Modelling of tooth trajectory and process geometry in ...eprint.iitd.ac.in/bitstream/2074/1499/1/raomod2005.pdf · Modelling of tooth trajectory and process geometry in peripheral

Modelling of tooth trajectory and process geometryin peripheral milling of curved surfaces

V.S. Rao, P.V.M. Rao*

Mechanical Engineering Department, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110 016, India

Received 20 July 2004; accepted 13 October 2004

Abstract

The paper presents modelling of tooth trajectory and process geometry in peripheral milling of curved surfaces. The paper differs fromprevious work in this area, in two respects. Firstly it deals with milling of variable curvature geometries unlike zero and constant curvaturegeometries dealt in the past. Secondly true tooth trajectories are considered for modelling process geometry in milling of curved surfacesinstead of simpler circular tooth trajectories. Mathematical expressions for, feed per tooth along cutter contact path, entry and exits angles oftooth, undeformed chip thickness and surface error are derived and effect of workpiece curvature on these variables is studied. As cuttingforces depend on these process variables, physical experiments were also performed to study the effect of workpiece curvature on cuttingforces. Process simulation experiments carried out show the need for modelling true tooth trajectories instead of circular tooth trajectoriesparticularly for curved geometries. Results also show that using simpler constant curvature models to variable curvature geometries for thepurpose of estimation of process geometry variables could be erroneous.

Keywords: Peripheral milling; Milling process geometry; Curved surfaces; True tooth trajectory

1. Introduction

Assuming the tool and workpiece to be rigid, the shaperealized in any milling process depends on three importantfactors namely, tool geometry, workpiece geometry andrelative motion between tool and the workpiece. In a processlike peripheral milling, the relative motion between tool andworkpiece has two components. One component of motionconsists of rotation of tool about its own axis and the secondis a feed motion of tool relative to workpiece. Acombination of these two motions together with thegeometry of tool and its interaction with the workpiecegeometry defines the final shape of workpiece. Accuratemodelling of tool and workpiece geometries and the relativemotion between the two is important for estimation of manymachining process related variables such as feed per tooth,entry and exit angles of tooth, undeformed chip thickness,surface error etc.

Surfaces generated in a peripheral milling process can beclassified into three major classes: zero curvature or straightsurfaces, constant curvature or cylindrical surfaces andvariable curvature or free-form surfaces. Estimation ofprocess geometry variables such feed per tooth along cuttercontact path, entry and exit angle of tooth and undeformedchip thickness in case of peripheral milling of straightsurfaces is trivial, and this has been the subject of researchfor many years now [1-25]. Some work has also beenreported on estimation of these variables in case ofperipheral milling of cylindrical surfaces [26]. Estimationof process geometry in case of variable curvature surfaces isnot straight forward as these variables are not constant andcontinuously vary with the workpiece curvature. Thisvariation also leads to change in cutting forces and surfaceerror generated.

Where as many efforts have been made in the past todevelop analytical models to estimate process geometry forzero and constant curvature surfaces, the same for variablecurvature surfaces has not received any attention. One hasan option to extend constant curvature models to variable

ARTICLE IN PRESSV.S. Rao, P.V.M. Rao / International Journal of Machine Tools & Manufacture xx (2004) 1-14

curvature surfaces by treating them as locally constantcurvature along the path of curve [26]. However,when curvature variations are high the error associated inusing constant curvature models to variable curvaturegeometries is expected to be high. A need was felt todevelop more correct variable curvature model to overcomethis. Secondly, modelling of true tooth trajectories in caseof zero curvature and constant curvature surfaces are trivial[1-26]. The same in the case of variable curvature surfacesis non-trivial. We found that modelling of true toothtrajectories in case of variable curvature surfaces has notbeen attempted in the past. The present work is an attempt tobridge these two gaps.

Firstly mathematical expressions were derived formachining variables, feed per tooth, entry and exit angleof tooth, undeformed chip thickness and surface error forfollowing three cases:

• Constant curvature model with circular tooth trajectory• Variable curvature model with circular tooth trajectory• Variable curvature model with true tooth trajectory

The above three models were compared by carrying outsimulation experiments of peripheral milling of curvedsurfaces. It has been demonstrated that modelling a variablecurvature workpiece as locally constant curvature one, leadsto significant errors in calculation of machining parametersand hence in modelling cutting forces. Surface errors due totrue tool path trajectories in peripheral milling of variablecurvature geometries has also been attempted in this workfor both convex and concave type of surfaces. Further,machining experiments were also conducted to estimate thevariation in cutting forces due to workpiece curvature.

Rest of this paper discusses different approaches forestimation of process geometry variables and surface errorsin peripheral milling of curved geometries. Sections 3-5discuss the modelling of cutting tooth trajectory and processgeometry using constant curvature model with circular toothtrajectory, variable curvature model with circulartooth trajectory and variable curvature model with truetooth trajectory, respectively. Results of comparison ofthree models are presented in Section 6 and experimentalresults were presented in Section 7. Finally conclusions ofthis study are discussed in Section 8.

2. Previous work

Our review is limited to some of the major contributionsdirectly related to this subject. Traditionally, peripheralmilling process models assume circular path trajectories fortooth cutting points. Martellotti [1,2] for the first timederived equations for true tooth trajectories for peripheralmilling of straight surfaces and showed that such atrajectory is a trochoid. It was concluded that in milling ofstraight surfaces the difference in chip thickness

calculations by both trajectories is negligible. You andEhmann [11,12] developed similar tooth trajectories for facemilling with ball nose end cutter. Subsequently, Spiewak[15,18] derived a more comprehensive model for cuttingtooth trajectory by homogenous transformation of coordi-nate systems associated with the motion between the cuttingtooth and workpiece in peripheral milling of straightsurfaces. More recently, Li and Li [24] gave an improvedmethod for estimation of undeformed chip thickness usingtrue tooth trajectories which was based on Taylor seriesapproximation.

The above mentioned [1,2,11,12,15,18,24] efforts tomodel true cutting tooth trajectories as trochoidal arerestricted to milling of straight surfaces only. There is noevidence of such work being carried out for peripheralmilling of curved surfaces.

Very recently an attempt has been made by Zhang et al.[26] to build a cutting force model for peripheral milling ofcurved surfaces. However, only constant curvature surfaces(cylindrical surfaces) are dealt in their work and cuttingtooth trajectories are considered as circular and nottrochoidal.

The present work differs from the work of Zhang et al. inthat true tooth trajectories are considered for calculation offeed per tooth, undeformed chip thickness, entry and exitangles as well as for estimating surface errors. Secondly themodel developed here is more generic and treats machinedgeometry as variable curvature curve and not locallyconstant curvature one.

3. Constant curvature model with circulartooth trajectory

In this section, a constant curvature model to estimateprocess geometry in peripheral milling is discussed byconsidering tooth trajectory to be circular. This has beendiscussed by Zhang et al. [26] earlier. They considered thegeometry of workpiece before a cut as reference geometryfor their calculations. Where as geometry of workpieceafter cut is taken as reference geometry for ourcalculations. This deviation is purely for convenienceand has no bearing on results. Moreover, the workpiecegeometry after a cut is often the target geometry in allfinishing operations.

The peripheral milling of variable curvature surface byan end mill is shown in Fig. 1(a). In the figure (Xw(t),Yw(t)),(X(t),Y(t)), (Xt(t),Yt(t)) are three parametric curves whichrepresent workpiece geometry before a cut is taken, work-piece geometry after a cut is taken and locus of tool center,respectively. (Xw(t),Yw(t)) and (Xt(t),Yt(t)) are offset curvesto the finished workpiece geometry (X(t),Y(t)) at offsetdistances equal to radial depth of cut (d) and tool radius (r),respectively [27,28,29]. This can be mathematically


(a) X,(t),Y«(t) (b)

X(t),Y(t)

PrecedingTooth Trajectory

Desired Workpiece Profile

CurrentTooth Trajectory

Current Tooth Trajectory(d)

Preceding ToothTrajectory

Fig. 1. (a) Constant curvature model with circular trajectory. (b) Surface error in constant curvature model. (c) Variable curvature model with circular toothtrajectory. (d) Variable curvature surface machining with true tooth trajectory.

expressed as:

XwðtÞ Z XðtÞ KdY0ðtÞ

XwðtÞ Z XðtÞ K

\/(X'(t)2 + Y'{t)2)'

dX0ðtÞ

\/(X'(t)2 + Y'{t)2)

rY0ðtÞ

\/(X'(t)2 + Y'{t)2)'

rX0ðtÞ

\/(X'(t)2 + Y'{t)2)

(1)

the radius of curvature of curve at point B0 and O is thecenter of corresponding osculating circle. Constant curva-ture model assumes that points A and B which lie on (X(t),Y(t)) have same osculating circle of radius R C r. Similarlypoints C and D which lie on (Xw(t), Yw(t)) have sameosculating circle having radius R C d.

qextcc is the tooth exit angle corresponding to the feedstation B and is equal to the OBD as shown in Fig. 1(a):

(2)Exit angle ðqextccÞ Z arccos

r2 C ðR C rÞ2 KðR C dÞ2

2rðR C rÞ

(3a)

Yt(ta)) and B(Xt(tb), Yt(tb)) are two successive toolpositions on a tool path curve at a distance of nominalfeed rate per tooth. Similarly C(Xw(tc),Yw(tc)) and D(Xw(td),Yw(td)) are the tooth exit points corresponding to toolpositions A and B, respectively. Referring to Fig. 1(a), R is

CD is the feed per tooth along cutter contact path and can beexpressed as

= VðR þ dÞ

ðR þ rÞNnt(3b)


The maximum undeformed chip thickness here is equal to

m a x ccZrKBC

From DBOC

BC2 Z ðR C dÞ2 C ðR C rf - 2ðR C dÞðR C rÞcosða K 2Þ

(4)

Referring to Fig. 1(a), angle AOC is equal to a, where

a Z arccosðR C df +(R C rf - r2

2ðR C dÞðR C rÞ(5)

arc AB subtends an angle at center O which is equal to? Z ðVf=ntNðR C rÞÞ, where Vf is the nominal feed rate; N,the spindle RPM and nt is number of tooth.

The surface error can be measured as a normal distancefrom intersection of current cutting tooth trajectory andprevious tooth trajectory to final workpiece geometry asshown in Fig. 1(b). Expression for surface error (ch) can bewritten in this case as

(6)

qentcc is the tooth entry angle corresponding to a feed stationB and is equal to OBE as shown Fig. 1(a).

Entry angle ðqentccÞ Z arccosr2 C ðR C rÞ2 KðR C chÞ2

2rðR C rÞ

(7)

4. Variable curvature model with circulartooth trajectory

In this section, a more generic approach to estimate theprocess geometry based on circular tooth trajectory ispresented. Unlike in the previous section, where a curvewith continuously varying curvature was dealt usingconstant curvature model with instantaneous radius ofcurvature, the model presented here accounts for curvaturevariation along the curve in order to estimate processgeometry.

Fig. 1(c) shows variable curvature model of peripheralmilling process. Here A(Xt(ta),Yt(ta) and B(Xt(tb),Yt(tb)) arethe successive tool positions at a distance of nominal feedrate per tooth along the tool path curve (the locus of toolcenter). Similarly C(Xw(tc),Yw(tc)) and D(Xw(td),Yw(td)) arethe tooth exit points corresponding to tool positions A and B,respectively. O(Xe(ta

0),Ye(ta0))is center of osculating circle of

the point A0 (X(ta0),Y(ta0)) on the finished workpiece

geometry. When the tool is at A, the corresponding toothexit point C should lie on the initial workpiece geometry asshown in Fig. 1(c). This can be mathematically expressed as

(Xt(O-Xw(tc))2 - Yw(tc)f = r2 (8)

= XðtcÞ K

YwðtcÞ Z YðtcÞ

dY0ðtcÞ

fiX0ðtcÞ2 C Y0ðtcÞ

2Þ

dX0ðtcÞ

cÞ2 C Y0ðtcÞ

2Þ

(9)

The intersection point (Q of the tooth trajectory and initialworkpiece geometry can be obtained by solving the aboveEqs. (8) and (9).

Similarly, the coordinates of successive tooth exit pointD corresponding to tool position B can be calculated. CD isequal to feed rate per tooth along the cutter contact path andit may not be equal to nominal feed rate per tooth (AB) as instraight surfaces machining.

fptvc = CD = ^J(XJtc)-XJtd))2 + (YJtc) - YJtd))

2

(10)

Referring to Fig. 1(c) Exit angle (qextvc) of a toothcorresponding to the feed station A and is equal to OAC

qextvc Z arccos

where

r2 C ðR C rÞ2 K OC2

2rðR C rÞ(11)

oc Z )2 C ðYwðtcÞ K

when the tool is at B, the maximum undeformed chipthickness can be expressed as equation given below:

max vc Z r K BC

where

BC = \J(Xt(th) - XJtc))2 + (Yt(th) - YJtJ)2

The intersection point F(Xi,Yi) of tooth trajectories corre-sponding to the tool position (points) A and B can beobtained by solving Eq. (12) given below

ðXiKXtðtaÞ2CðYiKYtðtaÞÞ2 Z r2

;

ðXiKXtðtbÞÞ2CðYiKYtðtbÞÞ2 Z(12)

The corresponding normal point (intersection of linepassing through F and normal to workpiece geometry withthe workpiece geometry) E(X(t),Y(t)) can be determined bysolving the Eq. (13) given below

i K YðtÞÞ C ðXi K XðtÞÞ Z 0

where b is equal to tangent angle at X(t),Y(t).Then the surface error

(13)

ð c hÞ Z EF Z ðXi K XðtÞÞ2 C ðYi K YðtÞÞ2 (14)


When the tool is at B, one can calculate corresponding toothentry angle (qentvc) as given below

= arccosr2 C ðR C rf - OF2

2rðR C rÞ(15)

parametrically as follows.

Xpa = *t(O + r sin(/3a), 7pa = 7t(fa) - r costf a) (16)

(17)Ypb Z YtðtbÞ K r cosðbb C qÞ

5. Variable curvature model with true tooth trajectory

In this section an attempt is made to model the true toothtrajectory by considering simultaneous rotational andtranslational motions of the tool and to predict processgeometry based on the same.

Fig. 1(d) shows a plan view of an instance of variablecurvature surface machining. Two rectangular coordinatesystems are attached to the center of the end mill. Onecoordinate system (X,Y) moves along with tool center andanother coordinate system (Xf,Yf) represents the instan-taneous feed and cross-feed directions.

When the tool is at A(Xt(ta),Yt(ta)) tooth i is normal tosurface at the start of cut, which correspond to parameter ta

equal to zero. When tooth i rotates by an angle of q withrespect to feed coordinate (Xf, Yf) frame, tool center movesto point B(Xt(tb),Yt(tb)) on tool path curve. During thisperiod (Xf, Yf) coordinate frame has rotated by an angle bb

with respect to XY coordinate frame as shown in Fig. 1(d).For tool being at positions A and B, the correspondingcutting tooth trajectory equations can be expressed

(18)

where DD is the distance traveled by the tool along the toolpath curve; Ft, the nominal feed rate per tooth along toothpath curve, nt, number of tooth.

The curve length between two feed stations A and B canbe written as

S Z 7/2(0) (19)

where, tb can be obtained by equating the distance traveledby tool along the tool path curve given by Eq. (18) to lengthof the curve segment AB Eq. (19).

5.1. Modelling of process geometry based on true toothtrajectory

In this section, modelling of process geometry isdiscussed based on the true tooth trajectory Eq. (17)developed above. Fig. 2(a) shows the true trajectory of

(a)

(c)

bol Path Curve

nital Workpiece curve

Final Workpiece Curve

Current Tooth Trajectory

Tool Path Curve

I nital Workpiece curveTrajectory Of Tooth i+1 T rue Trajectory of tooth i\

Final Workpiece Curve

Trajectory of Tooth i Final Workpiece Curve

Initial Workpiece Curve

True Trajectory of tooth i+1

Fig. 2. (a) Variable curvature surface machining true tooth trajectory exit point. (b) Variable curvature surface machining—intersection of successive teethtrajectory. (c) Variable curvature surface machining—undeformed chip thickness model.


tooth i. When the tooth i exit from the cut, the tool center isat A(Xt(ta),Yt(ta)) on tool path curve and the tooth trajectorypoint (Xp, Yp) should coincide with point C which lies oninitial workpiece geometry. This can be mathematicallyexpressed as

Xp Z Xt qiÞ Z Xw ðtcÞ

= YtðtaÞ K r cosðbi C qiÞ Z YwðtcÞ

ðXtðta Þ 0,) + (Yt(tJ

- rw(fc))sin(ft + /?,•) = 0

(20a)

(20b)

When the tool is at A, tooth i makes an angle of qi with thefeed coordinate system and instantaneous feed directionmakes an angle bi with the X, Y coordinate system as shownin Fig. 2(a).

The exit angle (qi) of tooth i can be obtained by solvingEq. (20b) by numerical procedure. Once qi is known,one can calculate the tooth exit point C(Xw(tc),Yw(tc)) fromthe Eq. (20a). The exit angle and the tooth exit pointD(Xw(td),Yw(td)) of the next tooth (i C 1) can be determinedin similar way. Once points C and D are known feed pertooth along cutter contact path can be calculated asdiscussed in Section 4.

Fig. 2(b) shows the intersection of two successivetooth (i, iC1) trajectories; where A(Xt(ta,i),Yt(ta,i)) andB(Xt(tb,iC1),Yt(tb,iC1)) are the tool center positions corre-sponding to these tooth trajectories. The points E and F aretrajectory intersection point and the corresponding normalpoint on final workpiece geometry, respectively.

The trajectory of tooth i C 1 can be expressed as

Xtðtb;iC1Þ K r cosðbb;iC1 C qb;iC1Þ;

Ytðtb;iC1Þ K r sinðbb;iC + 0W+1)(21)

When the tool is at B, tooth i C 1 makes an angle of qb,iC1with the feed coordinate system and instantaneous feeddirection makes an angle bb, iC1 with the X, Y coordinatesystem as shown in Fig. 2(b).

Similarly trajectory of tooth i, can be written as

+ r sin(/3a>1. + 6J),(22)

When the tool is at A, tooth i makes an angle of 0&i with thefeed coordinate system and instantaneous feed directionmakes an angle ba ,i with the X, Y coordinate system asshown in Fig. 2(b).

Point of intersection of the two successive tooth (i,iC 1)trajectories can be obtained by solving the given belowequation:

cosðba;iCq a ;

+ sinðba;iC - Yt(th4)

By solving for qb,iC1 from above equation, entry angle (qent)of tooth i C 1 can be obtained by following relation:

2ub,i+\

It can also be seen from the figure that EF is surface error,which can be calculated by Eqs. (13) and (14) as discussedin Section 4.

5.2. Instantaneous undeformed chip thickness model

Fig. 2(c) shows an instance of tooth trajectories oftwo successive teeth i, iC1. When the tool is atB(Xt(tb,iC1),Yt(tb,iC1)), tooth i C 1 trajectory point is atM whose coordinates can be expressed as

Xtðtb;iC1Þ C r

Yt(thJ+i) - Sb,i+\)(24a)

When the tool is at B, tooth i C1 makes an angle ofqb,i C 1 with the feed coordinate system and instantaneousfeed direction makes an angle /3b! + 1 with the X, Ycoordinate system as shown in Fig. 2(c).

Line MB intersecting with the workpiece surface (pathcurve) left by the previous tooth i at point N and thedistance MN is true value of the instantaneous unde-formed chip thickness. Then the tooth i trajectory point(N) to be determined. Assume that the tool is located at(Xt(ta,i),Yt(ta,i)) when the previous trajectory point oftooth i (Xp,j,Ypj) coincides with N and the trajectory pointcan be written as follows.

r sinðba;iC

r cosðba;

(24b)

When tool is at position A, tooth i makes an angle of 0&i

with the feed coordinate system and instantaneous feeddirection makes an angle ba,i with the X, Y coordinatesystem as shown in Fig. 2(c).

The distance traveled by the tool from position A to B isequal Dab and can be written as

' 'ab

where V Z tooth space angle Z 2p/nt.As the points M, N, B are located on the same line, hence

the following equation can be obtained based on thecollinearity condition.

Xt(tbi+i)(24c)

By substituting the Eqs. (24a) and (24b) in above Eq. (24c)and can be rewritten as

= 0 ð23Þ

tan(/3b > m + 0h

- ðXtðtb;iC1Þ K

r cosðba;iCq a ;iÞÞ

;iÞ Z 0 ð25Þ


Once d^i is calculated known from the above equation, theinstantaneous true undeformed chip thickness (hinst) can bedetermined from the given below equation:

5.3. Calculation of maximum undeformed chip thickness

The undeformed chip thickness will be the maximum,when the current cutting tooth i C1 passes through theexit point of previous cutting tooth i. Points B0 (Xt(tb, i C 1 ) ,t b , i C 1 ) ) , C(Xw(tc),Yw(tc)) and E(XpiC1, YpiC1) are thecurrent tool center position, previous cutting tooth exit pointand current tooth coordinates respectively as shownFig. 2(c). Then the Eq. (25) can be reduced to followingbelow given equation:

ð26Þ

tan(/3b>1.+1 + eh4+i)(Yt(th4+i) - YJtJ)

- ðXtðtb;iC1Þ K XwðtcÞÞ Z 0

(a)

-5

•1D

Tool Path

Final Workpiece Geometrynilal Workpiece Geometry

qb,i C 1 c a n be determined by solving above Eq. (26) bynumerical procedure. Once 6hi+i is known, cutting tooth(iC1) coordinates E can calculate by Eq. (24a) and themaximum undeformed chip thickness hmax can be deter-mined from the following expression:

hnmx = CE= J(Xw(tc)-Xpi+lf + (Yv(tc) - Ypi+lf

6. Results

Simulation studies were carried out to model theperipheral milling process for many curved geometries.This included estimation of feed per tooth-CC, entry andexit angle, maximum undeformed chip thickness andsurface error for machined geometries. Workpiece geo-metries selected include both convex and concave surfaces(Fig. 3). Effect of workpiece curvature on process

10 15 20 25 30 35 4D 45 50 55X - mm

(b)

Y-m

m

14

12

10

8

6

4

-True Trajectory

- \L

2 4 6

y"'e io 12

X- mm

• *

Tool Path

* ' \ ' * ' Desired Proiile

Workpiece

14 16 18 20

(c)

J-J

-10

Final Workpiece Geometry Inital Warkpiece Geometry

10 15

Tool Path

20 25 30 35

X mm

40 45 50

(d)1*3\£

10

8

6

£ 4

> 2

0

-2

.4

Desired Profile- — • — ^

A \

10 15 2D

..••V

\

I \ WorftpiO"

/ Tool Psih

AT̂rue Trajectory

25 X 35

X -mm

* • .

40

Fig. 3. (a) Initial and final workpiece geometry—convex. (b) True tooth trajectory for convex shape workpiece. (c) Initial and final workpiece geometry—concave. (d) True tooth trajectory for concave shape workpiece.

ARTICLE IN PRESSV.S. Rao, P.V.M. Rao /International Journal of Machine Tools & Manufacture xx (2004) 1-14

variables and surface error were studied using threedifferent models discussed in Sections 3-5. Summary ofthese simulation results are presented in this section.To discuss the results a representative geometry consisting

(a)

.2 4 5

2S

/Variable Curvature

Constant Curvature

Cl 02 004 006 008Curvature

0 1 0 12 0.14 016

(C)0 05

DD33

Variable Curvature

002 0.04 0.06 DOS 0 1 012 0 U 0.16Curvature

of logarithmic spiral is chosen here. This is a curvewhose curvature varies continuously along its path. Fig. 3shows the curve, tool path and tooth trajectory for thesame.

Variable Curvature

(b)

E

Rad

iili

e

S7=

xlt

*

LJ

066

064

062

06

058

056

0 5 4

as

0-5

0 48

(d)

003

0 029

IMa

6gooses

0025

0O34

0033

Censtant Curvature

DID 0 04 006 00B 01 017 014 0 16 016Curvature

DOS 0 04 ODE 0 08 01Curvature

0 12 0 14 0 16

t QOCCOB —

S

Fig. 4. (a) Entry angle vs curvature—convex. (b) Exit angle vs curvature—convex. (c) Feed per tooth along cutter contact path vs curvature—convex. (d)Maximum chip thickness vs curvature—convex. (e) Surface error vs curvature—convex.


The parametric representation of logarithmic spiral isgiven as:

XðtÞ Z a sinðtÞebt; 7 (0 = a cosðtÞebt (27)

6.1. Constant curvature model vs variable curvature model

As discussed earlier, machined geometries for whichcurvature varies continuously can be dealt by two differentapproaches for the purpose of calculation of processgeometry variables and surface error. In one case constantcurvature model is used along the path of machinedgeometry considering instantaneous curvature as discussedin Section 3. In the second case more generic variablecurvature model is used for calculation of process variablesand surface errors as discussed in Section 4. Fig. 4 shows thecomparison of these two models for convex and concavesurfaces. The results presented here by assuming tool radius(r) as 5 mm, radial depth of cut (d) as 1 mm, nominal feedrate per tooth as 0.05 mm, number of tooth to be 4 andspindle rpm to be 500.

Both entry and exit angles (Fig. 4(a) and (b)) decreaseswith workpiece curvature for convex type of surfaces andincreases for concave type surfaces (Fig. 5(a) and (b)). Asthe process chosen here is an up milling process thevariation in entry angles is insignificant using two models.However, the difference in exit angles by two models isconsiderable when the workpiece curvature is higher. Asimilar trend is observed in feed per tooth along cuttercontact path and maximum undeformed chip thickness asshown in Fig. 4(c) and (d) for convex and reverse trend in(Fig. 5(c) and (d)) concave surfaces. Figs. 4(e) and 5(e)shows the variation of surface error with workpiececurvature for both convex and concave type of surfaces.The error in the prediction of surface error by this model ishigher at low curvature portions of the surface. It can beobserved from the above discussion, the constant curvaturemethod either underestimate or over estimate the processvariables. Hence, it is suggested that the variable curvaturemodel is more suitable than the constant curvature model forpredicting the process variables and in estimating thecutting forces when machining curved geometries.

6.2. Trochoid vs circular path trajectories

Modelling of these two types of tooth trajectories werediscussed in Sections 4 and 5, respectively. Figs. 6(a),(b)and 7(a),(b) shows the variation of entry and exit angleswith workpiece curvature for convex and concave surfacesrespectively and both trajectories estimate the tooth exitangle very closely and deviation between them is negligible.The entry angle is decreasing with curvature, as in upmilling the entry angle is very small and negligible value.Figs. 6(c), (d) and 7(c), (d) shows the variation of feed pertooth along cutter contact path, maximum undeformedchip thickness for convex and concave type surfaces,

respectively, based on circular tooth trajectory and truetooth trajectories. From the Figs. 6(c) and 7(c) observed thatboth circular and true trajectories estimate the feed per toothalong cutter contact path approximately same value.Variation of maximum undeformed chip thickness isshown in Figs. 6(d) and 7(d) and the deviation betweenthe two approaches is increasing with the workpiececurvature. Plots 6(e) and 7(e) show the variation of surfaceerror for both convex and concave type surfaces and thedeviation is decreasing with workpiece curvature.

Results shown in figures indicate that modelling of truetooth trajectories is necessary for accurate modelling ofprocess variables in peripheral milling of curved geome-tries. It also shows that consideration of convex or concavetype of surface is important for process planning forperipheral milling process.

7. Experimental results

Cutting forces in any milling process depends onundeformed chip thickness. A better model for unde-formed chip thickness was discussed and derived inprevious sections. As the chip thickness was found todepend strongly on workpiece curvature values, avariation of cutting forces along cutter contact path canbe expected. To investigate this aspect, cutting forcemeasurement experiments were conducted for machiningcurved geometries, which included both convex andconcave type of surfaces (Fig. 3). Forces were measuredusing three-component pizeo-electric type of dynam-ometer. The work material chosen here was 7075-T6aluminum alloy. The experiments were performed usingan end mill of radius of 5 mm, four flutes, 50 mm inlength and having an helix angle of 308. Machining wasdone at constant spindle rpm 500, axial depth of cut of10 mm, radial depth of cut of 1 mm and feed per toothalong tool path as 0.05 mm were chosen. Unlike turningprocesses, in milling the instantaneous undeformed chipthickness varies periodically as a function of feed rateper tooth and tool engagement angle. The chip thicknesscan be expressed as

hðqÞ Z fpt ! sin q

The tangential (Ft), radial (Fr) cutting forces acting onworkpiece are expressed as a function of varying uncutchip thickness and given by

Ft(8) = Kth(0\ Ft(8) = KtFt(8)

The specific cutting pressure (Kt) and ratio of radial andtangential forces (Kr) are expressed as nonlinear func-tions of average chip thickness as follows

K = K h^ K = K h^

ARTICLE IN PRESS10 V.S. Rao, P.V.M. Rao / International Journal of Machine Tools & Manufacture xx (2004) 1-14

(a) 0022

002

0.T6

014

0.12

S 0.191

ooe

D,OE

0-O4

1 5

1.4

I «*,2

Variable Curvature

Constant Currature

ace o.a* ore o.oe o i o 12 B.H D.ISCurvature

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Fig. 5. (a) Entry angle vs curvature—concave. (b) Exit angle vs curvature—concave. (c) Feed per tooth along cutter contact path vs curvature—concave. (d)Maximum chip thickness vs curvature—concave. (e) Surface error vs curvature—concave.

where constants KT, KR,p and q are determined experimen-tally for a tool-workpiece material pair. The instantaneousresultant cutting force on the workpiece is given as

Fig. 8 shows the resultant cutting force in the cuttingplane. Resultant cutting force is increasing in machiningof convex surface geometry with decrease in curvatureand is well agreeing with the trends of increasing feed pertooth along cutter contact path and maximum uncut chipthickness (Figs. 4(c),(d), 5(c),(d), 6(c),(d) and 7(c),(d)).

ARTICLE IN PRESS

V.S. Rao, P.V.M. Rao / International Journal of Machine Tools & Manufacture xx (2004) 1-14 11

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Similarly a reverse trend was observed in resultant cuttingforce variation for concave surfaces which matches withthe simulation results for feed per tooth and uncut chipthickness.

8. Conclusions

This paper presents a study of the peripheral millingprocess for variable curvature surfaces. Based on the results


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Fig. 7. (a) Entry angle vs curvature—concave. (b) Exit angle vs curvature—concave. (c) Feed per tooth along cutter contact path vs curvature—concave. (d)Maximum chip thickness vs curvature—concave. (e) Surface error vs curvature—concave.

of the study, the following conclusions can be stated withrespect to peripheral milling of curved geometries:

• In modelling of process variables such as entry and exitangle, feed per tooth along cutter contact path,

maximum chip thickness and surface error, it isnecessary to use a model which considers variation ofworkpiece curvature. Extending constant curvaturemodels to variable curvature surface could be erroneous.

ARTICLE IN PRESSV.S. Rao, P.V.M. Rao / International Journal of Machine Tools & Manufacture xx (2004) 1-14 13

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Fig. 8. (a) Total cutting forces in convex surface machining. (b) Totalcutting forces in convcave surface machining.

• It is necessary to model true cutter tooth trajectoriescontrary to simple circular trajectories particularly forcurved geometries.

• Process planning of peripheral milling of curvedgeometries should consider convex or concave type ofregions in a machined surface. Process variables differconsiderably in both the cases for a chosen processparameter values such as speed, feed and depth of cut.

The results of this paper will be useful in processplanning for machining of curved workpieces. The results ofthis research show up to 10% variation in predicted valuesfor process variables by simple but less accurate modelsfound in literature as compared to an enhanced modelpresented in this paper. When machining curved geometriescorrect estimation of process variables will help in correctestimation of cutting forces as well as variation of cuttingforces along tool path.

The correct estimation of cutting forces is important forcalculating tool deflections and workpiece deflections incase of machining with slender tools and flexible work-pieces, respectively. This is particularly true in case ofmachining of aircraft structural components and die/moldmachining where machining of curved geometries is morecommon. An improved method to find variation of processvariables and hence cutting forces with workpiece curvature

will help in improving surface characteristics of machinedworkpieces.

References

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[25] B.W. Ikua, H. Tanaka, F. Obata, S. Sakamoto, Prediction of cuttingforces and machining error in ball end milling of curved surfaces-I.Theoretical analysis, Int. J. Precision Eng. Nanotechnol. 25 (2001)266-273.

[26] L. Zhang, L. Zheng, Z.-H. Zhang, Y. Liu, Z.-Z. Li, On cutting forcesin peripheral milling of curved surfaces, Proc. Inst. Mech. Engrs PartB 216 (2002) 1385-1397.

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