![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.](https://image.slidesharecdn.com/cuttingforcebasedonmodellingexperimentsduringbroachingoperation-150531150605-lva1-app6892/85/Cutting-force-based-on-modelling-experiments-during-broaching-operation-1-320.jpg)
![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:](https://image.slidesharecdn.com/cuttingforcebasedonmodellingexperimentsduringbroachingoperation-150531150605-lva1-app6892/85/Cutting-force-based-on-modelling-experiments-during-broaching-operation-2-320.jpg)
Cutting force based on modelling experiments during broaching operation
- 1. 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: [email protected] 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.
- 2. 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:
- 3. Maejo Int. J. Sci. Technol. 2009, 3(02), 343-351 345 + 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 .
- 4. Maejo Int. J. Sci. Technol. 2009, 3(02), 343-351 346 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
- 5. Maejo Int. J. Sci. Technol. 2009, 3(02), 343-351 347 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
- 6. Maejo Int. J. Sci. Technol. 2009, 3(02), 343-351 348 -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
- 7. Maejo Int. J. Sci. Technol. 2009, 3(02), 343-351 349 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
- 8. Maejo Int. J. Sci. Technol. 2009, 3(02), 343-351 350 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
- 9. Maejo Int. J. Sci. Technol. 2009, 3(02), 343-351 351 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 1. M. E. Merchant, “Mechanics of metal cutting process-_ orthogonal cutting and a Type-2 chip”, J. Appl. Phys., 1945, 16, 267-275. 2. J. W. Sutherland, E. J. Salisbury and F. W. Hoge, “A model for the cutting force system in the gear broaching process”, Int. J. Mach. Tools Manuf., 1997, 37, 1409-1421. 3. V. S. Belov and S. M. Ivanov, “Factors affecting broaching condition and broach life”, J. Stanki Instrum., 1974, 45, 31-33. 4. D. A. Axinte and N. Gindy, “Tool condition monitoring in broaching”, J. Wear, 2003, 254, 370- 382. 5. E. Kuljanic, “Cutting force and surface roughness in broaching”, Annual Rep. CIRP, 1975, 24, 77- 82. 6. E. H. Lee and B.W. Shaffer, “The theory of plasticity applied to a problem of machining”, J. Appl. Mech., 1951, 8, 405-413. 7. M. E. Merchant, “Basic mechanics of the metal cutting process”, J. Appl. Mech., 1954, 168, 175- 178. 8. S. P. Mo, D. Axinte, T. H. Hyde and N. N. Z Gindy, “An example of selection of the cutting conditions in broaching of heat-resistant alloys based on cutting forces, surface roughness and tool wear”, J. Mater. Process. Technol., 2005, 160, 382-389. 9. K. Sakuma and H. Kiyota, “Hole accuracy with carbide tipped reamers”, Bull. Japan Soc. Precis. Eng., 1986, 19, 89-95. 10. E. Usui, A. Hiota and M. Masuko, “Analytical predictions of three dimensional cutting process – part I: Basic cutting model an energy approach”, J. Eng. Ind. ASME Trans., 1978, 100, 222-228. 11. H. S. Kim and K. F. Ehmann, “A cutting force model for face milling operations”, Int. J. Mach. Tools Manuf., 1993, 33, 651-673. 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.