Determination of proportionality constants from cutting force

Maejo Int. J. Sci. Technol. 2009, 3(02), 343-351
Maejo International
Journal of Science and Technology
ISSN 1905-7873
Available online at www.mijst.mju.ac.th
Full Paper
Determination of proportionality constants from cutting force
modelling experiments during broaching operation
Raghavendra C. Kamath 1,*
and Appu K. K. Kuttan 2
1
Department of Mechanical Engineering, Manipal Institute of Technology (MIT), Manipal, India
2
Department of Mechanical Engineering, National Institute of Technology Karnataka (NITK),
Surathkal, India
*
Corresponding author, e-mail: mitian99@rediffmail.com
Received: 14 March 2009 / Accepted: 13 August 2009 / Published: 17 August 2009
Abstract: Mechanistic model assumes that the types of cutting force acting on the broach
teeth, namely axial force, normal force and lateral force, are proportional to chip-thickness
area. In this paper, the proportionality constants related to the cutting force and chip-
thickness area were obtained through experimentation. The shaping process was used to
determine the proportionality constants in terms of specific cutting energy constants. The
paper also includes static force modelling for broaching operation and graphical presentation
of the experimental and simulated results.
Keywords: metal cutting, cutting force, broaching operation
Introduction
Metal cutting is one of the basic operations in manufacturing industries to produce the parts of
desired dimensions and shape. Metal cutting constitutes a complex process involving the diversity of
physical phenomena such as large plastic deformation, frictional contact, thermo-chemical coupling, and
chip and burr formation mechanisms. A great deal of research [1-9] has been devoted to understanding
the mechanism of machining with the objective of obtaining more effective tools and manufacturing
operations.
Over the last several years, research has been carried out to develop mechanistic force models
for a variety of machining processes, including end milling, face milling, boring, turning and broaching.
These models have been employed in a number of designs, operation plannings and process control
settings to predict both the cutting force and the resulting machine surface error.

Maejo Int. J. Sci. Technol. 2009, 3(02), 343-351 344
In the literature, two different approaches have been adopted for the prediction of the cutting
force system. The first method is based on the work done by Merchant [1] and involves a study of the
cutting mechanics and the prediction of the shear angle in metal cutting. Both analytical and empirical
models for shear angle prediction have been attempted. Lee and Shaffer [6] applied slip line theory to
machining to develop the equation for predicting the shear angle. Usui et al. [10] developed a model
that was based on the minimum energy criterion for predicting the chip flow angle and empirical models
were used for predicting both the friction angle and the shear angle. This approach generally requires
experimentation of a more fundamental nature of cutting mechanics to achieve the measurement and
prediction of the shear angle.
In this paper, an attempt has been made to determine the specific energy constants
experimentally for mild steel, aluminium and cast iron. Shaping process was used for the experimental
purpose of determining specific energy constants and proportionality constants which are used in
mechanistic modelling during broaching operation. A mechanistic model has been developed to compute
static forces. The experimental results and simulated results by mechanistic modelling are presented
graphically.
Methods
Computation of specific cutting energy constant for predicting the cutting forces__
mechanistic
modelling approach
In the mechanistic modelling approach, for any machining process the basic equations that relate
the axial force Fa, normal force Fn and lateral force Ft to the chip-thickness area are given by [10]:
Fa = Ka Ac
Fn = Kn Ac (1)
Ft = Kt Ac
where Fa, Fn and Ft are the three-dimensional forces acting on the tool tip. Ac is the chip-thickness area
and Ka, Kn and Kt are the proportionality constants corresponding to three-directional cutting forces.
The proportionality constants depend on the chip thickness tc, cutting velocity vc and rake angle
a of the cutting tool. Mathematically [10],
0 31 2 4
0 31 2 4
0 31 2 4
logt logv logt logv logγa + aa + a + a+ c c c c aΚ = ea
logt logv logt logv logγb + b+ b + b + bc c c c aΚ = en
logt logv logt logv logγc + c+ c + c + cc c c c aΚ = et
(2)
The coefficients ai , bi and ci (i = 1, 2, 3, 4) are called the specific cutting energy constants.
These constants depend upon the tool, work piece material, range of cutting speed and chip thickness.
They are independent of the machining process. These constants are determined from calibration test for
a given tool and work piece combination and a given range of cutting conditions.
Keeping the rake angle and the velocity of the tool movement constant corresponding to the
broaching tool, the equations (2) reduce to:

+ logt
+ logt
+ logt
a a
Κ = ea
b b
Κ = en
c c
Κ = et
0 1
0 1
0 1
c
c
c
(3)
The values of specific cutting energy constants can be determined using a simple calibration
experiment. The experiment was conducted on a shaping machine and the specific cutting energy
constants were determined.
Experimental investigation
Several sets of shaping operation experiments were conducted in order to determine the specific
cutting energy constants. Three sets of experiments were conducted on each set of processes. The first
set of experiments was used to ascertain which tool and cutter geometry variables affected the
proportionality constants. The second set of experiments was used to develop an adequate model for
proportionality constants based on important tool and cutter geometry variables determined from the
first set of experiments. The third set of experiments was used to determine the specific cutting energy
constants. Experiments were repeated for aluminum, cast iron and mild steel at different depths of cut.
Chip-thickness area for calibration purpose was measured from the chip curl. The volume of the chip
was measured using water displacement method. After that the chip curl was heated and elongated.
The chip width (b) and length ( ) were measured using a micrometer and vernier calliper respectively.
Then the actual chip thickness (tc) was obtained by dividing the volume by the product of length and
width, i.e. tc = v / b , where v is the volume of chip curl. Knowing the chip thickness and width, the
chip-thickness area could be computed as the product of tc and b.
Static model
Figure 1 illustrates the static force measurement set-up during the broaching operation. The
cutting forces in three directions were measured using an accelerometre and the root mean square
(RMS) value was taken as the static cutting force. A static analysis calculated the effects of steady
loading conditions on a structure while ignoring inertia and damping effects such as those caused by
time-varying loads. A static analysis can, however, include steady inertia loads (such as gravity and
rotational velocity) and time-varying loads that can be approximated as static equivalent loads.
Chip-thickness area measurement
A tool-maker’s microscope was used to obtain the coordinates of the tooth profile of a
broaching tool. AutoCAD drawing of the broaching tool was drawn to procure the chip load area and
the dimensions of the tooth profile. Figure 2 shows the coordinates of a single tooth of the broach
obtained through the tool-maker’s microscope and AutoCAD drawing. Figure 3 shows three planar
angles of the cutting edge of a broach tooth where R is the rake angle and L is the release angle.
Figure 4 shows the coordinates of the full broach, which gives pitch = 6.78 mm, rake angle = 300
, rise
per tooth = 0.05 mm and width = 8 mm; hence, chip-thickness area = width x rise/tooth = 0.4 mm2
.

Figure 1. Experimental set-up to obtain the force pattern during the broaching process
Figure 2. Broach tooth profile as obtained by tool maker’s microscope
a) Normal b) Rotated c) Rotated d) Rotated
position. about z axis. about y axis about x axis
Figure 3. Planar angles of the cutting edge of broach tooth
(3.26, 5.20)
(4.97, 6.52)
(6.78, 8.52)
(8.02, 8.56)
(7.52, 6.00)
(8.39, 505)
Broach
Work material
Accelerometre
Fixture

Figure.4. AutoCAD drawing of a broach tool
Results
Figures 5, 6 and 7 illustrate the cutting force versus chip thickness during shaping operation for
mild steel, aluminium and cast iron respectively. In the plot, the x-axis is the logarithm of chip thickness
(marked as d) and y-axis is the logarithm of the cutting forces in Newton (marked as Kx, Ky and Kz)
along x, y and z directions respectively. Chip thickness corresponds to depth of cut which varied from
0.04 mm to 0.16 mm. The calibration test was performed for a small depth of cut to avoid error due to
impulsive cutting force coming on the work piece at greater depth of cut. A linear curve fitting was
made using Matlab software to determine the specific energy constants. Negative-slope linear curves
were obtained and the coefficients of the linear curve fitting gave the specific energy constants. The
proportionality constants Ka, Kn and Kt were determined using equation (3). Tables 1 and 2 give the
values of specific cutting energy constants and proportionality constants respectively.
Figure 5. Specific cutting energy constants for mild steel work material during shaping operation

-3.4 -3.2 -3 -2.8 -2.6 -2.4 -2.2 -2 -1.8
-1
0
1
2
3
4
5
6
y = - 1. 2*x + 2. 1
log(d)
log(Kz)
Wo rk : Alu min ium
c) Plot for determining c0 and c1
Figure 6. Specific cutting energy constants for aluminium work material during shaping operation

Figure 7. Specific cutting energy constants for cast iron work material during shaping operation
Table 1. Specific cutting energy constants obtained during shaping operation
Table 2. Proportionality constants determined from specific cutting energy constants
Material ao a1 bo b1 c0 c1
Mild steel 7.3 -0.46 7.2 -0.037 3.6 -0.71
Aluminium 7.4 -0.39 5.0 -0.68 2.1 -1.2
Cast iron 7.3 -0.29 5.9 -0.31 2.3 -1.0
Material Ka
(N/mm2
)
Kn
(N/mm2
)
Kt
(N/mm2
)
Mild steel 5732.3 1654.0 307.1
Aluminium 5370.0 1088.0 279.4
Cast iron 3562.0 924.4 199.5

0 20 40 60 80 100 120
0
2000
4000
6000
8000
10000
12000
Length (m m )
AxialForce(N)
Work : Mild steel
Depth : 40 mm
0 20 40 60 80 100 120
0
500
1000
1500
2000
2500
3000
3500
Length (mm)
NormalForce(N)
Work : Mild steel
Depth : 40 mm
0 20 40 60 80 100 120
0
100
200
300
400
500
600
700
Length (mm)
LateralForce(N)
Work : Mild steel
Depth : 4 0 mm
Static forces during broaching operation
A MatLab program was written to simulate the static forces during broaching for different
materials at different depths. The simulation results were plotted as shown in Figure 8 for mild steel
work material at 40-mm depth. The cutting force progressively increased to a steady-state force when
all the teeth were engaged in the work piece.
a) Axial force acting on the broach
b) Normal force acting on the broach
c) Lateral force acting on the broach
Figure 8. Comparison of measured and simulated static forces for mild-steel work material at 40-
mm depth
Simulated
Measured
Simulated
Measured
Simulated
Measured

Conclusions
This paper presents the determination of proportionality constants, which are a determining
factor in cutting force modelling. These proportionality constants were determined experimentally using
shaping as a fundamental calibration process. The experiment was repeated for different materials, i.e.
mild steel, aluminium and cast iron, and results are presented graphically. After determining the
proportionality constants, mechanistic modelling for static force computation was carried out for the
broaching operation. The comparison of the experimental and simulated results for static forces are
presented graphically and are closely in agreement.
Acknowledgements
The authors thank the Head of the Mechanical Department of NITK, Surathkal and the Director
of MIT, Manipal for providing the facilities to carry out the experiments.
References
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orthogonal cutting and a Type-2 chip”, J.
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12. R. R. Craig, “Structural Dynamics__
An Introduction to Computer Methods”, Wiley, New York,
1981.
© 2009 by Maejo University, San Sai, Chiang Mai, 50290 Thailand. Reproduction is permitted for
noncommercial purposes.

Determination of proportionality constants from cutting force

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