Numerical and Experimental Investigation for Stability Lobes Prediction in Thin Wall Machining

Transcription

1 Egieerig Leers, 7:4, EL_7_4_7 Numerial ad Exerimeal Ivesigaio for Sabiliy Lobes Prediio i Thi Wall Mahiig O. B. Adeoro, P. H. We, W. M. Sim, R. Vea Absra A Fiie Eleme Aalysis (FEA) ad Fourier rasform aroah o obai frequey resose fuio (FRF) is reseed i his aer. The aim i his aer is o elimiae he eed for he lassial ima exerimeal aroah used i exraig sruure s FRF. The umerial ad exerimeal FRFs have bee used o obai sable regios i mahiig of hi walled sruures, whih gives a good omariso. Examles are reseed ad omared wih exerimeal resuls wih a saisfaory agreeme. Idex Terms FEA, frequey resose fuio, disree Fourier rasform, sabiliy lobes, rasfer fuio. I. INTRODUCTION Eve afer suh a exesive researh io haer vibraio, i sill is (as saed by Taylor us over a eury ago) oe of he mos obsure ad deliae of all roblems faig he mahiis []. I eraily udermies ad redues roduiviy ad surfae qualiy i maufaurig. I ould also irease he os hrough ossible mahie or ool damage. I is beause of hese effes ha i has bee he oi of several sudies over he years. The sabiliy lobes/har aroah is more raial from he sae of a mahiis, while is exraio a be somewha edious. The auray of he redied sable regio relies o he rasfer fuio ideified a he uer-workiee oa zoe. The lassial aroah o obaiig he rasfer fuio is hrough ima es. However, his aer rooses a aleraive aroah whih uses fiie eleme mehod (FEM) modal aalysis o obai he rasfer fuio a seified uer-workiee oa zoes. While he rasfer fuios for he ool a be assumed o be osa, he workiee rasfer fuio/dyamis are osaly hagig as maerial is removed. Moreover, i Mausri reeived Marh 8, 9. This work was suored i ar by EPSRC ad Airbus UK uder Gra BS3456. O. B. Adeoro is wih Quee Mary Uiversiy of Lodo, Quee Mary, Uiversiy of Lodo, Mile Ed Road, Lodo E 4NS, UK ( o.adeoro@qmul.a.uk). * P. H. We is wih Quee Mary Uiversiy of Lodo, Quee Mary, Uiversiy of Lodo, Mile Ed Road, Lodo E 4NS, UK; (hoe: 44() , fax: 44() ; .h.we@qmul.a.uk). W. M. Sim is wih Airbus, New Filo House, Golf Course Lae, Filo BS34 7AR, UK ( WeiMig.Sim@airbus.om). R. Vea is wih Quee Mary Uiversiy of Lodo, Quee Mary, Uiversiy of Lodo, Mile Ed Road, Lodo E 4NS, UK ( r.vea@qmul.a.uk). hi wall mahiig, he workiee vibraio is sigifia omared o ha of he ool. Hee he rasfer fuio used mus be reise. I will be highly imraial o erform ima ess a mulile sages of mahiig, hee he eed for a offlie aroah o sabiliy lobes rediio. The rediio of sable odiios i he form of hars sared whe, Tobias [] ad Tlusy [3] simulaeously made he remarkable disovery ha he mos imora soures of self-exiaio, regeeraio ad mode oulig were assoiaed wih he sruural dyamis of he mahie ool-workiee sysem ad he feedbak resose bewee subseque us. A his early sage, he sabiliy lobes aroah ha is widely used by researhes o redi he sable margi was also esablished by []. While Tlusy ad Polaek [3] obaied a exressio for haer free axial deh of u usig he uig fore oeffiie, ad he real ar of he sruure s rasfer fuio i he direio ormal o he mahied surfae. This was laer imroved by Tlusy [4] o ilude he effe of he sidle seed o he haer frequey ( lobbig effe ). Oher sudies o he sabiliy of meal uig were reored Merri [5]. Though, a ioeerig researh, he sabiliy models by Tobias ad Tlusy are oly aliable o orhogoal meal uig where he direioal dyami millig oeffiies are osa ad o eriodi. This is quie he orary i millig due o he roaig uer wih mulile eeh. I order o aommodae his direioal dyami millig oeffiies, ime domai simulaio of he millig roess was irodued by Tlusy [6, 7]. Slaviek [8] ad Vaherk [9] made he assumio ha all he uer eeh have a osa direioal orieaio i heir sudy of he effe of irregular ih o he sabiliy. Sridhar e al. [, ] ad Hoh [] laer arried ou a i-deh sudy i whih, hey irodued ime-varyig direioal oeffiies i heir haer sabiliy aalysis. They used he sysem s sae rasiio marix i heir sabiliy model, whih hels o elimiae he eriodi ad ime delay erms. Oiz e al. [3, 4] used a average value of he eriodi direioal oeffiies i he aalysis. Tlusy [5] made a aem o aly he orhogoal model o millig roess by assumig he eeh of he ool had equal ih, was simulaeously i u ad ha he moio was reiliear wih osa deh of u. The Nyquis rierio was used by Miis ad Yaushevsky [6, 7] ad Lee e al. [8, 9] o obai he sabiliy limis. Lee e al. used he mea value mehod o relae he ime varyig direioal oeffiies by a osa. U uil his oi here exised o roosed aalyial aroah o rediig he sabiliy (Advae olie ubliaio: 9 November 9)

2 Egieerig Leers, 7:4, EL_7_4_7 margi for millig, whils reseig he varyig direioal dyami millig oeffiies. Followig he i-deh work by Budak, [], Alias ad Budak [,, 3] laer roosed a aalyi aroah i whih he zeroh order erm i he Fourier series exasio (sigle frequey soluio or zeroh order aroximaio) of he ime varyig oeffiies was adoed. A similar model was laer used by Alias e al. [4], where hey roosed a average sheme of he immersio agle, while he aalyial model was laer exeded o ilude hree direios by Alias [5], where he axial immersio agle was also assumed o be osa. Cama e al. [6] laer roosed a averagig aroah o alulaig he axial immersio agle i order o solve he sabiliy model aalyially. Adeoro e al. [7] reely roosed some modifiaios o he sabiliy lobe model by Alias [5]. The modifiaios allow for he ilusio of he oliear aure of he uig fore oeffiies ad he axial immersio agle alog he axial deh of u i he rediio of more aurae sable uig odiios. The resuls were obaied usig a umerial aroah To aalyially redi he sable regio he dyami arameers ideified a he uer-workiee oa zoe are used. The lassial aroah o obaiig he dyami arameers is hrough ima ess. Ulike i ool haer, he dyami arameers are o osa alog he workiee ad are osaly hagig as maerial is removed ad he geomery hages. Aems were made by Theveo [8] o use his varyig dyamis i hi wall mahiig o iiiae he variaio of he sidle seed alog he workiee i order o imrove surfae fiish. The edey i his aroah however is he edey for ew marks o be lef o he surfae due o he hage i uig odiios as see from heir exerimeal resuls. Budak osidered he variaios he dyamis of he uer ad he workiee alog he axial direio [, 3]. Seguy e al. [9] us reely arried ou a sudy o ilude he varyig dyamis alog a hi wall ad hi floor seio, alhough he resuls show erai disreaies whih ould have arise from he assumios made. I is however lear i hi wall ha i is isuffiie o assume he dyamis of he workiee are osa, whih has reviously bee he ase. This aer reses a umerial aroah o obaiig he sruures rasfer fuio, whih is required i he sabiliy model. This aroah aims o elimiae he eed for series of exerimeal ima esig a various ois o a hi walled workiee i order o obai he orresodig rasfer fuio a he oi. The oly exerimeal resul required would be he oe o obai he damig arameers of he sruure, whih a also be elimiaed by adaig he aroah o rediig he damig arameers by Adeoro e al. [35, 36]. However o obai he varyig dyamis alog he ool-ah, his aroah a be used o reve furher exerimeal ima esig. The full rasfer fuio marix i all he hree raslaioal direios a also be easily exraed. Comared o exerimeal mehods, his a rove diffiul i erai raslaioal direios. The aroah is reseed here wih he -D sabiliy model i [] ad a easily be adaed o he 3-D model i [5, 7] as show by Adeoro al. [3]. II. CHATTER STABILITY MODEL The sabiliy model used i his aer is he model roosed by Alias ad Budak [] as summarized below. The eriodi millig fores exie he uer ad he workiee ausig wo orhogoal dyami dislaemes x ad y i he global axis. vibraio marks lef by ooh () Figure Dyami Millig Model. This geeraes udulaios o he mahied surfae ad eah ooh removes he udulaios geeraed by he revious ooh (Figure ). Therefore leadig o a modulaed hi hikess whih a be exressed as h φ = siφ + υ υ υ υ () ( ) ( ) ( ), s s where is he feed er ooh, ( υ, ) ad ( υ, ) w υ w are w υ w he dyami dislaeme of he uer ad workiee a he revious ad rese ooh eriods reseively, φ = φ + Ω is he agular immersio of ooh ( ) for a uer ( Ω is he agular seed), wih osa ih agle φ = π N ( N is he umber of eeh). The dyami dislaemes i he hi hikess direio due o ool ad workiee vibraios are defied as υ = x siφ y osφ ( =, w), () where ad w idiae he uer ad workiee reseively, x, y ad x, y are he dyami dislaemes i he global axis for he urre ad revious ooh eriods reseively. By elimiaig he sai ar i (), he dyami hi hikess i millig is defied as h φ = Δxsiφ Δy osφ (3) y ( ), where, ( ) ( Δx = x x xw xw ), Δy = ( y y ) ( y y ), x w u, df ooh ( ) ooh (-) w v, F r Therefore, he dyami fores o ooh (usig Exoeial Fore Coeffiie Model, []) i he ageial ad radial direios a be defied as vibraio marks lef by ooh (-) ooh (-) (4) (Advae olie ubliaio: 9 November 9)

3 Egieerig Leers, 7:4, EL_7_4_7 ( φ) = Kah ( φ ), ( φ ) = K F ( φ), F (5) Fr r where a is he axial deh of u (ADOC), ad K ad K r are he ageial ad radial uig fore oeffiies reseively. For simliiy, like i oher sudies hese uig fore oeffiies have bee ake as osa here. However hey have bee show o affe he redied margi by Adeoro e al. [7]. This aroah here a easily be adaed o he modifiaios hey roosed. Therefore, by subsiuig (3) io (5) ad resolvig i he global direios, he followig exressio is obaied F a a Δx x F y where = a xy ak a xx yx a xy yy, Δy (6) are he eriodi direioal uig oeffiies ad deeds o he agular osiio of he uer ad he radial uig fore oeffiie K r, hereby makig (6) a fuio of ime { F() } ak [ A() ] Δ() { }, = (7) [ A () ] As meioed i revious seio, is eriodi a he ooh assig frequey ω = NΩ, herefore is Fourier series exasio is used for he soluio of he sysem. The average value i he Fourier series exasio (sigle frequey soluio) of he ime varyig direioal oeffiies is used i his aer. Hee, (7) redues o { F() } = ak [ A ]{ Δ() }, (8) where [ A ] is he ime ivaria, bu radial immersio deede direioal uig oeffiie marix. From he frequey resose fuio FRF ad he dyami fores, he dyami dislaeme veor i (8) a be solved. Usig he resose a rese ime ( ad he revious ooh eriod ( T ), equaio (8) a be exressed as [] iω iωt iω { F} e = ak [ A ]( e )[ G( iω )]{ F} e, (9) where { F} rereses he amliude of he dyami uig fore { F () }, [ G ( iω )] is he rasfer fuio marix. The rasfer fuio marix [ G( iω )] is he mai fous of his aer. I is defied as [ G ( iω )] = [ G ( iω )] + [ Gw ( iω )], () where [ G ( i )] ( iω ) G ( iω ) xy ( iω ) G ( iω ), G xx ω = ( =, w) () G yx yy Equaio (9) has a o-rivial soluio oly if is deermia is zero, [[] Λ[ G ( i )], de I ω = () + ) where [ ] [ A ][ G] G = The eigevalues is defied as N iωt = K a( e ), 4π Λ (3) Solvig () umerially will give eigevalues wih Λ = Λ +, ad from Euler s omlex ad real ars ( ) e iωt R iλ I = osωt i siωt formula,. Whe his is subsiued io (3), he omlex ar has o vaish (i.e. Λ I ( osωt ) = Λ R siωt ) beause he axial deh of u a is a real value. Therefore, Λ I siωt κ = = = aψ, (4) Λ R osωt where ψ is he hase shif of he eigevalues. From his exressio he relaioshi bewee he frequey ad he sidle seed is [] obaied ωt = ε + kπ, ε = π ψ, ψ = a κ, (5) 6 =, NT where ε is he hase differee bewee he ier ad ouer udulaios, k is a ieger orresodig o he umber of vibraio waves wihi a ooh eriod ad is he sidle seed (rm). Subsiuig (4) io (3) ad he fial exressio for haer free axial deh of u beomes πλ R alim = ( + κ ) (6) NK Therefore for a give haer frequey, ω he eigevalues are obaied from (), whih allows for he riial deh of u o be alulaed usig (6) ad fially he sidle seed usig (5) for differe umber of vibraio waves, k. This is reeaed for various frequeies aroud he sruures domia modes. III. THE SYSTEM S TRANSFER FUNCTION To obai he rasfer fuio of he sysem, he modal dyami aalysis o Abaqus was used. Beig a very well develoed model, he modal dyami aalysis gives he resose of a defied domai as a fuio of ime for a give ime deede loadig. This gives he liear resose of he sruure, whih a be very easily exraed oe he modes of he sysem are available. This is due o he modes beig orhogoal, hereby rederig he sysem as a mere ombiaio of sigle degree of freedom sysems. The modes are exraed i a frequey exraio aalysis, whih uilizes he Lazos algorihm. The free vibraio soluio of he equaio of moio akes he form { x } = { X } siω (7) Whe subsiued io equaio of moio, a eigevalue roblem is obaied as (Advae olie ubliaio: 9 November 9)

4 Egieerig Leers, 7:4, EL_7_4_7 ([ ] [ M ]){ X } =, [ ] K ω (8) where K is he siffess marix of he sysem, [ M ] is he mass marix, ω is he eigevalue or i his ase he udamed aural frequey of he sysem squared ad { X } is he eigeveor (he mode of vibraio or mode shae). The rasie modal dyami aalysis o Abaqus was used o solve he eigevalue roblem ad o redi he sysem s rasfer fuio. The modal dyami aalysis gives he resose of a defied domai as a fuio of ime for a give ime deede loadig. The resose obaied is he liear resose of he sruure, whih is easily exraed oe he modes of he sysem are available. The modes are exraed i a frequey exraio aalysis, whih uilizes he Lazos algorihm due o he size of he eigeroblem i equaio (4.9). The algorihm is deailed by Grimes e al. [3] ad i he Abaqus user maual [3]. Therefore, whe he model is roeed oo he eigemodes used for he sysem s dyami rereseaio (i.e. uoulig he sysem s siffess, mass ad damig maries usig he orhogoaliy roery exlaied earlier), is equaio of moio is uouled ad a exressio a ime is [3] obaied Δf q + ζ ω, q + ω, q = f Δ + Δ, (9) Δ where is he mode umber, q is he amliude of he resose of mode (i he geeralized oordiae ), ω, is he udamed aural frequey of mode, Δ f is he hage i f over he ime ireme, Δ assumig he exiaio varies liearly wihi eah ireme ad ζ is he damig raio for mode. The soluios is obaied [3] i he form q+δ d d q e e f = +, () q +Δ d d q e e f +Δ where i, l =,, d ad e are osas, whih are il deede o he hree differe ases of o-rigid body moio. These ases are based o he osillaio modes - uderdamed, riial damig ad overdamed. These osas are deailed i Abaqus user maual [3]. For he uderdamed ase, he osas are give as follows [3] ζω a = ex( ζω Δ) siωδ + osωδ (a) ω a = ex( ζω Δ) siωδ, (b) ω ω a = ex( ζω Δ) siωδ, () ζ ζω a = ex( ζω Δ) osωδ siωδ, (d) ω il b b b b u ζ ζ = ex( ζωδ) + si ωδ ω ω ω ωδ ζ ζ + + os Δ +, 3 3 ω ω ω Δ ωδ (a) ζ ζ = si os 3 ωδ + ωδ ω ωδ ω Δ ζ ex( ζωδ) +, 3 ω ω Δ = ω ζ + ωωδ ex ex ζ + 3 ωδ ( ω siωδ + ζω osωδ) ( ζω Δ), ζ = ω ωδ ζ 3 ω Δ N = X q, X ζ ωδ ω Δ (b) ( ω osωδ ζω siωδ) ( ζω siωδ ω osωδ) ( ω siωδ + ζω osωδ) ω Δ ( ζω Δ) +, () (d) Sie he ime iegraios is doe i geeralized oordiaes, he resose of he hysial variables are obaied hrough summaio (3) where are he eigeveor orresodig o he mode ad u is he aual odal dislaeme. From his he veloiy ad hee he odal aeleraio a be derived. The sysem s frequey resose fuio (FRF), is simly he raio of he Fourier rasform of he ouu over he iu (i he ase of a sysem wih sigle iu ad ouu). ( ω) G ( ω) = X ( =, w)(, = x, y) (4) F ω ( ) The disree Fourier rasform algorihm is adoed, whih is defied [33] as M Re [ ] πk H k = h[] i os, i= M Im H M = i= πk M [] k h[] i si, (5) (Advae olie ubliaio: 9 November 9)

5 Egieerig Leers, 7:4, EL_7_4_7 Re [] [ ] where k rus from o M, H k ad Im H k are he real ad imagiary ars of he frequey domai sigal ad h[] i is he ime domai sigal. The orresodig frequeies are defied as k f ω =, M (6) were ω is he frequey, f is he samlig frequey. IV. THE FINITE ELEMENT MODEL The workiee maerial used i he FEM model is Alumiium Alloy 7 T765. The maerial roeries required for geeraig he siffess ad mass maries are: 3 Desiy -.83 Kg m -3, Youg s Modulus GPa ad Poisso Raio Three differe yes of workiee were used i he fiie eleme aalysis (FEA). The dimesios are show i Figure ad he differe hikesses, (W) are show i ables, ad 3 reseively. The assumios made i he fiie eleme aalysis (FEA) are as follows: ) The workiee was boled a he bak surfae durig he ima ess ad i he FEM his was assumed o be lamed. ) The workiee was boled o he millig mahie durig he ima es ad i was assumed ha he aural frequeies of he mahie are very high omared o ha of he workiee, hee heir ifluee a be igored i he FEM aalysis. 3) The mass of he aeleromeer was assumed o be a oi mass added o he FEM model. R=5mm hikesses durig mahiig of hi wall seios. This aroah a be adaed wih he FEM aroah roosed i his aer as show i [36] o form a uified model ha would o require furher exerimes arried ou for differe wall hikesses. I he exerimeal ima ess, he workiee is exied usig a isrumeed hammer, whils he aeleromeer is laed o he oosie side of he ima oi, o measure he dire rasfer fuio. Usig a Fourier aalyser, he aelerae frequey resose fuio is exraed for eah ima es. This is simly he divisio of he Fourier rasform of he measured ime domai iu fore f ( ) ad aeleraio x ( ). X ( ω) A ω = (7) ( ) F( ω), where A ( ω) is he aelerae FRF, X ( ω) aeleraio sigal i frequey domai ad ( ω) is he ouu F is he iu fore sigal i frequey domai. The exerimeal measuremes are aalysed usig a modal aalysis sysem (CuPro was used for he soluios i his aer), whih sas he measured rasfer fuio ad fis a urve o he daa i order o obai he umerial values of aural frequey, damig [34]. Table Workiee A, W =.5mm MODE NUMBER NATURAL FREQUENCY, ω (HZ) DAMPING RATIO, ζ (%) E E E E E E E E-3 W 3mm 6mm Table Workiee B, W = 3.mm MODE NUMBER NATURAL FREQUENCY, ω (HZ) DAMPING RATIO, ζ (%) E E E E E-3 Figure Workiee dimesios. A. The Damig Raio 6mm The damig raios, ζ used i (9) here for 3mm demosraig he roosed aroah were ideified hrough ima ess (give i ables, ad 3). Adeoro e al. [35, 36] however reely roosed a aroah o rediig he damig arameers for differe wall Table 3 Workiee C, W = 4.5mm MODE NUMBER, NATURAL FREQUENCY, ω (HZ) DAMPING RATIO, ζ (%) E E E-3 V. RESULTS A. Exraig he Workiee Trasfer Fuio. For workiee A, he measured iu fore from he (Advae olie ubliaio: 9 November 9)

6 Egieerig Leers, 7:4, EL_7_4_7 ima es was used as he iu fore (i ime domai) i he FEM modal aalysis. The redied aeleraio (ime domai) is show i omariso o he exerimeal aeleraio from he aeleromeer (durig he ima es) i Figure 3. The redied FRF (usig he aroah i seio ) ad exerimeal FRF, are omared i Figures 4 a ad b reseively. The agreeme bewee he exerimeal resuls ad he rediios is saisfaory. For workiee B, he iu fore (i ime domai) used i he FEM modal aalysis was a Dira dela fuio. The redied ad exerimeal FRFs are omared i Figures 5 a ad b. The agreeme bewee he exerimeal resuls ad he rediios is saisfaory. Aeleraio, m s Predied Exerimeal Time, s Figure 3 Predied ad measured aeleraio for workiee A. Real, m s - N - (a) Real Imag, m s - N (b) Imag 5. Frequey, Hz Exerimeal Predied Exerimeal Predied Frequey, Hz Figure 4 Predied ad measured FRFs for workiee A, G. w yy Figure 6 omares he redied ad exerimeal FRFs for workiee C ad he agreeme has show o be good. Real, m s - N - (a) Real Imag, m s - N - (b) Imag Frequey, Hz Exerimeal Predied Exerimeal Predied Frequey, Hz Figure 5 Predied ad measured FRFs for workiee B, G. Real, m s - N - 8. Exerimeal 6. Predied Frequey, Hz (a) Real Imag, m s - N (b) Imag Exerimeal Predied Frequey, Hz w yy Figure 6 Predied ad measured FRFs for workiee C, G. w yy (Advae olie ubliaio: 9 November 9)

7 Egieerig Leers, 7:4, EL_7_4_7 B. Chaer Sabiliy Lobes. Usig boh he redied ad exerimeal FRFs, he sabiliy lobes were geeraed usig CuPro, for he differe yes of workiee usig he arameers lised i able 4. CuPro is a advaed aalyial ad ime-domai mahiig roess simulaio ommerial akage develoed by Alias. I has a i buil modal aalysis module ad also a sabiliy lobes module. The sabiliy lobes module a ake he rasfer fuio i all hree orhogoal direios for he workiee ad rasfer fuio i x, ad y direios for he ool. The uig odiios used durig he simulaios are deailed i able 4. The ageial uig fore oeffiie (TCFC) K ad he radial uig osa K r are give i his able. The radial uig osa is a raio of he radial uig fore oeffiie o he ageial uig fore oeffiie. The TCFC ad he radial uig osa are used i (5) o model he ageial ad radial uig fores reseively. The radial deh of u or radial immersio is also give i able 4, whih is used o alulae he ery ad exi agles of he uer. The ery ad exi agles are used as he limis for he elemes i he radial immersio deede marix, [ A ] required whe alulaig he orieed rasfer fuio [ G ] i (). The elemes of he marix [ A ] are deailed i [,, ad 4]. The redied ad exerimeal resuls are omared i Figures 7a, b & for he hree differe workiee. The omarisos show a saisfaory agreeme. The sligh disreay i he redied aural frequey (frequey a whih FRF real is zero ad imagiary is maximum) a be see i he sligh shif i he sidle seed alulaed i he sabiliy lobes. The aural frequey redied affes he sable ooh assig frequey alulaed i he sabiliy lobes, hee he sligh differees see i he sidle seeds. The redied sable axial deh of us i Figures 7 b ad are slighly higher ha he exerimeal sable ADOC ad his is due o he FEM model beig oo siff. This a be aused by he boudary odiio assumio saed i seio 3, where he bak surfae was assumed o be erfely lamed. I he FEM siffess marix formulaio, he elemes are herefore se o E+36 ad he degrees of freedom a his surfae are o iluded i he simulaio. A more aurae aroah would require kowledge of he friio a he boudary bewee he mahie ad he workiee. Table 4 Cuig Codiio ad Coeffiies WORKPIECE A WORKPIECE B WORKPIECE C K R K T (MPA) RADIAL DEPTH OF CUT, (mm).5.. Axial Deh of Cu, mm Exerimeal Predied Sidle Seed, rm (a) Workiee A Axial Deh of Cu, mm Measured Predied Sidle Seed, rm (b) Workiee B Axial Deh of Cu, mm Exerimeal Predied Sidle Seed, rm (b) Workiee C Figure 7 Sabiliy lobes omariso. C. Exerimeal Resuls To show he advaages of his aroah i hi-wall mahiig, a yial surfae fiish obaied for a hi walled seio is show i Figure 9. ad he uig fores show i Figure 8. This exerimeal resuls show ha he varyig dyamis alog he workiee ao be igored or assumed as osa. I he surfae fiish i is show ha he workiee is usable oly a he ed of he u, while his is ofirmed i he uig fore (F x ) lo. These resuls are show ad disussed i-deh i [3]. Therefore usig he aroah i his aer, he rasfer fuio alog he workiee a be easily exraed wihou reliae o exerimeal resuls. (Advae olie ubliaio: 9 November 9)

8 Egieerig Leers, 7:4, EL_7_4_7 Fore, Fx (N) Sar Ed Time (se) Figure 8 Exerimeal uig fores from [3]. (a) Par I (b) Par II Figure 9 Surfae fiish hi wall mahiig [3]. For omleeess, he full FRF marix i () is also required, however alyig he ima fore ad/or measurig he resose i erai direios exerimeally a rove diffiul. Usig he roosed aroah however, he full FRF marix i () a be obaied easily i all direios. This is doe by simly alyig he ima fore i he orresodig direios of ieres. VI. CONCLUSION Chaer sill udermies he effors of he mahiis by reduig surfae qualiy, roduiviy ad ireasig os i damage reair. I his aer, a aleraive aroah o exraig he rasfer fuio usig he FEM modal aalysis has bee reseed. The aroah is based o he Fourier rasform of he resuls obaied from he fiie eleme aalysis. The resuls are show o agree wih exerimeal resuls ad hee he rasfer fuio alulaed. Is auray is furher exlored by is use i sabiliy lobe rediios. This aroah a be used o solve differe roblems eouered hrough he use of ima es, iludig obaiig he frequey resose fuio i direios ha a rove diffiul exerimeally. ACKNOWLEDGEMENTS The auhors akowledge he suor give by EPSRC for fudig his roe ad also he immese suor give by Airbus alog wih Mr. Aliser Reyish (GKN Aerosae). REFERENCE [] F. W. Taylor, O he ar of uig meals, Trasaios of he Ameria Soiey of Mehaial Egieers, 8, (97), [] S. A. Tobias ad W. Fishwik, A Theory of Regeeraive Chaer, The Egieer Lodo, (958). [3] J. Tlusy, M. Polaek, The sabiliy of mahie ools agais self exied vibraios i mahiig, i: Proeedigs of he ASME Ieraioal Researh i Produio Egieerig, Pisburgh, USA, (963), [4] F. Koeigsberger ad J. Tlusy, Mahie Tool Sruures Sabiliy Agais Chaer, Pergamo Press,, (967). [5] H. E. Merri, Theory of Self-Exied Mahie Tool Chaer, Joural of Egieerig for Idusry, Trasaios of he ASME, 87, (965), [6] J. Tlusy ad P. Ismail, Basi Nolieariy i Mahiig Chaer, Aals of he CIRP, (), (98). [7] J. Tlusy ad P. Ismail, Seial Ases of Chaer i Millig, Joural of Vibraio, Aousis, Sress, ad Reliabiliy i Desig, 5, (983). [8] J. Slaviek, The Effe of Irregular Tooh Pih o Sabiliy of Millig, 6h MTDR Coferee Maheser, (965). [9] P. Vaherk, Ireasig Millig Mahie Produiviy by Use of Cuers wih No-Cosa Cuig Edge Pih, 8h MTDR Coferee Maheser, (967). [] R Sridhar, R. E. Hoh ad G. W. Log, Geeral Formulaio of he Millig Proess Equaio, Joural of Egieerig for Idusry, Trasaios of he ASME, (968), [] R. Sridhar, R.E. Hoh, ad G.W. Log. A Sabiliy Algorihm for he Geeral Millig Proess, Joural of Egieerig for Idusry, Trasaios of he ASME, (968), [] R.E. Hoh, R. Sridhar, ad G.W. Log. A Sabiliy Algorihm for a Seial Case of he Millig Proess, Joural of Egieerig for Idusry, Trasaios of he ASME, (968), [3] H. Oiz, Chaer Behaviour of Heavy Mahie Tools, Quarerly Tehial Reor No. AF 6 (5) 96 Researh ad Tehology Divisio Wrigh Paerso Air Fore Base OH. (968). [4] H. Oiz ad F. Berardi, Ivesigaio ad Calulaio of he Chaer Behaviour of Lahes ad Millig Mahies, Aals of he CIRP, 8, (97), [5] J. Tlusy ad F. Koeigsberger, Mahie Tool Sruures, Pergamo Press Oxford 5h Ed.,, (97). [6] I. Miis, T. Yaushevsky, R. Tembo ad R. Hoke, Aalysis of Liear ad Noliear Chaer i Millig, Aals of he CIRP, 39, (99), [7] I. Miis ad T. Yaushevsky, A New Theoreial Aroah for he Prediio of Mahie Tool Chaer i Millig, Joural of Egieerig for Idusry, Trasaios of he ASME, 5, (993),. -8. [8] A. C. Lee ad C. S. Liu, Aalysis of Chaer Vibraio i he Ed Millig Proess, Ieraioal Joural of Mahie Tool Desig ad Researh, 3(4), (99), [9] A. C. Lee, C. S. Liu ad S. T. Chiag, Aalysis of Chaer Vibraio i a Cuer Workiee Sysem, Ieraioal Joural of Mahie Tool Desig ad Researh, 3(), (99),. 34. [] E. Budak, Mehais ad dyamis of millig hi walled sruures. Ph.D. Thesis, The Uiversiy of Briish Columbia, Vaouver, B.C., Caada, (994). [] Y. Alias ad E. Budak, Aalyial rediio of sabiliy lobes i millig, CIRP Aals - Maufaurig Tehology, 44(), (995), [] E. Budak ad Y. Alias, Aalyial rediio of haer sabiliy i millig - Par I: Geeral formulaio, Proeedig of ASME 995 Ieraioal Mehaial Egieerig Coferee ad Exosiio, Sa Fraiso, USA, (995). [3] E. Budak, Aalyial Prediio of Chaer Sabiliy i Millig Par I: Geeral Formulaio, Joural of Dyami Sysems, Measureme ad Corol, Trasaios of he ASME,, (998),. 3. [4] Y. Alias, E. Shamoo, P. Lee ad E. Budak, Aalyial Prediio of Sabiliy Lobes i Ball-Ed-Millig, Joural of Maufaurig Siee ad Egieerig Trasaios of he ASME, (4), (999), (Advae olie ubliaio: 9 November 9)

9 Egieerig Leers, 7:4, EL_7_4_7 [5] Y. Alias, Aalyial rediio of hree dimesioal haer sabiliy i millig, JSME Ieraioal Joural, Series C: Mehaial Sysems, Mahie Elemes ad Maufaurig, 44(3), (), [6] F. J. Cama, L. N. Loez de Laalle, A. Lamikiz ad J. A. Sahez, Seleio of uig odiios for a sable millig of flexible ars wih bull-ose ed mills, Joural of Maerials Proessig Tehology, 9(-3), (7), [7] O. B. Adeoro, W. M. Sim, ad P. H. We, Sabiliy Lobes Prediio for Corer Radius Ed Mill usig Noliear Cuig Fore Coeffiies, Mahiig Siee ad Tehology (submied for ubliaio Augus 9). [8] V. Theveo, L. Araud, G. Dessei ad G. Cazeave-Larrohe, Ifluee of Maerial Removal o he Dyamis Behavior of Thi-walled Sruures i Periheral Millig. Mahiig Siee ad Tehology,, (6), [9] S. Seguy, F. J. Cama, L. N. Loez de Laalle, Toolah deede sabiliy lobes for he millig of hi-walled ars, Ieraioal Joural of Mahiig ad Mahiabiliy of Maerials, 4(4), (8), [3] O. B. Adeoro, W. M. Sim, ad P. H. We, Aurae Prediio of Sabiliy Lobes usig Noliear Thi Wall Dyamis, Joural of Maerial Proessig Tehology, (submied for ubliaio Augus 9). [3] R. G. Grimes, J. G. Lewis ad H. D. Simo, A Shifed Blok Lazos Algorihm for Solvig Sarse Symmeri Geeralized Eigeroblems, SIAM Joural o Marix Aalysis ad Aliaios, 5, (994), [3] Karlsso & Sorese, I. Hibbi, Abaqus Theory Maual, 8 Mai Sree Pawuke Rl USA, (6). [33] S. W. Smih, The Siei & Egieer's Guide o Digial Sigal Proessig, Califoria Tehial Pub. Sa Diego, s ediio, (997). Available: h:// [34] Y. Alias, Maufaurig Auomaio, Cambridge Uiversiy Press, New York, NY (). [35] O. B. Adeoro, W. M. Sim, P. H. We, R. Vea, Prediio of Proorioal Damig Parameers, Joural of Advaed Maufaurig Sysems, (submied for ubliaio Aril 9). [36] O. B. Adeoro, W. M. Sim, P. H. We, A New Damig Modellig Aroah ad is Aliaio i Thi Wall Mahiig, The Ieraioal Joural of Advaed Maufaurig Tehology (submied ubliaio May 9). (Advae olie ubliaio: 9 November 9)