Applying The Finite Element Method To Drill Design Based On Drill Deormations

A drilling experiment was conducted to verify whether the thick web drill with curved primary cutting edges would be effective in reducing the thrust force, the torque, and the tool wear

E L S E V I E R Finite Elements in Analysis and Design 26 (1997) 57-81

Keywords: Tool orthogonal rake angle; Curved primary cutting edge; Torsional rigidity; Drill deformation

I Introduction

Drilling is one of the most widely applied manufacturing operations and is, therefore, of considerable economic importance In m o d e r n industry, the present trend in machining is aimed at high speed, great accuracy, and a u t o m a t i c operation on n u m e r o u s new difficult-to-cut materials to

be drilled The conventional twist drill has some shortcomings of fundamental importance: the inability to drill diificult-to-cut materials and the difficulty of drilling deep holes The reason is that the torsional rigidity of the conventional twist drill appears to be insufficient F r o m a geometric standpoint of the conventional twist drill, the distribution of the tool orthogonal rake angle along the primary cutting edge is unreasonable for drilling work, since the tool o r t h o g o n a l rake angle varies from a b o u t 30 ° at the periphery of the drill to a b o u t - 30 ° near the chisel edge [1] This variation is m o r e obvious as the web gets thicker The design of the cross-sectional shape of a drill

b o d y is mainly determined by the manufacturers and is not easily altered b y the users Therefore, there has been much w o r k to improve the cutting performance from the standpoint of drill point 0168-874X/97/$17.00 c© 1997 Elsevier Science B.V All rights reserved

PII S0 1 6 8 - 8 7 4 X ( 9 6 ) 0 0 0 7 1-6

58 W.-C Chen / Finite Elements" in Analysis and Design 26 (1997) 5 ~ 8 1

shapes; out of this effort different types of drills such as spiral point drills [1], split point drills [2, 3], and multifacet point drills [4, 5] have been developed

The drill torsional rigidity also affects drill performance as pointed out by several workers [6-10] Spur and Masuha [6] investigated the influence of the cross-sectional shape on the torsional rigidity of the twist drill and showed that the design of the cross-sectional shape of a twist drill involves a proper balance between the size of the cross-sectional area and the area of the flute Wan and Suen [7] developed an optimum cross-section shape of the conventional twist drill based upon the torsional rigidity of the drill This allows the torsional rigidity of the drill to be improved through changing the cross-sectional shape of a drill

The web thickness is a significant geometry factor influencing the torsional rigidity of the drill It

is found that a thicker web drill increases the polar moment of the cross-sectional area, which in turn increases the torsional rigidity of the drill [11] On the other hand, a thicker web drill increases the chisel edge length Meanwhile, a greater amount of metal removal is done by extrusion rather than by cutting edges [12] As a result, higher thrust forces come into being The effect of the flute length on the drill life when cutting cobalt-based high-temperature alloy was investigated by Oxford in [9] The test results have shown that the drill life increased 80 times as the flute length decreases from 70 to 35 mm This may be explained by the fact that the drill has a shorter flute length and a greater torsional rigidity Applying statistical techniques to the test results the effect of the helix angle on drill performance has been analyzed by Lorenz in [10] The optimal helix angle

in the vicinity of 40 ° is expected to provide longer drill life in comparison with the conventional helix angle of 26 ° when drilling AISI P20 mold steel Therefore, the selection of the cross-sectional shape of a drill body (i.e., including the "primary" and "secondary" flute shapes, the web thickness, the flute length, and the helix angle) is of great importance in any drill design for it has a direct influence on the drilling process, the chip removal, the tool life, and the torsional strength of the drill The design of the twist drill with special cross-sectional shape based upon drill deformations and how the drill geometric parameters affected the torsional rigidity of the drill has so far not been systematically investigated by means of a three-dimensional finite element analysis in literature The purpose of this paper is to apply the finite element method to the calculation of the drill deformations for the design of the twist drill with special cross-sectional shape The effects of the drill geometric parameters on drill deformations are also investigated Secondly, the appropriate thick web drill with curved primary cutting edges based upon the calculated results using

a three-dimensional finite element analysis for the torsional rigidity of the drill was devised

A drilling experiment was conducted to verify whether the thick web drill with curved primary cutting edges would be effective in reducing the thrust force, the torque, and the tool wear, thus providing a better cutting ability and a longer tool life

2 Mathematical determination of the profile of the curved primary cutting edge

In order to improve the unfavourable distribution of the tool orthogonal rake angle along the straight primary cutting edge of a thicker web drill, an attempt was made to change the straight primary cutting edge to a curved one (i.e., to change the distribution of the tool orthogonal rake angle along the primary cutting edge) The mathematical determination of the profile of the curved primary cutting edge is described as follows

W.-C Chen / Finite Elements in Analysis and Design 26 (1997) 57-81 59

rx tan/3

tan ~ox R sin ~CRx

where ~ is the flute helix angle, R is the drill outer radius, 2STX is the tool cutting edge inclination angle on end view at radius rx, and XRX is the tool cutting edge angle which can be expressed as

sin 4' cos 2STX

cos 2sx where 2sx is the tool cutting edge inclination angle at radius rx

The relationship between 2sx and 2STX, as previously developed in [13], can be described by

60 W.-C Chen f Finite Elements in Analysis and Design 26 (1997) 57-H

Next, substituting Eqs (2) and (3) into Eq (2) yields

edge can then be determined, with both the values of the tool cutting edge inclination angle on end

The profile of the curved primary cutting edge on end view is selected on the basis of the favorable distribution of the tool orthogonal rake angle rox along the primary cutting edge The procedures employed to establish the profile of the curved primary cutting edge on end view are explained as follows

In order to establish the profile of the curved primary cutting edge, the tool cutting edge

parallel to the cross section of a drill body This figure indicates that rA is the radius of any point

A on the primary cutting edge and 6A is the angle between rA and the x-axis The lines of OA and

AP correspond to the values of rA and & and 1 sTA are known The radius of another point (i.e., point B) rB is given An arc is drawn with radius rg and center at 0, which intersects line AP at point

the entire profile of the curved primary cutting edge on end view is determined by computer-aided calculation

Accurate grinding of the drills with straight (and curved) primary cutting edges can be facilitated

by first determining the cross-sectional shape of the “primary” (or cutting edge) flute surface, which

the drill’s flute surface with flank surface, in which the latter is typically ground as part of a conical surface; hence, the primary cutting edge is located on this conical surface Therefore, the cross- sectional shape of the “primary” flute surface is determined, in which both the profile of the primary

Fig 2 Determination of the curved primary cutting shape in end view from the tool cutting edge inclination angle AsTA

W.-C Chen/Finite Elements in Analysis and Design 26 (1997) 57-81

61

Fig 3 The drill point geometry of a conventional twist drill with straight primary cutting edges

cutting edge on end view and the flank surface are known The procedures employed to determine the cross section ot" the "primary" flute surface are explained as follows

Fig 3 shows the point geometry of a conventional twist drill with straight primary cutting edges The distance from any point A on the primary cutting edge to the oz-axis is r, and the angle between the line OA and x-axis is 0o in Fig 3 With the chisel edge being approximated as lying on the

xoy-plane, the distance h from any point A to the xoy-plane can be calculated from

where r is the radius for any point A on the primary cutting edge, rc the half-web thickness, and

~b the chisel edge angle

Consider the point A rotating an angle 6 and moving an axial distance h along the drill axis to A*, the point A* lying in the xoy-plane In the xoy-plane, the intersecting angle ~ between OA and its projection OA* can be calculated from

6 h

where L is the lead of the helix flute equal to nD cot/3, and D is the drill diameter

Referring to Fig 3, for a given value of 0o which can be measured for a given value of 3 which can

be calculated from Eq (6) for any value of r, the angle 0 between OA* and x-axis can be calculated from

Next, the coordinates of the point A* can be determined, with both the values of angle 0, which can be calculated from Eq (7), and the radius r in the xoy-plane known In this manner, the cross-sectional shape of a "primary" flute surface is established by computer-aided calculation

62 W.-C Chen / Finite Elements in Analysis and Design 26 (1997) 57-81

3 Two-dimensional finite element analysis for the torsional rigidity of a drill

The selection of the web thickness and the "secondary" flute shape is of vital importance in any drill design procedure in case the balance between the torsional rigidity and the efficient chip disposal capacity of the drill is attempted The chip disposal capacity is limited if the "secondary" flute shape of a thicker web drill remained unchanged In m a k i n g the selection, the curved primary cutting edge profile discussed in the previous section must be kept constant

The effects of the web thickness and the "secondary" flute shape on the torsional rigidity of the drill based u p o n a two-dimensional finite element analysis are now discussed Six-node triangular isoparametric elements are selected in this study for meshing the cross section of the drill The parameters for calculating the torsional rigidity of the drill are the drill diameter D = 12 mm, the flute helix angle/3 = 35 °, the drill point angle 2~b = 130 °, the chisel angle ~ = 140 °, and the torque

T = 6320 N nm

3.1 Effect o f the web thickness on the torsional rigidity of a drill

The torsional rigidity and the ratio of the web thickness to the drill diameter are listed in Table 1 for a series of drills of a constant diameter with the conventional flute shape but with different web thicknesses These results indicate that as the ratio of the web thickness to the drill diameter increases from 0.15 to 0.36, the torsional rigidity of the drill increases by 149 % A thicker web drill leads to a longer chisel edge length, thereby producing a larger thrust force and affecting the chip disposal capacity Therefore, the appropriate value of the ratio of the web thickness to the drill diameter is selected within the range 0.25-0.35

3.2 Effect of the "secondary" flute shape on the torsional rigidity of a drill

The profile geometry of special twist drills has been analyzed by Spur and M a s u h a [6] Their investigation was indicated that the cross-sectional profile of special twist drills has a protuberance into the "secondary" flute space, which is characterized by three parameters W, H, and L, as shown

by curve II in Fig 4 P a r a m e t e r s L and H are the (x,y) coordinates of the top point of the protuberance P a r a m e t e r W is the y-coordinate of point A on the land The smaller values of

L indicate that the protuberance is located towards the center, i.e if the values of W and H are kept constant F o r W = 1 m m and H = 3.9 mm, the ratio of the web thickness to the drill diameter

Do/D = 0.3, and the area of the cross section kept constant, the ratio of the torsional rigidity to the cross-sectional area is listed in Table 2 for various values of L This table indicates that the closer

Table 1

Effect of the ratio of the web thickness to the drill diameter on the torsional rigidity of the drill

Ratio of web thickness to drill diameter 0.15 0.20 0.24 0.28 0.32 0.36 0.40 Torsional ridigity 17.60 19.77 26.29 31.99 36.29 43.75 50.05 ( x 106 Nmm2/rad)

W.-C Chen / Finite Elements in Analysis and Design 26 (1997) 57-81 63

Torsional rigidity (× 1,36 N mmZ/rad) 36.7 34.4 33.4 32,5

(Torsional rigidity)/(cross-sectional area) (× 104 N mmE/radmm z) 59.8 56.5 55.0 53.9

Table 3

Effect of the "secondary" flute shape on the ratio of the torsional rigidity to the cross-sectional area of the drill

(Torsional rigidity)/(cross-sectional area) (× 104N mmZ/rad mm z) 44.0 55.6

the p r o t u b e r a n c e is to the center, implies the large the torsional rigidity of the drill Next, the influence of the "secondary" (i.e., non-cutting edge) flute shape defined by R a d h a k r i s h n a n et al [-14] on the torsional rigidity of the drill, two different types of "secondary" flute shapes are investigated, as shown in Fig 4 The calculated values reveal that the ratio of the torsional rigidity

t the cross-sectional area of the flute shape finally used (curve II in Fig 4) is 26.3% larger than that for the flute shape of a conventional twist drill (curve I in Fig 4), as shown in Table 3 The

a p p r o p r i a t e "secondary" flute shape (i.e., the values of W, H and L), based u p o n b o t h the torsional rigidity of the drill and the space for chip removal without congestion, are selected as W = 2.4 mm,

H - 3.0 ram, and L 4.0 mm

Consequently, the final form of the b o d y cross-sectional shape of a thick web drill with curved primary cutting edges w o u l d be established, when the cross-sectional shapes of the "primary" and

64 W.-C Chen/Finite Elements in Analysis and Design 26 (1997) 57-81

the "secondary" flute surfaces, and the web thickness of the drill as developed from the foregoing analysis were known This cross-sectional shape of the drill is considered as the reference in computer-aided automatic meshing of the overall length of a drill

4 Three-dimensional finite element analysis for drill deformations

The two-dimensional finite element m e t h o d makes a comparison of the torsional rigidities of different cross sections relatively easy However, this m e t h o d is limited in that the effects of the flute helix angle, etc., on the torsional rigidity of the drill are not considered The distribution of loads and the constraint conditions are not consistent with the actual situation either Therefore,

a three-dimensional finite element analysis for the torsional rigidity of the drill is developed Calculating the torsional rigidity of the overall length of the drill is deemed necessary since all restraint nodes are on the drill shank Furthermore, a large number of elements on a personal computer can be utilized by developing the wavefront solution m e t h o d [-15] for determining the torsional rigidity of the drill The procedures employed to obtain the torsional rigidity of the drill are outlined as follows

4.1 Modeling and automatic meshing of drill

The mathematical models of the conically g r o u n d flank surfaces and the flute surfaces of

a conventional twist drill have been developed [16, 17] In this study, the conically ground flank surfaces, the "primary" and "secondary" flute surfaces, and the chisel edge of the drill are considered as sets of discrete points because the coordinates of these various points can be easily calculated via numerical methods These coordinates are actually used to facilitate the finite element m e t h o d for solving the torsional rigidity of the drill

The computer-aided automatic mesh generation of the overall length of the drill from the drill point requires that the nodal points of an 8-node isoparametric element and their coordinates on the reference cross section, as obtained from two-dimensional finite element analysis, be first

defined Defining solid elements is quite useful Both the nodal points on the reference cross-section profile of the thick web drill with curved primary cutting edges and the reference coordinate system are shown in Fig 5 (a) The coordinates of these nodal points, as determined in the previous section, are presented in Table 4 The reference cross section was automatically meshed by dividing the width in the x-direction into 8 segments initially using a straight line to go through the drill axis and then 4 circular arcs at each side where the ratio of the radius between two adjacent arcs is 1.2 The central points for these arcs were calculated from the k n o w n coordinates of the two reference nodal points and the radiuis of the arc, e.g., the central point for the arc which pass es through points R2 and R8' in Fig 5(a) were calculated from the coordinates of points R2 and R8' and the radius of the arc (i.e., the radius of this arc is equal to 0.5 x 1.2D) These arcs were then divided into

3 equal segments Consequently, each layer consisted of 24 elements with 36 nodal points The elements and the nodal points on the first layer of the thick web drill point are shown in Fig 5(b) and the coordinates of those nodal points can be automatically calculated from the coordinates of the nodal points on the reference cross-sectional profile

W.-C Chen / Finite Elements in Analysis and Design 26 (1997) 57-81 65

Fig 5 (a) The nodal pe,ints on the reference cross-sectional profile and the reference coordinate system (b) The elements and the nodal points on the first layer The elements are given numbers 1 e, 2 °, 24 ~ and the nodal points are given numbers 1, 2, , 36

Table 4

The coordinates of the nodal points on the reference cross-sectional profile

66 W.-C Chen/Finite Elements in Analysis and Design 26 (1997) 57 81

Mesh grading was used since the drill point is more complicated than any other part of the drill The coarser the layer becomes would initiate the farther it is away from the drill point This has the effect of reducing the total number of elements, and hence, the computer time without sacrificing the accuracy of the results The finite element model for total length of a drill consisted of 45 layers with 1080 solid hexahedronal elements and 1620 nodal points The computer-aided automatic mesh generation was carried out from the drill point over the total length of the drill

4.2 Drilling load and boundary conditions

A distribution model of cutting forces acting at three nodal points of an 8-node isoparametric element on the primary cutting edge was supplied by Singh and Miller [18] However, they neglected the drill deformation produced by the thrust force and the torque acting on the chisel edge, for the chisel edge of a conventional twist drill contributes some 50-60% of the total thrust forces [19] The percentage of the thrust force and the torque acting on the primary cutting edge and the chisel edge will be different from those for a conventional twist drill because the web thickness of a thick web drill will be larger than that of the conventional twist drill Table 5 shows the percentage of the total thrust force and torque acting on the primary cutting edge, the chisel edge, and the margin for two different types of drills Furthermore, the torsional rigidity of the drill can be analyzed only by applying the static loads corresponding to the cutting loads acting on the primary cutting edges and the chisel edge Young's modulus and Poisson's ratio for the drill material must be provided for the analysis

The distribution of the thrust force and the tangential force acting on the primary cutting edge is non-uniform due to the fact that the tool orthogonal rake angle varies along the primary cutting edge The relationships between the tool orthogonal rake angle and the correcting factors of thrust force and tangential force are first determined In the present study, the correcting factors of tangential force and thrust force reported in [20] are expressed by parameters V 1 and V2, respectively The relationship between the correcting factors and the tool orthogonal rake angle is developed here to yield the following empirical equations:

Table 5

The percentage of the total thrust forces and torque acting on the primary cutting edge, the chisel edge, and the margin for two different types of drills

Drill types Cutting forces Primary cutting edge Chisel edge Margin

W.-C Chen/Finite Elements in Analysis and Design 26 (1997) 57-81 6 7

The correcting factors of the tangential force and thrust force for the ith segment on the primary cutting edge are Vt (i) and V2(i), which can be calculated for a given value of Yox at specified radius rx

The distribution models for the torque, the thrust force, and the radial force acting on the nodal points on one of the primary cutting edges and one of the chisel edges are explained as follows (1) Changing the torque to the tangential force acting on the nodal points on the primary cutting edge and the chisel edge The tool orthogonal rake angle 7ox was assumed here to be equal to 15 °,

so that the tangential force per unit length acting on the primary cutting edge of FTp could be calculated from the equation

4

i = 1 where PTp is the percentage of the total torque acting on the primary cutting edge, Rp(i) and Ip(i) are the mean radius and the length of the ith segment on the primary cutting edge, respectively The tangential force acting on the ith segment on the primary cutting edge Fxp(i) was calculated from the equation

The tangential fi~rce was assumed here to be uniformly distributed on the segment Conse- quently, the tangential force acting on the nodal points (i.e., the nodal points were given numbers 1,

5, 9, 13, to 17 in Fig 5(b)) on the primary cutting edge could be determined, with the value of FTp(i) known

Furthermore, the tangential force was assumed here to be uniformly distributed on the chisel edge The tangential force acting on the nodal points (i.e., the nodal points were given numbers 17 and 18 in Fig 5(b)) on the chisel edge FTc was calculated from the equations

where PTC is the percentage of the total torque acting on the chisel edge and Lc is the length from one nodal point to the adjacent nodal point on the chisel edge

The tangential force acting on the nodal points (i.e., the nodal points were given number 1 or 36

in Fig 5 (b)) on the margin FTM was calculated from the equation

where Pa-M is the percentage of the total torque acting on the margin

(2) Thrust force TH acting on the nodal points on the primary cutting edge and the chisel edge The thrust force per unit length acting on the primary cutting edge FTnp was calculated from the equation

4-

i = 1 where PTnP is the percentage of the total thrust force acting on the primary cutting edge

68 W.-C Chen / Finite Elements in Analysis and Design 26 (1997) 57-81

The thrust force acting on the ith segment on the primary cutting edge FTHp(i) was calculated from the equation

The thrust force was assumed here to be uniformly distributed on the segment Thus, the thrust force acting on the nodal points on the primary cutting edge was determined, with the values of FTHp(i) known

Furthermore, the thrust force was assumed here to be uniformly distributed on the chisel edge The thrust force per unit length on the chisel edge FTHC was calculated from the equation

where PTHC is the percentage of the total thrust force acting on the chisel edge

The thrust force acting on the nodal points on the chisel edge was determined, with the values of FTHC known

(3) Radial force FR actin9 on the nodal points on the primary cuttin9 edoe The radial force was

obtained from the tangential force The radial force FR was assumed here to have acted on the nodal points on the primary cutting edge The radial force acting on the ith segment of the primary cutting edge FRp(i) was calculated from the equation

The radial force acting on the nodal points on the primary cutting edge can be determined, with the values of FRp(i) known

It follows from the proceding analysis that the radial force, the thrust force and the torque must

be distributed to the nodal points on the two primary cutting edges and the chisel edge can be determined The nodal point on the surface of the drill shank were assumed here to be fixed in three-dimensional space while determining the deformation of the drill

4.3 Drill deformations

The nodal point displacements in the x, y and z directions are represented by u, v and w, respectively These displacements are transformed into a radial displacement AR, an angular displacement A~0, and an axial displacement AZ The radial displacement AR represents the variation in drill size while cutting The angular displacement Aq~ represents the torsional rigidity

of a drill or the twist while cutting The values of AR, A~0 and AZ were obtained from the following formulae [18]: