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A hybrid model for analysis of 3-D machining operations
Journal of Manufacturing Systems Vol. 22/No. 4( 2003
A Hybrid Model for Analysis of 3-D Machining Operations Amir H. Adibi-Sedeh and Vis Madhavan, Dept. of Industrial and Manufacturing Engineering, Wichita State University, Wichita, Kansas, USA
Abstract
without delving into the physics of the process. These models require experimentation to find some calibration constants that help establish a good correlation between the model predictions and the measured values for different cutting parameters. Moreover, the extrapolative ability of these models is limited, as they do not consider the physics behind the process. For oblique cutting, Stabler (1951) determined experimentally that the chip flow angle is equal to the inclination angle of the cutting tool regardless of the rake angle, cutting speed, and material properties. Armarego and Brown (1969) evaluated Stabler's flow rule experimentally for a wide range of inclination and rake angles and concluded that Stabler's flow rule is a valid assumption only for small values of rake angles. Lin et al. (1982) extended Oxley's orthogonal cutting model to oblique machining along a straight cutting edge assuming Stabler's flow rule and independence of cutting and thrust forces upon inclination angle. Most of the other models proposed for the analysis of oblique cutting along a straight cutting edge (Morcos 1980; Shamoto and Altintas 1999) consider the material to be rigidplastic and ignore the effect of the variation of workpiece properties as a function of strain, strain rate, and temperature. To account for two cutting edges or cutting along a nose radius, Young, Mathew, and Oxley (1987) assumed the friction force for each element along the cutting edge to be normal to the cutting edge and proportional to the local uncut chip area. The chip flow direction was considered to be along the direction of the resultant friction force. To account for the size effect, Endres and Waldorf (1994) introduced a nonlinear dependence of specific friction and cutting forces on the chip thickness. Chandrasekharan, Kapoor, and DeVor (1995) extended this further by including the effect of normal
This paper describes the development of a physically based model for the analysis of commonly encountered 3-D machining processes using arbitrarily oriented cutting tools. This model consists of two modules, a generalized upper bound analysis module capable of handling any given cutting edge geometry, and a 2-D machining analysis module capable of using a wide range of constitutive equations to handle most commonly machined materials. The upper bound module is used for prediction of the chip flow angle and is followed by application of the extended Oxley's analysis of machining (Adibi-Sedeh, Madhavan, and Bahr 2003a) in the equivalent plane to obtain two components of the cutting force in the plane. The out-of-plane component is calculated by applying the constraint that the resultant cutting force should not have any component along the rake face in a direction perpendicular to the chip flow direction. The performance of the hybrid model is validated through extensive comparison with experimental data for different operations and materials.
Keywords: 3-D Machining, Chip Flow Angle, Cutting Forces, Drilling, Oxley's Model, Upper Bound Analysis
Introduction Knowledge of cutting forces is essential in the design of machining processes and cutting tools. Cutting forces can be related to the form and surface error, surface quality, tool wear and tool life, chatter, and machine capability. Most of the problems encountered with these issues are solved by identifying acceptable operating conditions, typically accomplished by trial and error. In many cases, the cutting conditions chosen are non-optimal from the point of view of production rate and manufacturing cost (Armarego 1998). Many models for the analysis of 3-D cutting processes are empirically based, semi-analytical models with the primary aim of predicting cutting forces This paper is an original work and has not been previously published except in the Transactions ofNAMRI/SME, Vol. 31, 2003.
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Journal of Manufacturing Systems Vol. 22/No. 4 2003
angle that results in minimum cutting force is found to be independent of the cutting velocity and vice versa. Such a weak coupling between the cutting speed and the chip flow angle has also been observed experimentally by many researchers (Stabler 1951; Russell and Brown 1966; Armarego and Brown 1969), who have proposed empirical equations that give fixed values of the chip flow angle for a given tool orientation regardless of the cutting speed. Due to this observed independence of the chip flow angle and the chip speed, the upper bound technique can be used to obtain the chip flow angle, equivalent rake angle, and so on, and can be followed by the application of the extended Oxley's analysis of machining (Adibi-Sedeh, Madhavan, and Bahr 2003a) in the equivalent plane to calculate the cutting forces. Consider the cutting tool geometry shown in Figure 1 (Arsecularatne, Mathew, and Oxley 1995), with the X axis along the velocity of the incoming workpiece material, the Y axis along the depth of cut (radial) direction, and the Z axis along the feed (axial) direction. The resultant force exerted by the workpiece on the tool is traditionally (Oxley 1989) resolved into three components, P~, P2, and P3, considered to be positive when acting in the direction of the cutting velocity (x), against the feed (-z), and radially outward (Y), respectively. P~ is the component of the resultant force that does most of the work. The shear surface can be obtained as the intersection of the cylindrical surface formed by sweeping the shear velocity vector along the length of engagement of cutting edge, and the uncut chip projection along the cutting velocity. The area of each element of the shear surface can be found from the area of
rake angle and cutting speed on the specific cutting forces. Arsecularatne, Mathew, and Oxley (1995) modeled cutting with nose radius tools as cutting along an imaginary equivalent cutting edge, for which the forces can be obtained using the approach proposed by Lin et al. (1982). Armarego and Samaranayake (1999) found that the application of Stabler's flow rule to the generalized cutting edge, the line joining the ends of the cutting area, leads to inaccurate predictions of the chip flow angle. Seethaler and Yellowley (1997) introduced an elegant approach to calculate the shear surface area from the uncut chip projection on the rake face of the tool (approximated by a combination of triangles and parallelograms) using the ratio of the shear velocity to the chip velocity. The total cutting power was minimized with respect to the chip flow angle and chip velocity. Adibi-Sedeh, Madhavan, and Bahr (2002, 2003b) extended Seethaler and Yellowley's approach to arbitrary cutting tool orientations and introduced a robust approach for calculation of the uncut chip area for all geometries of the tool-chip interface. This upper bound model was found to result in accurate predictions of the chip flow angle. However, the predictions for cutting forces were considerably lower than experimental values due to underestimation of the friction area by the orthogonal cutting model used. To improve the force prediction capability of the upper bound model, a hybrid model is introduced in this study. While preserving the capability of the upper bound model to capture the kinematics of the process accurately, the extended Oxley's orthogonal machining model (Adibi-Sedeh, Madhavan, and Bahr 2003a), which accounts for the dependence of material strength on strain, strain rate, and temperature and is applicable for a wide range of commonly machined materials, is used for cutting force predictions. The proposed hybrid model is validated through comparison of predictions for oblique cutting with a single straight cutting edge, turning with nose radius tools, and drilling with twist drills, for different cutting conditions and materials.
~~
•
Description of the Model In their studies of the upper bound model of machining, Adibi-Sedeh, Madhavan, and Bahr (2002, 2003b) found that the chip flow angle and chip speed are independent variables. In particular, the chip flow
Figure 1 Geometry of a Single-Point Cutting Tool (Adapted f r o m Arsecularatne, Mathew, and Oxley 1995)
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the corresponding element of the projection of the uncut chip area on the rake face using the ratio of the shear velocity to the chip velocity (Adibi-Sedeh, Madhavan, and Bahr 2002). Noting that the deformation of the material in the equivalent plane can be considered to be under plane strain, the length of contact along the tool-chip interface can be obtained from the length along the shear plane using the same factor as in orthogonal cutting. The friction area can be obtained by summing the elemental friction areas along the cutting edge. The total cutting power, comprised of the power required for shearing of material along the shear surface and the power consumed by friction along the tool-chip interface, is minimized with respect to the chip flow angle to obtain the predicted chip flow angle. For each lamina of work material parallel to the equivalent plane, the depth of cut, te, is expressed as t = L,, YzCsin~e e
Vc
Figure 2 Typical Variation of Length of Uncut Chip Projection o n Rake Face
(1) X
where Vs, V,, qbe, and L, are, respectively, the shear velocity, chip velocity, equivalent shear plane angle, and the length of each segment of the projection of uncut chip area on the rake face plane, parallel to the chip flow direction, as shown in Figure 2. The cutting and thrust forces for each lamina are obtained by the application of the extended Oxley's analysis (Adibi-Sedeh, Madhavan, and Bahr 2003a) for cutting with cutting speed V and rake angle % and scaling by the width of the lamina. Summing up the forces for each of the laminae, the components Pleq and P2eq of the total cutting force (see Figure 3) in the equivalent plane can be obtained. The unit vectors along which P~q and P2eq act are X and U r, where
P~q
~iiiiiit •
Figure 3
Xx(XxUc)
(P~,qX + ~,qU~) • U,c P3eq =
and U~ is the unit vector along the chip flow direction. U R, the unit vector along the out-of-plane component of the resultant cutting force, is obtained as U R = X x U r . The magnitude of the cutting force perpendicular to the equivalent plane, P3,q, is calculated assuming that the normal projection of the resultant force on the rake face should not have any component perpendicular to the chip flow direction on the rake
P 3,c.q
Cutting Force Components Acting on a Lamina Parallel t o t h e Equivalent Plane
-Xx(XxUc) Ur -
~1(~
U R *Unc
Therefore, the resultant cutting force components, P1, P 2 , and P3, can be obtained using the following transformation:
['l-1
face (UI1c). 311
Journal of Manufacturing Systems Vol. 22/No. 4
2003
~:~ .............................................................................................................................. !
~
i ~
~ere~ii~ ~.
ii
~::
i;
Figure 4 Comparison of Model Predictions for the Effect of Inclination Angle with Experimental Data for Cutting Along a Straight Cutting Edge (Pal and Koenigsberger 1968). Work material: Aluminum 6082-T6
Results The hybrid model described above is quite general and can be applied to a range of 3-D cutting processes of different materials. Model predictions have been compared with experimental results for a variety of machining operations. ire-
Oblique Cutting Along a Straight Cutting Edge
Figure 4 shows a comparison of the chip flow angle and cutting forces predicted by the model with experimental data for oblique cutting of aluminum 6082-T6 along a single cutting edge. The constants in the Johnson-Cook material model for this aluminum alloy are as follows (Jaspers 1999): Figure 5 Comparison of Predicted Cutting Force Components with Experimental Data (Watson 1984) for Different Cutting Speeds for Cutting Along a Straight Cutting Edge. Work materiah 0.58% carbon steel.
~ = ( A + B e " ) / l + C l n / ~ o / / I l l - ( I T "-T,.T-T" I '~) ( 2 )
Madhavan, and Bahr 2002). The total cutting force can be obtained by multiplying the area of the cutting cross section by the specific cutting force obtained for one lamina. As can be seen in Figures 4 a and 4b, the chip flow angle and cutting forces are predicted accurately over a wide range of inclination angles. Figure 5 shows a comparison of the predicted cutting forces with experimental data (Watson 1984) for oblique cutting of 0.58% carbon steel at different cutting speeds. The flow stress for this ma-
A = 250 MPa, B = 243.6 MPa, C = 0.00747,
n = 0 . 1 7 , m = 1.31, andT = 5 8 2 ° C //l
Note that in oblique cutting along a single cutting edge all the laminae parallel to the equivalent plane
have the same depth of cut, t~, obtained from Eq. (1) using L,, = f* cOSCs*/cos(c,* - "q~)where f* and c,* are the projections of the feed and side cutting edge angle on the rake face, respectively (Adibi-Sedeh,
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Journal of Manufacturing Systems Vol. 2 2 / N o . 4 2003
,.................................................................................................................................................. iFi
:.............................................................................................................................
! i
p,
~. 4
5
!
o
i ~":'~ ....
i
i
i
i
Figure 6 Comparison of Predicted Cutting Force Components with Experimental Data (Young, Mathew, and Oxley 1987) for Different Depths of Cut. Work material: 0.18% carbon steel.
terial is considered to be of the form proposed by Oxley (1989), • = (~l (Tmod) g nGm°d) , where the strength coefficient and the strain-hardening exponent are expressed as functions of the velocity modified temperature (Oxley 1989). It can be observed that the proposed model is able to predict the cutting forces very well. It is interesting to note that even though the magnitude of the cutting speed does not matter in the upper bound module of the hybrid model, the model is sensitive to the cutting speed. This is due to the sensitivity of the orthogonal cutting module to cutting speed.
theory for each of the laminae parallel to the equivalent plane. This makes the application of the model computationally complex. Instead, it was decided to use the average of the depth of cut in the laminae parallel to the equivalent planes along the cutting edge to calculate specific cutting forces. The average depth of cut can be obtained by first calculating the average length of the elements of the uncut chip projection area on the rake face (see Figure 2) as pO
02 - 01 Turning with Nose Radius Tools For cutting with nose radius tools, the uncut chip thickness varies along the cutting edge (Figure 2). This necessitates application of Oxley's machining
and using L in Eq. (1) to obtain te . Multiplying the specific cutting force for the average depth of cut by the area of the cutting cross section, the total cutting
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Journal of Manufacturing Systems Vol. 22/No. 4 2003
.........................................................................................
.........................
, , . . . . . . . . 3
x~
IL
m~ g~
i{
i ,2
~'? !
g
g
g,
~~.~,3., 1_,~.- . . . . . . . . . . . . . .
~ ...............
~'-'i
~grl~'~I" fzO,2Z~i'~N
I
,~,,~ I ~
Figure 7 Comparison of Predicted Cutting Force Components with Experimental Data (Arsecularatne, Mathew, and Oxley 1995) for Different (a) Inclination Angles, (b) Rake Angles, and (c) Nose Radii. Work material: AIS! 1022.
force can be obtained. Figure 6 compares the performance of the abovementioned approach based on the average depth of cut with the one in which 13 laminae along the cutting edge are considered individually and then summed up. It can be seen that the approach based on using the average value of the depth of cut in the equivalent plane along the cutting edge provides acceptable predictions of the cutting force components even for small ratios of depth of cut to nose radius (= 1) for which the variation of depth of cut along the cutting edge is the greatest. The averaging technique is used in all subsequent calculations. Figures 7 and 8 further illustrate the generality and robustness of the hybrid model through a compre-
hensive parametric study for nose radius tools. It can be clearly seen that the hybrid model predicts the magnitude and trends of the forces under all conditions accurately. Drilling with Conventional Twist Drills We have also studied drilling using conventional twist drills in which the inclination (i) and normal rake (%) angles vary along the cutting lips, as given below (Oxford 1955): i = sin -~ (W~rsin •)
/
0;,, = tan -~ 'Rsin K - w c o s l~tan [3
314
t/wco /
Journal of Manufacturing Systems Vo]. 22/No. 4 2003
The steady-state values of the thrust force and torque have been predicted using the hybrid model with 30 elements along the part of the cutting lips between the pilot hole and drill radius. The flow stress of the work material (aluminum 2024-T3) has been represented using the Johnson-Cook model [Eq. (2)], with the following constants (Johnson and Cook 1983):
l ",TX~
II
a
A = 325 MPa, B = 414 MPa, C = 0.015, n = 0.2, m = 1, and T,,, = 502 °C For the above cutting conditions the inclination angle decreases from 14.3 ° for the element at the pilot hole radius to 5.6 ° for the exterior element at the drill radius, and the rake angle is found to increase from 7.8 ° to 34.5 ° at the drill radius. It has also been observed that elemental specific thrust force obtained from the modified Oxley's model decreases with increase in radius along the cutting lip, whereas the elemental specific torque increases as the radius increases. Figure 9 shows the c o m p a r i s o n of the predicted and m e a s u r e d values of thrust force and torque. As can be seen, model predictions are within 15% of the experimental values.
Figure 8 Comparison of Predicted Cutting Force Components with Experimental Data (Young, Mathew, and Oxley 1987) for Different Side Cutting Edge Angles. Work material: 0.18% carbon steel.
In the above formulae, w is half the web thickness, R is the drill radius, K is one half of the point angle, [3 is the helix angle, and r is the local radial distance from the drill axis. The cutting lips of the drill can be divided into a number of oblique cutting elements with known inclination and rake angles. Applying the hybrid model to these elements, the local chip flow angle and the cutting forces can be obtained. The total thrust force can be obtained by summing up the elemental components of cutting forces along the drill axis. Similarly, summing up the contribution of elemental torques obtained by multiplying the elemental cutting forces along the cutting velocity direction by the corresponding radial distances from the axis of the drill, the total torque can be calculated. Due to the fact that the chisel edge of the drill extrudes material laterally rather than cutting it (Oxford 1955), the chisel edge performance is usually modeled as an indentation process using a wedge. To exclude the effect of the chisel edge, pilot holes are drilled into the workpiece so that only the cutting lips of the drill are involved in the cutting process. In this study, we have used experimental data (Durairajan 2003) obtained for drilling aluminum 2024-T3 using HSS twist drills with 15.9 m m diameter, 1.8 m m web thickness, 118 ° point angle, 33 ° helix angle, 130 ° chisel edge angle, and a pilot hole diameter of 3.04 mm.
Conclusions A hybrid model for analysis of 3-D cutting processes of different materials has been described in this paper. The model is comprised of the improved upper b o u n d m o d e l for determining the chip flow angle, and an extended Oxley's analysis of o r t h o g o n a l m a c h i n i n g for d e t e r m i n i n g forces. This model accurately predicts the chip flow angle as well as the cutting forces for a wide range of cutting conditions and materials. Moreover, the proposed m o d e l is a physically based model that minimizes the number of required empirical rules and calibration constants compared to previous models.
Acknowledgment T h e a u t h o r s w o u l d like to t h a n k Mr. Bala Durairajan and Dr. B e h n a m Bahr for providing the results of the drilling experiments used in this study.
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Arsecularatne, J.A.; Mathew, E; Oxley, EL.B. (1995). "Prediction of chip flow direction and cutting forces in oblique machining with nose radius tools." Proc. of the Institution of Mechanical Engineers, Part B (v209), pp305-315. Chandrasekaran, V.; Kapoor, S.G.; and DeVor, R.E. (1995). "A mechanistic approach to predicting the cutting forces in drilling: With application to fiber-reinforced composite materials." Journal of Engg. for h~dustl3, (v117), pp559-570. Durairajan, B. (2003). MS thesis. Wichita, KS: Wichita State Univ. Endres, W.J. and Waldorf. D.J. (1994). "The importance of considering size effect along the cutting edge in predicting the effective lead angle for turning." Trans. of NAMRI/SME (v22). Dearborn, MI: Society of Manufacturing Engineers, pp65-72. Jaspers, S.EF.C. (1999). "Metal cutting mechanics and material behaviour." PhD thesis. Technische Universiteit Eindhoven. Johnson, G.J. and Cook, W.H. (1983). "A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures." PJvc. of 7th Int'l Syrup. on Ballistics, pp541-547. Lin, G.C.I.; Mathew, P.; Oxley, P.L.B.; and Watson, A.R. (1982). "Predicting cutting forces for oblique machining conditions." Proc. of the Institution of Mechanical Engineers (v196), pp141-148. Morcos, W.A. (1980). "A slip line field solution of the free oblique continuous cutting problem in conditions of light friction at chiptool interface." Journal of Engg. for b2dustry (v102), pp310-314. Oxford, C.J., Jl: (1955). "On the drilling of metals- I. Basic mechanics of the process." Trans. of ASME (v77), pp102-114. Oxley, P.L.B. (1989). The Mechanics of Machining: An Analytical AppJvach to Assessing Machh~abili~. Chichester, England: E. Horwood. Pal, A.K. and Koenigsberger, F. (1968). "Some aspects of the oblique cutting process." bTt'l Journal of Machine Tool Design and Research (v8), pp45-57. Russell, J.K. and Brown, R.FL (1966). "The measurement of the chip flow direction." lnt'l Journal of Machine Tool Design and Research (v6), pp129-138. Seethaler, R.J. and Yellowley, I. (1997). "An upper-bound cutting model for oblique cutting tools with a nose radius?' hzt'l Journal of Machine Tools and Manufacture (v37), pp119-134. Shamoto, E. and Altintas, Y. (1999). "Prediction of shear angle in cutting with maximum shear stress and minimum energy principles:' Journal of Mfg. Science and Technology (v121), pp399-407. Stabler, G.V. (1951). "The fundamental geometry of cutting tools." Proc. of the Institution of Mechanical Engineers (v165), ppl4-121. Watson, A.R. (1984). "The theoretical prediction of torque and thrust in drilling?' PhD thesis. Univ. of New South Wales. Young, H.T.; Mathew, P.; and Oxley, P.L.B. (1987). "Allowing for nose radius effects in predicting the chip flow direction and cutting forces in bar turning," Proc. of the Institution of Mechanical Engineers, Part C (v201), pp213-226.
b
gf
7, .!
Figure 9 Comparison of Model Predictions with Experimental Data for Drilling (Durairajan 2003). Work material: Aluminum 2024-T3. Drill geometry: 15.9 m m diameter, 1.8 mm web thickness, 118 ° point angle, 33 ° helix angle, and 130 ° chisel edge angle.
References Adibi-Sedeh, A.H.; Madhavan, V.; and Bahr, B. (2002). "Upper bound analysis of oblique cutting with nose radius tools." Int'l JourT~al of Machine Tools and Manufacture (v42), pp1081-1094. Adibi-Sedeh, A.H.; Madhavan, V.; and Bahr, B. (2003a). "Extension of Oxley's analysis of machining to use different material models." ASME Journal of Mfg. Science and Engg. (v125), pp656-666. Adibi-Sedeh, A.H.; Madhavan, V.; and Bahr, B. (2003b). "Upper bound analysis of oblique cutting: Improved method of calculating the friction area?' Int'l Journal of Machine Tools and Manufacture (v43, n5), pp485-492. Armarego, E.J.A. (1998). "A generic mechanics of cutting approach to predictive technological performance modeling of the wide spectrum of machining operations?' Machining Science and Technology (v2), pp191-211. Armarego, E.J.A. and Brown, R.H. (1969). The Machining of Metals. Englewood Cliffs, NJ: Prentice Hall. Armarego, E.J.A. and Samaranayake, P. (1999). "Performance prediction models for turning with rounded corner plane faced lathe tools. I. Theoretical development." Machining Science and Technology (v3), pp143-172.
Authors' Biographies Amir H. Adibi-Sedeh received his PhD in mechanical engineering from Wichita State University in 2002. He is currently a research scientist in the Dept. of Industrial and Manufacturing Engineering at Wichita State University. His research interests are analysis of metalcutting processes, computer-aided design and manufacturing, applied solid mechanics and material modeling. Vis Madhavan obtained his PhD in industrial engineering from Purdue University in 1996. Since then he has been teaching in the Dept. of Industrial and Manufacturing Engineering at Wichita State University, where he is now an associate professor and Boeing Fellow. His research interests include the mechanics and tribology of machining and sheet metal forming, use of virtual reality in the design of manufacturing processes and lines, and use of virtual reality in engineering education.
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A Hybrid Model for Analysis of 3-D Machining Operations Amir H. Adibi-Sedeh and Vis Madhavan, Dept. of Industrial and Manufacturing Engineering, Wichita State University, Wichita, Kansas, USA
Abstract
without delving into the physics of the process. These models require experimentation to find some calibration constants that help establish a good correlation between the model predictions and the measured values for different cutting parameters. Moreover, the extrapolative ability of these models is limited, as they do not consider the physics behind the process. For oblique cutting, Stabler (1951) determined experimentally that the chip flow angle is equal to the inclination angle of the cutting tool regardless of the rake angle, cutting speed, and material properties. Armarego and Brown (1969) evaluated Stabler's flow rule experimentally for a wide range of inclination and rake angles and concluded that Stabler's flow rule is a valid assumption only for small values of rake angles. Lin et al. (1982) extended Oxley's orthogonal cutting model to oblique machining along a straight cutting edge assuming Stabler's flow rule and independence of cutting and thrust forces upon inclination angle. Most of the other models proposed for the analysis of oblique cutting along a straight cutting edge (Morcos 1980; Shamoto and Altintas 1999) consider the material to be rigidplastic and ignore the effect of the variation of workpiece properties as a function of strain, strain rate, and temperature. To account for two cutting edges or cutting along a nose radius, Young, Mathew, and Oxley (1987) assumed the friction force for each element along the cutting edge to be normal to the cutting edge and proportional to the local uncut chip area. The chip flow direction was considered to be along the direction of the resultant friction force. To account for the size effect, Endres and Waldorf (1994) introduced a nonlinear dependence of specific friction and cutting forces on the chip thickness. Chandrasekharan, Kapoor, and DeVor (1995) extended this further by including the effect of normal
This paper describes the development of a physically based model for the analysis of commonly encountered 3-D machining processes using arbitrarily oriented cutting tools. This model consists of two modules, a generalized upper bound analysis module capable of handling any given cutting edge geometry, and a 2-D machining analysis module capable of using a wide range of constitutive equations to handle most commonly machined materials. The upper bound module is used for prediction of the chip flow angle and is followed by application of the extended Oxley's analysis of machining (Adibi-Sedeh, Madhavan, and Bahr 2003a) in the equivalent plane to obtain two components of the cutting force in the plane. The out-of-plane component is calculated by applying the constraint that the resultant cutting force should not have any component along the rake face in a direction perpendicular to the chip flow direction. The performance of the hybrid model is validated through extensive comparison with experimental data for different operations and materials.
Keywords: 3-D Machining, Chip Flow Angle, Cutting Forces, Drilling, Oxley's Model, Upper Bound Analysis
Introduction Knowledge of cutting forces is essential in the design of machining processes and cutting tools. Cutting forces can be related to the form and surface error, surface quality, tool wear and tool life, chatter, and machine capability. Most of the problems encountered with these issues are solved by identifying acceptable operating conditions, typically accomplished by trial and error. In many cases, the cutting conditions chosen are non-optimal from the point of view of production rate and manufacturing cost (Armarego 1998). Many models for the analysis of 3-D cutting processes are empirically based, semi-analytical models with the primary aim of predicting cutting forces This paper is an original work and has not been previously published except in the Transactions ofNAMRI/SME, Vol. 31, 2003.
309
Journal of Manufacturing Systems Vol. 22/No. 4 2003
angle that results in minimum cutting force is found to be independent of the cutting velocity and vice versa. Such a weak coupling between the cutting speed and the chip flow angle has also been observed experimentally by many researchers (Stabler 1951; Russell and Brown 1966; Armarego and Brown 1969), who have proposed empirical equations that give fixed values of the chip flow angle for a given tool orientation regardless of the cutting speed. Due to this observed independence of the chip flow angle and the chip speed, the upper bound technique can be used to obtain the chip flow angle, equivalent rake angle, and so on, and can be followed by the application of the extended Oxley's analysis of machining (Adibi-Sedeh, Madhavan, and Bahr 2003a) in the equivalent plane to calculate the cutting forces. Consider the cutting tool geometry shown in Figure 1 (Arsecularatne, Mathew, and Oxley 1995), with the X axis along the velocity of the incoming workpiece material, the Y axis along the depth of cut (radial) direction, and the Z axis along the feed (axial) direction. The resultant force exerted by the workpiece on the tool is traditionally (Oxley 1989) resolved into three components, P~, P2, and P3, considered to be positive when acting in the direction of the cutting velocity (x), against the feed (-z), and radially outward (Y), respectively. P~ is the component of the resultant force that does most of the work. The shear surface can be obtained as the intersection of the cylindrical surface formed by sweeping the shear velocity vector along the length of engagement of cutting edge, and the uncut chip projection along the cutting velocity. The area of each element of the shear surface can be found from the area of
rake angle and cutting speed on the specific cutting forces. Arsecularatne, Mathew, and Oxley (1995) modeled cutting with nose radius tools as cutting along an imaginary equivalent cutting edge, for which the forces can be obtained using the approach proposed by Lin et al. (1982). Armarego and Samaranayake (1999) found that the application of Stabler's flow rule to the generalized cutting edge, the line joining the ends of the cutting area, leads to inaccurate predictions of the chip flow angle. Seethaler and Yellowley (1997) introduced an elegant approach to calculate the shear surface area from the uncut chip projection on the rake face of the tool (approximated by a combination of triangles and parallelograms) using the ratio of the shear velocity to the chip velocity. The total cutting power was minimized with respect to the chip flow angle and chip velocity. Adibi-Sedeh, Madhavan, and Bahr (2002, 2003b) extended Seethaler and Yellowley's approach to arbitrary cutting tool orientations and introduced a robust approach for calculation of the uncut chip area for all geometries of the tool-chip interface. This upper bound model was found to result in accurate predictions of the chip flow angle. However, the predictions for cutting forces were considerably lower than experimental values due to underestimation of the friction area by the orthogonal cutting model used. To improve the force prediction capability of the upper bound model, a hybrid model is introduced in this study. While preserving the capability of the upper bound model to capture the kinematics of the process accurately, the extended Oxley's orthogonal machining model (Adibi-Sedeh, Madhavan, and Bahr 2003a), which accounts for the dependence of material strength on strain, strain rate, and temperature and is applicable for a wide range of commonly machined materials, is used for cutting force predictions. The proposed hybrid model is validated through comparison of predictions for oblique cutting with a single straight cutting edge, turning with nose radius tools, and drilling with twist drills, for different cutting conditions and materials.
~~
•
Description of the Model In their studies of the upper bound model of machining, Adibi-Sedeh, Madhavan, and Bahr (2002, 2003b) found that the chip flow angle and chip speed are independent variables. In particular, the chip flow
Figure 1 Geometry of a Single-Point Cutting Tool (Adapted f r o m Arsecularatne, Mathew, and Oxley 1995)
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Journal of Manufacturing Systems Vol. 22/No. 4 2003
the corresponding element of the projection of the uncut chip area on the rake face using the ratio of the shear velocity to the chip velocity (Adibi-Sedeh, Madhavan, and Bahr 2002). Noting that the deformation of the material in the equivalent plane can be considered to be under plane strain, the length of contact along the tool-chip interface can be obtained from the length along the shear plane using the same factor as in orthogonal cutting. The friction area can be obtained by summing the elemental friction areas along the cutting edge. The total cutting power, comprised of the power required for shearing of material along the shear surface and the power consumed by friction along the tool-chip interface, is minimized with respect to the chip flow angle to obtain the predicted chip flow angle. For each lamina of work material parallel to the equivalent plane, the depth of cut, te, is expressed as t = L,, YzCsin~e e
Vc
Figure 2 Typical Variation of Length of Uncut Chip Projection o n Rake Face
(1) X
where Vs, V,, qbe, and L, are, respectively, the shear velocity, chip velocity, equivalent shear plane angle, and the length of each segment of the projection of uncut chip area on the rake face plane, parallel to the chip flow direction, as shown in Figure 2. The cutting and thrust forces for each lamina are obtained by the application of the extended Oxley's analysis (Adibi-Sedeh, Madhavan, and Bahr 2003a) for cutting with cutting speed V and rake angle % and scaling by the width of the lamina. Summing up the forces for each of the laminae, the components Pleq and P2eq of the total cutting force (see Figure 3) in the equivalent plane can be obtained. The unit vectors along which P~q and P2eq act are X and U r, where
P~q
~iiiiiit •
Figure 3
Xx(XxUc)
(P~,qX + ~,qU~) • U,c P3eq =
and U~ is the unit vector along the chip flow direction. U R, the unit vector along the out-of-plane component of the resultant cutting force, is obtained as U R = X x U r . The magnitude of the cutting force perpendicular to the equivalent plane, P3,q, is calculated assuming that the normal projection of the resultant force on the rake face should not have any component perpendicular to the chip flow direction on the rake
P 3,c.q
Cutting Force Components Acting on a Lamina Parallel t o t h e Equivalent Plane
-Xx(XxUc) Ur -
~1(~
U R *Unc
Therefore, the resultant cutting force components, P1, P 2 , and P3, can be obtained using the following transformation:
['l-1
face (UI1c). 311
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2003
~:~ .............................................................................................................................. !
~
i ~
~ere~ii~ ~.
ii
~::
i;
Figure 4 Comparison of Model Predictions for the Effect of Inclination Angle with Experimental Data for Cutting Along a Straight Cutting Edge (Pal and Koenigsberger 1968). Work material: Aluminum 6082-T6
Results The hybrid model described above is quite general and can be applied to a range of 3-D cutting processes of different materials. Model predictions have been compared with experimental results for a variety of machining operations. ire-
Oblique Cutting Along a Straight Cutting Edge
Figure 4 shows a comparison of the chip flow angle and cutting forces predicted by the model with experimental data for oblique cutting of aluminum 6082-T6 along a single cutting edge. The constants in the Johnson-Cook material model for this aluminum alloy are as follows (Jaspers 1999): Figure 5 Comparison of Predicted Cutting Force Components with Experimental Data (Watson 1984) for Different Cutting Speeds for Cutting Along a Straight Cutting Edge. Work materiah 0.58% carbon steel.
~ = ( A + B e " ) / l + C l n / ~ o / / I l l - ( I T "-T,.T-T" I '~) ( 2 )
Madhavan, and Bahr 2002). The total cutting force can be obtained by multiplying the area of the cutting cross section by the specific cutting force obtained for one lamina. As can be seen in Figures 4 a and 4b, the chip flow angle and cutting forces are predicted accurately over a wide range of inclination angles. Figure 5 shows a comparison of the predicted cutting forces with experimental data (Watson 1984) for oblique cutting of 0.58% carbon steel at different cutting speeds. The flow stress for this ma-
A = 250 MPa, B = 243.6 MPa, C = 0.00747,
n = 0 . 1 7 , m = 1.31, andT = 5 8 2 ° C //l
Note that in oblique cutting along a single cutting edge all the laminae parallel to the equivalent plane
have the same depth of cut, t~, obtained from Eq. (1) using L,, = f* cOSCs*/cos(c,* - "q~)where f* and c,* are the projections of the feed and side cutting edge angle on the rake face, respectively (Adibi-Sedeh,
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,.................................................................................................................................................. iFi
:.............................................................................................................................
! i
p,
~. 4
5
!
o
i ~":'~ ....
i
i
i
i
Figure 6 Comparison of Predicted Cutting Force Components with Experimental Data (Young, Mathew, and Oxley 1987) for Different Depths of Cut. Work material: 0.18% carbon steel.
terial is considered to be of the form proposed by Oxley (1989), • = (~l (Tmod) g nGm°d) , where the strength coefficient and the strain-hardening exponent are expressed as functions of the velocity modified temperature (Oxley 1989). It can be observed that the proposed model is able to predict the cutting forces very well. It is interesting to note that even though the magnitude of the cutting speed does not matter in the upper bound module of the hybrid model, the model is sensitive to the cutting speed. This is due to the sensitivity of the orthogonal cutting module to cutting speed.
theory for each of the laminae parallel to the equivalent plane. This makes the application of the model computationally complex. Instead, it was decided to use the average of the depth of cut in the laminae parallel to the equivalent planes along the cutting edge to calculate specific cutting forces. The average depth of cut can be obtained by first calculating the average length of the elements of the uncut chip projection area on the rake face (see Figure 2) as pO
02 - 01 Turning with Nose Radius Tools For cutting with nose radius tools, the uncut chip thickness varies along the cutting edge (Figure 2). This necessitates application of Oxley's machining
and using L in Eq. (1) to obtain te . Multiplying the specific cutting force for the average depth of cut by the area of the cutting cross section, the total cutting
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.........................................................................................
.........................
, , . . . . . . . . 3
x~
IL
m~ g~
i{
i ,2
~'? !
g
g
g,
~~.~,3., 1_,~.- . . . . . . . . . . . . . .
~ ...............
~'-'i
~grl~'~I" fzO,2Z~i'~N
I
,~,,~ I ~
Figure 7 Comparison of Predicted Cutting Force Components with Experimental Data (Arsecularatne, Mathew, and Oxley 1995) for Different (a) Inclination Angles, (b) Rake Angles, and (c) Nose Radii. Work material: AIS! 1022.
force can be obtained. Figure 6 compares the performance of the abovementioned approach based on the average depth of cut with the one in which 13 laminae along the cutting edge are considered individually and then summed up. It can be seen that the approach based on using the average value of the depth of cut in the equivalent plane along the cutting edge provides acceptable predictions of the cutting force components even for small ratios of depth of cut to nose radius (= 1) for which the variation of depth of cut along the cutting edge is the greatest. The averaging technique is used in all subsequent calculations. Figures 7 and 8 further illustrate the generality and robustness of the hybrid model through a compre-
hensive parametric study for nose radius tools. It can be clearly seen that the hybrid model predicts the magnitude and trends of the forces under all conditions accurately. Drilling with Conventional Twist Drills We have also studied drilling using conventional twist drills in which the inclination (i) and normal rake (%) angles vary along the cutting lips, as given below (Oxford 1955): i = sin -~ (W~rsin •)
/
0;,, = tan -~ 'Rsin K - w c o s l~tan [3
314
t/wco /
Journal of Manufacturing Systems Vo]. 22/No. 4 2003
The steady-state values of the thrust force and torque have been predicted using the hybrid model with 30 elements along the part of the cutting lips between the pilot hole and drill radius. The flow stress of the work material (aluminum 2024-T3) has been represented using the Johnson-Cook model [Eq. (2)], with the following constants (Johnson and Cook 1983):
l ",TX~
II
a
A = 325 MPa, B = 414 MPa, C = 0.015, n = 0.2, m = 1, and T,,, = 502 °C For the above cutting conditions the inclination angle decreases from 14.3 ° for the element at the pilot hole radius to 5.6 ° for the exterior element at the drill radius, and the rake angle is found to increase from 7.8 ° to 34.5 ° at the drill radius. It has also been observed that elemental specific thrust force obtained from the modified Oxley's model decreases with increase in radius along the cutting lip, whereas the elemental specific torque increases as the radius increases. Figure 9 shows the c o m p a r i s o n of the predicted and m e a s u r e d values of thrust force and torque. As can be seen, model predictions are within 15% of the experimental values.
Figure 8 Comparison of Predicted Cutting Force Components with Experimental Data (Young, Mathew, and Oxley 1987) for Different Side Cutting Edge Angles. Work material: 0.18% carbon steel.
In the above formulae, w is half the web thickness, R is the drill radius, K is one half of the point angle, [3 is the helix angle, and r is the local radial distance from the drill axis. The cutting lips of the drill can be divided into a number of oblique cutting elements with known inclination and rake angles. Applying the hybrid model to these elements, the local chip flow angle and the cutting forces can be obtained. The total thrust force can be obtained by summing up the elemental components of cutting forces along the drill axis. Similarly, summing up the contribution of elemental torques obtained by multiplying the elemental cutting forces along the cutting velocity direction by the corresponding radial distances from the axis of the drill, the total torque can be calculated. Due to the fact that the chisel edge of the drill extrudes material laterally rather than cutting it (Oxford 1955), the chisel edge performance is usually modeled as an indentation process using a wedge. To exclude the effect of the chisel edge, pilot holes are drilled into the workpiece so that only the cutting lips of the drill are involved in the cutting process. In this study, we have used experimental data (Durairajan 2003) obtained for drilling aluminum 2024-T3 using HSS twist drills with 15.9 m m diameter, 1.8 m m web thickness, 118 ° point angle, 33 ° helix angle, 130 ° chisel edge angle, and a pilot hole diameter of 3.04 mm.
Conclusions A hybrid model for analysis of 3-D cutting processes of different materials has been described in this paper. The model is comprised of the improved upper b o u n d m o d e l for determining the chip flow angle, and an extended Oxley's analysis of o r t h o g o n a l m a c h i n i n g for d e t e r m i n i n g forces. This model accurately predicts the chip flow angle as well as the cutting forces for a wide range of cutting conditions and materials. Moreover, the proposed m o d e l is a physically based model that minimizes the number of required empirical rules and calibration constants compared to previous models.
Acknowledgment T h e a u t h o r s w o u l d like to t h a n k Mr. Bala Durairajan and Dr. B e h n a m Bahr for providing the results of the drilling experiments used in this study.
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Arsecularatne, J.A.; Mathew, E; Oxley, EL.B. (1995). "Prediction of chip flow direction and cutting forces in oblique machining with nose radius tools." Proc. of the Institution of Mechanical Engineers, Part B (v209), pp305-315. Chandrasekaran, V.; Kapoor, S.G.; and DeVor, R.E. (1995). "A mechanistic approach to predicting the cutting forces in drilling: With application to fiber-reinforced composite materials." Journal of Engg. for h~dustl3, (v117), pp559-570. Durairajan, B. (2003). MS thesis. Wichita, KS: Wichita State Univ. Endres, W.J. and Waldorf. D.J. (1994). "The importance of considering size effect along the cutting edge in predicting the effective lead angle for turning." Trans. of NAMRI/SME (v22). Dearborn, MI: Society of Manufacturing Engineers, pp65-72. Jaspers, S.EF.C. (1999). "Metal cutting mechanics and material behaviour." PhD thesis. Technische Universiteit Eindhoven. Johnson, G.J. and Cook, W.H. (1983). "A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures." PJvc. of 7th Int'l Syrup. on Ballistics, pp541-547. Lin, G.C.I.; Mathew, P.; Oxley, P.L.B.; and Watson, A.R. (1982). "Predicting cutting forces for oblique machining conditions." Proc. of the Institution of Mechanical Engineers (v196), pp141-148. Morcos, W.A. (1980). "A slip line field solution of the free oblique continuous cutting problem in conditions of light friction at chiptool interface." Journal of Engg. for b2dustry (v102), pp310-314. Oxford, C.J., Jl: (1955). "On the drilling of metals- I. Basic mechanics of the process." Trans. of ASME (v77), pp102-114. Oxley, P.L.B. (1989). The Mechanics of Machining: An Analytical AppJvach to Assessing Machh~abili~. Chichester, England: E. Horwood. Pal, A.K. and Koenigsberger, F. (1968). "Some aspects of the oblique cutting process." bTt'l Journal of Machine Tool Design and Research (v8), pp45-57. Russell, J.K. and Brown, R.FL (1966). "The measurement of the chip flow direction." lnt'l Journal of Machine Tool Design and Research (v6), pp129-138. Seethaler, R.J. and Yellowley, I. (1997). "An upper-bound cutting model for oblique cutting tools with a nose radius?' hzt'l Journal of Machine Tools and Manufacture (v37), pp119-134. Shamoto, E. and Altintas, Y. (1999). "Prediction of shear angle in cutting with maximum shear stress and minimum energy principles:' Journal of Mfg. Science and Technology (v121), pp399-407. Stabler, G.V. (1951). "The fundamental geometry of cutting tools." Proc. of the Institution of Mechanical Engineers (v165), ppl4-121. Watson, A.R. (1984). "The theoretical prediction of torque and thrust in drilling?' PhD thesis. Univ. of New South Wales. Young, H.T.; Mathew, P.; and Oxley, P.L.B. (1987). "Allowing for nose radius effects in predicting the chip flow direction and cutting forces in bar turning," Proc. of the Institution of Mechanical Engineers, Part C (v201), pp213-226.
b
gf
7, .!
Figure 9 Comparison of Model Predictions with Experimental Data for Drilling (Durairajan 2003). Work material: Aluminum 2024-T3. Drill geometry: 15.9 m m diameter, 1.8 mm web thickness, 118 ° point angle, 33 ° helix angle, and 130 ° chisel edge angle.
References Adibi-Sedeh, A.H.; Madhavan, V.; and Bahr, B. (2002). "Upper bound analysis of oblique cutting with nose radius tools." Int'l JourT~al of Machine Tools and Manufacture (v42), pp1081-1094. Adibi-Sedeh, A.H.; Madhavan, V.; and Bahr, B. (2003a). "Extension of Oxley's analysis of machining to use different material models." ASME Journal of Mfg. Science and Engg. (v125), pp656-666. Adibi-Sedeh, A.H.; Madhavan, V.; and Bahr, B. (2003b). "Upper bound analysis of oblique cutting: Improved method of calculating the friction area?' Int'l Journal of Machine Tools and Manufacture (v43, n5), pp485-492. Armarego, E.J.A. (1998). "A generic mechanics of cutting approach to predictive technological performance modeling of the wide spectrum of machining operations?' Machining Science and Technology (v2), pp191-211. Armarego, E.J.A. and Brown, R.H. (1969). The Machining of Metals. Englewood Cliffs, NJ: Prentice Hall. Armarego, E.J.A. and Samaranayake, P. (1999). "Performance prediction models for turning with rounded corner plane faced lathe tools. I. Theoretical development." Machining Science and Technology (v3), pp143-172.
Authors' Biographies Amir H. Adibi-Sedeh received his PhD in mechanical engineering from Wichita State University in 2002. He is currently a research scientist in the Dept. of Industrial and Manufacturing Engineering at Wichita State University. His research interests are analysis of metalcutting processes, computer-aided design and manufacturing, applied solid mechanics and material modeling. Vis Madhavan obtained his PhD in industrial engineering from Purdue University in 1996. Since then he has been teaching in the Dept. of Industrial and Manufacturing Engineering at Wichita State University, where he is now an associate professor and Boeing Fellow. His research interests include the mechanics and tribology of machining and sheet metal forming, use of virtual reality in the design of manufacturing processes and lines, and use of virtual reality in engineering education.
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