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Prediction of contour accuracy in the end milling of pockets
Journal of Materials Processing Technology 113 (2001) 399±405
Prediction of contour accuracy in the end milling of pockets Kris M.Y. Law, A. Geddam* Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong
Abstract Many die and mold parts surface contours are produced by end milling operations. The contour accuracy of the milled pockets is strongly in¯uenced by the cutting tool de¯ection caused by the cutting forces. For milling a complex shape or a simple rectangular pocket, the accuracy of contours in corner cutting is mainly in¯uenced by the de¯ection of the end mill caused by the variation of cutting forces. This is believed to be due to variable radial depth of cut in corner cutting between the straight part and the corners resulting in variable contour accuracy. The way to improve contour accuracy in corner cutting is by decreasing the radial depth of cut to reduce the cutting forces and, consequently, the end mill de¯ection errors. During the cutting process the end mill cutter is subjected to radial and tangential de¯ections at different segments of the tool path. In order to compensate the tool path for improving the contour accuracy, it is necessary to predict the instantaneous radial depth of cut. A simple tool de¯ection model for predicting the de¯ection errors in straight segments and corners for rough milling with slot cutting as well as for ®nish milling with immersion cutting is presented with illustration of case studies. # 2001 Elsevier Science B.V. All rights reserved. Keywords: Cutting forces; End milling; Contour accuracy; Tool de¯ection errors
1. Introduction End milling is the most commonly used machining process for the manufacture of dies and moulds, as well as a variety of high precision aerospace components. The contour accuracy of the milled pro®les is dependent on the milling cutter de¯ection errors. Hence, the prediction of machining errors due to tool de¯ection with the cutting force modeling in the end milling process has been the subject of many investigations [1±5]. In the end milling process, the machining operations are carried out with slender helical cutters, and the cutting forces vary periodically. The slender helical end mill experiences both static and dynamic deformations causing contour and surface errors on the machined part. The instantaneous tool de¯ection is dependent on the static stiffness of the cutting tool and the instantaneous cutting force [6,7]. End milling of pockets includes the rough milling process consisting of slot cutting of straight segments and corners with circular concavities followed by the ®nishing milling process by immersion cutting [8]. Thus, during the cutting process the end mill cutter is subjected to radial and tangential de¯ections at different segments of the tool path. For milling a complex shape or simple rectangular pocket, the * Corresponding author. Tel.: 852-2788-7845; fax: 852-2788-8423. E-mail address: [email protected] (A. Geddam).
contour accuracy in corner cutting is mainly in¯uenced by the variation of the de¯ection of the end mill caused by the variation in the cutting forces [9,10]. This must be due to the variable radial depth of cut in corner cutting between the straight part and the corner resulting in variable contour accuracy. The results of Iwabe et al. [11] show that the radial depth of cut and the chip areas increase rapidly at the inside corners causing rapid increase in the cutting forces, resulting in deterioration of contour accuracy. In order to compensate the tool path for improving the contour accuracy, it is necessary to predict the instantaneous de¯ection by estimating the instantaneous cutting force by using the instantaneous radial depth of cut. This paper presents a simple tool de¯ection model for predicting contour errors in straight segments and corners for rough milling with slot cutting as well as ®nish milling with immersion cutting. Some case studies are presented to illustrate the results. 2. Estimation of cutting forces The cutting forces vary with the tool geometry and the machining conditions. There are other fundamental factors affecting the cutting forces such as the material hardness, friction and the stresses at the tool±chip interface. The end milling force model used here is based on the analytical and
0924-0136/01/$ ± see front matter # 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 0 1 ) 0 0 5 9 4 - 5
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K.M.Y. Law, A. Geddam / Journal of Materials Processing Technology 113 (2001) 399±405
experimental studies of Osta®ev et al. [12] which make use of the application of fundamental machining parameters [13]. The force model developed by the authors [14,15] is used to formulate the equations for the total tangential cutting force (P) and the torque (T) as given by cos g sin g t d (1) P 0:28Sk b 2:05r 0:55 m where Sk is the fracture stress determined by the true stress± strain tensile curve, b Sz sin ci the instantaneous width of cut during spindle rotation, t the depth of cut, r the chip ratio, d the contact length on the tool clearance surface without wear (d 0:1 mm for carbides, 0.05 mm for HSS [2]), and g the rake angle. The coef®cient of friction depends on the chip ratio r in the absence of a built-up edge and is given by [12]: m
22500 90
g
r 0:0015 90 2:46
g1:27
Therefore, the chip ratio can be obtained from !1=0:0015 90 g1:27 m 90 g2:46 r 22500
(2)
(3)
For end milling, the tool teeth inclination angle l and the relationship between the end mill diameter D, the number of teeth Z and the workpiece size should also be taken into account. Hence, the end milling torque may be expressed as: n D X ca sin c cc i 1 T P (4) 2 2 i1 where the angle of rotation of the cutter, c cos 1 D 2b=D; the angle between individual cutting edges, cc 360=Z; ca 114:65 t=D tan l; n is given by n c=cc (n should be an integer). 3. Estimation of tool de¯ection errors End milling cutter de¯ections can be predicted by considering the end mill as a cantilever. The de¯ection of the end mill is determined according to the cutting forces exerted on the tool, assuming that the force is acting at the center position at the tip of the cutter. The de¯ection d can be given by d
PL3 3EI
(5)
where P is the cutting force acting on the tool tip obtained from Eq. (4), L the effective length of the end mill cutter, E the value of Young's modulus of the tool material, and I the second moment of area. Substituting for P in Eq. (1), 3 cos g L d 0:28Sk b 20:5r 0:55 sin g t d (6) m 3EI
Fig. 1. Illustration of straight slot cutting.
3.1. De¯ection errors for straight slot cutting The basic de¯ection equation obtained in Eq. (5) is directly applicable only for straight slot cutting as shown in Fig. 1. During the slot cutting of a straight line, the radial width of cut is assumed to be a constant, i.e., the same as the diameter of the cutter. Therefore, for the straight part, the radial de¯ection may be assumed as 0 and the only cause of de¯ection is the force parallel to the feed direction, which is a constant. 3.2. De¯ection errors for corner slot cutting For right-angled corners, Fig. 2, both radial and tangential de¯ections will occur when the cutter is turned 908 and inserted into the material. As the cutter enters the material, the radial width of cut varies from a very small value until the width of cut becomes the same as the cutter diameter. With varying radial width of cut, the average cutting force varies. It is assumed that the radial width of cut changes during corner cutting as the cutter enters into the material and changes the direction of feed with a 908 angle until it is totally fed into the workpiece, as shown in Fig. 2. By simple geometrical calculation, as shown in Fig. 3, the instantaneous radial width of cut bi may be given as: bi 12 D 1 sin yi
(7)
where D is the cutter diameter, and yi the angle of the cutter that is immersed into the material at any instant i. In straight slot cutting, the cutting force P may be assumed to be constant, as the width of cut is constant. Assuming that the other cutting parameters are set to be constants, the average cutting force can be expressed as a product of a constant representing the cutting parameters and the radial width of cut. Thus, the instantaneous cutting force with varying width of cut at any instant can be determined as a ratio of the slot cutting force as given by bi Pi P (8) b
K.M.Y. Law, A. Geddam / Journal of Materials Processing Technology 113 (2001) 399±405
401
Fig. 2. A schematic illustration of the radial width of cut during corner slot cutting.
Since the de¯ection error d depends on the cutting force applied on the cutter and varies with the width of cut bi at that instant, the instantaneous de¯ection may be given by dr i
Pi L3 3EI
(9)
i.e.
Since
bi Pi P ; b then dr i
By integration, the radial de¯ection of cutting a 908 corner becomes !90 PL3 D 1 1 sin y3=2 p (11) dr 3EI 2b 3=2 2 0
bi
1 2 D 1
dr 0:431
sin yi
P L3 D yi 1 sin yi cos 45 b 3EI 2 2
(10)
PL3 D 3EI b
(12)
Similarly, the tangential de¯ection error against the feed direction dt, can be given by x P L3 D yi 1 sin yi dt i sin 45 (13) b 3EI 2 2 i.e. dt 1:1785
PL3 D 3EI b
The resultant de¯ection error q d d2r d2t
(14)
(15)
3.3. De¯ection errors at a circular concavity Fig. 3. A schematic illustration of the instantaneous radial width of cut at a corner.
For complex shaped pockets, the milling of circular concavities and corners are commonly performed. It is
402
K.M.Y. Law, A. Geddam / Journal of Materials Processing Technology 113 (2001) 399±405
Fig. 4. The mechanism for a slot path cut at a circular concavity.
believed that many complex shaped curves are formed by a series of circular curves; thus in these cases, the tool de¯ection error would vary as the cutter moves along the circular cutter path due to variation of the radial width of cut. This causes the cutter to de¯ect randomly and with rapidly increasing error values. From Fig. 4, a cutter is performing cutting at position A, with cutter rotation angle y referred to the line joining the center of curvature and the starting point of the circular curve. It is believed that the cutter would de¯ect towards the direction of position B due to the reaction forces exerted on the material, which is approximately proportional to the cutting forces, exerted by the cutter. In this case, the cutter is performing a slot cutting along the preset cutter path without cutting a rough cut, i.e. the radial depth of cut is equal to the cutter diameter as it cuts along the path. The required radius of curvature of the circular curve is not the same but is much greater than the cutter radius, therefore, the designed cutter path should follow the curve geometry with the offset of the cutter radius instead of a right-angled turn. At any instant, the radial width of cut b may be taken as equal to the cutter diameter D. The de¯ection will have two components, one along the X-axis (dx) and the other along the Y-axis (dy). The de¯ection component along the X-axis (dx) is given by: dx
d sin y
3.3.1. Case study For an example, it is proposed to estimate the de¯ection errors of a pocket of corner curvature 15 mm radius with respect to the cutter center as shown in Fig. 5. The material used and the cutting conditions are as follows: Material used Tensile strength of workpiece, Sk (N/mm2) Friction coefficient, m Tool diameter, D (mm) Rake angle, g Inclination angle, l Axial depth of cut, t (mm) Feed/tooth (mm)
Mild steel 800 0.76 10 188 17.58 3 0.25
By applying the slot cutting de¯ection equations for circular corners, the de¯ected cutter path is estimated. The calculated values of the de¯ection errors are shown in Table 1. The results of de¯ected cutter center path are plotted and compared with the desired cutter path as shown in Fig. 6. It can be seen that the cutter de¯ected away from the center of the pocket as it is approaching the corner. This
(16)
where sin y Yi Y0 =R, and Y0 and Yi are the cutter centers at the starting position of the corner and the position inside the corner at any instant: PL3 dx sin y (17) 3EI Similarly, the de¯ection error along the Y-axis, dy is PL3 cos y 3EI p where cos y R2 Yi Y0 2 =R: dy
(18) Fig. 5. Example of pocket milling at a corner curvature.
K.M.Y. Law, A. Geddam / Journal of Materials Processing Technology 113 (2001) 399±405
403
Table 1 De¯ection errors at circular concavities for slot cutting Cutting force, P (N)
138 138 138 138 137 138 137 138 139 138 138
Cutter center
Deflection errors
Deflected cutter
Xc
Yc
dx
dy
Xc
Yc
15 14 13 12 11 10 9 8 7 6 5
5 5.0501 5.2020 5.4606 5.8348 6.3397 7 7.8585 9 10.641 15
0 0.0132 0.0264 0.0396 0.0528 0.0660 0.0792 0.0924 0.1056 0.1188 0.1320
0.1320 0.1313 0.1293 0.1259 0.1210 0.1143 0.1056 0.0942 0.0792 0.0575 0
15 14.0132 13.0264 12.0396 11.0528 10.0660 9.0792 8.0924 7.1056 6.1188 5.1320
4.8679 4.9187 5.0726 5.3346 5.7138 6.2254 6.8943 7.7642 8.9207 10.5835 15
means that the de¯ection error values along the Y-axis are signi®cantly greater than the error values along the X-axis, and that the cutter de¯ected into the pocket as it left the corner due to the variation of radial width of cut. 3.4. Milling inside corners with small radial immersion The mechanism of milling at an inside corner is illustrated in Fig. 7. This situation occurs when ®nishing at corners with small radial immersion. Let points A and B represent the previous and current cutter center positions. The curves P1Q1 and P2Q2 are the previous and current concavities, respectively, machined by the cutter. The radius of the cutter is R, and Rc and Rr are the radii of curvature of the desired surface and the roughing surface referred to point O and Or respectively. Assume that the cutter center moves along the cutter path in anti-clockwise direction. The cut inside the corner is a circular arc of radius Rc minus the cutter radius R with the center at point O and the straight parts of the cutter path are paths parallel to the X- and Y-axis. During cutting along the straight parts, the cutter is cutting at a constant
Fig. 6. Comparison of the desired and de¯ected corner cutter path.
radial width of cut b0. As the cutter moves into the circular corner areas, the radial width changes as the cutter contact angle y varies and the actual radial width b becomes larger than the constant radial width b0. To calculate the actual radial width during the milling of an inside corner, it is required to obtain the points P1, P2, Q1 and Q2 in cartesian coordinates. P1 and Q1 are the previous positions of P2 and Q2, therefore, it is assumed that Q x; y1 i Q x; y2 i
1
(19)
The center of curvature of the inner and outer curve parts are Or(Xr, Yr) and O(X, Y) respectively and (Xc(i), Yc(i)) is the center of the cutter. Thus, the gradient of line OB, the line joining the center of curvature of the desired curve and the cutter center, is given by cot y
Yr Xr
Yc i Xc i
Fig. 7. Finishing at corners with small radial immersion.
(20)
404
K.M.Y. Law, A. Geddam / Journal of Materials Processing Technology 113 (2001) 399±405
If a is the angle between tool entry and tool exit, the actual
XQ2 i
q 2 Xc i H=cot y2 1 cot2 y=cot2 yXc i R2 Xc i H=cot y2
Xc i H=cot y
1 cot2 y=cot2 y
radial depth b will be a b P2 Q2 sin 2 Then, the distance between P2 and Q2 is given by
(21)
P2 Q22 XP2 i
(22)
XQ2 i 2 YP2 i
YQ2 i 2
2R2 cos a;
cos a 2 cos2
a 2
1;
a P2 Q 2 sin 2 2R Thus, the instantaneous radial width of cut is bi
XP2 i
XQ2 i 2 YP2 i 2R
(23)
YQ2 i 2
(24)
The X- and Y-coordinates of point P2 and Q2 can be obtained by q R2 XQ2 i Xc i 2 (25) YQ2 i Yc i YP2 i Yc i XP2 i Xc i
R2 Yc i
Y2
Xc i X2 Yc i Y 2 Xc i X YP2 i Yc i Yc i Y
; (26)
The point Q2(i) can be determined by Yr
YQ2 i 2 Xr
XQ2 i 2 R2r
(27)
Yc
YQ2 i 2 Xc
XQ2 i 2 R2
(28)
Subtracting Eq. (27) from Eq. (28), we have R2
R2r Xr2 Xc2 Yr2 2 Yr Yc i
Yc i 2
YQ2 i
R2
R2r Xr2 2 Yr
Xc2 Yr2 Yc i
2 Yc i
XQ2 i cot y (32)
Thus, the instantaneous cutting force and the de¯ection error can be obtained by using the calculated instantaneous radial width of cut. 3.4.1. Case study Assume that a pocket of corner curvature radius Rc 10 mm is machined by the method of roughing followed by ®nishing milling. Firstly, it is roughly milled to a smaller pocket of curvature radius Rr 7 mm with constant clearance to the desired dimension. This is followed by ®nishing milling with constant radial width along the straight sides with a cutter radius of R 5 mm. A schematic illustration is shown in Fig. 8. The left bottom corner is set as the origin of the coordinate system (0, 0), then for the left bottom corner, the center of curvature of the roughed and ®nished surface will be at Or(11, 11) and O(10, 10), respectively. The cutter enters the corner in a clockwise direction, i.e. the point of corner entrance and exit will be (10, 4) and (4, 10), respectively. By applying the radial width determination equation (24), for any cutter center position within the corner area, the radial width of cut can be found. From the calculated instantaneous cutting forces, the de¯ections can be calculated using the basic de¯ection equation (6).
XQ2 i YQ2 i cot y
(29) Substituting for YQ2(i) from Eq. (29) into Eq. (28): 2 2 R R2r Xr2 Xc2 Yr2 Yc i 2 XQ2 i Yc i 2 Yr Yc i cot y XQ2 i
Xc i 2 R2
Substituting by using: " 2 R R2r Xr2 Xc2 Yr2 H 2 Yr Yc i XQ22 i
1 cot2 y cot2 y
2 H2 Xc i
2 Yc i
# Yc i
H 2XQ2 i Xc i cot y
R2 0
(31)
and then substituting Eq. (31) into Eq. (30), we have
By the cosine rule, P2 Q22 2R2
By solving the quadratic equation
(30)
Fig. 8. A schematic illustration of pocket ®nishing.
K.M.Y. Law, A. Geddam / Journal of Materials Processing Technology 113 (2001) 399±405 Table 2 Tool de¯ection errors for ®nishing at corners with small radial immersion Position along X-axis (mm)
Position along Y-axis (mm)
Calculated width (mm)
Estimated deflection error (mm)
10 9.5 9.0 8.5 8.0 7.5 7.0 6.5 6.0 5.5 5.0 4.5 4.0
4.000000 4.020870 4.083920 4.190525 4.343146 4.545644 4.803848 5.126603 5.527864 6.031373 6.683375 7.602084 10.00000
4.000000 6.525127 6.408662 6.291377 6.170962 6.044600 5.908494 5.757003 5.580787 5.362141 5.307539 5.127739 4.000000
0.0880 0.0866 0.0877 0.0836 0.0826 0.0810 0.0801 0.0746 0.0750 0.0751 0.0754 0.0752 0.0751
milling or immersion cutting. Based on the methodology, the predicted results of the contour accuracy of pocket machining were veri®ed using CMM. The values of the predicted results closely match with those actually measured. References
The calculated de¯ection values for the following cutting conditions are shown in Table 2. The cutting conditions are as shown below: Material used Tensile strength of workpiece, Sk (N/mm2) Friction coefficient, m Tool diameter, D (mm) Rake angle, g Inclination angle, l Axial depth of cut, t (mm) Tool contact length used for HSS cutter, d (mm) Feed/tooth (mm)
405
Mild steel 800 0.76 10 188 17.58 3 0.05 0.25
4. Conclusions Pocket milling consists of the milling of straight segments and corners by roughing with slot cutting followed by ®nishing with immersion cutting. By determining the varying radial width of cut during corner cutting, the instantaneous cutting forces can be obtained and used in the de¯ection equations for corner milling with either slot
[1] J. Tlusty, P. Macneil, Dynamics of cutting forces in end milling, CIRP Ann. 24 (1) (1975) 21±25. [2] W.A. Kline, R.E. De Vor, J.R. Lindberg, The prediction of cutting forces in end milling with application to cornering cuts, Int. J. Mach. Tools Manuf. 22 (3) (1982) 52±680. [3] A. Ber, J. Rotberg, S. Zombach, A method for cutting force evaluation of end mills, CIRP Ann. 37 (1) (1988) 37±40. [4] E.J.A. Armarego, Computerized predictive cutting methods for forces in end milling including eccentricity affects, CIRP Ann. 35 (1) (1989) 45±49. [5] S. Smith, J. Tlusty, An overview of modelling and simulation of the milling process, J. Eng. Ind. 113 (1991) 169±175. [6] A.E. Bayoumi, G. Yucesan, L.A. Kendall, An analytical mechanistic cutting force model for milling operations: a theory and methodology, J. Eng. Ind. 116 (1994) 324±329. [7] B. Erhan, Y. Altintas, Peripheral milling conditions for improved dimensional accuracy, Int. J. Mach. Tools Manuf. 34 (7) (1994) 907± 918. [8] H. Cho, M. Yang, Precision pocketing in the ball end milling process, Int. J. JSPE 27 (4) (1993) 333±338. [9] Y. Li, H. Iwabe, Y. Fujii, Study on improvement of machining accuracy in cornering cut with end mill, JSME Int. J. III 35 (1) (1992) 168±173. [10] C.Y. Jung, J.H. Oh, Improvement of surface waviness by cutting force control in milling, Int. J. Mach. Tools Manuf. 31 (1) (1991) 9± 21. [11] H. Iwabe, Y. Fujii, K. Saito, T. Kishinami, Study on corner cut by end mill, Int. J. JSPE 28 (8) (1994) 218±223. [12] V. Osta®ev, K. Mahmudov, D. Osta®ev, Distribution of contact loads along rake and ¯ank surfaces, CIRP Ann. 43 (1) (1994) 61±63. [13] N.N. Zorev, Metal Cutting Mechanics, Pergamon Press, Oxford, 1966, 187 pp. [14] K.M.Y. Law, Process design for error compensation in end milling of pockets, M.Phil. Thesis, City University of Hong Kong, Hong Kong, 1998. [15] K.M.Y. Law, A. Geddam, V. Osta®ev, A process-design approach for error compensation in the end milling of pockets, J. Mater. Process. Technol. 89±90 (1999) 238±244.
Prediction of contour accuracy in the end milling of pockets Kris M.Y. Law, A. Geddam* Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong
Abstract Many die and mold parts surface contours are produced by end milling operations. The contour accuracy of the milled pockets is strongly in¯uenced by the cutting tool de¯ection caused by the cutting forces. For milling a complex shape or a simple rectangular pocket, the accuracy of contours in corner cutting is mainly in¯uenced by the de¯ection of the end mill caused by the variation of cutting forces. This is believed to be due to variable radial depth of cut in corner cutting between the straight part and the corners resulting in variable contour accuracy. The way to improve contour accuracy in corner cutting is by decreasing the radial depth of cut to reduce the cutting forces and, consequently, the end mill de¯ection errors. During the cutting process the end mill cutter is subjected to radial and tangential de¯ections at different segments of the tool path. In order to compensate the tool path for improving the contour accuracy, it is necessary to predict the instantaneous radial depth of cut. A simple tool de¯ection model for predicting the de¯ection errors in straight segments and corners for rough milling with slot cutting as well as for ®nish milling with immersion cutting is presented with illustration of case studies. # 2001 Elsevier Science B.V. All rights reserved. Keywords: Cutting forces; End milling; Contour accuracy; Tool de¯ection errors
1. Introduction End milling is the most commonly used machining process for the manufacture of dies and moulds, as well as a variety of high precision aerospace components. The contour accuracy of the milled pro®les is dependent on the milling cutter de¯ection errors. Hence, the prediction of machining errors due to tool de¯ection with the cutting force modeling in the end milling process has been the subject of many investigations [1±5]. In the end milling process, the machining operations are carried out with slender helical cutters, and the cutting forces vary periodically. The slender helical end mill experiences both static and dynamic deformations causing contour and surface errors on the machined part. The instantaneous tool de¯ection is dependent on the static stiffness of the cutting tool and the instantaneous cutting force [6,7]. End milling of pockets includes the rough milling process consisting of slot cutting of straight segments and corners with circular concavities followed by the ®nishing milling process by immersion cutting [8]. Thus, during the cutting process the end mill cutter is subjected to radial and tangential de¯ections at different segments of the tool path. For milling a complex shape or simple rectangular pocket, the * Corresponding author. Tel.: 852-2788-7845; fax: 852-2788-8423. E-mail address: [email protected] (A. Geddam).
contour accuracy in corner cutting is mainly in¯uenced by the variation of the de¯ection of the end mill caused by the variation in the cutting forces [9,10]. This must be due to the variable radial depth of cut in corner cutting between the straight part and the corner resulting in variable contour accuracy. The results of Iwabe et al. [11] show that the radial depth of cut and the chip areas increase rapidly at the inside corners causing rapid increase in the cutting forces, resulting in deterioration of contour accuracy. In order to compensate the tool path for improving the contour accuracy, it is necessary to predict the instantaneous de¯ection by estimating the instantaneous cutting force by using the instantaneous radial depth of cut. This paper presents a simple tool de¯ection model for predicting contour errors in straight segments and corners for rough milling with slot cutting as well as ®nish milling with immersion cutting. Some case studies are presented to illustrate the results. 2. Estimation of cutting forces The cutting forces vary with the tool geometry and the machining conditions. There are other fundamental factors affecting the cutting forces such as the material hardness, friction and the stresses at the tool±chip interface. The end milling force model used here is based on the analytical and
0924-0136/01/$ ± see front matter # 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 0 1 ) 0 0 5 9 4 - 5
400
K.M.Y. Law, A. Geddam / Journal of Materials Processing Technology 113 (2001) 399±405
experimental studies of Osta®ev et al. [12] which make use of the application of fundamental machining parameters [13]. The force model developed by the authors [14,15] is used to formulate the equations for the total tangential cutting force (P) and the torque (T) as given by cos g sin g t d (1) P 0:28Sk b 2:05r 0:55 m where Sk is the fracture stress determined by the true stress± strain tensile curve, b Sz sin ci the instantaneous width of cut during spindle rotation, t the depth of cut, r the chip ratio, d the contact length on the tool clearance surface without wear (d 0:1 mm for carbides, 0.05 mm for HSS [2]), and g the rake angle. The coef®cient of friction depends on the chip ratio r in the absence of a built-up edge and is given by [12]: m
22500 90
g
r 0:0015 90 2:46
g1:27
Therefore, the chip ratio can be obtained from !1=0:0015 90 g1:27 m 90 g2:46 r 22500
(2)
(3)
For end milling, the tool teeth inclination angle l and the relationship between the end mill diameter D, the number of teeth Z and the workpiece size should also be taken into account. Hence, the end milling torque may be expressed as: n D X ca sin c cc i 1 T P (4) 2 2 i1 where the angle of rotation of the cutter, c cos 1 D 2b=D; the angle between individual cutting edges, cc 360=Z; ca 114:65 t=D tan l; n is given by n c=cc (n should be an integer). 3. Estimation of tool de¯ection errors End milling cutter de¯ections can be predicted by considering the end mill as a cantilever. The de¯ection of the end mill is determined according to the cutting forces exerted on the tool, assuming that the force is acting at the center position at the tip of the cutter. The de¯ection d can be given by d
PL3 3EI
(5)
where P is the cutting force acting on the tool tip obtained from Eq. (4), L the effective length of the end mill cutter, E the value of Young's modulus of the tool material, and I the second moment of area. Substituting for P in Eq. (1), 3 cos g L d 0:28Sk b 20:5r 0:55 sin g t d (6) m 3EI
Fig. 1. Illustration of straight slot cutting.
3.1. De¯ection errors for straight slot cutting The basic de¯ection equation obtained in Eq. (5) is directly applicable only for straight slot cutting as shown in Fig. 1. During the slot cutting of a straight line, the radial width of cut is assumed to be a constant, i.e., the same as the diameter of the cutter. Therefore, for the straight part, the radial de¯ection may be assumed as 0 and the only cause of de¯ection is the force parallel to the feed direction, which is a constant. 3.2. De¯ection errors for corner slot cutting For right-angled corners, Fig. 2, both radial and tangential de¯ections will occur when the cutter is turned 908 and inserted into the material. As the cutter enters the material, the radial width of cut varies from a very small value until the width of cut becomes the same as the cutter diameter. With varying radial width of cut, the average cutting force varies. It is assumed that the radial width of cut changes during corner cutting as the cutter enters into the material and changes the direction of feed with a 908 angle until it is totally fed into the workpiece, as shown in Fig. 2. By simple geometrical calculation, as shown in Fig. 3, the instantaneous radial width of cut bi may be given as: bi 12 D 1 sin yi
(7)
where D is the cutter diameter, and yi the angle of the cutter that is immersed into the material at any instant i. In straight slot cutting, the cutting force P may be assumed to be constant, as the width of cut is constant. Assuming that the other cutting parameters are set to be constants, the average cutting force can be expressed as a product of a constant representing the cutting parameters and the radial width of cut. Thus, the instantaneous cutting force with varying width of cut at any instant can be determined as a ratio of the slot cutting force as given by bi Pi P (8) b
K.M.Y. Law, A. Geddam / Journal of Materials Processing Technology 113 (2001) 399±405
401
Fig. 2. A schematic illustration of the radial width of cut during corner slot cutting.
Since the de¯ection error d depends on the cutting force applied on the cutter and varies with the width of cut bi at that instant, the instantaneous de¯ection may be given by dr i
Pi L3 3EI
(9)
i.e.
Since
bi Pi P ; b then dr i
By integration, the radial de¯ection of cutting a 908 corner becomes !90 PL3 D 1 1 sin y3=2 p (11) dr 3EI 2b 3=2 2 0
bi
1 2 D 1
dr 0:431
sin yi
P L3 D yi 1 sin yi cos 45 b 3EI 2 2
(10)
PL3 D 3EI b
(12)
Similarly, the tangential de¯ection error against the feed direction dt, can be given by x P L3 D yi 1 sin yi dt i sin 45 (13) b 3EI 2 2 i.e. dt 1:1785
PL3 D 3EI b
The resultant de¯ection error q d d2r d2t
(14)
(15)
3.3. De¯ection errors at a circular concavity Fig. 3. A schematic illustration of the instantaneous radial width of cut at a corner.
For complex shaped pockets, the milling of circular concavities and corners are commonly performed. It is
402
K.M.Y. Law, A. Geddam / Journal of Materials Processing Technology 113 (2001) 399±405
Fig. 4. The mechanism for a slot path cut at a circular concavity.
believed that many complex shaped curves are formed by a series of circular curves; thus in these cases, the tool de¯ection error would vary as the cutter moves along the circular cutter path due to variation of the radial width of cut. This causes the cutter to de¯ect randomly and with rapidly increasing error values. From Fig. 4, a cutter is performing cutting at position A, with cutter rotation angle y referred to the line joining the center of curvature and the starting point of the circular curve. It is believed that the cutter would de¯ect towards the direction of position B due to the reaction forces exerted on the material, which is approximately proportional to the cutting forces, exerted by the cutter. In this case, the cutter is performing a slot cutting along the preset cutter path without cutting a rough cut, i.e. the radial depth of cut is equal to the cutter diameter as it cuts along the path. The required radius of curvature of the circular curve is not the same but is much greater than the cutter radius, therefore, the designed cutter path should follow the curve geometry with the offset of the cutter radius instead of a right-angled turn. At any instant, the radial width of cut b may be taken as equal to the cutter diameter D. The de¯ection will have two components, one along the X-axis (dx) and the other along the Y-axis (dy). The de¯ection component along the X-axis (dx) is given by: dx
d sin y
3.3.1. Case study For an example, it is proposed to estimate the de¯ection errors of a pocket of corner curvature 15 mm radius with respect to the cutter center as shown in Fig. 5. The material used and the cutting conditions are as follows: Material used Tensile strength of workpiece, Sk (N/mm2) Friction coefficient, m Tool diameter, D (mm) Rake angle, g Inclination angle, l Axial depth of cut, t (mm) Feed/tooth (mm)
Mild steel 800 0.76 10 188 17.58 3 0.25
By applying the slot cutting de¯ection equations for circular corners, the de¯ected cutter path is estimated. The calculated values of the de¯ection errors are shown in Table 1. The results of de¯ected cutter center path are plotted and compared with the desired cutter path as shown in Fig. 6. It can be seen that the cutter de¯ected away from the center of the pocket as it is approaching the corner. This
(16)
where sin y Yi Y0 =R, and Y0 and Yi are the cutter centers at the starting position of the corner and the position inside the corner at any instant: PL3 dx sin y (17) 3EI Similarly, the de¯ection error along the Y-axis, dy is PL3 cos y 3EI p where cos y R2 Yi Y0 2 =R: dy
(18) Fig. 5. Example of pocket milling at a corner curvature.
K.M.Y. Law, A. Geddam / Journal of Materials Processing Technology 113 (2001) 399±405
403
Table 1 De¯ection errors at circular concavities for slot cutting Cutting force, P (N)
138 138 138 138 137 138 137 138 139 138 138
Cutter center
Deflection errors
Deflected cutter
Xc
Yc
dx
dy
Xc
Yc
15 14 13 12 11 10 9 8 7 6 5
5 5.0501 5.2020 5.4606 5.8348 6.3397 7 7.8585 9 10.641 15
0 0.0132 0.0264 0.0396 0.0528 0.0660 0.0792 0.0924 0.1056 0.1188 0.1320
0.1320 0.1313 0.1293 0.1259 0.1210 0.1143 0.1056 0.0942 0.0792 0.0575 0
15 14.0132 13.0264 12.0396 11.0528 10.0660 9.0792 8.0924 7.1056 6.1188 5.1320
4.8679 4.9187 5.0726 5.3346 5.7138 6.2254 6.8943 7.7642 8.9207 10.5835 15
means that the de¯ection error values along the Y-axis are signi®cantly greater than the error values along the X-axis, and that the cutter de¯ected into the pocket as it left the corner due to the variation of radial width of cut. 3.4. Milling inside corners with small radial immersion The mechanism of milling at an inside corner is illustrated in Fig. 7. This situation occurs when ®nishing at corners with small radial immersion. Let points A and B represent the previous and current cutter center positions. The curves P1Q1 and P2Q2 are the previous and current concavities, respectively, machined by the cutter. The radius of the cutter is R, and Rc and Rr are the radii of curvature of the desired surface and the roughing surface referred to point O and Or respectively. Assume that the cutter center moves along the cutter path in anti-clockwise direction. The cut inside the corner is a circular arc of radius Rc minus the cutter radius R with the center at point O and the straight parts of the cutter path are paths parallel to the X- and Y-axis. During cutting along the straight parts, the cutter is cutting at a constant
Fig. 6. Comparison of the desired and de¯ected corner cutter path.
radial width of cut b0. As the cutter moves into the circular corner areas, the radial width changes as the cutter contact angle y varies and the actual radial width b becomes larger than the constant radial width b0. To calculate the actual radial width during the milling of an inside corner, it is required to obtain the points P1, P2, Q1 and Q2 in cartesian coordinates. P1 and Q1 are the previous positions of P2 and Q2, therefore, it is assumed that Q x; y1 i Q x; y2 i
1
(19)
The center of curvature of the inner and outer curve parts are Or(Xr, Yr) and O(X, Y) respectively and (Xc(i), Yc(i)) is the center of the cutter. Thus, the gradient of line OB, the line joining the center of curvature of the desired curve and the cutter center, is given by cot y
Yr Xr
Yc i Xc i
Fig. 7. Finishing at corners with small radial immersion.
(20)
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K.M.Y. Law, A. Geddam / Journal of Materials Processing Technology 113 (2001) 399±405
If a is the angle between tool entry and tool exit, the actual
XQ2 i
q 2 Xc i H=cot y2 1 cot2 y=cot2 yXc i R2 Xc i H=cot y2
Xc i H=cot y
1 cot2 y=cot2 y
radial depth b will be a b P2 Q2 sin 2 Then, the distance between P2 and Q2 is given by
(21)
P2 Q22 XP2 i
(22)
XQ2 i 2 YP2 i
YQ2 i 2
2R2 cos a;
cos a 2 cos2
a 2
1;
a P2 Q 2 sin 2 2R Thus, the instantaneous radial width of cut is bi
XP2 i
XQ2 i 2 YP2 i 2R
(23)
YQ2 i 2
(24)
The X- and Y-coordinates of point P2 and Q2 can be obtained by q R2 XQ2 i Xc i 2 (25) YQ2 i Yc i YP2 i Yc i XP2 i Xc i
R2 Yc i
Y2
Xc i X2 Yc i Y 2 Xc i X YP2 i Yc i Yc i Y
; (26)
The point Q2(i) can be determined by Yr
YQ2 i 2 Xr
XQ2 i 2 R2r
(27)
Yc
YQ2 i 2 Xc
XQ2 i 2 R2
(28)
Subtracting Eq. (27) from Eq. (28), we have R2
R2r Xr2 Xc2 Yr2 2 Yr Yc i
Yc i 2
YQ2 i
R2
R2r Xr2 2 Yr
Xc2 Yr2 Yc i
2 Yc i
XQ2 i cot y (32)
Thus, the instantaneous cutting force and the de¯ection error can be obtained by using the calculated instantaneous radial width of cut. 3.4.1. Case study Assume that a pocket of corner curvature radius Rc 10 mm is machined by the method of roughing followed by ®nishing milling. Firstly, it is roughly milled to a smaller pocket of curvature radius Rr 7 mm with constant clearance to the desired dimension. This is followed by ®nishing milling with constant radial width along the straight sides with a cutter radius of R 5 mm. A schematic illustration is shown in Fig. 8. The left bottom corner is set as the origin of the coordinate system (0, 0), then for the left bottom corner, the center of curvature of the roughed and ®nished surface will be at Or(11, 11) and O(10, 10), respectively. The cutter enters the corner in a clockwise direction, i.e. the point of corner entrance and exit will be (10, 4) and (4, 10), respectively. By applying the radial width determination equation (24), for any cutter center position within the corner area, the radial width of cut can be found. From the calculated instantaneous cutting forces, the de¯ections can be calculated using the basic de¯ection equation (6).
XQ2 i YQ2 i cot y
(29) Substituting for YQ2(i) from Eq. (29) into Eq. (28): 2 2 R R2r Xr2 Xc2 Yr2 Yc i 2 XQ2 i Yc i 2 Yr Yc i cot y XQ2 i
Xc i 2 R2
Substituting by using: " 2 R R2r Xr2 Xc2 Yr2 H 2 Yr Yc i XQ22 i
1 cot2 y cot2 y
2 H2 Xc i
2 Yc i
# Yc i
H 2XQ2 i Xc i cot y
R2 0
(31)
and then substituting Eq. (31) into Eq. (30), we have
By the cosine rule, P2 Q22 2R2
By solving the quadratic equation
(30)
Fig. 8. A schematic illustration of pocket ®nishing.
K.M.Y. Law, A. Geddam / Journal of Materials Processing Technology 113 (2001) 399±405 Table 2 Tool de¯ection errors for ®nishing at corners with small radial immersion Position along X-axis (mm)
Position along Y-axis (mm)
Calculated width (mm)
Estimated deflection error (mm)
10 9.5 9.0 8.5 8.0 7.5 7.0 6.5 6.0 5.5 5.0 4.5 4.0
4.000000 4.020870 4.083920 4.190525 4.343146 4.545644 4.803848 5.126603 5.527864 6.031373 6.683375 7.602084 10.00000
4.000000 6.525127 6.408662 6.291377 6.170962 6.044600 5.908494 5.757003 5.580787 5.362141 5.307539 5.127739 4.000000
0.0880 0.0866 0.0877 0.0836 0.0826 0.0810 0.0801 0.0746 0.0750 0.0751 0.0754 0.0752 0.0751
milling or immersion cutting. Based on the methodology, the predicted results of the contour accuracy of pocket machining were veri®ed using CMM. The values of the predicted results closely match with those actually measured. References
The calculated de¯ection values for the following cutting conditions are shown in Table 2. The cutting conditions are as shown below: Material used Tensile strength of workpiece, Sk (N/mm2) Friction coefficient, m Tool diameter, D (mm) Rake angle, g Inclination angle, l Axial depth of cut, t (mm) Tool contact length used for HSS cutter, d (mm) Feed/tooth (mm)
405
Mild steel 800 0.76 10 188 17.58 3 0.05 0.25
4. Conclusions Pocket milling consists of the milling of straight segments and corners by roughing with slot cutting followed by ®nishing with immersion cutting. By determining the varying radial width of cut during corner cutting, the instantaneous cutting forces can be obtained and used in the de¯ection equations for corner milling with either slot
[1] J. Tlusty, P. Macneil, Dynamics of cutting forces in end milling, CIRP Ann. 24 (1) (1975) 21±25. [2] W.A. Kline, R.E. De Vor, J.R. Lindberg, The prediction of cutting forces in end milling with application to cornering cuts, Int. J. Mach. Tools Manuf. 22 (3) (1982) 52±680. [3] A. Ber, J. Rotberg, S. Zombach, A method for cutting force evaluation of end mills, CIRP Ann. 37 (1) (1988) 37±40. [4] E.J.A. Armarego, Computerized predictive cutting methods for forces in end milling including eccentricity affects, CIRP Ann. 35 (1) (1989) 45±49. [5] S. Smith, J. Tlusty, An overview of modelling and simulation of the milling process, J. Eng. Ind. 113 (1991) 169±175. [6] A.E. Bayoumi, G. Yucesan, L.A. Kendall, An analytical mechanistic cutting force model for milling operations: a theory and methodology, J. Eng. Ind. 116 (1994) 324±329. [7] B. Erhan, Y. Altintas, Peripheral milling conditions for improved dimensional accuracy, Int. J. Mach. Tools Manuf. 34 (7) (1994) 907± 918. [8] H. Cho, M. Yang, Precision pocketing in the ball end milling process, Int. J. JSPE 27 (4) (1993) 333±338. [9] Y. Li, H. Iwabe, Y. Fujii, Study on improvement of machining accuracy in cornering cut with end mill, JSME Int. J. III 35 (1) (1992) 168±173. [10] C.Y. Jung, J.H. Oh, Improvement of surface waviness by cutting force control in milling, Int. J. Mach. Tools Manuf. 31 (1) (1991) 9± 21. [11] H. Iwabe, Y. Fujii, K. Saito, T. Kishinami, Study on corner cut by end mill, Int. J. JSPE 28 (8) (1994) 218±223. [12] V. Osta®ev, K. Mahmudov, D. Osta®ev, Distribution of contact loads along rake and ¯ank surfaces, CIRP Ann. 43 (1) (1994) 61±63. [13] N.N. Zorev, Metal Cutting Mechanics, Pergamon Press, Oxford, 1966, 187 pp. [14] K.M.Y. Law, Process design for error compensation in end milling of pockets, M.Phil. Thesis, City University of Hong Kong, Hong Kong, 1998. [15] K.M.Y. Law, A. Geddam, V. Osta®ev, A process-design approach for error compensation in the end milling of pockets, J. Mater. Process. Technol. 89±90 (1999) 238±244.