Abstract. Sigmund Max Young, Master of Science, In this thesis, dynamics of low immersion milling is explored through analytical

Transcription

1 Absrac Til of Thsis: Dynamics of Low Immrsion Milling Sigmund Ma Young, Masr of Scinc, 8 Dircd by: Profssor B. Balachandran Dparmn of Mchanical Enginring In his hsis, dynamics of low immrsion milling is plord hrough analyical and numrical mans. Using linar and nonlinar cuing forc modls, maps ar consrucd for singl dgr-of-frdom and wo dgr-of-frdom sysms whr h im spn cuing is "small" compard o h spindl roaion priod. Ths maps ar usd o sudy h possibiliis for diffrn nonlinar insabiliis and consruc sabiliy chars in h spac of cuing dph and spindl spd. Th analyical prdicions ar compard wih numrical rsuls as wll as prior primnal rsuls. Good agrmn amongs analyical, numrical, and primnal rsuls is sn. Limiaions of h analyical and numrical approachs ar discussd and nsions for fuur work ar suggsd.

2 Dynamics of Low Immrsion Milling by Sigmund Ma Young Thsis submid o h Faculy of h Gradua School of h Univrsiy of Maryland, Collg Park, in parial fulfillmn of h rquirmns for h dgr of Masr of Scinc 8 Advisory Commi: Profssor B. Balachandran, Chair/Advisor Profssor A. Baz, Mchanical Enginring Assisan Profssor N. Chopra, Mchanical Enginring

3 Copyrigh by Sigmund Ma Young 8

4 Ddicaion To my mohr Su, my fahr Charls, my sisr Sigrid, and my dar fris Silk and Edward. ii

5 Acknowldgmns I would lik o prss my dps apprciaion o Profssor B. Balachandran for his umos painc, ncouragmn, and guidanc during my im hr a Univrsiy of Maryland, Collg Park (UMCP). His wisdom and slflss suppor mad h complion of his hsis possibl. Much graiud gos o Profssor A. Baz and Profssor N. Chopra for aking im ou of hir busy schduls o srv on my commi and hlp wih h rfinmn of his hsis. Spcial hanks gos ou o Marclo Valdz for hlping m wih h numrical ingraion of h dlay diffrnial quaions and Profssor Xinhua Long of Shanghai Jiao Tong Univrsiy for hlping m wih h UMCP numrical sabiliy prdicion program. I would also lik o hank h Naval Surfac Warfar Cnr, Cardrock Division and h Cod 71 Signaur Marials Physics Branch for providing financial suppor o my cours work and hsis work. Finally, I would lik o hank my mohr, my fahr, my sisr, and my fris for hir suppor and undrsanding during h complion of his hsis. Thir insighful advic, comdic rlif, and coninuous ncouragmn, hlpd m immnsly whn facing h numrous challngs prsnd during h formaion of his hsis. iii

6 Tabl of Conns Chapr Inroducion and Background Inroducion Background Char and Rgnraiv Effcs Loss of Conac Effcs Cuing Forcs Rsarch Objcivs Thsis Organizaion... 8 Chapr... 1 Milling Modls Singl DOF Sysm Equaion of Moion Two DOF Sysm Equaions of Moion Singl DOF Sysm Map Two DOF Sysm Map Chapr Analyical Sabiliy Prdicions Inroducion Characrisic Equaion for h Singl DOF Sysm Cas 1: λ = +1 for h Singl DOF Sysm Cas : λ = -1 for h Singl DOF Sysm Cas 3: λλ = 1 for h Singl DOF Sysm Characrisic Equaion for h Two DOF Sysm Cas 1: λ = +1 for h Two DOF Sysm Cas : λ = -1 for h Two DOF Sysm Cas 3: λλ = 1 for h Two DOF Sysm Analyical Sabiliy Lob Prdicions... 9 Chapr Numrical Sabiliy Prdicions Numrical Vrificaion of h Analyical Prdicions UMCP Numrical Sabiliy Prdicion Program Modifid UMCP Numrical Sabiliy Prdicion Program for h 3/4 Rul Chapr Rsuls and Discussion Singl DOF Sysm Rsuls and Discussion Two DOF Sysm Rsuls and Discussion Chapr Conclusion Concluding Rmarks Suggsions for Fuur Work iv

7 Appi A Map Drivaions A.1 Drivaion of h Singl DOF Sysm Map A. Drivaion of h Two DOF Sysm Map Appi B MATLAB Programs for Sabiliy Compuaions B.1 Two DOF Sysm: Flip Bifurcaion Compuaions B. Two DOF Sysm: Nimark Sackr Bifurcaion Compuaions Appi C MATLAB Programs for DDE Numrical Compuaions C.1 Singl DOF Sysm: DDE Numrical Compuaions C. Two DOF Sysm: DDE Numrical Compuaions Appi D UMCP Numrical Sabiliy Prdicion Programs D.1 Malab Programs for Linar Cuing Forc Modl D. Malab Programs for 3/4 Cuing Forc Modl... 9 Rfrncs... 1 v

8 Lis of Tabls Tabl 5.1: Inpu paramrs from Davis al. () for analyical prdicions Tabl 5.: Inpu paramrs from Davis al. () for numrical calculaions Tabl 5.3: Inpu paramrs from Span al. (5) for analyical prdicions Tabl 5.4: Inpu paramrs from Span al. (5) for numrical calculaions Tabl 5.5: Inpu paramrs for wo DOF sysm analyical prdicions Tabl 5.6: Inpu paramrs for wo DOF sysm numrical calculaions vi

9 Lis of Figurs Figur 1.1: Milling diagram - scion viw Figur 1.: a) down-milling procss and b) up-milling procss.... Figur 1.3: Milling ool diagram... Figur 1.4: Rgnraiv ffcs of urning Figur 1.5: Loss of conac ffcs... 5 Figur 1.6: Sampl sabiliy char for milling... 7 Figur.1: Schmaic of a singl DOF milling configuraion... 1 Figur.: Schmaic of a wo DOF milling configuraion Figur 3.1: Bifurcaion yps - (a) cyclic fold bifurcaion, (b) flip bifurcaion, and (c) Nimark-Sackr bifurcaion Figur 4.1: Rspons and phas porrai diagrams for sabl cuing condiions Figur 4.: Rspons and phas porrai diagrams a sabiliy boundary Figur 4.3: Rspons and phas porrai diagrams for unsabl cuing condiions Figur 4.4: Schmaic of a four DOF milling configuraion Figur 4.5: Cylindrical mill wih infinisimal disk lmns Figur 5.1: Analyical prdicion and dlay diffrnial quaion numrical prdicion comparisons wih primnal daa for 5% immrsion... 4 Figur 5.: Analyical prdicion and UMCP numrical sabiliy program prdicion comparisons wih primnal daa for 5% immrsion... 4 Figur 5.3: Analyical prdicion and UMCP numrical sabiliy program prdicion comparisons wih primnal daa for 9% immrsion Figur Up-Milling Cas: Comparison of h UMCP numrical sabiliy prdicion program wih h analyical prdicion rsuls for 9% immrsion Figur 5.5: Analyical prdicion and UMCP numrical sabiliy program prdicion comparisons for 5% immrsion around h ool naural frquncy of 54.4 krpm Figur 5.6: Analyical prdicion and UMCP numrical sabiliy program prdicion comparisons wih primnal daa for 3% immrsion Figur Up-Milling Cas: Comparison of h UMCP numrical sabiliy prdicion program wih h analyical prdicion rsuls for 3% immrsion Figur 5.8: Analyical prdicion and UMCP numrical sabiliy prdicion comparisons for 5% immrsion around h ool naural frquncy of 8.81 krpm Figur Two DOF Sysm: Analyical prdicion and dlay diffrnial quaion numrical prdicion comparisons for ρ = Figur Two DOF Sysm: Analyical prdicion for ρ =.1 and UMCP numrical sabiliy program prdicion for 5% immrsion comparisons Figur Two DOF Sysm: Analyical prdicion for ρ =.1 and UMCP numrical sabiliy program prdicion for 1% immrsion comparisons... 5 vii

10 Figur Two DOF Up-Milling Cas: Comparison of h UMCP numrical sabiliy prdicion program for ρ =.1 wih h analyical prdicion rsuls for 1% immrsion Figur Two DOF Sysm: Analyical prdicion for ρ =.1 and UMCP numrical sabiliy program prdicion for 1% immrsion comparisons around h ool naural frquncy of 6.4 krpm Figur Two DOF Sysm: Analyical prdicion for ρ =.1 and UMCP numrical sabiliy prdicion for 1% immrsion comparisons around h workpic naural frquncy of 3. krpm viii

11 Chapr 1 Inroducion and Background 1.1 Inroducion Machining is an indusrial procss whr marial is rmovd o form a dsird shap on a workpic. Tradiional machining includs cuing opraions such as urning, boring, drilling, and milling as wll as abrading opraions such as grinding, polishing, and buffing. Thr ar non-radiional machining opraions ha includ chmical machining, abrasiv-j machining, lasr cuing, plasma cuing, and war-j cuing. Wih advancs of modrn machining, h manufacuring of larg, sculpurd pars hrough marial rmoval is fasr and mor conomical han h producion of a larg numbr of simpl pars (Hally, Hlvy, Smih, and Winfough, 1999). Th focus of his hsis will b on radiional machining mhods, spcially, ha of milling. 1. Background Milling is a procss in which a roaing cuing ool uss h h or flus on is dgs o rmov marial on a workpic. Figur 1.1: Milling diagram - scion viw. 1

12 Figur 1.: a) down-milling procss and b) up-milling procss. As dpicd in Figur 1.1, h workpic is fd a a fd ra f ino h ool of radius R, which spins a a high angular spd Ω. During h pass of ach ooh, a lil chip of marial is rmovd from h workpic. Marial rmoval can b achivd hrough an up-milling procss or a down-milling procss. During down-milling, h fd ra f is dircd along h sam dircion as h roaion of h ool; in up-milling, h fd ra f is dircd along h opposi dircion o h roaion of h ool. Th chip formaion in down-milling is opposi o ha sn in upmilling. Figur 1.3: Milling ool diagram.

13 Undr mos condiions, h amoun of marial rmovd pr ooh pass dps mosly on h fd ra and h aial dph of cu (ADOC), as shown in Figur 1.. Th ra of immrsion is dfind as whr RDOC is h radial dph of cu. ra of immrsion = RDOC R (1.1) In his hsis, h focus is on low immrsion milling, which is assumd o occur whn h RDOC is much lss han h radius of h ool R. Anohr way o prss h ra of immrsion is hrough ra of immrsion πρ N (1.) whr N is h numbr of h on h ool and ρ is dfind as h raio of im spn cuing o h oal spindl priod τ ; i is nod ha his assumpion only holds valid for saic cuing condiions, zro hli angl on h cuing ool, circular ool pahs, and small angls of ngagmn (Davis, Pra, and Durr, ). Thrfor, by making ρ a small paramr, h ra of immrsion also bcoms small. Low immrsion milling applis o modrn cuing opraions including h milling of hard-o-machin marials, conourd surfacs, and finishing opraions on flibl componns (Davis al., ). In gnral, low immrsion also occurs during high spd milling whn h spindl spd Ω is grar han 1 krpm. 3

14 1.3 Char and Rgnraiv Effcs During h milling procss, h cuing forcs on h ool caus rlaiv vibraions ha ar ihr sabl or unsabl. Sabl vibraions from h cuing forcs can b said o provid a n posiiv damping o h sysm ha allow h vibraions o dcay whil unsabl vibraions from h cuing forcs can b said o provid a n ngaiv damping o h sysm and hrfor inroduc nrgy o h sysm ha canno b dissipad (Tobias, 1965). Th associad loss of dynamic sabiliy is calld char. Primary sourcs of char includ h following: (1) rgnraiv insabiliis ha rsul from h undrcuing of a prviously cu surfac and () drivn oscillaions ha aris from h inrmin ngagmn bwn h workpic and h ool (.g., Davis and Balachandran, ). Th firs cas occurs mos commonly during full immrsion milling opraions whr a las on flu of h ool is ngagd wih h workpic a all ims. Th sabiliy analysis for his cas can b rad lik ha of a urning problm, whr h ool and workpic sysm is modld as a linar oscillaor and h cuing forc acing on h sysm is dpn on h prvious and prsn posiions of h ool. Figur 1.4: Rgnraiv ffcs of urning. 4

15 In Figur 1.4, h cuing forc of h ool is drmind from h diffrnc of h imdlayd ool posiion (-τ) wih h currn ool posiion () plus h fd h (), which is qual o h fd ra f muliplid by h oal spindl priod τ (). This kind of rgnraiv sysm was firs sudid by Arnold (1946), who usd a lah wih a siff workpic and flibl ool. Th ons of char vibraions dps on paramrs such as h fd ra, h RDOC, h aial dph of cu, and h spindl spd. Tlusy and Polack (1963) and Tobias (1965) lar showd ha h chip hicknss variaion along wih h dynamic cuing forc and is rgnraiv ffcs ar imporan mchanisms ha lad o char. 1.4 Loss of Conac Effcs Th scond primary sourc for char is significan during low immrsion opraions whr h cuing bcoms highly inrrupd. Whn his occurs, som of h cuing dgs ar no in conac wih h workpic for h majoriy of h im; his is also known as h loss of conac ffc (.g., Balachandran, 1). Figur 1.5: Loss of conac ffcs. 5

16 Thrfor, h acual cuing or conac of a ooh wih h workpic occurs during shor im inrvals; ha is ρ << 1. Through his small paramr assumpion, h cuing forc duraion shrinks owards zro making h non-cuing priod clos o h oal spindl priod. 1.5 Cuing Forcs Th cuing forc is mosly dpn on h following propris: i) cur gomry, ii) workpic gomry, iii) cuing condiions, iv) workpic marial propris, and v) rlaiv displacmn bwn h workpic and h ool. Whil Davis al. () amin a singl dgr-of-frdom (DOF) linar cuing forc modl, Szalai, Span, and Hogan (4) modify his linar modl by changing h cuing forc ino a nonlinar cuing forc funcion ha follows h 3/4 rul (Tlusy ). Zhao () and Long (6) modl h cuing ool as an ingrad s of hin disk lmns whil using linar dynamic uncu chip hicknss variaions. 1.6 Rsarch Objcivs Th moivaion for his sudy includs incrasing produciviy and lowring h cos of marial rmoval whil achiving a high qualiy surfac finish and kping h ool war low. Vibraions during h milling procss play a major rol in influncing h qualiy of h surfac finish, h war of h ool, and h marial rmoval ra. Whil aciv damping can rduc and sabiliz hs vibraions, nsiv modificaion of h sysm is rquird, which can b cosly. In addiion, hr ar limiaions o h n o which h damping lvls can b incrasd. Thrfor, a br approach is o look a h 6

17 opraing paramrs of h milling sysm. By obsrving cuing spds and aial dph of cus, idal opraing paramrs can b idnifid for produciv milling opraions. Through a horough aminaion of h dynamics on milling procsss, h paramrs for sablishing sabl char fr opraions can b drmind and graphd o cra a sabiliy char. By using a sabiliy char lik h on shown in Figur 1.6, a milling opraor can find h idal spindl spd and aial dph of cu o machin fficinly whil producing pars wih a high qualiy surfac finish. Knowldg of h dynamics and sabiliy bhavior of milling procsss is sough o improv h accuracy of sabiliy prdicion, and hrfor, nhanc milling prformanc. Figur 1.6: Sampl sabiliy char for milling. 7

18 From h arlir discussion, h cuing forcs during low immrsion milling can b rasonably modld wih loss of conac assumpions. Th non-cuing and cuing priods can b mappd hrough dynamic quaions ha incorpora rgnraiv ffcs. By conducing analyical and numrical sabiliy analysis wih hs maps, bifurcaion paramrs can b idnifid. Th rsuls of his analysis ar prsnd graphically hrough sabiliy chars. Th accuracy of hs rsuls is amind hrough comparison wih ising primnal rsuls. This hsis ffor has bn carrid ou wih h following objcivs: i) Dvlop singl and wo dgr-of-frdom sysm quaions of moion for low immrsion milling wih linar and 3/4 rul cuing forc modls ii) Consruc maps basd on cuing and non-cuing priods, and conduc sabiliy analysis prdicions by using h dvlopd maps iii) Us dlay diffrnial quaion numrical chniqus o vrify h analyical sabiliy prdicions iv) Us h UMCP numrical sabiliy prdicion program dvlopd by Zhao () and Long (6), and modify h linar dynamic uncu chip hicknss variaion in h program s cuing forc modl o incorpora h 3/4 rul v) Compar h prdicions obaind hrough analyical and numrical mans wih ha of ising primnal daa 1.7 Thsis Organizaion This hsis is organizd as follows. In h scond chapr, h singl dgr-of-frdom sysm quaions of moion wih linar and 3/4 rul cuing forcs ar summarizd from 8

19 Davis al. () and Szalai al. (4). Ths quaions of moion and h cuing forc modls ar hn pandd o a wo dgr-of-frdom modl. Dvlopmn of maps from h singl dgr-of-frdom and h wo dgr-of-frdom modls follow. In h hird chapr, analyical sabiliy prdicion quaions ar obaind for h singl dgr-of-frdom maps and h drivaions of h analyical sabiliy prdicion quaions for h wo dgr-of-frdom sysm maps ar daild. In chapr four, h dlay diffrnial quaion numrical vrificaion of h analyical prdicions is prsnd. Summaris dailing h UMCP numrical sabiliy prdicion program and is modificaion o includ h 3/4 rul cuing forc modl ar also includd. In h fifh chapr, sabiliy chars obaind by using h analyical modls ar compard wih h prdicions obaind from numrical simulaions as wll as daa akn from ising primnal rsuls found in h liraur (Davis al., and Span, Szalai, Mann, Bayly, Insprgr, Gradisk, and Govkar, 5). Concluding rmarks and suggsions for fuur work ar prsnd in h las chapr. Appics ar includd o provid dails of calculaions as wll as h cods usd for h numrical simulaions. Rfrncs ar lisd a h of h hsis. 9

20 Chapr Milling Modls.1 Singl DOF Sysm Equaion of Moion Th following figur dpics a on dgr-of-frdom milling configuraion. Figur.1: Schmaic of a singl DOF milling configuraion. Th govrning quaions of his sysm ar ( τ ) mq&& () + cq& () + kq() = g () F, () (.1) whr g() =, is h fr oscillaion priod τ 1, j < j+ 1 g() = 1, is h cuing priod τ, j+ 1 < j+ 1 1

21 and τ, h oal spindl im priod, qualsτ 1 + τ. Thrfor, a h sar of τ 1, from im j, o h of τ 1, a h im jus prior o j+ 1, h ool is no in conac wih h workpic; h quaion of moion in (.1) dos no dp on h cuing forc (, ( )) F τ and bcoms a linar, homognous diffrnial quaion. Thn a h sar of τ, from h im j 1, o h of + τ, a h im jus prior o j + 1, h ool conacs h workpic and hus, h quaion of moion in (.1) dps on h cuing forc (, ( )) F τ. A h im j + 1, a nw spindl priod bgins. In boh sas of τ 1 and τ, h quaion of moion of h ool includs displacmn q ( ), consan siffnss k, and consan damping c. Th ool roas wih a consan angular spd Ω and has N numbr of h. Th consan fd ra f of h workpic runs opposi o h major mod of moion q ( ). According o prvious rsarch as mniond in Long (6), a circular ooh pah is assumd, which hrfor cras h consan im dlayτ = π / NΩ. Through h dfiniion of ρ, τ = ρτ. Th linar cuing forc is givn by whil h 3/4 rul cuing forc is givn by ( τ ) [ τ ] F, ( ) = Kw q( ) q( ) + h (.) ( τ ) [ τ ] 3/4 F, ( ) = Kw q ( ) q ( ) + h (.3) whr K is h workpic marial consan, K is h modifid workpic marial consan and is rlad o K by K = K, (.4) 1/4 h w is h aial dph of cu or consan chip widh, and h is h fd, which is dfind as h = fτ. No ha boh h and τ ar dpn on. 11

22 . Two DOF Sysm Equaions of Moion Th following figur dpics a wo dgr-of-frdom milling configuraion. Figur.: Schmaic of a wo DOF milling configuraion. Th govrning quaions of his sysm ar ( τ ) ( τ ) mq&& () + cq& () + kq() = g () F, () mq&& () + cq& () + kq() = g () F, () u u u u u u u (.5) whr g() =, h fr oscillaion priod τ 1, j < g() = 1, h cuing priod τ, j j 1 j < + and τ, h oal im priod, qualsτ 1 + τ. As in h on dgr-of-frdom cas, a h sar of τ 1, from im j, o h of τ 1, a h im jus prior o j+ 1, h ool is no in conac wih h workpic, so h quaions of moion in (.5) do no dp on h 1

23 cuing forcs F (, τ ( ) ) and F (, τ ( ) ) u ; h sysm bcoms a s of linar, homognous diffrnial quaions. Thn a h sar of τ, from h im j+ 1, o h of τ, a h im jus prior o j 1, h ool conacs h workpic and h quaions of + moion in (.5) dp on h cuing forcs F (, τ ( ) ) and F (, ( ) ) u τ. A h im + 1, a nw spindl priod bgins. As in h singl dgr-of-frdom cas, h quaion of moion for h ool includs displacmn q ( ), consan siffnss k, and consan damping c. Th quaion of moion for h workpic includs displacmn qu ( ), consan siffnss k u, and consan damping c u. Th consan fd ra f of h workpic runs along h major mod of moion on h workpic qu ( ). Th linar cuing forc is givn by ( τ ) [ τ τ ] F, ( ) = K w q ( ) q ( ) + q ( ) q ( ) + h (.6) u u whil h 3/4 rul cuing forc is givn by ( τ ) [ τ τ ] 3/4 F, ( ) = Kw q( ) q( ) + qu( ) qu( ) + h (.7) j Applying Nwon's Third Law, h cuing forc (, ( ) ) opposi o h cuing forc (, ( ) ) moion bcom F τ acing on h ool is qual and F u τ acing on h workpic. Thus, h quaions of ( τ ) ( τ ) mq&& () + cq& () + kq() = g () F, () mq&& () + cq& () + kq() = g () F, () u u u u u u (.8) 13

24 .3 Singl DOF Sysm Map Th ky o drmining sabiliy is o find h appropria maps and hir soluions for h inrrupd cuing procss. For as of noaion, on manipulas h quaion of moion from (.1) ino g () q&& () + ζω q& () + ω q() = F(, τ() ) (.9) m whr ω = k / m is h naural frquncy for h ool and ζ = c / k m is h consan damping raio for h ool. Th fr vibraion non-cuing phas ( g ( ) = ) ha maps h sa ( q ), q& ( )) o h sa ( ), q ( )) ( j j q & is drivd o b ( j+ 1 j+ 1 q ( ( ) 1) q j+ j A = q ( ( ) j 1) q & + & j (.1) whr h sa-ransiion mari A is ζω 1 cos( ωdτ) + sin( ωdτ) sin( ωdτ) ωd ω d A = p( ζωτ ) ω ζ ω sin( ωdτ) cos( ωdτ) sin( ωdτ) ωd ωd (.11) and ω = ω 1 ζ is dfind o b h dampd naural frquncy for fr oscillaions d of h ool. No ha i has bn assumd ha τ 1 τ, as plaind n. In insancs whr h im priod of cuing is vry shor; ha is, τ, τ 1 can b approimad by τ (.g. Szalai al., 4). For h cuing priod τ, h spring and damping forcs can b nglcd (.g. Davis al., ) and h posiion of h ool can b assumd o princ a ngligibl chang during cuing, which lavs 14

25 Kw q && () q ( ) q ( ) + h, j+ 1 < j+ (.1) 1 j j 1 m + for h linar cuing forc modl and Kw 3/4 q && () q ( ) q ( ) + h, j+ 1 < j+ (.13) 1 j j 1 m + for h 3/4 rul cuing forc modl. Ingraing h abov quaion ovr h priod, ], on finds [ j+ 1 j+ 1 Kw q & ( ) = q & ( ) + τ q ( ) q ( ) + h (.14) j+ 1 j+ 1 j j 1 m + for h linar cuing forc modl and Kw 3/4 q & ( ) = q & ( ) + τ q ( ) q ( ) + h (.15) j+ 1 j+ 1 j j 1 m + for h 3/4 rul cuing forc modl. Combining quaions (.1) and (.14) oghr for h linar cuing forc modl, and quaions (.1) and (.15) oghr for h 3/4 rul cuing forc modl, and assuming ha h posiion of h oscillaing ool rmains consan during h inracion wih h workpic so ha q( j 1) q( j 1), on arrivs a + + q ( ) q ( ) j + 1 A j Kw q( 1) = j q( ) + + j τ [(1 A 11) q( j) A 1q( j) h ] & & & + m (.16) 15

26 and q ( ) q ( ) j + 1 A j Kw 3/4 q( 1) = j q( ) + + j τ [(1 A 11) q( j) A 1q( j) h ] & & & + m (.17) for h linar and 3/4 rul cuing forcs, rspcivly, whr A ij corrsponds o h lmns of A in (.11). By using fid poin and sabiliy chniqus such as hos found in Nayfh and Balachandran (1995), h maps in (.16) and (.17) ar found o hav h fid poins Kw A τ h 1 v = m [ 1 d(a) r(a) ] 1 A + 11 (.18) and 3/4 Kwτ A h 1 v = m [ 1 d(a) r(a) ] 1 A + 11 (.19) Th linarizd Jacobian mari for (.16) and (.17) is B = A11 A1 A + wˆ (1 A ) A wˆ A (.) whr h dimnsionlss chip widh for h linar cuing forc modl is wˆ Kwτ m = (.1) and h dimnsionlss chip widh for h 3/4 cuing forc modl is wˆ 3 Kwτ = (.) 4 m h 1/4 16

27 Linarizing around h fid poins, h gnralizd local dynamics is dscribd by q ( 1) ( ) j + q = + B j q& ( ) v q ( ) & j+ 1 j (.3) whr, v, and w ˆ ar basd on hir rspciv cuing forc modls..4 Two DOF Sysm Map As in h cas of h singl dgr-of-frdom map, on simplifis h noaion in h quaions of moion (.8) o yild g () q&& q& q F ( ) () + ζω () + ω () =, τ() m g () q&& q& q F ( ) u() + ζω u u u() + ωu u() =, τ() mu (.4) whr, as in h singl dgr-of-frdom cas, ω = k / m is h naural frquncy for h ool and ζ = c / k m is h consan damping raio for h ool. ω u = k u / mu is h naural frquncy for h workpic and ζ = c / k m is h consan damping u u u u raio for h workpic. Wih quaion (.4), h non-cuing phas ( g ( ) = ) ha maps h sa ( q ), q& ( ), q ( ), q& ( )) o h sa ( q ), q ( ), q ( ), q& ( )) drivd as ( j j u j u j & is ( j+ 1 j+ 1 u j+ 1 u j+ 1 q ( ) ( 1) q j+ j ( ( ) u j 1) qu q + j = C q& ( ( ) j 1) q& + j q& ( ( ) u j 1) qu + & j (.5) 17

28 whr h sa-ransiion mari C is ζωτ ζωτ ζω [cˆ s ˆ ] sˆ + ωd ωd ζuωuτ ζuωuτ ζω u u [cˆ s ˆ ] sˆ u+ u u ωud ωud ω ζωτ ζωτ ζ ω sˆ [cˆ s ˆ ] ωd ωd ωu ζuωuτ ζuωuτ ζuωu sˆ [cˆ s ˆ u u u] ωud ω ud (.6) and cˆ = cos( ω τ ), sˆ = sin( ω τ ), cˆ = cos( ω τ ), and sˆ = sin( ω τ ). d d u Ĉ is obaind from h sa-ransiion mari A of h ool in quaion (.11) ud u ud Ĉ A A C C = A = A1 A = C31 C 33 and C is h sa-ransiion mari of h workpic C C4 C = C4 C 44 (.7) whr again, ω = ω 1 ζ is h dampd naural frquncy for h frly oscillaing d ool and ω = ω 1 ζ is inroducd as h dampd naural frquncy for h frly ud u u oscillaing workpic. As in h singl dgr-of-frdom cas, wih τ, h spring and damping forcs can b nglcd and h posiion of h ool can b assumd o princ ngligibl chang during h cuing, which lads o 18

29 Kw q&& ( ) q ( ) q ( ) + q ( ) q ( ) + h j j 1 u j u j 1 m + + Kw q&& ( ) q ( ) q ( ) + q ( ) q ( ) + h u j j 1 u j u j 1 m + + u, j+ 1 < j+ 1 (.8) for h linar cuing forc modl and Kw q&& ( ) q ( ) q ( ) + q ( ) q ( ) + h j j 1 u j u j 1 m + + Kw q&& ( ) q ( ) q ( ) + q ( ) q ( ) + h u j j 1 u j u j 1 m + + u 3/4, j+ 1 < j+ 1 3/4 (.9) for h 3/4 rul cuing forc modl. Ingraing h abov quaion ovr h inrval, ] on finds [ j+ 1 j+ 1 Kw q& ( ) = q& ( ) + τ q ( ) q ( ) + q ( ) q ( ) + h j+ 1 j+ 1 j j 1 u j u j 1 m + + Kw q& ( ) = q& ( ) + τ q ( ) q ( ) + q ( ) q ( ) + h u j+ 1 u j+ 1 j j 1 u j u j 1 m + + u (.3) for h linar cuing forc modl and Kw q& ( ) = q& ( ) + τ q ( ) q ( ) + q ( ) q ( ) + h j+ 1 j+ 1 j j 1 u j u j 1 m + + Kw q& ( ) = q& ( ) + τ q ( ) q ( ) + q ( ) q ( ) + h u j+ 1 u j+ 1 j j 1 u j u j 1 m + + u 3/4 3/4 (.31) for h 3/4 rul cuing forc modl. 19

30 For h linar cuing forc modl, combining quaions (.5) and (.3) oghr yilds q( j+ 1) q( j) qu( 1) j qu( ) + j = C q& ( 1) j q& ( ) + j q& u( j+ 1) q& u( j) Kw + τ [(1 C 11) q( j) C 13q& ( j) + (1 C ) qu( j) C 4q& u( j) + h ] m Kw τ [(1 C 11) q( j) C 13q& ( j) + (1 C ) qu( j) C 4q& u( j) + h ] mu (.3) whil combining quaions (.5) and (.31) oghr for h 3/4 rul cuing forc modl yilds q( j+ 1) q( j) qu( 1) j qu( ) + j = C q& ( 1) j q& ( ) + j q& u( j+ 1) q& u( j) Kw + τ [(1 C 11) q( j) C 13q& ( j) + (1 C ) qu( j) C 4q& u( j) + h ] m Kw τ [(1 C 11) q( j) C 13q& ( j) + (1 C ) qu( j) C 4q& u( j) + h ] mu 3/4 3/4 (.33) whr C ij corrsponds o h lmns of C in quaion (.6). Th fid poins of quaion (.3) ar Kw C τ h 13 v = 1 d(c) ˆ r(c) ˆ m 1 C + 11 u Kw C τ h 4 w = 1 C mu 1+ d(c) r(c) (.34)

31 whil h fid poins of quaion of h sysm (.33) ar 3/4 Kwτ C h 13 v = 1 d(c) ˆ r(c) ˆ m 1 C /4 u Kwτ C h 4 w = 1 C mu 1+ d(c) r(c) (.35) Th linarizd Jacobian mari of quaions (.3) and (.33) can b formd as C11 C13 C C4 D= C ˆ ˆ ˆ ˆ 31 + w(1 C 11) w(1 C ) C33 wc 13 wc 4 w ˆ (1 C ) C + w ˆ (1 C ) wˆ C C wˆ C u 11 4 u u u 4 (.36) whr h dimnsionlss chip widhs for h linar cuing forc ar wˆ = Kw τ m wˆ u Kwτ m = (.37) u and h dimnsionlss chip widhs for h 3/4 rul cuing forc ar wˆ wˆ u 3 Kwτ = 4 m h 1/4 3 Kwτ = (.38) 4 m h u 1/4 An imporan paramr ha will appar during h sabiliy analysis is which quas mˆ = m / m (.39) u wˆ u = mw ˆ ˆ (.4) for boh cass. 1

32 From (.), ˆD = B = A11 A1 C11 C13 = A + wˆ (1 A ) A wˆ A C + wˆ (1 C ) C wˆ C and C C4 D = C ˆ ˆ 4 wu(1 C ) C44 wuc + 4 (.41) Wih h linarizaion around h fid poins, h gnralizd local dynamics for h wo dgr-of-frdom sysm is dscribd by q( j+ 1) ( ) q j qu( j 1) u qu( j) + = + D q& ( j+ 1) v q( j) q& ( ) w q& ( ) u j+ 1 u j (.4) whr, u, v, w, w ˆ, and w ˆu ar basd on h rspciv cuing forc modls.

33 Chapr 3 Analyical Sabiliy Prdicions 3.1 Inroducion Th sabiliy of h on or wo dgr-of-frdom sysm is drmind by h rspciv sysm's Jacobian mari givn by B (.) or D (.36). If h ignvalus li ousid h uni circl of h compl plan maning ha h magniud of ach ignvalu is grar han on, h ampliud of a prurbaion abou h sysm's fid poin will grow wihou bound on furhr iraions of h sysm map; in his scnario, h sysm is unsabl. Convrsly, if h ignvalus li wihin h uni circl of h compl plan, all subsqun prurbaions abou h fid poin of h sysm map will dcay o zro; in his scnario, h sysm is sabl, (.g., Nayfh and Balachandran, 1995). To drmin h sabiliy of h sysm, on mus look a h condiions whn on or mor ignvalus lav h uni circl. Figur 3.1: Bifurcaion yps - (a) cyclic fold bifurcaion, (b) flip bifurcaion, and (c) Nimark-Sackr bifurcaion. 3

34 According o sabiliy analysis (.g. Nayfh and Balachandran, 1995), hr ar hr condiions undr which h sysm can bcom unsabl: i) a ral-valud ignvalu ravling lf o righ hrough +1, which is a ncssary condiion for a cyclic fold bifurcaion ( s Figur 3.1 (a)) ii) a ral-valud ignvalu ravling righ o lf hrough -1, which is a ncssary condiion for a flip or priod doubling bifurcaion ( s Figur 3.1 (b)) iii) a pair of compl-conjuga ignvalus laving h uni circl hrough h compl boundary of his circl, which is a ncssary condiion for a Nimark- Sackr or scondary Hopf bifurcaion ( s Figur 3.1 (c)) For boh h on and wo dgr-of-frdom sysms, all hr possibiliis ar amind. 3. Characrisic Equaion for h Singl DOF Sysm Th ignvalus λ of h singl dgr-of-frdom map (.3) ar calculad from h characrisic quaion of h Jacobian mari B givn by (.), lading o λ r(b) λ+d(b)= (3.1) whr r(b) is h rac of B and d(b) is h drminan of B. Following som algbraic manipulaions, on can obain r(b) = r(a) wˆ A 1 d(b) = d(a) wˆ A 1 (3.) Wihou furhr analysis, immdia conclusions can b drawn by using (3.); for valus of paramrs in which A1 =, h characrisic quaion rducs o ha of a simpl, dampd harmonic oscillaing ool dscribd by mari A (.11); hus, h sysm is sabl 4

35 (Davis al., ). Sinc his is a limid spcial cas, h ohr cass ar now amind. 3.3 Cas 1: λ = +1 for h Singl DOF Sysm Sing λ = 1 in h characrisic quaion (3.1) producs 1 r(a) + d(a) = (3.3) This quaion is indpn of h aial dph of cu or chip widh and hrfor canno b a rou o insabiliy, whn h chip widh is usd as a bifurcaion conrol paramr. 3.4 Cas : λ = -1 for h Singl DOF Sysm Sing λ = 1 in h characrisic quaion (3.1), h criical dimnsionlss chip widh, which is h chosn conrol paramr, is drmind as wˆ d(a) + r(a) + 1 cosh( ζ ωτ) + cos( ω τ) = = ωd A1 sin( ωdτ) f d cr (3.4) A map princing a flip bifurcaion a is criical poin rsuls in h craion or dsrucion of wo branchs of priod-wo poins (.g. Nayfh and Balachandran, 1995). An iraion of a map iniiad a ihr of h priod-wo poins flips back and forh bwn hos poins if and 3 ar priod-wo criical bifurcaion poins of map M(); ha is, M ( ) = and M( ) = (3.5) 3 3 Thus, h sabiliy lobs formd from w ˆ f cr in (3.4) corrspond o flip bifurcaion poins. 5

36 3.5 Cas 3: λλ = 1 for h Singl DOF Sysm Sing λλ = 1, whr λ is h compl conjuga of λ, on arrivs a h gnral form i λ = φ i and λ = φ for Cas 3, whr φ is an angl ha nds o b solvd. For Cas 3, h gnral form of h characrisic quaion is drivd o b λ (cos φ) λ+ 1= (3.6) 1 Equaing lik radicals, h radicals of λ from (3.1) and (3.6) ar usd o solv for 1 r(b) φ = cos (3.7) By using h radicals of λ from quaions (3.1) and (3.6), h following condiion for Nimark-Sackr bifurcaions is obaind: d(b) = 1 (3.8) Applying his condiion, h criical dimnsionlss chip widh for his cas is drmind as ns d(a) 1 sinh( ωζτ ) wˆ cr = = ω d (3.9) A sin( ω τ ) 1 d Branchs of sabl priodic soluions a a criical poin ha is prior o h Nimark- Sackr bifurcaion spli as a pair of unsabl branchs afr h bifurcaion (.g. Nayfh and Balachandran, 1995). Th sabiliy lobs formd hrough w ˆ ns cr in (3.9) prdic h occurrnc of Nimark-Sackr bifurcaions along is boundary. As Davis al. () poin ou, h Nimark-Sackr bifurcaion producs unsabl, slf-cid vibraions wih frquncis ha ar no proporional o corrsponding spindl roaion frquncis. This yp of bifurcaion also appars in coninuous urning and givs ris o char frquncy. 6

37 3.6 Characrisic Equaion for h Two DOF Sysm Th ignvalus λ of h wo dgr-of-frdom map (.4) ar calculad from h characrisic quaion of h Jacobian mari D in quaion (.36), lading o 4 3 ˆ ˆ r(d) + [d(d) + d( D) + r(d)r( D) mw ˆˆ C13C 4] λ λ λ ˆ ˆ [r(d)d( D ) + r( D)d(D) mw ˆˆ C13C 4] λ d(d) = + (3.1) In his rgard, usful algbraic rlaionships includ h following: r(d) = r(c) C wˆ C d(d) ˆ = d(c) ˆ C d( D) = d(c) C wˆ 4 mw ˆˆ mw ˆ ˆ r(d)r( ˆ D) = r(c)r( ˆ C) r( C)C wˆ r(c)c ˆ mw ˆ ˆ + mw ˆ ˆ C C r(d)d( ˆ D) = r(c)d(c) ˆ d(c)c wˆ r(c)c ˆ mw ˆ ˆ + mw ˆ ˆ C C r( D)d(D) ˆ = r( C)d(C) ˆ r( C)C wˆ d(c)c ˆ mw ˆ ˆ + mw ˆ ˆ C C d(d) = d(c) d(c)c wˆ d(c)c ˆ mw ˆ ˆ (3.11) As in h on dgr-of-frdom cas, on can immdialy conclud by using (3.11) ha for valus of paramrs in which C13 = C4 =, h characrisic quaion rducs o ha of a simpl, dampd harmonic oscillaing ool and workpic dscribd by mari C (.6); hus, h sysm is sabl. Again, sinc his is a limid spcial cas, h ohr cass ar amind for h wo dgr-of-frdom sysm. 7

38 3.7 Cas 1: λ = +1 for h Two DOF Sysm Sing λ = 1 for h characrisic quaion in (3.1) and by using h rlaionships dfind in (3.11) lads o 1 r(c) + d(c) ˆ + d(c) + r(c)r(c) ˆ r(c)d(c) ˆ r(c)d(c) ˆ + d(c) = (3.1) Lik in h on dgr-of-frdom cas, his quaion is indpn o h aial dph of cu or chip widh, and hrfor his scnario canno b a rou o insabiliy. 3.8 Cas : λ = -1 for h Two DOF Sysm Sing λ = 1 for h characrisic quaion (3.1) and using h rlaionships dfind in (3.11), h criical dimnsionlss flip bifurcaion chip widh bcoms ˆ f w cr = 1 1+ r(c) + d(c) + r(c)r(c) ˆ + r(c)d(c) ˆ + r(c)d(c) ˆ + d(c) ˆ + d(c) (3.13) [1 + r(c) + d(c)]c + [1 + r(c) ˆ + d(c)] ˆ mˆ C cˆ + ω sˆ + cˆ + ω sˆ + cˆ + cˆ = [1+ c + ( c + s )] s + [1+ c + ( c + s )] s ζωτ ζωτ u u ζωτ ζωτ u u 1 ( ) ( u u u ) u ζωτ ζωτ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ u u ζωτ u u ζωτ u u ζωτ m ζωτ ˆ u u ωu u ω u ωd ωud ( cˆ + ω sˆ )( cˆ + ω sˆ ) + 4 cˆ cˆ [1+ c + ( c + s )] s + [1+ c + ( c + s )] s ( ζω + ζω u u) τ ( ζω + ζω u u) τ u u u u ζωτ ζωτ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ u u ζωτ u u ζωτ m u u ζωτ ζωτ ˆ u u ωu u ω u ωd ωud ( ζω + ζω u u) τ ( ζω + ζω u u) τ cˆ ( ˆ ˆ ) ˆ ( ˆ ˆ cu + ωu su + cu c + ω s ) ζωτ ζωτ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ u u ζωτ m u u ζωτ u u ζωτ ζωτ ˆ u u ωu u ω u ωd ωud 1 [1+ c + ( c + s )] s + [1+ c + ( c + s )] s whr cˆ = cos( ω τ ), sˆ = sin( ω τ ), cˆ = cos( ω τ ), and sˆ = sin( ω τ ). As in h on d d u dgr-of-frdom cas, h sabiliy lobs formd from w ˆ f cr in (3.13) corrspond o h flip bifurcaion poins. ud u ud 8

39 3.9 Cas 3: λλ = 1 for h Two DOF Sysm i Jus lik in h on dgr-of-frdom sysm analysis, on can us λ 1 = φ and λ1 = as h gnral form for h Nimark-Sackr bifurcaion cas. In ordr for h map in (.4) o princ a Nimark-Sackr bifurcaion, no only do wo compl conjuga ignvalus hav o lav h uni circl away from h ral ais such ha λ1λ 1 = 1, bu h ohr wo ignvalus mus b compl conjugas wihin h uni circl. Th gnral soluion of h scond pair of ignvalus is assumd o b λ = a+ bi and λ = a bi whr a and b ar ral valus. By using all four gnral forms of h ignvalus, h characrisic quaion in (3.1) bcoms i φ 4 3 λ ( a+ cos φλ ) + ( a + b + 4acosφ+ 1) λ = [( a b ) cos φ a] λ a b (3.14) Equaing h radicals of λ from (3.1) and (3.14), h following condiion for Nimark- Sackr bifurcaions is obaind d(d) = a + b (3.15) Evn whn quaing h rs of h radicals from (3.1) and (3.14), unlik h on dgrof-frdom cas, hr is no plici form for h criical dimnsionlss Nimark-Sackr bifurcaion chip widh. Thrfor, numrical mans nd o b usd o prdic h occurrnc of Nimark-Sackr bifurcaions along h sabiliy lob boundaris. A compuaional cod for hs soluions can b found in Scion B. of Appi B. 3.1 Analyical Sabiliy Lob Prdicions In ordr o compar h sabiliy lob boundaris wih ha of h numrical and primnal daa, h dimnsionlss chip widhs for boh h linar cuing forc modl 9

40 and 3/4 rul cuing forc modl mus b convrd ino chip widhs ha hav dimnsions. By using quaions (.1) and (.) from h on dgr-of-frdom map, rcalling from quaion (.4) ha 1/4 K = K h for h 3/4 rul cuing forc modl, and rmmbring ha τ = ρτ, whr τ = π / NΩ, h following gnralizd sabiliy prdicion quaions ar drivd in rms of hir corrsponding dimnsionlss chip widhs for h linar cuing forc modl w cr NΩm = wˆ cr (3.16) πk ρ and for h 3/4 rul cuing forc modl w cr 4 NΩm = wˆ cr (3.17) 3πK ρ By using quaions (.37), (.38), (.39), and (.4) for h wo dgr-of-frdom maps, on arrivs a h ac sam gnralizd sabiliy prdicion quaions of (3.16) and (3.17). I is nod ha hrough linarizaion, h linar cuing forc modl and h 3/4 rul cuing forc modl diffr only by a facor of 4/3. 3

41 Chapr 4 Numrical Sabiliy Prdicions Th condiions daild in Chapr 3 ar ncssary for bifurcaions o occur, and addiional analysis is ndd o vrify h occurrnc of a bifurcaion. In Szalai al. (4), asympoic analysis is usd o drmin h criicaliy of ach bifurcaion for h singl dgr-of-frdom sysm. In his scion, h analyical sabiliy prdicions ar numrically vrifid by im ingraion of h dlay diffrnial quaions. 4.1 Numrical Vrificaion of h Analyical Prdicions To numrically solv h quaions of moion givn by (.1) for h singl dgr-offrdom cas and (.5) for h wo dgr-of-frdom cas, h rspciv quaions of moion ar placd in sa spac form wih hir corrsponding cuing forc modls. Firs, on dfins q1 () q () q() q () = u q3() q& () q () q& () 4 u (4.1) Th singl dgr-of-frdom linar cuing forc modl is hn drivd o b q& () = q () 1 k c K w q& () = q () q () + g() [ q ( τ ) q () + fτ ] m m m (4.) 31

42 and h singl dgr-of-frdom 3/4 rul cuing forc modl is drivd o b q& () = q () 1 k c K w( fτ ) q& () = q () q () + g() [ q ( ) q () + fτ ] 1/4 1 1 τ m m m 1 3/4 (4.3) whil h wo dgr-of-frdom linar cuing forc modl is drivd o b q& () = q () 1 3 q& () = q () 4 k c K w q& () = q () q () + g() [ q ( τ ) + q ( τ) q () q () + fτ] m m m k c K w q& () = q () q () + g() [ q ( τ ) + q ( τ) q () q () + fτ] u u mu mu mu (4.4) and h wo dgr-of-frdom 3/4 rul cuing forc modl is drivd o b q& () = q () 1 3 q& () = q () 4 k c K w( fτ ) q& () = q () q () + g() [ q ( τ) + q ( τ) q () q () + fτ] 1/ m m m k c K w( fτ ) q& () = q () q () + g() [ q ( τ ) + q ( τ) q () q () + fτ] 1/4 u u mu mu mu 3/4 3/4 (4.5) Rcalling ha h cuing forc is only aciv (g() = 1) during h cuing priod τ and ha τ = ρτ, h non-cuing priod τ 1 can b found as (1 ρ) τ. Wih h cuing and non-cuing priods dfind, ach sa of ach quaion of moion is solvd, rspcivly, by using a usr dfind aial dph of cu and spindl spd, and is hn summd up o qual h oal spindl priod τ. Malab's dd3 solvr is usd o rsolv ach linar, homognous quaion during τ 1 as wll as ach non-linar, non-homognous dlay diffrnial quaion during τ. Th soluions ar hn summd up o form h compl soluion for τ. Iniial condiions ar carrid from h soluion a h of h currn 3

43 spindl priod and nrd ino h n priod for calculaion. This procss is irad for a usr dsignad numbr of spindl cycls. Th soluion gnrad across h chosn numbr of cycls is displayd in graphical form in rms of a rspons hisory and a phas porrai diagram. In Figur 4.1, an ampl of h rspons of a singl dgr-offrdom sysm is shown for sabl cuing condiions. Figur 4.1: Rspons and phas porrai diagrams for sabl cuing condiions. In Figur 4., an ampl of h rspons of h sam sysm is shown a h sabiliy boundary. Figur 4.: Rspons and phas porrai diagrams a sabiliy boundary. 33

44 In Figur 4.3, an ampl of h sysm rspons is shown for unsabl cuing condiions. Figur 4.3: Rspons and phas porrai diagrams for unsabl cuing condiions. Th Malab cod usd o find h numrical soluions of quaions (4.) o (4.5) can b found in Appi C. 4. UMCP Numrical Sabiliy Prdicion Program Th following figur dpics a four dgr-of-frdom milling configuraion. Figur 4.4: Schmaic of a four DOF milling configuraion. 34

45 Th govrning quaions of his sysm ar givn by ( τ ) ( τ ) ( τ ) ( τ ) mq&& () + cq& () + kq() = F ; (,, iz) mq&& () + cq& () + kq() = F ; (,, iz) y y y y y y y mq&& () + cq& () + kq() = F ; (,, iz) u u u u u u u mq&& () + cq& () + kq() = F ; (,, i z) v v v v v v v (4.6) Th quaions of moion for h ool includ, rspcivly, h horizonal and vrical displacmns q ( ) and q ( ), h consan siffnsss k and k y, and h consan y damping rms c and c y. Th quaions of moion for h workpic includ, rspcivly, h horizonal and vrical displacmns qu ( ) and qv ( ), h consan siffnsss k u and k v, and h consan damping rms c u and c v. Th consan fd ra f of h workpic is dircd along h major horizonal mod of moion on h workpic q () u. Basd on prvious work carrid ou by Zhao () and Long (6), h cur is modld as a sack of infinisimal disk lmns. A summary of h drivaion of h cuing forcs is providd hr. Th daild drivaion of h cuing forcs can b found in Chapr of Long (6). Figur 4.5: Cylindrical mill wih infinisimal disk lmns. 35

46 In Figur 4.5, ach disk lmn is locad a an aial disanc z along h ool from h boom of h mill up o h aial dph of cu. Th hli angl is η. For h ih ooh, h infinisimal cuing-forc componns on h disk lmn ar rprsnd by for h radial dircion, i Δ F for h angnial dircion, and i Δ F r i Δ F z for h aial dircion. Th cuing forcs along ach componn of h infinisimal disk lmn ar found from h dynamic uncu chip hicknss for h ih flu of h cur a im and high z as hi (,,z) = Ai (,,z)sin θ ( i,,z) + Bi (,,z)cos θ ( i,,z) + hsv (4.7) whr h saic uncu chip hicknss is Th rlaiv displacmns ar found o b 1 h f f R sv = τ sin θ + ( τ cos θ) (4.8) Ai (,, z) = q( ) q( τ( i,, z)) + q( ) q( τ( i,, z)) u u B(,, iz) = q() q( τ(,, iz)) + q() q( τ(,, iz)) y y v v (4.9) Th cuing forcs ar hn found o b i ΔF r 1 k i z F cosη sin η knk Δ Δ = h(, i, z) i cos η F Δ z sinη cos η μk(cosφn kncos φn) (4.1) Transforming h cuing forcs ino carsian coordinas, ingraing h infinisimal cuing forc componns from z(,) 1 i o z(,) i a h cuing zon, which is dfind o ' ' b θ < θ(, iz, ) < θ, and ignoring forcs along h z dircion, h cuing forcs ar s found as 36

47 z (,) i F i () ˆ i ˆ 11(, ) i k z k1(, z) Ai (,, z) z i = d F () ˆi ˆi y (,, ) z1 (,) i k1(, z) k(, z) Biz ˆ (, ) 1 + f sin ( iz,, ) fcos ( iz,, ) dz z (,) i i c1 z τ θ + ( τ θ ) i ˆ 1(,) (, ) R z i c z (4.11) whr ˆi ˆi k11(, z) k1(, z) sin θ( iz,, ) cos θ( iz,, ) kk 1 = [ sin (,, ) cos (,, )] ˆi ˆi θ iz θ iz k cos (,, ) sin (,, ) 1(, z) k(, z) θ iz θ iz kk (4.1) i cˆ sin (,, ) cos (,, ) 1(, z) θ iz θ iz kk 1 i = ˆ c cos (,, ) sin (,, ) (, z) θ iz θ iz kk kn k1 = cosη k = 1+ an η[ μ(cosφ k sin φ )] n n n (4.13) (4.14) and k is h spcific cuing nrgy, φ n is h normal rak angl, k n is a proporionaliy consan, and μ is h fricion cofficin. From Nwon's Third Law of Moion, h forcs acing on h workpic ar drmind as ( ; τ (,, )) ( ; τ (,, )) F i z ( ; τ (,, )) ( ; τ (,, )) F i z u = Fv i z Fy i z (4.15) Whn h ool is ousid of h cuing zon, hr is loss of conac, which maks h cuing forc componns qual o zro. Th drminaion of h cuing and non-cuing zons can b found in Chapr 3 of Zhao (). Th drivd sa spac quaions from (4.6), by using h cuing forcs dscribd in (4.11), can b found in Chapr of Long (6). Sabiliy analysis carrid ou by using h smi-discrizaion mhod is daild in Chapr 3 of Long (6). A copy of h UMCP numrical sabiliy prdicion program is givn in Scion D.1 of Appi D. 37

48 4.3 Modifid UMCP Numrical Sabiliy Prdicion Program for h 3/4 Rul Th dynamic uncu chip hicknss givn by quaion (4.7) is basd on a linar cuing forc modl. Applying h 3/4 rul o his prssion yilds 3/4 3/4 h (,,z) i = [ A(,,z)sin i θ(,,z) i + B(,,z)cos i θ(,,z) i + h sv ] (4.16) To incorpora his chang ino h UMCP numrical sabiliy prdicion program, h Taylor pansion is akn; ha is, h (,, i z) = [ h +Δh(,, i z)] 3/4 3/4 sv 3 3/4 3 Δhiz (,, ) 3[ Δhiz (,, )] 5 [ Δhiz (,, )] hsv h 3 h 18 h 1/4 5/4 9/4 sv sv sv (4.17) whr Δ hiz (,, ) = Aiz (,, )sin θ ( iz,, ) + Biz (,, )cos θ ( iz,, ) (4.18) h sv is prviously dfind in (4.8) and Ai (,, z ) along wih B( iz,, ) ar prviously dfind in (4.9). Th linarizd form of quaion (4.17) bcoms 3 Aiz (,, )sin θ (,, iz) + Biz (,, )cos θ (,, iz) h (,, i z) + h (4.19) 3/4 3/4 1/4 sv 4 hsv Making us of quaions (4.19) and (4.11), h cuing forcs ar found o b i F () i = Fy () + kˆ (, z) kˆ (, z) Ai (,, z) 3 1 [ f sin ( iz,, ) ( fcos ( iz,, )) ] dz ˆ (, ) ˆ τ θ + τ θ k z k (, z) Biz (,, ) 4 R z (,) i i i /4 i i z1 (,) i 1 ˆ (, ) z (,) i i c1 z 1 i ( ) 3/4 τ θ + τ θ ˆ z1 (,) c(, z) R i [ f sin (, i, z) f cos (, i, z) ] dz (4.) A copy of h modifid UMCP numrical sabiliy prdicion program is givn in Scion D. of Appi D. 38

49 Chapr 5 Rsuls and Discussion 5.1 Singl DOF Sysm Rsuls and Discussion For h analyical rsuls obaind for h singl dgr-of-frdom sysm wih h linar cuing forc, on uss quaions (3.4) and (3.16) o arriv a h flip bifurcaion prdicions, whil quaions (3.9) and (3.16) ar usd o obain h Nimark-Sackr bifurcaion prdicions. Th analyical rsuls obaind for h 3/4 rul cuing forc modl mak us of quaions (3.4) and (3.17) o arriv a h flip bifurcaion prdicions, whil quaions (3.9) and (3.17) ar usd o arriv a h Nimark-Sackr bifurcaion prdicions. As prviously sad, ngaiv and infini aial dph of cu rms do no mak physical sns and hy ar hrfor no includd in h rsuls. Th analyical rsuls ar compard wih h rsuls of h dlay diffrnial quaion (DDE) numrical simulaions daild in Scion 4.1. From Davis al. (), h following inpu paramrs ar usd for making h singl dgr-of-frdom modl prdicions. Propris Paramrs Unis m.431 kg ζ k 1.4E+6 N/m K 5.E+8 N/m N - f.1 m/ooh R.635 m Tabl 5.1: Inpu paramrs from Davis al. () for analyical prdicions. 39

50 1 1 8 ADOC (mm) Spindl Spd (krpm) Analyical Linar Cuing Forc - Flip Bifurcaion Analyical Linar Cuing Forc - N.S. Bifurcaion Analyical 3/4 Cuing Forc - Flip Bifurcaion Analyical 3/4 Cuing Forc - N.S. Bifurcaion DDE Numrical Linar Cuing Forc DDE Numrical 3/4 Cuing Forc Eprimnal Daa Figur 5.1: Analyical prdicion and dlay diffrnial quaion numrical prdicion comparisons wih primnal daa for 5% immrsion. By using quaion (1.), h immrsion ra is drmind from ρ =.14 o b approimaly 5%. Eprimnal daa from Figur 7 (b) of Davis al. () is also includd for comparison. For h n four graphs, h spindl spds rang from 1 o krpm. As pcd, h dlay diffrnial quaion numrical prdicions shown in Figur 5.1 follow h r of h analyical prdicions. Th 3/4 rul cuing forc modl prdics slighly highr rgions of sabiliy whn compard wih h rsuls obaind wih h linar cuing forc modl. Th pak in h sabiliy lob prdicd around 13.5 krpm, by using analyical mans, compars rasonably wll o h pak a 13 krpm sn in h primnal daa; h prdicions a 18 krpm, howvr, do no corrspond wll o h 4

51 primnal rsuls ha show a pak a 19 krpm. Ousid of hs rgions of highr sabiliy, h analyical rsuls and h numrical rsuls obaind by ingraing h dlay diffrnial quaions compar wll wih h primnal prdicions. Th following paramrs wr usd o run h UMCP numrical sabiliy prdicion program as wll as h modifid program for 3/4 rul cuing. Propris Paramrs Unis ProprisParamrs Unis m.431 kg m u 1.E+5 kg ζ ζ u 1 - k 1.4E+6 N/m k u 1.E+15 N/m m y 1.E+5 kg m v 1.E+5 kg ζ y 1 - ζ v 1 - k y 1.E+15 N/m k v 1.E+15 N/m Propris Paramrs Unis K 5.E+8 N/m K n.3 - R.635 m N - f.1 m/ooh hli angl dgrs rak angl 15 dgrs fricion. - Tabl 5.: Inpu paramrs from Davis al. () for numrical calculaions. In Figur 5., h 5% immrsion analyical prdicions ar shown along wih h prdicions mad from h UMCP numrical sabiliy programs for boh linar and 3/4 rul cuing forc modls; hs prdicions ar compard wih h 5% immrsion primnal daa. As shown in Figur 5.1, h analyical rsuls ar rasonably accura in prdicing h pak sn in h primnal rsuls a 13 krpm. Th UMCP programs, howvr, ar oo gnrous in hir sabiliy prdicions from 1 o 18.5 krpm. I is only 41

52 1 1 8 ADOC (mm) Spindl Spd (krpm) Analyical Linar Cuing Forc - Flip Bifurcaion Analyical Linar Cuing Forc - N.S. Bifurcaion Analyical 3/4 Cuing Forc - Flip Bifurcaion Analyical 3/4 Cuing Forc - N.S. Bifurcaion Numrical Linar Cuing Forc - Down Milling - Flip Bifurcaion Numrical Linar Cuing Forc - Down Milling - N.S. Bifurcaion Numrical 3/4 Cuing Forc - Down Milling - Flip Bifurcaion Eprimnal Daa Figur 5.: Analyical prdicion and UMCP numrical sabiliy program prdicion comparisons wih primnal daa for 5% immrsion. around 19 krpm, whr h UMCP program wih h linar cuing forc modl compars rasonably wll o h primnal daa. Th prdicions mad from h modifid 3/4 rul UMCP numrical sabiliy prdicion program, rsuls in gnrous opraing rgions for sabiliy ha mosly go byond h scal of h figur. This is mos likly du o h linarizaion of h cuing forc modl from h Taylor sris pansion. Facors such as rak angl and fricion ha ar includd in h UMCP numrical program ar no accound for in h analyical prdicions and hrfor, may b h caus of crain discrpancis. 4

53 1 1 8 ADOC (mm) Spindl Spd (krpm) Analyical Linar Cuing Forc - Flip Bifurcaion Analyical Linar Cuing Forc - N.S. Bifurcaion Analyical 3/4 Cuing Forc - Flip Bifurcaion Analyical 3/4 Cuing Forc - N.S. Bifurcaion Numrical Linar Cuing Forc - Down Milling - Flip Bifurcaion Numrical Linar Cuing Forc - Down Milling - N.S. Bifurcaion Numrical 3/4 Cuing Forc - Down Milling - Flip Bifurcaion Numrical 3/4 Cuing Forc - Down Milling - N.S. Bifurcaion Eprimnal Daa Figur 5.3: Analyical prdicion and UMCP numrical sabiliy program prdicion comparisons wih primnal daa for 9% immrsion. In Figur 5.3, a highr immrsion ra of 9% is now usd for ρ =.35 o s if h sabiliy prdicions brak down du o h loss of conac and low immrsion assumpions mad in h analyical modl. From h primnal rsuls, h high immrsion ra in Figur 5.3 causs h local paks sn a 13 krpm and 19 krpm o rduc in siz, compard wih Figur 5., hrby rducing h rgion of sabiliy. Th UMCP numrical sabiliy prdicion program shrinks h rgion of sabiliy as wll so ha h prdicions for h linar cuing forc compar wll wih h primnal daa from 14 krpm o 18 krpm. Howvr, lik h analyical prdicions, h UMCP numrical prdicions do no compar wll wih primnal daa, spcially around h rgions of 13 krpm and 19 krpm. As in h 43

54 prvious cas, h modifid UMCP program wih h 3/4 rul cuing forc dos no far wll in prdicing rsuls wih h assumd paramrs. Th UMCP numrical sabiliy prdicion program disinguishs bwn down-milling and up-milling procsss. During h up-milling procss, h flip and Nimark-Sackr bifurcaion boundaris ar diffrn from hos obaind during h down-milling procss. Th analyical prdicions do no disinguish bwn hs procsss and hy favor h numrical prdicions mad for h down-milling procss as shown in Figurs 5. and 5.3. This agrmn may b du o an assumpion in h analyical quaions ha placs h cuing priod a h sar or h of h spindl priod ADOC (mm) Spindl Spd (krpm) Analyical Linar Cuing Forc - Flip Bifurcaion Analyical Linar Cuing Forc - N.S. Bifurcaion Analyical 3/4 Cuing Forc - Flip Bifurcaion Analyical 3/4 Cuing Forc - N.S. Bifurcaion Numrical Linar Cuing Forc - Up Milling - Flip Bifurcaion Numrical Linar Cuing Forc - Up Milling - N.S. Bifurcaion Numrical 3/4 Cuing Forc - Up Milling - Flip Bifurcaion Numrical 3/4 Cuing Forc - Up Milling - N.S. Bifurcaion Figur Up-Milling Cas: Comparison of h UMCP numrical sabiliy prdicion program wih h analyical prdicion rsuls for 9% immrsion. 44

55 1 1 8 ADOC (mm) Spindl Spd (krpm) Analyical Linar Cuing Forc - Flip Bifurcaion Analyical Linar Cuing Forc - N.S. Bifurcaion Analyical 3/4 Cuing Forc - Flip Bifurcaion Analyical 3/4 Cuing Forc - N.S. Bifurcaion Numrical Linar Cuing Forc - Down Milling - Flip Bifurcaion Naural Frquncy - Tool Figur 5.5: Analyical prdicion and UMCP numrical sabiliy program prdicion comparisons for 5% immrsion around h ool naural frquncy of 54.4 krpm. Th naural frquncy of h ool a 54.4 krpm (or 97 Hz) is includd in Figur 5.5, along wih sabiliy lobs in is viciniy. As shown in his figur, h naural frquncy of h ool lis dircly in a high rgion of sabiliy prdicd by h analyical rsuls whil h UMCP program wih h linar cuing forc modl prdics a high rgion o h lf of h naural frquncy. Th sabiliy lobs prdicd by using boh mhods mus b discound in his rgion as h naural frquncy of h ool dicas a rgion of insabiliy around 54.4 krpm. Forunaly, milling opraions ar rarly carrid ou a such high spindl spds, hrby rducing h possibiliy of opraing around h naural frquncy of h ool. 45