1. Short Communication
Estimation and experimental validation of cutting forces in ball-end milling
of sculptured surfaces
Yuwen Sun , Fei Ren, Dongming Guo, Zhenyuan Jia
Key Laboratory for Precision and Non-Traditional Machining Technology of the Ministry of Education, Dalian University of Technology, Dalian 116024, China
a r t i c l e i n f o
Article history:
Received 29 April 2009
Received in revised form
31 July 2009
Accepted 31 July 2009
Available online 12 August 2009
Keywords:
Cutting forces
Chip thickness
Sculptured surface machining
Ball-end mill
a b s t r a c t
Chip thickness calculation has a key important effect on the prediction accuracy of accompanied cutting
forces in milling process. This paper presents a mechanistic method for estimating cutting force in ball-
end milling of sculptured surfaces for any cases of toolpaths and varying feedrate by incorporation into a
new chip thickness model. Based on the given cutter location path and feedrate scheduling strategy, the
trace modeling of the cutting edge used to determine the undeformed chip area is resulted from the
relative part-tool motion in milling. Issues, such as the selection of the tooth tip and the computation of
the preceding cutting path for the tooth tip, are also discussed in detail to ensure the accuracy of chip
thickness calculation. Under different chip thicknesses cutting coefficients are regressed with good
agreements to calibrated values. Validation tests are carried out on a sculptured surface with curved
toolpaths under practical cutting conditions. Comparisons of simulated and experimental results show
the effectiveness of the proposed method.
2009 Elsevier Ltd. All rights reserved.
1. Introduction
Ball-end milling is widely used in machining parts with curved
geometries such as die mould, propellers and turbine blades.
Regardless of the emergence of many advanced CAM systems,
machining of complicated surfaces is still identified as a challenge.
This partly contributes to high demand for tolerance, roughness or
productivity of machined parts and partly to the machinability of
difficult-to-cut materials. For this end, cutting force modeling has
become an essential step to understand the behavior of cutting
process and further to ensure the stability of machining system
and the optimization of process parameters.
Some strategies have been addressed for the prediction of
cutting forces. Kim et al. [1] analyzed the relationship between
undeformed chip geometry and the cutter feed inclination angle.
Cutting forces acting on the engaged cutting edge elements were
calculated using an empirical method. Then the resultant cutting
force was calculated by numerical integration of cutting forces
acting on the engaged cutting edge elements. Fontaine et al. [2]
researched the effect of tool–surface inclination on cutting forces
in ball-end milling, and presented a milling force model based on
a thermo-mechanical modeling of oblique cutting. Lazoglu [3]
presented a new mechanistic model, which has the ability to
calculate the workpiece/cutter intersection domain automatically,
for the prediction of cutting forces in ball-end milling. Further-
more, an analytical approach was used to determine the
instantaneous chip load and cutting forces. Lamikiz et al. [4]
estimated the cutting force in inclined surface machining based
on a semi-mechanistic force model. The undeformed chip for the
slope cutting was calculated as the same as the horizontal case by
means of a special reference system, composed by three
directions: feed direction, normal to machining surface and vector
cross-product of both. The coefficients of the semi-mechanistic
force model were obtained from horizontal slot cutting tests with
different cutting conditions. Imani et al. [5] developed a simula-
tion system for ball-end milling. A modified chip model was
represented based on the effect of vertical component of feed on
the chip thickness. And a commercial solid modeler was used to
automatically extract the critical geometric information required
for the physical simulation. Naserian et al. [6] introduced a static
rigid force model to estimate cutting forces of sculptured surface.
In the model, the approximated equation of chip thickness was
derived from the same fundamental basis as in [5]. Most of the
past researches are based on the premise that cutting force is
viewed as a product of a coefficient and undeformed chip
thickness. Some methods have been proposed for calibrating
milling force coefficients by different authors [7–9], and the chip
thickness is basically calculated with the classic approximation
formula tn ¼ fz sin c sin k.
The increasing number of researches [10–20] on simulation of
milling process highlights the importance of cutting force model
for machining process plan and optimization. Undeformed chip
thickness has become a critical factor of affecting the prediction
accuracy of cutting forces. Li et al. [21] found the classical chip
thickness model assumes that the tooth path is circular and thus
lacks accuracy. They developed a new model for the undeformed
ARTICLE IN PRESS
Contents lists available at ScienceDirect
journal homepage: www.elsevier.com/locate/ijmactool
International Journal of Machine Tools Manufacture
0890-6955/$ - see front matter 2009 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ijmachtools.2009.07.015
Corresponding author.
E-mail addresses: xiands@dlut.edu.cn (Y. Sun), renfei_dl@yahoo.com.cn (F. Ren).
International Journal of Machine Tools Manufacture 49 (2009) 1238–1244

2. ARTICLE IN PRESS
chip thickness in horizontal milling. A transcendental equation
was then derived to calculate the underformed chip thickness.
Kumanchik and Schmitz [22] also gave an analytic expression for
chip thickness while considering factors such as the cycloidal
motion of teeth, and uneven teeth spacing. In the model, line feed,
tool rotational speed, and radius associated with milling were
combined into a single, non-dimensional parameter. Sai et al. [23]
noted very little work has been done in research of modeling of
chip thickness in circular interpolation. They described a method
for calculating the instantaneous undeformed chip thickness in
face milling case of circular interpolation and it was compared
with the case of linear interpolation.
Serval improvements have been proposed in linear or arc
interpolation for some specificed ball-end milling cases. However,
there are few literaterures on modeling of chip thickness
for curved geometrics, varying feed and toolpath in parameteric
interploation. Despite the influence of factors in real milling,
from theoretical analysis the existing chip thickness models
inevitablely brings errors to the solution in ideal status in milling
freeform surface along curved path with adaptive feed. It is
essential to further improve the prediction accuracy of cutting
forces and algorithms to concrete implementions for sculptured
surface machining. Hence, we present an approach to estimating
cutting forces based on a new undeformed chip thickness model
derived from the relative tool-part motion analysis in milling.
2. Proposed method
The prediction of cutting forces consists of three steps in the
developed force model. First, a new chip thickness model is
proposed by analyzing the relative tool-part motion so that it is
able to handle the combined effects of toolpath pattern, feedrate
schedule and tool geometry. In the second step a special
procedure to Z-map model is applied for efficient extraction of
the engaged cutting edge. At last, differential cutting forces of
each engaged segment are obtained and integrated to determine
the resultant cutting force based on the calibrated cutting force
coefficients.
As shown in Fig. 1, according to the premise that the cutting
force is proportional to undeformed chip area, the tangential (dFt),
radial (dFr), axial (dFa) components of differential cutting force are
modeled as follows:
dFmðyÞ ¼ Km dAðcij; y; kÞ ð1Þ
where m ¼ t, r, a, Km (N/mm2
) denote the calibrated milling force
coefficients, dA can be calculated as
dAðcij; y; kÞ ¼ tnðcij; y; kÞ db ð2Þ
cijðy; zÞ ¼ y þ ði 1Þjc þ bjðzÞ ð2aÞ
jc ¼ 2p=Nf ð2bÞ
bjðzÞ ¼ zj tanði0Þ=R0 ð2cÞ
where zj denotes the z coordinate component of the jth segment of
the cutting edge in Tool Reference System, tn is the undeformed
chip thickness, and db is the height of axial disc segment.
2.1. Undeformed chip thickness calculation
At the moment during the tool-part engagement, as shown in
Fig. 2, the undeformed chip thickness can be calculated with the
following form by finding the intersection point Q between the
path left by the previous cutting edge and line segment CP
perpendicular to the cutter axis C–C and through the current
cutting point P
tn ¼ JQPJ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðXP XQ Þ2
þ ðYP YQ Þ2
q
ð3Þ
In this manner, two fundamental issues have to be dealt with.
One is the preceding trace modeling of cutting edge. The other is
the intersection computation of the trace with a line segment
defined by the point and related cutter axis vector. Following the
Nomenclature
(oT, x, y, z) Tool Reference System
(oG,X,Y,Z) Globe Reference System
i index of cutting edge
R0 tool radius
y cutter rotational angle
b lag angle
dA undeformed chip area
Kt, Kr, Ka cutting force coefficients
c rotational angle of a point on the cutting edge
dFt, dFr, dFa differential cutting forces in tangential, radial, and
axial directions system
j index of discrete element of cutting edge
i0 nominal helix angle
jc flute spacing angle
Nf number of cutting flute
F feed speed
N rotational speed of spindle
fz feed per tooth
k positioning angle between a point on the flute and
the z-axis in vertical plane
(XQ, YQ, ZQ) position of intersection point Q in globe reference
system
(XP, YP, ZP) position of current cutting point P in globe reference
system
Fig. 1. Geometrical model of ball-end milling process.
Y. Sun et al. / International Journal of Machine Tools Manufacture 49 (2009) 1238–1244 1239

3. ARTICLE IN PRESS
kinematics of the cutter with respect to the workpiece, the
trajectory surface of cutting edge can be derived which naturally
contains the combined effects of feedrate, toolpath patterns
and cutter geometry. The preceding trace at the z-plane is
just the intersection curve of the trajectory surface of the
preceding cutting edge and the plane. Then, the intersection
point is determined using the line segment/curve intersection
algorithm.
2.1.1. Trajectory of cutting edge in milling
Let rCL(u) ¼ {XCL(u),YCL(u),ZCL(u)} be a given cutter location
path along which the cutter performs the rotational and
translational motion. F(u) is the feed velocity of the cutter with
respect to path parameter u. According to the relative tool-part
motion analysis, the kinematic equation of an arbitrary point on
the cutting edge defined in Global Reference System is established
as follows:
rCE
ðtÞ ¼ rCLðuðtÞÞ þ BðyðtÞÞCðzÞ ð4Þ
where rCE
(t) represents the trajectory of the point at the
time moment t; B(y) is the rotation matrix of the cutter with
y ¼ 2pNt. In Tool Reference System, the position of the point is
given by
CðzÞ ¼
RðzÞ sin ði 1Þjc þ
z tan i0
R0
RðzÞ cos ði 1Þjc þ
z tan i0
R0
z
2
6
6
6
6
6
4
3
7
7
7
7
7
5
ð5Þ
where z is the z coordinate component of the point in Tool
Reference System, R(z) is the radius of the section circle of the
cutter at the z-plane perpendicular to the tool axis vector, as
provided in the following form:
RðzÞ ¼ R0 zZR0
RðzÞ ¼ R0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
z
R0
1
2
s
zoR0
8
:
ð6Þ
To guarantee the synchronization of two motion processes, it is
necessary to derive the relationship between parameter u and
time t. The mathematical formulation is given as
drCLðuÞ
dt
¼
drCL
du
du
dt
¼ FðuÞ
r0
CLðuÞ
jr0
CLðuÞj
ð7Þ
r0
CLðuÞ
du
dt
¼ FðuÞ
r0
CLðuÞ
jr0
CLðuÞj
ð8aÞ
jr0
CLðuÞjdu ¼ FðuÞdt ð8bÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðdXCL=duÞ2
þ ðdYCL=duÞ2
þ ðdZCL=duÞ2
q
FðuÞ
du ¼ dt ð8cÞ
2.1.2. Preceding trace for a point on the cutting edge
For a point on the current engaged cutting edge, there are two
necessary and sufficient conditions to determine those points on
the preceding trace: (I) satisfying Eq. (4) and (II) having the same Z
coordinate component as the specified point on the current
engaged cutting edge in Globe Reference System. Based on
Eqs. (4), (8a)–(8c) and Z coordinate component of the engaged
point, a mathematical equation can be derived to describe the
preceding trace for the point P whose coordinate components are
labeled as XP, YP and ZP in Global Reference System.
rPT
P ðt
Þ ¼ rCLðuðt
ÞÞ þ Bðyðt
ÞÞCðzðt
ÞÞ ð9Þ
where rPT
P ðt
Þ represents the preceding trace for P with regard to
parameter t*; z(t*) denotes the z coordinate component of the
point on the preceding trace in Tool Reference System.
zðt
Þ ¼ ZP ZCLðuðt
ÞÞ þ R0 ð10Þ
where ZCL(u(t*)) represents the Z coordinate component of the
cutter location point at t* in Global Reference System; t*A[t0,t], t
denotes the moment when the cutting point P is intersected with
the workpiece.
t0 ¼ t
i
N
ð11Þ
where i* represents the specified number of backward revolutions
of the cutter and satisfies i*A{1, 2, y}.
2.1.3. Intersection point calculation
As shown in Fig. 2, to calculate the undeformed chip thickness,
the intersection point Q between the preceding trace and the line
segment CP must be known first. Let rCE
P be a vector of the point P
intersected with the workpiece at t. rP
CLis the corresponding cutter
location point at the moment. Then the following expression can
be derived to construct the line segment
sðtÞ ¼ tfrP
CL ta½ðrP
CL rCE
P Þ tag þ ð1 tÞrCE
P ð12Þ
where s(t) represents the line segment with regard to t,tA[0,1]; ta
denotes the cutter axis vector and satisfies ta ¼ [0,0,1]T
in 3-axis
milling. Thus, the intersection point is derived as follows:
rPT
P ðt
Þ sðtÞ ¼ 0 ð13Þ
Due to the implicit relationship between t* and t, numerical
approach is needed to solve the equation. In this case, a geometric
transformation is performed on the preceding trace and the line
OG
X
Y
Z
Preceding trace
Current trace
Cutter geometry at the
cutting moment
Z = ZP section plane
C
P
Q
Current cutter axis
tn
Cutting Direction
Fig. 2. Schematic diagram of undeformed chip thickness calculation.
Y. Sun et al. / International Journal of Machine Tools Manufacture 49 (2009) 1238–1244
1240

4. ARTICLE IN PRESS
segment to convert the issue of curve/line segment intersection
into that of determining the root of an equation. Let b be the
vector angle between the direction vector of line segment CP and
the X component of Global Reference System, then
b ¼ cos1 wP rCE
P
jwP rCE
P j
!
ðvXÞ
#
0rbrp ð14Þ
with vX ¼ [0,0,1]T
. Since both preceding trace rPT
P ðtÞ and line
segment s(t̄) are located on the plane perpendicular to the Z-axis
of the Global Reference System in 3-axis milling, the intersection
point can be determined by solving the equation Y^
r
PT
P
¼ 0 with
subject to X^
r
PT
P
rjwP rCE
P j where X^
r
PT
P
and Y^
r
PT
P
are coordinate
components of a point on transformed preceding trace in Global
Reference System
^
r
PT
P ¼ RotðZ; bÞ ðrPT
P rP
CLÞ ð15Þ
with
RotðZ; bÞ ¼
cos ðbÞ sin ðbÞ 0
sin ðbÞ cos ðbÞ 0
0 0 1
2
6
4
3
7
5
where b ¼ b if the point P is in quadrant I; b ¼ bp in
quadrant II; b ¼ pb in quadrant III and b ¼ b in quadrant IV.
The iteration scheme can be used to solve Eq. (13). The
transformed point P and the negative X direction are selected
as starting iteration point and search direction, respectively. In
special cases, more than one intersection points may exist.
The Bezier clipping technology [24] can be used to find all the
intersection points (Q1, Q2, y) and a set of possible undeformed
chip thicknesses are obtained (tn
1
, tn
2
, y). The desired chip
thickness is determined by
tn ¼ mintl
nðl ¼ 1; 2; . . .Þ ð16Þ
2.2. Engaged cutting edge
Z-Map model is used to determine whether a differential
cutter element is intersected with the workpiece at the moment
of machining. The workpiece is meshed into small grids whose
projection into the XoGY plane is square. In general cases,
the engaged cutter element can be achieved according to the
difference between the cutter element and the projection of the
instantaneous workpiece height into the cutter element.
With consideration of adaptive feedrate schedule, varied work-
piece geometry feature and curved toolpath pattern, more
than one cutter element may cut a grid for each tool movement.
In this case, re-computing each projection of instantaneous
workpiece heights into cutter elements is time consuming.
To counteract the situation, in computing the mesh grids can
be viewed as a set of planes. For each mesh grid, the engaged
cutter elements are those not only they are below the related
plane, but also their projections are located into the correspond-
ing squares as shown in Fig. 3. Repeating the process, the lengths
of cutting edges and the workpiece surface topography can be
obtained.
2.3. Cutting force estimation
The tangential, radial and axial cutting forces of each cutter
disk element are calculated with Eq. (1). In Tool Reference System,
components of differential cutting forces are expressed as
ðdFx; dFy; dFzÞT
¼ A ðdFt; dFr; dFaÞT
ð17Þ
where the matrix A is defined as
A ¼
cos ðcÞ sin ðcÞsin ðkÞ sin ðcÞcos ðkÞ
sinðcÞ cosðcÞsinðkÞ cosðcÞcosðkÞ
0 cosðkÞ sinðkÞ
2
6
4
3
7
5
To obtain the resultant force, it is necessary to perform a
numerical integration along the cutting edge engaged in cutting
process. The engagement conditions of differential cutting
elements are used for the estimation of boundaries of the
integration. By summing up the differential cutting forces for all
in-cutting differential cutting elements, the total cutting forces
are finally determined.
2.4. Cutting coefficient calibration
Precondition to acquiring the cutting force is that the cutting
coefficients should be known. For a specified cutter, part material
and cutting conditions, the cutting coefficients can be calibrated
with experimental data. In this work, one planar milling test was
conducted in milling aluminum 2024-T6 with a vertical CNC
milling machine. Cutting parameters are appropriately selected to
ensure single cutting edge engagement during machining so that
it can guarantee the synchronization between measured cutting
forces and those predicted. At a given cutter rotational angles y*,
for each engaged differential cutter element (j ¼ 1, 2, y, q), the
proposed method is used to determine the instantaneous
undeformed chip thickness t*n(j). By means of instantaneous
average chip thickness, t
nis calculated and then used to calibrate
the coefficients with the corresponding measured instantaneous
cutting forces. According to Eqs. (1), (2) and (17), coefficients
oG
X
Y
Z
Meshed work-piece
CL-path
Discrete cutting
points
Points engaged
in cutting Projection of meshed
work-piece
Grids connecting
with incut edge
A
B
C
D
C'
D'
A'
B'
Fig. 3. Illustration of the Z-map model.
Y. Sun et al. / International Journal of Machine Tools Manufacture 49 (2009) 1238–1244 1241

5. ARTICLE IN PRESS
related to t
n can be obtained as follows:
½Ktðy
ÞKrðy
ÞKaðy
ÞT
¼
1
tndb
X
q
j¼1
A
0
@
1
A
1
½FXðy
ÞFY ðy
ÞFZðy
ÞT
ð18Þ
When the cutter rotates to kjc (k ¼ 1, 2, y, Nf), the angular
position of cutting edge is the same as the one at y*. Thus, the
average value of measured cutting forces can be used to reduce
random errors. Repeat the process for different cutter rotational
angles, relationship between cutter coefficients and the unde-
formed chip thickness is subsequently established.
3. Model validation
3.1. Comparison validation
To investigate the differences between the proposed under-
formed chip thickness model and the classical one, three
numerical cases are simulated where the cutter moves along the
given toolpaths with R0 ¼ 12 mm, N ¼ 600 rpm and fz ¼ 0.25 mm
feed per tooth. During the process, undeformed chip thickness of
specified points on the cutting edge is calculated using the two
methods. Position angles k of the points are selected as
k1 ¼ k3 ¼ 57.61 and k2 ¼ 34.21 where the subscript denotes cases
I, II and III, respectively. Errors of the existing method with respect
to the proposed one are illustrated in Fig. 4. From the figure it can
be seen that the error curves show different shapes and
magnitude ranges in the whole range of the rotational angle
cA[0,p] of the specified point. Case I simulates the simplest
cutting process and the error varies following a quasi-sinusoid
curve. In most region of the rotational angle c, the results
calculated by the existing method are larger than the one
calculated by the proposed method, which may be resulted from
the difference between the circular cutting trace used in the
existing method and the real trace generated by the relative tool-
part motion. In terms of varied feed directions, in case III the
distributions of errors dramatically change with respect to case I,
which shows that feed direction plays a vital role in defining
the chip geometry. The combined effects of the feed direction
and the sectional circle radius of the specified point are illustrated
in case II.
Designed toolpath Preceding trace on the plane perpendicular
to cutter axis from specified point
Designed toolpath
Position angle of specified points for
Case I, II and III
Tooth trajectory
Toolpath
Chip thickness in
one tool revolution
Case III
Tooth trajectory
Toolpath
Chip thickness in
one tool revolution
Approximation error patterns
0
-0.08
30 60 120 180
-0.06
0
0.02
0.04
tooth position angle (Deg.)
approximation
error
(mm)
-0.04
-0.02 Case I
Case III
Case II
Eap
tn
ap
tn
pr
Specified points for Case I and III
Specified point for
Case II
z
x
OT
57.6
34.2
G
o
X
Y
Z
Case II
Case III
Equation of the curve:
Tooth trajectory
Toolpath
Chip thickness in
one tool revolution
Case I
Case II
150
90
= -
Fig. 4. Error analysis between the existing and the proposed undeformed chip thickness model.
0 360
-800
-400
0
400
800
1200 Predicted by existing method
Measured
Predicted by proposed method
Cutting rotational angle (Deg.)
Cutting
force
(N)
Fz
Fy
Fx
60 120 180 240 300
Fig. 5. Comparisons of measured cutting forces and predicted cutting forces with
two methods.
Y. Sun et al. / International Journal of Machine Tools Manufacture 49 (2009) 1238–1244
1242

6. ARTICLE IN PRESS
Compared with the existing model, more accurate chip geometry
is derived by the proposed method. To validate this, a standard
horizontal slot cutting experiment is conducted. Measured cutting
forces and predicted cutting forces are shown in Fig. 5. It can be seen
that cutting forces predicted by the proposed method agrees well
with the measured results, and the maximum relative error of peak
cutting forces is less 5%. However, using the existing model, the
relative errors are about 12% and even more. It also shows the
necessity of using the proposed method in sculptured surface
machining for varying feedrate and curved toolpaths.
3.2. Curved surface milling validation
Validation tests are conducted to testify the proposed method
under real machining conditions. During cutting tests, down-
milling process are carried out without coolant. Workpiece
material is aluminum alloy 2024-T6. The spindle speed is set as
1000 rpm and the depth of cut is 1 mm. The cutter is made to
machine along sinusoidal type cutter paths on a curved surface
expressed as
rðu; vÞ ¼ ð125u; 20 cosðpvÞ; 10 sinðpvÞÞ where ðu; v 2 ½0; 1Þ ð19Þ
Fig. 6 shows the measured and predicted force signals for
sculptured surface machining as well as the detailed views of
cutting forces. From the figures of measured cutting forces,
it can be seen that there exist fluctuations of cutting forces
within a relatively low magnitude range, which are mainly
ascribed to the tool run-out, the dynamic characteristics of the
cutting system, possible influences from the adjacent working
machines in real workshop environment, and the uncertain
factors of the piezoelectric dynamometer. Regardless of these
error sources, results obtained from the experiments prove the
validity of the proposed model in different machining cases.
The magnitude and shape of predicted cutting forces have
good agreements with those of measured forces in validation
test, and the relative errors of peak cutting forces are controlled
within 10% in most milling regions. Unlike iso-parametric
path, this kind of wave-like path has obvious change of feed
direction that is helpful in investigating the influence of
feed direction on the prediction accuracy of cutting forces.
From tested results we can see that in some cutting moments
the chip thickness calculated by the proposed method is different
from each other, even under the same cutter rotational angle
and the engagement region of cutting edge and workpiece.
That is to say, except for the local geometry of part and feedrate,
cutting direction also affects the shape and magnitude of cutting
force. The proposed method has the ability to calculate the
difference of the chip thickness under varying machining
conditions.
Machined geometry
Detail
view
of
measured
and
predicted
cutting
forces
Cutting
force
(N)
z
F
y
F
x
F
Measured
Force
Predicted
Force
Cutting
force
(N)
Fig. 6. Measured cutting force, shape and predicted cutting forces, shapes in validation test.
Y. Sun et al. / International Journal of Machine Tools Manufacture 49 (2009) 1238–1244 1243

7. ARTICLE IN PRESS
4. Conclusions
This paper represents a mechanistic approach to estimate
cutting forces in ball-end milling of sculptured surfaces. On the
basis of driving the kinematic trace expression of cutting edge in
machining, an undeformed chip thickness model is established
which is able to handle cases with complex part geometry, varying
feedrate and various toolpath patterns. Cutting coefficients are
calibrated from the proposed chip thickness model. The relation-
ship between the cutting coefficient and the chip thickness is
established, and resultant cutting forces in milling are then
predicted. Comparison validation with the existing method and
curved surface milling experiments on machining aluminum
2024-T6 are reported. It is shown that the predictions of cutting
forces have good agreements with the experimental results, even
though different types of cutter location paths and curved
geometry are applied. Meanwhile, the accuracy improvement of
the proposed method is not accompanied with the obvious
increase of computing time. Although the proposed chip thickness
model is established for ball-end cutter, it can be easily applied to
other type general cutter such as cylindrical milling cutter. The
proposed approach is capable of using in the prediction of cutting
forces in sculptured surface machining with varying feedrate,
depth of cut and geometrically complex toolpath. It is a feasible
alternative especially in some cases that will result in the
imperfection of predicted cutting forces using the classic
undeformed chip thickness. However, mechanistic modelling of
5-axis milling and cutting force-based feedrate schedule have not
been considered currently. They need to be researched further.
Acknowledgements
This research is supported by NSFC (50775023), NCET
(NCET-8-0081) and National Basic Research Program of China
(2005CB724100).
References
[1] G.M. Kim, P.J. Cho, C.N. Chu, Cutting force prediction of sculptured surface
ball-end milling using Z-map, International Journal of Machine Tools and
Manufacture 40 (2000) 277–291.
[2] M. Fontaine, A. Moufki, A. Devillez, D. Dudzinski, Modelling of cutting forces
in ball-end milling with tool-surface inclination: Part I: Predictive force
model and experimental validation, Journal of Materials Processing Technol-
ogy 189 (2007) 73–84.
[3] I. Lazoglu, Sculptured surface machining, a generalized model of ball-end
milling force system, International Journal of Machine Tools and Manufacture
43 (2003) 453–462.
[4] A. Lamikiz, L.N. Lo’pez, J.A. de Lacalle, M.A. Salgado, Cutting force estimation
in sculptured surface milling, International Journal of Machine Tools and
Manufacture 44 (2004) 1511–1526.
[5] B.M. Imani, M.H. Sadeghi, M.A. Elbestawi, An improved process simulation for
ball-end milling of sculptured surfaces, International Journal of Machine Tools
and Manufacture 38 (1998) 1089–1107.
[6] R.S. Naserian, M.H. Sadeghi, H. Haghighat, Static rigid force model for 3-axis
ball-end milling of sculptured surfaces, International Journal of Machine Tools
and Manufacture 47 (2007) 785–792.
[7] E. Budak, Y. Altintas, Prediction of milling force coefficients from orthogonal
cutting data, Journal of Manufacturing Science and Engineering Transaction of
the ASME 118 (1996) 216–224.
[8] W.S. Yun, D.W. Cho, Accurate 3-D cutting force prediction using cutting
condition independent coefficients in end milling, International Journal of
Machine Tools and Manufacture 41 (2001) 463–478.
[9] J.H. Ko, D.W. Cho, Determination of cutting-condition-independent coeffi-
cients and runout parameters in ball-end milling, International Journal of
Advanced Manufacturing Technology 26 (2005) 1211–1221.
[10] R. Salami, M.H. Sadeghi, B. Motakef, Feed rate optimization for 3-axis ball-end
milling of sculptured surfaces, International Journal of Machine Tools and
Manufacture 47 (2007) 760–767.
[11] B. Ozturk, I. Lazoglu, H. Erdim, Machining of free-form surfaces. Part II:
calibration and forces, International Journal of Machine Tools and Manufac-
ture 46 (2006) 736–746.
[12] K.A. Desai, P.V.M. Rao, Effect of direction of parameterization on cutting forces
and surface error in machining curved geometries, International Journal of
Machine Tools and Manufacture 48 (2008) 249–259.
[13] L.N. López de Lacalle, A. Lamikiz, J.A. S
anchez, M.A. Salgado, Toolpath
selection based on the minimum deflection cutting forces in the program-
ming of complex surfaces milling, International Journal of Machine Tools and
Manufacture 47 (2007) 388–400.
[14] S. Doruk Merdol, Y. Altintas, Virtual cutting and optimization of three-axis
milling processes, International Journal of Machine Tools and Manufacture 48
(10) (2008) 1063–1071.
[15] H.S. Lu, C.K. Chang, N.C. Hwang, C.T. Chung, Grey relational analysis coupled
with principal component analysis for optimization design of the cutting
parameters in high-speed end milling, Journal of Materials Processing
Technology 209 (2009) 3808–3817.
[16] E. Budak, E. Ozturk, L.T. Tunc, Modeling and simulation of 5-axis milling
processes, CIRP Annals—Manufacturing Technology. doi:10.1016/j.cirp.2009.
03.044.
[17] Mohammad Malekian, Simon S. Park, Martin B.G. Jun, Modeling of dynamic
micro-milling cutting forces, International Journal of Machine Tools and
Manufacture 49 (2009) 586–598.
[18] S. Sun, M. Brandt, M.S. Dargusch, Characteristics of cutting forces and chip
formation in machining of titanium alloys, International Journal of Machine
Tools and Manufacture 49 (2009) 561–568.
[19] Min Wan, Wei-Hong Zhang, Systematic study on cutting force modelling
methods for peripheral milling, International Journal of Machine Tools and
Manufacture 49 (2009) 424–432.
[20] S. Sun, M. Brandt, M.S. Dargusch, Characteristics of cutting forces and chip
formation in machining of titanium alloys, International Journal of Machine
Tools and Manufacture 49 (2009) 561–568.
[21] H.Z. Li, K. Liu, X.P. Li, A new method for determining the underformed chip
thickness in milling, Journal of Materials Processing Technology 113 (2001)
378–384.
[22] L.M. Kumanchik, T.L. Schmitz, Improved analytical chip thickness model for
milling, Precision Engineering 31 (2007) 317–324.
[23] L. Sai, W. Bouzid, A. Zghal, Chip thickness analysis for different tool motions:
for adaptive feed rate, Journal of Materials Processing Technology 204 (2008)
213–220.
[24] T.W. Sederberg, T. Nishita, Curve intersection using Bezier clipping, Computer
Aided Design 22 (9) (1990) 538–549.
Y. Sun et al. / International Journal of Machine Tools Manufacture 49 (2009) 1238–1244
1244