International Journal of Machine Tools & Manufacture 51 (2011) 420–427

Contents lists available at ScienceDirect

International Journal of Machine Tools & Manufacture

0890-69

doi:10.1

n Corr

E-m

m.wan@

journal homepage: www.elsevier.com/locate/ijmactool

Short Communication

Effect of cutter runout on process geometry and forces in peripheral millingof curved surfaces with variable curvature

Yun Yang, Wei-Hong Zhang n, Min Wan

Key Laboratory of Contemporary Design and Integrated Manufacturing Technology, Ministry of Education, Northwestern Polytechnical University, P.O. Box 552, 710072 Xi’An,

Shaanxi, People’s Republic of China

a r t i c l e i n f o

Article history:

Received 2 October 2010

Received in revised form

12 January 2011

Accepted 14 January 2011Available online 21 January 2011

Keywords:

Peripheral milling

Curved surfaces

Instantaneous uncut chip thickness

Cutting force

Cutter runout

55/$ - see front matter & 2011 Elsevier Ltd. A

016/j.ijmachtools.2011.01.005

esponding author, Tel./fax: +86 29 88495774

ail addresses: zhangwh@nwpu.edu.cn (W.-H.

nwpu.edu.cn (M. Wan).

a b s t r a c t

This paper presents a novel method for cutting force modeling related to peripheral milling of curved

surfaces including the effect of cutter runout, which often changes the rotation radii of cutting points.

Emphasis is put on how to efficiently incorporate the continuously changing workpiece geometry along

the tool path into the calculation procedure of tool position, feed direction, instantaneous uncut chip

thickness (IUCT) and entry/exit angles, which are required in the calculation of cutting force.

Mathematical models are derived in detail to calculate these process parameters in occurrence of

cutter runout. On the basis of developed models, some key techniques related to the prediction of the

instantaneous cutting forces in peripheral milling of curved surfaces are suggested together with a

whole simulation procedure. Experiments are performed to verify the predicted cutting forces;

meanwhile, the efficiency of the proposed method is highlighted by a comparative study of the

existing method taken from the literature.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Peripheral milling is a widely used material removal process inautomobile, aerospace and die/mold manufacturing industries forroughing and finishing profiled components. In peripheral milling,many undesired issues such as cutter breakage, cutter wear,chatter and surface error are greatly dependent on cutting forces,whose prediction is thus an essential subject of milling simula-tion. Especially, in milling process of workpiece with complexgeometry, the cutting condition is process-dependent and it isimportant to simulate cutting forces under different conditions.

Base on the milling kinematics that the trochoidal path of cuttingpoint could be approximated by a circular path, as proposed byMartellotti [1], many attentions were paid on cutting force modelingmethods. Historically cutting force modeling of the milling processmight be attributed to Koenigsberger and Sabberwal [2]. Later, thechip shearing effect and the edge rubbing effect were consideredin different ways, either by a single coefficient [3–11] or by twoindependent coefficients [12–17]. Together with these typical mod-els, different methods were also proposed to determine the cuttingforce coefficients. For instance, Kline et al. [5], Budak et al. [12] andGradisek et al. [17] treated the cutting force coefficients as constantsthat were calibrated from the measured average cutting forces,whereas Larue and Anselmetti [6] did this using the measured cutter

ll rights reserved.

.

Zhang),

deflection. To reveal the influence of the cutting conditions, Altintasand Spence [7] treated the cutting force coefficients as exponentialfunctions of the average uncut chip thickness and a closed form wasderived. Shi and Tobias [11] expressed the cutting force coefficientsas cubic function of the chip thickness.

Under the above assumptions, it was however found that thedisagreement between the predicted cutting forces and themeasured ones may exist once the cutting condition is differentfrom that of the calibration test [18]. To remedy this, the conceptof instantaneous cutting force coefficients was introduced bymany researchers [19–23]. Among them, Ko et al. [19], and Koand Cho [20]expressed the cutting force coefficients as a Weibullfunction of the IUCT. Wan and Zhang [21] and Azeem et al. [22]treated the cutting force coefficients as an exponential function ofthe IUCT. Alternatively another exponential function was alsoadopted to bridge the cutting force coefficients and the IUCT [23].Besides, it is worth mentioning that by incorporating the effect ofvariable radii along the cutting edge into the calculation proce-dure of cutting forces, Ferry and Altintas [24] and Dombovariet al. [25] simulated the milling process for serrated cutters.

Apart from the above works [1–25] dedicated to the peripheralmilling of straight surfaces, much research effort was also focusedon the cutting force modeling for peripheral milling process ofcurved surfaces, e.g., cylindrical surfaces [26–29]. For instance, bytreating the feed per tooth as the length of the arc between twoneighboring entry points, Zhang and Zheng [26] studied thecutting force variations in peripheral milling of circular cornerprofiles. On the other hand, focus was also put on simulatingthe cutting forces in peripheral milling of variable curvature

Nomenclature

Ts a given sampling time intervalt sampling instant updated with a sampling time

intervalb0 cutter helix angleji,jðtÞ tooth positioning angle related to the jth axial cutting

disk of the ith flute at sampling instant t, which isdefined as the angle measured clockwise from YSðtÞ

direction to the middle point of the concerned diskelement regardless of the influence of b0

hi,jðtÞ instantaneous uncut chip thickness related to the jthaxial cutting disk element of the ith flute

hi,jðt,mÞ possible instantaneous uncut chip thickness relatedto the jth axial cutting disk element of the ith flute

Ki,j,TðtÞ,Ki,j,RðtÞ tangential and radial instantaneous cuttingforce coefficients related to hi,jðtÞ

r,l runout offset and runout location anglenf number of the flutesVf feed rate of the cylindrical end millS spindle speedap,ae axial and radial depth of cutft feed per toothu parameter variable of the theoretical tool pathv parameter variable of the workpiece boundaryRAD nominal cutter radiusRADi,j radius of the ith flute at the jth axial cutting disk

elementRADi�m,j radius of the (i–m)th previous tooth at the jth axial

cutting disk element, ma1 implies the existence ofcutter runout

FX total cutting force along the X direction in the XYZ

coordinate system

FY total cutting force along the Y direction in the XYZ

coordinate systemFXS

total cutting force along the XS direction in the XSYSZS

coordinate systemFYS

total cutting force along the YS direction in the XSYSZS

coordinate systemz axial coordinate of the jth axial disk element of the

ith fluteji,j,enðtÞ cutter entry angle related to the jth axial cutting disk

element of the ith flute at sampling instant t

ji,j,exðtÞ cutter exit angle related to the jth axial cutting diskelement of the ith flute at sampling instant t

KeðuðtÞÞ equivalent curvature of the actual tool path at thesampling instant t

feðuðtÞÞ equivalent feed direction of the cutter at the samplinginstant t

neðuðtÞÞ equivalent exterior normal direction of the actual toolpath at the sampling instant t

faðtÞ feed direction of the cutter at the sampling instant t

naðtÞ exterior normal direction of the actual tool path at thesampling instant t

paðtÞ ¼ ½XaðtÞ YaðtÞ 0�T tool position on the actual tool pathat the sampling instant t

peðuðtÞÞ ¼ ½XeðuðtÞÞ YeðuðtÞÞ 0�T equivalent tool position onthe theoretical tool path at the sampling instant t

pwðvÞ ¼ ½XwðvÞ YwðvÞ 0�T parametric equations of actualworkpiece boundary

LAC, LCD lengths of line segment AC and CDpst and pen start point and end point of a linear tool path

segmentoCTP and RCTP center point and radius of a circular tool path

segment

Y. Yang et al. / International Journal of Machine Tools & Manufacture 51 (2011) 420–427 421

surfaces [30–34]. Their research emphases were paid on effectivedetermination of the process-dependent feed per tooth, entry/exitangles and IUCT, etc.

It is worth noting that the above works [26–34] were con-ducted with the negligence of cutter runout. Actually, as cutterrunout occurs, the feed per tooth, the entry/exit angles and IUCTwill deviate from their nominal values. A literature review showsthat few results are available about the influence of cutter runouton curved surface milling operations although the latter wasinvestigated in-depth for straight surface milling [35–43]. Theuniquely available work is from Desai et al. [44] who incorporatedthe cutter runout into the simulation procedure of cutting forces.In their method, because IUCT, which is an implicit function of cutand workpiece geometry, is obtained by solving a set of nonlinearparametric equations, the computing time cost is high.

This paper presents an efficient method to reveal the effect ofcutter runout in curved surface milling. Key issues are investi-gated about how to efficiently determine the tool position andcurvature related to the tool position. Analytical expressions ofIUCT and exit angle are established systematically in terms ofcutting parameters and workpiece geometry. Experiments arealso performed to confirm the validity of the proposed cuttingforce model. Compared with the existing models and methods,the main contributions of this paper cover the following aspects:

(1)

The proposed method provides an efficient method for calculat-ing the cutting forces in curved surface without the requirementof large computing time.

(2)

Expression of IUCT is analytically derived to avoid solvingnonlinear equations.

(3)

The concepts of equivalent tool position and feed direction areintroduced to ensure that in the cutting force calculationprocedure, the curvature and angular position of feed direc-tion have a continuous variation along the tool path.

(4)

The effect of cutter runout on the exit angle at the engaging ordisengaging stage is taken into account.

2. Calculation of tool position and equivalent feed directionfor the curved surface milling

A typical peripheral milling of curved surfaces is shown inFig. 1. XYZ is a globally stationary coordinate system attached tothe table in which the workpiece boundary, the geometry ofdesired surface and the tool path are described. XSYSZS is a locallymoving coordinate system attached to the spindle of the machinetool with its origin OS at the center of the spindle. OSXS is alignedwith the instantaneous feed direction of the theoretical tool path.OSZS points upward along the spindle axis. OSYS is the normal tothe feed direction and follows the definition of right-handcoordinate system. Then XSYSZS will move as the tool movesalong the tool path. yðtÞ represents the angular location of theinstantaneous feed direction OSXS and is measured anti-clockwisefrom the positive direction of the X-axis. OTPðtÞ and RTPðtÞ denotethe center of curvature and the radius of curvature related to toolposition OSðtÞ on the theoretical tool path, respectively.

Generally speaking, the tool position depends upon the para-metric equation of theoretical tool path. However, if the theore-tical tool path is directly used to calculate the tool position, much

A (Os (t))

B

CD

hi, j (t)

OX

Y

YS (t)

XS (t)

RADi, j

RADi−m, j

Workpiece boundary

Desired surfaceRTP (t)

OTP (t)

( )SZ t

Z

Theoreticaltool path

� (t)

�i, j (t)

Fig. 1. Representation of typical peripheral milling of curved surfaces.

Y. Yang et al. / International Journal of Machine Tools & Manufacture 51 (2011) 420–427422

computing time must be spent in solving nonlinear equations.Furthermore, it is well known that the actual tool path generallyused in NC machining is made up of a series of straight linesegments and circular arc segments generated by the integratedCAM software, no matter how complex the theoretical tool pathis. One such approximation can easily be used to determine thetool position without solving nonlinear equations and the com-puting time is largely saved. Mathematically the actual toolposition paðtÞ is iteratively updated as

paðtþTsÞ ¼paðtÞþVf Tsfa for linear tool path segment

½R�ðpaðtÞ-oCTPÞþoCTP for circular tool path segment

(

ð1Þ

with

fa ¼pen�pst

pen�pst

�� �� , R½ � ¼

cosas �sinas 0

sinas cosas 0

0 0 1

264

375,

as ¼Vf Ts

RCTP

Theoretically the feed direction along the actual tool path mightbe easily determined by finding the tangential direction of theactual tool path. Nevertheless, if the tool position strides over thejoint of two adjacent tool path segments, e.g., the joint of astraight line and a circle, as stated by Wei et al. [34], the feeddirection, the exterior normal direction and the curvature of theactual tool path may be abruptly changed. In this paper, theproblem is avoided based on the concept of equivalent point

obtained by mapping the tool position from the actual tool pathto the theoretical one. Detailed procedures are presented below.

As shown in Fig. 2, peðuðtÞÞ is the equivalent tool positiondefined by the intersection point between the theoretical toolpath and the normal of the actual tool path at paðtÞ:

ðpeðuðtÞÞ�paðtÞÞ � naðtÞ ¼ 0 ð2Þ

where naðtÞ is the exterior normal of the actual tool path.

naðtÞ ¼

½0 0 1�T � fa for linear tool path segment

paðtÞ�oCTP for circular tool path segment ðconvexÞ

oCTP�paðtÞ for circular tool path segment ðconcaveÞ

8><>:

ð3Þ

Geometrically the feed direction feðuðtÞÞ can be defined as thetangential direction at peðuðtÞÞ. It can be easily obtained by meansof the value of uðtÞ solved from Eq. (2). With feðuðtÞÞ and uðtÞ theexterior normal direction neðuðtÞÞ and curvature KeðuðtÞÞ related topeðuðtÞÞ are thus calculated by

neðuðtÞÞ ¼ ½0 0 1�T � feðuðtÞÞ ð4Þ

KeðuðtÞÞ ¼Xeu ðuðtÞÞYe

00 ðuðtÞÞ�Xe00 ðuðtÞÞYeu ðuðtÞÞ

ððXeu ðuðtÞÞÞ2þðYeu ðuðtÞÞÞ

2Þ3=2

ð5Þ

In the following presentation, feðuðtÞÞ, neðuðtÞÞ and KeðuðtÞÞ, aretreated as the equivalent feed direction, equivalent exteriornormal direction and equivalent curvature for the actual toolposition paðtÞ.

Besides, the angular location of the feed direction, i.e., yðtÞ, iscalculated by [34]

yðtÞ ¼ arccosfeðuðtÞÞUi

feðuðtÞÞ�� ��

!ð6Þ

where i¼ ½1 0 0�T is the unit direction vector of the X-axis.

3. Cutting forces for the curved surface milling

3.1. Basic cutting force model

Suppose the cutter is discretized into disk elements with equalaxial length Da. The total cutting force in locally moving coordinatesystem can be evaluated by

FXSðtÞ ¼

Xi,j

½�Fi,j,TðtÞcosji,jðtÞ�Fi,j,RðtÞsinji,jðtÞ�

FYSðtÞ ¼

Xi,j

½Fi,j,TðtÞsinji,jðtÞ�Fi,j,RðtÞcosji,jðtÞ� ð7Þ

with

Fi,j,TðtÞ ¼ gðji,jðtÞÞKi,j,TðtÞhi,jðtÞDa

Fi,j,RðtÞ ¼ gðji,jðtÞÞKi,j,RðtÞhi,jðtÞDa

gðji,jðtÞÞ ¼1 if ji,j,enðtÞrji,jðtÞmod 2prji,j,exðtÞ

0 otherwise

(

Due to the mobility of XSYSZS, the transformation into the globallystationary XYZ coordinate system corresponds to

FXðtÞ ¼ FXSðtÞcosyðtÞ�FYS

ðtÞsinyðtÞFY ðtÞ ¼ FXS

ðtÞsinyðtÞþFYSðtÞcosyðtÞ ð8Þ

It is worth mentioning that the total cutting forces described inEq. (8) are influenced by many issues such as cutter runout,machining deflection, tool wear and tool positioning errors ofinsert cutters, etc. In the following content, only the influence ofcutter runout is taken into account.

Actual tool path segment

pstpen

Theoreticaltool path

Normal of the actual tool path na

Feed direction of the actual tool path fa

Equivalent tool position pe (t)

Actual tool position pa (t)

Actual tool path segment

pst pen

Theoreticaltool path

Normal of the actual tool path na (t)

Feed direction of the actual tool path fa (t)

Equivalent tool position pe (t)

Actual tool position pa (t)

Fig. 2. Interpolated tool position on the actual tool path: (a) linear tool path segment and (b) circular tool path segment.

Y. Yang et al. / International Journal of Machine Tools & Manufacture 51 (2011) 420–427 423

3.2. IUCT in the presence of cutter runout

In case of zero cutter runout, IUCT can be expressed as anexplicit function of feed per tooth and tooth positioning angleof the cutting point [5,7,8,12]. Oppositely IUCT will be greatlyredistributed in the presence of cutter runout and is generallycalculated as the distance between two points, i.e., the cuttingpoint related to the current circular path and the correspondingone at the previous circular path [3,35]. Explicit expressionsrelating the cutting parameters to cutter runout parameters werederived [2,3] only for straight surface milling. Here, an explicitexpression of IUCT including the effect of cutter runout is derivedfor the milling of curved surface with variable curvature. For themilling of convex surfaces shown in Fig. 1, suppose that thecurrent cutting point D related to the jth disk element of the ithflute, is removing the surface left by the mth previous tooth.At the mth circular tooth path, the cutting point related to D issymbolized by C. The tool positions related to D and C are denotedby A and B, respectively. By definition, IUCT related to the jth diskelement of the ith flute can be expressed as

hi,jðt,mÞ ¼ LCD ¼ RADi,j�LAC ð9Þ

with

RADi,j ¼ RADþrcos l�tanðb0Þ

RADz�

2ði�1ÞpNf

� �ð10Þ

Eq. (9) indicates that the value of hi,jðt,mÞ depends on LAC

whose calculation is as follows.According to the triangle geometry relationship in DABC,

LAC can be mathematically derived as follows:

LAC ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRADi�m,j

2� 2RTPðtÞsin

mft

2RTPðtÞ

� �cos

mft

2RTPðtÞþji,jðtÞ

� �� �2s

�2RTPðtÞsinmft

2RTPðtÞ

� �sin

mft

2RTPðtÞþji,jðtÞ

� �ð11Þ

Where RTPðtÞ stands for the curvature related to arc AB_

.

Theoretically the radii of curvature and the centers of curva-ture related to tool positions A and B may be different due to thevariable curvature of the theoretical tool path. Nevertheless, asthe feed per tooth used is relatively small in practical milling andthe curvature of arc between two adjacent tool positions A and Bhas a very mild variation, the curvature of the arc can be assumedto be a constant value equal to KeðuðtÞÞ and both A and B have thesame curvature center OTPðtÞ. Mathematically we have

RTPðtÞ ¼1

KeðuðtÞÞ�� �� ð12Þ

Note that the corresponding RTPðtÞ should be recalculated byEqs. (5) and (12) once the tool position changes.

With the aids of Eqs. (9), (11), the IUCT can be calculated by [44]

hi,jðtÞ ¼max 0, minm ¼ 1,:::,Nf

ðhi,jðt,mÞÞ

� �ð13Þ

Note that IUCT can be obtained in a similar way in the case ofmilling of concave surfaces.

3.3. Entry/exit angles in the presence of cutter runout

The presence of cutter runout not only influences IUCT but alsothe entry/exit angles in the milling of curved surface. Desaiet al. [44] firstly studied one such influence. In their computingof entry/exit angles, the intersection point between the tooth pathand the theoretical workpiece boundary is obtained as long asthe workpiece boundary is the parallel offset of the geometry ofdesired surface. However, when the workpiece boundary is notparallel to the geometry of desired surface, e.g., at the disengagingstage, results of entry or exit angles will be erroneous. Weiet al. [34] replaced the theoretical workpiece boundary with aset of straight line and circular arc segments, which are theparallel offset of tool path in pre-machining, i.e., the so-calledactual workpiece boundary. Unfortunately the influence of cutterrunout was not considered in their work.

In this contribution, improvements are made on the calcula-tion of exit angle at the engaging or disengaging stage includingthe influence of cutter runout. In the case of continuous

YS (t1)

XS (t1)Engagingstage

YS (t2)

XS (t2)

XS (t3)

YS (t3)Disengaging

stage

Continuous cutting stage

Convex

portion

Desired surface

Radial depth of cut

Conca

ve

porti

on

Feed

direction

Workpiece boundary

Tool path

Oi, jOi−1, jOi−2, jOi−3, j

ii-1i-2i-3

Tool position )b(

Oi, jOi−1, jOi−2, jOi−3, j

ii-1i-2i-3

exit point of ith tooth Workpiece

Current tooth path

Tool position)a(

exit point of ith tooth Workpiece

Currenttooth path

Two possibilities of tooth paths for a three-flute cutter at engaging stage with cutter runout

�en (t3)

�ex (t3)

�ex (t2)

�en (t2)

�en (t1)

�ex (t1)

Fig. 3. Entry/exit angles at different stages.

Y. Yang et al. / International Journal of Machine Tools & Manufacture 51 (2011) 420–427424

engagement, entry/exit angles are obtained using the method inRef. [44].

At the engaging stage, as shown in Fig. 3, the exit angle can beobtained by

ji,j,exðtÞ ¼ minm ¼ 1,:::,Nf

ðji,j,exðt,mÞ,ji,j,B,exðtÞÞ ð14Þ

in which ji,j,exðt,mÞ means the angle related to the intersectionpoint of the current tooth path and the mth previous tool pathcorresponding to the jth disk element of the ith flute. It can beobtained by adopting the exit angle calculating method inRef. [44]. ji,j,B,exðtÞ is the angle related to the possible exit point,which is the intersection of the current tooth path and the actualworkpiece boundary:

ji,j,B,exðtÞ ¼ arccosðpWðvi,j,B,exðtÞÞ�paðtÞÞUneðuðtÞÞ

9pWðvi,j,B,exðtÞÞ�paðtÞ9 neðuðtÞÞ�� ��

!ð15Þ

Although the rotation radius of the concerned disk elementwill deviate from its nominal value RAD to RADi,j due to the cutterrunout [3,35,44], the entry angle can still be obtained according tothe method in Ref. [44], as long as RAD is replaced by RADi,j.

3.4. Calculation procedure of instantaneous cutting forces for the

curved surface milling

The procedure for calculating the cutting forces in the peripheralmilling of curved surfaces is described as follows:

(1)

Set the initial parameters specifying the actual tool pathgenerated by CAM system, cutting conditions, cutter geome-try, sampling time interval, geometry of desired surface andworkpiece boundary.

(2)

Determine the theoretical tool path based on the cutterradius and geometry of desired surface.

angular position

Y. Yang et al. / International Journal of Machine Tools & Manufacture 51 (2011) 420–427 425

(3)

1

Calculate the actual tool position at sampling instant t withEq. (1), based on the actual tool path and given samplingtime interval.

0ad]

(4)

-1

� (t)

[r

Calculate the equivalent feed direction, the equivalent exter-ior normal direction, the equivalent curvature and theangular location of the feed direction with Eqs. (2) and(4)–(6).

(5)

Calculate the entry/exit angles related to the jth axial cuttingdisk of the ith flute at sampling instant t.

curvature

(6) 0.2

Calculate the IUCT related to the jth axial cutting disk of theith flute at sampling instant t with Eq. (13).

[-]

(7)

0.1

ure

Calculate the cutting forces related to the jth axial cuttingdisk of the ith flute at sampling instant t.

vat

(8) Repeat steps 5–7 for all disk elements of the cutter.

0 Air cut ur

(9) stage and

C

Calculate the cutting forces at sampling instant t withEqs. (7) and (8).

-0.1engaging

stage Continuous cutting stage

(10) Repeat steps 3–9 until the whole cutting process iscompleted.

20 4 6 8 10 12

Time [s]

Fig. 4. Angular position of feed direction and curvature along the entire tool path

in test 2.

2

0

200

400

600Fo

rces

[N]

Measured values

0 2 4 6 8 10 12 14

0

200

400

600

Time [s]

Forc

es [N

]

Predicted values

8.8 8.81 8.82 8.83 8.84 8.85 8.86 8.87 8.88 8.89 8.9

0100200300400500600

Time [s]

Forc

es [N

]

Colse-up of view measured valuespredicted values

Fig. 5. Comparison of the measured and predicted cutting forces in the Y-direction

vs. time for the entire tool path in test 2(ap¼10 mm, ae¼3 mm, S¼2000 rpm,

ft¼0.05 mm/tooth).

4. Simulation and experiment results

To verify the proposed method, the peripheral milling ofcurved surfaces is carried out without coolant on a three axisCNC milling machine center. A three-fluted cylindrical end millwith helix angle of 301 and diameter of 12 mm is used. Theworkpiece material is aluminum alloy 7050. Cutting forces aremeasured with Kistler dynamometer 9255B. Two tests of downmilling are carried out.

Test 1 (ap¼4 mm, ae¼6 mm, S¼2000 rpm, ft¼0.1 mm/tooth)is conducted for the milling of straight surface to calibrate thecutting force coefficients and the cutter runout parameters withthe help of the method of Wan et al. [23]. Results are

Ki,j,T ¼ 1239:20þ2474:65e�66:61hi,jðtÞ

Ki,j,R ¼ 746:46þ2501:16e�62:16hi,jðtÞ

r¼ 2:6mm

l¼ 31:81 ð16Þ

Test 2 is performed to validate the proposed method in thefinish milling of a curved surface with both concave and convexsegments as shown in Fig. 3. The curved geometry is described bya cubic parametric Bezier curve:

XðuÞ ¼ 20þ105uð1�uÞ2þ15u2ð1�uÞþ40u3,

YðuÞ ¼ 5þ90u2ð1�uÞþ30u3,uA ½0,1� ð17Þ

Correspondingly the tool path is produced by the PLM soft-ware system of UG within a tolerance of 0.005 mm. Note that thetool path is set to be circular tool path segment at the initialengaging and the final disengaging stage. As to the surface to bemachined, it is obtained by offsetting the desired one with adistance equaling the radial depth of cut.

Based on the model proposed in Section 2, the variations ofcurvature and angular position of feed direction along the entiretool path are plotted as a function of time in Fig. 4. Since thetheoretical tool path does not exist at the initial air cut stage andengaging stage, only the curvature and feed direction faðtÞ can becalculated based on the actual tool path. Therefore, the curvatureat the initial air cut stage and engaging stage is set to be thecurvature of the actual tool path. Meanwhile, faðtÞ is employed tocalculate the angular position of feed direction at the initial air cutstage and engaging stage by means of Eq. (6) instead of feðuðtÞÞ.Consequently following observations can be figured out. Firstlythe concave and convex portions have a positive and a negativecurvature, respectively. Secondly the curvature and angular

position of feed direction have a continuous variation along thetool path. Based on this, the predicted cutting forces in Y-directionare shown in Fig. 5 along the entire tool path where the outline of

Y. Yang et al. / International Journal of Machine Tools & Manufacture 51 (2011) 420–427426

the extreme value variations of the predicted cutting forcesindicates that no abrupt change occurs along the entire tool path.

For the purpose of comparison, the predicted cutting forces inY-direction of test 2 are shown in Fig. 5. Close-up of measured andpredicted cutting forces during 8.8–8.9 s are shown in Fig. 5.Following phenomena are observed:

�

The predicted cutting forces agree closely with the measuredone both in shape and magnitude. This means that theproposed method is general and effective for cutting forceprediction. � Abrupt changes of the measured cutting force appear occa-

sionally due to the feed changes caused by the accelerationand deceleration of the drives.

Besides, both numerical and experimental studies are alsoconducted under other cutting conditions similar to test 2. Thefollowing phenomena can be found:

�

If the same radial and axial depths of cut are concerned, thegreater the feed per tooth, the larger the cutting force. This isdue to the fact that a large feed per tooth will result in a largeIUCT and the cutting force. � If the same axial depth of cut and feed per tooth are concerned,

the peak point of the cutting force will shift against the time asthe radial depth of cut ae varies.

At the same time, the proposed method is compared with theapproach described in Ref. [44] to evaluate the efficiency indetermining the tool position and the IUCT. Computing codes ofboth methods are written in MATLAB and implemented on a PC(Intel Core (TM) 2 Duo Processor, 2.4 GHz, 2 GB). For the tool pathshown in Fig. 3, the proposed method can greatly improve thecomputing efficiency in the calculations of tool position (a timereduction of about 70 times) and IUCT (a time reduction of about100 times in 10 revolutions).

The above verification means that this work can be potentiallyextended to process planning in industry cases of peripheralmilling of curved surfaces. For example, the machining deflectionin peripheral milling of an aeroengine blade, which is required forprocess planning, can be obtained based on the cutting forcescalculated by the proposed method, without the requirement oflarge computing time. It is worth mentioning that the simulationprocedure can be further sped up by using appropriate programlanguages, e.g., FORTRAN.

5. Conclusions

In this paper, cutting force modeling is studied systematicallyfor the peripheral milling of curved surfaces including the cutterrunout effect. Attention is paid on how to improve the efficiencyof the cutting force modeling. Analytical expression of IUCT isderived in detail. Novel expressions for entry/exit angles aredescribed in-depth too. A whole simulation procedure is proposedto predict cutting forces. Finally the validity of the modelingmethod has been demonstrated by a series of cutting tests.It concludes that the following:

�

The proposed method is effective for modeling the cuttingforce in peripheral milling of curved surface with variablecurvature. At the same time, the computing efficiency can begreatly improved by the proposed method. � Based on the concepts of equivalent feed direction, equivalent

normal direction and equivalent curvature, the abrupt changeof predicted cutting forces due to the non-smoothing of the

actual tool path discussed in Ref. [34] can be avoided so that acontinuous variation of the predicted cutting forces can beensured.

� IUCT is influenced by the curvature of the theoretical tool

path in peripheral milling of curved surfaces. Locally, the arcrelated to two adjacent tool positions can be approximated bya circular arc in the cutting force modeling procedure withoutthe loss of accuracy.

� The developed equation of calculating IUCT is also suitable to

the milling case of circular surfaces.

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant nos. 10925212 and 51005182), theNatural Science Basic Research Plan in Shaanxi Province of China(Grant no. 2010JQ7011), the NPU Foundation for FundamentalResearch (Grant no. JC200810) and the Ao-Xiang Star Programof NPU.

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