Geometric modeling and validation of twist drills with a generic point profile

Applied Mathematical Modelling 36 (2012) 2384–2403

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Applied Mathematical Modelling

journal homepage: www.elsevier .com/locate /apm

Geometric modeling and validation of twist drills with a genericpoint profile

Kumar Sambhav a,1, Puneet Tandon b,⇑, Sanjay G. Dhande c,2

a Department of Mechanical Engineering, Indian Institute of Technology Kanpur, Kanpur 208 016, Uttar Pradesh, Indiab Mechanical Engg. & Design Programme, PDPM Indian Institute of Information Technology, Design and Manufacturing Jabalpur, Jabalpur 482 011,Madhya Pradesh, Indiac Department of Mechanical Engg. & Computer Sce. and Engg., Indian Institute of Technology Kanpur, Kanpur 208 016, Uttar Pradesh, India

a r t i c l e i n f o

Article history:Received 13 May 2011Received in revised form 8 August 2011Accepted 16 August 2011Available online 5 September 2011

Keywords:Drill point geometryGeneric definitionNURBS

0307-904X/$ - see front matter � 2011 Elsevier Incdoi:10.1016/j.apm.2011.08.034

⇑ Corresponding author. Tel.: +91 761 2632924; fE-mail addresses: [email protected] (K. Sambha

1 Tel.: +91 9455680200.2 Tel.: +91 512 2597258/2590763; fax: +91 512 25

a b s t r a c t

Traditionally, twist drills with a few specific point geometry, such as planar, conical, cylindri-cal, ellipsoidal or hyperboloidal, have been designed and adapted for specific applications.Using CAD, the point geometry can be given a generic definition which will enhance the free-dom to design drills with different point profiles and optimize them for multiple objectives.Such a definition can also be used for several downstream applications. This paper presents amethodology to model the twist drills with generic point geometry using NURBS. To beginwith, a detailed basic model for a fluted twist drill with sectional geometry made up of arcsand straight lines has been presented in terms of bi-parametric surface patches. The coordi-nates of cutting lips and chisel edge of the drill have been obtained as solution to a surface-curve intersection problem using optimization algorithm. Subsequently, the model has beengeneralized by employing NURBS to represent the curves whereby the cutting edges andangles can be altered simply by changing the control points or their respective weights. Usingthis methodology, the generic definitions of the conventional angles on the drill point havebeen derived and presented. The proposed model has been illustrated in MATLAB environ-ment and validated experimentally for a conical and an arbitrary point geometry. The exper-iments show a good conformity with the theoretical evaluations.

� 2011 Elsevier Inc. All rights reserved.

1. Introduction

The geometry of a cutting tool has a significant effect on the inputs as well as outputs of the machining process. Thusgeometric modeling of the tool is a very crucial part of tool design. Among the machining processes, drilling is an extensivelyused process [1]. Drills are multi-point rotary cutting tools with a variety of shank and point geometry. The most common ofthem is helically fluted twist drill for which the nomenclature has been shown in Fig. 1.

The objective of this work is to define geometric models of surface patches for helically fluted twist drills with genericsectional geometry and point profile. Such a generic CAD model will act as core model which can be adapted to develop drillsfor specific purposes or optimize the geometry for different objectives by suitably modifying the geometric and grindingparameters. Such a model can also be used for numerous downstream applications, thus opening up various probable areas

. All rights reserved.

ax: +91 761 2632 524.v), [email protected] (P. Tandon), [email protected], [email protected] (S.G. Dhande).

90260/259 7790.

Nomenclature

{} vector notation[] matrix notationk helix angleP pitchL length of fluteL0 length of drill shankR1 land surface patchR2 body clearance surface patchR3 flank surface patchR4 face surface patchbo angle of drill positioningd cone half angleHx, Hy offset of cone apex from the drill axis along x- and y-axes, respectivelyV1 . . .V5 vertices of segments on the drill sectionl1 width of the landl2 depth of body clearancer1, r2 radii of flank and face arcsrw web radiush1, /1, /2 angles on the flute cross-section in the basic modelc inclination angle of landDc drill diameter at cutting endDs shank diameter of drillC1, C2 center points of the two circular arcs on the cross-section in the basic model/ angle of sweepu angle of revolution[Ts] transformation matrixs, t parameters of variationR5, R6 relief surfaces~eabðtÞ generatrix of revolution½RAA1 � rotation matrix~piðsÞ curve vector joining Vi and Vi+1~piðs;/Þ parametric definition of surface of sweep~piðt;uÞ parametric definition of surface of revolution½Ty;da2

� translation along y-axis by da2

da3 distance along z-axisf inclination of axis of revolution with the z-axisb half point anglea relief angledi!jz distance at a chosen value of z

g superscript showing generic model~pg

i ðsÞ curves in the generic model

Rgi surfaces in the generic model

q(t) NURBS in parameter tw chisel edge angleap peripheral relief angle

pgi;tðt;uÞ unit tangent in the t-direction to surface pg

i ðt;uÞng

4 unit normal on Rg4

ng4p projection of ng

4

D~l element along the cutting lipcn normal rake angle

K. Sambhav et al. / Applied Mathematical Modelling 36 (2012) 2384–2403 2385

of work. For example, the surface definitions of the tool can be used to model mathematically the grinding process and theeffect of the grinding parameters on the tool geometry and machining can be studied for the generic case. This would enablethe entire grinding or sharpening process of the tool to be simulated on the computer and the results verified before materialremoval is done. It can further be used to predict drill wear and drill life analytically.

The drill point is the portion of drill that first indents into the workpiece and then initiates the cutting action of the drill.The drill performance is largely affected by the drill point geometry as it is the most important portion of the drill [2]. The

'L

L

LandFlute

Face

Shank

Lip relief surface

sD

Chisel edge

Primary cutting edge

Flank surface

cD

Body clearance

Helix angleHeel

Fig. 1. Projected geometry of drill.

2386 K. Sambhav et al. / Applied Mathematical Modelling 36 (2012) 2384–2403

point geometry is determined by flank and flute shapes as well as the grinding methodology and parameters. Galloway [3]mathematically modeled the conical grinding principle for the conventional point geometry and presented mathematicalrelations between the geometric factors. This was followed by works of several authors, [4–10], to name a few, who modeledor analyzed the drill geometry and present ways of improving the performance.

To explore other drill geometries than the conventional ones, Tsai and Wu [2] developed mathematical models for a somestandard geometries such as hyperboloidal and ellipsoidal drills in terms of the grinding factors. Stephenson and Agapiou[11] presented the parametric description of complex point geometries but they were not related to the grinding parameters.Chandrasekharan et al. [12,13] fitted polynomial equations to measured points on the cutting lip to determine the paramet-ric form of the edge. But this method did not relate the geometry to the grinding parameters again. Paul et al. [14,15] usedthe mechanistic model by Chandrasekharan and geometric model by Tsai and Wu to optimize the shape of the drill point forminimum thrust and torque for conical and quadratic grinding surfaces.

Lin et al. [16] developed a mathematical model for a helical drill point geometry and obtained curved cutting lip shapesfor different semi-point angles using helical grinding. In his work, he concluded that the model was more general than thecommonly used drill point models which were only special cases of this model. A literature review shows that few otherauthors like Galloway [3], Fugelso [6] and Fuh and Chen [7] have also advocated the design of drill points with curved cuttinglips the reason being that curved cutting lip shapes result in a longer drill life than a straight lip for various working condi-tions and working materials. Shi et al. [17] studied curved edged drills and varied the cutting edge shape to obtain a desiredrake angle distribution. Xiong et al. [18] presented methodology to design drills with arbitrarily given distribution of the cut-ting angles along the cutting edges. But he too did not relate the design to grinding parameters.

Tandon et al. [19] presented a model of drill with plane and conical grinding in terms of 3D rotational angles, followed bya forward and inverse mapping between conventional 2D angles on the drill point and rotational 3D grinding angles. A new3D nomenclature for geometric definition of the drill thus came up. A generalized approach to directly represent the sec-tional profile of the drill and the grinding profiles using NURBS was first attempted by Sambhav et al. [20]. Design usingNURBS enables the designer to alter the drill point geometry by just changing control points or the corresponding weights.This gives the freedom to develop drills of generic geometric profile and optimize them for various objectives. In this paper,the mathematical model for the generic drill has been elaborately presented. Starting from the basic profile, where the sec-tional geometry is made up of curves and straight lines, the drill point has been modeled. The cutting lips and the chisel edgehave been obtained by reducing the surface-surface intersection problem to surface-curve intersection problem and thenemploying the optimization technique. To generalize the drill model, the sectional geometry and the grinding profile havebeen represented by NURBS. This is followed by the generic definitions of the angles on the drill point that present the meth-od to evaluate the angles from point to point on the cutting edges. The design has been finally illustrated for a basic modeland a generic model. To validate the mathematical model, first a conical drill has been ground on a universal tool and cuttergrinder, and compared with the model. Then a drill has been ground with an arbitrarily curved generatrix and comparedwith the corresponding mathematical model. The comparisons show that the mathematical models are in good agreementwith the experimental results.

2. Geometric modeling of drill

In this paper, a right-cut solid twist drill with two flutes and straight shank has been modeled. This is the design mostprevalently used in the industry. Similar methodology can be used to design multi-flute drills too.

The geometry of the fluted shank of a twist drill is formed by sweeping helically the cross-section of the drill, while thepoint is formed by the intersection of the grinding surfaces with the fluted shank. The helix angle (k) (or the ratio of thelength of cylindrical fluted portion of the drill (L) to the pitch (P)) determines the sweep. The swept surface has eight surfacepatches, namely, land ðR1;R

01Þ, body clearance ðR2;R

02Þ, flank ðR3;R

03Þ and face ðR4;R

04Þ (Fig. 2). The angle of drill positioning

(bo), cone half angle (d) and the offset of the apex of the surface of revolution in grinding process from the drill axis, Hx and Hy

determine the point geometry of the twist drill (Fig. 3).Using above parameters, the drill is modeled using the methodology presented below (see Fig. 4).

Fig. 2. Surfaces on drill flute and point.

Fig. 3. Conical drill point.


2.1. Basic model of drill

The basic model of the drill is first created, followed by the generic model. In the basic model, the cross-section of thefluted portion is composed of lines and arcs and the grinding surface forms a circular cone about an inclined axis.

2.1.1. Drill blankThe drill blank is modeled as a straight cylinder of diameter Ds and length L0. Once the blank is modeled, the fluted portion

is represented using CAD. This is followed by Boolean operations to yield the complete model of the tool.

2.1.2. Sectional geometry of the basic modelThe geometry of a section of the fluted shank is a composition of curves formed by joining the vertices V1 . . .V5 in one half

of the section. Fig. 5 shows the four segments V1V2, V2V3, V3V4 and V4V5. Segment V1V2 and V2V3 are straight lines and formland and body clearance respectively, when swept helically. V3V4 and V4V5 are arcs of circles which on sweeping form flankand face, respectively. The sectional geometry is guided by these parameters: width of the land (l1), depth of body clearance(l2), radii of the flank (r1) and the face (r2), web radius (rw), angle subtended by flank about axis (h1), inclination angle of theland about axis (c) and diameter of cutting end of drill (Dc). Keeping the center of drill cross-section at the global origin (O) ofthe Cartesian system, position vectors of the end vertices of different curve sections (V1 . . .V5) and center points of the twocircular arcs (C1,C2) are given by the relations:

Fig. 4. Method for geometric modeling of drill.

Fig. 5. Half of the sectional geometry of the basic drill model.


V1 ¼ �Dc2 0 0 1

� �;

V2 ¼ �Dc2 þ l1 sin c

� �l1 cos c 0 1

� �;

V3 ¼ �Dc2 þ l1 sin cþ l2 cos c

� �ðl1 cos c� l2 sin cÞ 0 1

� �;

V4 ¼ �Dc2 þ l1 sin cþ ðl2 þ r2Þ cos c� r2 cosðcþ h2Þ

� �ðl1 cos c� ðl2 þ r2Þ sin cþ r2 sinðcþ h2ÞÞ 0 1

� �;

V5 ¼ Dc2 0 0 1� �

;

C1 ¼ �Dc2 þ l1 sin cþ ðl2 þ r2Þ cos c

� �ðl1 cos c� ðl2 þ r2Þ sin cÞ 0 1

� �;

C2 ¼ Dc2 � r1 cosð/1 þ /2Þ� �

r1 sinð/1 þ /2Þ 0 1� �

;

where; /1 ¼ cos�1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðV4x�V5xÞ2þðV4y�V5yÞ2

p2r2

� �; /2 ¼ tan�1 V4y�V5y

V5x�V4x

9>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>;

: ð2:1Þ

The curve vector having end points Vi and Vi+1 is termed as ~piðsÞ.Specifically, using Eq. (2.1),

~p3ðsÞ ¼ ðC1x � r1 cosðcþ ð1� sÞh1ÞÞðC1y þ r1 sinðcþ ð1� sÞh1ÞÞ 0 1� �

; ð2:2Þ~p4ðsÞ ¼ C2x þ r2 cos /1 þ /2 þ sðp� 2/1Þð Þð Þ C2y � r2 sin /1 þ /2 þ sðp� 2/1Þð Þ

� �0 1

� �; ð2:3Þ

where 0 6 s 6 1.


2.1.3. Constraints on the choice of the variablesThe above parameters have to obey certain geometric constraints. If the web radius is rw, r2 will depend on the value of rw.

To relate the two, it can be observed that

ðC2x � V4xÞ2 þ ðC2y � V4yÞ2 ¼ r22;

ðC2x � V5xÞ2 þ ðC2y � V5yÞ2 ¼ r22;

and; C22x þ C2

2y ¼ ðr2 þ rwÞ2:

9>>>=>>>; ð2:4Þ

The three simultaneous equations in terms of three unknowns C2x, C2y and r2 can be solved to get r2 in terms of rw. Alter-natively, r2 and rw are related as

rw ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC2

2x þ C22y

q� r2: ð2:5Þ

The value of r2 can be obtained iteratively using the above equation. The point of contact of segment V4V5 and the webcircle is found out using the fact,

xw ¼ C2x �rw

r2 þ rw; yw ¼ C2y �

rw

r2 þ rw: ð2:6Þ

At this point, s ¼ ðp�/o�/1�/2Þp�2/1

, where /o ¼ tan�1 C2y

C2x

.

2.1.4. Flute geometry of the basic modelThe flute geometry is obtained by sweeping the sectional curve helically. It consists of helicoidal surfaces R1, R2, R3 and

R4. The sweep matrix is given as:

½Ts� ¼

cos / sin / 0 0

� sin / cos / 0 0

0 0 1 0

0 0 P/2p 0

2666664

3777775; ð2:7Þ

where 0 6 / 6 2pLP .

The helicoidal surfaces R1 to R4 are formed on the basis of following sweep rules:

Land R1 ¼~p1ðsÞ � ½Ts� ¼~p1ðs;/Þ; ð2:8ÞBody clearance R2 ¼~p2ðsÞ � ½Ts� ¼~p2ðs;/Þ; ð2:9ÞFlank R3 ¼~p3ðsÞ � ½Ts� ¼~p3ðs;/Þ; ð2:10Þand; Face; R4 ¼~p4ðsÞ:½T� ¼~p4ðs;/Þ: ð2:11Þ

The flute geometry for the remaining part of the drill can be obtained by rotating the respective surfaces by 180� aboutthe Z-axis. The corresponding surfaces are termed as R01;R

02;R

03 and R04, respectively.

2.1.5. Point geometry of the basic modelFor the present model, where the drill has two flutes and the point is ground conically, two surface patches R5 and R6

called the lip relief surfaces, make the drill point. The two patches are parts of two cones respectively, each with half angled. The flank R3 is in contact with lip relief surface R5 at the end of the fluted shank. The primary cutting lips are formed bythe intersection of the helical surface of flute with the lip relief surface, while the chisel edge is formed by the intersection ofthe two lip relief surfaces. For the grinding cone shown in Fig. 3, the apex of cone has the coordinates (Hx,�Hy,Hz) where Hx,Hy, Hz are all positive. The generatrix given by AB revolves around the axis AA1 to form the lip relief surface R5.

The edge AB is denoted by~eabðtÞ where,

~eabðtÞ ¼

�Dc2 þ t Hx þ Dc

2

� �� Hy

t Hx þ Dc2

� �= tan bo

1

8>>><>>>:

9>>>=>>>;; 0 6 t 6 1: ð2:12Þ

To get R5, line AA1 is first translated along positive Y-direction by da2 so that it intersects with the Z-axis. Thus, hereda2 ¼ Hy as the axis AA1 is parallel to the Z � X plane. Then it is displaced along the Z-direction by da3 so that the axis AA1

on the Z � X plane passes through the origin (Fig. 3). da3 is given by:

da3 ¼ Hx= tan f� Hx þDc

2

� �= tan bo: ð2:13Þ

Table 1Approximate values of drill axis angles [21].

Relief angle a (degrees) Point angle 2b

60 70 90 118 140 160 180

Angle b0 (degrees)

6 30 35 45 59 70 79 8612 30 35 45 58 68 76 8018 30 35 44 57 66 71 7424 30 35 44 56 62 66 68


Angle bo is the angle of drill positioning and is influenced by point angle 2b and can be calculated from the Table 1. Next,AA1 is rotated about the Y-axis by-f and AB is swiveled about AA1 by angle u in the counterclockwise direction. Angle f isgiven by the relation

f ¼ bo � d: ð2:14Þ

Mathematically the process is represented by

R5 ¼~p5ðt;uÞ ¼ ½RAA1 �feabðtÞg; ð2:15Þ

where,

RAA1

� �¼ Ty;da2

h i�1� Tz;da3

h i�1� Ry;�f

� ��1 � Rz;u� �

� Ry;�f

� �� Tz;da3

h i� Ty;da2

h i; ð2:16Þ

Ty;da2

h i¼

1 0 0 00 1 0 da2

0 0 1 00 0 0 1

26664

37775; Tz;da3

h i¼

1 0 0 00 1 0 00 0 1 da3

0 0 0 1

26664

37775; Ry;�f

� �¼

cf 0 �sf 00 1 0 0sf 0 cf 00 0 0 1

26664

37775; Rz;u

� �¼

cu �su 0 0su cu 0 00 0 1 00 0 0 1

26664

37775:

Using Eqs. (2.15) and (2.16), the x, y and z coordinates of the surface R5 are given by the following relations [20]:

p5Xðt;uÞ ¼ Aðcos2 f cos uþ sin2 fÞ � Bðcos f sin f cos u� cos f sin fÞ; ð2:17Þp5Y ðt;uÞ ¼ A cos f sin u� B sin f sin u� da2; ð2:18Þp5Zðt;uÞ ¼ �Aðcos f sin f cos u� cos f sin fÞ þ Bðsin2 f cos uþ cos2 fÞ � da3; ð2:19Þ

where, A = �Dc/2 + t(Hx + Dc/2), B = t(Hx + Dc/2)/tanb0 + da3, 0 6 s 6 1Second lip relief surface R6 is formed by rotating R5 about the vertical axis by an angle of 180�. For an n-fluted drill, sur-

face patches R6 to R3+n can be formed by rotating the surface patch R5 about the axis by angles 2pm/n, respectively, wherem ? 1 . . . (n � 1).

2.1.6. Primary cutting lip in the basic modelThe primary cutting lip is obtained as the intersection of the face with the cone. It is the classical surface-surface inter-

section problem and the curve can be obtained by using Timmer’s method. In this paper, the problem has been reduced tofinding out the intersection between a curve and a surface and the points of intersection have been obtained using optimi-zation algorithms, as convergence was faster here.

At a point of intersection,

d1 !jz ¼~p04ðs;/Þjz �~p5ðt;uÞ ¼ 0; ð2:20Þ

where ~p04ðs;/Þjz is the curve traced on ~p04ðs;/Þ at any Z. ~p04ðs;/Þ is the helicoidal surface given by ~p04ðs;/Þ ¼ p04ðsÞ � ½Ts� andp04ðsÞ is the curve obtained by rotating p4(s) by 180�. d1

!jz is the distance between the curve and the surface at a chosen value

of Z.

2.1.7. Chisel edge in the basic modelThe chisel edge is obtained as the intersection of R5 and R6. The intersection of the two surfaces can be found out by find-

ing out the points where

d2 !¼~p6ðt;uÞ �~p5ðt;uÞ ¼ 0: ð2:21Þ

To simplify the problem, the contour of the surfaces at different values of z can be found out and points of intersectionobtained. This problem can again be solved using optimization technique.


d2 !jz ¼~p6ðt;uÞjz �~p5ðt;uÞjz ¼ 0: ð2:22Þ

The two extremities of the chisel edge are evaluated by finding the intersection of R4, R5 & R6 and R04;R5 and R6. The firstof these is obtained by solving the equation:

d23 ¼ j~p4ðt;uÞ �~p5ðt;uÞj2 þ j~p6ðs;/Þ �~p4ðt;uÞj2 þ j~p6ðs;/Þ �~p5ðt;uÞj2 ¼ 0: ð2:23Þ

Similarly, the second. The peak of the chisel edge will lie on the z-axis. This argument is used to find the peak of the chiseledge.

2.2. Generic model of the drill

Having modeled the basic profile, the generic model can be generated by replacing the arcs and lines by NURBS.

2.2.1. Sectional geometryThe sectional geometry is designed as shown in Fig. 6. Here, curves~p3ðsÞ and~p4ðsÞ of the basic model are defined as~pg

2ðsÞand ~pg

3ðsÞ between vertices V2, V3 and V3, V4, respectively, while ~pg1ðsÞ remains the same as ~p1ðsÞ. The superscript 0g0 denotes

the generic model. The body clearance forms a part of the flank now. The curves in this generic model are defined as

~pgj ðsÞ ¼

Pni¼0wiNm;mþiðsÞbi

!Pni¼0wiNm;mþiðsÞ

; j ¼ 2;3: ð2:24Þ

where, N1,i(s) = di such that di ¼ 1 for s 2 ½si�1; siÞ¼ 0; elsewhere

Nm;mþiðsÞ ¼s� si

smþi�1 � siNm�1;mþi�1ðsÞ þ

smþi � ssmþi � siþ1

Nm�1;mþiðsÞ ð2:25Þ

and, m is the order of the curve, wi are the weights and bi!

are the control points. N is the shape function.

2.2.2. Flute geometryThe flute geometry is obtained in a similar manner as for the basic model as a surface of sweep. In this case,

Rgi ¼~p

gi ðsÞ � ½Ts� ¼~pg

i ðs;/Þ; i ¼ 1;2;3: ð2:26Þ

2.2.3. Point geometryThe generatrix of revolution is a NURBS in this case. Here

~egabðtÞ ¼ qxðtÞ � Hy qzðtÞ1

� �; ð2:27Þ

where q(t) is a NURBS in terms of parameter t. The generatrix has been shown in Fig. 7 for two different cases. Using Eqs.(2.13)–(2.19), lip relief surfaces given by Rg

4 and Rg5 can be evaluated.

Surface Rg4 ¼~p

g4ðt;uÞ ¼ ½RAA1 �:~e

gabðtÞ where ½RAA1 � has a similar definition. Here, A = qx(t) and B ¼ qzðtÞ þ da3 .

Rg5 is formed by rotating Rg

4 about the axis by an angle of 180�.The cutting lips and the chisel edge are obtained in a similar manner as for the basic model.

Fig. 6. Sectional geometry of the generic drill model.

Fig. 7. Generatrix of grinding surface as a NURBS.


3. Generic definitions of angles on the drill point

This section provides the methodology to find out the conventional (2D) angles in terms of the parameters used in thepresent work. Table 2 shows the methodology of formation of conventional angles by various surface patches shaping thedrill. The angles of twist drills presented here are as per the conventional nomenclatures.

Formation of surfaces Rg1 to Rg

5 for the generic model has been explained earlier in the paper. The unit tangents and nor-mal to Rg

4 are obtained as follows:

Table 2Method

Conv

Half

Chis

Relie

Helix

Perip

pg4;tðt;uÞ ¼ @~pg

4ðt;uÞ=@t� �

= @~pg4ðt;uÞ=@t

� �� ; ð3:1Þpg

4;uðt;uÞ ¼ @~pg4ðt;uÞ=@u

� �= @~pg

4ðt;uÞ=@u� �� ; ð3:2Þ

ng4 ¼ pg

4;tðt;uÞ � pg4;uðt;uÞ: ð3:3Þ

The unit tangents and normal can be evaluated from point to point on the cutting edges using Eqs. (3.1)–(3.3).

Half point angle (bg): As shown in Table 2, ng4 is projected on ZX-plane and angle between this unit projected normal

vector ðng4pÞ and the unit vector normal to YZ-plane ðiÞ is half point angle bg.

Chisel edge angle (wg): The unit normal ng4 at any point on the chisel edge projected to XY-plane is given by ng0

4p. The scalarproduct of ng0

4p and unit vector j gives the chisel edge angle wg.Relief angle (ag): ng

4 is projected on YZ-plane given by ng00

4p. The relief angle ag is the angle between ng00

4p and unitvector k.

Helix angle (kg): Helix angle (kg) is formed by Rg1 and satisfies the relation

kg ¼ tan�1ðpDc=PÞ: ð3:4Þ

Peripheral relief angle (agp): This angle is formed by the surface patch Rg

1 with XY plane. The patch Rg1 is formed by heli-

cally sweeping parametric edge formed by joining vertices V1 and V2. The edge is defined as

~pg1ðsÞ ¼ ð�Dc=2þ sl sin cÞ sl cos c 0 1½ �:

ology to form the conventional angles.

entional angles Formed by the normal to With the plane Plane of projection

point angle, bg Rg4

YZ ZX

el edge angle, wg Rg4

ZX XY

f angle, ag Rg4

XY YZ

angle, kg Rg3

YZ ZX

heral relief angle, agp Rg

1YZ XY


The surface patch Rg1 formed by sweeping ~pg

1ðsÞ, is expressed by the following relation

~pg1ðs;/Þ ¼ ð�Dc=2þ sl sin cÞ cos /� sl cos c sin /ð Þðð�Dc=2þ sl sin cÞ sin /þ sl cos c cos /ÞðP/=2pÞ 1½ �

The tangents and normal to the surface patch at an arbitrary point are:

~pg1;sðs;/Þ ¼ l cosðc� /Þiþ l cosðc� /Þj;~pg

1;/ðs;/Þ ¼ Dc=2: sin /� sl cosðc� /Þð Þiþ �Dc=2: cos /þ sl sinðc� /Þð Þjþ P=2pk;

~ng1 ¼ Pl=2p cosðc� /Þi� Pl=2p sinðc� /Þj� ðDc=2:l sin c� sl2Þk:

The normal on projection to XY plane is given by ng1p ¼ cosðc� /Þi� sinðc� /Þj.

The scalar product of ng1p with unit vector i provides cosag

p ¼ cosðc� /Þ. At z = 0 plane, / = 0 and this leads to the mappingrelation ag

p ¼ c.Thus, we obtain the generic definitions for the half point angle, chisel edge angle, relief angle, helix angle and the periph-

eral relief angle.

3.1. Cutting angles on the drill point

To calculate the angles at a point on the cutting lip, the lip has to be divided into small cutting elements and the angles areevaluated at the midpoint of each element, which vary from point to point. Using the matrix of the cutting elements, thecutting lip element is defined as a vector D~l. The velocity vector is obtained as V

!¼ ~x�~r where ~r is the radius vector ofthe midpoint of the cutting lip and ~x is the angular velocity. The inclination angle (i) is obtained using the relation

cosð90þ iÞ ¼ Dl � bV ; ð3:5Þ

where the cap denotes the unit vector.Normal to the face is obtained as the normalized cross-product of the two parameters making the biparametric surface

patch:

ng3 ¼

~pg0

3;s �~pg0

3;/

~pg03;s �~p

g03;/

�� ; ð3:6Þ

where ~pg0

3;s and ~pg0

3;/ denote the derivatives of ~pg0

3 ðs;/Þ along s- and /- directions, respectively.Thus normal rake angle cg

n is obtained as:

cosp2� cg

n

¼ ng

3 �V!� D~l

V!� D~l�� : ð3:7Þ

4. Illustration of the model

Using the methodology presented above, geometry of the basic and the generic model are generated. The sectional geom-etry, the flute profile and the grinding cone for the basic model are shown in Figs. 8a, 8b, 8c, respectively. The cutting lips and

Fig. 8a. Sectional geometry of basic model.


the chisel edge are shown in Fig. 8d. The data used here given in Table 3. It can be seen from Fig. 8d that the cutting lip andchisel edge are curved for the basic model.

To obtain a generic profile, the sectional profile was modeled using NURBS to give a shape analogous to that of prevalentdrills. First this sectional profile was used for modeling drill point with a straight generatrix (as above), and then for a curvedgeneratrix. For the curved generatrix, the control points for NURBS of order 3 were chosen as follows:

ð�12:34;0;�2:54Þ; ð�8:82;0;�1:06Þ; ð�7:05;0; 0Þ; ð�5:29;0;1:91Þ; ð�3:53;0;2:97Þ; ð�1:76;0;3:60Þ; ð0;0;4:24Þf g:

Fig. 8b. Flute profile in the basic model as a sweep surface.

Fig. 8c. Grinding cones as surfaces of revolution.

Fig. 8d. Cutting lips and chisel edge for the basic model.

Table 3Basic drill profile data.

Dc = 12.5 mm P = 60 mm rw = 0.7 mmHx = 2.2 mm l1 = 1.5, l2 = 0.2 mm c = 12�b0 = 59� da2 ¼ 1 mm d = 35�r1 = 5.5 mm r2 = 4.37 mm h1 = 81.82�

Table 4Generic drill profile data.

Dc = 14 mm P = 68.3 mm rw = 0.7 mmHx = 2.2 mm l = 0.83 mm c = 10.9�b0 = 59� da2 ¼ 2 mm d = 50�


Using the parameters listed in Table 4, the surfaces of sweep and revolution were obtained. The cutting lips and the chiseledge were subsequently obtained through optimization algorithms using MATLAB. The flute geometry and the grinding sur-faces with a generic profile are shown in Figs. 9a and 9b respectively. The cutting lips and the chisel edge obtained for thestraight generatrix are shown in Fig. 9c, while those obtained for the curved generatrix are shown in Fig. 9d. It can be seenclearly from Fig. 9c that the cutting lip is curved and the chisel edge has a slightly curved S-shape.

5. Experimental validation and discussion

The model was validated for two different shapes of the drill point: a conical point and a curved point. The procedure isdescribed below.

First, a drill grinder was selected which follows the same grinding principle as described above. It is HMT universal tooland cutter grinder GTC-28T at IIT Kanpur. The parameters inherent to the grinding process were derived by measuring thedistances and angles on the machine (Fig. 10) while other parameters were set on the grinder. An HSS drill of Addison makeof diameter 12.5 mm was chosen. The relevant data are shown in Table 5. Using a flat Alumina grinding wheel, the drill wasground for a conical profile (Fig. 11a).

The drill section was captured on an Integrated Multi-process Machine Tool DT-110 by Mikrotools with a 300 micronprobe and modeled (Fig. 11b). Using the above formulation, the flute profile and the surfaces of revolution were modeled(Fig. 11c). The drill point profile was evaluated which gave the theoretical values of the coordinates of the primary cuttinglip, chisel edge and heel. The same were measured with the probe and registered with the model which is shown graphicallyin Fig. 11d. For registration, the peak of the drill was chosen as the origin. A comparison of the two data for the primary cut-ting lips are presented in Table 6a, for the two heels in Table 6b while Table 6c compares the data for the chisel edge. Tocompare the two data, coordinates were captured at the same z-values, and the x- and y-values were investigated. The pri-mary cutting lip and the chisel edge are in close proximity, while the lower part of the heel slightly deviates away from theanalytically obtained data. This is because the model uses a fixed generatrix for the grinding surface of revolution. But prac-tically, the generatrix provided by the grinding wheel with flat face down as the drill swivels about its axis and the line ofcontact recedes away with the swivel of the grinder arm.

Fig. 9b. Grinding cone with a generic profile.

Fig. 9a. Flute profile of the generic model.


To validate the geometric model for the generic case, the grinding wheel was first ground for a curved profile (Fig. 12a)with a diamond dresser. Then coordinates on the generatrix of the grinding wheel were captured on machine tool DT-110.These coordinates were taken as data points and used to calculate the control points to fit a B-spline curve through the datapoints using the following methodology:

Let the data points on any curve be~q0;~q1 . . .~qn. To fit them with a B-spline curve of order p 6 n, we select a set of param-eters s0, s1, . . .sn corresponding to each data point. If the unknown control points are represented by~ci; i ¼ 0 . . . n, we have

Fig. 9c. Two views of the cutting lips and chisel edge with a conical grinder.

Fig. 9d. The point geometry with a generically profiled grinder.

Fig. 10. HMT GTC-28T universal tool and cutter grinder.


Table 5Data used for validation of conical and generic drill.

Dc = 12.5 mm P = 80 mm rw = 1.25 mmHx = 15 mm l = 1.6 mm c = 11�b0 = 59� da2 ¼ 2 mm d = 35�

Fig. 11a. 12.5 mm drill ground with a flat faced grinding wheel.

Fig. 11b. 12.5 mm drill section geometry modeled using NURBS.


~cðuÞ ¼Xn

i¼0

Np;pþiðuÞ~ci: ð5:1Þ

Using the mapping of the data points and the parameters, we have,

~qk ¼~cðskÞ ¼Xn

i¼0

Np;pþiðskÞ~ci; k ¼ 0 . . . n

or,

fQg ¼

~q0

~q1

� � �� ~qn

26666664

37777775 ¼

Np;pðs0Þ Np;pþ1ðs0Þ Np;pþ2ðs0Þ . . . Np;pþnðs0ÞNp;pðs1Þ Np;pþ1ðs1Þ Np;pþ2ðs1Þ . . . Np;pþnðs1Þ� � �� Np;pðsnÞ Np;pþ1ðsnÞ Np;pþ2ðsnÞ . . . Np;pþnðsnÞ

26666664

37777775

~c0

~c1

� � �� ~cn

26666664

37777775 ¼ ½N�fCg ð5:2Þ

Fig. 11c. Flute geometry of the drill together with the conical grinding surfaces.

Fig. 11d. Comparison of experimental and theoretical results for a conical drill point.


or,

fCg ¼ ½N��1fQg ð5:3Þ

As the interpolating curve is in Z–X plane, each data point will have cartesian coordinates of the form (xs,0,zs) and thus thecontrol points.

Having defined the generatrix, the coordinates on the cutting lip, heel and chisel edge were calculated using a MATLABcode.

Next, the drill was ground and the drill point was generated (Fig. 12b). The coordinates on the cutting lip, heel and chiseledge were measured experimentally. The drill flute together with the curved grinding surfaces were modeled (Fig. 12c).

Table 6aComparison of coordinates of the two primary cutting lips for the conical drill.

S. No. Primary cutting lip 1 Primary cutting lip 2

Theoretical Experimental Theoretical Experimental

1 �0.35 �1.22 3.55 �0.4 �1.08 3.55 0.35 1.22 3.55 0.4 1.08 3.552 �1.06 �0.95 3.06 �1.06 �1.03 3.06 1.06 0.95 3.06 1.01 0.98 3.063 �1.54 �0.81 2.76 �1.77 �0.81 2.76 1.54 0.81 2.76 1.72 0.78 2.764 �2.02 �0.69 2.46 �2.26 �0.61 2.46 2.02 0.69 2.46 2.19 0.58 2.465 �2.66 �0.56 2.06 �2.7 �0.61 2.06 2.66 0.56 2.06 2.67 0.58 2.066 �3.15 �0.46 1.76 �3.26 �0.54 1.76 3.15 0.46 1.76 3.18 0.48 1.767 �3.8 �0.35 1.36 �3.78 �0.34 1.36 3.8 0.35 1.36 3.68 0.28 1.368 �4.15 �0.3 1.16 �4.23 �0.25 1.16 4.15 0.3 1.16 4.16 0.23 1.169 �4.45 �0.27 0.96 �4.66 �0.16 0.96 4.45 0.27 0.96 4.73 0.13 0.96

10 �4.94 �0.2 0.66 �5.04 �0.16 0.66 4.94 0.2 0.66 4.96 0.11 0.6611 �5.27 �0.15 0.46 �5.23 �0.09 0.46 5.27 0.15 0.46 5.17 0.08 0.4612 �5.75 �0.05 0.16 �5.7 �0.08 0.16 5.75 0.05 0.16 5.73 0.08 0.16

Table 6bComparison of coordinates of the two heels for the conical drill.

S. No. Heel 1 Heel 2


1 �0.35 �1.22 3.55 �0.4 �1.08 3.55 0.35 1.22 3.55 0.4 1.08 3.552 0.3 �1.8 2.91 0.36 �1.71 2.91 �0.3 1.8 2.91 �0.36 1.68 2.913 0.62 �2.35 2.51 0.66 �2.21 2.51 �0.62 2.35 2.51 �0.66 2.17 2.514 0.74 �3.26 2.06 0.86 �3.01 2.06 �0.74 3.26 2.06 �0.96 2.96 2.065 0.66 �3.8 1.86 0.79 �3.61 1.86 �0.66 3.8 1.86 �0.92 3.55 1.866 0.49 �4.32 1.66 0.66 �4.01 1.66 �0.49 4.32 1.66 �0.79 3.93 1.667 0.39 �4.56 1.56 0.61 �4.41 1.56 �0.39 4.56 1.56 �0.66 4.32 1.568 0.33 �4.69 1.52 0.41 �4.81 1.52 �0.33 4.69 1.52 �0.53 4.67 1.52

Table 6cComparison of coordinates of the chisel edge for the conical drill.

S. No. Theoretical Experimental

1 �0.35 �1.22 3.55 �0.40 �1.08 3.552 0.00 0.00 3.56 0.00 0.00 3.583 0.35 1.22 3.55 0.40 1.08 3.55

Fig. 12a. Grinding wheel with arbitrarily curved grinding surface.


These coordinates were compared with the evaluated ones. The comparison between the model and the ground drill isshown in Fig. 12d. The same has been tabulated in Table 7. As the primary cutting lips are curled along z-axis here, for com-parison, the data were captured for identical x-values and the y- and z-values were looked into. The comparison shows againa good proximity between the model and the generated profile, except near the lower part of the primary cutting lip and heelfor similar reason as mentioned above.

Fig. 12b. Drill of 12.5 mm diameter ground with curved grinding surface.

Fig. 12c. Flute geometry of the drill together with the curved grinding surfaces.


6. Downstream applications of the generic model

The geometric model can be used for multiple downstream applications such as force modeling, prediction of drill holequality, prediction of drill wear and drill life, study of drill dynamics, design modification for easy chip evacuation, etc. Thestudy of drilling performance in the micro-domain is another area where new shapes of drills need to be investigated. Andeach of these concerns can be used for the optimization of the drill geometry for different objectives. Such tasks have beentaken up by researchers in the past too. But the drill geometries reported in the earlier work have been limited to few stan-dard shapes. Using NURBS to model the drill geometry gives a new dimension to these tasks and gives a much higher degreeof freedom for predictions or optimization tasks. This generates a new scope for research in all these areas. Working in thisdirection, the authors have modeled the drilling normal and friction forces for generic drill point geometry mechanistically[20]. The mechanistic model assumes that the forces are proportional to the chip load and generates a semi-empirical modelthrough regression analysis of the forces as a function of the rake angle, feed and velocity.

Fig. 12d. Comparison of experimental and theoretical results for a generic drill point.

Table 7aComparison of coordinates of the two primary cutting lips for the generic drill.

S. No. Primary cutting lip 1 Primary cutting lip 2


1 �0.35 �1.26 2.35 �0.35 �1.15 2.39 0.35 1.26 2.35 0.35 1.15 2.392 �1.01 �0.88 1.67 �1.01 �0.85 1.89 0.88 0.95 1.79 0.88 0.73 1.993 �1.42 �0.7 1.3 �1.42 �0.69 1.44 1.26 0.77 1.45 1.26 0.63 1.614 �1.94 �0.48 0.91 �1.94 �0.47 1.08 1.78 0.54 1.01 1.78 0.35 1.095 �2.29 �0.18 0.45 �2.29 �0.34 0.87 2.35 0.34 0.67 2.35 0.15 0.686 �2.71 �0.23 0.51 �2.71 �0.28 0.65 2.82 0.2 0.48 2.82 �0.03 0.547 �3.21 �0.11 0.37 �3.21 �0.22 0.52 3.33 0.09 0.34 3.33 �0.05 0.358 �3.93 �0.01 0.24 �3.93 �0.24 0.35 3.73 0.03 0.28 3.73 �0.1 0.289 �4.31 0.01 0.2 �4.31 �0.21 0.29 4.31 �0.01 0.2 4.31 �0.03 0.23

10 �4.73 0.01 0.15 �4.73 �0.23 0.27 4.65 �0.01 0.16 4.65 0 0.1911 �5.88 �0.1 0.26 �5.88 �0.29 0.4 5.82 0.08 0.23 5.82 0.36 0.1612 �6.22 �0.19 0.39 �6.22 �0.42 0.39 6.06 0.15 0.33 6.06 0.31 0.21

Table 7bComparison of coordinates of the heels for the generic drill.

S. No. Heel 1 Heel 2


1 �0.35 �1.26 2.35 �0.35 �1.15 2.39 0.35 1.26 2.35 0.35 1.15 2.392 �0.09 �1.53 1.93 �0.06 �1.6 1.93 0.09 1.53 1.93 0.07 1.54 1.933 0.09 �1.77 1.66 0.07 �2 1.66 �0.09 1.77 1.66 �0.06 1.94 1.664 0.35 �2.37 1.15 0.38 �2.56 1.15 �0.35 2.37 1.15 �0.41 2.41 1.155 0.38 �2.5 1.06 0.46 �2.96 1.06 �0.38 2.5 1.06 �0.43 2.82 1.066 0.39 �3.12 0.7 0.41 �3.57 0.7 �0.39 3.12 0.7 �0.44 3.48 0.77 0.18 �3.96 0.33 0.12 �4.15 0.33 �0.18 3.96 0.33 �0.16 4.06 0.338 �0.3 �4.82 0.02 �0.16 �5.03 0.05 0.3 4.82 0.02 0.19 4.93 �0.13

Table 7cComparison of coordinates of the chisel edge for the generic drill.

S. No. Theoretical Experimental

1 0.35 1.26 2.35 0.35 1.15 2.392 0.17 0.58 2.43 0.15 0.46 2.453 0.00 0.00 2.52 �0.08 �0.11 2.494 �0.35 �1.26 2.35 �0.35 �1.15 2.39



7. Summary and conclusions

� The presented work employs CAD to design twist drills of generic shape. Starting from a simple basic model, the
drill flute and point are modeled using NURBS, which gives the design a generic profile where the drill shapeand cutting angles can be changed just by changing the control points or their respective weights. The cuttingedges obtained here are curved in space and not straight, and related to the grinding parameters.
� The calculation of cutting edge coordinates presents a surface-surface intersection problem. The problem has beensolved converting it to surface-curve intersection and employing optimization technique subsequently. The cuttinglip is obtained as the intersection of a sweep surface and a surface of revolution, while the chisel edge is obtained asthe intersection of two surfaces of revolution.

� Generic definitions of the conventional angles on the drill point are presented along the primary cutting lip and thechisel edge of the drill.

� The model has been illustrated in MATLAB environment and validated for a conical drill as well as a drill withcurved relief surface. The theoretical and experimental results are in good conformity.

� Further challenging applications of this generic model have been highlighted towards the end of the article.

In summary, the proposed methodology gives us a freedom to model drill point profiles of generic shape and apply fordifferent objectives.

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Geometric modeling and validation of twist drills with a generic point profile