ARTICLE IN PRESS

International Journal of Machine Tools & Manufacture 49 (2009) 1070–1076

Contents lists available at ScienceDirect

International Journal of Machine Tools & Manufacture

0890-69

doi:10.1

� Corr

E-m

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

Thrust force model for vibration-assisted drilling of aluminum 6061-T6

Simon S.F. Chang, Gary M. Bone �

McMaster Manufacturing Research Institute (MMRI), McMaster University, 1280 Main St. W., Hamilton, Ontario, Canada L8S 4L8

a r t i c l e i n f o

Article history:

Received 6 March 2009

Received in revised form

23 July 2009

Accepted 26 July 2009Available online 5 August 2009

Keywords:

Drilling

Metal cutting

Vibration assistance

Ultrasonic assistance

Vibration-assisted drilling

Ultrasonic assisted drilling

55/$ - see front matter & 2009 Elsevier Ltd. A

016/j.ijmachtools.2009.07.011

esponding author. Tel.: +1905 525 9140x2759

ail address: gary@mcmaster.ca (G.M. Bone).

a b s t r a c t

Vibration assistance has increasing applications in metal removal processes. This method induces high-

frequency and low-amplitude vibration in the feed direction during cutting, and has the potential to

reduce cutting forces leading to improved surface quality and reduced tool wear. Note that this cutting

process is distinct from ultrasonic machining. This paper presents a thrust force model to predict the

thrust force during vibration-assisted drilling of aluminum 6061-T6. This model incorporates plowing

force and strain rate-dependent shear strength to provide more accurate predictions than the existing

model. The results of 72 drilling experiments with TiN-coated standard twist drills are reported. The

predictions from the developed thrust force model are compared with the experimental results. The

comparison demonstrates that the maximum deviation between the predictions and the averaged

values of the experimental measurements is 20% using the existing model and only 7% using the

proposed model.

& 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Conventional metal cutting methods, such as drilling, producerelatively high cutting forces and low machined surface quality.High cutting forces generally increase tool wear, and reducemachined surface quality. This directly affects the post-processingefforts such as surface finishing and deburring, leading toincreased production cost. There are various methods to reducetool wear and improve surface finish. These include using aspecial coating on the tool; changing (typically reducing) thematerial removal rate (MRR); or even laser-assisted machining,which alters the mechanical properties of the workpiece material.One recent and promising technique is known as ultrasonicassisted or vibration-assisted machining.

Vibration-assisted machining is a pure mechanical process thatdoes not require sacrificing MRR or altering the mechanical pro-perties of the workpiece material. This technique typically induceshigh-frequency (41000 Hz) and low-amplitude (o0.015 mm)vibration in the feed direction of a cutting process. It has beenshown that this technique can reduce thrust force and improvesurface quality. One application of vibration-assisted machining isvibration-assisted drilling (VAD) [1–5]. We have previously shownthat under preferable vibration conditions, the thrust force can bereduced by VAD, while poor choice of vibration conditions canresult in increase in thrust force [1]. Modeling and predictingthrust force is important for finding these preferable conditions.

ll rights reserved.

1; fax: +1905 572 7944.

In general there are two methods to predict thrust force: finiteelement modeling and analytical modeling. This paper presentsthe development of an analytical thrust force model for VADthat extends the existing model and provides more accuratepredictions.

Analytical thrust force models for conventional drilling arewell established. They include work by Wiriyacosol and Armarego[6], Armarego and Wright [7], Watson [8,9], Elhachimi et al.[10,11], and Lopez de Lacalle et al. [12]. However, due to thedynamic nature of VAD, these models cannot be directly applied.Wang et al. [13] analyzed the instantaneous uncut chip thicknessto model the thrust force and torque in VAD under differentvibration frequencies. Zhang et al. [14] used a similar approach tostudy VAD under different vibration amplitudes.

Other related cutting force models include the dynamic cuttingforce model with the presence of regenerative vibrations orchatter vibrations. Altintas [15] presented the dynamics of thegeneral metal cutting process. Budak and Altintas [16,17] studiedand modeled the dynamic cutting forces for milling with chattervibrations. Their studies modeled the variation of uncut chipthickness by analyzing the instantaneous tool location and theprofile of the machined surface. Li and Li [18] modeled thedynamic cutting forces for milling by modeling the variation ofuncut chip thickness and strain rate-dependent shear strength.Roukema and Altintas [19–21] presented the modeling of thedynamic cutting forces for drilling by modeling the variation ofuncut chip thickness. Wu [22], Ismail et al. [23], Chandiramaniand Pothala [24], and Moufki et al. [25] modeled the dynamiccutting force with the presence of plowing. The plowing forces aremodeled analytically by determining the displaced volume of the

ARTICLE IN PRESS

Nomenclature

hf, hl axial/dynamic uncut chip thickness (mm)Tk axial location of segment k (mm)zðtÞ; _zðtÞ axial displacement/velocity (mm, mm/s)V cutting velocity (mm/s)D diameter of the drill (mm)DVi displaced volume (mm3)b0 drill helix angle (rad)2p drill point angle (rad)W drill web thickness (mm)lnd dynamic friction angle (rad)gnd dynamic normal rake angle (rad)fnd dynamic shear angle (rad)Zd dynamic feed angle (rad)ri effective radius of the ith element (mm)F feedrate (mm/rev)g flank clearance angle (rad)Dli length of each segment (mm)

2W0 length of the chisel edge (mm)mc mean coefficient of friction of the tool–work interfacezmax(y) maximum depth of removed material (mm)M number of elements on each cutting lipDdk plowing depth (mm)Fpt, Fpx plowing force in thrust and horizontal direction (N)Fl,C, Fl,T principle cutting force (N)DPi resultant thrust force for the ith element (N)y rotational angle of the drill (rad)Dy rotational difference between each segment (rad)ti shear strength (MPa)fsp specific plowing force (N/mm3)n spindle speed (rev/s)e; _e strain, strain rate (s�1)t time (s)k total number of segment on each elementDw width of each segment (mm)A vibration amplitude (mm)f vibration frequency (Hz)

S.S.F. Chang, G.M. Bone / International Journal of Machine Tools & Manufacture 49 (2009) 1070–1076 1071

workpiece. Lee and Altintas [26] and Wang and Zheung [27]modeled the plowing force empirically, and verified that plowingforce can be significant.

The prior model of VAD ignored both the plowing forcecomponent and the strain rate dependence of the material. Thispaper presents a novel thrust force model for VAD thatincorporates these two factors. Our modeling approach isconsistent with the approaches used in the modeling of dynamiccutting force in metal cutting with the presence of chattervibrations discussed above. It will be shown that by modelingthe dynamic uncut chip thickness alone as is done in the priorVAD force model, the model fails to predict the thrust force forVAD accurately, while the proposed model improved the accuracyof the force predictions and the prediction of the favorablevibration condition that minimizes thrust force. In Section 2, thetheoretical development of the model is presented. In Section 3,comparisons between model predictions and experimental mea-surements are presented. Conclusions are given in Section 4.

Fig. 1. Schematic of determining axial uncut chip thickness.

2. Thrust force model

2.1. Instantaneous axial uncut chip thickness

The effective (also known as dynamic) cutting geometries of adrill vary with radius. Therefore a drill is typically divided intoelements along the cutting lips direction and the elements areanalyzed individually using a conventional mechanistic cuttingmodel. The total thrust force is obtained by summing all of thethrust force components of the individual elements [6–12].Following the methodology of the dynamic cutting force modeling[13–27], the instantaneous uncut chip thickness for VAD can beestimated by studying the instantaneous displacement andvelocity of the tool. The displacement equals the summation ofthe displacement due to the feed, Fnt, and the displacement due tothe vibration, A sin(2pft), as follows:

zðtÞ ¼ A sinð2pftÞ þ Fnt ð1Þ

where A and f are the vibration amplitude (mm) and frequency(Hz), respectively; F is the feedrate (mm/rev); n is the spindlespeed (rev/sec); and t is the time (s). Similarly, the instantaneousvelocity is

_zðtÞ ¼ 2pfA cosð2pftÞ þ Fn ð2Þ

To determine the axial uncut chip thickness, which is critical toestimate cutting forces, it is necessary to monitor the maximumdepth of removed material at the rotational location of interest,zmax(y). Transforming the independent variable from time (t) torotational angle of the drill (y) by substituting y ¼ 2pnt intoEq. (1) gives

zðyÞ ¼ A sinFyn

� �þ

fy2p

ð3Þ

zmax(y) can be estimated by monitoring the axial location of thecutting lips prior to the current instant. For a two-flute drill, wherethere are two cutting lips at p radians away from each other, all theaxial locations of the cutting lips prior to current instant are equal toz(y�qp), where q is a positive integer (see Fig. 1). Therefore, themaximum axial location of the cutting lips prior to the currentinstant, i.e. the depth of the materials immediately in front of thecutting lips at the current instant zmax(y) equals

zmaxðyÞ ¼ zðy�mpÞ ð4Þ

where m is the minimum positive integer that satisfies theexpression

zðy�mpÞ4zðy� ½mþ 1�pÞ ð5Þ

ARTICLE IN PRESS

S.S.F. Chang, G.M. Bone / International Journal of Machine Tools & Manufacture 49 (2009) 1070–10761072

For the example shown in Fig. 2, zmax(y) ¼ z(y�p) and m ¼ 1.The axial uncut chip thickness hf therefore equals

hf ¼zðyÞ � zmaxðyÞ if zðyÞ4zmaxðyÞ0 otherwise

�ð6Þ

The dynamic uncut chip thickness hl can then be calculated byanalyzing the drill geometry as in [6–9]:

hl ¼ hf sinðpÞ cosðz=2Þ ð7Þ

where

z ¼ tan�1½tanðoÞ cosðpÞ� ð8Þ

o ¼ sin�1 W

ri

� �ð9Þ

In Eqs. (7)–(9), p is the drill point angle, W is the drillweb thickness, and ri is the effective radius of the ith element(see Fig. 3).

2.2. Plowing force model

Because of the oscillation of the drill during VAD, a wavymachined surface is produced after each half revolution of thedrill. When the cutting edges engage the workpiece again,plowing can occur, as shown in Fig. 4. If the volume of the

Fig. 3. End view of a drill showing its element on one of the cutting lips.

Fig. 4. (a) After the first cut (half revolution for a two-flute drill), a wavy machined

surface is formed. (b) In subsequent cut, the flank surface of the drill may come in

contact with the machined surface (even at multiple locations), causing plowing of

material.

Fig. 2. Schematic of determining zmax(y) of the tool.

workpiece displaced by the tool is known, the plowing force canbe estimated. Based on the analysis by Wu [22], the resultingforces for the ith element are

Fpt ¼ fspDVi ð10Þ

Fpx ¼ mcFpt ð11Þ

In Eqs. (10) and (11), Fpt and Fpx are the plowing force componentsin the thrust and horizontal directions, respectively; fsp is theexperimentally determined specific plowing force; DVi is thedisplaced volume; and mc is the mean friction coefficient of thetool–work interface. DVi will now be estimated by considering thetool profile and the maximum depth of the machined surface.

The tool profile can be modeled by dividing each drill elementinto small segments along the drill flank direction. Fig. 5 showsthe difference between the drill elements and the segments. Theaxial location of segment k on each drill element can be calculatedusing

Tk ¼ zðyÞ �kDli tanðgÞ

sinðpÞ cosðz=2Þð12Þ

where k refers to the kth segment, Dli is the length of eachsegment, and g is the flank clearance angle. The geometry of a drillelement is shown is Fig. 6. At any instant, the depth of themachined surface corresponding to each drill element is given by

zmax;k0 ¼ A sin

f ðy� jp� kDyÞn

� �þ

Fðy� jp� kDyÞ2p

k ¼ 1;2; . . . ; k ð13Þ

In Eq. (13), Dy is the rotational difference between each segment(see Fig. 5), k is the total number of segments on each element,and j is the minimum positive integer that satisfies the

ARTICLE IN PRESS

Fig. 6. Geometry of one element on the drill cutting edge.

Fig. 5. Geometry of drill elements and segments.

S.S.F. Chang, G.M. Bone / International Journal of Machine Tools & Manufacture 49 (2009) 1070–1076 1073

inequality:

A sinf ðy� jp� kDyÞ

n

� �

þFðy� jp� kDyÞ

2p 4A sinf ðy� ðjþ 1Þp� kDyÞ

n

� �

þFðy� ðjþ 1Þp� kDyÞ

2p ð14Þ

The theory behind Eq. (14) is same as the theory behind Eq. (5).The plowing depth of the kth segment on a drill element can thenbe calculated:

Ddk ¼Tk � z0max; k if Tk4z0max;k

0 otherwise

�ð15Þ

The displaced volume per segment can then be calculated bymultiplying Ddk with Dli and the width of the element, Dw:

DVi ¼Xk

k¼1

DdkDliDw ð16Þ

where

Dli ¼W þ ½W coso0 þ ðD=2Þcoso0 � ði� 1=2ÞDw�tanO2k

ð17Þ

Dw ¼ D coso0 � D0 coso0

2M sin pð18Þ

o0 ¼ tan�1 2W

D

� �ð19Þ

and

o0 ¼ sin�1 W

W 0

� �ð20Þ

where M equals the number of elements in one cutting lip, D is thediameter of the drill, and W0 equals half the length of the chiseledge (see Fig. 3).

2.3. Strain rate-dependent shear strength model

During metal cutting, the strain rate along the primary andsecondary shear zone reaches 103–106 s�1. These strain rates aresignificantly higher than the nominal strain rate (10�3–10�1 s�1)used to determine generic material properties. Therefore, insteadof using the generic value, the Johnson–Cook model will be usedto estimate the shear strength ti. The empirical constants used inthe Johnson–Cook model for aluminum 6061-T6 were reported byGuo [28]. These constants were fine-tuned slightly using some ofthe experimental results, which will be presented in Section 3. Inthe proposed model, ti is modeled by the following equations:

ti ¼1

2ð247:5þ 77:4e0:676Þ 1þ C ln

_e0:01

� �� �ð21Þ

C ¼ 0:058� 0:194hl � 0:003V ð22Þ

e ¼ cos gnd

sinfnd cosðfnd � gndÞð23Þ

_e ¼ V cos gnd

0:005 cosðfnd � gndÞð24Þ

In Eqs. (21)–(24), ti is the ultimate shear strength (MPa), e and _eare the strain and strain rate, respectively, V is the instantaneouscutting velocity (mm/s), and C is a unitless parameter. At a highervibration frequency, because of the increase in strain rate, theshear strength of the material will be increased, resulting in ahigher thrust force.

ARTICLE IN PRESS

Fig. 7. Vibration-assisted workpiece holder.

S.S.F. Chang, G.M. Bone / International Journal of Machine Tools & Manufacture 49 (2009) 1070–10761074

2.4. Thrust force model

The thrust force for VAD can be predicted by combining theresults of Sections 2.1–2.3 with a conventional mechanistic model.The principle cutting forces along the cutting lips can becalculated as discussed in [29]:

Fl;C ¼thlDw cosðlnd � gndÞ

sinfnd cosðfnd þ lnd � gndÞð25Þ

Fl;T ¼thlDw sinðlnd � gndÞ

sinfnd cosðfnd þ lnd � gndÞð26Þ

In Eqs. (25) and (26), lnd is the dynamic friction angle, gnd is thedynamic normal rake angle, and fnd is the dynamic shear angle.Based on the methodology from [6–9], these angles can becalculated using the following equations:

kr ¼ tan�1½�tanðpÞ cosðoÞ� ð27Þ

ri ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD coso0 � i�

1

2

� �Dw

� �2

þW2

sð28Þ

b ¼ tan�1 2ri tanðb0Þ

D

� �ð29Þ

gf ¼ tan�1 tanðbÞ sinðoÞsinðpÞ � cosðpÞ sinðoÞ tanðbÞ

� �ð30Þ

Zd ¼ tan�1_zðtÞ

2prin

� �ð31Þ

krd ¼ tan�1 sinðkrÞ

cosðkrÞ cosðZdÞ þ tanðlsÞ sinðZdÞ

� �ð32Þ

lsd ¼ sin�1½cosðpÞ sinðZdÞ þ sinðpÞ cosðoÞ cosðZdÞ� ð33Þ

gfd ¼ gf þ Zd ð34Þ

gnd ¼ tan�1tanðgfdÞ cosðlsdÞ

sinðkrdÞþ

sinðlsdÞ

tanðkrdÞ

� �ð35Þ

lnd ¼p6þgnd

2ð36Þ

fnd ¼p4þ gnd � lnd ð37Þ

In Eq. (29), b0 is the drill helix angle. Finally, the resultant thrustforce for each element can be calculated as

DPi ¼ Fl;T sin p cosZd þ Fl;C sinZd þ Fpt ð38Þ

The total thrust force can be found by adding all the elementalthrust force components together.

3. Experimental apparatus

3.1. Vibration-assisted workpiece holder

In order to perform experimental studies on VAD, it isnecessary to design an apparatus to produce the necessary axialvibrations between the tool and the workpiece that is compatiblewith conventional CNC machining center. In vibration-assistedturning, vibrations are typically generated on the tool. Designing avibration-assisted tool holder for drilling is not trivial because ofthe connection needed between the rotating spindle and theexternal power source of the vibration actuator. Moreover,

because of the length of the tool, vibrating the tool axially alsopotentially increases tool wobbling. In this study, it is moredesirable to produce axial vibrations on the workpiece. While therelative displacement between the tool and the workpieceremains unchanged, this approach provides a simple designsolution and reduces the potential of tool wobbling. A custom-designed piezoelectric actuated vibration-assisted workpieceholder was designed and tested for this purpose. Fig. 7 showsthe schematic and a photograph of the workpiece holder. Detailsof the mechanical and electrical design of this hardware arepresented in Chapter 5 of [30].

3.2. Experimental setup

Experiments have been conducted to verify the accuracy of thedeveloped model. Each test was performed on a Makino MC56-5XA horizontal CNC machine tool, with the vibration-assistedworkpiece fixture attached onto a table dynamometer, which wasattached onto the machine table. The vibration amplitude waschosen to be 0.002 mm, while the vibration frequency was variedfrom 4 to 12 kHz in 2 kHz increments. Note that the static stiffnessof a typical 5-axis horizontal machine center is 30 kN/mm [31],and the weight of the machine table is 58 kg. At the magnitude ofthe induced vibration force, the resultant vibration amplitude ofthe machine table at the lowest testing frequency (4 kHz) is2.76�10�6 mm, and therefore is negligible. The drills used were4.0 mm TiN-coated standard twist drills. The cutting conditionschosen were 4000 rpm spindle speed and 0.06 mm/rev drill feed.The workpieces were 3.18-mm-thick Al 6061-T6 plates(25 mm�25 mm). Four drilling experiments were performed foreach cutting and vibration conditions. Thrust forces weremeasured using Kistler Type 9255B table dynamometer withsampling frequency equal to 10 times the corresponding vibrationfrequency to avoid aliasing. The dynamometer has a naturalfrequency of 2 kHz and stiffness of 3000 kN/mm (Kistler Instru-mente AG [32]). At the magnitude of the induced vibration force,the resultant vibration amplitude of the dynamometer at thelowest testing frequency (4 kHz) is 1.22�10�5 mm, and againis negligible. The mean values of the thrust forces for eachcutting and vibration condition were calculated for subsequentcomparison with the mean values of the corresponding modelpredictions.

4. VAD results and discussion

The experimental mean thrust force results are compared withthe corresponding simulation results in Fig. 8. The solid datapoints represent simulation results (based on Section 2), andthe hollow circular data points represent experimental data. Themaximum deviation between experimental results and thesimulations was 10%. When the vibration frequency equals10 kHz, the thrust force is minimized, with a reduction of 16%.

ARTICLE IN PRESS

Fig. 9. Comparison between different simulation results.

Fig. 8. Comparison between simulations and experimental results.

S.S.F. Chang, G.M. Bone / International Journal of Machine Tools & Manufacture 49 (2009) 1070–1076 1075

Note that the thrust force reduction is dependent on the materialand cutting conditions.

Fig. 9 presents a comparison between the mean value of theexperimental thrust force measurements (averaged over each setof four tests) and simulation results with both the plowing forceand strain rate-dependent shear strength models (simulation),without the plowing force model (w/o Fpt), and without plowingforce and shear strength models (w/o Fpt and ti). Note that themodel w/o Fpt and ti represents the simulation results of the priormodels. The comparison shows that the mean thrust forcepredictions fall within 20% of the experimental measurementswhen using the prior models, and 7% using the proposed model.The plot also shows that at higher vibration frequencies includingti becomes more important. This is logical since increasing thevibration frequency increases the strain rate. Both experimentaland simulation results consistently showed that a favorablevibration frequency (10 kHz in this case) that minimize thrustforce exists. Note that we have obtained similar results for otherdrills and cutting conditions.

5. Conclusions

A novel analytical thrust force model for VAD has beenproposed. Instead of modeling only the variation of dynamicuncut chip thickness, the proposed model incorporates thepresence of plowing forces and the variation of shear strength

due to the changes in strain rate in VAD with the variation ofdynamic uncut chip thickness.

Comparisons between experimental results and model predic-tions have shown the reliability of the developed model. Themean values of the experimental thrust forces for all testedconditions fall within 77% of the model predictions, and theworst case error is 10%. The comparisons also demonstrated theimportance of modeling the plowing force and incorporatingstrain rate-dependent shear strength for producing accuratethrust force predictions. The proposed model predicted thevibration frequency where thrust force is minimized, within thefrequency resolution employed in our experiments (2 kHz).

This work is important for industrial applications becausereducing thrust force can reduce tool wear, reduce burr size, andincrease the quality of the machined surfaces. Moreover, becauseproducing the vibration may be difficult, the proposed model canalso assist the user to determine if the effort required to achievethe desired thrust force reduction is reasonable.

References

[1] S.F. Chang, G.M. Bone, Burr size reduction in drilling by ultrasonic assistance,Robot. Comput.-Integr. Manuf. 21 (2005) 442–450.

[2] X. Wang, L.J. Wang, J.P. Tao, Investigation on thrust in vibration drilling offiber-reinforced plastics, J. Mater. Process. Technol. 148 (2004) 239–244.

[3] K. Okamura, H. Sasahara, T. Segawa, M. Tsutsumi, Low-frequency vibrationdrilling of titanium alloy, Int. J. JSME C 49 (1) (2006) 76–82.

[4] J. Pujana, A. Rivero, A. Celaya, L.N. Lopez de Lacalle, Analysis of ultrasonic-assisted drilling of Ti6Al4V, Int. J. Mach. Tools Manuf. 49 (2009) 500–508.

[5] Y.S. Liao, Y.C. Chen, H.M. Lin, Feasibility study of the ultrasonic vibrationassisted drilling of Inconel superalloy, Int. J. Mach. Tools Manuf. 47 (2007)1988–1996.

[6] S. Wiriyacosol, E.J.A. Armarego, Thrust and torque prediction in drilling from acutting mechanics approach, Ann. CIRP 28 (1) (1979) 87–91.

[7] E.J.A. Armarego, J.D. Wright, Predictive models of drilling thrust and torque –a comparison of three flank configurations, Ann. CIRP 33 (1) (1984) 5–10.

[8] A.R. Watson, Geometry of drill elements, Int. J. Mach. Tools Des. Res. 25 (3)(1985) 209–227.

[9] A.R. Watson, Drilling model for cutting lip and chisel edge and comparison ofexperimental and predicted results. I—initial cutting lip model, Int. J. Mach.Tools Des. Res. 25 (4) (1985) 347–365.

[10] M. Elhachimi, S. Torbaty, P. Joyot, Mechanical modeling of high speed drilling.1: predicting torque and thrust, Int. J. Mach. Tools Manuf. 39 (1999) 553–568.

[11] M. Elhachimi, S. Torbaty, P. Joyot, Mechanical modeling of high speed drilling.2: Predicted and experimental results, Int. J. Mach. Tool. Manuf. 39 (1999)569–581.

[12] L.N. Lopez de Lacalle, A. Rivero, A. Lamikiz, Mechanistic model for drills withdouble point-angle edges, Int. J. Adv. Manuf. Technol. 40 (5–6) (2009)447–457.

[13] L.P. Wang, L.J. Wang, Z.J. Yang, Prediction and computer simulation ofdynamics thrust and torque in vibration drilling, Proc. Inst. Mech. Eng. 212 (B)(1998) 489–497.

[14] L.B. Zhang, L.J. Wang, X.Y. Liu, H.W. Zhao, X. Wang, H.Y. Luo, Mechanical modelfor predicting thrust and torque in vibration drilling fibre-reinforcedcomposite materials, Int. J. Mach. Tools Manuf. 41 (2001) 641–657.

[15] Y. Altintas, in: Manufacturing Automation, first ed., Cambridge UniversityPress, Cambridge, 2000.

[16] E. Budak, Y. Altintas, Analytical prediction of chatter stability in milling—part1: general formulation, J. Dyn. Syst. Meas. Control 120 (1998) 22–30.

[17] E. Budak, Y. Altintas, Analytical prediction of chatter stability in milling—part2: general formulation to common milling systems, J. Dyn. Syst. Meas. Control120 (1998) 31–36.

[18] H. Li, X. Li, Modeling and simulation of chatter in milling using a predictiveforce model, Int. J. Mach. Tools Manuf. 40 (2000) 2047–2071.

[19] J.C. Roukema, Y. Altintas, Time domain simulation of torsional-axialvibrations in drilling, Int. J. Mach. Tools Manuf. 46 (2006) 2073–2085.

[20] J.C. Roukema, Y. Altintas, Generalized modeling of drilling vibrations. Part I:Time domain model of drilling kinematics, dynamics and hole formation, Int.J. Mach. Tools Manuf. 47 (2007) 1455–1473.

[21] J.C. Roukema, Y. Altintas, Generalized modeling of drilling vibrations. Part II:chatter stability in frequency domain, Int. J. Mach. Tools Manuf. 47 (2007)1474–1485.

[22] D.W. Wu, A new approach of formulating the transfer function for dynamiccutting process, ASME Trans. J. Eng. Ind. 111 (1989) 37–47.

[23] F. Ismail, M.A. Elbestawi, R. Du, K. Urbasik, Generation of milled surfacesincluding tool dynamics and wear, J. Eng. Ind. 115 (1993) 245–252.

[24] N.K. Chandiramani, T. Pothala, Dynamics of 2-dof regenerative chatter duringturning, J. Sound Vib. 290 (2006) 448–464.

ARTICLE IN PRESS

S.S.F. Chang, G.M. Bone / International Journal of Machine Tools & Manufacture 49 (2009) 1070–10761076

[25] A. Moufki, A. Devillez, M. Segreti, D. Dudzinski, A semi-analytical model ofnon-linear vibrations in orthogonal cutting and experimental validation, Int.J. Mach. Tools Manuf. 46 (2006) 436–449.

[26] P. Lee, Y. Altintas, Prediction of ball-end milling forces from orthogonalcutting data, Int. J. Mach. Tools Manuf. 36 (1996) 1059–1072.

[27] J. Wang, C.M. Zheung, An analytical force model with shearing and ploughingmechanisms for end milling, Int. J. Mach. Tools Manuf. 42 (2002) 761–771.

[28] Y.B. Guo, An integral method to determine the mechanical behavior ofmaterials in metal cutting, J. Mater. Process. Technol. 142 (2003) 72–81.

[29] E.M. Trent, P.K. Wright, Metal Cutting, fourth ed., Butterworth Heinemann,Boston, 2000.

[30] S.F. Chang, Ultrasonic assisted drilling, M.A.Sc. Thesis, McMaster University,2003.

[31] D.A. Stephenson, J.S. Agapiou, in: Metal Cutting Theory and Practice, seconded., CRC Press, Boca Raton, FL, 2006.

[32] Kistler Catalogue—Force Sensors, Kistler Instrumente AG, 2005.