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Chisel edge and cutting lip shape optimization for improved twist drill point design
International Journal of Machine Tools & Manufacture 45 (2005) 421–431 www.elsevier.com/locate/ijmactool
Chisel edge and cutting lip shape optimization for improved twist drill point design Anish Paul, Shiv G. Kapoor*, Richard E. DeVor Department of Mechanical and Industrial Engineering, University of Illinois at Urbana-Champaign, 158 Mechanical Engineering Building, 1206 West Green Street, MC-244, Urbana, IL 61801, USA Received 6 October 2003; accepted 2 September 2004 Available online 25 November 2004
Abstract This paper investigates the optimization of twist drill point geometries in order to minimize thrust and torque in drilling. A point geometry parameterization based on the drill grinding parameters is used to ensure manufacturability of the optimized geometry. Three commonly used drill point geometries, namely, conical, Raconw and helical, are optimized for drilling forces while maintaining the inherent characteristics of each of the profiles. A significant reduction is shown in the drilling forces for the optimized drills. Drills with the optimized conical point profile are produced and tests run to validate the reduction in thrust and torque. q 2004 Elsevier Ltd. All rights reserved. Keywords: Drilling; Drill grinding; Thrust and torque; Shape optimization
1. Introduction The point geometry of the twist drill is the key element in determining two important drill performance characteristics, namely, the upward thrust along the drill axis and the drilling torque. A minimization of both these quantities leads to an improvement in performance by reducing drill deflection due to thrust loading, by lowering the power required for the drilling operation and by improving drill life. Even though the conical flank profile configuration is the most commonly used drill point geometry, and is the easiest to grind, it does not provide the most efficient cutting action in this regard. To overcome this, a number of different point shapes have been developed to provide inherent favorable characteristics such as the helical drill with an S-shaped chisel edge that provides a self-centering action and the Raconw drill that provides a reduction in exit burr formation. However, these geometries have tended to evolve from experience employing extensive experiments to evaluate and modify the design and are not optimal with respect to thrust and torque reduction. A methodology, rooted in the manufacture of the drill through * Corresponding author. Tel.: C1 217 333 3432; fax: C1 217 244 9956. E-mail address: [email protected] (S.G. Kapoor). 0890-6955/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2004.09.010
the grinding parameters, is thus required to arrive at a drill point geometry that maintains the favorable characteristics of the conical, Raconw and helical drills while optimizing their shape to minimize thrust and torque. There has been considerable work in the area of drill performance improvement by modifying the drill point geometry using flank surfaces made up of multiple facets. The first drill with a multifacet flank shape was developed experimentally around 1953 for drilling a special alloy steel [1]. Bhattacharyya et al. [2] used a modification to the chisel edge to reduce thrust. Liu [3] developed a new multifacet drill geometry with an emphasis on thrust. Wang et al. [4] used a computer-aided analysis method to improve the drill point design on multifaceted drills by analyzing the rake and relief angle distributions along the cutting edges of different drills. Chen et al. [5] studied the effect of notch angle on split-point drill design. However, little work can be found that addresses the design optimization of single facet arbitrary flank surfaces and that evaluates the shape of the chisel edge and cutting lips as a variable. Further, optimization of drill parameters in terms of the drill angles directly does not provide the grinding parameters needed for its manufacture, so any approach to drill point geometry should be based on the grinding parameters.
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In this paper, a methodology is described to obtain the optimum shape for the chisel edge and cutting lips of drills with arbitrary point geometries. In order to ensure manufacturability, the grinding profile parameters described by Tsai and Wu [6] are used to describe the point geometry of the drills. These parameters are optimized for the conical, Raconw and helical drill points using a real-coded genetic algorithm, the objective being the minimization of thrust or torque or both. The chisel edge forces are evaluated using the method described in Ref. [7], while the cutting lip forces are evaluated as described by Chandrasekharan et al. [8]. It may be noted that optimization is carried out only on the point grinding profile while the flute grinding profile is maintained constant and is assumed to follow the flute equation given by Galloway [9]. The paper evolves in the following manner. First, a point geometry shape parameterization is described. Then, the optimization procedure is discussed, including the objective and the constraints imposed. Benchmark drill performance is then established for standard conical, Raconw and helical drills as a point of comparison for the optimized drills. The optimization results in terms of the optimized cutting edge shapes along with the corresponding predicted optimized thrust and torque are then presented and interpreted. The drilling performance in terms of the drilling torque and thrust for the optimized shape of the conical profile is then validated through actual experiments.
parameters are assumed so that the optimized drill points can be reground on a standard twist drill. In order to evaluate the thrust and torque of a drill point described by the grinding parameters, these parameters must first be linked to the shape of the cutting edge. The method used is an extension of the method described in Paul [7] to include both the cutting lips and the chisel edge modeling using flank and flute contour intersections. The chisel edge is formed by the intersection of the flank surfaces of the drill, while the intersection of the flute surface with the flank surface forms the cutting lips. The flank contours equations are obtained from Tsai and Wu [6]. Eqs. (1) and (2) represent the symmetrical flank contours for a conical drill in the cylindrical coordinate system (r, b, z), viz. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 r cos b cos fCz sin fC d2 tan2 qKS2 Cðr sin bKSÞ2 Kðz cos fKr cos b sin fCdÞ2 tan2 q Z0; ð1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 Kr cos b cos fCz sin fC d 2 tan2 qKS2 CðKr sin bKSÞ2 Kðz cos fCr cos b sin fCdÞ2 tan2 q Z0: ð2Þ The flute equation given by Galloway, [9] can then be written as " # 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w 2 K1 w 2 b Zsin r K C tanðh0 Þcot r 2r r0 2
2. Point geometry shape parameterization C [1] shows coordinate systems and the four drill-grinding parameters q, f, d and S for the conical drill based on the work of Tsai and Wu [6]. q is the half angle of the cone, f is the angle between the axis of the cone and the drill axis measured in the x–z plane and d and S are the translations required to move along the grinding surface axis from the coordinate system at the apex of the cone (x 0 , y 0 , z 0 ) to the coordinate system at the drill tip (x, y, z). The distance S is measured along the y-axis, while d is measured along z 0 -axis. [2] and [3] show the coordinate systems and the five drillgrinding parameters a, c, f, d and S for ellipsoidal and hyperboloidal drills. a and c determine the form of the grinding surfaces while d equals C1 for the ellipsoidal drill and K1 for the hyperboloidal drill. The hyperboloidal grinding profile creates the helical drill point, while the ellipsoidal profile creates the Raconw drill point. The grinding parameters determine the flank surface of the drill. The cutting edge is made up of the chisel edge and the cutting lips of the drill and is determined not only by the flank surface but also by the flute surface [9], which is characterized by the flute grinding parameters. The flute grinding parameters are the helix angle, web thickness and manufacturers point angle. In this work, standard flute
z tanðh0 Þ; r0
(3)
where r0 is the radius of the drill, w is the web thickness, h0 is the helix angle and r is the manufacturers point angle. Fig. 4 shows the drill point sliced by planes 1, 2 and 3 normal to the drill axis. Plane 1 passes through the center of the drill point and has a z elevation of zero (zZ0). The cutting lips and the chisel edge intersect in plane 2. This is the plane in which the flank contours intersect with each other as well as with the flute contour for that elevation as shown in Fig. 5. Solving Eqs. (1)–(3) for the unknowns r, b and z gives the cylindrical coordinates of the point that marks this intersection of the chisel edge and cutting lips and hence yields the z coordinate for plane 2. The chisel edge lies between planes 1 and 2. Flank contours at various z elevations between these planes, shown as dotted lines in Fig. 5, intersect at coordinates that yield a discretized model of the chisel edge. The coordinates on the chisel edge are obtained by solving for the contour intersections using Eqs. (1) and (2). Plane 3 passes through the outer most point on the cutting lip at a radius of r0. Solving Eqs. (1) and (3) or Eqs. (2) and (3) at rZr0 yields the z coordinate for plane 3. The cutting lips lie between planes 2 and 3. Flank contours between these planes, shown as dotted lines in Fig. 6, intersect at coordinates that yield a discretized model of the cutting lips.
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These coordinates on the cutting lips are obtained by solving for the contour intersections using Eqs. (1) and (3) and Eqs. (2) and (3) for the two cutting lips. A similar approach can be used to obtain coordinates on the cutting edges of the Raconw and helical drills. The equivalent of Eqs. (1) and (2) for the ellipsoidal (Raconw) and hyperboloidal (helical) drills is given as " rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi#2 1 a2 ðr cos b cos fCz sin fÞC a2 Kd 2 d2 KS2 2 a c C
1 d ðr sin bKSÞ2 C 2 ðz cos fKr cos b sin fCdÞ2 Z1; a2 c ð4Þ
"
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi#2 1 a2 ðKr cos b cos fCz sin fÞC a2 Kd 2 d 2 KS2 2 a c C
1 d ðKr sin bKSÞ2 C 2 ðz cos fCr cos b sin fCdÞ2 Z1; 2 a c ð5Þ
where dZC1 for the ellipsoidal drill and dZK1 for the hyperboloidal drill. The flute equation (Eq. (3)) is the same for all the drills. The coordinates obtained represent a discretized model of the cutting edges, which can be used to calculate the thrust and torque created by these drill points. The mechanistic modeling technique is then used to calculate the chisel edge and cutting lip forces. The model used for the chisel edge is as described in Ref. [7]. It uses the drill grinding parameters to obtain a discretized model of the chisel edge and then models the cutting action at each element as that of oblique cutting. The model used for the cutting lip forces is as described in Ref. [8]. However, instead of using an approximate polynomial representation to obtain coordinates on the cutting lip, the cutting lip coordinates were obtained from the grinding profiles using the contour intersection technique described via Eqs. (1)–(3). These coordinates are then used to discretize the cutting lip into n linear elements and the forces for each individual element are calculated and summed to obtain the total cutting lip forces. 3. Optimization procedure
Fig. 1. Conical grinding profile.
conditions and the drill and workpiece material. The optimization is carried out keeping the process conditions and the drill and workpiece material constant. The drill grinding parameters, viz. q, f, d and S for the conical drill and a, c, f, d and S for the ellipsoidal and hyperboloidal drills, are thus the optimization variables, which must be optimized under certain geometric constraints to ensure viable values. The constraints that can be directly applied to the conical grinding parameter values (q, f, d and S) are given in Ref. [6] 0! q! 90;
(7)
where q is the half cone angle and d! 0;
(8)
so that the drill point lies within the grinding cone (Fig. 1). Another known constraint is the bounds on S given as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (9) w! S! d 2 tan2 q; where w is the web thickness, which is a function of the basic flutedesignandisnoteffectedbythegrindingparameters.IfS! w, the clearance angle becomes negative. The upper limit on S also exists as (d2 tan2qKS2) must be positive in Eqs. (1) and (2). For the ellipsoidal drill, the direct constraints on the grinding parameters a, c, f, d and S are aO r0 ;
(10)
so that the drill point lies completely within the ellipsoid (Fig. 2) and 0! d! c;
(11)
The objective of the optimization is to minimize thrust and torque. The objective function is defined as a weighted linear combination of thrust and torque and is given as F Z w1 Th C w2 To;
(6)
where w1 and w2 are the weights such that w1Cw2Z1. These weights allow the optimization of either thrust or torque or a weighted combination of both. Th and To are the total drilling thrust and torque, respectively, which are calculated using the force models for the chisel edge and the cutting lips. The thrust and torque are functions of the point geometry (therefore the drill grinding parameters), the process
Fig. 2. Ellipsoidal grinding profile.
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certain parameter combinations yield valid drill geometries. A valid geometry is one that is manufacturable. Grinding parameter sets that do not yield real solutions when solving for the cutting edge coordinates are used to identify invalid geometries. To make sure that the invalid geometries are removed from the search space during the optimization, a method of penalties is used. Invalid parameter sets are penalized by increasing the objective function value to a very high quantity.
Fig. 3. Hyperboloidal grinding profile.
4. Establishing a benchmark for drill point optimization
so that the drill point lies in the upper half of the ellipsoid (Fig. 2). The bounds on S are similar to those of the conical drill profile and are given as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 w! S! a2 K d2 2 d 2 ; (12) c as ða2 K d2 ða2 =c2 Þd 2 K S2 Þ must be positive in Eqs. (4) and (5). For the hyperboloidal drill, the constraints on the grinding parameters a, c, f, d and S are a! r0 ;
(13)
so that the drill point intersects with the hyperboloidal grinding surface and d is small as point O is near the middle of the hyperboloid where the steeper grinding surface provides a sharper drill point center (Fig. 3). The bounds on S are same as that for the ellipsoidal profile as is given in Eq. (12). The parameter f is determined as f Z r K q;
(14)
for the conical profile, and a f Z r K tanK1 ; c
For each of the three drill geometries, the force simulation was first run for standard, commercially available drills. Experiments were also run using these drills and force data collected in order to validate the force model while providing a benchmark against which the improvement resulting from the optimized geometry could be measured. The drilling tests were carried out on a Mori Seiki TV-30 CNC vertical machining center. The workpiece was mounted on a four-channel dynamometer (Kistler model No. 9273), which was used to measure the torque and thrust. The drills used as a benchmark had a diameter of 12.69 mm, a helix angle of 338 and a web thickness of 1.39 mm. The process conditions used were a speed of 1000 rpm and a feed of 0.1 mm/rev. The drill material was TiN-coated HSS and the workpiece material was 1018 steel. The experiments were conducted with a pilot hole. Table 1 shows the predicted and experimental values of the total thrust and torque as well as the cutting lip and chisel edge components. The error is seen to be within 6% for all three geometries and this indicates that the force model being used for the optimization predicts forces for a conical drill with a good accuracy (Figs. 4–6).
(15)
for the ellipsoidal and hyperboloidal profiles, where r is the manufacturers point angle. Even when the grinding parameters lie within the specified range defined by the direct constraints, only
5. Optimization results The optimization was carried out using genetic algorithms. This technique differs from the more traditional
Table 1 Predicted and experimental torque and thrust 1000 rpm, 0.1 mm/rev
Experimental Chisel edge
Predicted Cutting lip
Total
Chisel edge
Conical grinding parameters fZ41.48, qZ16.68, dZK9.56 mm, SZ1.12 mm Thrust (N) 1309 656 1965 1249 Torque (N cm) 180 510 690 170 Raconw grinding parameters fZ348, aZ21 mm, cZ4.7 mm, dZ3 mm, SZ0.2 mm Thrust (N) 1275 323 1598 1223 Torque (N cm) 16 470 486 14 Helical grinding parameters fZ218, aZ2.57 mm, cZ2.59 mm, dZ0 mm, SZ1.17 mm Thrust (N) 701 229 1000 674 Torque (N cm) 150 430 580 140
%Error
Cutting lip
Total
617 500
1866 670
5.03 2.89
310 460
1533 474
4.06 2.46
276 410
950 550
5.00 5.17
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Fig. 4. Cutting edge discretization.
optimization techniques in that it is population based, working with many candidate solutions rather than trying to progress from a single point in the search space. Genetic algorithms use only the value of the objective function, instead of derivatives and other auxiliary knowledge. This, combined with its simplicity of operation and computational efficiency, makes genetic algorithms an ideal method for drill point optimization using a model that can predict the objective function value. The population size used for the genetic algorithm was 50. Further details of the genetic algorithm used can be found in Ref. [7]. 5.1. Conical drill optimization results The optimization was run for the conical drill under the same process conditions, diameter and flute geometry as that used to establish the benchmark in Section 4. Table 2 shows the optimized conical drill grinding parameters obtained for each of the three objectives, namely, minimization of thrust, torque and a combination of both. The corresponding drill profiles in terms of their x–y and x–z projections are shown in Figs. 7–9. Also shown in the figures with the dotted line is the profile of the conical drill used as a benchmark. The design parameters corresponding to the optimized grinding parameters are shown in Fig. 10.
Fig. 6. Cutting lip contour intersections.
The drill half-point angle at the periphery is calculated as 0 K1 vx K1 Xn K XnK1 r Z tan ztan ; (16) vz Zn K ZnK1 where Xn and Zn are the coordinates of the outer edge of the peripheral cutting element (as discussed in Section 2) and XnK1 and ZnK1 are the coordinates of the inner edge of that element. It can also be seen that the point angles for the optimized geometries are in keeping with the intuitively expected values of the point angle, i.e. as point angle decreases the thrust is reduced. Table 3 shows the optimized thrust and torque for the conical drill. The reduction in thrust compared to the benchmark conical drill is 44%. The reduction in torque is 43.9%. Using an equal tradeoff, the thrust and torque are reduced by 41.7 and 32.6%, respectively. It is important to note that even though a significant reduction in thrust is seen for the drill optimized for thrust, the variation in the shape and orientation of the cutting edge compared to that of the benchmark drill is very subtle (Fig. 7). The reduction in thrust can be explained on the basis of the rake angle distribution for the drills. The normal rake angles for the chisel edge and cutting lips of the benchmark and optimized conical drills are shown in Figs. 11 and 12, respectively. In both figures, the rake angles are much higher for the optimized drills compared to the benchmark drills, especially at the inner edge. Consequently, the cutting forces are lower. This trend is clearly noticeable when the thrust values for the cutting Table 2 Optimized conical drill grinding parameters
Fig. 5. Chisel edge contour intersections.
Objective
f (deg)
q (deg)
d (mm)
S (mm)
Thrust Torque Thrust and torque
51.37 24.01 46.46
7.63 34.99 12.54
K18.33 K35.37 K38.97
126 17.35 5.92
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Fig. 7. Thrust-optimized conical (a) x–y profile, (b) x–z profile.
Fig. 8. Torque-optimized conical (a) x–y profile, (b) x–z profile.
lip and the chisel edge (Table 3) are compared to the corresponding rake angles shown in Figs. 11 and 12. The thrust value is used for comparison here as it is independent of the radial distance and depends only on the cutting force, which in turn depends on rake angles. The torque, on the other hand, is not just a function of the cutting force but depends on radial distance as well. Thus, it cannot be directly compared with the cutting force or the rake angles.
5.2. Raconw drill optimization The optimization was carried out for the Racon drill for the same process conditions, diameter and flute geometry as that of the Racon drill that was used to establish the benchmark in Section 4. Table 4 shows the optimized Raconw drill grinding parameters obtained for each of the three objectives. The corresponding drill profiles in terms
Fig. 9. Thrust and torque-optimized conical (a) x–y profile, (b) x–z profile.
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Fig. 10. Geometry of the conical drills optimized for (a) thrust and torque, (b) torque and (c) thrust.
of their x–y and x–z projections are shown in Figs. 13 and 14. These were obtained along the same lines as Figs. 7–9, but were combined to provide a clear comparison. The x–z profiles show that the optimized drill points have an overall form that would be intuitively expected. The thrustoptimized drill has the sharpest drill point, while the torqueoptimized drill has the most blunt point and, consequently, has the shortest cutting lip length. The thrust and torque values and their components for the optimized geometries are shown in Table 5. The reduction in thrust compared to Raconw drill used Table 3 Optimized thrust and torque for a conical drill Objective
Thrust Torque Both Benchmark
Chisel edge
Cutting lip
Thrust (N)
Thrust (N)
Torque (N cm)
Thrust (N)
Torque (N cm)
366.17 252.46 243.539 656
417 333 378 510
1053.66 1369.67 1169.63 1965.00
615 414 508 690
Torque (N cm)
687.48 180 1117.21 81 926.08 130 1309.00 180
as a benchmark in Section 4 is 42.05%. The reduction in torque is, however, only 6.3%. Since the improvement in torque in this case is relatively low, it may not warrant the change in shape. Using an equal tradeoff, the thrust and torque are reduced by 36 and 2.88%, respectively. On the benchmark Raconw drill, the chisel edge force is seen to be predominant and is quite high, while the cutting lip force is relatively low. The drastic reduction in chisel edge thrust in the optimized drill is a result of lower negative rake angles as can be seen from Fig. 15. Even though the cutting lip thrust for the optimized drills is higher than that of the benchmark drills, the reduction in chisel edge thrust more than compensates for this, resulting in a reduced total
Total
Fig. 12. Cutting lip normal rake angle comparison for the conical drill. Table 4 Optimized Raconw drill grinding parameters
Fig. 11. Chisel edge normal rake angle comparison for the conical drill.
Objective
a (mm)
c (mm)
f (deg)
Thrust Torque Thrust and torque
11.41 7.90 12.85
18.7 37.33 29.55
27.61418 9.80 47.05257 33.77 35.50094 5.59
d (mm)
S (mm) 1.21 1.56 0.827
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Fig. 13. Optimized Raconw drill x–y profiles.
Fig. 15. Chisel edge normal rake angle comparison for the Raconw drill.
Fig. 14. Optimized Raconw drill x–z profiles.
Fig. 16. Cutting lip normal rake angle comparison for the Raconw drill.
thrust for the optimized drills. It must be noted that the chisel edge and cutting lip shapes and angles are not independent of each other as they both result from the same flank grinding profile. The normal rake angle distribution along the cutting lips of the Raconw drills is shown in Fig. 16. As was the case with the conical drill, the thrust is seen to have an inverse relation with the normal rake angle. Table 5 Optimized thrust and torque for a raconw drill Objective
Thrust Torque Both Benchmark
Chisel edge
Cutting lip
Total
Thrust (N)
Thrust (N)
Torque (N cm)
Thrust (N)
Torque (N cm)
590.15 425.39 657.37 323
532 406 472 470
925.96 1524.81 1021.78 1598
534 454 475 486
Torque (N cm)
335.80 2 1099.41 48 364.40 3 1275 16
5.3. Helical drill optimization Table 6 shows the optimized helical drill grinding parameters obtained for each of the three objectives. The corresponding drill profiles in terms of their x–y and x–z projections are shown in Figs. 17 and 18. Also shown is the profile of the helical drill being used as a benchmark. The thrust and torque values and their components for the optimized drills are shown in Table 7. The reduction in thrust compared to the helical drill used as a benchmark is Table 6 Optimized helical drill grinding parameters Objective
a (mm)
c (mm)
f (deg)
Thrust Torque Thrust and torque
6.17 4.80 5.30
7.62 13.29 9.04
20.00757 1.48 39.14418 0.27 28.62156 2.14
d (mm)
S (mm) 3.45 2.36 3.23
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Fig. 17. Optimized helical drill x–y profiles.
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Fig. 19. Chisel edge normal rake angle comparison for the helical drill.
cutting forces seen in the optimized drills, with much less negative values along the chisel edge.
6. Validation of optimum conical geometry
Fig. 18. Optimized helical drill x–z profiles.
46.13%. The reduction in torque is 44.48%. Using a tradeoff, the thrust and torque are reduced by 41.94 and 28.10%, respectively. The thrust-optimized drill shows a sharp drill point, while the torque-optimized drill shows a blunt drill point as is expected. The rake angles distributions for the helical drill are shown in Figs. 19 and 20. As was the case with the Raconw drills, the rake angle distributions explain the reduction in
The parameters for the optimized conical flank profiles were used to regrind a standard 1188 twist drill with a helix angle of 338. The grinding operation was performed on a Walter grinder at the S.M. Wu Manufacturing Research Center at the University of Michigan at Ann Arbor. The manufactured drills along with the measured point angles are shown in Fig. 21. The cutting edge shape was checked using a Brown and Sharpe Coordinate Measuring Machine. The drills matched the x–z profiles of the cutting edges quite well. However, there was a minor variation in the x–y profiles. This could be attributed to setting errors when trying to set the center of the drill point in the zZ0 plane.
Table 7 Optimized thrust and torque for a helical drill Objective
Thrust Torque Both Benchmark
Chisel edge
Cutting lip
Total
Thrust (N)
Torque (N cm)
Thrust (N)
Torque (N cm)
Thrust (N)
Torque (N cm)
269.41 392.91 229.36 701
13 12 11 150
269.22 423.86 351.17 299
457 310 406 430
538.63 816.78 580.54 1000
470 322 417 580
Fig. 20. Cutting lip normal rake angle comparison for the helical drill.
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Fig. 21. Images of the conical drills optimized for (a) thrust and torque, (b) torque, (c) thrust. Table 8 Optimized thrust and torque for a conical drill Objective
Thrust Torque Both Benchmark
Chisel edge
Cutting lip
Total
Thrust (N) predicted (experimental)
Torque (N cm) predicted (experimental)
Thrust (N) predicted (experimental)
Torque (N cm) predicted (experimental)
Thrust (N) predicted (experimental)
Torque (N cm) predicted (experimental)
687.48 (769.85) 1117.21 (1170.09) 926.08 (935.1) (1309.00)
180 (127) 81 (88) 130 (121) (180)
366.17 (330.25) 252.46 (300.4) 243.539 (210.1) (656)
417 (452) 333 (299) 378 (344) (510)
1053.66 (1100.1) 1369.67 (1471.3) 1169.63 (1145.2) (1965.00)
615 (579) 414 (387) 508 (465) (690)
Drilling tests were then run using the ground optimized conical drills under the same process conditions as that used when establishing the benchmark. The experimental setup described earlier was used for these tests. The predicted and experimental values of the thrust and torque for the optimized conical drills are also shown in Table 8. As predicted, the reduction in thrust and torque was over 40% compared to the benchmark drill. The error in the predicted values of total thrust and torque was 4.2 and 6.2%, respectively, for the thrust-optimized drill. For the torqueoptimized drills, the errors in total thrust and torque were both around 6.9%. For the drill optimized for both thrust and torque, the error was 2.1 and 9.2% for the total thrust and torque, respectively.
7. Conclusions A methodology is described that allows the regrinding of a standard twist drill point into a drill point of a desired profile with the grinding parameters optimized so as to reduce thrust and torque Three grinding profiles are studied, namely, the conical, Raconw and helical profiles. More specifically, 1. The drill point geometry was parameterized using the flank grinding profile parameters. Conical and
quadratic grinding models were then used to arrive at the chisel edge and cutting lip shape, which were in turn used to calculate the drilling thrust and torque that made up the objective function for optimizing the drilling forces. 2. The optimization results indicate a reduction in thrust and torque of around 40% over the benchmark conical drill. The optimized Raconw drill shows a marked reduction in thrust, but only a marginal improvement in torque indicating that the benchmark drill is well optimized with respect to torque. The optimized helical drill reduces torque and thrust individually by over 40%; thus, showing a large improvement over the benchmark helical point geometry. 3. The rake angle distributions indicate that the optimized drills have increased rake angles and show reduced forces. This is seen in the rake angle distributions for both the cutting lips and the chisel edge. The chisel edge variations in rake angle distributions are the most pronounced. The optimized drills have increased rake angles; thus, reducing the high forces caused due to large negative rake angles. 4. Drills were ground using the optimized conical flank grinding parameters. Force data from drilling tests using these drills validated the reduction predicted. The error
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in predicted thrust and torque was seen to be within 10%.
Acknowledgements The authors gratefully acknowledge the support of the National Science Foundation and the UIUC Center for Machine-Tool Systems Research. The authors would also like to acknowledge Dr Xuewen Lin at the S.M. Wu Manufacturing Research Center at the University of Michigan at Ann Arbor for his help in grinding the optimized drills.
References [1] S.M. Wu, J.M. Shen, Mathematical model for multifacet drills, ASME Journal of Engineering for Industry 105 (1983) 173–182.
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[2] A. Bhattacharyya, A.B. Chatterjee, L. Ham, Modification of drill point for reducing thrust, Journal of Engineering for Industry 1971; 1073–1078. [3] T.L. Liu, Development of new drill geometry and its performance evaluation, Journal of Shaping Technology 9 (1991) 161–169. [4] J. Wang, J. Sha, J. Ni, Further improvement of multifacet drills, S.M. Wu Symposium 1 (1994) 115–122. [5] W. Chen, K. Fuh, C. Wu, B. Chang, Design optimization of a splitpoint drill by force analysis, Journal of Materials Processing Technology 58 (2/3) (1996) 314–322. [6] W.D. Tsai, S.M. Wu,, A mathematical model for drill point design and grinding, ASME Journal of Engineering for Industry 101 (1979) 333– 340. [7] A. Paul, Force-Based Modeling and Optimization of Arbitrary Drill Point Geometry Using the Grinding Parameters, Masters Thesis, University of Illinois at Urbana-Champaign, 2003. [8] V. Chandrasekharan, S.G. Kapoor, R.E. DeVor, Mechanistic model to predict the cutting force system for arbitrary drill point geometry, Journal of Manufacturing Science and Engineering, Transactions of the ASME 120 (3) (1998) 563–570. [9] D. Galloway, Some experiments on the influence of various factors on drill performance, Transactions of the ASME 79 (1957) 191–231.
Chisel edge and cutting lip shape optimization for improved twist drill point design Anish Paul, Shiv G. Kapoor*, Richard E. DeVor Department of Mechanical and Industrial Engineering, University of Illinois at Urbana-Champaign, 158 Mechanical Engineering Building, 1206 West Green Street, MC-244, Urbana, IL 61801, USA Received 6 October 2003; accepted 2 September 2004 Available online 25 November 2004
Abstract This paper investigates the optimization of twist drill point geometries in order to minimize thrust and torque in drilling. A point geometry parameterization based on the drill grinding parameters is used to ensure manufacturability of the optimized geometry. Three commonly used drill point geometries, namely, conical, Raconw and helical, are optimized for drilling forces while maintaining the inherent characteristics of each of the profiles. A significant reduction is shown in the drilling forces for the optimized drills. Drills with the optimized conical point profile are produced and tests run to validate the reduction in thrust and torque. q 2004 Elsevier Ltd. All rights reserved. Keywords: Drilling; Drill grinding; Thrust and torque; Shape optimization
1. Introduction The point geometry of the twist drill is the key element in determining two important drill performance characteristics, namely, the upward thrust along the drill axis and the drilling torque. A minimization of both these quantities leads to an improvement in performance by reducing drill deflection due to thrust loading, by lowering the power required for the drilling operation and by improving drill life. Even though the conical flank profile configuration is the most commonly used drill point geometry, and is the easiest to grind, it does not provide the most efficient cutting action in this regard. To overcome this, a number of different point shapes have been developed to provide inherent favorable characteristics such as the helical drill with an S-shaped chisel edge that provides a self-centering action and the Raconw drill that provides a reduction in exit burr formation. However, these geometries have tended to evolve from experience employing extensive experiments to evaluate and modify the design and are not optimal with respect to thrust and torque reduction. A methodology, rooted in the manufacture of the drill through * Corresponding author. Tel.: C1 217 333 3432; fax: C1 217 244 9956. E-mail address: [email protected] (S.G. Kapoor). 0890-6955/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2004.09.010
the grinding parameters, is thus required to arrive at a drill point geometry that maintains the favorable characteristics of the conical, Raconw and helical drills while optimizing their shape to minimize thrust and torque. There has been considerable work in the area of drill performance improvement by modifying the drill point geometry using flank surfaces made up of multiple facets. The first drill with a multifacet flank shape was developed experimentally around 1953 for drilling a special alloy steel [1]. Bhattacharyya et al. [2] used a modification to the chisel edge to reduce thrust. Liu [3] developed a new multifacet drill geometry with an emphasis on thrust. Wang et al. [4] used a computer-aided analysis method to improve the drill point design on multifaceted drills by analyzing the rake and relief angle distributions along the cutting edges of different drills. Chen et al. [5] studied the effect of notch angle on split-point drill design. However, little work can be found that addresses the design optimization of single facet arbitrary flank surfaces and that evaluates the shape of the chisel edge and cutting lips as a variable. Further, optimization of drill parameters in terms of the drill angles directly does not provide the grinding parameters needed for its manufacture, so any approach to drill point geometry should be based on the grinding parameters.
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In this paper, a methodology is described to obtain the optimum shape for the chisel edge and cutting lips of drills with arbitrary point geometries. In order to ensure manufacturability, the grinding profile parameters described by Tsai and Wu [6] are used to describe the point geometry of the drills. These parameters are optimized for the conical, Raconw and helical drill points using a real-coded genetic algorithm, the objective being the minimization of thrust or torque or both. The chisel edge forces are evaluated using the method described in Ref. [7], while the cutting lip forces are evaluated as described by Chandrasekharan et al. [8]. It may be noted that optimization is carried out only on the point grinding profile while the flute grinding profile is maintained constant and is assumed to follow the flute equation given by Galloway [9]. The paper evolves in the following manner. First, a point geometry shape parameterization is described. Then, the optimization procedure is discussed, including the objective and the constraints imposed. Benchmark drill performance is then established for standard conical, Raconw and helical drills as a point of comparison for the optimized drills. The optimization results in terms of the optimized cutting edge shapes along with the corresponding predicted optimized thrust and torque are then presented and interpreted. The drilling performance in terms of the drilling torque and thrust for the optimized shape of the conical profile is then validated through actual experiments.
parameters are assumed so that the optimized drill points can be reground on a standard twist drill. In order to evaluate the thrust and torque of a drill point described by the grinding parameters, these parameters must first be linked to the shape of the cutting edge. The method used is an extension of the method described in Paul [7] to include both the cutting lips and the chisel edge modeling using flank and flute contour intersections. The chisel edge is formed by the intersection of the flank surfaces of the drill, while the intersection of the flute surface with the flank surface forms the cutting lips. The flank contours equations are obtained from Tsai and Wu [6]. Eqs. (1) and (2) represent the symmetrical flank contours for a conical drill in the cylindrical coordinate system (r, b, z), viz. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 r cos b cos fCz sin fC d2 tan2 qKS2 Cðr sin bKSÞ2 Kðz cos fKr cos b sin fCdÞ2 tan2 q Z0; ð1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 Kr cos b cos fCz sin fC d 2 tan2 qKS2 CðKr sin bKSÞ2 Kðz cos fCr cos b sin fCdÞ2 tan2 q Z0: ð2Þ The flute equation given by Galloway, [9] can then be written as " # 1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w 2 K1 w 2 b Zsin r K C tanðh0 Þcot r 2r r0 2
2. Point geometry shape parameterization C [1] shows coordinate systems and the four drill-grinding parameters q, f, d and S for the conical drill based on the work of Tsai and Wu [6]. q is the half angle of the cone, f is the angle between the axis of the cone and the drill axis measured in the x–z plane and d and S are the translations required to move along the grinding surface axis from the coordinate system at the apex of the cone (x 0 , y 0 , z 0 ) to the coordinate system at the drill tip (x, y, z). The distance S is measured along the y-axis, while d is measured along z 0 -axis. [2] and [3] show the coordinate systems and the five drillgrinding parameters a, c, f, d and S for ellipsoidal and hyperboloidal drills. a and c determine the form of the grinding surfaces while d equals C1 for the ellipsoidal drill and K1 for the hyperboloidal drill. The hyperboloidal grinding profile creates the helical drill point, while the ellipsoidal profile creates the Raconw drill point. The grinding parameters determine the flank surface of the drill. The cutting edge is made up of the chisel edge and the cutting lips of the drill and is determined not only by the flank surface but also by the flute surface [9], which is characterized by the flute grinding parameters. The flute grinding parameters are the helix angle, web thickness and manufacturers point angle. In this work, standard flute
z tanðh0 Þ; r0
(3)
where r0 is the radius of the drill, w is the web thickness, h0 is the helix angle and r is the manufacturers point angle. Fig. 4 shows the drill point sliced by planes 1, 2 and 3 normal to the drill axis. Plane 1 passes through the center of the drill point and has a z elevation of zero (zZ0). The cutting lips and the chisel edge intersect in plane 2. This is the plane in which the flank contours intersect with each other as well as with the flute contour for that elevation as shown in Fig. 5. Solving Eqs. (1)–(3) for the unknowns r, b and z gives the cylindrical coordinates of the point that marks this intersection of the chisel edge and cutting lips and hence yields the z coordinate for plane 2. The chisel edge lies between planes 1 and 2. Flank contours at various z elevations between these planes, shown as dotted lines in Fig. 5, intersect at coordinates that yield a discretized model of the chisel edge. The coordinates on the chisel edge are obtained by solving for the contour intersections using Eqs. (1) and (2). Plane 3 passes through the outer most point on the cutting lip at a radius of r0. Solving Eqs. (1) and (3) or Eqs. (2) and (3) at rZr0 yields the z coordinate for plane 3. The cutting lips lie between planes 2 and 3. Flank contours between these planes, shown as dotted lines in Fig. 6, intersect at coordinates that yield a discretized model of the cutting lips.
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These coordinates on the cutting lips are obtained by solving for the contour intersections using Eqs. (1) and (3) and Eqs. (2) and (3) for the two cutting lips. A similar approach can be used to obtain coordinates on the cutting edges of the Raconw and helical drills. The equivalent of Eqs. (1) and (2) for the ellipsoidal (Raconw) and hyperboloidal (helical) drills is given as " rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi#2 1 a2 ðr cos b cos fCz sin fÞC a2 Kd 2 d2 KS2 2 a c C
1 d ðr sin bKSÞ2 C 2 ðz cos fKr cos b sin fCdÞ2 Z1; a2 c ð4Þ
"
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi#2 1 a2 ðKr cos b cos fCz sin fÞC a2 Kd 2 d 2 KS2 2 a c C
1 d ðKr sin bKSÞ2 C 2 ðz cos fCr cos b sin fCdÞ2 Z1; 2 a c ð5Þ
where dZC1 for the ellipsoidal drill and dZK1 for the hyperboloidal drill. The flute equation (Eq. (3)) is the same for all the drills. The coordinates obtained represent a discretized model of the cutting edges, which can be used to calculate the thrust and torque created by these drill points. The mechanistic modeling technique is then used to calculate the chisel edge and cutting lip forces. The model used for the chisel edge is as described in Ref. [7]. It uses the drill grinding parameters to obtain a discretized model of the chisel edge and then models the cutting action at each element as that of oblique cutting. The model used for the cutting lip forces is as described in Ref. [8]. However, instead of using an approximate polynomial representation to obtain coordinates on the cutting lip, the cutting lip coordinates were obtained from the grinding profiles using the contour intersection technique described via Eqs. (1)–(3). These coordinates are then used to discretize the cutting lip into n linear elements and the forces for each individual element are calculated and summed to obtain the total cutting lip forces. 3. Optimization procedure
Fig. 1. Conical grinding profile.
conditions and the drill and workpiece material. The optimization is carried out keeping the process conditions and the drill and workpiece material constant. The drill grinding parameters, viz. q, f, d and S for the conical drill and a, c, f, d and S for the ellipsoidal and hyperboloidal drills, are thus the optimization variables, which must be optimized under certain geometric constraints to ensure viable values. The constraints that can be directly applied to the conical grinding parameter values (q, f, d and S) are given in Ref. [6] 0! q! 90;
(7)
where q is the half cone angle and d! 0;
(8)
so that the drill point lies within the grinding cone (Fig. 1). Another known constraint is the bounds on S given as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (9) w! S! d 2 tan2 q; where w is the web thickness, which is a function of the basic flutedesignandisnoteffectedbythegrindingparameters.IfS! w, the clearance angle becomes negative. The upper limit on S also exists as (d2 tan2qKS2) must be positive in Eqs. (1) and (2). For the ellipsoidal drill, the direct constraints on the grinding parameters a, c, f, d and S are aO r0 ;
(10)
so that the drill point lies completely within the ellipsoid (Fig. 2) and 0! d! c;
(11)
The objective of the optimization is to minimize thrust and torque. The objective function is defined as a weighted linear combination of thrust and torque and is given as F Z w1 Th C w2 To;
(6)
where w1 and w2 are the weights such that w1Cw2Z1. These weights allow the optimization of either thrust or torque or a weighted combination of both. Th and To are the total drilling thrust and torque, respectively, which are calculated using the force models for the chisel edge and the cutting lips. The thrust and torque are functions of the point geometry (therefore the drill grinding parameters), the process
Fig. 2. Ellipsoidal grinding profile.
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certain parameter combinations yield valid drill geometries. A valid geometry is one that is manufacturable. Grinding parameter sets that do not yield real solutions when solving for the cutting edge coordinates are used to identify invalid geometries. To make sure that the invalid geometries are removed from the search space during the optimization, a method of penalties is used. Invalid parameter sets are penalized by increasing the objective function value to a very high quantity.
Fig. 3. Hyperboloidal grinding profile.
4. Establishing a benchmark for drill point optimization
so that the drill point lies in the upper half of the ellipsoid (Fig. 2). The bounds on S are similar to those of the conical drill profile and are given as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 w! S! a2 K d2 2 d 2 ; (12) c as ða2 K d2 ða2 =c2 Þd 2 K S2 Þ must be positive in Eqs. (4) and (5). For the hyperboloidal drill, the constraints on the grinding parameters a, c, f, d and S are a! r0 ;
(13)
so that the drill point intersects with the hyperboloidal grinding surface and d is small as point O is near the middle of the hyperboloid where the steeper grinding surface provides a sharper drill point center (Fig. 3). The bounds on S are same as that for the ellipsoidal profile as is given in Eq. (12). The parameter f is determined as f Z r K q;
(14)
for the conical profile, and a f Z r K tanK1 ; c
For each of the three drill geometries, the force simulation was first run for standard, commercially available drills. Experiments were also run using these drills and force data collected in order to validate the force model while providing a benchmark against which the improvement resulting from the optimized geometry could be measured. The drilling tests were carried out on a Mori Seiki TV-30 CNC vertical machining center. The workpiece was mounted on a four-channel dynamometer (Kistler model No. 9273), which was used to measure the torque and thrust. The drills used as a benchmark had a diameter of 12.69 mm, a helix angle of 338 and a web thickness of 1.39 mm. The process conditions used were a speed of 1000 rpm and a feed of 0.1 mm/rev. The drill material was TiN-coated HSS and the workpiece material was 1018 steel. The experiments were conducted with a pilot hole. Table 1 shows the predicted and experimental values of the total thrust and torque as well as the cutting lip and chisel edge components. The error is seen to be within 6% for all three geometries and this indicates that the force model being used for the optimization predicts forces for a conical drill with a good accuracy (Figs. 4–6).
(15)
for the ellipsoidal and hyperboloidal profiles, where r is the manufacturers point angle. Even when the grinding parameters lie within the specified range defined by the direct constraints, only
5. Optimization results The optimization was carried out using genetic algorithms. This technique differs from the more traditional
Table 1 Predicted and experimental torque and thrust 1000 rpm, 0.1 mm/rev
Experimental Chisel edge
Predicted Cutting lip
Total
Chisel edge
Conical grinding parameters fZ41.48, qZ16.68, dZK9.56 mm, SZ1.12 mm Thrust (N) 1309 656 1965 1249 Torque (N cm) 180 510 690 170 Raconw grinding parameters fZ348, aZ21 mm, cZ4.7 mm, dZ3 mm, SZ0.2 mm Thrust (N) 1275 323 1598 1223 Torque (N cm) 16 470 486 14 Helical grinding parameters fZ218, aZ2.57 mm, cZ2.59 mm, dZ0 mm, SZ1.17 mm Thrust (N) 701 229 1000 674 Torque (N cm) 150 430 580 140
%Error
Cutting lip
Total
617 500
1866 670
5.03 2.89
310 460
1533 474
4.06 2.46
276 410
950 550
5.00 5.17
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Fig. 4. Cutting edge discretization.
optimization techniques in that it is population based, working with many candidate solutions rather than trying to progress from a single point in the search space. Genetic algorithms use only the value of the objective function, instead of derivatives and other auxiliary knowledge. This, combined with its simplicity of operation and computational efficiency, makes genetic algorithms an ideal method for drill point optimization using a model that can predict the objective function value. The population size used for the genetic algorithm was 50. Further details of the genetic algorithm used can be found in Ref. [7]. 5.1. Conical drill optimization results The optimization was run for the conical drill under the same process conditions, diameter and flute geometry as that used to establish the benchmark in Section 4. Table 2 shows the optimized conical drill grinding parameters obtained for each of the three objectives, namely, minimization of thrust, torque and a combination of both. The corresponding drill profiles in terms of their x–y and x–z projections are shown in Figs. 7–9. Also shown in the figures with the dotted line is the profile of the conical drill used as a benchmark. The design parameters corresponding to the optimized grinding parameters are shown in Fig. 10.
Fig. 6. Cutting lip contour intersections.
The drill half-point angle at the periphery is calculated as 0 K1 vx K1 Xn K XnK1 r Z tan ztan ; (16) vz Zn K ZnK1 where Xn and Zn are the coordinates of the outer edge of the peripheral cutting element (as discussed in Section 2) and XnK1 and ZnK1 are the coordinates of the inner edge of that element. It can also be seen that the point angles for the optimized geometries are in keeping with the intuitively expected values of the point angle, i.e. as point angle decreases the thrust is reduced. Table 3 shows the optimized thrust and torque for the conical drill. The reduction in thrust compared to the benchmark conical drill is 44%. The reduction in torque is 43.9%. Using an equal tradeoff, the thrust and torque are reduced by 41.7 and 32.6%, respectively. It is important to note that even though a significant reduction in thrust is seen for the drill optimized for thrust, the variation in the shape and orientation of the cutting edge compared to that of the benchmark drill is very subtle (Fig. 7). The reduction in thrust can be explained on the basis of the rake angle distribution for the drills. The normal rake angles for the chisel edge and cutting lips of the benchmark and optimized conical drills are shown in Figs. 11 and 12, respectively. In both figures, the rake angles are much higher for the optimized drills compared to the benchmark drills, especially at the inner edge. Consequently, the cutting forces are lower. This trend is clearly noticeable when the thrust values for the cutting Table 2 Optimized conical drill grinding parameters
Fig. 5. Chisel edge contour intersections.
Objective
f (deg)
q (deg)
d (mm)
S (mm)
Thrust Torque Thrust and torque
51.37 24.01 46.46
7.63 34.99 12.54
K18.33 K35.37 K38.97
126 17.35 5.92
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Fig. 7. Thrust-optimized conical (a) x–y profile, (b) x–z profile.
Fig. 8. Torque-optimized conical (a) x–y profile, (b) x–z profile.
lip and the chisel edge (Table 3) are compared to the corresponding rake angles shown in Figs. 11 and 12. The thrust value is used for comparison here as it is independent of the radial distance and depends only on the cutting force, which in turn depends on rake angles. The torque, on the other hand, is not just a function of the cutting force but depends on radial distance as well. Thus, it cannot be directly compared with the cutting force or the rake angles.
5.2. Raconw drill optimization The optimization was carried out for the Racon drill for the same process conditions, diameter and flute geometry as that of the Racon drill that was used to establish the benchmark in Section 4. Table 4 shows the optimized Raconw drill grinding parameters obtained for each of the three objectives. The corresponding drill profiles in terms
Fig. 9. Thrust and torque-optimized conical (a) x–y profile, (b) x–z profile.
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Fig. 10. Geometry of the conical drills optimized for (a) thrust and torque, (b) torque and (c) thrust.
of their x–y and x–z projections are shown in Figs. 13 and 14. These were obtained along the same lines as Figs. 7–9, but were combined to provide a clear comparison. The x–z profiles show that the optimized drill points have an overall form that would be intuitively expected. The thrustoptimized drill has the sharpest drill point, while the torqueoptimized drill has the most blunt point and, consequently, has the shortest cutting lip length. The thrust and torque values and their components for the optimized geometries are shown in Table 5. The reduction in thrust compared to Raconw drill used Table 3 Optimized thrust and torque for a conical drill Objective
Thrust Torque Both Benchmark
Chisel edge
Cutting lip
Thrust (N)
Thrust (N)
Torque (N cm)
Thrust (N)
Torque (N cm)
366.17 252.46 243.539 656
417 333 378 510
1053.66 1369.67 1169.63 1965.00
615 414 508 690
Torque (N cm)
687.48 180 1117.21 81 926.08 130 1309.00 180
as a benchmark in Section 4 is 42.05%. The reduction in torque is, however, only 6.3%. Since the improvement in torque in this case is relatively low, it may not warrant the change in shape. Using an equal tradeoff, the thrust and torque are reduced by 36 and 2.88%, respectively. On the benchmark Raconw drill, the chisel edge force is seen to be predominant and is quite high, while the cutting lip force is relatively low. The drastic reduction in chisel edge thrust in the optimized drill is a result of lower negative rake angles as can be seen from Fig. 15. Even though the cutting lip thrust for the optimized drills is higher than that of the benchmark drills, the reduction in chisel edge thrust more than compensates for this, resulting in a reduced total
Total
Fig. 12. Cutting lip normal rake angle comparison for the conical drill. Table 4 Optimized Raconw drill grinding parameters
Fig. 11. Chisel edge normal rake angle comparison for the conical drill.
Objective
a (mm)
c (mm)
f (deg)
Thrust Torque Thrust and torque
11.41 7.90 12.85
18.7 37.33 29.55
27.61418 9.80 47.05257 33.77 35.50094 5.59
d (mm)
S (mm) 1.21 1.56 0.827
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Fig. 13. Optimized Raconw drill x–y profiles.
Fig. 15. Chisel edge normal rake angle comparison for the Raconw drill.
Fig. 14. Optimized Raconw drill x–z profiles.
Fig. 16. Cutting lip normal rake angle comparison for the Raconw drill.
thrust for the optimized drills. It must be noted that the chisel edge and cutting lip shapes and angles are not independent of each other as they both result from the same flank grinding profile. The normal rake angle distribution along the cutting lips of the Raconw drills is shown in Fig. 16. As was the case with the conical drill, the thrust is seen to have an inverse relation with the normal rake angle. Table 5 Optimized thrust and torque for a raconw drill Objective
Thrust Torque Both Benchmark
Chisel edge
Cutting lip
Total
Thrust (N)
Thrust (N)
Torque (N cm)
Thrust (N)
Torque (N cm)
590.15 425.39 657.37 323
532 406 472 470
925.96 1524.81 1021.78 1598
534 454 475 486
Torque (N cm)
335.80 2 1099.41 48 364.40 3 1275 16
5.3. Helical drill optimization Table 6 shows the optimized helical drill grinding parameters obtained for each of the three objectives. The corresponding drill profiles in terms of their x–y and x–z projections are shown in Figs. 17 and 18. Also shown is the profile of the helical drill being used as a benchmark. The thrust and torque values and their components for the optimized drills are shown in Table 7. The reduction in thrust compared to the helical drill used as a benchmark is Table 6 Optimized helical drill grinding parameters Objective
a (mm)
c (mm)
f (deg)
Thrust Torque Thrust and torque
6.17 4.80 5.30
7.62 13.29 9.04
20.00757 1.48 39.14418 0.27 28.62156 2.14
d (mm)
S (mm) 3.45 2.36 3.23
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Fig. 17. Optimized helical drill x–y profiles.
429
Fig. 19. Chisel edge normal rake angle comparison for the helical drill.
cutting forces seen in the optimized drills, with much less negative values along the chisel edge.
6. Validation of optimum conical geometry
Fig. 18. Optimized helical drill x–z profiles.
46.13%. The reduction in torque is 44.48%. Using a tradeoff, the thrust and torque are reduced by 41.94 and 28.10%, respectively. The thrust-optimized drill shows a sharp drill point, while the torque-optimized drill shows a blunt drill point as is expected. The rake angles distributions for the helical drill are shown in Figs. 19 and 20. As was the case with the Raconw drills, the rake angle distributions explain the reduction in
The parameters for the optimized conical flank profiles were used to regrind a standard 1188 twist drill with a helix angle of 338. The grinding operation was performed on a Walter grinder at the S.M. Wu Manufacturing Research Center at the University of Michigan at Ann Arbor. The manufactured drills along with the measured point angles are shown in Fig. 21. The cutting edge shape was checked using a Brown and Sharpe Coordinate Measuring Machine. The drills matched the x–z profiles of the cutting edges quite well. However, there was a minor variation in the x–y profiles. This could be attributed to setting errors when trying to set the center of the drill point in the zZ0 plane.
Table 7 Optimized thrust and torque for a helical drill Objective
Thrust Torque Both Benchmark
Chisel edge
Cutting lip
Total
Thrust (N)
Torque (N cm)
Thrust (N)
Torque (N cm)
Thrust (N)
Torque (N cm)
269.41 392.91 229.36 701
13 12 11 150
269.22 423.86 351.17 299
457 310 406 430
538.63 816.78 580.54 1000
470 322 417 580
Fig. 20. Cutting lip normal rake angle comparison for the helical drill.
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Fig. 21. Images of the conical drills optimized for (a) thrust and torque, (b) torque, (c) thrust. Table 8 Optimized thrust and torque for a conical drill Objective
Thrust Torque Both Benchmark
Chisel edge
Cutting lip
Total
Thrust (N) predicted (experimental)
Torque (N cm) predicted (experimental)
Thrust (N) predicted (experimental)
Torque (N cm) predicted (experimental)
Thrust (N) predicted (experimental)
Torque (N cm) predicted (experimental)
687.48 (769.85) 1117.21 (1170.09) 926.08 (935.1) (1309.00)
180 (127) 81 (88) 130 (121) (180)
366.17 (330.25) 252.46 (300.4) 243.539 (210.1) (656)
417 (452) 333 (299) 378 (344) (510)
1053.66 (1100.1) 1369.67 (1471.3) 1169.63 (1145.2) (1965.00)
615 (579) 414 (387) 508 (465) (690)
Drilling tests were then run using the ground optimized conical drills under the same process conditions as that used when establishing the benchmark. The experimental setup described earlier was used for these tests. The predicted and experimental values of the thrust and torque for the optimized conical drills are also shown in Table 8. As predicted, the reduction in thrust and torque was over 40% compared to the benchmark drill. The error in the predicted values of total thrust and torque was 4.2 and 6.2%, respectively, for the thrust-optimized drill. For the torqueoptimized drills, the errors in total thrust and torque were both around 6.9%. For the drill optimized for both thrust and torque, the error was 2.1 and 9.2% for the total thrust and torque, respectively.
7. Conclusions A methodology is described that allows the regrinding of a standard twist drill point into a drill point of a desired profile with the grinding parameters optimized so as to reduce thrust and torque Three grinding profiles are studied, namely, the conical, Raconw and helical profiles. More specifically, 1. The drill point geometry was parameterized using the flank grinding profile parameters. Conical and
quadratic grinding models were then used to arrive at the chisel edge and cutting lip shape, which were in turn used to calculate the drilling thrust and torque that made up the objective function for optimizing the drilling forces. 2. The optimization results indicate a reduction in thrust and torque of around 40% over the benchmark conical drill. The optimized Raconw drill shows a marked reduction in thrust, but only a marginal improvement in torque indicating that the benchmark drill is well optimized with respect to torque. The optimized helical drill reduces torque and thrust individually by over 40%; thus, showing a large improvement over the benchmark helical point geometry. 3. The rake angle distributions indicate that the optimized drills have increased rake angles and show reduced forces. This is seen in the rake angle distributions for both the cutting lips and the chisel edge. The chisel edge variations in rake angle distributions are the most pronounced. The optimized drills have increased rake angles; thus, reducing the high forces caused due to large negative rake angles. 4. Drills were ground using the optimized conical flank grinding parameters. Force data from drilling tests using these drills validated the reduction predicted. The error
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in predicted thrust and torque was seen to be within 10%.
Acknowledgements The authors gratefully acknowledge the support of the National Science Foundation and the UIUC Center for Machine-Tool Systems Research. The authors would also like to acknowledge Dr Xuewen Lin at the S.M. Wu Manufacturing Research Center at the University of Michigan at Ann Arbor for his help in grinding the optimized drills.
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