Peripheral1

1. Simulation of cutting process in peripheral milling by predictive cutting force
model based on minimum cutting energy
Takashi Matsumura n
, Eiji Usui
Department of Mechanical Engineering, Tokyo Denki University, 2-2 Kanda-Nishiki-cho, Chiyoda-ku, Tokyo 101-8457, Japan
a r t i c l e i n f o
Article history:
Received 7 October 2009
Received in revised form
24 January 2010
Accepted 26 January 2010
Available online 1 February 2010
Keywords:
Cutting
Ball end mill
Cutting force
Chip flow
Cutting energy
Peripheral cutting
a b s t r a c t
The cutting force and the chip flow direction in peripheral milling are predicted by a predictive force
model based on the minimum cutting energy. The chip flow model in milling is made by piling up the
orthogonal cuttings in the planes containing the cutting velocities and the chip flow velocities. The
cutting edges are divided into discrete segments and the shear plane cutting models are made on the
segments in the chip flow model. In the peripheral milling, the shear plane in the cutting model cannot
be completely made when the cutting point is near the workpiece surface. When the shear plane is
restricted by the workpiece surface, the cutting energy is estimated taking into account the restricted
length of the shear plane. The chip flow angle is determined so as to minimize the cutting energy. Then,
the cutting force is predicted in the determined chip flow model corresponding to the workpiece shape.
The cutting processes in the traverse and the contour millings are simulated as practical operations and
the predicted cutting forces verified in comparison with the measured ones. Because the presented
model determines the chip flow angle based on the cutting energy, the change in the chip flow angle
can be predicted with the cutting model.
& 2010 Elsevier Ltd. All rights reserved.
1. Introduction
Machine shops in the mold manufacturing industries perform
milling operations to finish sculptured surfaces with ball end
mills. The cutter paths should be determined by evaluating the
cutting forces because the machining errors largely depend on
the cutting forces. Fine surface quality and machining accuracy
are also required in finishing the mold. Because chips sometimes
scratch the finished surfaces, coolants have been supplied for
control of the chip flow as well as cooling and lubrication.
However, the usage of coolants has recently been restricted from
the environmental point of view. Therefore, chip control should
be considered in the determinations of cutting parameters and
cutter path.
Many researches have been made on force models in milling
processes. Smith and Tlusty [1] reviewed many works in the
modeling of milling processes. Ehamann et al. [2] also reviewed
mechanistic models in milling. Koenigsberger and Sabberwal [3]
developed a mechanistic model for slab milling and face milling
operations based on the cutting force coefficients. Kline et al. [4]
developed a model to predict milling forces based on the chip
load. Armarego and Deshpande [5,6] predicted cutting forces in
end milling based on the oblique cutting model. Liu et al. [7]
presented a model in the peripheral milling with associated
machining error. Ratchev et al. [8] performed simulations to
control the cutting load.
Recently, so many ball end mills have been used to finish
sculptured surfaces in machine shops. Many force models have
been presented since the work done by Yang and Park [9].
Bayoumi et al. [10] presented a mechanistic force model of the
profile end mill. Most of them predicted the cutting force based on
the cutting coefficient [11] or the oblique cutting mechanism [12].
These force models in the ball end milling have been applied to
more practical operations. Imani et al. [13] developed the process
simulation for ball end milling taking into account the workpiece
shape. Fontaine et al. [14] presented a force model in a wavelike
form machining process. Kim et al. [15] applied a force model of
the ball end mill to the sculptured surface cutting using Z-map.
Lazoglu [16] presented a generalized model for machining of the
sculptured surface. However, the change in chip flow direction
with cutter path has not been predicted in their models. Although
the force models based on the oblique cutting consider the chip
flow directions, chip flow angles were given uniquely by an
assumption based on the local edge inclination or with the
approximation equation. However, the chip flow angle depends
not only on the edge geometry but also on friction on the rake face
of the tool and material properties.
FE analysis is an effective approach to review the cutting
processes with chip formations [17]. Commercial softwares have
recently been available to simulate the milling and the drilling
ARTICLE IN PRESS
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journal homepage: www.elsevier.com/locate/ijmactool
International Journal of Machine Tools & Manufacture
0890-6955/$ - see front matter & 2010 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ijmachtools.2010.01.007
n
Corresponding author. Tel.: +81 3 5280 3391; fax: +81 3 5280 3568.
E-mail address: tmatsumu@cck.dendai.ac.jp (T. Matsumura).
International Journal of Machine Tools & Manufacture 50 (2010) 467–473

2. ARTICLE IN PRESS
processes. However, it requires a long time for simulating changes
in cutting processes in practical operations such as contour
milling.
Usui et al. [18] presented a force model to predict cutting
forces with the chip flow model based on the cutting energy. Later
Matsumura and Usui [19] presented a force model for cutting
with the complex-shaped end mills and applied the model to
simulations of slotting operations. Many peripheral milling
operations are also performed in the mold machining. In the
simulation for the peripheral milling, the cutting or the non-
cutting process should be considered when the chip flow model is
determined corresponding to the workpiece shape.
This paper presents a force model based on the minimum
cutting energy for the peripheral milling process. The cutting
force and the chip flow direction are predicted taking into account
the change in the workpiece shape. Some case studies in the
peripheral millings with a ball end mill are shown to verify the
presented force model.
2. Force model in the peripheral milling process
2.1. Outline of the force model based on the minimum cutting energy
A force model for complex-shaped end milling was presented
with a detail procedure for analysis in Ref. [19]. This force model
is briefly described here. Fig. 1(a) shows the chip flow on the rake
face of a ball end mill. The force model predicts the cutting force
in the direction of X-, Y- and Z-axis designated in Fig. 1 along with
the chip flow direction. The chip flow in milling is interpreted as a
piling up of orthogonal cuttings in the planes containing the
cutting velocities V and the chip flow velocities Vc. Thus, the
cutting edges are divided into discrete edge segments and ortho-
gonal cutting models are made on the segments. The orthogonal
cutting models are obtained by the following equation:
f ¼ fða; V; t1Þ
ts ¼ gða; V; t1Þ
b ¼ hða; V; t1Þ
9
>=
>;
ð1Þ
where f, ts and b are the shear angle, the shear stress on the shear
plane and the friction angle in the orthogonal cutting; a, V and t1
are the rake angle, the cutting velocity and the uncut chip
thickness. Eq. (1) is obtained from the orthogonal cutting tests.
The cutting energy, which is the sum of the shear energy on the
shear plane and the friction energy on the rake face, is calculated
in the cutting model. Because the cutting model changes with
chip flow angle, the chip flow angle is determined to minimize the
cutting energy in the chip flow model. Then, the cutting force
loaded on the tool can be predicted in the determined chip flow
model.
2.2. Cutting model in peripheral milling
Cutting models on the segmented cutting areas should be
made taking into account the workpiece shape in the peripheral
milling. When the cutting point is near the workpiece surface, the
shear plane cannot be completely determined, i.e., the shear
planes to be determined in the cutting models are restricted by
the workpiece surface. Therefore, the lengths of the shear planes
have to be calculated to estimate the cutting energy in the cutting
models.
Fig. 1(b) shows the coordinate systems in the analysis. X–Y–Z is
the reference system; X0
–Y0
–Z0
rotates with the cutting edge at
angular velocity o. Because the cutting velocity VRE is the
resultant of the circumferential velocity VP and the feed rate f,
as shown in Fig. 1(b), the orthogonal cutting model is determined
in the X00
–Y00
–Z00
coordinate system based on the direction of the
cutting velocity. The Y00
-axis is defined in the velocity direction
and the X00
-axis is perpendicular to the Y00
-axis.
The cutting model in the peripheral milling depends on the
cutting position relative to the workpiece surface. Fig. 2 shows
the orthogonal cutting model at a cutting point on an edge. Plane
PCGEF contains the cutting velocity and the chip flow velocity at
Point P. The rake face of the tool PACBD inclines at the radial rake
angle aR
00
and the axial rake angle aA
00
in X00
–Y00
–Z00
. The orthogonal
cutting model is made in PCGEF based on Eq. (1), as shown in
Fig. 2(a), when the shear plane completely occurs in the material.
Point Q is the end of the shear plane in the cutting model. When
the shear plane to be made in the cutting model crosses the
workpiece surface as shown in Fig. 2(b), the end of the shear plane
is regarded as Point Q0
. PCGEF is divided into the inside and the
outside of the workpiece.
The surface of the workpiece to be removed is expressed by a
combination of finite discrete surfaces in X–Y–Z:
Siðx; y; zÞ ¼ 0 ð2Þ
where i is the index of the discrete surface. The presence of
material with respect to Eq. (2) is switched by a cutting manner.
Si(x,y,z)Z0 is associated with the inside of the material in the up-
cut milling. The inside of the material in the down-cutting is given
by Si(x,y,z)r0.
The coordinates of P and Q in X00
–Y00
–Z00
are transformed to
those of the rotating coordinate system X0
–Y0
–Z0
by the following
VRE
VC
Chip
Rake face
Surface finished by
the previous cutter
Orthogonal cutting plane
Cutting area on
orthogonal cutting plane
Rotation axis
Y
Z
Feed
Chip flow angle
X
Cutter path
O
P
Y
Y"
X"
VP
VRE
f
Feed direction Z
Z
Y
X
X
Z"
RP
ft
t-
Fig. 1. Chip flow model in analysis. (a) Chip flow in milling with a ball end mill. (b)
Coordinate systems in analysis.
T. Matsumura, E. Usui / International Journal of Machine Tools & Manufacture 50 (2010) 467–473468

3. ARTICLE IN PRESS
equations:
x0
¼ x00
cosYÀy00
sinY
y0
¼ x00
sinYþy00
cosY
z0
¼ z00
9
>=
>;
ð3Þ
Y is the wedge angle between the direction of the circumferential
velocity and that of the resultant cutting velocity at P, as shown in
Fig. 1(b), and is given by
tanY ¼
fsinðotÀgÞ
RPoþfcosðotÀgÞ
ð4Þ
where RP is the radius of rotation at P. The coordinates of P and Q
in X0
–Y0
–Z0
are transformed to those of the reference coordinate
system X–Y–Z by the following equations:
x ¼ x0
sinðotÀgÞÀy0
cosðotÀgÞþft
y ¼ x0
cosðotÀgÞþy0
sinðotÀgÞ
z ¼ z0
9
>=
>;
ð5Þ
where t, f and g are time, feed rate and delay angle of P with
respect to the bottom of the edge, respectively.
When the coordinates (xQ, yQ, zQ) of Q satisfy the following
conditions, the end point of the shear plane Q exists in the
material:
SiðxQ ; yQ ; zQ ÞZ0; up-cutting
SiðxQ ; yQ ; zQ Þr0; down-cutting
(
ð6Þ
Therefore, the orthogonal cutting model can be formulated by
Eq. (1) without considering the workpiece surface as shown in
Fig. 2(a).
When the coordinates (xP, yP, zP) of P satisfy the following
conditions, P exists outside the material:
SiðxP; yP; zPÞo0; up-cutting
SiðxP; yP; zPÞ40; down-cutting
(
ð7Þ
Therefore, no cutting force is loaded on the tool because the
cutting point P does not remove the material.
When the coordinates of P and Q are in the following
conditions, P exists inside, whereas Q exists outside the material:
up-cutting :
SiðxP; yP; zPÞZ0
SiðxQ ; yQ ; zQ Þo0
(
ð8Þ
down-cutting :
SiðxP; yP; zPÞr0
SiðxQ ; yQ ; zQ Þ40
(
ð9Þ
A
B
O
α
''
R
''
A
c
X
''
Cutting edge
e
n
Vs
Rake face
Orthogonal
cutting plane
Rotation axis
Y
''
(VREP)
e
Z''
Vc
D
G
P
Q
E
C
F
A
B
O
''
R
''
A
c
X
''
Cutting edge
e
n
Vs Q
Rake face
Orthogonal
cutting plane
Workpiece
surface
Rotation axis
Y
''
(VRE)
e
Z
''
Vc
D
G
P
Q
E
C
F
0
0
P
Fig. 2. Orthogonal cutting model in peripheral milling. (a) Orthogonal cutting
model in the material. (b) Orthogonal cutting model restricted by workpiece
surface.
Z
X
Y
Feed
2.0mm
7.5mm
Workpiece
Tool
Workpiece
Tool
Z
Y
Fig. 3. Peripheral cutting operation with repeating pick feed. (a) Cutter path. (b)
Initial position of ball end mill in the Y–Z plane.
T. Matsumura, E. Usui / International Journal of Machine Tools & Manufacture 50 (2010) 467–473 469

4. ARTICLE IN PRESS
Q0
is the intersection of the shear plane PQ and the workpiece
surface as shown in Fig. 2(b). The shear energy is estimated as the
energy consumed in the shear plane PQ0
. PQ is expressed by the
following equation with parameter z:
xÀxP
xQ ÀxP
¼
yÀyP
yQ ÀyP
¼
zÀzP
zQ ÀzP
¼ z ð10Þ
For the coordinates of Q0
, the parameter zQ0 can be determined by
substituting Eq. (10) in Eq. (2). Then, the length of PQ0
can be
given by
PQ0 ¼ zQ0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðxQ ÀxPÞ2
þðyQ ÀyPÞ2
þðzQ ÀzPÞ2
q
ð11Þ
When the chip flow angle is assumed, the chip flow model is
constructed by piling up the orthogonal cuttings on the
segmented cutting areas. The shear energy on each segmented
area is estimated corresponding to the length of the shear plane
given by the above procedure. Then, the cutting energy can be
estimated as the sum of the friction energy on the rake face and
the shear energy on the shear plane. Because the cutting model
changes with the chip flow angle, the chip flow angle is
determined to minimize the cutting energy consumed in the chip
flow model. Finally, the cutting force can be predicted in the chip
flow model at the minimum cutting energy.
3. Case study
3.1. Traverse cutting with repeating pick feed
The presented model was verified for traverse cutting, where
peripheral cutting was repeated with the pick feed as shown in
Fig. 3(a). The workpiece shape to be machined changes with the
pick feed in the operation. The tool geometry and the cutting
conditions are shown in Table 1.
The orthogonal cutting models were determined by the
following equation corresponding to Eq. (1) for a combination of
0.45% carbon steel and carbide tool:
f ¼ expð0:01022V þ28671:2t1 þ0:07482aÀ0:48355Þ
ts ¼ expðÀ0:44485VÀ18569:8t1 þ0:62798aþ20:39367Þ
b ¼ expð0:73741V þ29600:4t1À0:78319aÀ0:93674Þ
9
>>=
>>;
ð12Þ
Fig. 3(b) shows the initial position of the ball end mill in the
Y–Z plane. The distance between the end face of workpiece and
the center of the cutter is 7.5 mm. The pick feed given by the
radial depth of cut is 5 mm. The end shape of the workpiece in the
nth path operation is mathematically expressed by
ðyþpÞ2
þðzÀRÞ2
¼ R2
½zZhcŠ
yÀfRÀðnþ1Þpg ¼ 0 ½zohcŠ
9
>=
>;
ð13Þ
where p is the pick feed and hc the lowest height of the curved
surface machined by the previous cutter path:
hc ¼
RÀ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2RðnÀ1ÞpÀðnÀ1Þ2
p2
q
½ðnÀ1ÞprRŠ
0 ½ðnÀ1Þp4RŠ
8
<
:
ð14Þ
The cutting processes are predicted in the first and the second
paths. Fig. 4 shows the schematic removal shapes viewed in the
feed direction and the height of the cutting area during a rotation
Table 1
Cutting conditions.
Workpiece 0.45% Carbon steel Spindle speed 985 rpm
Tool Ball end mill
(insert type)
Feed rate 197 mm/min
Number of edges 2 Axial depth of cut 2 mm
Tool radius 25 mm Lubrication Dry
Radial rake angle 0 deg
Axial rake angle 01
0.0
0.50
1.0
1.5
2.0
2.5
3.0
0 0.01 0.02 0.03 0.04 0.05
Edge 1
Edge 2
Heightmm
Time s
0.0
0.50
1.0
1.5
2.0
2.5
3.0
0 0.01 0.02 0.03 0.04 0.05
Edge 1
Edge 2
Heightmm
Time s
0
0.5
1
1.5
2
2.5
3
-4 -2 0 2 4 6 8
Cutting edge
Workpiece shape
zmm
y mm
Removed
area
0
0.5
1
1.5
2
2.5
3
-4 -2 0 2 4 6 8
Cutting edge
Workpiece shape
zmm
y mm
Removed
area
Fig. 4. Cutting areas during a rotation of cutter. (a)First path. (b) Second path.
T. Matsumura, E. Usui / International Journal of Machine Tools & Manufacture 50 (2010) 467–473470

5. ARTICLE IN PRESS
of the cutter. The cutting edge is divided into 16 discrete
segments and the time division is 64 in a rotation. Compared
with the removal area in the first path, the removal area in the
second path enlarges. Then, the upper cutting area in the second
path reduces in the latter of the cutting period of an edge due to
the workpiece shape machined in the first path. Fig. 5 shows the
predicted and measured cutting forces. The predicted cutting
forces can be verified in comparison with the measured cutting
forces. The cutting force in the second path changes with the
workpiece shape machined by the previous path.
Fig. 6(a) shows the shear plane length and the uncut chip
thickness during a rotation of the cutter in the second path. The
shear plane length changes with uncut chip thickness. Fig. 6(b)
shows the chip flow angle (Zc+Z0) in Fig. 2, which is the wedge
angle between the chip flow direction and the projected axial
direction onto the rake face. The chip flow inclines toward the
radial direction with increasing chip flow angle when the shear
plane length increases with uncut chip thickness. According to
Fig. 4(b), the cutting area does not change in a certain period after
edge’s penetration into the workpiece. It should be noted that the
chip flow angle changes even though the material is removed in
the same cutting area. In the presented model, the chip flow
angle is controlled not only by the local edge inclination but also
by the cutting energy based on the shear plane cutting model.
Then, the upper cutting area reduces after the cutting edge
removes the workpiece surface machined by the first cutter path.
Because the local edge inclination with respect to the radial direc-
tion decreases with the height of the cutting position on the
ball end mill, the chip flows upward, reducing the upper cutting
area.
-1000
-500
0
500
1000
-0.01
X measured
Y measured
Z measured
X simulated
Y simulated
Z simulated
CuttingforceN
Time s
-1000
-500
0
500
1000
-0.01
X measured
Y measured
Z measured
X simulated
Y simulated
Z simulated
CuttingforceN
Time s
0.01 0.02 0.03 0.04 0.05 0.060
0 0.01 0.02 0.03 0.04 0.05 0.06
Fig. 5. Cutting forces in peripheral operations with pick feeds. (a) First path.
(b) Second path.
0
5
10
15
20
25
30
35
40
-0.01
Edge 1
Edge 2
Chipflowangledeg
Time s
-0.01
0.00
0.01
0.02
0.03
0.04
0.05
0.06
-0.01
Edge 1, shear plane length
Edge 1, uncut chip thickness
Edge 2, shear plane length
Edge 2, uncut chip thickness
Shearplanelengthmm
Uncutchipthicknessmm
Time s
0 0.01 0.02 0.03 0.04 0.05 0.06
0 0.01 0.02 0.03 0.04 0.05 0.06
Fig. 6. Cutting model and chip flow angle in the second path of peripheral milling.
(a) Shear plane length and uncut chip thickness at the center of cutting area.
(b) Chip flow angle.
-20-1001020
-25
0
25
50
75
100
Y mm
Xmm
Cutter pathSurface finish line
Workpiece
Reference
position
Fig. 7. Cutter path in contour milling.
T. Matsumura, E. Usui / International Journal of Machine Tools & Manufacture 50 (2010) 467–473 471

6. ARTICLE IN PRESS
3.2. Contour milling
As a more practical application, the cutting process was
simulated for contour milling with changing feed direction and
the conditions in Table 1. Eq. (12) was employed to make the
orthogonal cutting model in the chip flow. Fig. 7 shows the cutter
path in the X–Y plane with workpiece. The center of the tool
moves along the solid line. The coordinates (x, y) of the cutter path
were given by the following harmonic function:
y ¼
R
2
sin
x
4R
p
ð15Þ
where R is the cutter radius. When the axial depth of cut is 2 mm,
the maximum radius of the cutting area is 6.78 mm. The surface is
to be finished along the dotted line.
The feed direction changes continuously with cutter location.
In the force model, the tool is fed in the X direction in the
reference coordinate system, as shown in Fig. 1(b). Therefore,
the simulation was performed in an equivalent cutting manner as
shown in Fig. 8. The end face of the workpiece rotates
corresponding to the feed direction. The end face is inclined at
the angle y given by the differential coefficient of Eq. (15). Then,
the coordinates (x, y) in X–Y–Z are transformed to (xn
, yn
) in the
coordinate system inclined at the angle y as follows:
xÃ
yÃ
#
¼
cosy siny
Àsiny cosy
x
y
#
ð16Þ
Because the end face of workpiece contains the origin of the
coordinate system, the plane equation of workpiece can be
expressed as
xÃ
sinyþyÃ
cosy ¼ 0 ð17Þ
Fig. 9 shows the predicted and measured cutting forces at
distances of 0, 25 and 50 mm from the reference position
designated in Fig. 7. The predicted cutting forces agree with the
measured ones. Fig. 10 shows the chip flow angle. The change in
the chip flow angle can be reviewed in the simulation as well as
the cutting force.
4. Conclusion
A force model based on the minimum cutting energy was
applied for prediction of the cutting force and the chip flow
direction in peripheral milling. Three-dimensional chip flow in
Cutter path
Surface line to
be finished
Cutter path
Surface line to
be finished
X
Y
X*
Y*
Fig. 8. Cutting process with changing feed direction. (a) Original cutter path. (b)
Equivalent manner.
-1000
-500
0
500
1000
0
X measured
Y measured
Z measured
X simulated
Y simulated
Z simulated
CuttingforceN
Time s
-1000
-500
0
500
1000
7.55
X measured
Y measured
Z measured
X simulated
Y simulated
Z simulated
CuttingforceN
Time s
-1000
-500
0
500
1000
15.22
X measured
Y measured
Z measured
X simulated
Y simulated
Z simulated
CuttingforceN
Time s
0.01 0.02 0.03 0.04 0.05
7.56 7.57 7.58 7.59 7.6 7.61
15.23 15.24 15.25 15.26 15.27 15.28
Fig. 9. Cutting forces in contour milling. (a) Cutting position: 0 mm. (b) Cutting
position: 25 mm. (c) Cutting position: 50 mm.
T. Matsumura, E. Usui / International Journal of Machine Tools Manufacture 50 (2010) 467–473472

7. ARTICLE IN PRESS
milling is interpreted as a piling up of orthogonal cuttings in the
planes containing the cutting velocities and the chip flow
velocities. The chip flow angle is determined to minimize the
cutting energy. Then, the cutting force is predicted for the
determined chip flow model. The conclusions of this paper are
summarized as follows:
(1) In peripheral milling, the shear plane to be determined in the
cutting model is restricted by the workpiece shape when
the cutting edge removes the material near the end face of
the workpiece. Then, the shear plane length is analyzed
mathematically. The shear energy in the cutting model is
estimated as the energy consumed in the restricted shear
plane length. As a result, the chip flow model can be made
taking into account the change in the shear plane length.
(2) The presented model was verified for traverse milling with
repeated pick feed and contour milling. The model predicts
the cutting force and the chip flow direction well correspond-
ing to the workpiece shape to be removed and feed direction.
The cutter path and the cutting parameters can be optimized
by reviewing the cutting process in simulations based on the
presented model.
(3) The presented model determines the chip flow angle based
on the cutting energy. Therefore, the chip flow angle depends
not only on the local edge inclination of the milling tool but
also on parameters of the cutting models such as shear plane
length.
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0
5
10
15
20
25
30
35
40
0
Edge 1
Edge 2
Chipflowangledeg
Time s
0
5
10
15
20
25
30
35
40
7.55
Edge 1
Edge 2
Chipflowangledeg
Time s
0
5
10
15
20
25
30
35
40
15.22
Edge 1
Edge 2
Chipflowangledeg
Time s
0.01 0.02 0.03 0.04 0.05
7.56 7.57 7.58 7.59 7.6 7.61
15.23 15.24 15.25 15.26 15.27 15.28
Fig. 10. Chip flow angles in contour milling. (a) Cutting position: 0 mm.
(b) Cutting position: 25 mm. (c) Cutting position: 50 mm.
T. Matsumura, E. Usui / International Journal of Machine Tools Manufacture 50 (2010) 467–473 473