cutter. The modal parameters of domnant mode of the mllng system are ω = 900 Hz, ζ x = 0.0, k x = 5 0 6 /m; ω = 900 Hz, ζ y = 0.0, k y = 5 0 6 /m. The tangental and radal specfc cuttng forces are K t = 7.96 0 8 /m and K r =.68 0 8 /m. The SLDs for varous a e are shown n Fg.. It s shown that when a e s very low,.e. the process s hghly ntermttent, the SLDs are more accurately predcted by usng SDM than by ZOA (see Fg. 3). The accuracy of the SDM s lmted only by ts samplng nterval and the computaton tme depends on the number of modes n the system and samplng nterval. The approxmatn parameter for the SDM s m = 40, the same as Refrences []. Upon the same.4 GHz Intel Core (TM) Duo CPU, the computaton tme for determnng SLD was 0 s usng the mproved algorthm vs 43 s usng the Insperger s orgnal program. It s shown that wth the mproved algorthm, the total computaton tme was reduced greatly, and thus makes t possble to apply t n shop-floor applcaton. (7) 4
Concluson Fg. Flow chart of obtanng stablty lobe dagram for mllng process wth SDM An mproved SDM algorthm was used to predct the SLDs of mllng process n ths paper. On one hand, as t consders the tme varaton of drectonal factors at each dscrete samplng nterval, t can predct the stablty lobes at a speed and have better predcton accuracy than ZOA, especally when the radal depth of cut s very small. On the other hand, as dchotomy search other than sequental search was used to calculate the crtcal cuttng depth, both the cycles and the computng tme have been reduced greatly, and thus make t possble to be appled n engneerng applcatons. Acknowledgement Ths work was fnancally spported by the atonal atural Scence Foundaton of Chna (537560, 53756), atonal Major Scence and Technology Specal Projects of Chna (0ZX040-0). (a) a e /D = (b) a e /D = 0. 5 43
(c) a e /D = 0. (d) a e /D = 0.05 Fg. Stablty lobe dagram for DOF down mllng wth dfferent radal depth of cut (a) SDM (b) ZOA Fg.3 Comparson of stablty lobe dagrams obtaned wth dfferent method (a e = 0.D) References [] R. Srdhar, R.E. Hohn and G.W. Long: J. Eng. Ind. Vol. 90(968): p. 330 [] S.A. Tobas: Machne tool vbratons (Blacke and Sons Ltd., Glasgow 965) [3] Y. Altntas and E. Budak: Annals of the CIRP, Vol. 44(995), p. 357 [4] F. Ismal and E. Solman: Int J. Mach. Tool Manu. Vol.37 (997): p. 763 [5] Y. Altntas and E. Budak: CIRP Annals Manuf. Techn. Vol. 44(995): p. 357 [6] E. Budak and Y. Altntas: J. Dyn. Syst-T ASME Vol. 0(998): p. [7] S.D. Merdol and Y.Altntas: J. Manuf. Sc. Eng. Vol. 6(004): p. 459 [8] T. Insperger and G. Stepan: Int. umer. Meth. Eng, Vol. 55 (00): 503 58. [9] Y. Dng, L.M. Zhu, X. J. Zhang and H. Dng: Int J. Mach. Tool Manu. Vol. 50(00): p.50 44