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Jeffrey C. Mellinger

Graduate Research Assistant

O. Burak Ozdoganlar

Post Doctoral Research Associate

Richard E. DeVor

Professor

Shiv G. Kapoor

Professor

Department of Mechanical and Industrial

Engineering,

University of Illinois at Urbana-Champaign,

Urbana, IL 61801

Modeling Chip-Evacuation Forces

and Prediction of Chip-Clogging

in Drilling

One of the fundamental difﬁculties of the drilling process is the evacuation of the chips

from the drilled hole. As the hole depth increases, the chips tend to cluster together and

clog the ﬂutes, causing increased forces, poor hole quality, elevated drill temperatures,

and drill breakage. In this paper, a model for chip evacuation has been developed to

predict the force and torque arising from the evacuation of the discontinuous chips. The

model considers the pressure on a differential chip section being created by the forces

required to push the chips out of the hole. The two coefﬁcients of friction required by the

model are established via a calibration procedure. The effectiveness of both the calibra-

tion and the force models has been assessed via a set of validation experiments. The

model can be used to predict the depth when chip-clogging occurs, indicating the need for

a pecking cycle, and the depth where the drill experiences an excessive amount of torque,

which may result in drill breakage. 关DOI: 10.1115/1.1473146兴

1 Introduction

Chip evacuation represents one of the fundamental difﬁculties

associated with the drilling process because the chips generated at

the cutting lips are conﬁned by the hole wall and the drill ﬂute. As

the depth of the hole increases, an increased amount of chips ﬁll

the ﬂutes, leading to chip-clogging, and eventually to tool break-

age 关1,2兴. For deep-hole drilling, pecking 共periodic drill retrac-

tion兲is one method used to alleviate chip-clogging. While effec-

tive for this purpose, pecking greatly reduces productivity. The

determination of proper pecking cycles becomes a critical factor

in process planning. Therefore, understanding the factors affecting

chip evacuation and predicting the occurrence of chip-clogging

becomes important, especially when drilling larger depth to diam-

eter ratios.

Once the entire drill is engaged 共i.e., when the depth exceeds

the drill-point height兲, the torque and thrust force reach steady-

state values arising from the cutting process. For the drilling pro-

cesses that produce discontinuous chips, such as those performed

on materials of low ductality, e.g., cast aluminum alloys, it has

been observed that the torque and thrust continue to increase with

increasing depth. As the process progresses, the rate of increase

increases until the torque exerted on the drill exceeds its torsional

limit, causing drill breakage 关1兴. This phenomenon has been at-

tributed to chip-clogging, and constitutes the major limitation of

the drilling process as it increases the forces during the process,

increases the drilling temperature, lowers the quality of the hole,

and accelerates tool wear and breakage 关3兴.

To avoid these adverse effects, researchers have sought to de-

tect and control the initiation of the chip-clogging. Using the fact

the cutting forces reﬂect the development of chip-clogging, con-

trol schemes that monitor the cutting forces 共especially the torque兲

have been employed. These schemes call for the alteration of the

cutting conditions to compensate for the torque increase 关1,2,4–

6兴, which usually involves decreasing the spindle speed and/or the

feed. Another method used to detect the onset of chip-clogging is

the use of drill temperatures 关7兴. Although the drill temperatures

gradually increase with hole depth when there is no chip-clogging,

in the presence of chip-clogging, the rate of the temperature in-

crease has been observed to rapidly grow.

The depth where the torque increases more abruptly has been

considered as the point of ﬂute clogging and used as a perfor-

mance criterion for chip-evacuation capability. This provides the

means to make comparisons of the chip-evacuation performance

of different tooling parameters 共drill geometry, substrate material,

coating兲and cutting conditions 关8兴.

To improve the evacuation of the chips, the mechanism respon-

sible for moving the chips through the ﬂutes must be understood.

Although a fair amount of experimental research exists, only a

few works have attempted to model chip evacuation. One method

to model chip evacuation through the ﬂutes is with a kinematic

approach 关9兴. This approach considered the discontinuous chip as

a particle, and applied a force balance to this particle when it is

moving through the ﬂutes. The differential equation is solved to

predict the chip velocity, which can be used as a performance

criterion considering that a faster moving chip has a better chance

of leaving the drilled hole without initiating clogging. This model

can be used to compare the chip velocities of different drill geom-

etries and cutting conditions, however, it cannot be used to predict

the onset of chip-clogging, since the model is independent of

depth. Chip velocity is difﬁcult to measure experimentally, so

only qualitative validation experiments were performed to quali-

tatively verify the chip velocity model.

The objective of this work is to develop a model that is capable

of predicting the chip-evacuation force and torque in drilling with

discontinuous chips. The purpose of the chip-evacuation model is

to establish a relationship between the hole depth and the chip-

evacuation force/torque as a function of the process parameters

and ﬂute geometry. The model can be used to ﬁnd the depth where

the drill is susceptible to drill breakage, and the location where a

pecking cycle is warranted because of the development of ﬂute

clogging.

2 Modeling Chip Evacuation

The model presented here is concerned with drilling processes

that produce discontinuous chips of considerably smaller dimen-

sions when compared to the ﬂute cross-sectional area, such as

those produced when drilling cast aluminum alloys or cast iron.

Thus, the chips can be classiﬁed as granular solids, which are

characterized as a group of particles that are of roughly the same

size 关10兴. Granular solids have characteristics of both ﬂuids and

solids. They occupy the shape of the container in which they ﬁll,

exert pressure on the container boundaries, and ﬂow through

openings like ﬂuids. However, like solids, they possess cohesive

strength, are capable of having nonisotropic stress distributions,

Contributed by the Manufacturing Engineering Division for publication in the

JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript received

March 2001; revised December 2001. Associate Editor: J. Hu.

Journal of Manufacturing Science and Engineering AUGUST 2002, Vol. 124 Õ605

Copyright ©2002 by ASME

and have shear stresses that are proportional to the normal stress

关10兴. The model presented here, in parallel to the properties of the

granular solids, assumes that the chips 共1兲exert pressure on the

ﬂute surface of the drill body and hole walls, 共2兲ﬂow through and

ﬁll the ﬂutes, 共3兲remain in contact with each other, 共4兲exhibit

nonisotropic stress distributions, and 共5兲have friction forces pro-

portional to the normal forces.

Modeling the movement of granular solids has been investi-

gated in the solids conveying zone of a screw extruder that is used

for polymer processing 关10–14兴. The techniques employed in sol-

ids conveying to model the movement of granular solids will

serve as the foundation for the chip-evacuation model presented in

this paper.

In the following analysis, a straight-ﬂuted drill is used to reduce

the geometric complexity that arises from the helix angle of twist

drills. Straight-ﬂuted drills are generally used in horizontal drill-

ing applications for materials that produce broken chips 共such as

cast iron or cast aluminum兲. These drills are chosen to produce

better roundness and straightness properties than those of twist

drills for moderate to high depth-to-diameter ratios.

2.1 Cutting and Chip-Evacuation Forces. The force in-

crease experienced during drilling as a function of hole depth can

be considered as representative of the chip-evacuation perfor-

mance. Figures 1 and 2 illustrate how the torque and thrust change

with the nondimensional depth-to-diameter ratio for a typical

drilling process that exhibits discontinuous chips. The initial rapid

increase of the forces reﬂects increasing cutting forces due to the

engagement of the drill point. Once the drill point is fully en-

gaged, the thrust and torque reﬂect those that are required for chip

formation, which will be referred to here as the cutting force and

cutting torque, respectively. It can be observed from the ﬁgures

that the thrust and the torque keep increasing after this point.

Considering the cutting forces to be constant, the force increase

can be attributed solely to the higher force required to move the

increasing amount of chips that ﬁll the ﬂutes. It is natural, then, to

refer to this force as the chip-evacuation force, and associated

torque as the chip-evacuation torque. This chip-evacuation force

acts in the direction of the drill helix angle. At some depth, the

chip-evacuation force and torque begin to experience an increas-

ing rate of increase, which will be referred to as the critical depth.

This is the depth that deﬁnes the onset of chip-clogging, and if the

process continues, the excessive chip-evacuation force and torque

can cause the torsional capacity of the drill to be exceeded, caus-

ing drill breakage.

2.2 Chip-Evacuation Force Model. The behavior of the

chip-evacuation force can be explained by considering the pres-

sure distribution and associated force balance on a section of chips

in the ﬂute. Figure 3 shows the straight-ﬂuted drill in the work-

piece and the differential chip section used for the force balance.

The drill cross-section can be seen in Fig. 4, where the shaded

region represents the 共half兲ﬂute area. An enlarged view of the

differential chip section and associated forces are shown in Fig. 5.

The differential chip section has three surfaces on the ﬂute and

one surface on the hole wall. In these ﬁgures, A0is the cross-

sectional area of the ﬂute, z⬘is the distance from the workpiece

surface to the current location of the drill point, and ⍀is the

Fig. 1 Typical plot of the torque varying with depth

Fig. 2 Typical plot of the thrust force varying with depth

Fig. 3 Straight-ﬂuted drill during cutting

Fig. 4 Flute cross-section

606 ÕVol. 124, AUGUST 2002 Transactions of the ASME

direction of the drill rotation. The dimensionless depth, z, is used

in the analysis and is related to the depth by the following trans-

formation

z⫽z⬘

D, (1)

where Dis the drill diameter.

To determine the chip-evacuation force per ﬂute, Fc, a force

balance on the chip section in Fig. 5 is performed in the axial

direction resulting in

0⫽Fc⫺共Fc⫹dFc兲⫹Fff⫹Fhf⫹Frf⫹Ffwi, (2)

where Fff,Fhf , and Frf are the axial friction forces from the

ﬂute face, ﬂute heel, and ﬂute root, respectively, and Ffwi is the

friction force resulting from the interaction between the chips and

the hole wall. In the force balance, the inertial effects are assumed

to be negligible. The friction force from the wall Fwf does not

appear in Eq. 共2兲because it does not act in the axial direction. It is

taken to be in the tangential direction considering the fact that the

magnitude of the tangential velocity, which results from the rota-

tional speed of the drill, is several orders of magnitude greater

than the axial velocity, which results from the feed rate. The chip-

evacuation force, Fc, causes an axial pressure, P, in the differen-

tial section, which is aligned with the chip-evacuation force for

the straight-ﬂuted drill.

The chip-evacuation force is the product of the axial pressure

and the ﬂute cross-sectional area, viz.,

Fc⫽A0P. (3)

The axial pressure compresses the differential chip section, forc-

ing the chips to expand laterally in the plane of the ﬂute cross-

section. Since the chips are contained on all sides of the cross

section 共from the ﬂute and hole wall兲, a lateral pressure develops

on the chips in the plane of the ﬂute cross-section. This lateral

pressure is considered to be proportional to the axial pressure via

a constant k. Thus, the lateral pressure can be given as kP. There

will be a tendency for this pressure to vary somewhat; however,

following the methodology employed for solids conveying, the

lateral pressure is assumed uniform. The normal forces on the

outer surfaces of the differential section can then be calculated by

multiplying the lateral pressure by the affected area. The resulting

normal forces on the chip section are:

Ffn⫽kPSfDdz, (4)

Fhn⫽kPShDdz, (5)

Frn⫽kPSrDdz, (6)

and

Fwn⫽kPSwDdz, (7)

where Ffn,Fhn ,Frn , and Fwn are the normal forces from the

ﬂute face, ﬂute heel, ﬂute root, and hole wall, respectively. Sf,

Sh,Sr, and Sware the lengths of contact and are deﬁned in Fig.

4. The friction forces at each surface, which act in the opposite

direction of the relative velocity between the contacting surfaces,

are related to the normal force at that surface by a coefﬁcient of

friction. The friction forces on the chip section from the ﬂute are:

Fff⫽

fkPSfDdz, (8)

Fhf⫽

fkPShDdz, (9)

and

Frf⫽

fkPSrDdz, (10)

while the friction force on the chip section from the hole wall is

Fwf⫽

wkPSwDdz, (11)

where

fis the coefﬁcient of friction between the chips and ﬂute

and

wis the coefﬁcient of friction between the chips and hole

wall. Fwf is the friction force from the hole wall.

When calculating the normal force from the hole wall, the ‘‘ef-

fective’’ area, that is the area component of the wall interface

perpendicular to the normal force, must be considered. The con-

cept of ‘‘effective area’’can be explained by considering the sec-

tion between the shaded plane and the hole wall 共Fig. 5兲interface

as a separate entity. For this entity, the sum of the forces on the

Fig. 5 Force balance on a differential section of chips

Journal of Manufacturing Science and Engineering AUGUST 2002, Vol. 124 Õ607

shaded plane must be equivalent to those on the hole wall to

satisfy the equilibrium. Since the normal force Fwn is considered

to have a line of action passing through the center of the drill, the

shaded area shown in Fig. 5 is taken as the effective area with the

length Sw. The resulting friction force from the wall is taken to be

in the tangential direction as previously discussed.

The friction force from the hole wall, Fwf , causes the chips to

be pushed against the ﬂute face, thus increasing the normal force

on the ﬂute face. While this would have an impact on the normal

forces developing on the ﬂute heel and rake, it is assumed to be

negligible. This normal force creates a friction force, Ffwi, and

can be expressed as

Ffwi⫽

fFwf sin

, (12)

where

is the angle between the ﬂute face and the shaded plane.

Substituting the appropriate forces into Eq. 共2兲results in

0⫽A0P⫺A0共P⫹dP兲⫹

fkPSfDdz⫹

fkPShDdz

⫹

fkPSrDdz⫹

f

wkPSwDsin

dz, (13)

which, after simplifying, becomes

dP

P⫽kD

A0共Sf

f⫹Sh

f⫹Sr

f⫹Sw

w

fsin

兲dz. (14)

Deﬁning Bas

B⫽共Sf

f⫹Sh

f⫹Sr

f⫹Sw

w

fsin

兲, (15)

and integrating Eq. 共14兲, the pressure in the ﬂute can be expressed

as

P共z兲⫽P0e共kBD/A0兲z, (16)

where P0is the initial axial pressure at z⫽0.

Equation 共16兲deﬁnes how the axial pressure on the chips

changes from the workpiece surface to the drill point as a function

of ﬂute geometry and two coefﬁcients of friction,

fand

w.

Assuming that the cross-sectional area remains constant through-

out the ﬂute, the chip-evacuation force at any depth can be deter-

mined by multiplying Eq. 共16兲by the area A0to give

Fc共z兲⫽Fc共0兲e共kBD/A0兲z, (17)

where Fc(0) is the initial chip-evacuation force at z⫽0. The

change in torque due to the chip-evacuation force, which will be

deﬁned as the chip-evacuation torque, can be calculated from the

friction force on the wall, Fwf as

dM⫽RFwf , (18)

where Ris the radius of the drill. Substituting Eq. 共11兲for Fwf and

Eq. 共16兲for the pressure, P, the chip-evacuation torque per ﬂute

can be found as

M⫽

冕

0

zR

wkSwDFc共0兲

A0e共kBD/A0兲

d

, (19)

and integrating results in

M⫽R

wSwFc共0兲

B共e共kBD/A0兲z⫺1兲. (20)

3 Model Calibration Procedure

Coefﬁcients of friction depend on many factors, such as veloc-

ity between the contacting surfaces, temperature, surface rough-

ness, and applied load 关15兴. These parameters are affected by the

cutting conditions of the process, since they dictate the velocities

of the contacting surfaces, temperatures, and chip thickness.

Therefore, it is expected that the coefﬁcients of friction are a

function of the cutting conditions, which can be determined via a

set of calibration experiments and associated regression coefﬁ-

cients of the calibration equations. Here, a power law model 关16兴

is proposed relating these process parameters to the coefﬁcients of

friction, viz.,

ln

f⫽a0⫹a1ln f⫹a2ln N⫹a3ln fln N, (21)

ln

w⫽b0⫹b1ln f⫹b2ln N⫹b3ln fln N, (22)

where fis the feed and Nis the spindle speed. In order to make the

coefﬁcients dimensionless, the parameters can be coded between

⫺1 and 1. The coefﬁcients 共a"and b"兲for a given workpiece

material and tooling combination can be determined via a 22fac-

torial design of experiments with feed and spindle speed as the

variables. A replicated factorial design can be used to obtain the

statistical signiﬁcance of the coefﬁcients and develop a conﬁdence

interval on the predicted coefﬁcients of friction.

The calibration procedure is described in the ﬂow chart in Fig.

6. The coefﬁcients of friction for each experiment can be found

from the measured chip-evacuation force and torque via a nonlin-

ear least-squares curve ﬁt as

min

兺

z

冋

共FModel⫺FExp兲2⫹

冉

MModel⫺MExp

R

冊

2

册

, (23)

⫽关

f

w兴, (24)

where FModel and MModel are deﬁned in Eqs. 共17兲and 共20兲, and

FExp and MExp are the chip-evacuation force and torque estab-

lished from the experimentation, respectively. The chip-

evacuation torque residual is divided by the radius to make the

two residuals have the same dimensions. The last step is to per-

form a linear regression to determine a"and b"from Eqs. 共21兲and

共22兲.

Fig. 6 Calibration procedure ﬂow chart

Table 1 Experimental design for model calibration

608 ÕVol. 124, AUGUST 2002 Transactions of the ASME

4 Experimentation for Calibration and Validation

Calibration and validation experiments have been conducted to

validate the model presented above. Experiments were performed

on a Mori Seiki SH-400 Horizontal Machining Center. A Kistler

9272 4-component drilling dynamometer was used to collect the

forces. Flood coolant 共Master Chemical Trim-Sol兲was used in all

the experiments. The workpiece material was aluminum 356-T6.

A 3.175 mm diameter two-ﬂuted solid carbide drill was used with

a zero degree helix angle.

4.1 Calibration Experiments. The ﬁrst step in calibrating

the chip-evacuation model is to identify the ﬂute geometry inputs

to the model. The cross-section of the ﬂute, shown in Fig. 4, was

measured with a Precision Twist Drill, Co. Tool Analyzer 共model

560兲. The ﬂute area is 1.5419 mm2. The lengths of contact on the

hole wall, Sw, is 2.1539 mm. The length of contact on the ﬂute

face, ﬂute heel, and ﬂute root are 1.1379 mm, 1.2649 mm, and

0.2718 mm, respectively. The angle,

, is 37.8 deg.

The constant k, which is the ratio between the lateral and axial

pressures, is a material property that has received a limited

amount of research. In the polymer literature, determination of

this property remains poorly understood 关17兴. To determine this

Fig. 7 Chip-evacuation force for calibration test 1

Fig. 8 Chip-evacuation torque for calibration test 1

Fig. 9 Chip-evacuation force for calibration test 4

Fig. 10 Chip-evacuation torque for calibration test 4

Fig. 11 Fitted chip-evacuation forces for the calibration

experiments

Table 2 Coefﬁcients of friction for the calibration experiments

Table 3 Friction model coefﬁcients

Table 4 Cutting conditions for the validation experiments

Journal of Manufacturing Science and Engineering AUGUST 2002, Vol. 124 Õ609

ratio, specially designed compaction cells have been used 关17兴,

which have been employed for a selected set of solid polymers. In

this work, the ratio of the lateral and axial pressures was found to

be strongly dependent on temperature and vary between 0.3–0.9

for different materials. Determination of this property for metals

has only been done with metal powders, where initial discoveries

found that the ratio takes the value of Poisson’s ratio at low axial

pressures 关18兴. Following this work on granular powders, kis

assumed to be 0.33, which is the value of Poisson’s ratio for

aluminum 关19兴.

Table 1 provides the cutting conditions for the calibration ex-

periments. For each case, the chip-evacuation force and torque

were calculated by subtracting the cutting force and cutting torque

from the total observed data, respectively, and dividing by the

number of ﬂutes. Figures 7–10 show the chip-evacuation forces

and torques obtained from two of the four calibration experiments.

In the ﬁgures, the abrupt increase in the force and torque is rec-

ognized as the onset of chip-clogging. Further, the variation of the

chip-evacuation force and torque increase after the onset of chip-

clogging. Once the chip-evacuation force and the torque are de-

termined, the coefﬁcients of friction values can be found from the

least-squares method given in Eq. 共23兲and are given in Table 2.

Using the coefﬁcient of friction values of Table 2, a linear re-

gression procedure was followed to ﬁnd the coefﬁcients of the

friction model given in Eqs. 共21兲and 共22兲. The results are shown

in Table 3. From the replicated calibration tests, a conﬁdence in-

terval can be established to ascertain the variation of the coefﬁ-

cients of friction. At a 95% conﬁdence, the interval for

fis

⫾0.0253, and

whas an interval of ⫾0.0316. The conﬁdence

intervals demonstrate that the cutting conditions do affect the co-

efﬁcients of friction signiﬁcantly. To observe the effectiveness of

Fig. 12 Experimental and predicted chip-evacuation force for

validation test 5

Fig. 13 Experimental and predicted chip-evacuation torque for

validation test 5

Fig. 14 Experimental and predicted chip-evacuation force for

validation test 9

Fig. 15 Experimental and predicted chip-evacuation torque for

validation test 9

Table 5 Results of the validation experiments

610 ÕVol. 124, AUGUST 2002 Transactions of the ASME

this procedure, these coefﬁcients of friction were used to calculate

the chip-evacuation force and torque from Eqs. 共17兲and 共20兲.

Figure 11 shows the ﬁtted chip-evacuation force curve for each

calibration test and visually reinforces the variation of the coefﬁ-

cients of friction with the cutting conditions.

4.2 Validation Experiments. A set of validation experi-

ments were conducted to assess the effectiveness of the chip-

evacuation model. The cutting conditions were chosen within the

calibration range as seen in Table 4.

First, an assessment of the calibration model is made by com-

paring the predicted and experimental coefﬁcients of friction. The

predicted coefﬁcients of friction are calculated from the friction

models in Eqs. 共21兲and 共22兲with the friction model coefﬁcients

given in Table 3. The experimental coefﬁcients of friction are

determined in the same manner as the calibration experiments by

using the nonlinear least-squares method given in Eq. 共23兲. The

results are given in Table 5, with the percent errors. The average

error for the coefﬁcient of friction,

f, is 4.09%. The coefﬁcient

of friction,

w, has an average error of 8.20%.

The chip-evacuation model in Eqs. 共17兲and 共20兲was then used

to predict the chip-evacuation force and torque. Figures 12–15

illustrate the experimental and predicted chip-evacuation force

and torque for two of the validation experiments. To determine the

effectiveness of the chip-evacuation model in capturing the trends

of the experimental data, a normalized error was used to quantify

the quality of the curves. The normalized error is deﬁned as

Normalized Error⫽

兺共XModel⫺XExp兲2

兺XExp

2, (25)

where Xrepresents either the chip-evacuation force, F,orthe

chip-evacuation torque, M, and XExp is from the experimental data

and XModel is from the chip-evacuation model. The calculated nor-

malized errors for the validation experiments are in Table 5. For

each validation experiment, the normalized error is determined for

both the chip-evacuation force and torque. The average error for

the chip-evacuation torque and force was seen to be 15.75% and

14.50%, respectively, for validation tests 1 through 8. It should be

noted in examining Figs. 14 and 15 for validation test 9 that the

chip-evacuation process becomes much more erratic in nature past

the point of chip-clogging. However, it is clear from Figs. 12–15

that, even in the worst-case scenario, the chip-evacuation model

predicts the evacuation force and torque quite well to the onset of

chip-clogging, which is the primary point of interest for deep-hole

drilling.

5 Determination of the Critical Depth in Drilling

The chip-evacuation model can be used to determine two

depths of interest in drilling: the depth that can be drilled prior to

subjecting the drill to a torque limit based on the drill breakage

torque and a desired factor of safety, and the depth at which chip-

clogging begins to occur indicating the need for a pecking cycle.

These two critical depths can be used jointly during process plan-

ning. For a given speciﬁed depth, the chip-evacuation model can

be used to determine if the hole can be drilled in one pass without

exceeding the chosen torque limit. If the hole cannot be drilled

without violating the limit, the depth at which to perform a peck-

ing cycle can be determined. This depth is determined from chip-

clogging, since once chip-clogging becomes substantial, pecking

will be ineffective in removing the chips.

5.1 Drill Breakage-Based Criterion. The failure mode

共breakage兲of drills is torsional 关20兴. Based on this consideration,

researchers 关20,21兴have developed various criteria that rely on

the breaking torque of the drill, Mb, and a desired factor of safety,

FS. Thus, the depth of the hole that can be drilled prior to violat-

ing this threshold torque, Mth , can be determined, and will be

deﬁned as the threshold depth zt. The threshold torque is deﬁned

as Mth⫽Mb/FS.

Drill breakage results from the total torque exerted on the drill.

Thus, in addition to the chip-evacuation torque, the cutting torque,

Mcut , is also required for the prediction of the threshold depth.

Therefore, the total torque on a sharp drill can be written as

Mtotal⫽Mcut⫹nfM, (26)

where nfis the number of ﬂutes. The chip-evacuation torque per

ﬂute, M, can be determined from the chip-evacuation model in Eq.

共20兲. At the threshold depth, the total torque on the drill will be

Mth⫽Mcut⫹nfM共zt兲, (27)

where M(zt) represents the chip-evacuation torque per ﬂute

evaluated at the threshold depth. After substituting the chip-

evacuation torque from Eq. 共20兲, the threshold depth becomes

zt⫽A0

kBD ln

冉

B

nfR

wSw共Mth⫺Mcut兲⫹1

冊

. (28)

In this study, the cutting torque, Mcut , at each cutting condition

is predicted with the use of the mechanistic model developed in

关16兴. This model divides the cutting edge into small elements. For

each element, the forces on the rake face, the normal and friction

forces, are determined by multiplying the elemental chip areas

with speciﬁc force coefﬁcients, which can be given as

ln Kn⫽a0⫹a1ln tc⫹a2ln V⫹a3ln共1⫺sin

␣

n兲⫹a4ln tcln V

(29)

Table 6 Coefﬁcients for the mechanistic force model

Table 7 Experimental and predicted cutting torques and threshold depths

Journal of Manufacturing Science and Engineering AUGUST 2002, Vol. 124 Õ611

ln Kf⫽b0⫹b1ln tc⫹b2ln V⫹b3ln共1⫺sin

␣

n兲⫹b4ln tcln V,

(30)

where Knand Kfare the normal and frictional force constants,

respectively, tcis the elemental chip thickness, Vis the elemental

cutting 共surface兲speed, and

␣

nis the elemental normal rake angle.

The elemental torque can then be calculated by coordinate trans-

formation and the torque can be found by summing all the el-

emental torques. From validation experiments, the mechanistic

model predicted the drilling forces with errors less than 10%.

The same calibration experiments used for the chip-evacuation

model are used to determine the mechanistic force model speciﬁc

force coefﬁcients. Therefore, the validity range of feeds and

speeds for the mechanistic force model is 0.0508–0.1270 mm/rev

and 4000–7000 rpm, respectively. The resulting values of the spe-

ciﬁc force coefﬁcients can be seen in Table 6. The coefﬁcients a3

and b3in Table 6 deal with the normal rake angle of the cutting

element, and therefore take the value of zero because the straight-

ﬂuted drill used here has a constant zero degree rake angle along

the cutting lips. The predicted cutting torque, via the model of

关16兴, and experimental cutting torque are within 12%, as shown in

Table 7.

In this analysis, the breaking torque of the drills must be deter-

mined. For the purpose of demonstration here, a drill was delib-

erately broken during an experiment, resulting in the breaking

torque of 175 Ncm. In an actual application, two or more drills

might be broken to obtain a more precise estimate. This value

could be determined analytically as well. Afactor of safety of 5 is

chosen here 关20兴, which makes the threshold torque 35 Ncm.

Figures 16 and 17 show the total torque measured from the

experiments 共Calibration Test 1 and Validation Test 2 in Tables 1

and 4, respectively兲and those predicted by the model in Eq. 共26兲,

along with the threshold torque and the location of the threshold

depth. The experimental and predicted threshold depths are also

provided in Table 7. The predicted threshold depth, zt, is calcu-

lated from Eq. 共28兲, using the threshold torque, cutting torque, and

the predicted coefﬁcients of friction. The experimental threshold

depth is determined when the mean experimental torque exceeds

the threshold torque. As seen in Table 7, the average error between

the predicted and experimental threshold depth is 5.73%.

5.2 Flute-Clogging Based Criterion. When drilling holes

with large depth-to-diameter ratios, pecking cycles 共periodic drill

retractions兲are generally used. The goal of the pecking cycle is to

clear the chips from the ﬂutes. After a certain depth, which has

been deﬁned here as the critical depth, chips may become so

severely compacted that pecking will be ineffective. When chip-

clogging occurs, the forces begin to rapidly increase, thus the

gradient of the chip-evacuation force can be used as the criterion

to determine the critical depth.

The gradient of the modeled chip-evacuation force is obtained

by differentiating Eq. 共17兲as

dFc

dz ⫽kBD

A0e共kBD/A0兲z. (31)

Therefore, the critical depth, z*, can be given as

z*⫽A0

kBD ln

冋

A0

kBD

dFc*

dz

册

, (32)

Fig. 18 Experimental and predicted chip-evacuation force for

calibration test 1

Fig. 19 Experimental and predicted chip-evacuation force for

calibration test 3

Fig. 16 Experimental and predicted total torque for calibration

test 1

Fig. 17 Experimental and predicted total torque for validation

test 2

612 ÕVol. 124, AUGUST 2002 Transactions of the ASME

where dFc*/dz is the gradient of the chip-evacuation force at

which the critical depth occurs.

The selection of dFc*/dz can be made by analyzing the cali-

bration data. From each calibration experiment, the depth where

the chip-evacuation force rapidly increases can be identiﬁed as the

observed critical depth, as seen in Figs. 18 and 19. This depth is

chosen by observing the experimental data and selecting the depth

at which the forces begin to rapidly increase. While somewhat

subjective, this technique is shown here to be effective. Substitut-

ing this observed critical depth in Eq. 共31兲for z, the gradient of

the chip-evacuation force at this depth can be found from the

model. This gradient is illustrated in Figs. 18 and 19. Table 8

provides the observed critical depths and associated gradients de-

termined from the calibration experiments of Table 1. Using the

gradients thusly obtained, an equation to predict the gradient for

different cutting conditions can be developed. For the sake of

simplicity, a linear relationship is used, which resulted in the fol-

lowing equation,

dFc*

dz ⫽23.950⫹0.617f⫺1.189N⫺3.168fN. (33)

Table 9 provides the predicted gradient and the observed and

predicted critical depths for the validation experiments. The pre-

dicted gradient, dFc*/dz, is calculated from Eq. 共33兲with the

corresponding coded cutting conditions. Using the predicted gra-

dient and coefﬁcients of friction for each validation experiment of

Table 4, the predicted critical depth can be obtained from Eq. 共32兲.

The observed critical depth is determined in the same manner as

in the calibration experiments. It can be seen from the difference

between the observations and predictions that the critical depth

can almost be predicted within a depth less than one drill radius

and has an average error of 4.60%.

6 Conclusions

The speciﬁc conclusion of this work are:

1 A chip-evacuation model that predicts the chip-evacuation

force and torque has been developed for drilling processes with

discontinuous chips. The model is based on the force balance on a

differential chip section, and considers straight-ﬂuted drills, which

are common for deep-hole drilling processes.

2 A calibration procedure has been proposed to determine the

two coefﬁcients of friction required by the model. The friction

models considered are power functions with feed and spindle

speed as the variables.

3 A set of calibration and validation experiments has been

completed to assess the effectiveness of both the calibration pro-

cedure and the chip-evacuation model. It has been seen that the

calibration model is capable of accurately predicting the coefﬁ-

cients of friction in the applicable range. The coefﬁcient of fric-

tion,

fis predicted with an average error of 4.09%, while the

coefﬁcient of friction,

whas an average error of 8.20%. A nor-

malized error criterion is used to compare the predicted and ex-

perimental chip-evacuation forces and torques. For the validation

experiments, the chip-evacuation force and torque had an average

normalized error of 14.50% and 15.75%, respectively.

4 Two drilling critical depths are predicted by the chip-

evacuation model using a torque-limit based criterion, and a chip-

clogging criterion. The former imposes an upper limit on the

torque exerted on the drill with a factor of safety to prevent break-

age, and the latter speciﬁes a limit on the slope of the chip-

evacuation forces to determine the pecking depth. Validation ex-

periments have been used to prove the effectiveness of the chip-

evacuation model in predicting these depths. The torque-limit

based critical depth has an average error of 5.73%, while the

chip-clogging critical depth is predicted with an average error of

4.60%.

Acknowledgment

The authors wish to state their appreciation to the National

Science Foundation Industry/University Cooperative Center for

Machine Tool Systems Research and Delphi Automotive Systems

Co. for support of this research. The authors many conversations

with Dr. R. Khetan of DelphiAutomotive Systems Co. are greatly

appreciated. The authors are grateful to Mr. Shiva Kalidas and Mr.

Mike Vogler for their valuable comments and suggestions.

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614 ÕVol. 124, AUGUST 2002 Transactions of the ASME