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International Journal of Mechanical Engineering and Robotics Research Vol. 7, No. 1, January 2018
© 2018 Int. J. Mech. Eng. Rob. Res.doi: 10.18178/ijmerr.7.1.16-21
Prediction of Cutting Force in a Circular
Peripheral Milling Process
Kamel Mehdi1,2
1URMSSDT, Engineering National High School of Tunis (ENSIT), University of Tunis (UT), 5 Avenue Taha Hussein
P.B. 56 Bab Mnara 1008 Tunis, Tunisia 2Preparatory Institute for Engineering Studies El Manar (IPEIEM), University of Tunis EL Manar (UTM), PB. 244,
2092 Tunis, Tunisia
Email: [email protected]
Abstract—In this paper, we present a two-dimensional
machining model that allows the simulation of cutting force
including the cutting process damping in a dynamic circular
milling process. The cutting forces obtained through the
simulation model are compared with experimental results,
and a discussion on the effect of various milling parameters
on cutting force, such as cutting speed, feed rate, radial
depth of cut, tool diameter, and tool helix angle, in a
circular peripheral milling process will be presented.
Index Terms—cutting force, cutting damping process,
circular peripheral milling, numerical simulation.
I. INTRODUCTION
Prediction of cutting force has been the subject of
different research works for a long time [1–4]. The study
of the dynamic behavior of the cutting system (machine,
tool, and workpieces) can be analyzed from the responses
of the cutting forces to the fluctuations of the cutting
parameters (cutting speed and uncut chip thickness).
Montgomery and Altintas [5] studied the mechanism of
cutting force and surface generation in dynamic milling by
presenting a simulation model for dynamic milling. Huang
and Wang [6] proposed an analytical model of the milling
process including damping effects. Two cutting
mechanisms (shearing and ploughing mechanisms) and
two process damping effects (directional and magnitude
effects) are included. In 2012, Mehdi and Zghal [7]
proposed a numerical model that allows the prediction of
cutting forces in a linear peripheral milling process. They
studied the effects of tool parameters (diameter, helix
angle, and number of teeth) on cutting process damping
and cutting force distributions. In 2015, Li et al. [8]
confirmed the importance of taking into account the
cutting damping process in their research on the
machining of thin-walled workpieces.
All these traditional researches on the cutting force
model are usually focused on linear milling and do not
take into consideration other cutting conditions, especially
in circular milling processes.
Manuscript received July 1, 2017; revised December 21, 2017.
In this paper, we present a cutting force model for
circular peripheral milling based on the linear peripheral
milling force model of Mehdi and Zghal [7]. In the first
section, a geometric model of cutting forces, including the
cutting process damping in circular peripheral milling, is
presented. In this model, the regenerative chatter is
considered. In the second section, the simulated cutting
force components are compared to experimental results,
and a discussion on the effects of cutting and tool
parameters (cutting speed, feed rate, axial and radial depth
of cut, tool diameter, tool helix angle, and number of
tool’s teeth) on process damping is presented.
II. MECHANICS OF THE CIRCULAR PERIPHERAL
MILLING PROCESS
The circular peripheral milling process is assumed to be
a linear peripheral milling process for a length equal to
pR (Fig. 1). In the model, this length is considered to
be equal to the feed rate per tool rotation tN f in the x
direction. Hence, t
p
N f
R , where pR is the circular
workpiece profile radius, f is the feed rate per tooth,tN
is the number of teeth, and is the tool’s angular
change position vis-à-vis the workpiece for each feed rate
per tool rotation.
From Fig. 1, we can see that the feed rate per tooth in
the x direction can be expressed in the two directions
0 0( , )x y by
0 0cos sin ,xf f x y (1)
where 1
p
i
, in which p is the pth position of the
cutter center.
In the case of the linear peripheral milling and
according to the previous work of [7], the instantaneous
total cutting force vector ( )F t acting by the tool on the
workpiece in the ( , )x y directions is given by
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International Journal of Mechanical Engineering and Robotics Research Vol. 7, No. 1, January 2018
© 2018 Int. J. Mech. Eng. Rob. Res.
1
( , )
1 ( , )
( )( )
,( )
( )
t
j
t
j
N
x
jx
Ny x y
y
j x y
F tF t
F tF t
(2)
According to [7], ( )F t is the sum of the shearing
cutting force ( )sF t and the damping cutting force ( )dF t
due to the ploughing mechanism. The damping cutting
force ( )dF t is arranged to be calculated by
( , ) ( , )
( ) ( )( ) ,
( ) ( )
dx x
d c
dy yx y x y
F t u tF t C
F t u t
(3)
where ( ) ( )
( ) ( )
xx xy
c
yx yy
C t C tC
C t C t
is the cutting damping
matrix with
,
1 1
( ) ( ( )) [ ( ) ( )]cos ( ) sin ( ) ,t d
j j
N N
xx k j j t r j j
j k
C t t c t c t t t
(4)
2 2
,
1 1
( ) ( ( )) [ ( ) cos ( ) ( )sin ( )] ,t d
j j
N N
xy k j j t j r j
j k
C t t c t t c t t
(5)
2 2
,
1 1
( ) ( ( )) [ ( ) cos ( ) ( )sin ( )] ,t d
j j
N N
yx k j j r j t j
j k
C t t c t t c t t
(6)
( ) ( ) ,yy xxC t C t (7)
where jrc ( t ) and )(tc
jt represent the cutting damping
factors in the thrust and tangential directions [7] and
( , )
( )
( )
x
y x y
u t
u t
is the vibrating velocity vector of the
workpiece cutting point in the two directions ( , )x y .
The two components of the cutting force in the case of
the circular peripheral milling can be calculated according
to
0
00 0
( , )( , )
( ) ( ),
( )( )
x x
Rotyy x yx y
F t F tM
F tF t
(8)
where cos sin
sin cosRotM
is the rotation matrix
from ( , )x y to 0 0( , )x y .
The cutting process system of the circular peripheral
milling is modeled with two degrees of freedom, in which
the workpiece characteristics are represented by the mass
matrix M , stiffness matrix K , and viscous damping
matrix C . The tool is composed of dN elementary
cutting disks of thickness dz . By neglecting the training
and Coriolis inertia effects, the milling process simulation,
in the ( , )x y directions, is summed up by
( ) ( ) ( ) ( ) ,M U t C U t K U t F t (9)
( )U t where is the displacement vector of the
workpiece cutting point over the tool and ( )U t and ( )U t
are, respectively, the vibrating velocity and acceleration
vectors of this cutting point.
Figure 2 shows a nondetailed flowchart adopted for
the resolution of Eq. (9) and the determination of the
cutting force components in the ( , )x y directions, from
which we can deduce their values in the 0 0( , )x y
directions [Eq. (8)].
The algorithm of the program is divided essentially
into three procedures. These procedures have been
developed using the programming language of Maple. As
a first procedure, formal expressions for the calculation of
cutting forces were defined. These expressions are based
on the relative vibrations between the cutter and the
workpiece in the ( , )x y directions. Subsequently, the
second procedure allows data entry representing the
geometric characteristics of the cutter (helix angle, rake
angle, diameter, rigidity, number of flutes, etc.), the
cutting parameters (feed rate, cutting speed, radial and
axial depths of cut, etc.), and the material parameters of
the workpiece (tangential, radial, and specific pressures).
As a result of data entry, the third procedure provides the
numerical solution of the system motion equations
presented by Eq. (9). The resolution is assessed using the
Maple dsolve function, taking the components of the
displacement vector ( )U t as unknown. The components
of this vector are defined by Maple as procedures that
consider the time t as variable. The knowledge of
( ), ( ), ( ), ( )x y x yu t u t u t u t allows the determination of the
cutting force vector components ( )F t for each tool
rotation.
( )U t
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International Journal of Mechanical Engineering and Robotics Research Vol. 7, No. 1, January 2018
© 2018 Int. J. Mech. Eng. Rob. Res.
Figure 1. Dynamic cutting model scheme in the case of circular peripheral milling.
Figure 2. Flowchart of the curvilinear peripheral milling process.
III. EXPERIMENTAL VALIDATION AND DISCUSSION
The objective of this section is to experimentally
validate the proposed simulation model. For this, a
comparison between the simulated and the experimental
cutting efforts will be first investigated, using the same
cutting and tool parameters. Second, the error between
simulation and experimental results will be determined. In
order to give more credibility to this work, we chose to
Evaluation
; ; 0; 2 ;90
t
s e
p
N fd
R
t
Evaluation
( ); ( ); ( ); ( )x y x yu t u t u t u t
Evaluation ( ); ( )x yF t F t
0
0
( ) ( )
( )( )
x x
Rotyy
F t F tM
F tF t
e
d
Yes
ultime
; 2s e e e
End
Yes
s Second procedure: Data entry
Tool features; Cutting parameters; Material
characteristics
Third procedure: Resolution
( ) ( ) ( ) ( )dsolve M U t C U t K U t F t
Computational procedures
( ); ( ); ( ); ( )x y x yu t u t u t u t
First procedure: Formal Maple expression of
cutting forces
O : Workpiece center’s
0x
Workpiece
dFtj
dFrj
j
Theoretical profile
= t R
Cutting profile
ae
x
0y
cy
ky
kx
cx
Cutter Nt f=Rp
Machine Table
xf
0xf
0yf
xf f x
y
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International Journal of Mechanical Engineering and Robotics Research Vol. 7, No. 1, January 2018
© 2018 Int. J. Mech. Eng. Rob. Res.
measure the cutting forces in two different cutting zones
of the workpiece.
A. Experimental Device
The experimental device is shown in Fig. 3.
Acceleration is measured using an accelerometer (three-
axis AC115 CTC) along the three axes. The accelerometer
is positioned on the spindle and connected to an
acquisition card with a function margin of 1,600 Hz to
over 20,000 Hz.
Figure 3. The experimental device for the acquisition of cutting forces
and acceleration values.
Efforts are measured by a piezoelectric dynamometer
bridled on the machine table (platinum Kistler no. 9257A).
The dynamometer loads forces signals along the three
orthogonal axes. The dynamometer is connected to
amplifier signal converters delivering voltage signals
compatible with the used capture card (NI 4470 ± 10 V).
The card has eight synchronized inputs, and the maximum
sampling frequency is 100 kHz. This workpiece was
inspired by “Piece NASA,” which is known in the world
of machining, fixed directly on the dynamometer. This
workpiece, since its invention by NASA, has often served
as a master part to control the precision of machines (Fig.
4).
The machined material is A60 steel (or E335) in a
square section of 50 × 50 mm. Table I lists the
characteristics of this material.
The tool is a milling four-tooth tool that is
manufactured by Mitsubishi (ref. APX 3000), and it has a
20 mm diameter (Fig. 5).
Figure 4. (a) Real part from NASA; (b) machined part in CAD.
Figure 5. Platelets used for experimental tests.
TABLE I. MATERIAL CHARACTERISTICS OF THE WORKPIECE.
Material propriety Value
Steel material A60
Yield strength, Re (MPa) 335
Young’s modulus, E (MPa) 200000
Density, ρ (kg/m3) 7850
Poisson’s ratio 0.29
B. Cutting Force Measurements
Figure 6 shows the measured cutting forces along the
two main directions 0 0( , )x y during machining of the
circular milling profile of the NASA workpiece. Table II
lists the cutting parameters used for this test. From Fig. 6,
we can see that the measured cutting forces progress in
stages. This is can be attributed to the tool movement
(back and forth) during machining. The cutting effort in
the z-direction is negligible compared to the other two
components, which can confirm the choice of our model.
TABLE II. CUTTING PARAMETERS USED IN THE NUMERICAL
SIMULATION AND EXPERIMENTAL TESTS.
Cutting parameters Value
Tangential cutting force pressure, Kt (N/mm²) 2100
Radial cutting force pressure, Kr (N/mm²) 1200
Helix angle, β (°) 20
Rake angle, γ (°) 12
Tool diameter, D (mm) 20
Number of teeth, Nt 4
Cutting speed, Vc (m/min) 51
Axial depth of cut, ae (mm) 0.5
Radial depth of cut, ap (mm) 0.5
Feed rate per tooth, f (mm/tooth) 0.0325
Figure 6. Measured cutting forces circular milling (experimental).
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International Journal of Mechanical Engineering and Robotics Research Vol. 7, No. 1, January 2018
© 2018 Int. J. Mech. Eng. Rob. Res.
Figure 7. Curve cutting forces: (a) simulation, (b) experimental.
Figure 7 shows the simulated and experimental
components Fx0 and Fy0 of the cutting force for one
revolution of the tool. From this figure, we can see that the
simulated components are calculated with a good
approximation. In fact, the measured error between the
simulated and experimental cutting forces on the x-axis is
less than 1.5 N, and on the y-axis, it is lower than 1 N. In
consequence, we can consider that there is good harmony
between the experimental tests and simulation results.
C. Discussion: The Effect of Some Cutting and Tool
Parameters on the Cutting Force
1) Effect of the Tool Diameter
Three tool diameters (30, 20, and 12 mm) were
considered to study their effect on the cutting force
components (Fx0 and Fy0). The results of this simulation
(Fig. 8) show that the average values of Fx0 and Fy0
components decrease when the tool diameter increases.
This can be attributed firstly to the tool frequency rotation,
and secondly to the exit angle of cut arccos(1 )e
e
a
R .
Consequently, the involved number of elementary cutting
disks Nd is more considerable for a tool with D = 12 mm
than for a tool with D = 20 and 30 mm.
Figure 8. Effect of tool diameter on cutting force.
2) Effect of the Tool Helix Angle
In order to study the influence of the tool helix angle
on the cutting force components, three helix angle values
were considered (β = 10°, 20°, and 30°). The results of
simulation are plotted in Fig. 9. From this figure, we can
see that the cutting force increases when the tool helix
angle decreases. Consequently, the machining process of
the workpiece can be more stable when the tool helix
angle, β, is equal to 30°.
3) Effect of the Radial Depth of Cut and Feed Rate
Figure 10 shows the evolution of cutting force
components (Fx0 and Fy0) with the three considered radial
depths of cut (2, 1.5, and 1 mm). From this figure, we can
see that the average values of Fx0 and Fy0 increase when
the radial depth of cut increases.
4) Effect of Feed Rate
Three feed rates (2, 1.5, and 1 mm) were considered to
study their effect on cutting force components (Fx0 and
Fy0). The results of this simulation show that the average
values of Fx0 and Fy0 components increase when the feed
rate increases.
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International Journal of Mechanical Engineering and Robotics Research Vol. 7, No. 1, January 2018
© 2018 Int. J. Mech. Eng. Rob. Res.
Figure 9. Effect of helix angle on cutting force
Figure 10. Effect of radial depth of cut.
IV. CONCLUSION
In this paper, a two-dimensional machining model that
allows the simulation of cutting force, including the
cutting process damping in a dynamic circular milling
process, was presented.
In this model, the regenerative chatter is considered
and the total cutting force is obtained through numerical
integration of the local forces. A comparison between the
experimental and the simulated components of the cutting
force shows that there is good harmony between them.
A discussion on the effect of some cutting and tool
parameters (tool diameter, tool helix angle, radial depth of
cut, and feed rate) on cutting force components has been
made. This discussion revealed that the amplitude of the
cutting force increases when the tool diameter and/or the
tool helix angle decreases. The cutting force increases
with increasing the radial depth of cut and/or the feed rate.
Our future work will focus on the study of the stability
criteria process of circular milling. A comparison between
linear milling and the circular milling stability areas can
be helpful to understand the difference between the two
machining modes.
REFERENCES
[1] J. Tlusty, “Dynamics of high speed milling,” Transactions of the ASME, Journal of Engineering for Industry, vol. 108, pp. 59–67, May 1986.
[2] F. Ismail, M. A. Elbestawi, R. Du, and K. Urbasik, “Generation of milled surface includingtool dynamics and wear,” Transactions of the ASME: Journal of Engineering for Industry, vol. 115, August 1993, pp. 245–252.
[3] K. Mehdi, J. F. Rigal, and D. Play, “Dynamic behavior of a thin wall cylindrical Workpiece during the turning process part I: cutting process simulation,” Journal of Manufacturing Science and Engineering, vol. 124, no. 3, pp. 562–568, August 2002.
[4] K. Mehdi, J. F. Rigal, and D. Play, “Dynamic behavior of a thin wall cylindrical WorkPiece during the turning process part II: experimental approach and validation,” Journal of Manufacturing Science and Engineering, vol. 124, no. 3, pp. 569–580, August 2002.
[5] D. Montgomery and Y. Altintas, “Mechanism of cutting force and surface generation in dynamic milling,” Transactions of the ASME: Journal of Engineering for Industry, Vol. 113, pp. 160–168, May 1991.
[6] C. Y. Huang and J. J. Wang, “Mechanistic modeling of process damping in peripheral milling,” Transactions of the ASME, Journal of Manufacturing Science and Engineering, vol. 129, pp. 12–20, Feb. 2007.
[7] K. Mehdi. A. Zghal, “Modelling cutting force including thrust and tangential damping in peripheral milling process,” International Journal of Machining and Machinability of Materials, vol. 12, no. 3, pp. 236–251, 2012.
[8] X. Li, W. Zhao, L. Li, N. He, and S. W. Chi. Modeling and application of process damping in milling of thin-walled workpiece made of titanium alloy. Hindawi Publishing Corporation Shock and Vibration. vol. 2015, Article ID 431476. p. 12. [Online]. Avavilable: http://dx.doi.org/10.1155/2015/431476
Kamel Mehdi was graduated as a
Mechanical Engineer from ENIS, Tunisia, in
1989. He received his Ph.D. degree in mechanical engineering in 1995 from INSA
of Lyon, France, and his HDR diploma in
2008 from ENIS, Tunisia. His research interests are machining and manufacturing
processes, concurrent engineering, and
computer integrated design of mechanical systems. He is currently an Associate
Professor in mechanical engineering at the
Preparatory Institute for Engineering Studies El Manar (IPEIEM), University of Tunis EL Manar (UTM), Tunis, and he is a Researcher at
the Mechanical Laboratory of Solids, Structures and Technological
Development of the Engineering National High School of Tunis (ENSIT),
University of Tunis (UT), Tunisia. His research works have been
published in the Transactions of the ASME (Journal of Manufacturing
Science and Engineering), International Journal of Vehicle Design (IJVD), Journal of Machining and Forming Technology (JoMFT), Int.
Journal of Engineering Simulation (IJES), Journal of Decision Systems,
Applied Mechanics and Materials, Advanced Materials Research (JDS), and Int. J. Machining and Machinability of Material and in many
international conferences. He is a member of Editorial Review Board of
the Journal «Materials Forming and Machining Processes [IJMFMP]» and member of the scientific committees of many national and
international conferences in mechanical engineering.