International Journal of Machine Tools & Manufacture 54–55 (2012) 34–45

Contents lists available at SciVerse ScienceDirect

International Journal of Machine Tools & Manufacture

0890-69

doi:10.1

n Corr

E-m

URL

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

Unified cutting force model for turning, boring, drilling andmilling operations

M. Kaymakci, Z.M. Kilic, Y. Altintas n

Manufacturing Automation Laboratory, The University of British Columbia, Department of Mechanical Engineering, 2054-6250 Applied Science Lane Vancouver,

B.C., Canada V6T 1Z4

a r t i c l e i n f o

Article history:

Received 29 July 2011

Received in revised form

16 December 2011

Accepted 16 December 2011Available online 26 December 2011

Keywords:

Mechanics

Drilling

Turning

Milling

Boring

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

016/j.ijmachtools.2011.12.008

esponding author.

ail address: altintas@mech.ubc.ca (Y. Altintas

: http://www.mal.mech.ubc.ca (Y. Altintas).

a b s t r a c t

A unified cutting mechanics model is developed for the prediction of cutting forces in milling, boring,

turning and drilling operations with inserted tools. The insert and its orientation on a reference tool

body are mathematically modeled by following ISO tool definition standards. The material and cutting

edge geometry-dependent friction and normal forces acting on the rake face are transformed into

reference tool coordinates using a general transformation matrix. The forces are further transformed

into turning, boring, drilling and milling coordinates by simply assigning operation specific parameters.

The unified model is validated in cutting experiments.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

The aim of the current research is to develop process modelsthat can be used to predict and optimize the machining opera-tions ahead of costly physical trials. The process models, whichcombine the material properties, cutting mechanics, tool geome-try, process kinematics and structural dynamics, are used topredict force, torque, power, form errors and vibrations duringmetal cutting operations. The simulation allows the processplanners to forecast whether the operation is feasible for themachine tool and part, or to optimize the cutting conditions andtool geometry for higher material removal rates.

There have been valuable contributions in the past in estab-lishing process mechanics models in turning, drilling, boring andmilling. Mechanistic process models consider the cutting forcesacting on the tool edge as a function of chip area and empiricallycutting force coefficients, which are calibrated from machiningtests. The force distribution along the cutting edges is modeledand summed to predict the total load acting on the machine. Anexemplary application of the mechanistic approach in face millingis presented by Fu et al. [1], and a comprehensive review ofmechanistic force models has been presented by Ehmann et al. [2].Armarego [3] proposed that the cutting force coefficients could bepredicted from the average shear stress, shear angle and frictioncoefficient by applying orthogonal to oblique transformation [4]. The

ll rights reserved.

).

oblique transformation method is most useful when modelingsolid tools with continuously varying edge and chip geometry [5].The mechanistic and cutting mechanics based methods publishedin the literature are reviewed by Luttervelt et al. [6] andAltintas [7]. Since 2000, the research moved more towardspredictive modeling of the metal cutting process using numericalmethods. The cutting force coefficients are predicted from finiteelement and slip line field models, and used in predicting cuttingforces [8–10]. The numerical models are completely based on thestrain, strain rate and temperature-dependent flow stress of thematerial and friction coefficient. In summary, it is possible topredict the process forces with acceptable accuracy usingmechanistic, cutting mechanics and numerical models.

The current process models are dedicated to individual toolfamilies and operations. For example, process models are indivi-dually tailored to helical end mills, indexed cutters, ball end mills,turning tools, drilling tools and boring with single or multipleinserts [11,12]. However, the fundamental mechanics of cutting atthe primary and secondary zones are the same for all chipremoval operations while the geometry and kinematics differ ineach process. Engin and Altintas have presented generalizedmathematical models of solid end mills [13] and inserted cutters[14,15], and predicted cutting forces for helical, ball, tapered andinserted cutters. Although their generalized model was the firstattempt in unifying the milling process models, the detailedgeometric features of indexed cutters such as chamfer edge, noseradius and wiper edge were not included.

This paper presents a unified geometric, kinematic andmechanics model that allows the prediction of turning, milling,

Nomenclature

CRP cutting reference pointer tool included (nose) angleiW insert widthDc cutter diameterL insert lengthkr cutting edge angle of the insertkn

r true cutting edge angle along the cutting edgeRDR,R0R transformation matrices used for determining true

tool anglesRTA radial-tangential-axialfpn pitch angle between preceding insertsTRU,T0R matrices used in force transformationsts shear stressfn normal shear angleFu,Fv friction and normal forces on the rake faceFx,Fy,Fz cutting forces in machine coordinate systemdAc differential chip loaddS differential cutting edge lengthVc cutting speedb width of cut

bs wiper edge lengthgf radial rake anglegp axial rake angleZ chip flow angleyi lag anglere corner radiusv0,v3 cutting velocity vector represented in design and

reference framesls inclination (helix) anglegn normal rake anglevedge cutting edge vectorcj orientation angle of the insertf0 initial orientation angle at current level of cutterp parameter to include type of operationba average friction angleKuc,Kvc friction and normal cutting force coefficients on the

rake faceKue,Kve friction and normal edge cutting force coefficientso angular speed of rotating toola depth of cuth uncut chip thickness

M. Kaymakci et al. / International Journal of Machine Tools & Manufacture 54–55 (2012) 34–45 35

boring and drilling operations. The friction and normal cuttingforces on the rake face are first modeled using an orthogonalto oblique transformation model. True angles of oblique cuttingedge are adopted from international tool geometry standardsISO 13399 [16,17]. The rake forces are transformed to severalintermediary coordinate frames in order to be representedin machine tool coordinates. The unified model has beenexperimentally validated in turning, boring, milling and drillingoperations.

2. Generalized geometric model of inserted cutters

There are various insert geometries used on indexed cutters,which need to be considered by the generalized process model.While 17 insert shapes are defined in ISO 13399 standards [17],there are a number of additional, custom designed, nonstandardtools as well. The generalized geometric model of inserted tools ispresented starting with the insert, placement of the insert on thecutter body, identification of oblique tool angles needed by thecutting mechanics model and the kinematics of cuttingoperations.

Fig. 1. Design points on a parallelogram insert.

2.1. Mathematical model of insert

The geometry of an insert is defined in its local coordinatesystem analytically. A parallelogram shaped insert is shown inFig. 1 as a sample to illustrate the modeling process. Since thechip breaking and lubrication grooves are not defined in catalogs,the rake face is assumed to be flat. ISO standards define insertshape (ISO-L parallelogram), length (L), wiper edge length (bs),corner radius (re), nose angle (er) and insert width (iW). A, B, D andE are control points of the cutting edge. The Cutting ReferencePoint (CRP) is the origin for specific dimensions and rotations,which are used for placing the insert on the cutter. The CRP of aninsert is located on the circumference of the cutter for drilling,milling and boring heads. Point F is the theoretical sharp corner ofthe insert.

The control points of the insert are derived as functions ofinsert parameters as follows (Fig. 1):

CRPðx,y,zÞ ¼ ð0,0,0Þ

Aðx,y,zÞ ¼ ð�bsþreðcot kr�csc krÞ,0,0Þ

Bðx,y,zÞ ¼ ðreðcot kr�csc krÞ,0,0Þ

Cðx,y,zÞ ¼ ðreðcot kr�csc krÞ,0,reÞ

Dðx,y,zÞ ¼ re cos kr tankr

2,0,reð1�csc krÞ

� �Eðx,y,zÞ ¼ ðAxþLcoskr�cosðerþkrÞcsc erðbssinkr

�ðrecoskr�1ÞÞ,0,Lsinkr�csc ersin

�ðerþkrÞðbssinkr�ðrecoskr�1ÞÞ

Iðx,y,zÞ ¼ Axþ1

2ðLcoskrþ iWcosðerþkrÞ

�

�2cosðerþkrÞcscerðbssinkr�reðcos kr�1ÞÞÞ,

M. Kaymakci et al. / International Journal of Machine Tools & Manufacture 54–55 (2012) 34–4536

�0,1

2ðLsinkrþ iWsinðerþkrÞ�2csc er

�sinðerþkrÞðbssinkr�reðcoskr�1ÞÞÞ

�ð1Þ

The insert can be oriented on cutter pockets by transformingthe control points [2] by the amount of axial rake (gp), radial rake(gf) and the cutting edge (kr) angles around CRP. The updatedcontrol points are then connected by lines (wiper and maincutting edges) and arcs (corner radius or round insert) by varyingthe distance parameter (s) in small increments along the insertedges:

Lines : P!¼ P!

i�1þð P!

i� P!

i�1Þs, 0rsr1

Arcs : P!¼ Rcoss u

!þRsinsð n

!� u!Þþ C!

, Si�1rsrSi ð2Þ

where P!

is any point between P!

i and P!

i�1, which are twoconsecutive control points; R is the radius of the arc; u

!is a unit

vector drawn from the center of the arc to any point on thecircumference; n

!is a unit vector perpendicular to the plane of

the arc; and C!

is the center vector of the arc. Si�1 and Si

represents the start and end angles (boundary) of the arc.

Fig. 2. Schematics of rotations of insert to be located on cutter. (gf: Radial rake angle

Fig. 3. Schematics of rotations of

2.2. Placing insert on a tool holder

The insert is placed on the turning tool, boring head, drillingand milling tool holders as shown in Fig. 2. The sequence oforientation of the inserts on the cutter is illustrated in Fig. 3. Thecoordinate system of the insert is used as the reference Frame R

(XRYRZR). The positive radial (gf) and axial (gp) rake angles of thetool are imposed by rotating the insert reference frame counterclockwise (CCW) around ZR, followed by a clockwise (CW) rota-tion around the resulting X1 axis, respectively. The insert is tiltedaround the resulting Y2 axis to impose the local cutting edge angle(kr). The successive orientations shown in Fig. 3 lead to the designcoordinate system (D) of the insert placed in the cutter’s pockets.The coordinates of the insert in the design coordinate system areexpressed from the successive rotational transformations as:

XD

YD

ZD

8><>:

9>=>;¼ ½RDR�

XR

YR

ZR

8><>:

9>=>; ð20Þ

where the equivalent transformation matrix [RDR] is:

½RDR� ¼

coskr 0 sinkr

0 1 0

�sinkr 0 coskr

264

375

1 0 0

0 cosgp �singp

0 singp cosgp

264

375 cosgf �singf 0

singf cosgf 0

0 0 1

264

375

; gp: Axial rake angle; kr: Insert cutting edge angle; X,Y,Z: Machine coordinates).

insert to be located on cutter.

Fig. 4. Tool-in-hand planes and motion directions (adapted from ISO 3002) and the definition of normal rake angle. (a) Definition of tool-in-hand planes in ISO 3002.

(b) Definition of normal rake (gn) and inclination (oblique) angle of cut (ls). (c) Definition of true cutting edge angle (knr ) in the insert reference Frame R.

M. Kaymakci et al. / International Journal of Machine Tools & Manufacture 54–55 (2012) 34–45 37

¼

coskrcosgf þsinkrsingpsingf �coskrsingf þsinkrsingpcosgf sinkrcosgp

cosgpsingf cosgpcosgf �singp

�sinkr cosgf þcoskrsingpsingf sinkrsingf þcoskrsingpcosgf coskrcosgp

264

375ð3Þ

2.3. Normal rake (gn), cutting edge (kn

r ) and oblique (ls) angles

Normal rake, cutting edge and oblique angles, which areneeded in modeling the mechanics of cutting, are evaluated fromthe tool geometry and cutting velocity direction. The obliquecutting mechanics model is based on the tool-in-hand planesdefined by ISO 3002 [16] as shown in Fig. 4a. The tool referenceplane (Pr) is oriented perpendicular to the primary motion(cutting velocity) vector and parallel to the tool axis. Tool cuttingedge plane (Ps) is tangential to the cutting edge at the selectedpoint and perpendicular to Pr. The cutting edge normal plane (Pn)is perpendicular to the selected point on the cutting edge andperpendicular to plane (Ps). The cutting velocity vector (v0) istransformed to design coordinates (Frame D) after the insert isoriented on the cutter body:

vD ¼ RDRUv0 ¼

singf Ucoskr�cosgf UsingpUsinkr

�cosgf Ucosgp

�singf Usinkr�cosgf UsingpUcoskr

8><>:

9>=>; ð4Þ

Normal rake angle (gn) is defined in the design Frame (D) fromFig. 4b as the angle between the projection of velocity vector (vD)

and the normal of the rake face (YD) measured in YDZD frame;

gn ¼ arctanvDz

vDy

� �¼ arctan

singf Usinkrþcosgf UsingpUcoskr

cosgf Ucosgp

!

ð5Þ

The true cutting edge angle (knr ) is defined only when the

insert is placed on the cutter, hence it is initially defined in thedesign frame. The cutting edge vector (vedge,D ¼ 1 0 0

� �T ) is

aligned with the XD axis in the design frame and transformed to areference insert frame using the cutting edge angle (kr) as:

vedge,R ¼ R�1RDUvedge,D ¼

cosgf Ucoskrþsingf UsingpUsinkr

�singf Ucoskrþcosgf UsingpUsinkr

cosgpUsinkr

264

375 ð6Þ

The true cutting edge angle (knr ) however is measured in the

XRZR plane of the insert reference frame and defined as the anglebetween vedge,R and the XR axis (Fig. 4c).

kn

r ¼ arctanðvedge,Rz=vedge,RxÞ

¼ arctancosgpUsinkr

cosgf Ucoskrþsingf UsingpUsinkr

!ð7Þ

The angle of inclination or the oblique angle of cut (ls) isdefined between the cutting edge (XD) and the tool referenceplane Pr (XRZR plane) measured in the tool cutting edge plane Ps

(Cut surface), as shown in Fig. 4b. The cutting edge of the insert onthe cutter body can be obtained by rotating the reference insert’scutting edge [10] around YR by the amount of cutting edge

M. Kaymakci et al. / International Journal of Machine Tools & Manufacture 54–55 (2012) 34–4538

angle (kr).

vedge,s ¼

coskr 0 sinkr

0 1 0

�sinkr 0 coskr

264

375vedge,R

¼

cosgf Ucos2krþsingf UsingpUsinkrþcosgpUsin2kr

�singf Ucoskrþcosgf UsingpUsinkr

�sinkrUcoskrUcosgf�sin2krUsingf UsingpþsinkrUcoskrUcosgp

8>><>>:

9>>=>>;ð8Þ

The inclination or oblique angle of the tool (ls) is defined as:

ls ¼ arctanðvedge,sy=vedge,sxÞ

Fig. 6. Transformation from RTA (XIYIZI) to

Fig. 5. Mechanics of oblique cutting [18].

¼ arctan�singf Ucoskrþcosgf UsingpUsinkr

cosgf Ucos2krþsingf UsingpUsinkrþcosgpUsin2kr

!

ð9Þ

In summary, the normal rake (gn), true cutting edge (knr )

and inclination (ls) angles are derived as functions of thecutting velocity, cutting edge and rake face orientation angles(gf,kr,gp) specified in the cutter design. Indexed cutters withmultiple inserts are modeled by placing inserts on the mathema-tically-described cutter body similar to the model presentedin [14].

3. Generalized modeling of cutting forces

3.1. Oblique mechanics model

The generalized mechanics model requires the evaluation ofthe friction force on the rake face (Fu), which is aligned with thechip flow angle (Z), and the normal force (Fv) described in chipflow coordinates as shown in Fig. 5.

A differential cutting edge that produces a chip with an area(dAc) and length (dS) creates the following friction and normalforces:

dFujðiÞ ¼ KucjðiÞdAcjðiÞþKuejdSjðiÞ

dFvjðiÞ ¼ KvcjðiÞdAcjðiÞþKvejdSjðiÞ ð10Þ

where j is the insert number for a tool with N edges. If the inserthas a varying geometry along the edge, which leads to varyingtrue rake, true cutting edge and oblique angles, it is divided into K

number of differential segments with an index label i. Kuc and Kvc

are the friction and normal cutting force coefficients, and Kue andKve are the edge force coefficients in oblique cutting for eachinsert (j) and differential segment (i). The chip area (dAc) andwidth (dS) are evaluated as dAcj(i)¼hj(i)Ubj(i)U and dSjðiÞ ¼ bjðiÞ=

sinkn

rj where hj(i) and bj(i) are local chip thickness and edgeheight. Cutting coefficients in friction and normal directions canbe evaluated using either mechanistic [2] or orthogonal to obliquetransformation [4] methods. The following oblique transforma-tion method is used here for each insert segment to generalize theforce equations [18].

insert reference coordinates (XRYRZR).

Fig. 7. Schematics of (a) milling/boring cutter, (b) drilling cutter and (c) turning tool showing insert reference coordinates (XRYRZR), machine coordinates (X0Y0Z0) and the

orientation angle (ci,j).

M. Kaymakci et al. / International Journal of Machine Tools & Manufacture 54–55 (2012) 34–45 39

Kuc ¼ts

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�tan2Zsin2bn

qcoslssinfn

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficos2ðfnþbn�gnÞþtan2Zsin2bn

q sinbn

Kvc ¼ts

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�tan2Zsin2bn

qcoslssinfn

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficos2ðfnþbn�gnÞþtan2Zsin2bn

q cosbn ð11Þ

The shear stress (ts), shear angle (fn), average friction angle(ba) and edge coefficients (Kue and Kve) are evaluated fromorthogonal turning tests as a function of chip thickness (h),cutting speed (v0) and rake angle (gn). The projection of thefriction angle on the cutting edge normal plane (Pn) isbn ¼ tan�1ðtanbaUcosZÞ: Alternatively, the cutting force coeffi-cients (Kuc, Kvc) can be predicted from material based FiniteElement [8,9] or mechanistic methods [2,7] when the rake facehas chip breaking grooves, the cutting edge has chamfers, or toolhas a wear. The tool wear would affect edge force coefficients(Kue and Kve) that represent the flank friction most. The proposedtransformation method is independent of any method that is usedto predict the cutting force coefficients, which are the startingpoint in the unified kinematic model of the cutting forces.

The two dimensional differential forces are transformed from thechip flow (U,V) coordinates [11] to the Radial-Tangential-Axial (RTA)coordinates of the tool. First, the differential forces are projected tothe cutting edge (YIII or YII) and plane (XIIIZIII) as in Fig. 5:

XIII

YIII

ZIII

8><>:

9>=>;¼

0 1

�sinZ 0

cosZ 0

264

375 U

V

ð12Þ

where YIIIZIII is on the rake face. Next, the forces are transformed tothe cutting edge (YII) and normal plane (Pn¼XIIZII) by rotating (XIIIZIII)around YIII at the amount of normal rake (�gn):

XII

YII

ZII

8><>:

9>=>;¼

cosgn 0 singn

0 1 0

�singn 0 cosgn

264

375

XIII

YIII

ZIII

8><>:

9>=>; ð13Þ

This rotation superimposes YIIZII on the cutting edge plane (Ps).The forces are transformed to Radial-Tangential-Axial (RTA)coordinates (XIYIZI) by rotating (XIIYII) around ZII at an amount ofinclination angle (ls):

XI

YI

ZI

8><>:

9>=>;¼

0 0 1

cosls �sinls 0

sinls cosls 0

264

375

XII

YII

ZII

8><>:

9>=>; ð14Þ

This rotation aligns the cutting velocity (v0) with the tangen-tial direction (YI or T), and the XIZI plane with the insert referenceplane (Pr). Finally the forces are transformed from RTA to insertreference coordinates (XRYRZR) using the cutting edge angle ðkn

r Þ,see Fig. 6:

XR

YR

ZR

8><>:

9>=>;¼

cos knr þpU p2

� �0 �sin kn

r þpU p2� �

0 1 0

sin knr þpU p2

� �0 cos kn

r þpU p2� �

264

375

XI

YI

ZI

8><>:

9>=>; ð15Þ

where p¼0 for turning/boring/drilling, and p¼1 for milling(Fig. 7). The transformation from (UV) to the reference coordinatesystem can be summarized by substituting Eqs. (13)–(15) intoEq. (12):

XR

YR

ZR

8><>:

9>=>;¼

cos knr þpU p2

� �0 �sin kn

r þpU p2� �

0 1 0

sin knr þpU p2

� �0 cos kn

r þpU p2� �

264

375

�

cosgncosZ �singn

sinlssinZþcoslssingncosZ coslscosgn

�coslssinZþsinlssingncosZ sinlscosgn

264

375

T

U

V

¼ TRUUU

V

ð16Þ

Transformation from the chip flow (UV) to tool reference (R)coordinates is common for all stationary and rotating cutters, anddepends only on the insert geometry.

M. Kaymakci et al. / International Journal of Machine Tools & Manufacture 54–55 (2012) 34–4540

Each insert is placed on a rotating or a stationary tool body byapplying the orientation angle (ci,j) of the insert, which isconstant for stationary tools (turning and single point boring),and which changes periodically as a function of spindle rotationfor rotating tools (drilling, milling and indexed boring heads). Theinsert coordinates are transformed from its reference frame(XRYRZR) to operation specific machine coordinates (X0Y0Z0) asfollows:

X0

Y0

Z0

8><>:

9>=>;¼

sincj �coscj 0

coscj sincj 0

0 0 1

264

375

XRþDcj=2

YR

ZR

8>><>>:

9>>=>>;¼ T0RU

XRþDcj=2

YR

ZR

8>><>>:

9>>=>>;ð17Þ

where Dcj is the radial distance of the insert from the rotating

Table 1Orthogonal cutting parameters for two work materials used in experiments. H¼uncut

Source: CUTPRO [19].

AISI-1045 Steel Al7050-T

Shear stress ts (MPa) 450.3þ0.4Vcþ227.5h 266.804þ

Shear angle fn (deg.) atan ð0:4þ0:6hÞcosgn

1�ð0:4þ0:6hÞsingn

19.4þ42

Friction angle ba (deg.) 26.8�0.031Vcþ11.77h 25.877�

Kve (N/mm) 0:0103Vc2þ1:985Vc�55:431 10

Kue (N/mm) 0:0983Vc2þ10:299Vc�294:96 10

Table 2Turning test conditions. Width of cut: 4 mm, cutting speed V¼150 m/min and feed ra

Insert #1 (Square ISO S-SNMA 12 04 08)

L 12 mm kr 751

iW – gf �61

bs – gp �61

re 0.8 mm er 901

Holder DSBNL 2020K 12

Feedrate [mm/rev]

0.12 0.16 0.20 0.24

16.2% 15.7% 14.2% 19.2%

15.7% 7.9% 4.2% 5.6%

15.6% 22.2% 7.2% 9.0%

0.12 0.16 0.2 0.241000

1500

2000

0.12 0.16 0.2 0.241500

2500

3500

F [N

]X

F Y [N

]

0.12 0.16 0.2 0.241500

2500

3500

Feed Rate [mm/rev]

F Z [N] Sim

Exp

SimExp

SimExp

Fig. 8. Measured and predicted cutting forces in turning AISI 1045 steel. See Table 2 for

error for each case.

cutter axis. Finally, the cutting forces in chip flow coordinates aretransformed from the reference insert to machine (i.e. forcemeasurement) coordinates as:

dFXj

dFYj

dFZj

8>><>>:

9>>=>>;

i

¼ T0RUTRUU

dFuj

dFvj

( )i

gijðci,jÞ ð18Þ

where gij¼1 when the insert is in cut and otherwise gij¼0. Theorientation angle (ci,j) is measured from the (Y0) axis in a CWdirection. The orientation matrix (T0R) is specific for each opera-tion as follows (Fig. 7):

a)

chi

745

17

.017

128

te ra

I

L

iW

b

r

H

F X [N]

F Y [N

]F Z [N

]

inse

Turning and single point boring: cj¼�p/2 and Dcj¼0

p thickness [mm]; normal rake gn (deg.); cutting speed Vc (m/min).

1 Al6061-T6

4.128h�0.043Vcþ0.896gn 205.928þ204.038hþ0.016Vc�0.056gn

hþ0.02Vcþ0.384gn 13.866þ62.929hþ0.005Vcþ0.259gn

3h�0.007Vcþ0.181gn 20.835�4.901h�0.007Vcþ0.291gn

50.94�0.004Vcþ0.039gn

94.255�0.018Vc�0.8451gn

nge: 0.12–0.24 mm/rev. Material: AISI1045 steel.

nsert #2 (Rhombic ISO C-CNMA 12 04 08)

12 mm kr 751

– gf �81

s – gp �81

e 0.8 mm er 801

older DCKNL 2020K 12

Feedrate [mm/rev]

0.12 0.16

2.5% 10.7%

3.0% 8.6%

0.20 0.24

5.2% 6.4%

2.6% 4.7%

3.1% 8.8% 2.8% 4.8%

0.12 0.16 0.2 0.241000

1500

2000

0.12 0.16 0.2 0.241500

2500

3500

0.12 0.16 0.2 0.241500

2500

3500

Feed Rate [mm/rev]

SimExp

SimExp

SimExp

rt dimensions and cutting conditions. Tables below the plots display the mean

M. Kaymakci et al. / International Journal of Machine Tools & Manufacture 54–55 (2012) 34–45 41

b)

TabCut

T

(a(b

Drilling-indexed boring: ci,j ¼f0þPN

j ¼ 1 fpjþoUtP

c) Milling: ci,j ¼f0þ

Nj ¼ 1 fpj�yiþoUt

where f0 is the reference insert’s angular position measuredfrom axis (Y0); fpj is the pitch angle of segment i insert j on thecircumference of the cutter; yi is the lag angle caused by thehelical edge; o(rad/s) is the angular spindle speed; and t is thetime. The total forces can be evaluated by summing the differ-ential forces contributed by all teeth (N) and edge segments (K),which are in cut at time t. The process is simulated by rotating thetool for one revolution (oUt¼2p) for variable pitch and variablehelix, and fp¼2p/N for cutters with constant pitch angles [18].

Fig. 9. Illustration of axial (ef) and radial (er) runouts on a boring head.

le 3ting conditions used in boring tests.

est Cuttingspeed(m/min)

Borediameter(mm)

Feed rate[mm/rev]

Axialrunout(mm)

Radialrunout(mm)

Nominaldepth ofcut (mm)

) 150 59.7 0.06 0.09 0.2 1.285

) 175 66.2 0.055 0.14 0.18 0.920

0 0.1 0.2 0.3 0.4

-50

0

50

100

150

Cut

ting

Forc

es [N

]

Tim

FX

FZ

Fig. 10. Experimental validations for a multiple insert boring head with axial and

4. Experimental results

The generalized mechanics model for inserted tools, which isused to predict cutting forces in milling, turning and drillingoperations, has been experimentally validated. The cutting forcecoefficients are evaluated from the orthogonal parameters of thethree materials given in Table 1 according to oblique transforma-tion model given in [13]. While the turning tests have beenperformed on a CNC turning center, the drilling, indexed boringand milling tests have been performed on a CNC machining centerunder chatter free conditions. Kistler 9121 turning and Kistler9257B table dynamometers are attached to CUTPRO MALDAQdata acquisition software in measuring the cutting forces. Thetests have been conducted under dry and chatter free conditions.

4.1. Turning tests

Turning and single point boring operations have the samemechanics. The model has been validated in turning experimentsconducted with square and rhombic inserts having re¼0.8 mmnose radius as outlined in Table 2. The workpiece material is AISISteel 1045 bar with 255HB hardness and 38.1 mm diameter. Theproposed model is able to predict the measured cutting forcesreasonably well for both inserts as shown in Fig. 8. The meanerror between the mathematical model and the experimentaldata is 12.7%, 4.48% and 5.12% in X, Y and Z directions,respectively.

4.2. Boring tests with multiple inserts

A Valenite VPB head with two adjustable inserts has been usedto validate the boring process. The cutter has two identicalrhombic inserts (Valenite CCGT432-FH) with a 951 cutting edgeangle, and �51 axial and radial rake angles. The inserts aresymmetrically distributed along the cutter axis to cancel the

0 0.1 0.2 0.3 0.4

-40

-20

0

20

40

60

80

100

120

e [s]

FX

FZ

Exp

Sim

radial runouts. Cutting conditions and tool geometry are shown in Table 3.

Fig. 11. Geometric parameters of the drill used in cutting tests (from [20]). (a) Frontal view. Red: Cutting edge of central insert. Green: Cutting edge of peripheral insert.

(b) Cutting zones of each insert and overlapped cutting zones. (c) Geometric model of the rectangular drill insert. Red line represents cutting edge. (d) Drill Geometry:

Sandvik R416.2-0210L25-21 drill with 2 LCMX04 inserts. Axial run-out: 116 mm. (For interpretation of the references to color in this figure legend, the reader is referred to

the web version of this article.)

0 0.1 0.2 0.3 0.4 0.5-500-250

0250500

F X [N

]

0 0.1 0.2 0.3 0.4 0.5-500-250

0250500

F Y [N]

0 0.1 0.2 0.3 0.4 0.5300350400450500

Time [sec]

F Z [N]

SimExp

SimExp

SimExp

0 0.1 0.2 0.3 0.4 0.5-1000-500

0500

1000

F X [N

]

0 0.1 0.2 0.3 0.4 0.5-1000-500

0500

1000

F Y [N]

0 0.1 0.2 0.3 0.4 0.5400

600

800

Time [sec]

F Z [N]

SimExp

SimExp

SimExp

Fig. 12. Measured and predicted drilling forces. Material: AL7050-T7451. Pilot hole diameter: 6 mm, drilled hole diameter: 21 mm. (a) Feedrate¼0.050 mm/tooth,

Speed¼600 rev/min. (b) Feed rate¼0.100 mm/tooth, Speed¼700 rev/min.

M. Kaymakci et al. / International Journal of Machine Tools & Manufacture 54–55 (2012) 34–4542

Fig. 13. Indexable cutter with 32 mm diameter, 4 inserts per flute. (a) Actual CAD model of insert. (b) Geometric model of insert. (c) Actual and approximated cutter edge

by the geometric model. (d) Insert parameters – Sandvik R390-11 T3 08E.

Fig. 14. Variables along indexed mill’s cutting edge. (a) Schematics representing insert locations on cutter body. (b) Local radius along cutting edge. (c) True cutting edge

angle. (d) Normal rake angle. (e) Helix angle. (f) Lag angle of the cutting edge segment.

M. Kaymakci et al. / International Journal of Machine Tools & Manufacture 54–55 (2012) 34–45 43

Fig. 15. Cutting force experiment plots. (a) Slotting, f¼0.100 mm/tooth, N¼700 rpm, a¼5 mm, (b) Slotting, f¼0.200 mm/tooth, N¼700 rpm, a¼5 mm, (c) Quarter

immersion down milling, f¼0.200 mm/tooth, N¼1000 rpm, a¼20 mm and (d) Quarter immersion down milling, f¼0.200 mm/tooth, N¼350 rpm, a¼30 mm.

M. Kaymakci et al. / International Journal of Machine Tools & Manufacture 54–55 (2012) 34–4544

cutting forces in lateral (X,Y) directions. However, a radial andaxial runout is applied using the adjustable slides and shims toone of the inserts to model the process faults in boring [21] asshown in Fig. 9. The proposed model is validated in boringaluminum Al6061-T6 (Table 1) under two different cutting con-ditions with different run-out values (Table 3).The predictedforces are in reasonable agreement as shown in Fig. 10. Sinceplanar cutting forces (FX and FY) have same magnitudes with a 901phase shift, only the FX direction is plotted.

4.3. Drilling tests

Drilling experiments have been conducted with a Sandvikindexable drill with two inserts. The work material was Al7050-T4751. Axial run-out between inserts and the diameter of pilothole (Dp) are used to evaluate cutting edge zones of the inserts asshown in Fig. 11a. The chip thickness varies on each insert alongthe overlapped cutting zone due to the presence of axial run out(Fig. 12b). The digitization of the cutting edge into differential (i)elements automatically considers any run-out and geometricdeviations along the inserts. The geometric parameters of theinserts and tool holder are given in Figs. 13c and 14d. A pilot holewith a 6 mm diameter is opened ahead of drilling tests in order toeliminate the indentation effect at zero and the low speed zoneclose to centerline of cutter. The diameter of the finished hole is

21 mm, thus the width of cut was 7.5 mm. The predicted andexperimentally measured forces are in good agreement as shownin Fig. 12. The proposed transformation model can be used ondrilling operations without the pilot hole by separating the chiselpenetration and shearing zones. The chisel penetration coeffi-cients are best predicted either mechanistically from experiments[2] or through Finite Element methods [9].

4.4. Milling tests

An indexable cutter with rectangular inserts, which are dis-tributed along two identical helical flutes, is used in millingAl7050-T7451. The insert has chip breaking grooves on the rakeface and the cutting edge has undefined, nonstandard helical stylecurves (Fig. 13a), which is approximated as a flat faced tool(Fig. 13b) by the generalized mathematical model. The actual andapproximated edges are compared in Fig. 13c with a maximumgeometric error of 50 mm. However, the force prediction requiresonly the chip thickness distribution along the cutting edge, whichis still correctly evaluated.

Each flute has 4 inserts with 11 mm length spread along the36 mm flute length as modeled in Fig. 14. Modeled local radius,normal rake angle, helix angle and true cutting edge angle along aflute are presented in Fig. 14a–f. The cutter has a 0.75 mm cornerradius, which leads to quadratic variation of edge parameters at

M. Kaymakci et al. / International Journal of Machine Tools & Manufacture 54–55 (2012) 34–45 45

the tip. The corners of the successive inserts at higher elevationsoverlap; hence true cutting edge, normal rake and helix anglesconverge to 87.11, 8.51 and 19.41, respectively. The helix angle ofthe flutes and angular offsets between inserts create a varying lagangle along the cutting edge, which is modeled as shown inFig. 14f.

A series of slot and quarter immersion down milling tests havebeen conducted, and the predicted and measured forces areshown in Fig. 15. While the prediction of peak forces have abouta 20% error in low feed rates (Fig. 15a), the error reduces to lessthan 15% in higher feed rates (Fig. 15b–d). The orthogonal database was created with sharp orthogonal tools with flat rake faces,while the inserts used in the tests have variable rake faces anddifferent edge preparation, which leads to the higher predictionerrors in low chip loads. The prediction errors can be minimizedusing mechanistically calibrated cutting force coefficients for eachinsert [1], which is still accommodated by the generalized modelpresented here.

5. Conclusion

The prediction of turning, drilling, boring and milling forces isunified in one generalized mathematical model. The inserts aremathematically defined using ISO tool standards, and placed onthe tool holders through geometry-dependent transformationmatrixes. The material and insert geometry-dependent frictionand normal forces are transformed into a common, referenceframe followed by the operation specific machine coordinates. Itis shown that one unified model is capable of predicting forces formultiple metal cutting operations. The generalized modeling ofmetal cutting operations allows the simulation of part machiningwith multiple operations and various tools. The model will beextended to develop unified chatter stability laws for multipleoperations.

Acknowledgment

This research is supported by an NSERC CANRIMT (www.nserc-canrimt.org) grant, and the cutting tests have been conducted on aMori Seiki NMV5000 CNC machining center loaned by MTTRF(www.mttrf.org). The cutters are provided by Sandvik Coromant.

References

[1] H.,J. Fu, R.E. DeVor, S.G. Kapoor, A mechanistic model for the prediction of theforce system in face milling operations, Journal of Engineering for Industry106/1 (1984) 81.

[2] K.F. Ehmann, S.G. Kapoor, R.E. DeVor, I. Lazoglu, Machining process modeling:a review, Transactions of ASME, Journal of Manufacturing Science andEngineering 119 (4B) (1997) 655–663. November 1997.

[3] E. Armarego, The unified-generalized mechanics of cutting approach—a steptowards a house of predictive performance models for machining operations,Machining Science and Technology 4/3 (2000) 319–362.

[4] E. Budak, Y. Altintas, E. Armarego, Prediction of milling force coefficients fromorthogonal cutting data, Journal of Manufacturing Science and Engineering118 (1996) 216.

[5] S.D. Merdol, Y. Altintas, Mechanics and dynamics of serrated cylindrical andtapered end mills, Journal of Manufacturing Science and Engineering 126 (2)(2004) 317.

[6] C.A. Luttervelt, T.H.C. van, Childs, I.S. Jawahir, F. Klocke, P.K. Venuvinod,Progress report of the clrp working group ‘ modelling of machining opera-tions’, Annals of the ClRP 47 (2) (1998) 588–626.

[7] Y. Altintas, Modeling approaches and software for predicting the perfor-mance of milling operations at MAL-UBC, International Journal of MachiningScience and Technology 4/3 (2000) 445–478.

[8] S.M. Afazov, S.M. Ratchev, J. Segal, Modelling and simulation of micro-millingcutting forces, Journal of Materials Processing Technology 210 (2010)2154–2162.

[9] T. Ozel, M. Sima, A.K. Srivastava, B. Kaftanoglu, Investigations on the effects ofmulti-layered coated inserts in machining Ti-6Al-4V alloy with experimentsand finite element simulations, Annals of CIRP 59/1 (2010) 77–80.

[10] Y. Altintas, X. Jin, Mechanics of micro-milling with round edge tools, Annalsof CIRP 60/1 (2011).

[11] D. Montgomery, Y. Altintas, Mechanism of cutting force and surface genera-tion in dynamic milling, Transactions of ASME, Journal of Engineering forIndustry 113 (1991) 160–168.

[12] J. Ko, Y. Altintas, Time domain model of plunge milling operation, Interna-tional Journal of Machine Tools and Manufacture 47 (9) (2007) 1351–1361.

[13] S. Engin, Y. Altintas, Mechanics and dynamics of general milling cutters. PartI: helical end mills, International Journal of Machine Tools and Manufacture41 (15) (2001) 2195–2212.

[14] S. Engin, Y. Altintas, Mechanics and dynamics of general milling cutters. Part II:inserted cutters, International Journal of Machine Tools and Manufacture 41 (15)(2001) 2213–2231.

[15] F. Atabey, I. Lazoglu, Y. Altintas, Mechanics of boring processes–Part I,International Journal of Machine Tools and Manufacture 43 (5) (2003)463–476.

[16] ISO 3002, 1982, Basic quantities in cutting and grinding.[17] ISO 13399, 2006, Cutting Tool Data Representation and Exchange—Part I.[18] Y. Altintas, Manufacturing Automation, Cambridge University Press, 2000.[19] CUTPRO–Advanced Machining Simulation Software (www.malinc.com).[20] J. Roukema, Mechanics and dynamics of drilling, The University of British

Columbia, 2006.[21] F. Atabey, I. Lazoglu, Y. Altintas, Mechanics of boring processes–Part

II—multi-insert boring heads, International Journal of Machine Tools andManufacture 43 (2003) 477–484.