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

FINITE ELEMENTS
IN ANALYSIS
A N D DESIGN

ELSEVIER

Finite Elements in Analysis and Design 26 (1997) 57-81

Applying the finite element method to drill design based
on drill deformations
Wen-Chou Chen
Department of Mechanical Engineering, Tahua College of Technology and Commerce, Hsinchu, Taiwan

Abstract

The design of the twist drill with special cross-sectional shape using the finite element method based upon drill
deformations was presented in this paper. In order to improve the unfavorable distribution of the total orthogonal rake
angle along the primary cutting edge of a thick web drill, a thick web drill with curved primary cutting edges was devised.
The profile of the curved primary cutting edge was mathematically determined by changing the distribution of the tool
orthogonal rake angle along the primary cutting edge. A three-dimensional finite element analysis based upon the drill
deformations was applied to obtain the "secondary" flute shape and to specifythe web thickness, the helix angle, and the
flute length. The effect of these geometric parameters on drill deformations was also investigated. Experiments were
conducted to evaluate the drill's cutting performance. Experimental results indicated that the thick web drill with curved
primary cutting edges was effectivein reducing the thrust force, the torque, and the tool wear, a strength that definitely
provided a better cutting drill and longer tool life for drilling the JIS $50C heat-treated steel.
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 modern industry, the present trend in machining is aimed at

high speed, great accuracy, and automatic operation on numerous 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 orthogonal 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 more 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 by the users. Therefore,
there has been much work 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

,IB

~ ~
°B+I

i

View from B

ection
p._p.

e'

View from C

Fig. I. Relationships for the conventional twist drill with straight primary cutting edges.

Fig. I illustrates the tool orthogonal rake angle 7ox, the tool cutting edge angle XRX,the half-drill
point angle ~, the tool cutting edge inclination angle 2sx and the tool cutting edge inclination angle
on the end view 2srx at radius rx to any point p on the straight primary cutting edge. From the
geometric analysis of a drill, the tool orthogonal rake angle 7ox varies along the primary cutting
edge. The major factors that govern this variation of the tool orthogonal rake angle 7ox at any
point p are the flute helix angle fl, the tool cutting edge angle XRX, the half-drill point angle ~b,

the web thickness De, and the tool cutting edge inclination angle on end view 2svx. Tool
orthogonal rake angle 7ox at radius rx to any point p on a straight primary cutting edge can be
developed as
rx tan/3
tan ~ox R sin ~CRx

tan "~STXCOS/£RX,

(1)

where ~ is the flute helix angle, R is the drill outer radius, 2STXis 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 XRX --

sin 4' cos 2STX
,
cos 2sx

(2)

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
sin 2sx = - sin ~bsin 2srx.

(3)


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

60

Next, substituting

Eqs. (2) and (3) into Eq. (2) yields

tan yox = 2 tan /I

1’

cos24(1 + tan21ZsTX) iI2
(1 + cos’ 4 tan2 AsTxP2 _ tan A
sin 4
STX[ 1 + cos2 4 tan’ llsTx

The total orthogonal rake angle yox at radius rx to any point p on the straight primary cutting
edge can then be determined, with both the values of the tool cutting edge inclination angle on end
view IZsTX,R, /?, and 4 known.
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
inclination angles on end view AsTXare calculated from Eq. (4) utilizing Newtons method for given
values of R, 4, /3 and appropriate values of yox to be first determined. In Fig. 2, the x and y axes are
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 1sTAare 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
B, another point on the primary cutting edge. Curve AB, which is a part of the primary cutting
edge, may be represented by the tangent line at point A if point B is close to point A. In this manner,

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
is defined by Radhakrishnan et al. [14]. The primary cutting edge is formed by the intersection of
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 crosssectional 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
z

Y

(a)

(b)

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
h = (Ircos0ol - rccot ~)cot qS,

(5)

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
2n - L '

(6)

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
0 = 0o + 3.

(7)

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 making 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 upon 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 of 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. Parameters L and H are the (x,y) coordinates of the top point of the

protuberance. Parameter 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. For 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
Torsional ridigity
( x 106 Nmm2/rad)

0.15
17.60

0.20

0.24

19.77 26.29

0.28

0.32

0.36

0.40

31.99

36.29

43.75

50.05


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

63

!

Ht

m--X

O

Fig. 4. The influences of the various "secondary" flute section shapes on the torsional rigidity of the drill. Curve I is the
section shape of the conventional twist drill and Curve II of the drill used in this experiment.

Table 2
Effect of the location of proturberance on the ratio of the torsional rigidity to the cross-sectional area of the drill
L (mm)
Torsional rigidity (× 1,36 N mmZ/rad)
Cross-sectional area (ram2)
(Torsional rigidity)/(cross-sectional area) (× 104 N mmE/radmm z)

1.2

1.6

2.1

3.0

36.7
61.4
59.8

34.4
60.9
56.5

33.4
61.3
55.0

32,5
60,4
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
"Secondary" flute shape types

Curve I

Curve II

Torsional rigidity (× 106N mm//rad)
Cross-sectional area (ramz)
(Torsional rigidity)/(cross-sectional area) (× 104N mmZ/rad mm z)

26.1
59.3
44.0

34.0
61.1
55.6

the protuberance 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 Radhakrishnan 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
appropriate "secondary" flute shape (i.e., the values of W, H and L), based upon both 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 would 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 method makes a comparison of the torsional rigidities of
different cross sections relatively easy. However, this method 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 method [-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 ground 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 method 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 known 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

Y

R

l o

R6

I~ X

~

1

8

0

°- ¥

X

D
(a)

(b) viewfromD

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 1e, 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
Reference nodal points

R1

R2

R3

R4

R5

R6

R7

R8

R9

0

1.58

2.85

4.16

5.58

1.80

0.95

1.34

1.97

2.27

Coordinates

X

- 5.51

- 4.00

- 2.25

- 1.00

(mm)

Y

2.37

2.95

2.90

2.38


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:
1.2 - 0.01547ox + 1.13 x 10-47Zox,

(8a)

V2 = 1.8 - 0.06087ox + 4.24 × 10-472x.

(8b)

V1 =

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

Conventional twist drill

Thust force

40

57

3

Torque

80

8

12

Thrust force

30

70

Torque

68

20

Thick web drills

Margin

12


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

67

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

FTp = PTPT/2 ~ V1(i)Rp(i)Le(i),

(9)

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
FTp(i)

=

FTpVl(i)Lp(i), where i = 1, 2, 3, 4.

(10)

The tangential fi~rce was assumed here to be uniformly distributed on the segment. Consequently, 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
FTC(17) = 2pTcT/9Lc,

(1 la)

FTc(18) = PTcT/3Lc,

(1 lb)

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
FTM = PTM T/2D,

(12)

where Pa-Mis 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-

FTnp = pxnpTI:I/2 ~', Vz(i)Lp(i),
i=1

where PTnP is the percentage of the total thrust force acting on the primary cutting edge.

(13)

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

68

The thrust force acting on the ith segment on the primary cutting edge FTHp(i) was calculated
from the equation
FTHp(i) = FTHpV2(i)Lp(i)

where i = 1, 2, 3, 4.

(14)

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
FTHC = P T H c T H / 3 ,

(15)

where PTHCis 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

FRp(i) =

FTp(i)cot q~ where i = 1, 2, 3, 4.

(16)

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]:
AR = [(Rcos tp + u) 2 -~- (Rsin ~o -- v)2] 1/2,

(17a)

( R s i n q~ - v )
Aq~ = ~o - tan - l \ R c o s q ~ + u '

(17b)

AZ = w,

(17c)

where R is the drill radius and the value of q~ can be calculated, with the coordinates of the nodal
point known.


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

69

Both the displacements of the outermost node and the m a x i m u m stresses at the cutting edge on
the drill point are of interest as they are directly related to the drill's cutting performance. A m o n g
these stresses, the value of the shear stress is the largest and hence the isogram of m a x i m u m shear
stress is analyzed using the Surfer Access System. In order to assess the influences of the drill
geometric parameters on drill deformations (i.e., the torsional rigidity of the drill), calculations were
carried out for the following drill properties: the drill diameter D = 12 mm, the drill point angle
2~b = 130 °, the chisel edge angle ~k = 125 °, th relief angle at the periphery ct = 9 °, Young's modulus
E = 2 x 105 MPa, Poisson's ratio /~ = 0.25, the total drill length LI~ = 190 mm, the torque
T = 6320 N m m , and the thrust force T H = 1814 N.

5. Calculated results and discussion
5.1. Effect o f the web thickness on drill deformations
The various ratios of the web thickness to the drill diameter, e.g., from 0.2 to 0.45, are calculated
in this analysis to evaluate the effect of the web thickness on drill deformations. F o r the flute length
Le = 130 m m and the helix angle fl = 35 °, the values of m a x i m u m radial displacement ARMAx and
angular displacement Aq~ for these various ratios of the web thickness to the drill diameter are listed
in Table 6. This table indicates that as the ratio of the web thickness to the drill diameter increased
from 0.2 to 0.45, the angular displacement Aq~ decreased by 123% and the m a x i m u m radial
displacement ARMAx decreased by 135%. The isograms of m a x i m u m shear stress for the various

ratios of the web thickness to drill diameter are shown in Fig. 6. This figure indicates that the peak
value of the m a x i m u m shear stress decreased and the position of the m a x i m u m shear located
towards the drill periphery as the ratio of the web thickness to the drill diameter increased.
Calculated results indicated that both the torsional rigidity and the strength of a drill and the
quality of drilled holes increased on increasing the ratio of the web thickness to the drill diameter.

5.2. Effect o f theflute length on drill deformations
When the ratio of the web thickness to drill diameter Dc/D = 0.3 and the helix angle fl = 35 °, the
influences of the flute length variations on drill deformations are as listed in Table 7. This table
indicates that the flute length decreased from 240 to 80 mm, the m a x i m u m radial displacement
ARMAX decreased by 74.6% and the angular displacement A~0 decreased by 64.5%. Calculated
Table 6
Effect of the various ratios of the web thickness to drill diameter on drill deformation
Ratio of the web thickness to drill diameter

0.20

0.28

0.36

0.45

Web thickness (ram)
Maximum radial displacement ARMAx(~xn)
Angular displacement A0 (deg)

2.40
6.35
2.27

3.36
4.44
1.68

4.32
3.21
1.24

5.40
2.70
1.02


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

70

lao

(a) DJD=0.20

d,o

(b) D~/D=0.28

13.o !o

(d) De~D=0.45

(c) D~/D=0.36

Fig. 6. Isograms of maximum shear stress for the various ratios of the web thickness to drill diameter.

Table 7
Effect of the flute length variations on drill deformations
Flute length

80

Max. radial displacement ARMAx (~tm) 2.00
angular displacement A0 (deg)
0.95
Max. axial displacement AZMAx (ktm) 5.40

120

160

200

240

2.80
1.35
6.40

3.80
1.75
7.20

5.20
2.15
8.00

7.40
2.55
9.00

results indicated that the torsional rigidity of a drill and quality of drilled holes may be decreased
by increasing the flute length of a drill.
The isograms of maximum shear stress for two different flute lengths are shown in Fig. 7. This
figure indicates that the effect of the flute length on maximum shear stress was insignificant on the
drill web, but the maximum shear stress increased at the drill periphery as the flute length
increased.


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

71

(a) L s : 80mm

i z o ~~30fz°~v''-~'
~30

t30

(b)

Ls = 2 2 0 r a m

Fig. 7. Isograms of maximum shear stress for two different lengths of the drill flute.

Table 8
Effect of the helix angle variations on drill deformations
Flute length

20

25

30

35

40

45

Max. radial displacement ARMAx (gm)
angular displacement AO (deg)
Max. axial displacement AZ~,AX({am)

3.60
1.80
7.50

3.40
1.65

6.25

3.30
1.55
5.75

3.20
1.45
6.40

3.10
1.36
7.55

2.90
1.25
10.25

5.3. Effect of the helix angle on drill deformations
When the ratio of the web thickness to drill diameter Dc/D = 0.3 and the flute length
Le -- 220 mm, the influence of the helix angle variations on drill deformations is listed in Table 8.
This table indicates that the values of the helix angle increased from 20 ° to 45 °, the angular
displacement Aq~ decreased by 30.5% and the maximum radial displacement ARMAxdecreased by
19.4%. The maximum axial displacement AZMAxreached a minimum value when the helix angle
was equal to 30 °. The maximum axial displacement AZMAxincreased rapidly when the value of the
helix angle was larger than 30 °.
The isograms of maximum shear stress for the various helix angles are shown in Fig. 8. From this
figure, it was found that the values of the maximum shear stress increased with increasing helix
angle and the maximum shear stress occurred on the cutting edge near the web.

72

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

(a) Helixangle,8=25°
1,9"

(b) Helixanglefl =35°

(c) Helixanglefl =45°
Fig. 8. Isograms of maximum shear stress for various helix angles.

Calculated results indicated that the values of Ago and ARMAx decreased with increasing the helix
angle, but the value of AZMAx increased with increasing the helix angle for helix angles larger than
30 °. Therefore, it is unfavorable to choose a very large value of the helix angle. The helix angle m a y
be chosen to be 35 ° for the thick web drill in order to improve the tool orthogonal rake angle and
the chips removal capacity. In practical operation, for drilling easy-to-cut materials (e.g., aluminum
alloys), the helix angle m a y be chosen even larger.


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

73

Table 9
Effect of the various protuberance locations on drill deformations
Location of protuberance L (mm)

1.00

2.50

3.35

Max. radial displacement ARMAx(~tm)
Angular displacement Atp (deg)

3.67
1.37

3.78
1.44

3.86
1.52

5.4. Effect of the "~econdary" flute shape on drill deformations
When the ratio of the web thickness to drill diameter Dc/D = 0.3, the flute length Lf = 130 mm,
the helix angle fl = 35 °, W = 2.4 mm and H --- 3.0 mm, the effect of various protuberance locations
L on drill deformations is listed in Table 9. In this calculation, the cross-sectional area of the drill
was kept constant. From this table, it was found that the values of L decreased from 3.35 to
1.00 mm, the angu]iar displacement Atp decreased by 5% and the maximum radial displacement
ARMAx decreased by 11%. The isograms of maximum shear stress for various values of L are shown
in Fig. 9. It is observed that the closer the protuberance is located towards the center, the smaller
the drill deformations and maximum shear stress, i.e., the larger the torsional rigidity and the
strength of the drill.
Finally, in order to investigate the effect of the cross-sectional shape of a drill body on the drill
torsional rigidity, the displacements of two different types of drills are presented in Table 10. This
table indicated that the displacements A~0 and ARMAxof the conventional twist drill increased more

rapidly as the flute length becomes larger. The trend is even more obvious for the drill having
a longer flute length. In case the flute length Lf = 220 mm, the isograms of maximum shear stress
for two different types of drills are shown in Fig. 10. This figure indicated that the peak value of
maximum shear stress of the thick web drill with straight primary cutting edges is smaller than that
of conventional twist drills. In other words, the strength of a thick web drill with straight primary
cutting edges is larger than that of a conventional twist drill.

6. Experimental procedure
Three types of drills were prepared in this investigation to assess the effect of the cross-sectional
shape of a drill body on the cutting performance. The first two were conventional twist drills with
straight primary cutting edges but different web thicknesses, while the third was a thick web drill
with curved prima'~ry cutting edges. The distributions of the tool orthogonal rake angle Yox along
the primary cutting edge (as proposed in the present study) were compared with that for the
conventional twist drill, as shown in Fig. 11. The drills used in this experiment were ground and
supplied by Shanghai Tool Works, Shanghai, China. Among those drills, the chisel edges of the
thick web drills with straight (and curved) primary cutting edges need to be thinned in order to
reduce the thrust force. The web-thinning at that point (i.e., the design parameters of the split point
drill such as the notch angle fiN, the notch rake angle/?R and the splitting angle ~ were described in
[2, 3]) was ground by Taiwan Precision Mfg. Co. Ltd., Hsinchu, Taiwan. Fig. 12 shows the


74

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

(a) L=3.35mm

(b) L=2.5Omm

(c) L = 1.00ram

Fig. 9. Isograms of maximum shear stress for various protuberance locations in the "secondary" flute shape.
cross-sectional shapes of the three different types of drills. Table 11 lists the characteristics of the
thick web used in the present experiment.
The thrust force, the torque, and the drill life were taken as indices for the assessment of the drill's
cutting performance. T w o series of drilling tests were carried out. The details are given below.
6.1. Drilling tests to measure torque and thrust force
Drilling tests were carried out on a 3 H P x 4P drilling machine in which the torque and the thrust
force during drilling operations were measured with a Kistler model 9273 four-component


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

75

Table 10
Effect of the flute length on drill deformations for two different types of drills
Flute length (mm)

80
140
220

Conventional twist drill

Thick web drill with straight
primary cutting edges

ARiA x
(p.m)

A0
(deg)

ARMAx
(,am)

A0
(deg)

2.14
4.75
13.71

1.41
2.27
3.94

1.95
3.33
6.37

0.90
1.54
2.40

(a) Conventiomdtwist drill

I~°~
(b) Thick web drillwith straightprimarycutting edges

]Fig. 10. Isograms of maximum shear stress for two different types of drills.

piezoelectric dynamometer. The dynamometer had a fixture mounted on it to hold the workpieces.
The drilling torque and the thrust force signals were transmitted to Kistler model 5007 charge
amplifiers to magnify the signals, which were subsequently recorded on a KYOWA-RTP-670-A
data recorder.


76

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

30

~,

2o

I

-t0

~

/

o
o

f

-30

~
~

Thick web drill with curved prbmwy curt/rag edges
Convendmud twist drill

40

Drm

radius (nun)

Fig. 11. The proposed distribution of the tool orthogonal rake angle ~ox along the curved primary cutting edge in
comparison with the conventional twist drill.
Table 11
Characteristics of the thick web drills used
Diameter (mm)
Point angle (deg)
Chisel edge angle (deg)
Total drill length (mm)
Drill flute length (mm)
Clearance angle at periphery (deg)
Helix angle (deg)
Ratio of web thickness to drill diameter
Web thickness after splitting (mm)
Splitting angle ~ (deg)
Notch angle Br~(deg)
Notch rake angle BR(deg)

12
130
125
190
101
9
35
0.3
0.9
130
57
9

Materials

Molybdenum-tungsten high-speed steel

In this series of tests, the spindle speed was fixed at 485 rpm with a feed rate of 0.12 mm/rev. The
hole depth was kept constant at 36 mm except during the investigation of the influence of the
primary cutting edge profile on the cutting forces for pilot hole drilling.
6.2. Drilling tests f o r .measuring drill life

Drilling tests were carried out using a Y C M - V A M - 6 0 A vertical machining center. Blind holes
were drilled in steel blocks and the hole depth was kept constant at 36 mm. In this series of tests, the


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

(a)

(b)

77

(c)

Fig. 12. The cross-sectionalshapes of three differenttypesof the drills:(a) conventionaltwist drill;(b) thick web drill with
straight primary cutting edges; and (c) thick web drill with curved primary cutting edges.
feed rate and the spindle speed were fixed at 0.12 mm/rev and 485 rpm, respectively. After every two
holes were drilled, the drilling process was stopped and the drill was removed from the spindle
chuck to measure the wear widths developed on the flank surfaces. The wear on the flank was
measured using a Rilox Hi-Scope Compact-Micro Vision System Model KH-2200 at 50 x
magnification. Drill life was determined by comparing the numbers of holes drilled before the
maximum flank wear width VBMAx reached 0.3 mm.
The workpiece materials, i.e., 140 x 140 x 40 mm blocks of JIS $50C heat-treated steel, were used
to measure the drill life, the thrust force, and the torque. The workpiece materials had a hardness of
243 ,,~ 253BHN and a Charpy V-notch impact toughness of 2.2 MJ/m 2. All drilling tests were
performed dry, i.e., without any coolant.

7. Experimental re,,mlts and discussion
7.1. Experimental results o f thrust forces and torque

Table 12(a) lists the average values of the thrust force and torque for the three different types of
drills working on tile JIS $50C heat-treated steel. This table indicates that the thick web drill with
curved primary cutting edges required the least thrust force and torque among the three different
types of drills. However, the effect of the primary cutting edge profile on the thrust force and torque
is of great interest. This table also indicated that thick web drill with curved primary cutting edges
required 7.2% less thrust force and 7.9% less torque than the thick web drill with straight cutting
edges for the normal drilling of JIS $50C heat treated steel.

Pilot hole drilling was next conducted to further evaluate the influence of the primary cutting
edge profile on cutting force performance. The average values of the thrust force and the torque of
thick web drills wJith straight (and curved) primary cutting edges (when drilling the JIS $50C
heat-treated steel in pilot hole drilling) are listed in Table 12(b). This table indicated that the thick
web drill with curved primary cutting edges resulted in 19.6% less thrust force and 11% less torque
than the thick web drill with straight primary cutting edges for pilot hole drilling of JIS $50C
heat-treated steel. This difference was attributed to the different distribution of the tool orthogonal
rake angle along the primary cutting edge.


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

78

Table 12
Experimental results of the thrust force and the torque for three different types of drills (drill diameter
12 mm, spindle speed 485 rpm, feed rate 0.12 mm/rev, and pilot hole diameter 4.7 ram)
Drill types

Thrust force (N)

Torque (N mm)

(a) Normal drilling
Thick web drill with curved primary cutting edges
Thick web drill with straight primary cutting edges
Conventional twist drill

1245
1338

1848

7600
8200
8400

(b) Pilot hole drilling
Thick web drill with curved primary cutting edges
Thick web drill with straight primary cutting edges

387.5
481

6120
6890

7.2. Experimental results of drill life
The tool life of three different types for the drilling of JIS $50C heat-treated steel is shown in
Fig. 13 (a). This figure indicated that the thick web drill with curved primary cutting edges has the
longest tool life among those drills. In order to explain the reason why the thick web drill with
curved primary cutting edges has the longest tool life than the other two types of drills with straight
primary cutting edges but different web thicknesses, the widths of flank wear versus the numbers of
holes drilled are investigated by drilling JIS $50C heat-treated steel. In Fig. 13(b), it was found that
the conventional twist drill wears at the fastest rate and the thick web drill with curved primary
cutting edges wears at the slowest rate for the drilling of JIS $50C heat-treated steel. Additionally,
the conventional twist drills were blunted after drilling 24 holes. The major reason for this could be
accounted for by the fact that the conventional twist drills appeared to be inadequate to drill JIS
$50C heat-treated steel.
From the test results, therefore, we can infer that the thick web drill with curved primary cutting
edges has a more reasonable distribution of the tool orthogonal rake angle along the primary

cutting edge. This drill decreases the chip deformation during the drilling operation, reduces the
friction between the upward moving chip and the drill, and helps to diminish the thrust force, the
torque, and the tool wear.

8. Conclusions
The present study has resulted in the development of a three-dimensional finite element method
based upon the drill deformations for the design of the twist drill with special cross-sectional shape.
The usefulness of this method is demonstrated by solving the drill torsional rigidity involving
variation of a range of design parameters. From the calculated and test results of this investigation,
the following conclusions can be drawn:
1. From the calculated results, it was shown that the web thickness is a predominant factor
affecting the torsional rigidity of a drill. Both the torsional rigidity and the strength of the drill
increased with increasing the ratio of the web thickness to drill diameter. The appropriate value for


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

79

140

1120
IITYPE-1

Conventional twist drm
Thick web drill with straight primary cutting edges
Thlckweb drill with curved primary cut~ag edges

• TYPE-2
100

q=

d

56

~

36

4o
24

20

TYP~I

(a)

T~E-2

T~E-3

D~types

0,5

0.45

~

0.4

~

Conventional twist drill
Thick web drm with straight prhnary cutting edges
Thickweb drill with curved primary cutting edges

0.35

i
i

qll.3
0,,25
,11.2
0.15
0.1

0.05
0
0

(b)

4

8

12

1$

20

24

28

32

36

40

44

48

52

M

No. of holes drilled

Fig. 13. (a) Tool life and (b) the width of flank wear versus the numbers of holes drilled for three different types of drills.


80

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

the ratio of the web thickness to the drill diameter is within 0.25-0.35. However, the chisel edge of
a thick web drill needs to be thinned in order to reduce the thrust force.
2. The torsional rigidity of a drill and quality of drilled holes may be decreased by increasing the
flute length of a drill.
3. The torsional rigidity of a drill increased with increasing helix angle, but the value of the axial
displacement AZraAX increased with increasing the helix angle for helix angles larger than 30 °.
Therefore, it is unfavorable to choose too large a value for the helix angle. In the present study, the
helix angle was chosen to be 35 ° for the thick web drill in order to improve the distribution of the
tool orthogonal rake angle along the primary cutting edge and the chips removal capacity.
4. The effect of the protuberance location on the torsional rigidity of the drill is insignificant
compared to the effects of the ratio of the web thickness to the drill diameter, the flute length, and
the helix angle on the torsional rigidity of the drill.
5. It was evident from the test results that in drilling the JIS $50C heat-treated steel the drill with
curved primary cutting edges resulted in 7% less thrust force and 8.5% less torque in normal
drilling. It produced 19.4% less thrust force and 11% less torque in drilling with a pilot hole, but
yielded 1.6 times longer tool life than did the drill with straight primary cutting edges. This was
because the former improves the distribution of the tool orthogonal rake angles along the primary
cutting edge. This drill reduces the chip deformation and the friction between the upward moving
chips and the tool during a drilling operation, as well as diminishes the torque, the thrust force, and
the tool wear. Additionally, the conventional twist drills appeared to be inadequate to drill JIS
$50C heat-treated steel in this study.

References
[1] H. Ernst and W.A. Haggerty, "The spiral point drill - a new concept in drill point geometry", Trans. Soc. Mech.
Eng. 80, pp. 1059-1072, 1958.
[2] T. Radhakrishnan, S.M. Wu and C. Lin, "A mathematical model for split point drill flanks", Trans. A S M E J. Eng.
lnd 105, pp. 137-142, 1983.

[3] Kuang-Hua Fuh, Wen-Chou Chen and Chih-Fu Wu, "A force model for split point drill", Proc. 8th Nat. Conf. on
Mech. Eng. CSME, Taiwan, ROC, pp. 951-958, 1991.
[4] S.M. Wu and J.M. Shen, "Mathematical model for multifacet drills", Trans. A S M E J. Eng. Ind. 105, pp. 173-182,
1983.
[5] S.M. Wu and L.H. Chen, "Further investigation of multifacet drills - mathematical model, methods of grinding, and
computer plotting', Trans. A S M E J. Eng. Ind. 106, pp. 313-324, 1984.
[6] G. Spur and J.R. Masuha, "Drilling with twist drills of different cross section profile", Ann. CIRP 30, pp. 31-35, 1981.
[7] Wan Guang-min and Suen Tung-min, "Study on high torsional rigidity twist drill with special cross section shape",
4th CSMCU, Nanjing, China, pp. 305-309, 1991 (in Chinese).
[8] Yang Jian and Jin Chih-yuan, "Study on the torsional rigidity of the conventional twist drills', 4th CSMCU,
Nanjing, China, pp. 332-336, 1991 (in Chinese).
[9] C.J. Oxford Jr., "A review of some recent development in the design and application of the twist drill", Proc. 8th Int.
M T D R Conf., pp. 854-861, 1967.
[10] G. Lorenz, "Helix angle and drill performance", Ann. CIRP 28, pp. 83-86, 1979.
[11] R.H. Thornley, A.B. E1Wahab and J.D. Maiden, "Some aspects of twist drills design", Int. J. Mach. Tools Manufac.
27, pp. 383-397, 1987.
[12] D.F. Galloway, "Some experiments on the influence of various factors on drill performance", Trans. Amer. Soc.
Mech. Eng. 79, pp. 191-213, 1957.


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

81

1-13] Editorial Board of Nanjing Engineering Institute and Wuxiao Light Industry Institute, Principal of Metal Cutting,
Fujian Science artd Technology Press, Ch. 4, p. 200, 1984 (in Chinese).
1-14] T. Radhakrishnan, R.K. Kawlra and S.M. Wu, "A mathematical model of the grinding wheel profile required for
a specific twist drill flute", Int. J. Mach. Tool Des. Res. 22, pp. 239-251, 1982.
1-15] B.M. Irons, "A frontal solution problem for finite element analysis", Int. J. Numer. Methods Eng. 2, pp. 5-32, 1970.
1,16] W.D. Tsai and S.M Wu, "Computer analysis of drill point geometry", Int. J. Mach. Tool. Des. Res. 19, pp. 95-108,

1979.
1-17] W.D. Tsai and S.M. Wu, "A mathematical model for drill point design and grinding", Trans. ASME J. Eng. Ind.
101, pp. 333-340, 1979.
[18] U.P. Singh and P.P. Miller, "Finite element analysis of drill point geometry", Ann. CIRP 37, p. 69-72, 1988.
[19] M.C. Shaw and C.J. Oxford Jr., "On the driling of metals - II. The torque and thrust in drilling", Trans. Amer. Soc.
Mech. Eng. 79, pp. 139-148, 1957.
[20] Hsing Ay and Shih-kang Hsiao, Metal Cutting Handbook, Mechanical Industry Press, Beijing, China, p. 166, 1984
(in Chinese).