GEEALIZED MODELIG OF MILLIG MECHAICS AD DYAMICS: PAT I - HELICAL ED MILLS Seafettin Engin Gaduate Student <engin@mech.ubc.ca> Yusuf Altintas Pofesso and ASME Fellow <altintas@mech.ubc.ca> The Univesity of Bitish Columbia Depatment of Mechanical Engineeing 34 Main Mall, Vanocuve, B.C., V6T Z4, CAADA tel: (64) 8 8, fax: (64) 8 4 3 ABSTACT Vaiety of helical end mill geomety is used in industy. Helical cylindical, helical ball, tape helical ball, bull nosed and special pupose end mills ae widely used in aeospace, automotive and die machining industy. While the geomety of each cutte may be diffeent, the mechanics and dynamics of the milling pocess at each cutting edge point ae common. This pape pesents a genealied mathematical model of most helical end mills used in industy. The end mill geomety is modeled by helical flutes wapped aound a paametic envelope. The coodinates of a cutting edge point along the paametic helical flute ae mathematically expessed. The chip thickness at each cutting point is evaluated by using the tue kinematics of milling including the stuctual vibations of both cutte and wokpiece. By integating the pocess along each cutting edge, which is in contact with the wokpiece, the cutting foces, vibations, dimensional suface finish and chatte stability lobes fo an abitay end mill can be pedicted. The pedicted and measued cutting foces, suface oughness and stability lobes fo ball, helical tapeed ball, and bull nosed end mills ae povided to illustate the viability of the poposed genealied end mill analysis. OMECLATUE: P : A cutting point on cutting edge,y,z : Global stationay coodinate sys. as shown in Figue 3 a : Axial depth of cut d : Diffeential height of the chip segment h : Valid cutting edge height fom tool tip h( ψ, φ, ) : Chip thickness at a cutting point specified by ( ψ, φ, ) i() : Helix angle lead : Lead value fo constant lead n : Spindle speed () : adial coodinate of a cutting edge point α, β : Paametic angles of the end mill φ : otation angle of cutting edge ψ () : Cutting edge position angle at level on the Y plane F x, F y, F : Foce components in, Y and Z diections D,,, : Paametic adial dimensions of the end mill M, : adial offsets of the end mill pofile fo points M and M, : Axial offsets of the end mill pofile fo points M and f : umbe of flutes K tc, K c, K ac : Cutting foce coefficients in tangential, adial and axial diections K te, K e, K ae : Edge foce coefficients in tangential, adial and axial diections df, df t, df a : Diffeential tangential, adial and axial foces () : Vecto fom tool cente to cutting edge s t : Feed pe tooth fo tooth feedate φ p st n π x, y,, : Coodinates of point P which is in cutting φ () : Total angula otation of flute at level on the Y pl. φ : Pitch angle of flute p [T] : Tansfomation matix fo the cutting foces. ITODUCTIO Vaiety of helical end mill geomety is used in milling opeations. Simple cylindical helical end mills ae used in peipheal milling of pismatic pats. Staight and helical ball end mills ae widely used in machining sculptued die and aeospace pat sufaces, and bull nosed cuttes poduce peiphey of pats meeting with the bottom floo with fillets. Tapeed helical end mills ae used in five axis machining of et engine compessos, and fom cuttes ae used to open complex pofiles such as tubine blade caie ings. A classical appoach in the liteatue has been to develop milling mechanics models fo each cutte shape, theefoe, mechanics and dynamics models developed individually fo face (Fu et al., 984), cylindical (Sutheland et al. 986, Bayoumi et al. 994; Budak et al.,996; Spiewak, 994), ball end (Laoglu et al., 997; Altintas et al., 998; Yucesan et al., 996) and tapeed ball end mills (amaa et al., 994) have been epoted in the liteatue. Ehmann et al. (997) summaied the oveview of past eseach in mechanics and dynamics of milling, which wee mainly specific to standad end and face milling cuttes. A genealied
mechanics and dynamics model that can be used to analye any cutte geomety is theefoe equied in ode to analye vaiety of end mill shapes used in manufactuing industy. The cutte geomety has two geometic components. The envelope o oute geomety of the cutte is used in geneating C tool paths on CAD/CAM systems. Moeove, the envelope of the cutte is used in identifying the intesection of cutte and wokpiece geomety, which is equied in simulating the mateial emoval pocess and in dynamically updating the blank geomety fo gaphical C tool path veification (Leu et al., 997; Gu et al., 997; Spence et al., 994). The geometic model must also include the cutting edge geomety along the flutes fo analying the mechanics and dynamics of the milling pocess. The pediction of the cutting foces and vibations equie the coodinates, as well as ake, helix, cleaance angles of the cutting edge point on the flute (Budak et al., 996). A genealied model of end mill geomety and cutting flutes is intoduced in this pape. The envelope of the geomety is defined simila to the paametic epesentation used by APT (Childs, 973) and CAD/CAM softwae systems. The fomulation of cutting edge coodinates is pesented. It is shown that a vast vaiety of helical end mill geomety can be designed using the poposed geometic model of genealied cuttes. The modeled cutting edge can be boken into small incements, whee the cutting constant may be diffeent at each location. As an example, helical ball, tapeed helical ball and bull nose cuttes ae povided. It is expeimentally poven that the model can be used in pedicting the cutting foces, vibations, dimensional suface finish, as well as chatte stability lobes fo any cutte geomety.. GEEALIZED GEOMETIC MODEL OF MILLIG CUTTES APT and CAD/CAM systems define the envelope of milling cuttes by seven geometic paametes (Childs, 973): CUTTE/ D,,,, α, β, h whee the cutte paametes D,,,, α, β and h ae shown in Figue. The genealied paametic statement can define a vaiety of face and helical end mill shapes used in industy as shown in Figue. These seven geometic paametes ae independent of each othe, but with geometic constains in ode to ceate mathematically ealiable shapes. A helical cutting edge is wapped aound the end mill envelope as shown in Figue 3. The mechanics of cutting equie the identification of coodinates, the local cutting edge geomety, chip load, and the thee diffeential cutting foces (df a, df, df t ) at cutting points (i.e. P in Figue 3) along the cutting edge. Point P has elevation, adial distance () on Y plane, axial immesion angle κ () and adial lag angle ψ (). The axial immesion angle is defined as the angle between the cutte axis and nomal of helical cutting edge at point P (Figue 3). The adial lag angle is the angle between the line, which connects the point P to the cutte tip on the Y plane and the cutting edge tangent at the tip of the cutte. The coodinates of P ae defined by vecto () in cylindical coodinates. The peiphey of the milling cutte is divided into thee ones (Figue ): ( ) fo one OM tanα ( ) ( ) + fo one M () D ( ) u + tan β, u ( tanα tan β ) fo one S Cylindical End Mill D¹,,D/,ab,h¹ Tape End Mill D¹,,D/,a,b¹,h¹ Cone End Mill D¹,, D/ ¹,a¹,b¹,h¹ O D a M u M T C Figue. Geneal tool geomety. Ball End Mill D¹, D/, ab,h¹ Tape Ball End Mill D¹, ¹, a,b¹,h¹ ounded End Mill D¹,¹,, ¹ a,b,h¹ Figue. End mill shapes. b S L M h Bull ose End Mill D¹, D/4 ab,h¹ Geneal End Mill D¹,¹, ¹, ¹ a¹,b¹,h¹ Inveted Cone End Mill D¹,¹, ¹, ¹ a¹,b¹,h¹ whee () is the adius of the cutte at elevation, u is the distance between the cutte tip and the point at which the S line intesects the Y plane. An ac with the cente at point C, a adial offset and ac adius of, is tangent to o intesects the tape lines OL and LS at points M and, espectively (Figue ). The adial and axial offsets of points M and fom the cutte axis and tip ae found, espectively, as:
Z Cylindical End Mill Ball End Mill Bull ose End Mill n k() () P df t S Z Cutting edge Cutte envelope n O Y f+ df df a M f Y A k() O Y S h(y,f,k) A d db D¹,,D/,ab,h¹ Tape End Mill D¹, D/, ab,h¹ Tape Ball End Mill D¹, D/4 ab,h¹ Geneal End Mill Cutting edge O () f () f s t sinf D¹,,D/,a,b¹,h¹ Cone End Mill D¹, ¹, a,b¹,h¹ ounded End Mill D¹,¹, ¹, ¹ a¹,b¹,h¹ Inveted Cone End Mill P f p Section A-A y() M Figue 3. Geometic model of the geneal end mill. tanα + M M tanα and + ( ) tan α + fo α < 9 tan α + tanα ( u) tanβ + ( ) tan β + ( u) tanβ ( u) u + tanβ fo β < 9 tan β + + + () (3) In Figue, the lines OM and S ae not necessaily tangent to the cone ac at the points M and. If the lines ae tangent to the ac, oute suface of the cutte will be continuous though out the thee segments, othewise the suface will be discontinuous so as the helix angle. The adial offset at elevation and the axial immesion angle in thee ones ae (Figue, Figue 3), fo along line OM: ( ψ ) ( ), κ () α tanα fo along ac M: ( ) ( ( ψ )) ( ) +, κ ( ) sin fo along line S: π ( ) + ( ( ψ ) ) tan β, κ () β (4) D¹,, D/ ¹,a¹,b¹,h¹ D¹,¹,, ¹ a,b,h¹ Figue 4. Helical cutting edges on the tool. D¹,¹, ¹, ¹ a¹,b¹,h¹. Genealied Geometic Model of the Helical Cutting Edges Helical flutes can be wapped aound the cuttes as shown in Figue 4. A vecto dawn fom cutte tip (O) to any point (P) on a helical flute can be expessed as (Figue 3), x i + y + k ( φ ) sinφ i + cosφ + ( φ ) (5) ( ) k whee φ is the adial immesion angle of point P on flute numbe. The adial immesion angle ( φ ) vaies as a function of otation angle, flute position, and the local helix angle at point P. The fist flute () is consideed to be a efeence edge, and its otation angle at the elevation is φ. The immesion angle fo flute at axial location is expessed as: φ ( ) φ + φ ψ ( ) (6) p n whee φ p is the pitch angle between the peceding flutes. It should be noted that this geneal fomulation allows vaiable pitch cuttes as well. The adial lag angle ψ () due to the local helix angle i. Since the cutte diamete may be diffeent along the axis, the helix and thus lag angles vay along the flute as well, and they ae evaluated in the following subsection. An infinitesimal length (ds) of a helical cutting edge segment can be given as follows, ( ( φ) ) + ( ( φ) ) dφ ds d ( φ) + d( φ) ( φ), dφ d( φ) ( φ) dφ (7)
The chip thickness changes with both adial ( φ ) and axial immesion (κ ) as the following, h ( φ ) s t sinφ sinκ (8) whee st is the feed pe tooth fo tooth. ote that the effective feed fo evey tooth may be diffeent when vaiable pitch cuttes ae used. The lag angle ψ () has a diffeent expession fo each one along the flute, which is wapped aound geneal cutte geomety. Zone OM ( M ): The helix angle is assumed to be constant at this usually small cone pat, i.e. i ( ) io. amaa (994) gives the diffeential equation and its solution fo a helical flute spialed on a cone as, d cosα ( cosα / tan io )ψ and theefoe e (9) dψ tanio Since the spial adius is eo at the tool tip, the lag angle condition becomes ψ (). Instead, the simulations wee stated with a small adial offset fom the tip, i.e. s M / which gives a stating lag angle of ψs ln s tanio / cosα. The vaiation of lag angle along the helical flute is evaluated fom Eq. (9), ( cotα ) ln tani ) o s ψ ( cosα ψ () The final lag angle at the point M becomes ln M tan io ψe ψs. cosα If the cone one (OM) does not exist (i.e. bull nose and ball end cuttes), α, ψ s and ψ e will be eo. Ac Zone M ( M ) < : Due to changing adial offset fom the cutte axis, the helix angle vaies along the flute fo constant lead cuttes as, i( ) tan ( ( ) ) tanio () Since the ac is not a full quate due to tangency to the cone, thee is a discontinuity on the helix angle at point M. This leads to the following lag angle expession, ( + ) tan io ψ ( ) ψ as + ψe () whee ψ e and ψ as ae the final lag angles at point M fomed by the cone and ac, espectively. The lag angle ( ψ as ) fomed by the ac at point M is, ( + M ) tanio ψ as (3) The final lag angle at the end point of the ac is: ( + ) tani ψe (4) o ψ ae ψ as + if the ac is missing fom the cutte geomety ψ, ψ ae ψe. as Tape Zone S ( < ) : The tape one of the cuttes is gound eithe with a constant lead o constant helix. The constant lead, which leads to vaiable helix angle along the flute, is pefeed by the cutte gindes in ode to save fom the mateial duing e-ginding opeation. Howeve, cutting mechanics ae moe unifom with constant helix cuttes, which equie vaying lead. Both methods ae modeled hee. Constant helix: The helix angle is constant, and the lag angle changes along the flute: i ( ) i o ln( ( ) tan β ) tani o ψ ( ) ψ s + ψ ae if β sin β (5) ( ) tanio ψ ( ) ψ s + ψ ae if β whee ψ ae and ψ s ae final lag angle at the point fom the ac and initial lag angle geneated by the tape one (S) at point, espectively. The initial value ψ s is, ln( ) tanio ψ s if β sin β (6) ψ s if β The final value ψ e at point S is, ( ( h) tan β ) ln tanio ψ e ψ s + ψ sin β ( h ) tani ψ +ψ ae if β o e ae if β (7) Constant Lead: The lead of the helical flute is constant, and the helix vaies along the flute. The nominal helix ( i s ) and lead (lead) of the of the tapeed flute is defined at point, π tan is (8) lead cos β which leads to a vaiable helix expession ( ψ ψ ae) ( ) i( ) tan (9) The vaiation of the lag angle ψ () is given by, ( ) tanis ψ ( ) + ψ ae fo h () 3. MODELIG OF CUTTIG FOCES The diffeential tangential ( df t ), adial ( df ) and axial ( df a ) cutting foces acting on infinitesimal cutting edge segment ae given by (Altintas and Lee, 998), dft Kte ds + Ktc h( φ, κ ) db df Ke ds + Kc h( φ, κ ) db () df K ds + K h(, ) db a ae ac φ κ
whee h ( φ, κ ) is the uncut chip thickness nomal to the cutting edge and vaies with the position of the cutting point and cutte otation. Subindices (c) and (e) epesent shea and edge foce components, espectively. The edge cutting coefficients K te, K e and K ae ae constants and elated to the cutting edge length ds given in Eq (7). The shea foce coefficients K tc, K c, K ac ae identified eithe mechanistically fom milling tests conducted at a ange of feed ate (Fu et al., 984; Yucesan and Altintas 996) o a set of othogonal cutting tests using an oblique tansfomation method pesented by Budak et al. (996). db ( db d / sinκ ) is the poected length of an infinitesimal cutting flute in diection along the cutting velocity. The chip thickness h( φ, κ ) is evaluated using the tue kinematics of milling (Matellotti, 945) as well as the vibation of both the cutte and wokpiece. The cutte is otated at a spindle speed and the wokpiece is fed with the given feed using a small discete time inteval. The positions of cutting points along the flute ae evaluated using the geometic model pesented in section. The location of the same flute point on the cut suface is identified using both the igid body kinematics as well as stuctual displacements of cutte and wokpiece. The dynamic chip thickness is evaluated by subtacting the pesent coodinate of the cutting point fom the pevious suface geneated by the peceding tooth. The mathematical model and the pocedue to evaluate dynamic chip load ae well explained fo the helical cylindical and ball end mills in the pevious publications (Montgomey and Altintas, 99; Altintas and Lee, 998), and not epeated hee due to similaity of the appoaches. Once the chip load is identified and cutting constants ae evaluated fo the local edge geomety, the cutting foces in Catesian coodinate system can be evaluated as, o { } [ T ]{ } df () xy df ta d sinφ sinκ cosφ d cosφ sinκ sinφ df cosκ sinφ cosκ df cosφ cosκ dft sinκ df a (3) The total cutting foces fo the otational position φ can be found integating Eq. (3) along the axial depth of cut fo all cutting flutes which ae in contact with the wokpiece. F ( φ) x F ( φ) y F ( φ) f x f y f F F F [ φ() ] [ φ() ] [ φ() ] f [ df sinφ sinκ dft cosφ dfa sinφ cosκ ] d f [ df cosφ sinκ dft sinφ dfa cosφ cosκ ] d (4) f F [ df cosφ dfa sinκ ] d whee f is the numbe of flutes on the cutte. and ae the contact boundaies of the flute within the cut and can be found fom the geometic model of each one given in section. The cutte is axially digitied with small disk elements with a unifom diffeential height of d. The diffeential cutting foces ae calculated along the full contact length fo all flutes which ae in cut, and digitally summed to find the total cutting foces F x(φ ), F y (φ ) and F (φ ) at a given otation angle φ Ω dt whee Ω( ad / s) is the spindle speed and dt is the diffeential time inteval fo digital integation. The stuctual dynamics of both cutte and wokpiece ae measued at the tool tip, and the modes ae identified using expeimental modal analysis technique. The dynamic chip thickness, cutting foces, vibations, suface finish, toque and moments geneated by the inteaction of cutting foces and stuctual dynamics ae simulated in the time domain using a technique simila to one pesented fo ball end milling by Altintas and Lee (998). The chatte stability lobes ae pedicted both using time domain simulations and fequency domain analytical chatte stability pediction method pesented peviously by Altintas et al. (995, 998, 999). The eades ae efeed to the pevious liteatue fo the mathematical details of the time and fequency domain solutions of the dynamic milling pocess. 4. SIMULATIO AD EPEIMETAL ESULTS Moe than 3 cutting tests with vaious cutte geomety and mateial wee conducted using the genealied method pesented hee. A sample of helical and inseted end mills ae pesented hee to demonstate the flexibility of the poposed model. Tapeed Helical Ball End Mill: These cuttes ae mainly used in five axis milling of et engine compessos made of Titanium Ti 6 Al 4 V alloys. Because of lage axial depth of cuts and poo machinability of Titanium, the chip loads ae small and the cutting speed is low in ode to avoid shank beakage and edge chipping. Chatte vibations ae also most fequently expeienced due to heavy cuts with slende end mills. The poposed model is applied to the design and vitual analysis of the tapeed cuttes fo an aicaft et engine company. The obective was to optimie the cutting tool geomety fo stength, and identify the chatte stability lobes to avoid self-excited vibations duing milling compessos and integal bladed otos. The cutting constants fo Ti 6 Al 4 V ae identified using othogonal to oblique cutting tansfomation method and given in (Budak et al., 996). One of the paticula cutte geomety geneated by the poposed geometic model is given in Figue 5. Although the tests wee conducted at vaious depth of cuts and feeds, two sample pedictions and expeimental validations ae given in Figue 6. The tests cove both the ball end and tapeed ones. The highe axial depth of cuts poduced sevee chatte vibations on ou machining cente, which does not have a spindle as igid as the five axis machining centes used in ou industial patne s shop. Even though the othogonal cutting database is used, the pedicted and measued cutting foces ae in satisfactoy ageement. Ball End Cuttes: Ball end mills ae used mostly in die and mold machining industy. The suface finish, static fom eos, chatte vibations, and tool life ae the main constains in ball end milling of dies and molds. The pediction of ball end milling foces and cutting constants (Yucesan and Altintas, 996), chatte stability (Altintas et al., 999), and suface finish (Altintas and Lee, 998) wee pesented befoe using specific geometic model of ball end mills. Slot ball end milling tests and pedicted cutting foces ae shown in Figue 7. The cutte mateial was WC coated with TiAlO, the wok mateial was GGG7 spheoidal gaphite cast ion with 5-83HB hadness, and the cutting conditions ae given in Figue 7.
i 6 5 4 Ball End Mill - GGG7 Simulated Measued Axial Depth of Cut. mm Feed ate.8 mm/tooth Speed ev/min lead i Cutting Foces [] 3 - - F -3 lead -4 3 4 5 6 7 otation Angle [deg] Constant lead Constant helix Figue 5. Tape helical ball end mill. Constant lead and constant helix on the tape. Cutting Foces [] 5-5 - 5 F Tape Helical Ball End Mill - Ti6Al4V Simulated Measued Axial Depth of Cut. mm Feed ate.5 mm/tooth Speed 5 ev/min 3 4 5 6 7 otation Angle [deg] Cutting Foces [] 8 6 4 - -4 F Simulated Measued Axial Depth of Cut 4. mm Feed ate.6 mm/tooth Speed ev/min -6 3 4 5 6 7 otation Angle [deg] Figue 7. Measued and pedicted cutting foces fo slot cutting. Cutting conditions: ake angle, elief angle, tool type is ball end mill, D. mm, 6. mm, mm, α, β, h 6. mm, f flutes, cutte mateial was WC coated with TiAlO, wokpiece mateial GGG7 spheoidal gaphite, Cutting coefficients: K tc 7., K c848.9, K ac -75.7 /mm, K te 7.9, K e 7.79, K ae -6.63 /mm. CUTTE MODEL Bull ose End Mill EAL CUTTE KEAMETAL End Mill with Inset Tool umbe: KIP5-P3 Inset umbe: PGB-35 Simulated Measued Cutting Foces [] 5-5 - -5 F Axial Depth of Cut. mm Feed ate. mm/tooth Speed 5 ev/min 3 4 5 6 7 otation Angle [deg] Figue 6. Measued and pedicted cutting foces fo slot cutting. Cutting conditions: ake angle, elief angle, tool type is tape helical ball end mill, D 6. mm, 3. mm, mm, 3. mm, α, β 4., h 38. mm, lead 5. mm, f flute, cabide cutte, see efeence (Budak et al., 996) fo Ti 6Al 4V othogonal cutting coefficients. Cutte D5/4", h3/6" 7/6", ab o 3D CUTTE MODEL Z Wokpiece Cutting edge Cutting Conditions: Spindle speed (n) 7 ev/min Axial Depth of Cut (a) 4. mm, Ti 6 Al 4V Feed ate (s t ). mm/tooth, slotting Only one inset is used duing the cutting test. Y CUTTIG FOCES Expeimental Simulation 5 [] 5.3.4.5.6.7.8.9 5 [] 5.3.4.5.6.7.8.9 6 F [] 4.3.4.5.6.7.8.9 Time [sec] OTE: Cutting Coefficients ae obtained fom Othogonal Cutting Expeiments. Figue 8. Geneal tool geomety application fo inseted end mill. See efeence (Budak et al., 996) fo the cutting coefficients.
Axial depth of cut [mm] Stability Lobes fo Bull ose Cutte and Al775 6 5 4 3 Analytical Time domain (95 pm) (4.7 mm ) (4 pm) (4.7 mm ) 4 6 8 4 6 Spindle speed [ev/min] [] 5 Expeimental esultant Foce [] 5 Expeimental esultant Foce 5 5 3 4 5 6 7 Tool otation Angle [deg] [] Simulated esultant Foce 5 5 3 4 5 6 7 Tool otation Angle [deg] [] Simulated esultant Foce 5 5 3 4 5 6 7 Tool otation Angle [deg] 3 4 5 6 7 Tool otation Angle [deg] Expeimental Suface oughness Expeimental Suface oughness max mm max mm 5 5 5 5-5 3-5 3.5.5.5 3 Z [mm] (axial diection).5.5.5 3 Z [mm] (axial diection) [mm] [mm] (feed diection) (feed diection) [Amp.] FFT fo esultant Foce [Amp.] FFT fo esultant Foce Tooth Passing 8 Chatte 8 Fequency 6 Fequency 6 (467 H) (448 H) 4 4 4 6 8 4 6 8 4 6 8 4 6 8 Fequency [H] Fequency [H] Y [mm] Figue 9. Stability lobes fo Bull nose cutte (see Figue 8). Cutting conditions: half immesion down milling, f flutes, feed ate.5 mm/tooth. See Table fo the tansfe function paametes. Aveage cutting coefficients fo Al-775: K tc 39.4, K c 788,83, K ac 48.75 /mm, K te9.65, K e 6.77, K ae.5 /mm. Y [mm] Bull osed Cutte: A bull nosed cutte with two coated cicula insets is used in milling Titanium Ti 6 Al 4 V and Aluminum Al- 775 alloys. The cutte was fist designed using the poposed geneal geometic model as shown in Figue 8. The cutting foces ae pedicted fo slot milling of Ti 6 Al 4 V with one inset. The othogonal cutting database was used in evaluating the aveage cutting constants. The simulated and measued cutting foces wee found to be in good ageement as shown in Figue 8. The same cutte was also tested on Al-775. The tansfe function of the cutte attached to tape 4 spindle with a mechanical chuck is given in Table. The tansfe function model has the following stuctue:
x F [ Φ xx ( s)] [ + s] K x x k k s + ζ x, kω x, ks + whee x, F ae the vibation and foce in the feed diection, espectively. ς x, k,ω x, k ae the damping atio and natual fequency fo mode k, and K is the numbe modes. The modal paametes ae evaluated fom estimated complex mode esidues ( σ k ± iυk ) as x, k ( ζ x, kωnx, kσ k ωdx, kυk ),, k σ k. Simila teminology is used in the nomal diection (Y). The chatte stability lobes of the system ae pedicted both in time and fequency domains, see Figue 9. The analytical stability solution is accuate, computationally fast, and agees well with the stability lobes pedicted by time consuming, iteative time domain solutions (Altintas and Budak, 995). The stability diagam indicates chatte fee milling speed at 4 ev/min with an axial depth of cut 4.7 mm. The same axial depth is pedicted to poduce significant chatte at the lowe spindle speed of 95 ev/min. The simulated and expeimentally measued cutting foces, simulated suface finish, foced and chatte vibation fequencies indicate the coectness of the poposed model. At 4 ev/min spindle speed, thee is no chatte, the cutting foces have egula static pulsation at tooth passing fequency, and the pedicted suface finish is smooth. At 95 ev/min, thee is chatte at 448 H which coincides to second bending mode of the spindle in the feed diection, the foce magnitudes ae at least twice lage than chatte fee machining test, and the suface is quite wavy. Table. Measued modal paametes of the bull nose inseted cutte on a machining cente. Diection σ ± iυ Mode ω n [H] ζ [%] Mode esidue [m/] ) Y 45.77 448.53 56.7 47.64.37.65.43 3.4 ω x, k ( k k (9.966 i 86.95). -6 (-4.856 i 34.36). -6 (-.399 i 7.539). -6 (4.555 i 36.888). -6 The authos conducted vaious cutting tests with helical ball and cylindical helical end mills and obtained simila esults. The obective of the eseach has been to design a vitual milling simulation system which can handle a vaiety of diffeent end mills fo impoved cutte design o pocess planning in industy. 5. COCLUSIO A genealied mathematical model of abitay end mills is pesented. The model allows paametic design and epesentation of vaiety of end mill shapes and helical flutes. Sample design examples include cylindical, ball, tapeed helical end mills as well as inseted bull nosed cuttes. The model allows evaluation of local cutting edge geomety along the flute. Using peviously developed exact kinematics of dynamic milling, the chip thickness and the coesponding cutting foces, vibations, and dimensional suface finish geneated by end mills with abitay geomety. The mathematical models ae suppoted by a numbe of expeiments conducted with helical tapeed end, ball end and bull nosed cuttes. The poposed appoach allows the design and analysis of vaiety of milling opeations used in industy. ACKOWLEDGEMETS This eseach is suppoted SEC, Geneal Motos and Boeing unde Coopeative eseach and Development eseach Gant (SEC 893). EFEECES Altintas, Y. and Lee, P., 998, Mechanics and Dynamics of Ball End Milling, Tansaction of ASME, Jounal of Manufactuing Science and Engineeing, Vol., pp. 684-69. Altintas, Y., and Budak, E., 995, Analytical Pediction of Stability Lobes in Milling, Annals of CIP, Vol. 44(o.), pp. 357-36. Altintas, Y., Shamoto, E., Lee, P., and Budak, E., 999, Analytical Pediction of Stability Lobes in Ball End Milling, Tansactions of ASME Jounal of Manufactuing Science and Engineeing, (in pess). Bayoumi A. E., Yucesan, G., and Kendall, L. A., 994, An Analytic Mechanistic Cutting Foce Model fo Milling Opeations: A Theoy and Methodology, Tansaction of the ASME, Vol. 6, pp. 34-33. Budak, E., Altintas, Y. and Amaego, E.J.A., 996, Pediction of Milling Foce Coefficients fom Othogonal Cutting Data, Tansactions of ASME, Vol. 8, pp. 6-4. Childs, J. 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