STABILITY OF UP- AND DOWN-MILLING USING CHEBYSHEV COLLOCATION METHOD

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1 Proceedigs of IDETC/CIE 2005 ASME 2005 Ieraioal Desig Egieerig Techical Cofereces & Comuers ad Iformaio i Egieerig Coferece Seember 24-28, 2005, Log Beach, Califoria, USA DETC STABILITY OF UP- AND DOWN-MILLING USING CHEBYSHEV COLLOCATION METHOD Eric A. Bucher Dearme of Mechaical Egieerig Uiversiy of Alaska Fairbaks Fairbaks AK Pravee Niduarla Dearme of Mechaical Egieerig Uiversiy of Alaska Fairbaks Fairbaks AK Ed Bueler Dearme of Mahemaical Scieces Uiversiy of Alaska Fairbaks Fairbaks AK ABSTRACT The dyamic sabiliy of he millig rocess is ivesigaed hrough a sigle degree-of-freedom model by deermiig he regios where chaer (usable) vibraios occur i he woarameer sace of sidle seed ad deh of cu. Dyamic sysems like millig are modeled by delay-differeial equaios (DDEs) wih ime-eriodic coefficies. A ew aroximaio echique for sudyig he sabiliy roeries of such sysems is reseed. The aroach is based o he roeries of Chebyshev olyomials ad a collocaio rereseaio of he soluio a heir exremum ois, he Chebyshev collocaio ois. The sabiliy roeries are deermied by he eigevalues of he aroximae moodromy marix which mas fucio values a he collocaio ois from oe ierval o he ex. We check he resuls for covergece by varyig he umber of Chebyshev collocaio ois ad by simulaio of he rasie resose via he DDE23 MATLAB rouie. The millig model used here was derived by Iserger e al. [14]. Here, he secific cuig force rofiles, sabiliy chars, ad chaer frequecy diagrams are roduced for u-millig ad dow-millig cases for oe ad four cuig eeh ad 25 o 100 % immersio levels. The usable regios due o boh secodary Hof ad fli (erioddoublig) bifurcaios are foud which agree wih he revious resuls foud by oher echiques. A i-deh ivesigaio i he viciiy of he criical immersio raio for dow-millig (where he average cuig force chages sig) ad is imlicaio for sabiliy is reseed. INTRODUCTION Oe of he mos imora maufacurig rocesses is he millig rocess. The sigle degree-of-freedom model of he millig rocess leads o a delay differeial equaio (DDE) wih ime-eriodic coefficies due o he ime-varyig aure of he forces o he cuig ool eeh. Alhough several aalyical mehods o fid he sabiliy boudaries for DDEs wih cosa coefficies exis, he sabiliy crieria of he millig sysem cao be give i a closed form. A aroximaio mehod is eeded, which aroximaes he ifiie dimesioal moodromy oeraor wih a fiie dimesioal marix. Therefore, he sabiliy ma of he millig rocess as a fucio of he cuig arameers ca be aroximaely deermied. Miis ad Yaushevsky [1] used Fourier series exasios for eriodic erms ad deermied he Fourier coefficies of relaed arameric rasfer fucios. Alias ad Budak [2] used a similar mehod exce ha hey reaied oly he cosa erm i each Fourier series exasio of a eriodic erm. Davies e al. [3] ad Zhao ad Balachadra [4] examied how he eriodic moios los sabiliy durig arial immersio millig oeraios. Davies e al. [5] reseed exerimeal resuls for millig oeraios wih log, sleder edmills, which idicae ha he cosideraio of regeeraive effecs aloe may o be sufficie o exlai loss of sabiliy of eriodic moios for cerai arial immersio oeraios. Davies e al. [6] aalyically showed he exisece of erioddoublig isabiliy lobes alog wih he radiioal Hof isabiliy lobes i machiig. The resuls were cofirmed 1 Coyrigh #### by ASME

2 ideedely by Corus ad Edres [7], ad by Iserger ad Sea [8,9]. These mehods are o resriced o ifiiesimal imes i he cu. Bayly e al. [10,11] exeded he revious aroaches by he use of ime fiie eleme aalysis. This aroach also led o sabiliy aalysis of a discree ma, bu he requireme of small ime i he cu was relaxed. Aalyical ad exerimeal resuls were obaied for a 1-DOF sysem. Mos of he sabiliy resuls obaied by usig he above meioed aroximaio mehods ad he mehods used by he researchers agree wih each oher. Iserger e al. [12] also erformed a frequecy aalysis o obai he sabiliy codiios of ime-eriodic DDEs from which hey discovered ha chaer frequecies (secodary Hof bifurcaio ad eriod doublig bifurcaio) occur a he sabiliy boudaries. They also aalyzed he sabiliy codiios of u- ad dow-millig oeraios [13,14] usig he semi-discreizaio mehod [15] ad he emoral fiie eleme mehod. The sudy was resriced o a 1-DOF millig model ha has he cuig ool carryig a sigle flue. Bayly e al. [16] exeded he revious work o a wo degreeof-freedom model. The rese work rereses he imlemeaio of a ew aroximaio echique based o Chebyshev collocaio. I solves he ime eriodic liear DDEs wih mulile ieger delays ad iecewise smooh coefficies [17]. This mehod evolved from he mehods develoed by Siha ad Wu [18] o solve eriodic ODEs usig he Chebyshev olyomial aroximaio ad by Bucher e al. [19] o obai he moodromy marix for ime-eriodic DDEs wih smooh coefficies by Chebyshev olyomial exasio of he soluio. The collocaio mehod is show i [17] o be secrally accurae o iiial value roblems. I gives a aroximaio o he comac moodromy oeraor of he DDE, whose eigevalues coverge secrally o he exac Floque muliliers. The mehod geeralizes ad exeds o he eriodic coefficies case he liear muli-se mehods ad seudosecral echiques iroduced i [20,21], ad leads o exoeially fas covergece abou he Floque muliliers. I is flexible for sysems wih mulile degrees of freedom ad i roduces sabiliy chars wih high seed ad accuracy i a give arameer rage. I his work, sabiliy chars ad frequecy diagrams are roduced for u-millig ad dowmillig cases of several cuig eeh ad 25 o 100 % immersio levels usig he Chebyshev collocaio mehod. The usable regios due o boh secodary Hof ad fli bifurcaios are foud which agree wih he resuls foud by oher echiques i he lieraure. A ivesigaio i he viciiy of he criical immersio raio for dow-millig (where he average cuig force chages from egaive o osiive) ad is imlicaio for sabiliy is reseed. MECHANICAL MODEL OF MILLING We use he same sigle degree-of-freedom millig model as i [14], o which he reader is referred for addiioal deails i he derivaio. The ool is assumed o be flexible i he feed direcio oly. A summaio of forces acig o he ool i ha direcio roduces he equaio of moio 2 F() && x () + 2 ζω x &() + ω x () =, m where m is he modal mass, ζ is he damig raio, ω is he aural agular frequecy, ad F () is he oal cuig force i he feed direcio o all egaged cuig eeh. The force o he h ooh is give by F () = g ()( F ()cos θ () F ()si θ ()) (2) where g () acs as a swichig fucio. I is equal o oe if he h ooh is acive ad zero if i is o cuig. θ () is he cuer agle of he h ooh as i roaes. The cuig force comoes are he roduc of he ageial ad ormal liearized cuig coefficies K ad K, resecively, he omial deh of cu b, ad he chi widh w () as where F () = K bw (), F () K bw () (1) = (3) w () = f si θ () + [ x() x( τ)]si θ () (4) deeds o he feed er ooh f, he curre ad delayed osiio of he ool, ad θ (). Here, τ = 60 / NΩ[s] is he ooh ass eriod, Ω is he sidle seed give i rm, ad N is he umber of eeh. A summaio over he oal umber N of cuig eeh, ad he subsiuio of equaios (3-4) io equaio (2) yields bh() bf () 2 0 && x () + 2 ζω x & () + ω x () = [ x () x ( τ )] m (5) where N h () = g ()[ K cos θ () + K si θ ()]si θ () (6) = 1 is he τ eriodic secific cuig force variaio, N f () = g 0 ()[ K cos θ () + K si θ ()] f si θ () (7) = 1 ad he agular osiio of he ool is θ ( ) = (2 π Ω / 60) + 2 π / N, where Ω is give i rm. A soluio o equaio (5) is assumed of he form m 2 Coyrigh #### by ASME

3 x() = x () + ξ () (8) where x() = x( + τ ) is he uerurbed τ -eriodic moio, ad ξ () is he erurbaio which vaishes whe o regeeraive chaer vibraios are rese. Subsiuio of equaio (8) io equaio (5) yields bh() 2 && ξ () + 2 ζω & ξ () + ω ξ () = [ ξ () ξ ( τ )] (9) m This is he liear variaioal DDE model used i his aer (ad i [14]). Sabiliy of he ξ () =0 soluio i equaio (9) imlies he sabiliy of he ideal (chaer-free) moio x (). UP-MILLING AND DOWN-MILLING The relaioshi bewee he direcio of ool roaio ad he feed defies wo yes of arial immersio millig oeraios: he u-millig ad dow-millig oeraios. Boh oeraios work i a similar way exce ha he roaio of he cuig ool is i he oosie direcio. However he dyamics ad sabiliy roeries are differe. Parial immersio millig oeraios are characerized by he umber N of eeh ad he radial immersio raio a/d, where a is he radial deh of cu, ad D he diameer of he ool. We ca differeiae u-millig from dow-millig by kowig he agles of coac made by a aricular ooh iside he workiece. The secific cuig force variaio h() i equaio (6) deeds o he scree fucio for eer he h ooh which is defied as g () = 1 if θ < θ () exi < g = 0 oherwise. The ery ad exi agles θ ad () eer θ ca be foud from he figure below [14] as = 0 ad exi 1 θ = cos (1-2a/D) for u-millig, while for dow-millig eer exi he agles are θ = (2a/D-1) ad 1 cos θ = π. chars for 1,2,4, ad 8 eeh. Here we show u- ad dowmillig resuls for 1 ad 4 eeh for he above immersio raios. CHEBYSHEV COLLOCATION APPROXIMATION The Chebyshev collocaio aroximaio mehod used o solve he millig roblem ad obai he sabiliy diagrams is based o he roeries of he Chebyshev olyomials. The sadard formula o obai he Chebyshev olyomial of degree, which is deoed by T () is T () = cosθ, θ = arccos( ), 1 1 (10) The Chebyshev collocaio ois are uevely saced i he give domai corresodig o exreme ois of he Chebyshev olyomial. We ca visualize hese ois as he roecios o he domai [-1,1] of equisaced ois o he uer half of he ui circle as = cos( π /( m 1)), = 0, 1,, m-1 (11) A secral differeiaio marix for m Chebyshev collocaio ois is obaied by ierolaig a olyomial hrough he fucio values a he collocaio ois, differeiaig ha olyomial, ad he evaluaig he resulig olyomial a he collocaio ois. As show i [22], he differeiaio marix D has he followig form: 2 2 2( m 1) + 1 2( m 1) + 1 D11 =, Dmm =, 6 6 D D i =, 2(1 ) i+ ci ( 1) =, c ( ) i 1 1 c i 2, = 1, =2,, m-1, (12) i, i, = 1,, m i = 1, m oherwise The dimesio of D is m m where m is he umber of Chebyshev ois. If I is he x ideiy, he we also defie a dimesio m x m differeiaio marix usig he Kroecker roduc oeraio as The secific cuig force for u- ad dow-millig for immersio raios of 0.25, 0.5, 0.75, ad 1.0 are show i Figures 1-4 for he cases of oe ad four cuig eeh. While he sabiliy chars for u- ad dow-millig for a sigle cuig ooh were reseed i [14], we have roduced ^ D = D I, (13) Now cosider a liear, ime eriodic sysem of DDEs wih fixed delay τ > 0, 3 Coyrigh #### by ASME

4 x &() = Ax () () + Bx () ( τ ) x() φ() τ (14) =, 0 0 x is a 1 where () sae vecor, A () = A ( + T) ad B() = B( + T) are eriodic marices, φ () is a 1 iiial vecor fucio i he ierval [ τ,0]. Assumig he delay is equal o he eriod ( τ = T, which is o a ecessary assumio for he rocedure), if Φ () is he fudameal soluio marix o he o-delay ar of (14) ad Ψ () is he fudameal soluio marix for he adoi sysem such ha 1 T Φ () =Ψ (), he ifiiedimesioal moodromy oeraor for a eriodic DDE sysem ca be defied as [23] T ( Ux)() =Φ (){ x( τ ) + Ψ( s) B() s x( s) ds} (15) which mas coiuous fucios from he ierval [0,T] back o he same ierval, i.e., U : C[0, T] C[0, T]. If he maximum of he modulus of he eigevalues (Floque muliliers) of he moodromy oeraor U is less ha 1, he he sysem is said o be sable. I is imossible o umerically fid all he eigevalues of he ifiie dimesioal U marix. However, we use he Chebyshev collocaio aroximaio mehod o reduce he size of he U marix o a fiie dimesio, whose secral radius decides he sabiliy. Because of he comacess of he U marix, all of he egleced eigevalues are guaraeed o be clusered abou he origi ad hus do o ifluece he sabiliy. Solvig he umerical aroximaio of equaio (14) usig he Chebyshev collocaio mehod will give a aroximaio o he moodromy oeraor i equaio (15) [17]. Fidig he aroximae soluio by kowig he fucio values a differe ois i a give ierval is he basic idea of collocaio. Firs, le { φ } ad { v }, =1,,m be ses of fucio values a shifed Chebyshev collocaio ois i he ierval [0,T] where he ois are ordered righ o lef as i equaio (11) ad [22]. The { φ } are give values of he iiial fucio φ() i he ormalized ierval [-T,0] ad he v } are values of he soluio x() o be foud i he { ormalized ierval [0,T]. Noe ha he machig codiio a =0 requires ha φ 1 = v m. The by he mehod of ses we ca obai he { v } as { v } = U{ φ } for a fiie marix U which aroximaes he moodromy oeraor. To obai U, we wrie equaio (14) i he collocaio aroximaio form as 0 D { v } = M { v } + M { φ } (16) ^ ^ ^ A B ^ The marix D is obaied from D by modifyig he las rows as [0 0 0 I ] where 0 ad I are x ull ad ideiy marices ad he scalig o accou for he shif [ 1,1 ] [0, T ] by mulilyig he resulig marix by 2/T. ^ ^ The aers of he M A, M B marices are A ( 1) A ( 2) A ( ^ 3) M A =, (17). A ( m 1) ad B ( 1) B ( 2) B ( ^ 3) M B = (18). B ( m 1) I h where A ( i ) meas A() evaluaed a he i shifed collocaio oi. Here he ha ^ ex o he oeraors refers o he modificaio of he las rows o accou for he machig codiios bewee successive iervals (icludig he modificaio o D ^ above). Therefore, we ge he aroximaio o he moodromy oeraor as ^ 1 ^^ ^ ^ U= D M M A B (19) If m is he umber of collocaio ois ad is he size of he DDE sysem, he he size of he U marix will be m m. We ca achieve higher accuracy by icreasig he value of m. I [17] i is show i a a oseriori sese ha if A() ad B() are sufficiely smooh he he aroximae eigevalues (Floque muliliers) of U coverge o he exac eigevalues of U i equaio (15) a a exoeial rae. STABILITY CHARTS AND FREQUENCY DIAGRAMS Sabiliy chars are deermied by usig he Chebyshev collocaio mehod o aalyze he moodromy oeraor as i deeds o arameers. We will cosider a series of millig rocesses like u-millig ad dow-millig, varyig immersio raios, ad varyig umber of cuig eeh. Sice he secific cuig force variaio h() is ideede of he sidle seed of he ool, we assume he sidle seed Ω is 4 Coyrigh #### by ASME

5 3300 rm o lo h(). Exerimeally ideified arameers give i [15] are used o cosruc he sabiliy chars: m = kg, ζ = , ω = Hz, K = N/m 2 8 ad K = N/m 2. Sabiliy chars are cosruced wih arameers beig he sidle seed Ω (ragig from 2000 o rm) ad he chi hickess b (ragig from 0 o 5 mm). MATLAB sofware is used for roducig he sabiliy char usig he collocaio mehod. The dimesio of he moodromy oeraor for all cases is However i is ossible o roduce he same sabiliy char by akig a lower dimesioal (e.g ) moodromy marix for mos of he cases (exce for he criical immersio raio cases). We chose grid ois i he arameer lae. The comuaioal secificaios used o ru he MATLAB rograms are: Iel Peium IV, rocessor seed 1.5 GHz, RAM 1.02 GB. I akes aroximaely miues o obai each sabiliy diagram. The sabiliy aalysis is based o he deermiaio of he releva characerisic mulilier usig he collocaio mehod. We use bifurcaio heory o exlai he ye of isabiliy. For he µ =1 (fold bifurcaio) case, i ca be show ha his bifurcaio cao occur i he millig equaio. For secodary Hof bifurcaio, µ =1 ad = i λ ω is urely imagiary where ω =Im(l µ )/τ. I his case, he chaer frequecies are deermied from ω, which is also he osiive agle made by he characerisic mulilier i he comlex lae. Sice he comlex exoeial fucio is eriodic, he logarihmic fucio is o uique i he lae of comlex umbers. This raises he ossibiliy of mulile chaer frequecies. These chaer frequecies ca be observed while doig exerimes o he millig machies. These frequecies ca be measurable ad comarable wih he heoreical resuls. Chaer frequecy diagrams are cosruced i Figures (1-4) by cosiderig he characerisic muliliers obaied a he sabiliy boudary ad usig he formulaio i [12]. However, if he characerisic muliliers are foud usig he collocaio mehod, he he equaios for Hof frequecies i [12] mus be alered by dividig by he facorτ due o he ormalizaio used i he collocaio mehod. Therefore, he Hof frequecies are give as f H ω NΩ = ( ± + ) [Hz], =,-1, 0, 1, 2π 60 (20) where τ is give i sec. ad Ω i rm. For he eriod doublig case ( µ = -1), he characerisic exoe is λ = (l( 1)) / τ or we subsiue agle ω = π io (20) as 1 N Ω f = ( + ) PD [Hz], =,-1, 0, 1, (21) 2 60 We check he resose a some of he arameer ois i he sabiliy chars usig he MATLAB rouie DDE23 [24]. If he soluio of he give sysem decays as ime goes o ifiiy, he he sysem is said o be sable a he give arameer ois; oherwise he sysem is usable. Usig his coce we ick hree arameer ois from he sabiliy chars show i Figures 1 ad 4 ad, usig DDE23, we check wheher hose arameer ois are sable or usable. The resuls show i Figures 5-6 agree wih he sabiliy chars obaied by he collocaio mehod (see he locaios of characerisic muliliers obaied by usig he collocaio mehod). DISCUSSION For some cases i he millig rocess, we ca oice a drasic chage i he sabiliy chars us by chagig he immersio raio. Cosider he sabiliy chars of he dow millig sigle ooh case show i Figure 2, where he order of Hof bifurcaio sabiliy lobe ( U shaed) ad fli bifurcaio sabiliy lobe ( V shaed) is swiched by chagig he immersio raio. Sabiliy chars draw for differe immersio raios bewee 0.62 o 0.71are show i Figure 7, o illusrae wha really haes o he sabiliy diagrams bewee hese immersio raios. We ca see ha he millig case wih immersio raios 0.63 o 0.68 has larger sabiliy regio comared o ay oher millig case for he give sidle seed Ω ad is fully sable for he sidle seed rage of 9000 o r..m. We ca also oice ha he immersio raios above have a osiive average secific cuig force variaio h() value, whereas for lower immersio raios, he value is egaive. This is oe of he reasos exlaiig he drasic chage i he sabiliy codiios ear he criical immersio raio. For he egaive deh of cu case, wih immersio raios less ha he criical immersio raio, he corresodig sabiliy diagrams reveal iformaio abou obaiig he sabiliy regio for osiive chi hickess by kowig he usable regio for egaive chi hickess. Noe ha he above heory is alicable o oly Hof ye sabiliy lobes (i.e., he fli ye lobes do o chage drasically). I Figures 1-4, he similariies ad differeces bewee umillig ad dow millig ca be clearly observed. The fli (eriod doublig) lobes, for examle, vary i size bu are locaed more or less a he same sidle seed rage (aroud o r..m). This is o rue for Hof lobes. For low immersio umillig, he Hof lobes are locaed o he lef of fli lobes, while he dowmillig cases show his secial dualiy or mirror symmery for immersio raios wih 0.5 or less. A exlaaio for hese ieresig resuls is as follows: The fli lobes are relaed o he imac effecs of eerig ad leavig he workiece maerial. While hese are more or less ideede of he sese (u or dowmillig) of he millig, 5 Coyrigh #### by ASME

6 his is o he case for he Hof lobes. These fli lobes occur for lower immersios ad lower umber of cuig eeh cases. High seed millig oeraios ca be sabilized simly by chagig o dowmillig from umillig a cerai wide high seed arameer domais (9000 o rm). This is where he criical immersio raio rage of 0.63 o 0.68 for dowmillig is so imora, because i has a higher sabiliy regio ha ay oher case. For he mulile cuig eeh umillig case (Figure 3), he resece of idle ime (i.e., if h() is zero) leads o fli lobes i he sabiliy char. Accordig o he assumio made earlier, h ha ay ooh follows he same cuig rofile as he firs cuig ooh, leads us o he coclusio ha for all eve umbers of eeh wih full immersio (exce N = 2), we will have cosa secific cuig force variaio ha makes his millig case look similar o he urig oeraio. Also he sabiliy chars for millig ad urig cases look he same. The DDE23 resuls (Figures 5-6) ad Chebyshev collocaio resuls agree wih each oher. For he secific cuig force variaio (Figures 1-4), he aroximaio of h() usig Chebyshev ois gives reasoably good resuls wih similar relaive errors comared o he oher mehods which use equisaced ois. The umber of Chebyshev ois should be large eough o ge reasoably accurae sabiliy chars. Thus, also he Chebyshev collocaio mehod is exoeially coverge for smooh coefficies [17], he resece of discoiuiies i he secific cuig force variaio leads o a higher miimum umber of ois for he millig roblem ha wha would ormally be execed. (The suggesed miimum umber for m is 20, while ear he criical immersio raio i Figure 7 we use m = 80.) ACKNOWLEDGMENTS This aer is based uo work suored by he Naioal Sciece Foudaio uder Gra No REFERENCES [1] I. Miis, R. Yaushevsky, 1993, A ew heoreical aroach for he redicio of machie ool chaer i millig, ASME Joural of Egieerig for Idusry 115, 1-8. [2]Y. Alias, E. Budak, 1995, Aalyical redicio of sabiliy lobes i millig, Aals of he CIRP 44, [3] M. A. Davies, J. R. Pra, B. Duerer, T.J. Burs, 2002, Sabiliy redicio for low radial immersio millig, J. Maufacurig Sciece Egieerig 124, [4] M. X. Zhao, B. Balachadra, 2001, Dyamics ad sabiliy of millig rocess. Il. J. Solids ad Srucures, Vol. 38, [5] M. A. Davies, B. Duerer, J. R. Pra, A. Schau, 1998, O he dyamics of high seed millig wih log, sleder edmills, Aals of he CIRP, Vol. 47, [6] M. A. Davies, J. R. Pra, B. Duerer, T. J. Burs, 2000, Ierrued machiig- a doublig i he umber of sabiliy lobes, J. Maufacurig Sciece Egieerig. [7] W. T. Corus, W. J. Edres, 2000, A high order soluio for he added sabiliy lobes i iermie machiig, MED-Vol. 11, Proc. ASME Maufac. Egg. Div., [8] T. Iserger, G. Sea, 2000, Sabiliy of he millig rocess, Periodica Polyechica 44 (1), [9] T. Iserger, G. Sea, 2000, Sabiliy of high seed millig, Proc. ASME 2000 IMECE, Nov. 5-10, 2000, Orlado, FL [10] P. V. Bayly, M. A. Davies, J. E. Halley, 2000, Sabiliy Aalysis of ierrued cuig wih fiie ime i he cu, Proc. ASME 2000 IMECE, Nov. 5-10, Orlado, FL. [11] P. V. Bayly, J. E. Halley, B. P. Ma, M. A. Davies, 2001, Sabiliy of ierrued cuig by emoral fiie eleme aalysis, Proc. ASME 2001 DETC, Pisburgh, PA, aer o. DETC2001/VIB (CD-ROM). [12] T. Iserger, G. Sea, P.V. Bayly, B.P. Ma, 2003, Mulile chaer frequecies i millig rocesses, Joural of Soud ad Vibraio 262, [13] T. Iserger, B.P. Ma, G. Sea, P.V. Bayly, 2003, Sabiliy of u-millig ad dow-millig, ar 1: aleraive aalyical mehods. Ieraioal Joural of Machie Tools ad Maufacure 43, [14] B.P. Ma, T. Iserger, P.V. Bayly, G. Sea, 2003, Sabiliy of u-millig ad dow-millig, ar 2: exerimeal verificaio. Ieraioal Joural of Machie Tools ad Maufacure 43, [15] T. Iserger, G. Sea, 2004, Udaed semi-discreizaio mehod for eriodic delay-differeial equaios wih discree delay, I. J. Num. Meh. Egg. 61, [16] Bayly, P. V., Ma, B. P., Schmiz, T. L., Peers, D. A., Sea, G., Iserger, T., 2002, Effecs of radial immersio ad cuig direcio o chaer isabiliy i ed-millig, Proc. ASME IMECE, New Orleas, aer o. IMECE (CD-ROM). [17] E. Bueler, 2004, Chebyshev collocaio for liear, eriodic ordiary ad delay differeial equaios: a oseriori esimaes, arxiv:mah.na/ [18] S. C. Siha, D. -H. Wu, 1991, A efficie comuaioal scheme for he aalysis of eriodic sysems, Joural of Soud ad Vibraio, Vol. 151, [19] E. A. Bucher, H. Ma, E. Bueler, V. Averia, Z. Szabo, 2004, Sabiliy of liear ime-eriodic delay-differeial equaios via Chebyshev olyomials, I. J. Num. Meh. Egg. 59, [20] K. Egelborghs, D. Roose, 2002, O sabiliy of LMS mehods ad characerisic roos for delay differeial equaios, SIAM J. Num. Aal. 40(2), [21] D. Breda, S. Mase, ad R. Vermiglio, Pseudosecral differecig mehods for characerisic roos of delay differeial equaios, SIAM J. Sci. Comu., o aear. [22] L. N. Trefehe, Secral Mehods i MATLAB, SIAM, Sofware- Evirome-Tools Series, Philadelhia, [23] J. K. Hale, M. V. Luel, 1993, Iroducio o fucioal differeial equaios, New York: Sriger. [24] L.F. Shamie, S. Thomso,, 2001, Solvig delay differeial equaios wih DDE23, Al. Numer. Mah. 37(4), Coyrigh #### by ASME

7 Figure 1. U-millig, umber of cuig eeh N = 1, secific cuig force variaio diagrams, frequecy diagrams ad sabiliy diagrams for varyig immersio raios a/d=0.25, 0.5, 0.75, 1 Figure 2. Dow-millig, umber of cuig eeh N = 1, secific cuig force variaio diagrams, frequecy diagrams ad sabiliy diagrams for varyig immersio raios a/d=0.25, 0.5, 0.75, 1 7 Coyrigh #### by ASME

8 Figure 3. U-millig, umber of cuig eeh N = 4, secific cuig force variaio diagrams, frequecy diagrams ad sabiliy diagrams for varyig immersio raios a/d=0.25, 0.5, 0.75, 1 Figure 4. Dow-millig, umber of cuig eeh N = 4, secific cuig force variaio diagrams, frequecy diagrams ad sabiliy diagrams for varyig immersio raios a/d=0.25, 0.5, 0.75, 1 8 Coyrigh #### by ASME

9 Figure 5. DDE23 resuls for he arameer ois A, B ad C icked from he sabiliy diagram of u-millig wih N = 1, a/d = 1 ad collocaio resuls for fidig he locaios of characerisic muliliers Figure 6. DDE23 resuls for he arameer ois A, B ad C icked from he sabiliy diagram of dow-millig wih N = 4, a/d = 0.25 ad collocaio resuls for fidig he locaios of characerisic muliliers 9 Coyrigh #### by ASME

10 Figure 7. Criical immersio raios for dow-millig, N = 1, collocaio ois m = 80, arameer lae grid ois 10 Coyrigh #### by ASME