accurate 3-d cutting force prediction using cutting condition independent coefficients in end...

International Journal of Machine Tools & Manufacture 41 (2001) 463–478

Accurate 3-D cutting force prediction using cutting conditionindependent coefficients in end milling

Won-Soo Yun, Dong-Woo Cho*

Department of Mechanical Engineering, Pohang University of Science and Technology, San 31 Hyoja-dong, Nam-gu, Pohang, Kyungbuk 790-784, South Korea

Received 30 May 2000; accepted 25 September 2000

Abstract

The instantaneous uncut chip thickness and specific cutting forces have a significant effect on predictionsof cutting force. This paper presents a systematic method for determining the coefficients in a three-dimen-sional mechanistic cutting force model—the cutting force coefficients (two specific cutting forces, chipflow angle) and runout parameters. Some existing models have taken the approach that the cutting forcecoefficients vary as a function of cutting conditions or cutter rotation angle. This paper, however, considersthat the coefficients are affected only by the uncut chip thickness. The instantaneous uncut chip thicknessis estimated by following the movement of the position of the center of a cutter. To consider the sizeeffect, the present method derives the relationship between the re-scaled uncut chip thickness and thenormal specific cutting force,Kn with respect to the cutter rotation angle, while the other two coefficients—frictional specific cutting force,Kf and chip flow angle,qc—remain constant. Subsequently, all the coef-ficients can be obtained, irrespective of cutting conditions. The proposed method was verified experimen-tally for a wide range of cutting conditions, and gave significantly better predictions of cutting forces.2001 Published by Elsevier Science Ltd.

Keywords:Cutting force; Constant cutting force coefficients; Uncut chip thickness; Size effect

1. Introduction

The enhancement of productivity and the reliability of manufacturing systems have becomemore and more important in modern industry. Adequate prediction of machining performance canimprove selection of correct machining conditions, save operation time, reduce waste, and

* Corresponding author. Tel.:+82-562-279-2171; fax:+82-562-279-5899.E-mail address:[email protected] (D.-W. Cho).

0890-6955/01/$ - see front matter 2001 Published by Elsevier Science Ltd.PII: S0890-6955(00)00097-3

464 W.-S. Yun, D.-W. Cho / International Journal of Machine Tools & Manufacture 41 (2001) 463–478

Nomenclature

f cutting edge location angletc uncut chip thicknessDa height of az-axis axial disk elementft feed per toothi,j,k flute, cutter rotation, and axial disk element index, respectivelyq(j) cutter rotation angle atjfc flute spacing angleDq increment of cutter rotation angleqh helix anglear rake angleR,D cutter radius and diameter, respectively

dF→

rb,dF→

r normal and frictional force acting on the rake surface, respectivelyFx,Fy,Fz orthogonal force components in Cartesian coordinatesKn,Kf,qc cutting force coefficients—normal specific cutting force, frictional specific cutting

force, and chip flow angle, respectivelyda,dw depth of cut and width of cut, respectivelytcn re-scaled uncut chip thicknesstcg,max,tcg,min maximum and minimum uncut chip thickness, respectively, from the

geometric uncut chip thickness modelr,arun radial runout offset and its location angle, respectivelyrcri upper limitation for a radial runout offset, 20µm in this studynf,nz number of a flute and an axial disk element, respectivelynq 360°/Dθftx,fty feed per tooth inx,y directions, respectively

decrease the time required for test cuts in NC program verification. An accurate model for thecutting forces is essential to analysis and prediction of machining performance.

The cutting forces in end milling have often been considered as the product of a specific cuttingforce and uncut chip thickness. The uncut chip thickness can be separated into purely geometricaland physical aspects. The basic uncut chip thickness relationship based purely on geometry wasgiven by Eq. (1) in Martelloti’s work [1,2].

tc(f)5ft sinf (1)

The physical aspects of uncut chip thickness include the effects of cutter deflection, runout,vibration, and so forth. Kline et al. [3] and Kline and DeVor [4] presented a mechanistic cuttingforce model considering the effect of cutter runout. Sutherland and Devor [5] added the effect ofcutter deflection to the uncut chip model taken from Refs. 3 and 4 and predicted the machinedsurface error as well as cutting forces. These mechanistic cutting force and uncut chip modelshave also been applied to the ball end milling process [6,7].

465W.-S. Yun, D.-W. Cho / International Journal of Machine Tools & Manufacture 41 (2001) 463–478

On the other hand, numerous investigators have studied and modeled specific cutting forces.Kline et al. [3], Kline and DeVor [4], and Sutherland and Devor [5] assumed that the specificcutting forces are functions of cutting conditions, and the values were calculated using the averagecutting forces during one revolution of a cutter. Yucesan et al. [8] calibrated the specific cuttingforces and chip flow angle relationships using the forces measured at many discrete angular pos-itions of the cutter. Budak et al. [9] suggested a method that predicts the coefficients from orthog-onal cutting data.

In addition to the research related to uncut chip thickness and specific cutting forces, the sizeeffect should also be included in a mechanistic cutting force model, to obtain more accurateprediction results [10]. The size effect refers to the increase in a specific cutting force at smalluncut chip thickness. Boothroyd and Knight [11] explained the size effect as the forces acting onthe rounded edge or the flank face of a cutter, referred to collectively as the plowing force.Elbestawi et al. [12] explained the plowing force as a damping force generated at thetool/workpiece interface. To consider the size effect, the relationship between a specific cuttingforce and uncut chip thickness in Eq. (2) was proposed by Sabberwal [13] and later used by manyresearchers [7,10,14,15].

KT5b(tc)p (2)

whereKT is the specific cutting force andb andp are the constants that depend on the workpiecematerial and the milling cutter.

The cutting forces during machining operations are often predicted from empirical equations.The required constants or parameters for these equations are determined experimentally. Thesetechniques are useful and necessary, but the resulting equations and parameters are often limitedto the particular operation and conditions tested. In die and mold machining, where the cuttingconditions vary widely, a prohibitive amount of test cuts may be needed to determine the para-meters.

This paper presents a systematic procedure by which the coefficients in the empirical equationsfor a given workpiece and cutter can be determined, regardless of cutting conditions. The instan-taneous cutting forces and uncut chip thickness are used to determine the coefficients, and foraccurate cutting force prediction the size effect is considered.

2. Model formulation

2.1. Mechanistic cutting force model

The end milling cutter can be divided into a finite number of disk elements and the totalx-,y-, and z-force components acting on a flute at a particular instant are obtained by numericallyintegrating the force components acting on an individual disk element. Finally, a summation overall flutes engaged in cutting yields the total forces acting on the cutter at that time.

Fig. 1 shows schematic views of an end milling process geometry and coordinate system. Thecutter geometry and coordinate systems adopted in the present study are shown in Fig. 2. Theangular position of thekth axial disk element of theith flute at thejth angular position of thecutter is given by Eqs. (3) and (4):


Fig. 1. Schematic views of the basic end milling process geometry and coordinate.

f(i,j,k)5q(j)1(i21)fc1(kDa1Da/2)tanqh

R(3)

q(j)52jDq (4)

From previous works [8,16], the three orthogonal force components in Cartesian coordinates

can be derived as follows from the expressions of the normal (dF→

n(f)) and frictional (dF→

r(f))forces as follows:

Fx(i,j,k)5[C1Kncos(f2ar)1KfKnC3cosf2KfKnC4sin(f2ar)]tc(f)B1

Fy(i,j,k)5[C1Knsin(f2ar)1KfKnC3sinf1KfKnC4cos(f2ar)]tc(f)B1 (5)

Fz(i,j,k)5[2C2Kn1KfKnC5]tc(f)B1

where

C15cosqh/sinqtk, C25sinqh/sinqtk, C35sinqh(sinqc2cosqccotqtk), C45cosqc/sinqtk,

C55cosqh(sinqc2cosqccotqtk), B15cosar(Da/cosqh), cosqtk5sinarsinqh.

Subsequently, the cutting force components at an arbitrary rotation angle can be evaluated byEqs. (6)–(8).


Fig. 2. Cutter geometry, coordinate system, and unit vectors on the rake surface.

Fx(j)5Ok

Oi

Fx(i,j,k)5B1C1K1Ok

Oi

cos(f2ar)tc(f)1B1K2Ok

Oi

cosftc(f) (6)

2B1K3Ok

Oi

sin(f2ar)tc(f)

Fy(j)5Ok

Oi

Fy(i,j,k)5B1C1K1Ok

Oi

sin(f2ar)tc(f)1B1K2Ok

Oi

sinftc(f) (7)

1B1K3Ok

Oi

cos(f2ar)tc(f)


Fz(j)5Ok

Oi

Fz(i,j,k)52B1C2K1Ok

Oi

tc(f)1B1K2cotqhOk

Oi

cosftc(f) (8)

whereK1=Kn, K2=KnKfC3 andK3=KnKfC4.The above equations can be rewritten in a matrix form as:

5Fx(j)

Fy(j)

Fz(j)653A11 A12 A13

A21 A22 A23

A31 A32 A3345K1(j)

K2(j)

K3(j)6 (9)

2.2. Uncut chip thickness model

In this paper the uncut chip thickness will be calculated as a function of the position of thecenter of the cutter for eachz-axis disk element. The position of the center of the cutter deviatesfrom its nominal positions due to numerous factors, such as cutter deflection, runout, servo error,volumetric error, thermal error, wear, etc. Of all the factors, cutter deflection and runout accountfor the most deviation. Thus, the actual position of the center of the cutter, (xa,ya), can be realisti-cally represented as:

xa(j,k)5xn(j)1xr(j)1xd(j,k) (10)

ya(j,k)5yn(j)1yr(j)1yd(j,k)

This representation can be used to effectively predict the cutting force in contour machiningoperations that occur frequently in end milling processes.

2.2.1. Nominal cutter position (xn,yn)The nominal position of the cutter in thex and y directions in terms of the nominal feed per

tooth is given by:

xn(j)5xn(j21)1ftxnf

nq(jDq) (11)

yn(j)5yn(j21)1ftynf

nq(jDq)

2.2.2. Deviation caused by cutter runout (xr,yr)The deviation of the position of the center of the cutter caused by cutter runout is defined as:

xr52r sin(arun2q(j))

yr52r cos(arun2q(j)) (12)


The runout offset,r, can be easily measured by a dial gauge or other suitable device, whereasits location angle,arun, is difficult to measure. This angle determines the initial state of the pre-dicted cutting forces.

2.2.3. Deviation caused by cutter deflection (xd,yd)The cutter deflection model is based on cantilever beam theory. Young’s modulus and the

moment of inertia can be taken asE=217 GPa andI=D4/48, respectively [17].The uncut chip thickness can therefore be estimated from the position of the center of the cutter

[5]. The uncut chip thickness is the distance between the path that the current tooth of interestis generating, and the exposed workpiece surface generated by the previous tooth. If the runoutparameters are known, theAij components in Eq. (9) can be determined. Thus,Kn, Kf andqc cannow be calculated with respect to a cutter rotation angle by substituting the measured cuttingforces in the left-hand side of Eq. (9), as will be elaborated in the next section.

3. Determination of the cutting force coefficients

3.1. Pre-process for calculating the coefficients

To calculate the cutting force coefficients, the cutting forces for one revolution are needed frommeasured data to substitute in the left-hand side of Eq. (9). However, different samples of meas-ured cutting force data will give different coefficient values, because the simulation begins topredict the cutting forces when the first bottom tooth is in the zero angular position. Thus, toobtain unique cutting force coefficients, cutting forces sampled at the same angular positions asthose in the simulation are needed, which is here termed ‘synchronization’.

On the other hand, the runout parameters are needed to calculate the uncut chip thickness.Considering thatKn, Kf andqc can conceptually be defined as constant values for a given cutterand workpiece, the runout parameters can be determined by choosing values that best maintainKn, Kf andqc constant in the mechanistic cutting force model. The algorithm for estimating therunout parameters is outlined in Fig. 3. After calculating the runout parameters using the synchron-ized cutting forces,Kn, Kf and qc can be obtained. A detailed description for the pre-process isgiven in [16].

3.2. Size effect in end milling

3.2.1. Size effectTo determine the cutting force coefficients and verify the model, a set of 20 steady state cutting

tests was carried out; the relevant cutting conditions are listed in Table 1. All test cuts wereperformed with high-speed steel (HSS) end mills with four flutes, 30° helix angle, 11° rake angleand 10 mm diameter in a vertical type machining center (Daewoo Heavy Industries Ltd., ACE-V30). The workpiece material was aluminum 2014-T6. A tool dynamometer (Kistler, type 9257B)was used to measure the three cutting force components. The cutting forces measured in Test 17will be used to determine the cutting force coefficients, and others will be used to verify theproposed scheme. After the pre-process, the calculated values ofKn, Kf and qc with respect to


Fig. 3. Algorithm for the selection of runout parameters.

the cutter rotation angle for three test cuts are shown in Figs. 4(a)–(c), respectively. In thesefigures, the values ofKn, Kf andqc for some test cuts have a similar shape and magnitude evenunder different cutting conditions. The cutting conditions can therefore be considered not to haveany significant effect upon the values of cutting coefficients.

In order to investigate the relationship between the uncut chip thickness andKn, their valuesobtained from Test 3 and Test 17 are presented in Figs. 5(a) and (b), respectively. The resultsshow thatKn tends to become large for small uncut chip thickness values, and this is referred toas the size effect.


Table 1Cutting conditions for test cuts

Depth of cut Width of cut Feed Spindle rpmda (mm) dw (mm) ft (mm/tooth) N (rev/min)

Test 1 2.0 5.0 0.03750 1000Test 2 4.0 5.0 0.03750 1000Test 3 5.0 5.0 0.03750 1000Test 4 6.0 5.0 0.03750 1000Test 5 8.0 5.0 0.03750 1000Test 6 5.0 1.0 0.03750 1000Test 7 5.0 2.5 0.03750 1000Test 8 5.0 4.0 0.03750 1000Test 9 5.0 6.0 0.03750 1000Test 10 5.0 7.5 0.03750 1000Test 11 5.0 9.0 0.03750 1000Test 12 5.0 5.0 0.02500 1000Test 13 5.0 5.0 0.03000 1000Test 14 5.0 5.0 0.03500 1000Test 15 5.0 5.0 0.04000 1000Test 16 5.0 5.0 0.04500 1000Test 17 5.0 5.0 0.05000 1000Test 18 5.0 5.0 0.03750 800Test 19 5.0 5.0 0.03750 1200Test 20 5.0 5.0 0.03750 1500

3.2.2. Re-scaling of uncut chip thicknessTo strictly consider the size effect requires the relationship between the cutting force coef-

ficients and uncut chip thickness at everykth axial disk element of theith flute at thejth angularposition of the cutter in Eq. (5). Althoughtc(i,j,k), which is the same astc(f), can be evaluatedfrom the uncut chip model, the values ofFx(i,j,k), Fy(i,j,k) and Fz(i,j,k) can not be obtained.Cutting forces measured using a tool dynamometer can not be decomposed intoFx(i,j,k),Fy(i,j,k) and Fz(i,j,k), and are available only in the formFx(j), Fy(j) and Fz(j). This makes the‘strict’ approach practically impossible.

As an approximate approach, it is reasonable to facilitatetc(j) to evaluate the cutting coef-ficients, since a relationship betweentc(j) andKn(j) is obvious in the results of Figs. 5(a) and (b).

Direct application of the relation of Eq. (2), however, causes variation ofKn(j) with cuttingconditions, sincetc(j) in the end milling process is the sum of the uncut chip thickness removedby each disk element at cutter rotation angleq(j). This is not desirable, and we have been endeav-oring to obtain coefficient values that do not depend on cutting conditions, but depend only oncutter geometry (rake angle, helix angle, etc.) and the material properties of the cutter and theworkpiece. The feasibility of this endeavor can be ascertained by the results in Figs. 4(a)–(c).The effects of cutting conditions can be excluded by re-scalingtc(j) as follows:

tcn(j)5tc(j)

(tcg,max−tcg,min)(13)


Fig. 4. Comparison of the cutting coefficients and uncut chip thickness for some test cuts: (a) comparison ofKn, (b)comparison ofKf, and (c) comparison ofqc obtained from different test cuts.

where tcn(j) is the re-scaled uncut chip thickness and thetcg,max and tcg,min are derived from thepurely geometrical uncut chip thickness model of Eq. (1) as follows:

tcg,min5minFOk

Oi

ft sin(f)G tcg,max5maxFOk

Oi

ft sin(f)G (14)

3.2.3. Relationship between tcn(j) and Kn(j)The relationship betweentcn(j) and log scaledKn(j) can readily be derived in the form of the

Boltzman function of Eq. (15), which is a nonlinear curve fitting function:

ln(Kn)5A1−A2

1+e(tcn−x0)/dx1A2 (15)

The fitted results for Test 17 are as follows:

A1512.9122,A256.73768,x0520.62399, dx50.44168

Fig. 6 shows ln(Kn(j)) individually calculated from the measured cutting forces with respect


Fig. 5. Relation between uncut chip thickness andKn: (a) for Test 3; (b) for Test 17.

to tcn(j), together with the fitted ln(Kn(j)) for Test 17. It can be seen that ln(Kn(j)) and tcn(j) havea mutual relationship and that the Boltzman function fits the values in Fig. 6 well.

On the other hand,Kf(j) andqc(j) with respect totcn(j) are shown in Figs. 7(a) and (b), respect-ively. No obvious relationships are observed when compared with the case ofKn(j). Thus, itwould not be unduly determined ifKf(j) and qc(j) be assumed to be constant. In this study, theaverage value ofKf(j) and qc(j) for Test 17 were taken as the constantsKf and qc. The Kf andqc so obtained and the runout parameters (the offset and its location angle) estimated by the pre-process are listed in Table 2.

The fitted values ofKn(j) together with the original values with respect to rotation angle arepresented in Fig. 4(a). The determinedKf andqc are shown together with the original values inFigs. 4(b) and (c), respectively.

4. Prediction results of cutting forces

To verify the proposed method of determining cutting coefficients, predictions of cutting forceswere made for various test cuts. In all the predictions, the values listed in Table 2 were used for


Fig. 6. Comparison ofKn values calculated from the pre-process and those fitted by Boltzman function with respectto the re-scaled uncut chip thickness.

the values ofKf, qc and the runout parameters, andKn(j) was derived for each test from Eq. (15).The x-, y-, andz-cutting force components were obtained from Eqs. (6)–(8) using these values.Fig. 8(a) shows the results predicted by the two different methods and the cutting forces measuredin Test 3. The force predicted considering the size effect is presented together with that predictedusing the constantKn. The former is in better agreement than the latter with the measured valuesof all three of the force components. Figs. 8(b) and (c) compare the predicted and measuredresults for different widths of cut (Test 7) and for different depths of cut (Test 1), respectively.In Fig. 8(d), the results for higher feed rates are shown. Fig. 8(e) compares the predicted andmeasured results for different cutting speeds (spindle rpm). From the figures it can be seen thatthe predicted forces are in good agreement with the measured values in both magnitude and shape,regardless of change in cutting conditions.

The last example is to illustrate the feasibility of the proposed method in transient cuttingprocess. Fig. 9 shows a schematic diagram of the transient cut occurring when a cutter engageswith and disengages from a workpiece, which is a common process in real cutting. The cuttingconditions are identical with those in Test 3 except thatdw varies as the cutter proceeds. All theparameters and cutting coefficients used in the simulation are the same as the previous test cuts.The start position of the cutter center (x0,y0) was set as (0, 0) in the simulation. The measuredand predicted cutting forces for the full cutting range are presented in Figs. 10 and 11, respectively.The predicted forces are again in good agreement with the measured ones in both shape andmagnitude, regardless of continuous changes in the width of cut.

Overall, the test results indicate that the proposed method is effective and accurate. This wasalso confirmed by measuring cutter runout offset using a dial gauge. The measured value was inthe range 5–6µm, which corresponded remarkably well with that estimated (6µm).


Fig. 7. Kf andqc with respect to the re-scaled uncut chip thickness. (a) ForKf and (b) forqc.

Table 2Estimated cutting force coefficients and runout parameters

Kf qc (rad.) r (µm) arun (deg.)

0.8287 0.471 6 95

5. Conclusions

We have developed a systematic method for estimating the 3-D cutting force coefficients andrunout parameters. By implementing the pre-process, the cutting force coefficients can be con-sidered as constant, except for the size effect. The size effect was effectively considered by exclud-ing the effect of cutting conditions. To do so, the uncut chip thickness were re-scaled using thedeviation of the maximum and minimum uncut chip thickness values calculated from the purelygeometric uncut chip model. The relationship betweenKn(j) and the re-scaled uncut chip thicknesswas derived in the form of the Boltzman function, whereasKf(j) andqc(j) were simply averagedfor use as the respective constants. Comparison of the predicted and measured cutting forces


Fig. 8. Comparison of measured and predicted cutting forces: (a) for Test 3; (b) for Test 7; (c) for Test 1; (d) forTest 17; (e) for Test 18.

shows that the proposed method gives more accurate results than previous methods and wellpredicts the cutting forces even in transient cut, where the cutting conditions continuously change.It can subsequently be inferred that the cutting force coefficients can be determined as the materialproperties independent of cutting conditions for a given combination of workpiece and cutter.


Fig. 9. Schematic diagram of the transient cut occurring when a cutter engages with and disengages from a workpiece.

Fig. 10. Measured cutting forces for the full cutting range of Fig. 9.


Fig. 11. Predicted cutting forces for the full cutting range of Fig. 9.

Acknowledgements

The authors would like to thank the Science and Technology Policy Institute (STEPI) of Koreafor supporting this work through the International Joint Project Research Fund (98-I-01-03-A-023).

References

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