Improving Milling Process Using Modeling

E. BudakFaculty of Engineering and Natural Sciences

Sabanci University, Istanbul, Turkey

AbstractComplex part surfaces such as dies and molds can be generated using CAD/CAM systems andprecision machine tools. In many cases, however, part quality and productivity deteriorate duringmachining due to excessive cutting forces and chatter vibrations. Milling force and stability models canbe very effective in optimizing the process by improving part quality and productivity. Process modeling isparticularly important for low volume production such as die and mold machining as the opportunity fortesting is very limited whereas in mass production process can be somewhat improved over time by trialand error. In this paper, analytical force and stability models developed by the author are presented withapplications. An analytical milling force model and its simplified version for feedrate optimization arereviewed with applications. The application of the analytical stability model to spindle speed optimizationfor chatter suppression is demonstrated. As an alternative approach to chatter suppression, optimaldesign of variable pitch cutters is also presented. The paper is concluded with some recommendationsfor effective implementation of the models in production environment which can be used in die and moldmachining.

Keywords: Milling, Cutting Force, Chatter

1 INTRODUCTIONMachining time and quality of dies and molds are verycritical in many industries as they may be the limitingfactors for development of production processes.Reduction of milling time and improvement of surfacequality in machining can directly affect lead time and costof product and process development cycle. Althoughmodern CAD/CAM systems are very efficient ingenerating tool paths for complex die surfaces forvarious machine tool configurations and cuttergeometries, material removal rate (MRR) and surfacequality are usually limited due to the machining process.High cutting forces, chatter vibrations and tooldeflections limit depth of cut, spindle speed and feed ratecausing reduced productivity, and may also result in poorsurface finish and dimensional quality. Machiningprocess modeling can be very effective in improvingproductivity and quality. Therefore, in this paper, millingforce and stability models that can be applied to die andmold machining are briefly discussed with someapplications.

Modeling of milling process has been the subject ofmany studies and an overview is given in [1]. The focusof these studies have been mostly on the modeling ofcutting force and stability and prediction of part quality.In several applications, cutting force models have beenused in order to increase material removal rate byoptimizing feed rate [2,3]. Milling forces have beeninvestigated using different approaches. Koenigsbergerand Sabberwal [4] developed equations for milling forcesusing mechanistic modeling where the cutting forcecoefficients which relate the chip area to the tangential,radial and axial forces are calibrated through forcemeasurements. The mechanistic approach has beenwidely used for the force predictions and also extendedto predict associated machine component deflections orsurface geometrical errors [5-6]. Another approached is

to use mechanics of cutting in determining milling forcecoefficients as used by Armarego et al. [7]. In thisapproach, an oblique cutting force model together withan orthogonal cutting database are used to predictmilling force coefficients eliminating the need for millingtests as different tool and cutting geometries can behandled by the oblique model [8]. Once the cutting forcecoefficients are known, the milling forces can bedetermined by integrating the forces along the cuttingedges. Altintas et al. [9] also demonstrated theapplication of this approach to complex milling cuttergeometries.

Tobias [10] and Tlusty [11] identified the most powerfulsource of self-excitation, regeneration, which isassociated with the structural dynamics of the machinetool and the feedback between the subsequent cuts onthe same cutting surface. Self-excited chatter vibrationsin milling develop due to dynamic interactions betweenthe cutting tool and workpiece which results inregeneration of waviness on the cutting surfaces, andthus modulation in the chip thickness [12]. Under certainconditions the amplitude of vibrations grows and thecutting system becomes unstable. Especially for highlyflexible machining systems, such as long slender endmills which are also used in die and mold machining,chatter may develop even at very slow cutting speedswhich are commonly used to suppress vibrations inmetal cutting. In general, additional operations, mostlymanual, are required to clean the chatter marks left onthe surface. Thus, chatter vibrations result in reducedproductivity, increased cost and inconsistent productquality.The stability analysis of milling is complicated due to therotating tool, multiple cutting teeth, periodical cuttingforces and chip load directions, and multi-degree-of-freedom structural dynamics, and has been investigatedusing experimental, numerical and analytical methods. Inthe early milling stability analysis, Tlusty [12] used his

orthogonal cutting model considering an averagedirection for the cut. Later, however, Tlusty et al. [13]showed that the time domain simulations would berequired for accurate stability predictions in milling.Sridhar et al. [14-15] performed a comprehensiveanalysis of milling stability which involved numericalevaluation of the dynamic milling system's statetransition matrix. Minis et al. [16-17] used Floquet'stheorem and the Fourier series for the formulation of themilling stability, and numerically solved it using theNyquist criterion. Budak [18] developed a stabilitymethod which leads to analytical determination ofstability limits. The method was verified by experimentaland numerical results, and demonstrated to be very fastfor the generation of stability lobe diagrams [19-20]. Thismethod was also applied to the stability of ball-endmilling [21]. These methods can be used to generatestability diagrams from which stable cutting conditions,and spindle speeds resulting in much higher stability canbe determined. Stability diagrams have been extensivelyused in high speed milling applications, to utilize thelarge stability pockets [22]. If the required high spindlespeeds are not available on machine tools, then anotheralternative is to use special geometry cutters to suppresschatter. Milling cutters with irregular tooth spacing, orvariable pitch, can be used to improve the stability. Theeffectiveness of variable pitch cutters in suppressingchatter vibrations in milling was first demonstrated bySlavicek [23]. He assumed a rectilinear tool motion forthe cutting teeth, and applied the orthogonal stabilitytheory to irregular tooth pitch. By assuming analternating pitch variation, he obtained a stability limitexpression as a function of the variation in the pitch.Opitz et. al. [24] considered milling tool rotation usingaverage directional factors, however they too consideredalternating pitch with only two different pitch angles.Their experimental results and predictions showedsignificant increase in the stability limit using cutters withalternating pitch. Vanherck [25] considered differentpitch variation patterns in the analysis by assumingrectilinear tool motion. His computer simulations showedthe effect of pitch variation on stability limit. Tlusty et al.[26] analyzed the stability of milling cutters with specialgeometries such as irregular pitch or serrated edges,using numerical simulations. These studies mainlyconcentrated on the effect of pitch variation on thestability limit, however they do not address the cuttingtool design, i.e. determination of optimal pitch variation.Altintas et. al. [27] adapted the analytical milling stabilitymodel to the case of variable pitch cutters which can beused more practically to analyze the stability withvariable pitch cutters. Budak [3,28] recently developedan analytical method for the design of pitch angles forgiven chatter frequencies and spindle speeds.

Both cutting force and stability models can be used toimprove the machining operations on dies and molds.Therefore, cutting force and stability models will besummarized in sections 2 and 3, and applications of themodels will be presented in section 4. The paper will beconcluded with brief summary of results and suggestionsfor the implementation of the models.

2 MILLING FORCES

2.1 Analytical Force ModelMilling forces can be modeled for given cutting and cuttergeometry, cutting conditions, and work material. Forcemodeling for cylindrical milling cutters, which are used inroughing of dies and molds, will be given in the following.The model can be extended to ball end mills aspresented in [29-32] for the analysis of finishingoperations.

φj

y

x

b

dFrj

dFtj

!"#$&%' (*) +, -. /0/2131456. 789:. 8 8 . 4 ;</2=3>+1:>=9:?=4 14*52@AConsider the cross sectional view of an end mill shown inFigure 1. For a helical milling cutter, the radial (b) andaxial depth of cut (a), number of teeth (N), cutter radius(R) and helix angle (β) determine what portion of a toothis in contact with the workpiece for a given angularorientation of the cutter B φCED For a point on the F jth C cuttingtooth, differential milling forces in the tangential F dFt C andradial direction F dFr C and can be given as

),(),(

),(),(

zdFKzdF

dzzhKzdF

jj

j

trr

jtt

φφ

φφ

=

= FHGICJ,KLML φ N OQP KL N R'R LM ON STVUT WX L R L UYO*Z ML[V\M SR]P K L^ SYO*N PN _ L y `*ab cd`Ycdcefg"hib hQjb klmnpoqrsentlh u*l*v:ueb wxyYz |+ ~":~:Yz x2~Y'Y

)(sin),( zfzh jtj φφ = 2 , ft 2Y 2Y φ (z¡ ¢ £¥¤¦ §¢ ¨'¨§©£¢ ª«¬ « ­® §¯ª©¤6¦§'¯® °*¤2§¥± j ² ¬ ¤ ¬³ ¢ ¬ ®*´ª£¢ ¤6¢ ª« z µ¶(·¸:¹2º:¹6»¸¼»¸½ ¾ ¿+À½Á ÃÄ2ÅÇÆÈÄ6ÉÅËÊ Ì'ÌÅÍÎÊ ÏÐÒÑÐ Ó ÅÕÔ*É ÑÐÓYÅÎ Ñ ÏÐÓÒÄ6ÉÅÖÑ×Ê ÑÂØ Ê ÍÅÔÄÊ ÏÐ<ÑYÎ

zR

jz pjβφφφ tan

)1()( −−+= ÙÚÛ

where the pitch angle is defined ÜÝ φp=2π/N Þ Milling forcecoefficients Kt Üß à Kr can be expressed as exponentialfunctions of the average chip thickness as follows

qaRr

paTt hKKhKK −− == ; áãâä

Constants KT, KR, p and q are usually determined usingmechanistic models where linear regression onmeasured milling forces is performed for differentfeedrates. Therefore, they depend on the workpiecematerial and cutting tool geometry. The average chipthickness is defined as the ratio of the chip volumeproduced in one revolution of the cutter to the exposedchip area:

Φ=−−

= tstex

exstta ffh

φφφφ coscos åæ ç

Where φè é and φêìë are the start and exit angles of the toothto and from the cut, respectively. A more generalapproach is to use mechanics of milling method aspresented in [7]. An oblique cutting force model is usedto relate the shear stress in the shear plane, shearangle, friction coefficient on the rake face, rake angle,

chip flow angle and helix angle to the cutting forcecoefficients. The cutting data are obtained fromorthogonal cutting tests as explained in

The tangential and radial forces can be resolved in thefeed, x, and normal, y, directions as follows

jrjty

jrjtx

jjj

jjj

dFdFdF

dFdFdF

φφ

φφ

cossin

sincos

−=

−−=

By integrating the differential forces given in equation (6)within the portion of the cutting flute that is in contact withthe workpiece, the forces contributed by one tooth of thecutter can be determined as [6]

( )[ ]

( )[ ] )(

)(

)(

)(

)(2cos)(2sin)(2tan4

)(

)(2sin)(22costan4

)(

φφ

φφ

φφφβ

φ

φφφβ

φ

ju

jlj

ju

jlj

z

zjrjjtt

y

z

zjjrjtt

x

zKzzRfK

F

zzKRfK

F

+−−=

−+−=

where zjl(φ) and zju(φ) are the lower and higher limits ofthe contact for the tooth (j). The total milling forces onthe cutter in both directions can then be determined as

∑∑==

==N

jyy

N

jxx jj

FFFF11

)()(;)()( φφφφ

Milling forces for a more complicated tool geometry suchas ball end mills can be determined similarly [29-31]. Asit can be seen from the above equations, prediction ofmilling forces require that the tool geometry, cuttingcoefficients, cutting conditions and tool-workpieceintersection geometry, i.e. axial and radial depth of cuts,be known accurately. If the required information isavailable, then the model can be used to optimize orschedule feedrate in a cutting cycle for a target cuttingforce as shown in [2] or maximum allowed tool deflectionas demonstrated in [6,32-33]. However, for many casesthere is a need for a more practical model as all of therequired data may not be available for productionimplementation. This simplified model is presented next.

2.2 Simplified Force Model For FeedrateOptimization

As shown by equation (7), milling forces vary periodicallyas the cutter rotates. In order to maximize the feedrates,the peak resultant force in the

x,y

plane, F which cause

bending stresses in the tool, has to be kept undercontrol. The resultant force can be expressed as

)()()( 22 φφφ yxR FFF +=

and the peak resultant force as

[ ] πφφ 20:;)(max −= RFF From equations (1-10), F can also be described as

),,,,,( rtt KbaRNCfKF β= where C is function of tool and cutting geometry, and Kr Substituting equations "!#$ into %&&')(

),,,,,,(1 qKbaRNCfKF Rpp

tT β−− Φ= %&+*'If equation (12) is written for two different feedrates , ft1

and ft2 , corresponding to two peak forces , F1 and F2 , andthey are divided side by side (neglecting the effect offeed on Kr for simplicity), the following is obtained:

p

t

t

f

f

F

F−

≈

1

2

1

2

1 %&-'

Equation (13) can be used to optimize feedrates tomaintain the peak force constant in a cycle. For this, thepeak resultant milling force, FM , using the originalfeedrate, ft , need to be measured. Then, the optimalfeedrate, fo , for a reference target force, Fref , can bedetermined from equation (13) as

p

ref

Mto F

Fff

−

=

1

1

%&/.0'

The feed rate override FOR can also be determined as

p

ref

M

t

o

F

F

f

fFOR

−

==

1

1

%&1'

Equation (14) can be used to optimize feedrates in amilling cycle provided that the milling forces are known.A force dynamometer can be used to measure themilling forces. Therefore, this method is applicable to thecases where more than one die need to be machined.

3 MILLING STABILITY

3.1 Analytical Model for Stability Lobes

Another very important limitation in milling is the selfexcited chatter vibrations which cause poor surface finishand tool life resulting in reduced productivity. In thissection, the stability model developed by Budak [18-20]will be reviewed. In this analysis, both milling cutter andworkpiece are considered to have two orthogonal modaldirections as shown in Figure 2. Milling forces excite bothcutter and workpiece causing vibrations which areimprinted on the cutting surface. Each vibrating cutting

Figure 2. Chatter model for milling.

tooth removes the wavy surface left from the previoustooth resulting in modulated chip thickness which can beexpressed as follows:

)()(sin)(wcwc jj

oj

ojjtj vvvvsh −−−+= φφ (16)

where the feed per tooth st represents the static part ofthe chip thickness, and φj=(j-1)φp+φ is the angularimmersion of tooth (j) for a cutter with constant pitchangle φp=2π/N as shown in Figure 2. φ=Ω.t is the angularposition of the cutter measured with respect to the firsttooth and corresponding to the rotational speed Ω (rad/sec). vj and vj

o are the dynamic displacements dueto tool and workpiece vibrations for the current andprevious tooth passes, for the angular position φj, andcan be expressed in terms of the fixed coordinate systemas follows:

),(cossin wcpyxv jppj jp

=−−= φφ (17)

where c and w indicate cutter and workpiece,respectively. The static part in equation (1), (stsinφj), isneglected in the stability analysis. If equation (17) issubstituted in equation (16), the following expression isobtained for the dynamic chip thickness in milling:

]cossin[)( jyjxjh φφφ ∆+∆= (18)

where

)()(

)()(oww

occ

oww

occ

yyyyy

xxxxx

−−−=∆

−−−=∆ (19)

where (xc, yc) and (xw, yw) are the dynamic displacementsof the cutter and workpiece in x and y directions,respectively. Similar to the static force analysis, dynamiccutting forces can be obtained using the dynamic chipthickness as

∆∆

=

y

x

aa

aataK

F

F

yyyx

xyxx

y

x

2

1 (20)

where the directional coefficients are given as:

∑

∑

∑

∑

=

=

=

=

++−−=

+−−−=

++−=

−+−=

N

jjrjxx

N

jjrjyx

N

jjrjxy

N

jjrjxx

Ka

Ka

Ka

Ka

1

1

1

1

)2cos1(2sin

2sin)2cos1(

2sin)2cos1(

)2cos1(2sin

φφ

φφ

φφ

φφ

(21)

The directional coefficients depend on the angularposition of the cutter which makes equation (20) time-varying :

[ ] )()(2

1)( ttAaKtF t ∆= (21)

[A(t)] is periodic with the tooth passing frequency ofω=NΩ and with the corresponding period of T=2π/ω. Ingeneral, the Fourier series expansion of the periodicterm is used for the solution of the periodic systems [34].Budak and Altintas [18-20] have shown that the higherharmonics do not affect the accuracy of the predictions,and it is sufficient to include only the average term in theFourier series expansion of [A(t)]:

[ ] [ ]dttAT

A

T

∫=0

0 )(1

(22)

As all the terms in [A(t)] are valid within the cutting zonebetween start and exit immersion angles (φst, φex),equation (22) reduces to the following form in the angulardomain:

[ ] [ ]

== ∫

yyyx

xyxx

p

ex

st

NdAA αα

ααπ

φφφ

φ

φ2

)(1

0 (23)

where the integrated, or average, directional coefficientsare given as:

[ ]

[ ]

[ ]

[ ] ex

st

ex

st

ex

st

ex

st

rryy

ryx

rxy

rrxx

KK

K

K

KK

φφ

φφ

φφ

φφ

φφφα

φφφα

φφφα

φφφα

2sin22cos2

1

2cos22sin2

1

2cos22sin2

1

2sin22cos2

1

−−−=

++−=

+−−=

+−=

(24)

Substituting equation (23), equation (21) reduces to thefollowing form:

[ ] )(2

1)( 0 tAaKtF t ∆= (25)

Substituting harmonic functions for dynamic forces andvibrations, the characteristics equation is obtained as

[ ] [ ][ ] 0)(det 0 =Λ+ ciGI ω (26)

where [I] is the unit matrix, and the oriented transferfunction matrix is defined as:

[ ] [ ][ ]GAG 00 =

[ ] [ ] [ ])()()( cwccc iGiGiG ωωω += (27)

[ ] ),( wcpGG

GGG

yyyx

xyxx

pp

ppp =

=

and the eigenvalue (Λ) in equation (26) is given as

( )Tit

ceaKN ω

π−−−=Λ 1

4 (28)

If the eigenvalue Λ is known, the stability limit can bedetermined from equation (28). Λ can easily becomputed from equation (26) numerically. However, ananalytical solution is possible if the cross transferfunctions, Gxy and Gyx, are neglected in equation (26):

−±−=Λ 0211

0

42

1aaa

a (29)

where

( )

)()(

)()(

1

0

cyyyycxxxx

yxxyyyxxcyycxx

iGiGa

iGiGa

ωαωα

ααααωω

+=

−= (30)

T

T

c

c

R

I

ωωκ

cos1

sin

−=

ΛΛ=

The above can be solved to obtain a relation betweenthe chatter frequency and the spindle speed [19-20]

NTn

kTc

60

tan;2

2

1

=

=−=

+=

− κψψπε

πεω

(31)

where ε is the phase difference between the inner andouter modulations, k is an integer corresponding to thenumber of vibration waves within a tooth period, and n isthe spindle speed (rpm). After the imaginary part inequation (28) is vanished, the following is obtained forthe stability limit [19-20]:

( )2lim 1

2κ

π+

Λ−=

t

R

NKa (32)

Equations (31-32) can be used to determine the stabilitylimit and corresponding spindle speed. When thisprocedure is repeated for a range of chatter frequenciesand number of vibration waves, k, the stability lobediagram for a milling system is obtained.

3.2 Stability of Variable Pitch CuttersIn general, stability diagrams can be used to increasethe chatter free material removal rate particularly at highcutting speeds where the stability lobes are larger.However, this approach may not be very effective if ahigh speed spindle is not available. Variable pitch cutterscan be very effective in suppressing chatter even at slowspeeds. The stability limit in terms of axial depth of cut(a) with equal pitch cutters can be put in the followingform

TcN

I

tKa

ω

π

sin

4lim

Λ−= (33)

The analytical stability model was used in [28] for thestability of variable pitch cutters and the stability limit wasobtained as

S

I

tK

vpa

Λ−=

π4lim (34)

where

∑=

=N

j

jcTS1

sinω (35)

Note that S in equation (34) replaces NsinωcT in equation(33) as the tooth period T, and thus the phase between

subsequent vibration waves, ε, is different for each toothdue to nonequal pitch angles:

)..,,1( NjT jcj == ωε (36)

The fundamental idea behind the variable pitch cutters isto alter the phase difference ε which is the main sourceof regenerative chatter. It is possible to increase thestability by selecting the pitch angles in such a way thatthe cumulative effect of the phase delay is minimized,i.e. S=0 in equation (34). The phase difference inequation (36) can be re-written as:

),..,2(1 Njjj =∆+= εεε (37)

where ∆εj is the phase difference between tooth j andtooth (1) corresponding to the difference in the pitchangles for these teeth. For an introduced variation inpitch, ∆P, the corresponding change in the phasedifference can be determined from the geometry as [28]

Ps

c ∆=∆ωω

ε (38)

where ωs=2πn/60 and n is the spindle speed in (rpm).Although different pitch variation patterns can be used,Budak [28] shows that the linear variation gives thewidest range of high stability area where

,...3,2,,: 0000 PPPPPPPPitch ∆+∆+∆+ (39)

Then, S takes the following form

...)2sin(sincos

...)2coscos1(sin

1

1

+∆+∆++∆+∆+=

εεεεεεS

(40)

S=0 can be satisfied for

)1,...,2,1(2 −==∆ NkN

kπε (41)

This is the amount of additional phase that has to beintroduced to satisfy S=0. The corresponding optimalpitch variation can be determined from equation (38). Asthe sum of all pitch angles must be equal to 2π (rad), P0

can be determined as

2

)1(20

PN

NP

∆−−= π (42)

Figure 3. Variation in the stability limit with the phasedifference ∆ε introduced by a variable pitch cutter withlinear pitch variation.

0

2

4

6

8

10

0 0.25 0.5 0.75 1∆∆ εε

r

ε1=π/4 ε1=3/2πε1=πx2π

In order to determine the stability increase with thevariable pitch cutters, the ratio of the stability limit withvariable pitch cutters, as given in equation (34), to theminimum, or absolute, stability limit for equal pitchcutters, i.e. equation (33) with (sin ωcT=1), is considered

S

N

a

ar

cr

vp

== lim (43)

Figure 3 shows the variation of the stability gain (r) for a4-tooth milling cutter. As ε1 in general depends on thecutting conditions and the dynamic characteristics of themilling system, r is plotted for 3 different values of ε1. Asit can be seen from figure 2 the stability increase ismaximized for the specific values of ∆ε as also predictedby equation (9). It should be noted that variations in thechatter frequency and speed may cause ∆ε and thus r tovary. However, at least 4 times stability improvement isguaranteed for a cutter with 4 teeth for

.2/32/ πεπ <∆< Therefore ∆P should be selected suchthat 2/32/ πεπ <∆< must be satisfied alwaysconsidering possible variations in chatter frequency.

4 APPLICATIONS

The applications of the models presented in the previoussections will be demonstrated through several examplesin the following.

4.1 Feedrate Optimization in RoughingThe optimization of the feedrates in a 5-axis slottingoperation is considered. The work material is titaniumalloy (Ti6Al4V) and a taper-ball-end-mill is used in thisheavy roughing operation where the productivity islimited due to cutter breakage. Therefore, feedrates canbe optimized to keep the cutting forces at a safe level.The variation of the peak cutting force determined fromthe measured milling forces is shown in Figure 4. As itcan be seen from the figure, the peak force in this cycleincreases 3 times due to changes in the axial depth ofcut and the tool orientation with respect to the feeddirection. The target force of 900 N for feed optimizationis selected from a database which contains safe forcelevels for different cutter sizes. Finite element analysiscan also be used to determine the cutting load capacityof end mills as presented in [35]. The parameter p hasbeen identified as 0.58 from the cutting force data. TheFOR determined using equation (15) is shown in Figure5. FOR is limited to maximum of 3 as it corresponds tothe maximum chip thickness for this cutter. The modifiedFOR is also shown in the figure.The forces in this slottingoperation were measured again, this time using the newfeedrates. The simulated forces from equation (15) andthe measured forces before and after the feed rateoptimization are shown in Figure 6. Although themeasured forces are close to the target force of 900 Nfor most of the cycle, there are some discrepanciesespecially in the beginning of the cycle. This is mainlydue to the neglected effect of feedrate on Kr in theformulation which is amplified in the beginning of thecycle due to very high FOR values. Dynamic effects andthe changes in the orientation of the cutter alsocontribute to the differences. The feedrates can beupdated again by using the new forces with the newfeedrates to get closer to the target force. However, thiswas not pursued in this case since the difference is onlyabout 100 N as it can be seen from the figure. Theimportant point to note here is that the cycle time hasbeen reduced by about 35% with the new feedrates.Depending on the force variation in a cutting cycle higherreductions can also be obtained.

Figure 4. Variation of the peak forces in a 5-axis millingoperation

Figure 5. Variation of the calculated and modified feedoverride values in the example slotting operation.

Figure 6. Peak force variations with the original andoptimized feedrates. Simulated forces are also shown.

4.2 Stability LobesDemonstration of the method will be done on a 2-DOFend milling example given in [20]. The dynamicproperties of the 3-flute end mill in two orthogonaldirections were identified using modal analysis as [36]follows

ωn (Hz) k (kN/m) ζX 603 5600 0.039

Y 666 5700 0.035

0

250

500

750

1000

0.00 10.00 20.00 30.00 40.00 50.00 60.00Time

0

250

500

750

1000

0,00 10,00 20,00 30,00 40,00 50,00 60,00

Time (sec)

Pea

k F

orce

(N

)

Before

After

Simulation

0

2

4

6

8

0.00 20.00 40.00 60.00Time (sec)

FO

R

FOR calculated

FOR modified

0

2

4

6

8

10

0 4000 8000 12000 16000 20000 24000

Spindle Speed (rpm)

Axi

al d

epth

of c

ut (

mm

)

Chatter

Stable

Time domain

Analytical

Figure 7. Analytical and numerical stability diagrams forthe example case considered.

The aluminum workpiece is considered to be rigidcompared to the cutter. The stability diagram for thissystem has been generated using the analytical modeland is shown in Figure 7. As it can be seen from thefigure, the stability limit predictions using the analyticalmethod and the numerical time-domain solutions arevery close. It should be noted that the analytical stabilitydiagram can be generated in a few seconds whereastime domain simulations usually take several hours (upto a full day) depending on the precision required. In timedomain simulations, dynamic system equations have tobe simulated over several tool rotations using very smalltime steps. As it can be seen from the figure, the chatter-free depth-of-cut could be increased about 8 times byusing about 12500 rpm compared to, for example, 8000rpm. With the increased spindle speed, this would resultin more than 10 times increase in chatter-free materialremoval rate.

4.3 Chatter Suppression with Variable Pitch Cutters

The variable pitch cutter design is demonstrated througha finishing operation on a very flexible workpiece(compressor blade). The original process with equalpitch cutter was highly unstable, and the sound spectrumshown in Figure 8 was used to determine the chatterfrequency. The depth of cut varies throughout the cuttingcycle, maximum being more than 100 mm. A taper-ball-end mill made out of carbide with length-to-averagediameter ratio of over 10 and 6 flutes is used to flank millthe blade. The chatter in this operation was so severethat a very slow spindle speed, 300 rpm, was being usedto reduce the intensity. However, the surface finish wasstill very poor, and additional finishing cuts and manualfinishing were necessary to remove the chatter marks.The optimal variable pitch cutter with linear pitchvariation can be designed using equations (38-42). Forthe chatter frequency of 420 Hz (as shown in Figure 8),spindle speed of 300 rpm and ∆ε=π, the pitch variation isdetermined as ∆P≈20 and P0=55. Note that althoughequation (41) gives multiple solutions, ∆ε should bechosen as π when N is even, and (N±1)π/N when N isodd, since the high stability region is the widest close tothe center, as it can also be seen from Figure 3. Thevariable pitch cutter with pitch angles (55, 57, 59, 61, 63,65) suppressed the chatter completely in this operation,at 300 rpm. The improvement in surface finish is shownin Figure 9. The cutting tool life was also improvedsignificantly, by about a factor of 2.

Figure 8. Sound spectrum of the blade finish-millingoperation using a 6-fluted equal pitch end mill.

Figure 9. Improvement of surface finish using thevariable pitch cutter.

5 SUMMARYIn this paper, milling process models which can be usedin the optimization of die and mold machining arereviewed. The cutting force models can be used infeedrate optimization of roughing operations. Forcomplete analytical modeling of the forces, cuttinggeometry and the material dependent milling forcecoefficients have to be known. If multiple parts are to bemachined, feedrates can be optimized using thesimplified force model based on the force measurementsperformed on the first part. For chatter suppression,stability lobes can be used in order to determinepreferred spindle speeds with high chatter-free materialremoval rate when a high speed machine is available.Stability lobes can be generated using the analyticalmodel which requires the modal data for the cuttingsystem. Variable pitch cutters, on the other hand, can beused to improve stability at slow speeds as well. Chatterfrequency must be known for the design of these cuttersusing the analytical model presented in this paper.Therefore, force and stability models can be used toimprove productivity and surface finish quality inmachining of dies and molds.

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0

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