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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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