Int. J. Mach. Tool. Des. Res. Vol. 19, pp. 95 108. Pergamon Press Ltd. 1979. Printed in Great Britain.

COMPUTER ANALYS IS OF DRILL POINT GEOMETRY

W. D. TSAI* and S. M. Wu*

(Received 7 December 1978)

Abstract The difficulties in drill point geometry analysis are aggravated by the process of drill point design which now includes the hyperboloidal drill and ellipsoidal drill. This paper provides a tool based upon mathematical models for drill point and drill flute geometries which enables drill point geometry to be analyzed accurately and conveniently by computer. The explicit equations for the drill flank contour and the drill angles are derived and shown to be directly calculated from the grinding parameters. This methodology also enables one to calculate the shapes of cutting edges and chisel edges and their relationships with the grinding parameters. A flute contour is designed for the ellipsoidal drill to improve the excessive inclination angles of cutting edges near outside comers. The chisel edge shape is defined so that it can be measured quantitatively as well as qualitatively.

INTRODUCTION

Tim PERFORMANCE of the twist drill is considerably affected by the drill point geometry, yet the accurate measurement of the drill point geometry is not easy because of the complexity of the geometry. In the conventional method of drill point measurement, the clearance angle is measured empirically or is calculated from the measured values of the point angle and the chisel edge angle. This approach lacks accuracy and is not sufficient to capture the more complex geometries of drill points other than the conventional conical twist drill. In an effort to improve twist drill performance, the ellipsoidal and hyperboloidal drill point designs have been developed [7]. For these drill designs, the included angles of the cutting edges as well as the clearance angle are functions of the radial distance and hence are more difficult to measure accurately. Because of these innovations in drill point design, the drill angles become insufficient to specify the drill point geometry. The drill flank contour and the shapes of the cutting edge and the chisel edge are also important design features which have significant effects on the drill's performance. Hence, a mathematical tool to accurately measure complex drill geometries and reveal how these geometries may be generated through their relationship to the grinding parameters is needed.

The purpose of this paper is to provide an analytical tool to calculate the drill point geometry based upon the mathematical drill point model. The mathematical model facilitates the use of digital computer and hence makes the analysis easy and accurate. The relationships between the drill point geometry and the drill point grinding parameters are developed. Explicit equations for the included angle of the cutting edges, the chisel edge angle and the clearance angle are derived so that these angles can be calculated directly from the grinding parameters by computer. The shape of the chisel edge is defined quantitatively, so that it may be easily generated graphically and precisely measured. For the ellipsoidal drill, the non-straight cutting edge projection on the plane perpendicular to the drill axis results in an excessive inclination angle near periphery. A flute contour design is derived to produce straight cutting edge on the ellipsoidal drill.

In the first section of the paper, the equation of the drill flank contour is derived and the effects of the grinding parameters on the drill flank contour are discussed. In section 2, the drill flank contour and the flute contour are used to generate and compare the shapes of the cutting edges for the conical, hyperboloidal and ellipsoidal drills. The equations of the cutting edges and the flute shape design for the ellipsoidal drill are also given in section 2. In section 3, the explicit equations for the included angle of cutting edges, the chisel edge angle and the clearance angle are derived. In section 4, the influence of the grinding parameters on the shape of the chisel edge and the measurement of the shape of chisel edge are discussed.

* Department of Mechanical Engineering, University of Wisconsin, Madison, WI 53706, U.S.A.

95

96 W. D. TSA! and S. M. Wu

1. DR ILL FLANK

The drill point geometry is determined by the drill flank configuration, the flute shape and the helix angle. Since the flute shape and the helix angle are designed by the manufacturer and not alterable, the drill flank configuration becomes the principal factor in determining a variable drill point geometry and hence the focus of drill modification by drill point grinding to improve drilling performance.

In a previous paper, [7], the mathematical model of the drill point was derived and found to be

~[(xcscp+zs incp)+( a2- t~a2d2 c 2 -- $2)1/212

1 1 (z = +~(y-S)2+ -cz cos~b-xs inq~+d) z 1. (1)

Equation (1) describes the drill flank configuration as a quadratic surface in the three dimensional space represented by the Cartesian coordinates (x, y, z). a, c, d, ~b, and S are the grinding parameters : a and c determine the shape of the quadratic surface and 4) determines the direction of the drill axis with respect to the axis of the quadratic surface. 6 equals + 1 for the ellipsoidal drill and - 1 for the conical drill and the hyperboloidal drill, d and S determine the location of the drill point on the grinding surface.

1.1 Drill flank contour The drill flank contour is a set of elliptical curves obtained from the intersection of cutting

planes orthogonal to the drill axis and the drill flank represented by the mathematical model (1). By considering z = z~ to be the elevation of the ith orthogonal cutting plane, equation (1) becomes a quadratic equation in terms of x and y

Ax 2 + By 2 + Cx + Dy + E = 0, (2)

where a 2

A - cosZ~b + 6 ~- sin2th, (3a)

B = 1, (3b)

sin cos + (a = - c2 cos q~ - 26 ~- (z~ cos ~b + d)sin ~b, (3c) D = -2S, (3d)

E=z2s inZcp+(a 2-6a2dEc 2 -S2)+2( a2 -6a2d2-S2)z , a

a 2 + S z -a 2 +6~-(z icos~b+d) . (3e)

A, B, and D are independent of the elevation zl, while C and E are functions ofzi. Equation (2) may be conveniently rewritten in the familiar form for the equation of an ellipse :

(X -Xo) z (y -yo) 2 + - 1, (4)

k 2 k2 2

where (xo, Yo) is the center of the ellipse and

x o = C/2A,

Yo = D/2B = D/2 = S,

k 1 = {[(C2/4A) + (D2/4) -- E]/A} 1/2,

k2 = [(CZ/4A) + (D2/4) - E] '/2.

(5a)

(5b)

(5c)

(Sd)

Computer Analysis of Drill Point Geometry 97

~SE

FIG. 1. Drill flank contour ellipses.

While Y0 is independent of zi, x0 is linearly related to zi, hence the center of the contour line shifts along a straight line on the plane y = S and the shape of the contour line changes accordingly as the elevation level z~ changes. If the directional angle of the drill ~b is zero, Xo becomes independent of z~, i.e.

T = - c2 - (6) and the contour lines become concentric circles. Equation (4) is graphically illustrated in Fig. 1. As the elevation becomes smaller, the contour ellipse becomes larger and the center of the ellipse shifts toward the + x direction.

1.2 Effects of grinding parameters on drill flank contour In drill point grinding, the grinding parameters determine the relative motion between the

drill and the grinding wheel surface and hence determine the drill flank configuration to be generated.

a. Effect of a and c. As the ratio a/c increases, the space between the neighboring contour lines of the drill flank increases, the drill flank is less steep and the cutting edges have a smaller included angle. If a is larger, the curvature of the contour line becomes smaller.

b. Effect ofc~. An increase in the parameter ~b elongates the contour ellipse which in effect increases the chisel edge angle. When ~b decreases and approaches zero, the contour ellipse reduces to a circle.

c. Effect ofd. An increase in the magnitude of parameter d increases the size of the contour ellipse passing through the drill point center and in effect decreases the drill clearance angle and the chisel edge angle.

d. Effect of S. The parameter S is the distance between the drill axis and the center of the elliptic contour. This distance provides appropriate clearance angle on the drill flank. The clearance angle is large ifS is large. As S approaches zero, the clearance angle also approaches zero. S is positive, because a negative value of S will result in negative value of clearance angle.

e. Effect of types of drills. Since the conical, the hyperboloidal, and the ellipsoidal drills can be represented by the same quadratic mathematical model, their drill flanks are influenced by the grinding parameters in the same way. The major difference among the drill flank contours of these three types of drills is the space between the neighboring contour lines. If the difference of elevation between the neighboring contour lines is constant, the contour lines are evenly spaced for the conical drill. For the hyperboloidal drill the space between two neighboring contour lines is smaller near the drill point center and increases toward the periphery; while for the ellipsoidal drill, the space is large near the drill pointer center and decreases toward the periphery.

98 W.D. TSA1 and S. M. Wu

2. CUTT ING EDGE AND THE FLUTE SHAPE DES IGN FOR ELL IPSOIDAL DRILL WITH STRAIGHT CUTT ING EDGE

The cutting edge is formed by the intersection of the drill flank and the drill flute. The drill flank is represented by the quadratic drill point model, equation (1); while the drill flute is represented by the mathematical model (Appendix)

2 = sin-l(W/2r) + ~/r 2 - (W/2) 2 tan ho cot p + - - tan ho. (7)

ro

By superimposing contours for each of these surfaces at various levels of elevation (zi) their intersection can be easily realized and hence the shape of the cutting edges defined. This is shown graphically in Figs. 2(a)-2(c) for the conical, hyperboloidal, and ellipsoidal drills. The flute shape and drill flank contours of the same elevation intersect at the points which are connected by the dashed lines to form the cutting edges.

2.1 Determination of the cutting edges for different types of drills It can be seen from Figs. 2(a) and 2(b) that the cutting edges of the conical drill and the

hyperboloidal drill are straight, while the cutting edges of the ellipsoidal drill as shown in Fig. 2(c) are not straight. Figures 2(a)-2(c) also show the cross-sectional view of the drill point formed by the drill flank contour and the flute contour if the drill point is cut by the orthogonal cutting plane z = zi. The cross-sections of the drill point are mathematically obtained from equations (1) and (7) and the cutting planes z = z i, i = 1, 2 . . . . . Since the cutting edge is formed by the drill flute and the drill flank, the shape of the cutting edge is dependent on the web thickness W, the drill radius, ro the helix angle ho, the manufacturer- designed point angle and the grinding parameters a, c, d, q~, and S. If the drill point angle actually ground (2Pat0 given by

2pact = 2(0 + ~b), (8)

is equal to the manufacturer-designed point angle, the projection of the straight cutting edge on the plane perpendicular to the z axis can be expressed by

q; = sin- l(W/2r). (9)

Otherwise it can be approximately expressed by

~k = sin-l(W/2r) + x / r 2 - (W/2) 2 tan h0(- cot ]9ac t + cot p)/ro, (10)

which is concave or convex depending on Pact being greater than or smaller than p. Equation (10) is obtained from equation (7) by setting z = 0 and adding the term

- [r E - (W/2)2] 1/2 tan h 0 cot Pact.

By setting z = 0, equation (7) becomes a flute contour. The term added is the angle between the flute contour and the projection of the cutting edge on a plane perpendicular to the z axis and therefore transforms the former into the latter.

2.2 Flute shape design for the ellipsoidal drill The ellipsoidal drill has convex cutting edges and large inclination angle near the

periphery which tends to increase the specific cutting energy and the interface temperature. An ellipsoidal drill will produce straight cutting edges as viewed from the drill point and small inclination angles near periphery provided that the flute contour is designed as given by (Appendix)

= sin-l(W/2r) + x / r 2 - - (W/2) 2 tan ho cot[A(r)], (11)

where A(r) is the angle between the drill axis and the line connecting the drill point center and the point on the cutting edge at radial distance r, i.e.

A(r) tan-1 [- - x /~ -- $2 1

Computer Analysis of Drill Point Geometry

(8.) CONICAL DRILL

0 =55 :24 d = - .55 in(--I.40o~) S = 19'. (.48cm)

99

5 4

I 2

- 2 \

(b) HYPERBOLOIDAL DRILL 8 = .21 ~. 053 cm)

c = .22i( .56 ~m)

@=21 d = O~(Ocm) B = I0 ~. (.25 ~)

( ) ELLIPSQIDAL DRILL

a = .75 ,. (h91 c,-.-, ) c = 1.68~n (4.27cm) =34 d = 1.19+~, (3.02~m) S = .15,~ ( .58 ~m)

FIG. 2. Computer analysis of drill point geometry. (a) Conical drill. (b) Hyperboloidal drill. (c) Ellipsoidal drill.

tan,[ S2] f (x ,S ) ' (12)

= tan- - $2

f(x, y) is the elevation of the drill flank z at the horizontal position (x, y) and can be derived from equation (1) by solving the equation for z in terms of x and y.

If the values of the grinding parameters of the ellipsoidal drill are given, A(r) can be calculated using equations (1) and (12) for any value of r and then the flute shape can be obtained from equation (11) such that the ellipsoidal drill point ground using the given grinding parameters will have straight cutting edges. An example of the flute contour obtained this way for the ellipsoidal drill with the included angle of the cutting edges equal to 118 near the drill point center and 92 near the periphery is compared with the conventional

100 W.D. TSAI and S. M. Wu

- - FLUTE CONTOUR FOR ELLIPSOIDAL

DRILL.

. . . . . CONVENTIONAL FLUTE CONTOUR

(a t : l iB )

. . . . . . . CONVENTIONAL FLUTE LONTO I R

(2.?= 92 )

FIG. 3. A flute contour design for ellipsoidal drill as compared with the conventional flute contour.

flute contours for the conical twist drills with point angles 118 and 92 respectively as shown in Fig. 3. The flute contour for the ellipsoidal drill has larger curvature and lies between the two conventional flute contours.

3. DRILL ANGLES

Conventional methods use inspection apparatus to measure drill angles. With the mathematical model, drill angles can be derived and explicitly expressed in terms of the grinding parameters.

The rake angle, the helix angle and the inclination angle are mainly determined in the flute shape design and are affected by grinding to a lesser degree. The included angle of the cutting edges, the chisel edge angle and the clearance angle are determined by drill point grinding. The relationships between the drill angles and the grinding parameters are derived as follows.

3.1 Included angle of cutting edges The included angle of the cutting edges is the angle between the two tangent lines to the two

cutting edges of the drill as shown in Fig. 4. The slopes of the tangent lines intersecting with

Z _/INCLUDED ANGLE 2(1)p

@Z ~-)

3,Z)

AT p _ elZ - eX

Y=O

I - J ~Z Tr 2(1)p = 2[TAN (~- - ) - ~--1

FIG. 4. Included angle of cutting edges.

Computer Analysis of Drill Point Geometry 101

the cutting edges at radial distance r are given by ___m"

_ _ \c~x/ix:, y=O

: +

-\ ~3x /i;,=, y=O

Hence, the point angle or the included angle of the cutting edges 2q~ is given by

2q)~ = 2 tan- 1 (m) - ~,

= 2tan- i (c3 f (x 'Y )~ _ \ c~x /Ix=, re, (14)

.v=0

which is a function of r. For the conical drill

tan -t (Oftx' Y)~ =0 + q~ (15) \ Gx /1~:,

y=O

and hence,

2~p = 2(0 + oh).

The hyperboloidal drill resembles the conical drill near the periphery, while the ellipsoidal drill resembles the conical drill near the drill point center. Therefore, 2~p approaches 2(0 + ~b) near the periphery for the hyperboloidal drill and near the drill point center for the ellipsoidal drill.

3.2 Chisel edye angle

The chisel edge is obtained from the tangent line to the contour ellipses passing through the point (x ,y,z) = (0,0,0) as shown in Fig. 5. The chisel edge angle is given by

~=~-F

=r~- tan- 1 (d~-xY) , (16) (x,y,z)=(O,O,O)

dy where (dx) ~x.y.:) =~o.0.m is the slope of the tangent line.

FLANK CONTOUR ELLIPSE / ~ TANGENT /

r

- I ~ /K | I '~ , CHISEL E"I~E | co..,.oEo E X" , q ,,, /

FK;. 5. Derivation of chisel edge angle.

"ql I)R I L) ] I1

102 W. D. TSA! and S. M. Wu

dX R

I

TANGENT ~/

CLEARANCE ANGLE

FIG. 6. Clearance angle.

Since the contour ellipses are given by equation (2), from which we have

dxx (~,y,z)=~o,o,o) ~- L2By + D (~,y,z)=(o,o,o) C ,

- c2 - cos4~-~ds in~b

= (17) S

Substituting equation (17) into equation (16), one can obtain the chisel edge angle

(a 2 6 a2 d 2 S2) 1/2 a 2 - - C2 - - COS ~b - 6 ~- d s in ~b

= rt - tan -1 S (18)

For example consider the one-inch-diameter drill with grinding parameters given by

a = 0.7 in. (1.78 cm),

c = 1.56 in. (3.96 cm),

d = 1.11 in. (2.82cm),

q5 = 24 ,

S = 0.18 in. (0.46 cm).

The drill is ellipsoidal (6 = + 1), as indicated by the magnitudes of a and c which are larger than the drill radius and by the positive value of d. Substituting these parameters into equation (18) one can obtain the chisel edge angle ~ = 125 . As another example, consider the same size drill with grinding parameters given by

a = 0.21 in. (0.53 cm),

c = 0.22 in. (0.56 cm),

d = -0 .01 in. ( -0 .03 cm),

Computer Analysis of Drill Point Geometry 103

4= 12 ,

S = 0.11 in. (0.28 cm).

The drill is hyperboloidal (6 = - 1) as indicated by the values of a and c which are smaller than the radius of the drill and by the small magnitude of d. Substituting these parameters into equation (18), one can obtain the chisel edge angle ~ = 127 . The computer program based upon equation (18) can be used to calculate the chisel edge angles of different types of drills.

3.3 Clearance angle The clearance angle at a point P on the cutting edge is given by

n l fdY. ' ] clearance angle = ~ - tan- \d-~RJ'

as shown in Fig. 6, where

dXn dYR

1 (dXn'~ = tan- \~ j , (19)

-y*

02 x* cos(@p - ~b) + ~-z* sin(@p - ~b)

(20)

Equation (20) is derived from the drill point model, equation (1) by transforming the (x, y, z) coordinates into the (XR, YR, ZR) coordinates

[il rco x ]l 21 [sin~bp 0 cos~bp] Zn

and then taking the derivative. (x , y , z ) are the coordinates of point P related with (x, y, z) by the transformation

Ii I ICo~)O-si~qb[Ix*-a2-t~a2d2-S21/2( c 2 ) = 1 y* + S (22)

[sin~b 0 cos~bJ z* -d

If the radial distance of point P from the drill axis is r, the (x, y) coordinates of point P will be

(x, y) = (x/r E - (W/2) 2, W/Z). (23)

20

15

CD Z <

10

W z ~5 LP

0

ro= .5 J,,(L27,m)

~ 30 cr~ 0 -35 m =24 ~ d =-.55 in (-I.40 cm ) S =.19 m ( .4~ c= )

i I I I

0 .I .2 .3 .4 .5 r(,~)

FK}. 7. Variation of clearance angle across cutting edge of conical twist drill.

104 W.D. TSAI and S. M. Wu

The corresponding z coordinate can be calculated from equation (1). (x*, y*, z*) can then be obtained from (x, y, z) using equation (22) to calculate the clearance angle by equation (19). As an example consider the conical drill with r 0 = 0.5 in. (1.27 cm), W = 0.12 in. (0.30 cm), 0 = 35 , ~b = 24 , d = -0.55 in. ( -1 .40 cm) and S = 0.19 in. (0.48 cm). By using equations (19), (20), (22) and (23) for various values of r, the clearance angles calculated are plotted in Fig. 7 as a function of r.

The other drill angles are mainly dependent on the manufacturer's design. The grinding parameters can also affect the inclination angle, the side rake angle and the back rake angle in some degree. But the effects are insignificant in the sense of drill's performance as compared with the effects of the grinding parameters on the included angle of the cutting edges, the chisel edge angle and the clearance angle.

4. GENERATION OF THE S-SHAPED CHISEL EDGE

The chisel edge which divides the drill flanks and connects the cutting edges of the drill is generated by the intersection of the two symmetrical quadratic grinding surfaces for the two flanks. The intersection of the two symmetrical quadratic grinding surfaces is an S-shaped curve and the chisel edge is the central portion of this curve between the points of intersection of this curve and the cutting edges. The length of the chisel edge depends upon its proportion to this S-shaped curve. For the hyperboloidal drill, this proportion is large and hence the hyperboloidal drill has an S-shaped chisel edge. For the conical drill and the ellipsoidal drill, this proportion is small and therefore the conical drill and the ellipsoidal drill appear to have straight chisel edges.

4.1 Different shapes of chisel edges In drill point grinding, the drill flank generated by the quadratic surface has elliptic

contour (recall equation 14 and Fig. 1). Consider the conical drill with the grinding parameters given by:

0= 35 ,

= 24 ,

d = -0.55 in. ( - 1.40 cm),

S = 0.19 in. (0.48 cm).

The elliptical drill flank contour for the conical drill is shown in Fig. 8. The intersections of the contour lines with the same elevations are shown by dots connected by the S-shaped curve. The chisel edge is a central portion of the S-shaped curve between the cutting edges AA'. Since this portion is small compared with the size of the S-shaped curve, it can be approximated by a straight line.

- SI-F~D INTERSEGTION 111(3 SU I~rz~CES

TTING E

DRILL FLANK C

FIG. 8. Computer analysis of drill flank contour and chisel edge.

Computer Analysis of Drill Point Geometry

~QHISEL

FIG. 9. Computer analysis of the chisel edge of the hyperboloidal drill.

105

The drill flank contour and the chisel edge of the hyperboloidal drill with the grinding parameters

a = 0.21 in. (0.53 cm),

c = 0.22 in. (0.56 cm),

~= 12 ,

d = -0.01 in. (-0.03 cm),

S = 0.11 in. (0.28 cm),

can be obtained in the same way, and the central part is amplified and shown in Fig. 9. The chisel edge connecting the intersections of the contour lines of the same levels is S-shaped. For the hyperboloidal drill, the chisel edge has a large proportion to the S-shaped curve generated by the intersection of the two symmetric grinding surfaces not because the chisel edge is longer, but because the S-shaped intersection is shorter for the hyperboloidal drill. The S-shaped intersection can be reduced by decreasing the magnitude of d or 0 for the conical or hyperboloidal drill and by increasing the magnitude ofd for the ellipsoidal drill or by decreasing parameter a for all types of drills. The curvature of the chisel edge can be improved by adjusting these parameters. The large curvature of the chisel edge of the hyperboloidal drill is due to d ~ 0. The shape of the chisel for the ellipsoidal drill is similar to that for the conical drill since the ellipsoidal drill resembles the conical drill near the drill point center.

4.2 Characterization and design of the shape of the conical edge Strictly speaking, all drills produced by the model equation (1) have S-shaped chisel edges.

However, three important questions remain: (1) how can we precisely measure the shape of

F~6. 10. Measurement of shape of chisel edge.

106 W.D. TSAI and S. M. Wu

the chisel edge, (2) what is the optimum shape of chisel edge and (3) how can we produce the desirable shape of chisel edge? To answer these three questions, a measure of the shape of the chisel edge is proposed and the drill point model ]-equation (1)] is used for the analysis.

The shape of the chisel edge can be measured by the ratio (:

AB - CD' (24)

where AB is the length of the chisel edge and CD is the length of the straight edge tangent to the chisel edge and passing the drill point center 0 as shown in Fig. 10. ( value is one if the chisel edge is straight and greater than one otherwise.

If the value of ( is large, the cutting edges of the drill are connected by a smoothly curved chisel edge which tends to reduce the thrust force, but the longer chisel edge means shorter cutting edges which tend to reduce the effectiveness of cutting. On the other hand if the value is too small, the cutting edge and the chisel edge form a sharper corner but provide longer cutting edges which tend to increase the effectiveness of cutting. A compromise between the length of the chisel edge and the length of the cutting edge can be obtained to improve the drill's performance.

The drill point grinding parameters can be adjusted to produce the desirable value oft. As a increases or c decreases the quadratic grinding surfaces are elongated to make the S-shaped intersection shorter and hence ( becomes larger. As d decreases, the two symmetric grinding surfaces are flatter to make a longer S-shaped intersection of which the chisel edge becomes a relatively smaller portion and hence ( becomes smaller. As ~b or S is increased, both AB and CD are increased, while their ratio ( is not significantly changed. The effects of the grinding parameters on these measures (, AB and CD are summarized in Table 1. For the chisel edge shown in Fig. 8, ( = 1 and for the chisel edge shown in Fig. 9, the ( value computed is 1.35.

CONCLUSION

The analysis, design, and generation through grinding of twist drill point geometries is encumbered by the need for more complex geometries such as the ellipsoidal and hyperboloidal configurations and the absence of a comprehensive mathematical model to describe them. A mathematical model has recently been proposed to characterize the twist drill point geometry for conical, hyperboioidal and ellipsoidal configurations. In this paper this general model has been used to more accurately and precisely describe the twist drill point geometry and demonstrate how this geometry can be designed and generated via the grinding parameters. In particular,

(1) The drill flank contour equation has been obtained from the drill point mathematical model and the relationship of the geometrical measures of the drill flank to the grinding parameters has been discussed.

(2) The drill flank contours and a model for the drill flute contours have been employed to develop a mathematical model for the cutting edges. The conical, hyperboloidal and

TABLE I. EFFECTS OF GRINDING PARAMETERS ON AB, CD AND

Increment of gr inding Effect on Effect on Effect on

parameter AB CD (

Aa increase decrease increase Ac decrease increase decrease Ad decrease decrease decrease

no A~b increase increase significant

effect

no AS increase increase significant

effect

Computer Analysis of Drill Point Geometry 107

el l ipsoidal geometr ies are shown to have different shapes for their cutt ing edges. A flute contour for the el l ipsoidal drill is designed which produces a straight cutt ing edge, as viewed from the dril l point. The purpose is to reduce the excessive inc l inat ion angle near the outs ide corner of the cutt ing edge.

(3) Explicit equat ions for the inc luded angle of cutt ing edges, the chisel edge angle and the c learance edge are derived. By using these equat ions the dril l angles of the conical drill, the hyperbo lo ida l drill and the el l ipsoidal dril l can be calculated directly from the gr ind ing parameters. The c learance angle for all types of dril ls and the inc luded angle of cutt ing edges for the hyperbo lo ida l dril l and the el l ipsoidal drill are variables dependent on the radial distance. The mathemat ica l equat ions can be used to calculate these angles at any radial distance.

(41 The chisel edge shape is def ined quant i tat ive ly so that it can be measured and specified by a numer ica l value to facil itate the design and analysis by computer . The shape of the chisel edge is determined by the parameters 0, a and d. For the hyperbo lo ida l dril l d ~ 0, hence the chisel edge is S-shaped.

Acknowledgements - The authors are indebted to Professor R. E. DeVor for his comments and help in preparing the manuscript. Use of the facilities of the Engineering Computing Laboratory at the University of Wisconsin, Madison is gratefully acknowledged.

REFERENCES [1] S. FUJII, M. F. DEVRIES and S. M. Wu, J. Eng. Industry, Trans. ASME, B 92(3), 647 (1970). [2] S. FUJU, M. F. DEVRIES and S. M. Wu, J. Eng. Industry, Trans. ASME, B 92(3), 657 (1970). ['3] S. FUJI1, M. F. DEVRIES and S. M. Wu, J. Eng. Industry, Trans. ASME, B 93(4), 1093 (1971). ['4] S. FUJU, M. F. DEVRIES and S. M. WE, J. Eng. Industry, Trans. ASME, B 94(4) (1972). [5] D. F. GALLOWAY, Trans. ASME 79, 192 (1957). ['6] W. D. TSAI, Ph.D. Thesis, University of Wisconsin, Madison (1977). [7] W. D. TSAI and S. M. Wu, Paper No. 78-WA/PROD-35. ASME.

APPENDIX MATHEMATICAL MODEL OF THE DRILL FLUTE SHAPE

As shown in Fig. A1, the flute shape can be represented by the polar coordinates (r,~k). At point P, the radial distance is r and the angle ff is given by

~b = ~t + fl, (A1) where

W = sin- 1 __ (A2)

2r and

Hence

fl = [x/r 2 - (W/2)2 /ro]tan h o cot p. (A3t

w ~k = sin -1 ~r + [-x/r2 - (W/2)2/r]tan ho cot p. (A41

In equation (A4), (r,~k) are the polar coordinates, h o is the periphery helix angle, p is half of the manufacturer- designed point angle, ro is the radius of the drill and W is the web thickness. Equation (A4) is the mathematical model for the flute cross-section on the cutting plane perpendicular to the drill axis.

The mathematical representation of the drill flute cross-section was given by Galloway in parametric form [-5]. Equation (A 1) is developed from Galloway's parametric equations. From equation (1), the mathematical model of the flute shape can be obtained as :

- tanho = sin- 1 W + x/r z (W/2)2 tan ho cot p + z (A5) 2r r 0 r 0

by adding the third term to take care of the rotation of the flute with the elevation z. The flute shapes in different cross-sections perpendicular to the drill axis are the same, but the directions are rotated by the helix angle. The flute shapes on two cross-sections at distance one pitch apart are rotated by 360 relative to each other. The peripheral helix angle is given by

ho = tan- 1 2rcr I ' (A6)

where I = pitch of the helix.

108 W.D. TSAI and S. M. Wu

I/2 r'~

.,.._fr, v] Co~ r .v..,~---~, P

. ,.j,,, i / i / ii ' rr'W >I ' , " I < ' I II I

I I I I

I

FIG. A1. Flute shape.

Equation (A5) is the mathematical model of the drill flute represented in the cylindrical coordinates (r, , zl. The geometrical interpretation of equation (A4) is illustrated below.

The drill flute contour is designed so that if the drill point angle is 2p, the point P of the drill flute before grinding becomes the point Q on the straight cutting edge.

OP = OQ = r, (A7}

as shown in Fig. AI. The polar coordinates of the point P is (r,0}. is composed of angle a and angle '8. The difference of elevation between points P and Q is t, which is given by

t = x / r 2 - - (W/2 J 2 cotp. (A8)

The drill flute is rotated by the angle 8 since from point P to point Q. The elevation is changed by t. The angle/7 is proportional to z and will equal 2n when t = l, hence

2zt tan ho ' 8 = t = - - - t . (A9)

I rtl

By substituting equation (A8) into equation (A9}, equation (A3} is obtained. Equation (A4) is the drill flute cross-

sectionforz=0.1fz:/: 0, the drill flute is rotated by the angle (!a~ h'', z),hencethedrillflutecross-sectionforz-0

is given by equation (A5).