Especial

ARTICLE IN PRESS

International Journal of Machine Tools & Manufacture 49 (2009) 586–598

Contents lists available at ScienceDirect

International Journal of Machine Tools & Manufacture

0890-69

doi:10.1

� Corr

E-m

simon.p

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

Modeling of dynamic micro-milling cutting forces

Mohammad Malekian a, Simon S. Park a,�, Martin B.G. Jun b

a Micro Engineering Dynamics Automation Laboratory (MEDAL), Department of Mechanical and Manufacturing Engineering,

University of Calgary, 2500 University Dr. NW, Calgary, Alberta, Canada T2N 1N4b Laboratory for Advanced Multi-scale Manufacturing, Department of Mechanical Engineering, University of Victoria,

PO Box 3055 STN CSC, Victoria, British Columbia, Canada V8W 3P6

a r t i c l e i n f o

Article history:

Received 14 May 2008

Received in revised form

10 February 2009

Accepted 14 February 2009Available online 9 March 2009

Keywords:

Micro-machining

Cutting forces

Receptance coupling

Ploughing

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

016/j.ijmachtools.2009.02.006

esponding author. Tel.: +14303 220 6959; fax

ail addresses: [email protected] (M. Mal

[email protected] (S.S. Park), [email protected]

a b s t r a c t

This paper investigates the mechanistic modeling of micro-milling forces, with consideration of the

effects of ploughing, elastic recovery, run-out, and dynamics. A ploughing force model that takes the

effect of elastic recovery into account is developed based on the interference volume between the tool

and the workpiece. The elastic recovery is identified with experimental scratch tests using a conical

indenter. The dynamics at the tool tip is indirectly identified by performing receptance coupling analysis

through the mathematical coupling of the experimental dynamics with the analytical dynamics. The

model is validated through micro end milling experiments for a wide range of cutting conditions.

& 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Highly accurate miniaturized components that are made upof a variety of engineering materials play key roles in thefuture development of a broad spectrum of products [1]. Manyinnovative products require higher functionality with significantlydecreased size; however, conventional fabrication methods usingphotolithographic fabrication methods are not applicable to allengineering materials, and the processes are slow and expensiveand limited to essentially planar geometries [2]. To overcome thechallenges, micro-mechanical machining processes can be utilizedto remove materials mechanically using a miniature tool to createcomplex three-dimensional shapes using a variety of engineeringmaterials [3,4]. Micro-mechanical machining techniques bringmany advantages to the fabrication of micro-sized features. Theycan produce micro-components cost-effectively because there isno need for expensive photolithographic masks. The flexibility andefficiency of micro-machining processes using miniature cuttingtools allows for the economical fabrication of smaller batch sizescompared with other processes [5].

Due to the miniature nature of the mechanical removalprocess, micro-machining operations are susceptible to excessivetool wear, noise, and poor productivity. Thus, the modelingand understanding of micro-cutting processes are important toimprove the machined part quality and increase productivity.

ll rights reserved.

: +1403 282 8406.

ekian),

(M.B.G. Jun).

However, the conventional mechanistic modeling approachcannot be applied to micro-scale cutting. In micro end milling,the cutting edge radius of the end mill is comparable in size to thechip thickness [6]. As a result, no chip is formed when the chipthickness is below the minimum chip thickness [7,8]; instead, partof the work material plastically deforms under the edge of thetool, and the rest elastically recovers. This change in the chipformation process, known as the minimum chip thickness effectand the associated material elastic recovery, causes increasedcutting forces [9] and surface roughness [10] at low feed rates.Furthermore, when the chip actually forms during cutting with afinite edge radius tool, ploughing under the edge contributes to anincrease in the specific energy, also known as size effect.

Many researchers have investigated the effect of ploughing onthe size effect. Armarego and Brown [11] suggested that thegreater relative contribution of the ploughing forces with a blunttool is responsible for the increase in the specific cutting energy.Similarly, Lucca et al. [12] showed that the ploughing and elasticrecovery, which were used to explain the increase in the cuttingforce, of the workpiece along the flank face of the tool play asignificant role in micro-machining. Komanduri [13] studied theploughing mechanism experimentally by using sharp tools withextremely negative rake angles to replace the rounded-edge tools.

In order to understand the ploughing mechanisms, ploughingforce models have been developed by many researchers. Vogleret al. [9] made the first attempt at incorporating the effect ofminimum chip thickness into a micro end milling force model.They used the slip-line plasticity model developed by Waldorfet al. [14]. More complicated slip-line plasticity models thataccount for elastic–plastic deformation and elastic recovery have

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Nomenclature

Ap ploughed area (mm2)dx, dy dynamic tool deflection (mm)e errorft feed rate (mm/flute)Ft tangential force (N)Fr radial force (N)Fexp experimental force (N)Ftheo theoretical force (N)h chip thickness (mm)hc minimum chip thickness (mm)her height of elastic recovery (mm)Krc, Ktc radial and tangential cutting coefficients (N/mm2)

Kre, Kte radial and tangential edge coefficients (N/mm)Krp, Ktp radial and tangential ploughing coefficients (N/mm3)N number of flutespe elastic recovery (%)re edge radius (mm)r0 tool run-out (mm)Vp ploughed volume (mm3)Xc, Yc location of the tool centre (mm)G receptance for the assembled structure (m/N)H receptance for substructures (m/N)D rpm of the spindle (rev/min)y immersion angle (rad)ce clearance angle (rad)ct, cs geometric angles (rad)

M. Malekian et al. / International Journal of Machine Tools & Manufacture 49 (2009) 586–598 587

been developed by Jun et al. [15]. Fang [16] also developed auniversal slip-line model for rounded-edge tools. The finiteelement model approach has also been utilized by manyresearchers to model the micro-cutting process and to understandsize effect [17], machining stresses [18,19], and the influence ofcutting edge radius on wear resistance [20]. However, the majorityof these methods require many assumptions, and the parametersused in the model are difficult to estimate. There are a fewmechanistic models developed for micro end milling processes[21–24], but these models do not consider the effects of edgeradius, minimum chip thickness, elastic recovery, and tooldynamics together.

The objective of this paper is to develop a novel mechanisticmicro-milling cutting force model, based on the shearing andploughing-dominant cutting force regimes, that considers differ-ent effects, such as elastic recovery, run-out and dynamics.The mechanistic approach for cutting force modeling has beenvery effective for parameter estimation, force prediction, processmonitoring and control, and understanding of the cutting process.Therefore, development of a new mechanistic micro end millingforce model is important and will be useful for process under-standing and monitoring/control.

Micro end mills have small tool tip diameters; therefore,impact hammer testing cannot be applied directly to the microend mills, making it difficult to predict the tool tip dynamics.To overcome this, the receptance coupling (RC) technique isemployed to mathematically couple the spindle/micro-machineand arbitrary micro end mills with different geometries in theprediction of dynamic forces and vibrations.

We first identify the critical chip thickness, based on theedge radius and the experimentally obtained forces vs. feedrate curve. For the shearing-dominant cutting regime, i.e. chipthickness greater than the critical thickness, we use the conven-tional sharp-edge theorem to identify the cutting constants byperforming curve fittings from the experimental data. Whenthe chip thickness is smaller than the critical chip thickness,we consider a model for the ploughing-dominant cutting regime.We introduce the ploughing coefficient based on the ploughedarea. The elastic recovery rate of the workpiece is experimentallyidentified using a conical indenter and by observing the elasticdeformation after the scratch tests. The cutting force model isverified for Aluminum 6061 (Al6061).

The organization of the paper is as follows: Section 2depicts the experimental setup. Section 3 describes the metho-dology for predicting shearing and ploughing-dominant cuttingforces through mechanistic modeling, including the effectsof the tool tip dynamics, elastic recovery, and run-out andcompares the simulation with the experimentally obtainedforces. Section 4 illustrates the assumptions and limitations

associated with the model, and Section 5 concludes with thecontributions.

2. Experimental setup

We have utilized an ultra precision vertical CNC millingmachine (Kern Micro 2255) with a spindle that can rotate from60,000 to 160,000 rev/min (rpm). The base of the machine ispolymer concrete, which damps out external vibrations. Unlikemany micro CNC systems, the CNC machine used in this studyutilizes hybrid ball bearings, which provide higher stiffnessand linearity, and an elaborate lubrication system that allowsfor temperature stability during the high-speed rotations. Theaccuracy of the stage is 1mm. The experimental setup for thisstudy is depicted in Fig. 1.

The micro-tools used in this study were uncoated tungstencarbide (WC) micro end mills with 500mm diameter flat microend mills (PMT TS-2-0200-S) with the helix and clearance anglesof approximately 301 and 101, respectively. The tool overhanglength was 15 mm from the collet; and, this value remainedconstant so that the dynamics were not changed during theexperiments. The scanning electron microscopy (SEM) picture ofthe tip of the 500mm diameter carbide end mill is shown in Fig. 2.The edge radius of the tools was measured from the SEM picturesand observed to be approximately 2mm.

Several sensors, such as a table dynamometer, an acousticemission sensor, accelerometers and capacitance sensors, and anoptical vision system were used to capture various signals andmonitor the cutting processes. The piezo-electric table dynam-ometer (Kistler 9256C2) with an accuracy of 0.002 N measuredthe micro-cutting forces. The charge signals generated from theforce sensor were fed into the charge amplifiers (Kistler 9025B),which converted the charge signals into voltage signals. Thecalibration of the table dynamometer was performed using bothmodal impact hammer tests (Dytran 5800SL) and a force gauge(Omega DFG51-2) to verify the force measurement. The sensitivityof the dynamometer was 26 pC/N for X and Y directions. The noiselevel was approximately 0.005 N which was insignificant com-pared to the cutting forces. The frequency bandwidth of thedynamometer was found to be approximately 1500 Hz (Fig. 3)from the impact hammer tests.

The zero point in the Z direction was found by movingthe rotating tool down very slowly and looking at the acousticemission (AE) signal carefully. As soon as the tool touched theworkpiece, a sudden jump in the AE signal was observed, andthe position was set to zero. The forces were preprocessed bysubtracting air cutting forces from the measured cutting forcesthrough synchronization at each revolution of the spindle using

ARTICLE IN PRESS

Interface

Spindle

Capacitance Sensor

Workpiece

Accelerometer

AE Sensor

Table Dynamometer

XY Stage

MiteeTM Grip

X

Z

Y

Fig. 1. Experimental setup: (a) CNC micro-machine and (b) setup.

Fig. 2. SEM pictures of the tool.

00

5

10

15

Frequency [Hz]

Mag

nitu

de [N

/N]

Bandwidth

1

1000 2000 3000 4000 5000 6000 7000

Fig. 3. Frequency bandwidth of the miniature table dynamometer in the X

direction.

M. Malekian et al. / International Journal of Machine Tools & Manufacture 49 (2009) 586–598588

capacitance sensors (Lion Precision C3-D, RD20-2) with anaccuracy of 10 nm. The measured force signals while the spindlewas rotating without material removal (i.e. air cutting) weresubtracted from the cutting forces during the material removalafter synchronizing the two signals based on the capacitancesensor measurements. An AE sensor (Physical Acoustics Nano30)was used for capturing high-frequency vibrations. The acceler-ometers (Kistler 8778A500) were attached to the workpiece to

measure vibration signals in both the X and Y directions. Theworkpiece material was Al6061-T6 with a hardness of 95 HB.The workpiece was attached with the aid of Mitee-GripTM to aplate that could interface with the dynamometer.

3. Cutting force model

A mechanistic model has been developed to predict micro-milling forces for both shearing and ploughing-dominant cuttingregimes. The model assumes that there is a critical chip thicknessthat determines whether a chip will form or not. When machiningis performed at high feed rates, the effects of ploughing and elasticrecovery are insignificant enough to ignore, and the cuttingmechanism is considered to be shearing [10]. However, at lowerfeed rates, these effects are substantial and need to be taken intoaccount. As a result, two different cutting regimes have beendefined. Since the cutting forces in the axial direction are smallcompared with the planar directions, only the forces in the X andY directions have been considered in this study. The model hasbeen verified using experimental data for full and half immersioncutting conditions.

In micro-machining, the edge radius of the tools is consider-ably large compared to the uncut chip thickness; as a result, theso-called minimum chip thickness phenomenon occurs in micro-machining. Thus, when the uncut chip thickness is less than theminimum chip thickness (hc), no chip formation occurs and onlyploughing/rubbing takes place. Material separation occurs when

ARTICLE IN PRESS

Tooth Path 1

Surface generatedfrom previous tooth path

h er

h

h c

Surface generatedfrom current tooth path

Ploughing Dominant Regime

Shearing Dominant Regime

Fig. 5. Chip thickness in the ploughing-dominant regime.

Ci

Ci+1

Fi

h

Tooth Path 1

Surface to be generatedfrom tooth path 1

Surface generated

from tooth path 0 Si+1

Fi+1

hIi+1

h er

1

1

1

1

Si1

1

Surface at the ith rotational angle

E

D G

0

Si0

Ii0

θ i θi+1

Fig. 6. Chip thickness model considering elastic recovery [15].

M. Malekian et al. / International Journal of Machine Tools & Manufacture 49 (2009) 586–598 589

the uncut chip thickness is greater than the critical minimum chipthickness (hc), or at what is sometimes referred to as thestagnation point [25,26], when the material above the minimumchip thickness forms a chip and the material below the minimumchip thickness deforms under the edge with a partial elasticrecovery, resulting in material ploughing. The edge radius of thetool was 2mm, and it has been shown that the critical chipthickness for aluminum is approximately 0.3 of the edge radius[27,28]. As a result, the critical chip thickness is considered to beapproximately 0.7mm in this study.

We considered a helix angle of 301 when we discretized theaxial slices in the axial direction. Fig. 4 shows the ploughingprocess at the ith rotational angle in micro end milling, whenthe chip thickness is less than the minimum chip thickness for anarbitrary axial slice, where yi represents the angles at the ithrotational angle, h is the uncut chip thickness, her is the heightof elastic recovery, re is the edge radius, ce is the clearance angle,and Ap is the ploughed area (represented by the hatched area)at the rotational angle. The shaded area represents the ploughedmaterial. The ploughed volume, Vp, at the rotational angle, yi, canbe obtained by summing up the ploughed areas (Ap) of all theaxial slices along the cutting edge.

When the chip thickness, h, increases to greater than theminimum chip thickness, hc, the ploughing becomes negligible,and the elastic recovery drops to zero (Fig. 5). Thus, in micro endmilling, each flute goes through different material removalmechanisms in a single path, and the cutting mechanisms switchback and forth from the ploughing-dominant regime to theshearing-dominant regime [28], depending on the uncut chipthickness value, as shown in Fig. 5.

A comprehensive chip thickness model was developed in [15]to compute the correct chip thickness, including the effects of thetrochoidal tool path, minimum chip thickness, elastic recovery,and tool vibrations. Fig. 6 shows the surface generation and chipthickness computation in the presence of elastic recovery, whichis represented as the shaded region, for an arbitrary axial slice.Points C and F represent the tool centre and cutting edge locations,respectively. The superscript denotes the tooth pass number, andthe subscript represents the rotational angle. Point I is found atthe intersection between the previously generated surface fromthe previous tooth pass and the line connecting C and F for thecurrent tooth pass. The chip thickness can be formulated as [15]

h ¼maxð0; kCjiF

jik � kC

jiI

j�1i kÞ. (1)

The mechanistic micro-milling forces are obtained by consider-ing both the ploughing and shearing effects when the feed rate isless than the critical chip thickness. We first determine the

Current Tooth

Previous Tooth Path

Micro-Endmillh

θi

Ploughed Material

Fig. 4. Ploughing due to the finite ed

shearing cutting coefficients based on the experimental data.In the ploughing cutting, we introduce ploughing coefficients,which are the ploughing forces per unit of ploughed volume.The elastic recovery is identified using the instrumented scratchtests. The model also considers other effects, such as tool run-outand the dynamics of the tool at the tool tip that can be obtainedusing a capacitance sensor. Since experimental modal analysiscannot be performed directly on the tip of micro-tools and the

Path

h

reψe

Tool at θi

Ploughed Area Ap

her

ge radius in micro end milling.

ARTICLE IN PRESS

−101234

x 10−5

Rea

l [m

/N]

ExperimentalCurve fitted

M. Malekian et al. / International Journal of Machine Tools & Manufacture 49 (2009) 586–598590

spindle dynamics cannot be neglected, the receptance couplingtechnique is used to find the tool dynamics.

3.1. Tool dynamics – receptance coupling

Micro-tools are very small in diameter; therefore, tool deflec-tion can be significant during the cutting operations. This canresult in excessive tool vibration and cutting forces. In order toaccurately model micro-machining operations, it is important topredict the deflection of the tool subjected to the cutting forces. Inconventional machining, this can be done by accurate measure-ment of the dynamics, i.e. the frequency response function (FRF)at the tool tip, using an instrumented hammer and a displacementsensor. However, impact hammer testing cannot be directlyapplied in the determination of the dynamics of the micro endmills, since the diameter of micro end mills is very small, and thetools are fragile. Therefore, receptance coupling of the spindle/machining centre and the micro-tools is employed to extract thedynamics at the tool tip [29,30]. The RC method allows for themathematical coupling of the experimentally and analyticallyobtained substructure dynamics to predict the overall assembledsystem dynamics. Knowing the coupled FRF at the tool tip,the deflection of the tool due to the cutting forces, and its effectson the chip formation and surface generation can be furtherinvestigated.

The receptance of a structure is the relationship betweenthe applied force and the displacement of the structure in thefrequency domain. Knowing the approximate geometry ofthe micro end mills, finite element (FE) analysis can be employedto determine the receptance at the tool tip for different boundaryconditions, such as a cantilever beam condition. However, thismethod is not accurate in obtaining the actual FRF of the tool tip,since the tool is not perfectly clamped and the structuraldynamics of the machine contribute to the FRF of the tool tip[31]. The receptance coupling method overcomes this challengeby combining experimentally obtained spindle/machine dynamicswith arbitrary tool dynamics, which are determined with FEanalysis. Fig. 7 shows two substructures: Substructure A consistsof the lower portion of the micro end mill, and Substructure Bincludes the remainder of the tool and spindle. The aim of thereceptance coupling, in this case, is to obtain the FRF at the tool tip(i.e. point 1):

G11 ¼X1

F1¼ H11 �H12ðH22 þH33Þ

�1H21 (2)

where G and H denote the assembled and substructure dynamics,respectively. Since the rotational dynamics cannot be neglected,the dynamics can be re-represented as

x1

y1

" #¼

h11;ff h11;fm

h11;mf h11;mm

" #=

f 1

M1

" #) fX1g ¼ ½H11�fF1g (3)

θ1

θ2θ3

M2M3

f1f2

f3

x1

x2

x3

Sub. A

Sub. B

Fig. 7. Receptance coupling of a spindle and an arbitrary end mill [29].

where X is a vector with the components of translational (x) androtational (y) displacements, and F is a vector with force (f) andmoment (M) components.

The dynamics of Substructure A is obtained using FE analysis,while the dynamics of Substructure B is determined throughexperimental modal analysis. The author [32] has shown that,for an accurate prediction of the tip dynamics, the rotationaldynamics of the substructures cannot be ignored. Since theexperimental measurement of the rotational dynamics at Location3 is very difficult, the indirect method as outlined in [29] is usedbased on the dynamics measurements using two blank gaugecylinders (i.e. overhang lengths of 10 and 15 mm from the collet)of the same material as the tool.

For performing experimental modal analysis to find thedynamics of Substructure B, a miniature impact hammer (Dytran58008L) was used to excite the system. The displacementwas measured using a capacitance sensor (Lion Precision C3D).The signals were acquired, filtered, and transformed into thefrequency domain using the discrete Fourier transformation. Sincethe method is very sensitive to the signal noise, the impacthammer tests were repeated multiple times and averaged. Also,the resulting data were curve fitted to minimize the effect ofnoise.

In this study, the parameters used to formulate the dynamicsof Substructure A with the FE analysis of the tungsten carbide endmill were: density ¼ 14,300 kg/m3, Young’s modulus ¼ 580 GPa,Poisson’s ratio ¼ 0.28, and damping ratio E0.01. The dampingratio was obtained experimentally by performing the impacttest of blank cylinder with free-free boundary conditions. Theassembled dynamics at the tool tip were obtained using the RCmethod in Eq. (2), with the indirectly obtained rotationaldynamics, as depicted in Fig. 8. Two modes were observed inthe frequency range below 10,000 Hz. Therefore, the dynamics ofthe system in this range can be written as

G11ðjoÞ ¼X2

i¼1

o2n;i=ki

o2n;i �o2 þ 2zion;ijo

(4)

The modal parameters of the system, i.e. natural frequencies,stiffness and damping coefficients for the first and second modeswere obtained through the curve fitting method, as shown inTable 1. The curve fitting were performed based on the

−2

0−3

−2

−1

0

1

2x 10−5

Frequency [Hz]

Imag

inar

y [m

/N] Experimental Curve fitted

1000 2000 3000 4000 5000

5000

6000

0 1000 2000 3000 4000 6000

Fig. 8. Receptance coupled dynamics (G11) at the tool tip for the 500mm tool.

ARTICLE IN PRESS

Table 1Dynamical parameters for the tool tip.

First mode Second mode

on (Hz) 4035 5163

z 0.016 0.038

k (MN/m) 2.1425 0.5397

00

0.5

1

1.5

2

2.5

3

3.5

F [N

]

Shearing Dominant

Plo

ughi

ng D

omin

ant

Tran

sitio

n

Feed Rate [µm/flute]2 4 6 8 10

Fig. 9. Micro-milling resultant forces vs. feed rates at full immersion with depth of

cut of 100mm.

M. Malekian et al. / International Journal of Machine Tools & Manufacture 49 (2009) 586–598 591

minimization of the error between the theoretical and experi-mentally obtained FRFs using the steepest descent method. Thefirst and second modes occur at approximately 4035 and 5163 Hz,respectively. This dynamic model of the tool allows us to take thedeflection of the tool into account during the chip formationprocess. If the frequency of the forces are close the naturalfrequencies of the system, the forced vibration can be significantand immensely affect the micro-cutting operation and surfacegeneration. It has been assumed that the dynamics of the tool areequivalent in the X and Y directions, since the tool and spindlehave cylindrical symmetry.

The dynamics of the tool can be utilized to find the deflectionof the tool due to the cutting forces. This deflection affects thesurface generation and the cutting forces. If the cutting operationoccurs only in the X direction, the coordinates of the centre of theend mill can be obtained from:

XCi¼ f tN

D60

ti þ r0 sin yi þ dxi; YCi

¼ r0 cos yi þ dyi(5)

where ft is the feed rate, N is the number of flutes, D is the rpm ofthe spindle, y is the rotation angle, r0 is the spindle run-out, and dx

and dy are the dynamic tool deflections as a result of cutting forcesin the X and Y directions, respectively. Tool deflections that resultfrom the dynamics are obtained using convolution integrals:

dx ¼

Z t

0FxðxÞg11ðt � xÞdx; dy ¼

Z t

0FyðxÞg11ðt � xÞdx (6)

where Fx and Fy are the cutting forces in the X and Y directions,respectively, and g11 is the impulse response at the tool tip. Bothmodes were considered for the formulation of micro-cuttingforces.

The RC method was utilized to identify the dynamics of thetool tip for micro end mills, which cannot be acquired directlyusing the hammer testing. The method mathematically combinesthe dynamics of the micro-tool and the spindle, which wereobtained using FE method and experimental modal analysis,respectively. The joint rotational dynamics could not be ignoredand were identified indirectly using blank cylindrical tools.The deflection of the tool as a result of tool dynamics, whichcan significantly affect the micro-milling forces and the chipformation process, is considered in the developed force model.

3.2. Force model development in the shearing-dominant regime

When the chip thickness is bigger than the critical value,the cutting mechanism is assumed to be similar to the conven-tional cutting mechanism that considers the shearing and edgecoefficients. In milling operations, the tangential (dFts) and radial(dFrs) shearing cutting forces acting on a differential flute elementwith height dz, as shown in Fig. 4, can be modeled as follows,when the uncut chip thickness is greater than the minimum chipthickness value (h4hc) [33]:

dFrs ¼ ½KrchðyiðzÞÞ þ Kre�dz

dFts ¼ ½KtchðyiðzÞÞ þ Kte�dz (7)

where dz is the height of the differential flute element; Krc and Ktc

are the radial and tangential cutting coefficients, respectively;

and, Kre and Kte are the radial and tangential edge coefficients,respectively. The cutting coefficients represent shearing of theworkpiece, and the edge components represent friction betweenthe tool and the workpiece. In order to come up with the micro-cutting force model for the shearing-dominant regime, theidentification of cutting constants is imperative. These coefficientsare obtained from the experimental data; therefore the accuratemeasurement of the cutting forces is important. Since thefrequency bandwidth of the table dynamometer is not sufficientfor high-speed cutting operations (i.e. 1500 Hz), the measuredcutting forces are need to be compensated for the unwanteddynamics of the table dynamometer. A Kalman filter method isemployed as outlined in [34] to accurately measure the high-speed cutting forces based on the dynamics of the sensor asshown in Fig. 3.

Since the uncut chip thickness is usually bigger than thecritical chip thickness when the feed rate is higher than a criticalvalue, the cutting mechanism is mainly shearing, and the effectof ploughing can be neglected. This critical feed rate canbe identified from the forces in this shearing-dominant regime.Fig. 9 shows the root mean square (RMS) value of the resultantforce vs. feed rate. As can be observed, for feed rates larger thanapproximately 2.5mm/flute, the force has a linear trend. A linearcurve fit of the forces in this region is also shown in this figure. Inthe ploughing-dominant regime, the forces have increased valuesand vary smoothly, but do not follow the same linear behaviourand are bigger than the predicted values by the extrapolatedlinear curve fit (dashed line in Fig. 9). A transition region betweenthe ploughing and shearing regions has been defined, in whichthe forces do not follow a smooth curve. The force data in theshearing-dominant regime can be used to obtain the cutting andedge coefficients. Since the micro end mills generally have twoflutes, the average of the forces is nearly zero, and the ordinarymethod [33] of finding the coefficient does not work. Therefore,the cutting coefficients are found via a nonlinear curve fitting. Forthis purpose, the following error is minimized through a steepestdescent algorithm [35]:

e ¼Xn

i¼1

Xm

j¼1

ðFexpi;j� FtheoÞ

2 (8)

where n is the number of feed rates considered in the shearing-dominant region, m is the number of samples, Fexp is theinstantaneous experimental forces data (stars in Fig. 9), and Ftheo

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M. Malekian et al. / International Journal of Machine Tools & Manufacture 49 (2009) 586–598592

is the theoretical force obtained from the conventional sharp-edgetheorem using arbitrary initial values for the unknown coeffi-cients. Full immersion cutting data were used for this purpose,since they showed better agreement with the theoretical data. Toobtain the shearing-dominant cutting coefficients, 8 different feedrates (n) with 500 samples (m) for each feed rate were used.Experimental and theoretical forces were synchronized before theoptimization. The coefficients that minimized the error in Eq. (8)are shown in Table 2.

As can be observed in Table 2, the cutting coefficients werebigger than those of the conventional end mills, which can be aresult of round edges making the effective rake angle smaller oreven negative, whereas the edge coefficients for the micro endmills were much less. This is perhaps due to the smaller contactarea between the tool and the workpiece compared with macromilling operations.

Fig. 10 shows the experimental and theoretical forces obtainedusing the identified coefficients for the feed rate of 9mm/flute andthe full immersion cutting condition in X and Y directions basedon Eq. (7) with the run-out. It can be seen that the conventionalcutting theory with the coefficients shown in Table 2 can predictthe forces in the shearing-dominant regime properly and withgood accuracy. Also, the forces for the feed rate of 9mm/flute andthe half immersion cutting condition are shown in Fig. 11.Although the agreement between the experiments and simula-tions was not as good as what was observed for full immersion,there was still a good match between the data. The frequencycontent of the forces also shows good agreement between theexperiments and the simulation.

The modeling of the cutting forces in micro end milling, similarto the conventional end milling with the new shearing and edge

Table 2The cutting constants for the shearing dominant regime.

Parameter Ktc (N/mm2) Krc (N/mm2) Kte (N/mm) Kre (N/mm)

Al 6061 3650 2420 0.9 0.7

0−4−3−2−1012

Angle [Degree]

Fx [N

]

02468

1012

Mag

nitu

de

Experiment

Experiment

Simulation

Simulation

200 400 600 800 1000 1200 1400 1600 1800

0Frequency [Hz]

2000 4000 6000 8000 10000

Fig. 10. Comparison of forces between experiments and sim

coefficients, can appropriately predict the forces in the shearing-dominant region. However, it is unable to account for theincreased forces at lower feed rates, and another model for thisploughing-dominant region needed to be developed.

3.3. Force model development in the ploughing-dominant regime

In the ploughing-dominant regime, the micro-mill undergoesboth ploughing and shearing during the removal process. Chipformation does not occur when the uncut chip thickness is lessthan the minimum chip thickness; instead, there is ploughing andpartial elastic recovery of the material. The ploughing forces aremodeled as proportional to the volume of interference betweenthe tool and the workpiece. Many researchers have employed thisprocedure in the modeling of the forces due to interferencebetween the clearance face of the tool and the workpiece fororthogonal cutting [36,37], turning [38,39], and micro end millingprocesses [8]. However, none of these modeling approaches haveconsidered the effect of the elastic recovery during the ploughingprocess, as full elastic recovery was assumed for all materials. Ithas been shown that the effect of the elastic recovery on the microend milling process is substantial [40]. Thus, in this paper, insteadof simply assuming full elastic recovery, the ploughing forces aremodeled as proportional to the volume of interference betweenthe tool and the workpiece, considering the effect of the elasticrecovery.

Fig. 2 shows two different cases: the height of elastic recovery(her) is greater than or equal to re(1�cosce), i.e. heXre(1�cosce);and the height of elastic recovery (her) is less than re(1�cosce),i.e. herore(1�cosce). The elastic recovery height can be ex-pressed as her ¼ peh, where pe is the elastic recovery rate.The shaded area represents the ploughed area for both cases.Angle ap is the angle that the point on the rounded edge at theminimum chip thickness makes with respect to the y-axis, and itis given by

ap ¼ cos�1 1�h

re

� �(9)

0Frequency [Hz]

−4

−2

0

2

Fy [N

]

0

5

10

15

Mag

nitu

de

ExperimentSimulation

Simulation Experiment

0Angle [Degree]

200 400 600 800 1000 1200 1400 1600 1800

2000 4000 6000 8000 10000

ulation at full immersion and a feed rate of 9mm/flute.

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0−4

−3

−2

−1

0

1

2

Angle [Degree]

Fx [N

]

00

2

4

6

8

10

Frequency [Hz]

Mag

nitu

de

Experiment Simulation

Simulation

Experiment

−3

−2

−1

0

1

2

Fy [N

]0

1

2

3

4

5

6

Mag

nitu

de Simulation

Simulation

Experiment

Experiment

200 400 600 800 1000 1200 1400 1600 1800 0Angle [Degree]

200 400 600 800 1000 1200 1400 1600 1800

2000 4000 6000 8000 10000 0Frequency [Hz]

2000 4000 6000 8000 10000

Fig. 11. Comparison of forces between experiments and simulation at half immersion and a feed rate of 9mm/flute: (a) X and (b) Y.

hA

B

CD

reαp

Tool at θi

her

ψe

hA

B

C D

re

αp

ψs

Tool at θi

her

ψeψt

Fig. 12. Ploughing area for two different cases: (a) herXre(1�cosce) and (b) herore(1�cosce).

M. Malekian et al. / International Journal of Machine Tools & Manufacture 49 (2009) 586–598 593

For both cases, point B represents the centre of the circle or therounded edge, and point D the end of the arc. Line lBD is at thesame angle as the clearance angle from the vertical line. Point A isat the height of the elastic recovery.

When herXre(1�cosce), the ploughed area, Ap, indicated by theshaded area in Fig. 12(a), can be obtained as

Ap ¼ ABCD þ AABD � AABC (10)

where ABCD is the area of the arc segment, and AABD and AABC arethe areas of the triangles connecting the corresponding points.Area ABCD can be obtained by

ABCD �12r2

e ðap þceÞ (11)

The area AABD is given by

AABD ¼re

2

her � reð1� cos ceÞ

sin ce

� �¼ 1

2relAD

where lAD ¼her � reð1� cos ceÞ

sin ce

� �(12)

The area AABC can be obtained by

AABC ¼12relAB sinðap þ ce þ ctÞ (13)

where

lAB ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2

e þ l2AD

qct ¼ tan�1 lAD

re

� �(14)

Thus, the ploughed area Ap can be expressed as

Ap ¼12r2

e ðap þ ceÞ þ12relAD �

12relAB sinðap þ ce þ ctÞ (15)

When herore(1�cosce), the ploughed area, Ap, indicated by theshaded area in Fig. 12(b) can be obtained by

Ap ¼12r2

e ðap þ cs � sinðap þ csÞÞ (16)

where

cs ¼ cos�1 1�her

re

� �(17)

For a discretized disk element of a micro end mill flute, theploughing volume can be expressed as Vp ¼ Apdz, where dz is thethickness of the disk element. Since the ploughing forces can bemodeled as proportional to the volume of interference betweenthe tool and the workpiece, both radial and tangential ploughing

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forces can be written as a multiplication of a constant and theploughed volume. Therefore, the ploughing forces for eachdifferential flute element can be computed as

dFrp ¼ ðKrpAp þ KreÞdz

dFtp ¼ ðKtpAp þ KteÞdz(18)

where Krp and Ktp are the radial and tangential ploughing coeffi-cients, respectively, and Ap can be determined from Eqs. (15) or(16). The ploughing coefficients represent ploughing of theworkpiece material. Kre and Kte are the same edge coefficientsas in Eq. (7). Since the edge coefficients represent friction as theuncut chip thickness goes to zero, the edge components inthe ploughing-dominant regime must be the same as those in theshearing-dominant regime. The unit for the ploughing coefficientsis N/mm3.

A

B

C

a

b

c

Fig. 13. Finding elastic recovery through an indentation test using a conical

indenter.

0−0.8−0.6−0.4−0.2

00.20.4

Angle [Degree]

Fx [N

]

00

0.5

1

1.5

2

Frequency [Hz]

Mag

nitu

de

Simulation

Experiment

Simulation Experiment

200 400 600 800 1000 1200 1400 1600 1800

2000 4000 6000 8000 10000

Fig. 14. Comparison of forces between experiments and simu

Table 3The cutting constants and estimated parameter values for the ploughing-dominant

regime.

Parameter Ktp (kN/mm3) Krp (kN/mm3) ro (mm) pe (%)

Al 6061 790 1210 0.2 10

In order to develop a micro-cutting force model for theploughing-dominant regime, the identification of ploughingconstants (Krp, Ktp) is essential. The ploughing coefficients in theploughing-dominant regime can be identified in the same manneras the shearing coefficients in the shearing-dominant regime,based on a nonlinear curve fitting with minimized error through asteepest descent algorithm. In the identification of the ploughingcoefficients, edge coefficients Kre and Kte are set to the valuesdetermined for the shearing-dominant regime. Therefore, theradial and tangential forces acting on a discretized disk elementcan be expressed as

dFt ¼ðKtchþ KteÞdz when hXhc ðshearingÞ

ðKtpAp þ KteÞdz when hohc ðploughingÞ

(

dFr ¼ðKrchþ KreÞdz when hXhc ðshearingÞ

ðKrpAp þ KreÞdz when hohc ðploughingÞ

((19)

where h ¼ f(ft, y, her, dx, dy, ro), Ap ¼ f(h, re, her), ft is the feed pertooth, and y is the rotational angle of the tool. The computedforces are summed among all the engaged axial slices over all thecutting flutes to obtain the total tangential and radial forces,which are then transformed to forces in the planar directions,with respect to the global coordinate system. The proposed modelsuggests cutting and ploughing coefficients that inherentlycontain different aspects of plastic deformation, such as strainhardening and strain gradient effects. The friction forces areconsidered constant for different conditions and modeled withthe edge coefficients.

When the uncut chip thickness is smaller than the criticalvalue, an elastic recovery occurs that can affect the chipformation, cutting forces, and surface generation during machin-ing operations. The elastic recovery is different for variousmaterials and should be identified in order to accurately modelthe micro-machining operations. It has been shown [41] that theelastic recovery rate of the material can be identified directlyusing instrumented conical scratch tests. In this method, theremaining grooves from the scratch tests are inspected using a

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

Fy [N

]

00

0.5

1

1.5

2

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Mag

nitu

de

Simulation Experiment

ExperimentSimulation

0Angle [Degree]

200 400 600 800 1000 1200 1400 1600 1800

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

lation at full immersion and a feed rate of 0.5mm/flute.

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00.20.4

Fx [N

]

0

0.5

1

1.5

2

2.5

3

Mag

nitu

de

ExperimentSimulation

SimulationExperiment

−0.4

−0.2

0

0.2

0.4

0.6

Fy [N

]

0

0.5

1

1.5

2

Mag

nitu

de

Simulation Experiment

Simulation Experiment

0Angle [Degree]

0Frequency [Hz]

200 400 600 800 1000 1200 1400 1600 1800 0Angle [Degree]

200 400 600 800 1000 1200 1400 1600 1800

2000 4000 6000 8000 100000Frequency [Hz]

2000 4000 6000 8000 10000

Fig. 15. Comparison of forces between experiments and simulation at half immersion and a feed rate of 0.5mm/flute.

00

0.02

0.04

0.06

0.08

0.1

0.12

Frequency [Hz]

Mag

nitu

de

2000 4000 6000 8000 10000

Fig. 16. FFT of the AE signal for full immersion and a feed rate of 0.5mm/flute.

00

1

2

3

4

Fx [N

]

00

1

2

3

Fy [N

]

Experimental DataLinear Curve FitSimulation

2 4 6 8 10

Feed Rate [μm/flute]2 4 6 8 10

Fig. 17. Comparison of the RMS of data between experiments and simulation for

full immersion.

M. Malekian et al. / International Journal of Machine Tools & Manufacture 49 (2009) 586–598 595

surface profilometer. The surface profile of the groove can bemeasured with the aid of the profilometer, as shown in Fig. 13.The area of the groove under the original surface (‘abc’ in Fig. 13)is then calculated. Knowing this area and the projected area of thetool in the vertical direction (‘ABC’ in Fig. 13), the elastic recoveryrate of the material is acquired from:

pe ¼SABC � Sabc

SABC(20)

The scratch tests were performed utilizing conical tools with anapex angle of 901. The nominal depths of the grooves were chosento be 5, 10 and 15mm; and, the surface profile was measuredat three different points for each groove. It was found that theaverage elastic recovery, pe, of Aluminum 6061 was approximately10%. This acquired elastic recovery was used as one of theparameters of the developed model for micro-machining.

In order to obtain the ploughing coefficients in Eq. (19), thesame method used for the identification of the coefficients inthe shearing-dominant regime was utilized. The experimental and

theoretical forces were synchronized for full immersion, andthe defined error between the forces (Eq. (8)) was minimized for10 different feed rates and 500 samples for each feed rate throughthe steepest descent method. To assure that the global minimumwas obtained, the initial values of the parameters were variedseveral times. As a result, the ploughing coefficients wereidentified as shown in Table 3.

Fig. 14 shows the experimental and theoretical forces obtainedusing the ploughing-dominant formulation (Eq. (19)) and theidentified coefficients in Tables 2 and 3. It can be observedthat there is good agreement between the simulation and theexperiment; and, the proposed model can properly model themore complex force profiles in the ploughing-dominantregime. Also, the comparison between the simulations and the

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0

0.5

Fx [N

]

−0.5

0

0.5

Fy [N

]

Experiment Simulationr0 = 0.2

Simulationr0 = 0 r0 = 0.2 r0 = 1

r0 = 0.2 r0 = 1r0 = 0.2 r0 = 0Simulation SimulationExperiment

−0.5

0

0.5

Fx [N

]

−0.5

0

0.5

Angle [Degree]Angle [Degree]

Fy [N

]

Simulation

SimulationExperiment

SimulationExperiment

Simulation

0 200 400 600 800 1000 1200 1400 1600 1800

0 200 400 600 800 1000 1200 1400 1600 1800

0 200 400 600 800 1000 1200 1400 1600 1800

0 200 400 600 800 1000 1200 1400 1600 1800

Fig. 18. Comparison of forces between experiments and simulation at full immersion and a feed rate of 0.5mm/flute for different run-outs: (a) r0 ¼ 0 and 0.2mm and (b)

r0 ¼ 0.2 and 1mm.

−0.5

0

0.5

Fx [N

]

−0.5

0

0.5

Angle [Degree]

Fy [N

]

Experiment

Experiment

Simulation Pe=10%

Simulation Pe=0%

Simulation Pe=10%

Simulation Pe=0%

−0.5

0

0.5

Fx [N

]

−0.5

0

0.5

Angle [Degree]

Fy [N

]

SimulationPe=20%

SimulationPe=10%

SimulationPe=20%

SimulationPe=10%

Experiment

Experiment

0 200 400 600 800 1000 1200 1400 1600 1800 0 200 400 600 800 1000 1200 1400 1600 1800

0 200 400 600 800 1000 1200 1400 1600 1800 0 200 400 600 800 1000 1200 1400 1600 1800

Fig. 19. Comparison of forces between experiments and simulation at full immersion and a feed rate of 0.5mm/flute for different elastic recoveries: (a) Pe ¼ 0% and 10% and

(b) Pe ¼ 10% and 20%.

M. Malekian et al. / International Journal of Machine Tools & Manufacture 49 (2009) 586–598596

experimental data for half immersion are shown in Fig. 15.The cutting force predictions in the Y direction show averagedeviations of approximately 15% and this can be attributed bytransient vibrations due to the intermittent nature of the halfimmersion cutting.

The frequency content of the forces shows that, for low feedrates, there is more energy at the spindle frequency (1000 Hz),compared to the forces obtained at higher feed rates because theeffects of run-out and tool imperfections are more significantwhen the feed rate is lower. Also, there is a strong energycomponent (especially for forces in the Y direction) near the firstmode of the system (4000 Hz), which indicates that the secondharmonic of the tooth passing frequency can excite the toolsignificantly and cause forced vibration. This is even more evident

for half immersion conditions, since the second harmonics arestronger in this situation and cause more excitation at thisfrequency. Also, inspection of the frequency component of the AEsensor (Fig. 16) shows that there is a large frequency componentnear the first mode of the system, which confirms that most of thevibration happens at the first natural frequency of the combinedtool/machining centre.

Fig. 17 shows the experimental and theoretical RMS valuesof the forces for both the X and Y directions with a linear curvefit of the experimental data for the shearing-dominant region. Thetheoretical values were obtained using the developed model withthe identified parameters. It can be observed that the proposedmodel can appropriately predict the linear behaviour in theshearing-dominant regime, as well as the nonlinear behaviour,

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M. Malekian et al. / International Journal of Machine Tools & Manufacture 49 (2009) 586–598 597

and increased forces in the ploughing-dominant regime. The smalldeviation in the data is mainly due to the instabilities that occurduring machining operations.

0x 104

0

0.05

0.10

0.15

0.20

0.25

Spindle Speed [rev/min]

a lim

[mm

]

2 4 6 8 10

Fig. 20. Regenerative chatter stability lobes for full immersion micro-milling of

Al6061.

4. Discussions

There are several assumptions and limitations associated withthe proposed micro-milling force model. The model in thispaper assumes that the ploughing force is proportional to theploughed volume of the material, which is a commonly acceptedassumption by many researchers [8,36]. Also, it is assumed in thispaper that the edge force component is the same, regardlessof the cutting regimes, i.e. shearing and ploughing-dominantregimes. Thus, the same edge cutting coefficients are used in themechanistic models for both the shearing and ploughing-dominant regimes. An increase in the friction force due to thenonlinear interaction between ploughing and rubbing is notconsidered in the modeling. In this study, we performed theexperimental tests using uncoated tools, and the frictional forcesmay be different for coated tools than for uncoated tools. Also,the workpiece is assumed to be uniform. Furthermore, tooldeflection results in tilted tool conditions that may affect differentparameters, such as rake angle and tool engagement. Theseeffects, along with thermal effects, have not been considered inthis study.

Utilizing the model developed in this study, the effect ofdifferent parameters, such as elastic recovery and tool run-out,could be further investigated. Fig. 18 shows the actual andsimulated forces for three different run-outs in full immersionand a 0.5mm feed rate. As can be observed in Fig. 18(a), whenthere is no tool run-out (r0 ¼ 0), the maximum of forces isunderestimated but not significantly, and the agreement betweenthe forces becomes worse. For a run-out of 1mm (Fig. 18(b)), thereis a flat part in the force simulation, indicating that only one fluteis engaged. The run-out significantly affects the forces when thefeed rate is small. However, after a certain feed rate, only one fluteis engaged, and the effect of increasing run-out on the cuttingforces is not significant.

Fig. 19 shows the effect of different elastic recovery for fullimmersion cutting at the feed rate of 0.5mm. It can be observedthat no elastic recovery (0% in Fig. 19(a)) exhibit smaller forceswith higher deviations from the experimental results; while, forbigger elastic recovery (20% in Fig. 19(b)), slightly higher forcesare predicted. The effects of run-out and elastic recovery in theshearing-dominant regions are minimal and can be neglected.The experimental and theoretical forces were also comparedfor different depths of cut, which showed good agreement andverified the validity of the model for different cutting conditions.

The experimental cutting conditions were selected in order tohave chatter-free conditions, by simulating the frequency domainchatter stability analysis [33] using the mechanistically obtainedcutting parameters. Fig. 20 depicts the chatter stability lobesof ploughing-dominant full immersion micro-milling operations.We have selected the chatter-free region at 60,000 rev/min with adepth of cut of 0.1 mm (shown in Fig. 20). In order to accuratelycapture the cutting forces, regenerative chatter-free machiningconditions are imperative [30]. The adverse effect of running thespindle at the 60,000 rev/min is that forced vibrations may beexperienced.

We have found that the accuracy of the micro-tool fabrications,with respect to eccentricity, is important to minimize errors [42].Further studies include investigations of micro-cutting forcesusing a single fluted cutter and for various workpiece materials,since different materials behave differently, especially related towork hardening and size effects.

5. Conclusion

The accurate prediction of micro-milling forces is important inthe determination of the optimal machining parameters, in orderto prevent excessive tool wear and poor surface finish whilemaintaining high productivity. Unlike macro machining opera-tions, ploughing occurs in micro-machining operations when thechip thickness is less than the critical chip thickness. There havebeen attempts by other researchers to include the effects ofploughing, elastic recovery and the minimum chip thickness,based on slip-line plasticity or finite element modeling. However,these models are very complex, and the estimation of the manyparameters in the models is difficult. In this paper, a mechanisticforce model is developed for the ploughing and shearing-dominant regimes for micro end milling operations, consideringrun-out, dynamics and the effects of the elastic recovery ofmaterial commonly encountered during micro-machining. Sincethe direct measurement of tool tip dynamics is not feasible, wehave employed the receptance coupling method to indirectlyobtain the dynamics. The mechanistic force model has beenverified with experimental cutting force measurements ofAluminum 6061.

Acknowledgements

This research was supported by the Natural Sciences andEngineering Research Council of Canada (NSERC), Auto21, andCanada Foundation for Innovation (CFI). The authors thank Mr.Abe from Mitsubishi Materials, Japan and Mr. Aarts from Jabro(Seco) Tools, Netherlands for their generous support.

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