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Geometric modeling and validation of twist drills with a generic point profile
Applied Mathematical Modelling 36 (2012) 2384–2403
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Geometric modeling and validation of twist drills with a generic point 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, India Mechanical Engg. & Design Programme, PDPM Indian Institute of Information Technology, Design and Manufacturing Jabalpur, Jabalpur 482 011, Madhya Pradesh, India c Department of Mechanical Engg. & Computer Sce. and Engg., Indian Institute of Technology Kanpur, Kanpur 208 016, Uttar Pradesh, India b
a r t i c l e
i n f o
Article history: Received 13 May 2011 Received in revised form 8 August 2011 Accepted 16 August 2011 Available online 5 September 2011 Keywords: Drill point geometry Generic definition NURBS
a b s t r a c t Traditionally, twist drills with a few specific point geometry, such as planar, conical, cylindrical, 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 freedom 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 a methodology to model the twist drills with generic point geometry using NURBS. To begin with, a detailed basic model for a fluted twist drill with sectional geometry made up of arcs and straight lines has been presented in terms of bi-parametric surface patches. The coordinates of cutting lips and chisel edge of the drill have been obtained as solution to a surfacecurve intersection problem using optimization algorithm. Subsequently, the model has been generalized by employing NURBS to represent the curves whereby the cutting edges and angles can be altered simply by changing the control points or their respective weights. Using this methodology, the generic definitions of the conventional angles on the drill point have been derived and presented. The proposed model has been illustrated in MATLAB environment and validated experimentally for a conical and an arbitrary point geometry. The experiments show a good conformity with the theoretical evaluations. Ó 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. Thus geometric modeling of the tool is a very crucial part of tool design. Among the machining processes, drilling is an extensively used process [1]. Drills are multi-point rotary cutting tools with a variety of shank and point geometry. The most common of them 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 generic sectional geometry and point profile. Such a generic CAD model will act as core model which can be adapted to develop drills for specific purposes or optimize the geometry for different objectives by suitably modifying the geometric and grinding parameters. Such a model can also be used for numerous downstream applications, thus opening up various probable areas
⇑ Corresponding author. Tel.: +91 761 2632924; fax: +91 761 2632 524. 1 2
E-mail addresses: [email protected] (K. Sambhav), [email protected] (P. Tandon), [email protected], [email protected] (S.G. Dhande). Tel.: +91 9455680200. Tel.: +91 512 2597258/2590763; fax: +91 512 2590260/259 7790.
0307-904X/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2011.08.034
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Nomenclature {} [] k P L L0
vector notation matrix notation helix angle pitch length of flute length of drill shank R1 land surface patch R2 body clearance surface patch R3 flank surface patch R4 face surface patch bo angle of drill positioning d cone half angle Hx, Hy offset of cone apex from the drill axis along x- and y-axes, respectively V1 . . . V5 vertices of segments on the drill section l1 width of the land l2 depth of body clearance radii of flank and face arcs r1, r2 rw web radius h1, /1, /2 angles on the flute cross-section in the basic model c inclination angle of land Dc drill diameter at cutting end Ds shank diameter of drill C1, C2 center points of the two circular arcs on the cross-section in the basic model / angle of sweep u angle of revolution [Ts] transformation matrix s, t parameters of variation R5, R6 relief surfaces ~ eab ðtÞ generatrix of revolution ½RAA1 rotation matrix ~ curve vector joining Vi and Vi+1 pi ðsÞ ~ pi ðs; /Þ parametric definition of surface of sweep ~ pi ðt; uÞ parametric definition of surface of revolution ½T y;da2 translation along y-axis by da2 da3 distance along z-axis f inclination of axis of revolution with the z-axis b half point angle a relief angle ! distance at a chosen value of z di jz g superscript showing generic model ~ curves in the generic model pgi ðsÞ
Rgi
q(t)
w ap
surfaces in the generic model NURBS in parameter t chisel edge angle peripheral relief angle
^gi ðt; uÞ ^gi;t ðt; uÞ unit tangent in the t-direction to surface p p ^ g4 n
unit normal on Rg4
^ g4p n
^ g4 projection of n
D~l
element along the cutting lip normal rake angle
cn
of work. For example, the surface definitions of the tool can be used to model mathematically the grinding process and the effect of the grinding parameters on the tool geometry and machining can be studied for the generic case. This would enable the entire grinding or sharpening process of the tool to be simulated on the computer and the results verified before material removal 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
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L' L
Face
Heel
Flank surface
Helix angle
Chisel edge
Dc Shank Primary cutting edge
Lip relief surface
Land
Flute
Ds
Body clearance
Fig. 1. Projected geometry of drill.
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 mathematical relations between the geometric factors. This was followed by works of several authors, [4–10], to name a few, who modeled or 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 some standard 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 parametric form of the edge. But this method did not relate the geometry to the grinding parameters again. Paul et al. [14,15] used the mechanistic model by Chandrasekharan and geometric model by Tsai and Wu to optimize the shape of the drill point for minimum 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 shapes for different semi-point angles using helical grinding. In his work, he concluded that the model was more general than the commonly used drill point models which were only special cases of this model. A literature review shows that few other authors like Galloway [3], Fugelso [6] and Fuh and Chen [7] have also advocated the design of drill points with curved cutting lips the reason being that curved cutting lip shapes result in a longer drill life than a straight lip for various working conditions and working materials. Shi et al. [17] studied curved edged drills and varied the cutting edge shape to obtain a desired rake angle distribution. Xiong et al. [18] presented methodology to design drills with arbitrarily given distribution of the cutting 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 by a forward and inverse mapping between conventional 2D angles on the drill point and rotational 3D grinding angles. A new 3D nomenclature for geometric definition of the drill thus came up. A generalized approach to directly represent the sectional profile of the drill and the grinding profiles using NURBS was first attempted by Sambhav et al. [20]. Design using NURBS 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 sectional geometry is made up of curves and straight lines, the drill point has been modeled. The cutting lips and the chisel edge have been obtained by reducing the surface-surface intersection problem to surface-curve intersection problem and then employing the optimization technique. To generalize the drill model, the sectional geometry and the grinding profile have been represented by NURBS. This is followed by the generic definitions of the angles on the drill point that present the method to evaluate the angles from point to point on the cutting edges. The design has been finally illustrated for a basic model and a generic model. To validate the mathematical model, first a conical drill has been ground on a universal tool and cutter grinder, and compared with the model. Then a drill has been ground with an arbitrarily curved generatrix and compared with the corresponding mathematical model. The comparisons show that the mathematical models are in good agreement with 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 most prevalently 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 the point is formed by the intersection of the grinding surfaces with the fluted shank. The helix angle (k) (or the ratio of the length of cylindrical fluted portion of the drill (L) to the pitch (P)) determines the sweep. The swept surface has eight surface patches, namely, land ðR1 ; R01 Þ, body clearance ðR2 ; R02 Þ, flank ðR3 ; R03 Þ and face ðR4 ; R04 Þ (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).
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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 the fluted portion is composed of lines and arcs and the grinding surface forms a circular cone about an inclined axis. 2.1.1. Drill blank The 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 model The 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 form land and body clearance respectively, when swept helically. V3V4 and V4V5 are arcs of circles which on sweeping form flank and 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 the land about axis (c) and diameter of cutting end of drill (Dc). Keeping the center of drill cross-section at the global origin (O) of the Cartesian system, position vectors of the end vertices of different curve sections (V1 . . . V5) and center points of the two circular arcs (C1, C2) are given by the relations:
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Fig. 4. Method for geometric modeling of drill.
Fig. 5. Half of the sectional geometry of the basic drill model.
9 0 0 1 ; > > > > c > þ l1 sin c l1 cos c 0 1 ; V2 ¼ > > 2 > D > > c > þ l1 sin c þ l2 cos c ðl1 cos c l2 sin cÞ 0 1 ; V3 ¼ 2 > Dc > > > > V4 ¼ þ l sin c þ ðl þ r Þ cos c r cosð c þ h Þ ðl cos c ðl þ r Þ sin c þ r sinð c þ h ÞÞ 0 1 ; 1 2 2 2 2 1 2 2 2 2 = D 2 c : V5 ¼ 2 0 0 1 ; > c > > > C 1 ¼ D þ l sin c þ ðl þ r Þ cos c cos c ðl þ r Þ sin c Þ 0 1 ; ðl 1 2 2 1 2 2 > > 2 > > > C 2 ¼ D2c r1 cosð/1 þ /2 Þ r 1 sinð/1 þ /2 Þ 0 1 ; > > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi >
2 2 > ðV 4x V 5x Þ þðV 4y V 5y Þ > 1 V 4y V 5y 1 > ; ; / ¼ tan where; /1 ¼ cos 2 2r 2 V 5x V 4x
V1 ¼
Dc
2D
ð2:1Þ
The curve vector having end points Vi and Vi+1 is termed as ~ pi ðsÞ. Specifically, using Eq. (2.1),
~ p3 ðsÞ ¼ ðC 1x r 1 cosðc þ ð1 sÞh1 ÞÞðC 1y þ r 1 sinðc þ ð1 sÞh1 ÞÞ 0 1 ; ~ p4 ðsÞ ¼ ðC 2x þ r 2 cos ð/1 þ /2 þ sðp 2/1 ÞÞÞ C 2y r2 sin ð/1 þ /2 þ sðp 2/1 ÞÞ 0 1 ; where 0 6 s 6 1.
ð2:2Þ ð2:3Þ
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2.1.3. Constraints on the choice of the variables The 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
9 ðC 2x V 4x Þ2 þ ðC 2y V 4y Þ2 ¼ r 22 ; > > > = 2 2 2 ðC 2x V 5x Þ þ ðC 2y V 5y Þ ¼ r 2 ; > > > ; and; C 22x þ C 22y ¼ ðr 2 þ r w Þ2 :
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. Alternatively, r2 and rw are related as
rw ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C 22x þ C 22y r 2 :
ð2:5Þ
The value of r2 can be obtained iteratively using the above equation. The point of contact of segment V4V5 and the web circle is found out using the fact,
xw ¼ C 2x
rw ; r2 þ rw
yw ¼ C 2y
rw : r2 þ rw
/1 /2 Þ At this point, s ¼ ðp/po2/ , where /o ¼ tan1 1
ð2:6Þ
C 2y C 2x
.
2.1.4. Flute geometry of the basic model The 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:
2
cos / sin / 6 6 sin / cos / 6 ½T s ¼ 6 6 0 0 4 0
0
0 0 1 P/ 2p
0
3
7 07 7 7; 07 5
ð2:7Þ
0
where 0 6 / 6 2pP L. The helicoidal surfaces R1 to R4 are formed on the basis of following sweep rules:
Land R1 ¼ ~ p1 ðsÞ ½T s ¼ ~ p1 ðs; /Þ; Body clearance R2 ¼ ~ p2 ðsÞ ½T s ¼ ~ p2 ðs; /Þ; Flank R3 ¼ ~ p3 ðsÞ ½T s ¼ ~ p3 ðs; /Þ;
ð2:10Þ
and; Face; R4 ¼ ~ p4 ðsÞ:½T ¼ ~ p4 ðs; /Þ:
ð2:11Þ
ð2:8Þ ð2:9Þ
The flute geometry for the remaining part of the drill can be obtained by rotating the respective surfaces by 180° about the Z-axis. The corresponding surfaces are termed as R01 ; R02 ; R03 and R04 , respectively. 2.1.5. Point geometry of the basic model For 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 angle d. The flank R3 is in contact with lip relief surface R5 at the end of the fluted shank. The primary cutting lips are formed by the intersection of the helical surface of flute with the lip relief surface, while the chisel edge is formed by the intersection of the 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 ~ eab ðtÞ where,
9 8 Dc þ t Hx þ D2c > > 2 > > > > < H = y ~ eab ðtÞ ¼ ; D c > > > > t Hx þ 2 = tan bo > > : ; 1
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, here da2 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:
Dc da3 ¼ Hx = tan f Hx þ = tan bo : 2
ð2:13Þ
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Table 1 Approximate values of drill axis angles [21]. Relief angle a (degrees)
Point angle 2b 60
70
90
118
140
160
180
45 45 44 44
59 58 57 56
70 68 66 62
79 76 71 66
86 80 74 68
Angle b0 (degrees) 6 12 18 24
30 30 30 30
35 35 35 35
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 is given by the relation
f ¼ bo d:
ð2:14Þ
Mathematically the process is represented by
R5 ¼ ~ p5 ðt; uÞ ¼ ½RAA1 feab ðtÞg;
ð2:15Þ
where,
i1 h i1 i h i h 1 h RAA1 ¼ T y;da2 T z;da3 Ry;f Rz;u Ry;f T z;da3 T y;da2 ; 2
3 1 0 0 0 h i 60 1 0 d 7 a2 7 6 T y;da2 ¼ 6 7; 40 0 1 0 5 0
0
0
2
3 1 0 0 0 h i 60 1 0 0 7 6 7 T z;da3 ¼ 6 7; 4 0 0 1 da3 5 0
1
0
0
ð2:16Þ
2
3 cf 0 sf 0 6 0 07 60 1 7 Ry;f ¼ 6 7; 4 sf 0 cf 0 5
1
0
0
0
1
2
3 cu su 0 0 7 6 6 su cu 0 0 7 Rz;u ¼ 6 7: 4 0 0 1 05 0
0
0
1
Using Eqs. (2.15) and (2.16), the x, y and z coordinates of the surface R5 are given by the following relations [20]: 2
p5X ðt; uÞ ¼ Aðcos2 f cos u þ sin fÞ Bðcos f sin f cos u cos f sin fÞ;
ð2:17Þ
p5Y ðt; uÞ ¼ A cos f sin u B sin f sin u da2 ;
ð2:18Þ 2
2
p5Z ðt; uÞ ¼ Aðcos f sin f cos u cos f sin fÞ þ Bðsin f cos u þ cos fÞ da3 ;
ð2:19Þ
where, A = Dc/2 + t(Hx + Dc/2), B = t(Hx + Dc/2)/tanb0 + da3, 0 6 s 6 1 Second lip relief surface R6 is formed by rotating R5 about the vertical axis by an angle of 180°. For an n-fluted drill, surface patches R6 to R3+n can be formed by rotating the surface patch R5 about the axis by angles 2pm/n, respectively, where m ? 1 . . . (n 1). 2.1.6. Primary cutting lip in the basic model The primary cutting lip is obtained as the intersection of the face with the cone. It is the classical surface-surface intersection problem and the curve can be obtained by using Timmer’s method. In this paper, the problem has been reduced to finding out the intersection between a curve and a surface and the points of intersection have been obtained using optimization algorithms, as convergence was faster here. At a point of intersection,
! d1 jz ¼ ~ p04 ðs; /Þjz ~ p5 ðt; uÞ ¼ 0; ~ p04 ðs; /Þjz
ð2:20Þ ~ p04 ðs; /Þ
~ p04 ðs; /Þ
~ p04 ðs; /Þ
p04 ðsÞ
where is the curve traced on at any Z. is the helicoidal surface given by ¼ ½T s and
! p04 ðsÞ 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 model The chisel edge is obtained as the intersection of R5 and R6. The intersection of the two surfaces can be found out by finding out the points where
! d2 ¼ ~ p5 ðt; uÞ ¼ 0: p6 ðt; uÞ ~
ð2:21Þ
To simplify the problem, the contour of the surfaces at different values of z can be found out and points of intersection obtained. This problem can again be solved using optimization technique.
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! d2 jz ¼ ~ p6 ðt; uÞjz ~ p5 ðt; uÞjz ¼ 0:
ð2:22Þ 0 4;
The two extremities of the chisel edge are evaluated by finding the intersection of R4, R5 & R6 and R R5 and R6 . The first of these is obtained by solving the equation: 2
d3 ¼ j~ p4 ðt; uÞ ~ p5 ðt; uÞj2 þ j~ p6 ðs; /Þ ~ p4 ðt; uÞj2 þ j~ p6 ðs; /Þ ~ p5 ðt; 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 chisel edge. 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 geometry The sectional geometry is designed as shown in Fig. 6. Here, curves ~ p3 ðsÞ and ~ p4 ðsÞ of the basic model are defined as ~ pg2 ðsÞ and ~ pg3 ðsÞ between vertices V2, V3 and V3, V4, respectively, while ~ pg1 ðsÞ remains the same as ~ p1 ðsÞ. The superscript 0 g0 denotes the generic model. The body clearance forms a part of the flank now. The curves in this generic model are defined as
! Pn wi Nm;mþi ðsÞbi ~ ; pgj ðsÞ ¼ Pi¼0 n i¼0 wi N m;mþi ðsÞ where, N1,i(s) = di such that
Nm;mþi ðsÞ ¼
j ¼ 2; 3:
ð2:24Þ
di ¼ 1 for s 2 ½si1 ; si Þ ¼ 0; elsewhere
s si smþi s Nm1;mþi1 ðsÞ þ Nm1;mþi ðsÞ smþi siþ1 smþi1 si
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 geometry The flute geometry is obtained in a similar manner as for the basic model as a surface of sweep. In this case,
Rgi ¼ ~ pgi ðsÞ ½T s ¼ ~ pgi ðs; /Þ;
i ¼ 1; 2; 3:
ð2:26Þ
2.2.3. Point geometry The generatrix of revolution is a NURBS in this case. Here
~ egab ðtÞ ¼ qx ðtÞ Hy
qz ðtÞ1 ;
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 Rg4 and Rg5 can be evaluated. Surface Rg4 ¼ ~ pg4 ðt; uÞ ¼ ½RAA1 :~ egab ðtÞ where ½RAA1 has a similar definition. Here, A = qx(t) and B ¼ qz ðtÞ þ da3 . Rg5 is formed by rotating Rg4 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.
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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 the present work. Table 2 shows the methodology of formation of conventional angles by various surface patches shaping the drill. The angles of twist drills presented here are as per the conventional nomenclatures. Formation of surfaces Rg1 to Rg5 for the generic model has been explained earlier in the paper. The unit tangents and normal to Rg4 are obtained as follows:
g g ^g4;t ðt; uÞ ¼ @~ p4 ðt; uÞ=@t = @~ p4 ðt; uÞ=@t ; p g g ^g4;u ðt; uÞ ¼ @~ p4 ðt; uÞ=@ u = @~ p4 ðt; uÞ=@ u ; p
ð3:2Þ
^g4;t ðt; uÞ p ^g4;u ðt; uÞ: ^ g4 ¼ p n
ð3:3Þ
ð3:1Þ
The unit tangents and normal can be evaluated from point to point on the cutting edges using Eqs. (3.1)–(3.3). ^ g4 is projected on ZX-plane and angle between this unit projected normal Half point angle (bg): As shown in Table 2, n ^ g4p Þ and the unit vector normal to YZ-plane ð^iÞ is half point angle bg. vector ðn 0 ^ g4 at any point on the chisel edge projected to XY-plane is given by n ^ g4p . The scalar Chisel edge angle (wg): The unit normal n g0 ^ 4p and unit vector ^j gives the chisel edge angle wg. product of n 00 00 ^ g4 is projected on YZ-plane given by n ^ g4p . The relief angle ag is the angle between n ^ g4p and unit Relief angle (ag): n ^ vector k. Helix angle (kg): Helix angle (kg) is formed by Rg1 and satisfies the relation
kg ¼ tan1 ðpDc =PÞ: Peripheral relief angle (a
ð3:4Þ g p ):
g 1
g 1
This angle is formed by the surface patch R with XY plane. The patch R is formed by helically sweeping parametric edge formed by joining vertices V1 and V2. The edge is defined as
~ pg1 ðsÞ ¼ ½ðDc =2 þ sl sin cÞ sl cos c 0 1: Table 2 Methodology to form the conventional angles. Conventional angles
Formed by the normal to
With the plane
Plane of projection
Half point angle, bg
Rg4 Rg4 Rg4 Rg3 Rg1
YZ
ZX
Chisel edge angle, wg Relief angle, ag Helix angle, kg Peripheral relief angle, agp
ZX
XY
XY
YZ
YZ
ZX
YZ
XY
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The surface patch Rg1 formed by sweeping ~ pg1 ðsÞ, is expressed by the following relation
~ pg1 ðs; /Þ ¼ ½ððDc =2 þ sl sin cÞ cos / sl cos c sin /ÞððDc =2 þ sl sin cÞ sin / þ sl cos c cos /ÞðP/=2pÞ 1 The tangents and normal to the surface patch at an arbitrary point are:
~ pg1;s ðs; /Þ ¼ l cosðc /Þ^i þ l cosðc /Þ^j; ^ ~ pg1;/ ðs; /Þ ¼ ðDc =2: sin / sl cosðc /ÞÞ^i þ ðDc =2: cos / þ sl sinðc /ÞÞ^j þ P=2pk; 2 ^ ~ ng1 ¼ Pl=2p cosðc /Þ^i Pl=2p sinðc /Þ^j ðDc =2:l sin c sl Þk:
^ g1p ¼ cosðc /Þ^i sinðc /Þ^j. The normal on projection to XY plane is given by n g ^ ^ 1p with unit vector i provides cos agp ¼ cosðc /Þ. At z = 0 plane, / = 0 and this leads to the mapping The scalar product of n relation agp ¼ c. Thus, we obtain the generic definitions for the half point angle, chisel edge angle, relief angle, helix angle and the peripheral 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 are evaluated at the midpoint of each element, which vary from point to point. Using the matrix of the cutting elements, the ! ~ ~ cutting 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 of ~ is the angular velocity. The inclination angle (i) is obtained using the relation the midpoint of the cutting lip and x
b; cosð90 þ iÞ ¼ D^l V
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: 0
0
~ pg3;/ pg ~ ^ g3 ¼ 3;s0 ; n 0 ~g pg3;/ p3;s ~ 0
0
ð3:6Þ 0
where ~ pg3;s and ~ pg3;/ denote the derivatives of ~ pg3 ðs; /Þ along s- and /- directions, respectively. Thus normal rake angle cgn is obtained as:
cos
!
V D~l : ^ g3 ! cgn ¼ n 2 V D~l
p
ð3:7Þ
4. Illustration of the model Using the methodology presented above, geometry of the basic and the generic model are generated. The sectional geometry, 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.
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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 and chisel 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 prevalent drills. First this sectional profile was used for modeling drill point with a straight generatrix (as above), and then for a curved generatrix. For the curved generatrix, the control points for NURBS of order 3 were chosen as follows:
fð12:34; 0; 2:54Þ; ð8:82; 0; 1:06Þ; ð7:05; 0; 0Þ; ð5:29; 0; 1:91Þ; ð3:53; 0; 2:97Þ; ð1:76; 0; 3:60Þ; ð0; 0; 4:24Þg:
Fig. 8b. Flute profile in the basic model as a sweep surface.
Fig. 8c. Grinding cones as surfaces of revolution.
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Fig. 8d. Cutting lips and chisel edge for the basic model.
Table 3 Basic drill profile data. Dc = 12.5 mm Hx = 2.2 mm b0 = 59° r1 = 5.5 mm
P = 60 mm l1 = 1.5, l2 = 0.2 mm da2 ¼ 1 mm r2 = 4.37 mm
rw = 0.7 mm c = 12° d = 35° h1 = 81.82°
Table 4 Generic drill profile data. Dc = 14 mm Hx = 2.2 mm b0 = 59°
P = 68.3 mm l = 0.83 mm da2 ¼ 2 mm
rw = 0.7 mm c = 10.9° d = 50°
Using the parameters listed in Table 4, the surfaces of sweep and revolution were obtained. The cutting lips and the chisel edge were subsequently obtained through optimization algorithms using MATLAB. The flute geometry and the grinding surfaces with a generic profile are shown in Figs. 9a and 9b respectively. The cutting lips and the chisel edge obtained for the straight generatrix are shown in Fig. 9c, while those obtained for the curved generatrix are shown in Fig. 9d. It can be seen clearly 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 is described below. First, a drill grinder was selected which follows the same grinding principle as described above. It is HMT universal tool and cutter grinder GTC-28T at IIT Kanpur. The parameters inherent to the grinding process were derived by measuring the distances and angles on the machine (Fig. 10) while other parameters were set on the grinder. An HSS drill of Addison make of diameter 12.5 mm was chosen. The relevant data are shown in Table 5. Using a flat Alumina grinding wheel, the drill was ground 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 micron probe 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 cutting lip, chisel edge and heel. The same were measured with the probe and registered with the model which is shown graphically in Fig. 11d. For registration, the peak of the drill was chosen as the origin. A comparison of the two data for the primary cutting lips are presented in Table 6a, for the two heels in Table 6b while Table 6c compares the data for the chisel edge. To compare the two data, coordinates were captured at the same z-values, and the x- and y-values were investigated. The primary cutting lip and the chisel edge are in close proximity, while the lower part of the heel slightly deviates away from the analytically obtained data. This is because the model uses a fixed generatrix for the grinding surface of revolution. But practically, the generatrix provided by the grinding wheel with flat face down as the drill swivels about its axis and the line of contact recedes away with the swivel of the grinder arm.
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Fig. 9a. Flute profile of the generic model.
Fig. 9b. Grinding cone with a generic profile.
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 data points 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 parameters s0, s1, . . . sn corresponding to each data point. If the unknown control points are represented by ~ ci ; i ¼ 0 . . . n, we have
K. Sambhav et al. / Applied Mathematical Modelling 36 (2012) 2384–2403
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.
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K. Sambhav et al. / Applied Mathematical Modelling 36 (2012) 2384–2403 Table 5 Data used for validation of conical and generic drill. Dc = 12.5 mm Hx = 15 mm b0 = 59°
P = 80 mm l = 1.6 mm da2 ¼ 2 mm
rw = 1.25 mm c = 11° 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 ¼
n X
Np;pþi ðuÞ~ ci :
ð5:1Þ
i¼0
Using the mapping of the data points and the parameters, we have,
~ qk ¼ ~ cðsk Þ ¼
n X
Np;pþi ðsk Þ~ ci ;
k ¼ 0...n
i¼0
or,
3 2 32 3 ~ ~ Np;p ðs0 Þ Np;pþ1 ðs0 Þ Np;pþ2 ðs0 Þ . . . Np;pþn ðs0 Þ c0 q0 7 7 6 ~ 7 6 N ðs Þ N 6 ~ 6 q1 7 6 p;p 1 p;pþ1 ðs1 Þ N p;pþ2 ðs1 Þ . . . N p;pþn ðs1 Þ 76 c1 7 76 7 6 7 6 7 7 7 6 6 6 fQg ¼ 6 7 ¼ 6 76 7 ¼ ½NfCg 76 7 6 7 6 54 5 45 4 ~ ~ Np;p ðsn Þ Np;pþ1 ðsn Þ Np;pþ2 ðsn Þ . . . Np;pþn ðsn Þ cn qn 2
ð5:2Þ
K. Sambhav et al. / Applied Mathematical Modelling 36 (2012) 2384–2403
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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 ½N1 fQ g
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 the control points. Having defined the generatrix, the coordinates on the cutting lip, heel and chisel edge were calculated using a MATLAB code. Next, the drill was ground and the drill point was generated (Fig. 12b). The coordinates on the cutting lip, heel and chisel edge were measured experimentally. The drill flute together with the curved grinding surfaces were modeled (Fig. 12c).
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Table 6a Comparison of coordinates of the two primary cutting lips for the conical drill. S. No.
Primary cutting lip 1
Primary cutting lip 2
Theoretical 1 2 3 4 5 6 7 8 9 10 11 12
0.35 1.06 1.54 2.02 2.66 3.15 3.8 4.15 4.45 4.94 5.27 5.75
Experimental 1.22 0.95 0.81 0.69 0.56 0.46 0.35 0.3 0.27 0.2 0.15 0.05
3.55 3.06 2.76 2.46 2.06 1.76 1.36 1.16 0.96 0.66 0.46 0.16
0.4 1.06 1.77 2.26 2.7 3.26 3.78 4.23 4.66 5.04 5.23 5.7
1.08 1.03 0.81 0.61 0.61 0.54 0.34 0.25 0.16 0.16 0.09 0.08
Theoretical 3.55 3.06 2.76 2.46 2.06 1.76 1.36 1.16 0.96 0.66 0.46 0.16
0.35 1.06 1.54 2.02 2.66 3.15 3.8 4.15 4.45 4.94 5.27 5.75
Experimental 1.22 0.95 0.81 0.69 0.56 0.46 0.35 0.3 0.27 0.2 0.15 0.05
3.55 3.06 2.76 2.46 2.06 1.76 1.36 1.16 0.96 0.66 0.46 0.16
0.4 1.01 1.72 2.19 2.67 3.18 3.68 4.16 4.73 4.96 5.17 5.73
1.08 0.98 0.78 0.58 0.58 0.48 0.28 0.23 0.13 0.11 0.08 0.08
3.55 3.06 2.76 2.46 2.06 1.76 1.36 1.16 0.96 0.66 0.46 0.16
Table 6b Comparison of coordinates of the two heels for the conical drill. S. No.
Heel 1
Heel 2
Theoretical 1 2 3 4 5 6 7 8
0.35 0.3 0.62 0.74 0.66 0.49 0.39 0.33
Experimental 1.22 1.8 2.35 3.26 3.8 4.32 4.56 4.69
3.55 2.91 2.51 2.06 1.86 1.66 1.56 1.52
0.4 0.36 0.66 0.86 0.79 0.66 0.61 0.41
1.08 1.71 2.21 3.01 3.61 4.01 4.41 4.81
Theoretical 3.55 2.91 2.51 2.06 1.86 1.66 1.56 1.52
0.35 0.3 0.62 0.74 0.66 0.49 0.39 0.33
Experimental 1.22 1.8 2.35 3.26 3.8 4.32 4.56 4.69
3.55 2.91 2.51 2.06 1.86 1.66 1.56 1.52
0.4 0.36 0.66 0.96 0.92 0.79 0.66 0.53
1.08 1.68 2.17 2.96 3.55 3.93 4.32 4.67
3.55 2.91 2.51 2.06 1.86 1.66 1.56 1.52
Table 6c Comparison of coordinates of the chisel edge for the conical drill. S. No.
Theoretical
1 2 3
0.35 0.00 0.35
Experimental 1.22 0.00 1.22
3.55 3.56 3.55
0.40 0.00 0.40
1.08 0.00 1.08
3.55 3.58 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 is shown in Fig. 12d. The same has been tabulated in Table 7. As the primary cutting lips are curled along z-axis here, for comparison, the data were captured for identical x-values and the y- and z-values were looked into. The comparison shows again a good proximity between the model and the generated profile, except near the lower part of the primary cutting lip and heel for similar reason as mentioned above.
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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 hole quality, prediction of drill wear and drill life, study of drill dynamics, design modification for easy chip evacuation, etc. The study of drilling performance in the micro-domain is another area where new shapes of drills need to be investigated. And each of these concerns can be used for the optimization of the drill geometry for different objectives. Such tasks have been taken up by researchers in the past too. But the drill geometries reported in the earlier work have been limited to few standard shapes. Using NURBS to model the drill geometry gives a new dimension to these tasks and gives a much higher degree of freedom for predictions or optimization tasks. This generates a new scope for research in all these areas. Working in this direction, 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 model through regression analysis of the forces as a function of the rake angle, feed and velocity.
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Fig. 12d. Comparison of experimental and theoretical results for a generic drill point.
Table 7a Comparison of coordinates of the two primary cutting lips for the generic drill. S. No.
Primary cutting lip 1
Primary cutting lip 2
Theoretical 1 2 3 4 5 6 7 8 9 10 11 12
0.35 1.01 1.42 1.94 2.29 2.71 3.21 3.93 4.31 4.73 5.88 6.22
Experimental 1.26 0.88 0.7 0.48 0.18 0.23 0.11 0.01 0.01 0.01 0.1 0.19
2.35 1.67 1.3 0.91 0.45 0.51 0.37 0.24 0.2 0.15 0.26 0.39
0.35 1.01 1.42 1.94 2.29 2.71 3.21 3.93 4.31 4.73 5.88 6.22
1.15 0.85 0.69 0.47 0.34 0.28 0.22 0.24 0.21 0.23 0.29 0.42
Theoretical 2.39 1.89 1.44 1.08 0.87 0.65 0.52 0.35 0.29 0.27 0.4 0.39
0.35 0.88 1.26 1.78 2.35 2.82 3.33 3.73 4.31 4.65 5.82 6.06
Experimental
1.26 0.95 0.77 0.54 0.34 0.2 0.09 0.03 0.01 0.01 0.08 0.15
2.35 1.79 1.45 1.01 0.67 0.48 0.34 0.28 0.2 0.16 0.23 0.33
0.35 0.88 1.26 1.78 2.35 2.82 3.33 3.73 4.31 4.65 5.82 6.06
1.15 0.73 0.63 0.35 0.15 0.03 0.05 0.1 0.03 0 0.36 0.31
2.39 1.99 1.61 1.09 0.68 0.54 0.35 0.28 0.23 0.19 0.16 0.21
Table 7b Comparison of coordinates of the heels for the generic drill. S. No.
Heel 1
Heel 2
Theoretical 1 2 3 4 5 6 7 8
0.35 0.09 0.09 0.35 0.38 0.39 0.18 0.3
Experimental 1.26 1.53 1.77 2.37 2.5 3.12 3.96 4.82
2.35 1.93 1.66 1.15 1.06 0.7 0.33 0.02
0.35 0.06 0.07 0.38 0.46 0.41 0.12 0.16
1.15 1.6 2 2.56 2.96 3.57 4.15 5.03
Theoretical 2.39 1.93 1.66 1.15 1.06 0.7 0.33 0.05
0.35 0.09 0.09 0.35 0.38 0.39 0.18 0.3
Experimental 1.26 1.53 1.77 2.37 2.5 3.12 3.96 4.82
2.35 1.93 1.66 1.15 1.06 0.7 0.33 0.02
0.35 0.07 0.06 0.41 0.43 0.44 0.16 0.19
1.15 1.54 1.94 2.41 2.82 3.48 4.06 4.93
2.39 1.93 1.66 1.15 1.06 0.7 0.33 0.13
Table 7c Comparison of coordinates of the chisel edge for the generic drill. S. No.
Theoretical
1 2 3 4
0.35 0.17 0.00 0.35
Experimental 1.26 0.58 0.00 1.26
2.35 2.43 2.52 2.35
0.35 0.15 0.08 0.35
1.15 0.46 0.11 1.15
2.39 2.45 2.49 2.39
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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 shape and cutting angles can be changed just by changing the control points or their respective weights. The cutting edges 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 been solved converting it to surface-curve intersection and employing optimization technique subsequently. The cutting lip is obtained as the intersection of a sweep surface and a surface of revolution, while the chisel edge is obtained as the intersection of two surfaces of revolution. Generic definitions of the conventional angles on the drill point are presented along the primary cutting lip and the chisel edge of the drill. The model has been illustrated in MATLAB environment and validated for a conical drill as well as a drill with curved 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 for different objectives. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
[11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
H.K. Tonshoff, W. Spintig, W. Konig, A. Neises, Keynote paper, machining of holes: developments in drilling technology, Ann. CIRP 43 (2) (1994). W.D. Tsai, S.M. Wu, A mathematical model for drill point design and grinding, ASME J. Eng. Indust. 101 (1979) 330–340. D.F. Galloway, Some experiments on the influence of various factors on drill performance, Trans. ASME 79 (1956) 191–231. E.J.A. Armarego, C.Y. Cheng, Drilling with flat rake face and conventional twist drills. I- Theoretical investigations and II- experimental investigations, Int. J. Machine Tool Design Res. 12 (1972) 17–54. E.J.A. Armarego, D. Kang, Computer aided modeling of the fluting process for twist drill design and manufacture, CIRP Ann. 47 (1) (1998) 259–264. M.A. Fugelso, Conical flank twist drill points, Int. J. Machine Tools Manufact. 30 (2) (1990) 291–295. K.H. Fuh, W.C. Chen, Cutting performance of thick web drills with curved primary cutting edges, Int. J. Machine Tools Manufact. 35 (7) (1995) 975–991. S. Fujii, D.F. DeVries, S.M. Wu, Analysis of the chisel edge and effect of the d-theta relationship on drill point geometry, ASME J. Eng. Indust. 93 (4) (1971) 1093–1105. J.F. Hsieh, P.D. Lin, Mathematical model of multiflute drill point, Int. J. Machine Tools Manufact. 42 (2009) 1181–1193. A.R. Watson, Drilling model for cutting lip and chisel edge and comparison of experimental and predicted results. I- Initial cutting lip model, II- Revised cutting lip model, III- Drilling model for chisel edge, IV- Drilling tests to determine chisel edge contribution to thrust and torque, Int. J. Machine Tool Design Res. 25 (4) (1985) 347–404. D.A. Stephenson, J.S. Agapiou, Calculation of main cutting edge forces and torque for drills with arbitrary point geometries, Int. J. Machine Tools Manufact. 32 (1992) 521–538. V. Chandrasekharan, S.G. Kapoor, R.E. DeVor, A mechanistic model to predict the cutting forces in drilling: with application to fiber reinforced composite materials, ASME J. Eng. Indust. 117 (1995) 559–570. V. Chandrasekharan, S.G. Kapoor, R.E. DeVor, Mechanistic model to predict the cutting force system for arbitrary drill point geometry, J. Manufact. Sci. Eng. 120 (1998) 563–570. A. Paul, S.G. Kapoor, R.E. DeVor, A chisel edge model for arbitrary drill point geometry, J. Manufact. Sci. Eng. 127 (2005) 23–32. A. Paul, S.G. Kapoor, R.E. DeVor, Chisel edge and cutting lip shape optimization for improved twist drill point design, Int. J. Machine Tools Manufact. 45 (2005) 421–431. C. Lin, S.K. Kang, K.F. Ehmann, Helical micro-drill point design and grinding, ASME J. Eng. Indust. 117 (1995) 277–287. H.M. Shi, H.S. Zhang, L.S. Xiong, A study on curved edge drills, ASME J. Eng. Indust. 116 (1994) 267–273. L. Xiong, N. Fang, H. Shi, A new methodology for designing a curve-edged twist drill with an arbitrarily given distribution of the cutting angles along the tool cutting edge, Int. J. Machine Tools Manufact. 49 (2009) 667–677. P. Tandon, P. Gupta, S.G. Dhande, Geometric modeling of fluted cutters, J. Comput. Inform. Sci. Eng. 8 (2008) 1–15. K. Sambhav, P. Tandon, S.G. Dhande, CAD based mechanistic modeling of forces for generic drill point geometry, Comput. Aided Design Appl. 7 (6) (2010) 809–819. S. Popov, L. Dibner, A. Kamenkovich, Sharpening of Cutting Tools, Mir Publishers, Moscow, 1988.
Contents lists available at SciVerse ScienceDirect
Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Geometric modeling and validation of twist drills with a generic point 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, India Mechanical Engg. & Design Programme, PDPM Indian Institute of Information Technology, Design and Manufacturing Jabalpur, Jabalpur 482 011, Madhya Pradesh, India c Department of Mechanical Engg. & Computer Sce. and Engg., Indian Institute of Technology Kanpur, Kanpur 208 016, Uttar Pradesh, India b
a r t i c l e
i n f o
Article history: Received 13 May 2011 Received in revised form 8 August 2011 Accepted 16 August 2011 Available online 5 September 2011 Keywords: Drill point geometry Generic definition NURBS
a b s t r a c t Traditionally, twist drills with a few specific point geometry, such as planar, conical, cylindrical, 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 freedom 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 a methodology to model the twist drills with generic point geometry using NURBS. To begin with, a detailed basic model for a fluted twist drill with sectional geometry made up of arcs and straight lines has been presented in terms of bi-parametric surface patches. The coordinates of cutting lips and chisel edge of the drill have been obtained as solution to a surfacecurve intersection problem using optimization algorithm. Subsequently, the model has been generalized by employing NURBS to represent the curves whereby the cutting edges and angles can be altered simply by changing the control points or their respective weights. Using this methodology, the generic definitions of the conventional angles on the drill point have been derived and presented. The proposed model has been illustrated in MATLAB environment and validated experimentally for a conical and an arbitrary point geometry. The experiments show a good conformity with the theoretical evaluations. Ó 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. Thus geometric modeling of the tool is a very crucial part of tool design. Among the machining processes, drilling is an extensively used process [1]. Drills are multi-point rotary cutting tools with a variety of shank and point geometry. The most common of them 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 generic sectional geometry and point profile. Such a generic CAD model will act as core model which can be adapted to develop drills for specific purposes or optimize the geometry for different objectives by suitably modifying the geometric and grinding parameters. Such a model can also be used for numerous downstream applications, thus opening up various probable areas
⇑ Corresponding author. Tel.: +91 761 2632924; fax: +91 761 2632 524. 1 2
E-mail addresses: [email protected] (K. Sambhav), [email protected] (P. Tandon), [email protected], [email protected] (S.G. Dhande). Tel.: +91 9455680200. Tel.: +91 512 2597258/2590763; fax: +91 512 2590260/259 7790.
0307-904X/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2011.08.034
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Nomenclature {} [] k P L L0
vector notation matrix notation helix angle pitch length of flute length of drill shank R1 land surface patch R2 body clearance surface patch R3 flank surface patch R4 face surface patch bo angle of drill positioning d cone half angle Hx, Hy offset of cone apex from the drill axis along x- and y-axes, respectively V1 . . . V5 vertices of segments on the drill section l1 width of the land l2 depth of body clearance radii of flank and face arcs r1, r2 rw web radius h1, /1, /2 angles on the flute cross-section in the basic model c inclination angle of land Dc drill diameter at cutting end Ds shank diameter of drill C1, C2 center points of the two circular arcs on the cross-section in the basic model / angle of sweep u angle of revolution [Ts] transformation matrix s, t parameters of variation R5, R6 relief surfaces ~ eab ðtÞ generatrix of revolution ½RAA1 rotation matrix ~ curve vector joining Vi and Vi+1 pi ðsÞ ~ pi ðs; /Þ parametric definition of surface of sweep ~ pi ðt; uÞ parametric definition of surface of revolution ½T y;da2 translation along y-axis by da2 da3 distance along z-axis f inclination of axis of revolution with the z-axis b half point angle a relief angle ! distance at a chosen value of z di jz g superscript showing generic model ~ curves in the generic model pgi ðsÞ
Rgi
q(t)
w ap
surfaces in the generic model NURBS in parameter t chisel edge angle peripheral relief angle
^gi ðt; uÞ ^gi;t ðt; uÞ unit tangent in the t-direction to surface p p ^ g4 n
unit normal on Rg4
^ g4p n
^ g4 projection of n
D~l
element along the cutting lip normal rake angle
cn
of work. For example, the surface definitions of the tool can be used to model mathematically the grinding process and the effect of the grinding parameters on the tool geometry and machining can be studied for the generic case. This would enable the entire grinding or sharpening process of the tool to be simulated on the computer and the results verified before material removal 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
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L' L
Face
Heel
Flank surface
Helix angle
Chisel edge
Dc Shank Primary cutting edge
Lip relief surface
Land
Flute
Ds
Body clearance
Fig. 1. Projected geometry of drill.
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 mathematical relations between the geometric factors. This was followed by works of several authors, [4–10], to name a few, who modeled or 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 some standard 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 parametric form of the edge. But this method did not relate the geometry to the grinding parameters again. Paul et al. [14,15] used the mechanistic model by Chandrasekharan and geometric model by Tsai and Wu to optimize the shape of the drill point for minimum 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 shapes for different semi-point angles using helical grinding. In his work, he concluded that the model was more general than the commonly used drill point models which were only special cases of this model. A literature review shows that few other authors like Galloway [3], Fugelso [6] and Fuh and Chen [7] have also advocated the design of drill points with curved cutting lips the reason being that curved cutting lip shapes result in a longer drill life than a straight lip for various working conditions and working materials. Shi et al. [17] studied curved edged drills and varied the cutting edge shape to obtain a desired rake angle distribution. Xiong et al. [18] presented methodology to design drills with arbitrarily given distribution of the cutting 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 by a forward and inverse mapping between conventional 2D angles on the drill point and rotational 3D grinding angles. A new 3D nomenclature for geometric definition of the drill thus came up. A generalized approach to directly represent the sectional profile of the drill and the grinding profiles using NURBS was first attempted by Sambhav et al. [20]. Design using NURBS 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 sectional geometry is made up of curves and straight lines, the drill point has been modeled. The cutting lips and the chisel edge have been obtained by reducing the surface-surface intersection problem to surface-curve intersection problem and then employing the optimization technique. To generalize the drill model, the sectional geometry and the grinding profile have been represented by NURBS. This is followed by the generic definitions of the angles on the drill point that present the method to evaluate the angles from point to point on the cutting edges. The design has been finally illustrated for a basic model and a generic model. To validate the mathematical model, first a conical drill has been ground on a universal tool and cutter grinder, and compared with the model. Then a drill has been ground with an arbitrarily curved generatrix and compared with the corresponding mathematical model. The comparisons show that the mathematical models are in good agreement with 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 most prevalently 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 the point is formed by the intersection of the grinding surfaces with the fluted shank. The helix angle (k) (or the ratio of the length of cylindrical fluted portion of the drill (L) to the pitch (P)) determines the sweep. The swept surface has eight surface patches, namely, land ðR1 ; R01 Þ, body clearance ðR2 ; R02 Þ, flank ðR3 ; R03 Þ and face ðR4 ; R04 Þ (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).
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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 the fluted portion is composed of lines and arcs and the grinding surface forms a circular cone about an inclined axis. 2.1.1. Drill blank The 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 model The 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 form land and body clearance respectively, when swept helically. V3V4 and V4V5 are arcs of circles which on sweeping form flank and 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 the land about axis (c) and diameter of cutting end of drill (Dc). Keeping the center of drill cross-section at the global origin (O) of the Cartesian system, position vectors of the end vertices of different curve sections (V1 . . . V5) and center points of the two circular arcs (C1, C2) are given by the relations:
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Fig. 4. Method for geometric modeling of drill.
Fig. 5. Half of the sectional geometry of the basic drill model.
9 0 0 1 ; > > > > c > þ l1 sin c l1 cos c 0 1 ; V2 ¼ > > 2 > D > > c > þ l1 sin c þ l2 cos c ðl1 cos c l2 sin cÞ 0 1 ; V3 ¼ 2 > Dc > > > > V4 ¼ þ l sin c þ ðl þ r Þ cos c r cosð c þ h Þ ðl cos c ðl þ r Þ sin c þ r sinð c þ h ÞÞ 0 1 ; 1 2 2 2 2 1 2 2 2 2 = D 2 c : V5 ¼ 2 0 0 1 ; > c > > > C 1 ¼ D þ l sin c þ ðl þ r Þ cos c cos c ðl þ r Þ sin c Þ 0 1 ; ðl 1 2 2 1 2 2 > > 2 > > > C 2 ¼ D2c r1 cosð/1 þ /2 Þ r 1 sinð/1 þ /2 Þ 0 1 ; > > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi >
2 2 > ðV 4x V 5x Þ þðV 4y V 5y Þ > 1 V 4y V 5y 1 > ; ; / ¼ tan where; /1 ¼ cos 2 2r 2 V 5x V 4x
V1 ¼
Dc
2D
ð2:1Þ
The curve vector having end points Vi and Vi+1 is termed as ~ pi ðsÞ. Specifically, using Eq. (2.1),
~ p3 ðsÞ ¼ ðC 1x r 1 cosðc þ ð1 sÞh1 ÞÞðC 1y þ r 1 sinðc þ ð1 sÞh1 ÞÞ 0 1 ; ~ p4 ðsÞ ¼ ðC 2x þ r 2 cos ð/1 þ /2 þ sðp 2/1 ÞÞÞ C 2y r2 sin ð/1 þ /2 þ sðp 2/1 ÞÞ 0 1 ; where 0 6 s 6 1.
ð2:2Þ ð2:3Þ
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2.1.3. Constraints on the choice of the variables The 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
9 ðC 2x V 4x Þ2 þ ðC 2y V 4y Þ2 ¼ r 22 ; > > > = 2 2 2 ðC 2x V 5x Þ þ ðC 2y V 5y Þ ¼ r 2 ; > > > ; and; C 22x þ C 22y ¼ ðr 2 þ r w Þ2 :
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. Alternatively, r2 and rw are related as
rw ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C 22x þ C 22y r 2 :
ð2:5Þ
The value of r2 can be obtained iteratively using the above equation. The point of contact of segment V4V5 and the web circle is found out using the fact,
xw ¼ C 2x
rw ; r2 þ rw
yw ¼ C 2y
rw : r2 þ rw
/1 /2 Þ At this point, s ¼ ðp/po2/ , where /o ¼ tan1 1
ð2:6Þ
C 2y C 2x
.
2.1.4. Flute geometry of the basic model The 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:
2
cos / sin / 6 6 sin / cos / 6 ½T s ¼ 6 6 0 0 4 0
0
0 0 1 P/ 2p
0
3
7 07 7 7; 07 5
ð2:7Þ
0
where 0 6 / 6 2pP L. The helicoidal surfaces R1 to R4 are formed on the basis of following sweep rules:
Land R1 ¼ ~ p1 ðsÞ ½T s ¼ ~ p1 ðs; /Þ; Body clearance R2 ¼ ~ p2 ðsÞ ½T s ¼ ~ p2 ðs; /Þ; Flank R3 ¼ ~ p3 ðsÞ ½T s ¼ ~ p3 ðs; /Þ;
ð2:10Þ
and; Face; R4 ¼ ~ p4 ðsÞ:½T ¼ ~ p4 ðs; /Þ:
ð2:11Þ
ð2:8Þ ð2:9Þ
The flute geometry for the remaining part of the drill can be obtained by rotating the respective surfaces by 180° about the Z-axis. The corresponding surfaces are termed as R01 ; R02 ; R03 and R04 , respectively. 2.1.5. Point geometry of the basic model For 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 angle d. The flank R3 is in contact with lip relief surface R5 at the end of the fluted shank. The primary cutting lips are formed by the intersection of the helical surface of flute with the lip relief surface, while the chisel edge is formed by the intersection of the 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 ~ eab ðtÞ where,
9 8 Dc þ t Hx þ D2c > > 2 > > > > < H = y ~ eab ðtÞ ¼ ; D c > > > > t Hx þ 2 = tan bo > > : ; 1
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, here da2 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:
Dc da3 ¼ Hx = tan f Hx þ = tan bo : 2
ð2:13Þ
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Table 1 Approximate values of drill axis angles [21]. Relief angle a (degrees)
Point angle 2b 60
70
90
118
140
160
180
45 45 44 44
59 58 57 56
70 68 66 62
79 76 71 66
86 80 74 68
Angle b0 (degrees) 6 12 18 24
30 30 30 30
35 35 35 35
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 is given by the relation
f ¼ bo d:
ð2:14Þ
Mathematically the process is represented by
R5 ¼ ~ p5 ðt; uÞ ¼ ½RAA1 feab ðtÞg;
ð2:15Þ
where,
i1 h i1 i h i h 1 h RAA1 ¼ T y;da2 T z;da3 Ry;f Rz;u Ry;f T z;da3 T y;da2 ; 2
3 1 0 0 0 h i 60 1 0 d 7 a2 7 6 T y;da2 ¼ 6 7; 40 0 1 0 5 0
0
0
2
3 1 0 0 0 h i 60 1 0 0 7 6 7 T z;da3 ¼ 6 7; 4 0 0 1 da3 5 0
1
0
0
ð2:16Þ
2
3 cf 0 sf 0 6 0 07 60 1 7 Ry;f ¼ 6 7; 4 sf 0 cf 0 5
1
0
0
0
1
2
3 cu su 0 0 7 6 6 su cu 0 0 7 Rz;u ¼ 6 7: 4 0 0 1 05 0
0
0
1
Using Eqs. (2.15) and (2.16), the x, y and z coordinates of the surface R5 are given by the following relations [20]: 2
p5X ðt; uÞ ¼ Aðcos2 f cos u þ sin fÞ Bðcos f sin f cos u cos f sin fÞ;
ð2:17Þ
p5Y ðt; uÞ ¼ A cos f sin u B sin f sin u da2 ;
ð2:18Þ 2
2
p5Z ðt; uÞ ¼ Aðcos f sin f cos u cos f sin fÞ þ Bðsin f cos u þ cos fÞ da3 ;
ð2:19Þ
where, A = Dc/2 + t(Hx + Dc/2), B = t(Hx + Dc/2)/tanb0 + da3, 0 6 s 6 1 Second lip relief surface R6 is formed by rotating R5 about the vertical axis by an angle of 180°. For an n-fluted drill, surface patches R6 to R3+n can be formed by rotating the surface patch R5 about the axis by angles 2pm/n, respectively, where m ? 1 . . . (n 1). 2.1.6. Primary cutting lip in the basic model The primary cutting lip is obtained as the intersection of the face with the cone. It is the classical surface-surface intersection problem and the curve can be obtained by using Timmer’s method. In this paper, the problem has been reduced to finding out the intersection between a curve and a surface and the points of intersection have been obtained using optimization algorithms, as convergence was faster here. At a point of intersection,
! d1 jz ¼ ~ p04 ðs; /Þjz ~ p5 ðt; uÞ ¼ 0; ~ p04 ðs; /Þjz
ð2:20Þ ~ p04 ðs; /Þ
~ p04 ðs; /Þ
~ p04 ðs; /Þ
p04 ðsÞ
where is the curve traced on at any Z. is the helicoidal surface given by ¼ ½T s and
! p04 ðsÞ 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 model The chisel edge is obtained as the intersection of R5 and R6. The intersection of the two surfaces can be found out by finding out the points where
! d2 ¼ ~ p5 ðt; uÞ ¼ 0: p6 ðt; uÞ ~
ð2:21Þ
To simplify the problem, the contour of the surfaces at different values of z can be found out and points of intersection obtained. This problem can again be solved using optimization technique.
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! d2 jz ¼ ~ p6 ðt; uÞjz ~ p5 ðt; uÞjz ¼ 0:
ð2:22Þ 0 4;
The two extremities of the chisel edge are evaluated by finding the intersection of R4, R5 & R6 and R R5 and R6 . The first of these is obtained by solving the equation: 2
d3 ¼ j~ p4 ðt; uÞ ~ p5 ðt; uÞj2 þ j~ p6 ðs; /Þ ~ p4 ðt; uÞj2 þ j~ p6 ðs; /Þ ~ p5 ðt; 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 chisel edge. 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 geometry The sectional geometry is designed as shown in Fig. 6. Here, curves ~ p3 ðsÞ and ~ p4 ðsÞ of the basic model are defined as ~ pg2 ðsÞ and ~ pg3 ðsÞ between vertices V2, V3 and V3, V4, respectively, while ~ pg1 ðsÞ remains the same as ~ p1 ðsÞ. The superscript 0 g0 denotes the generic model. The body clearance forms a part of the flank now. The curves in this generic model are defined as
! Pn wi Nm;mþi ðsÞbi ~ ; pgj ðsÞ ¼ Pi¼0 n i¼0 wi N m;mþi ðsÞ where, N1,i(s) = di such that
Nm;mþi ðsÞ ¼
j ¼ 2; 3:
ð2:24Þ
di ¼ 1 for s 2 ½si1 ; si Þ ¼ 0; elsewhere
s si smþi s Nm1;mþi1 ðsÞ þ Nm1;mþi ðsÞ smþi siþ1 smþi1 si
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 geometry The flute geometry is obtained in a similar manner as for the basic model as a surface of sweep. In this case,
Rgi ¼ ~ pgi ðsÞ ½T s ¼ ~ pgi ðs; /Þ;
i ¼ 1; 2; 3:
ð2:26Þ
2.2.3. Point geometry The generatrix of revolution is a NURBS in this case. Here
~ egab ðtÞ ¼ qx ðtÞ Hy
qz ðtÞ1 ;
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 Rg4 and Rg5 can be evaluated. Surface Rg4 ¼ ~ pg4 ðt; uÞ ¼ ½RAA1 :~ egab ðtÞ where ½RAA1 has a similar definition. Here, A = qx(t) and B ¼ qz ðtÞ þ da3 . Rg5 is formed by rotating Rg4 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.
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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 the present work. Table 2 shows the methodology of formation of conventional angles by various surface patches shaping the drill. The angles of twist drills presented here are as per the conventional nomenclatures. Formation of surfaces Rg1 to Rg5 for the generic model has been explained earlier in the paper. The unit tangents and normal to Rg4 are obtained as follows:
g g ^g4;t ðt; uÞ ¼ @~ p4 ðt; uÞ=@t = @~ p4 ðt; uÞ=@t ; p g g ^g4;u ðt; uÞ ¼ @~ p4 ðt; uÞ=@ u = @~ p4 ðt; uÞ=@ u ; p
ð3:2Þ
^g4;t ðt; uÞ p ^g4;u ðt; uÞ: ^ g4 ¼ p n
ð3:3Þ
ð3:1Þ
The unit tangents and normal can be evaluated from point to point on the cutting edges using Eqs. (3.1)–(3.3). ^ g4 is projected on ZX-plane and angle between this unit projected normal Half point angle (bg): As shown in Table 2, n ^ g4p Þ and the unit vector normal to YZ-plane ð^iÞ is half point angle bg. vector ðn 0 ^ g4 at any point on the chisel edge projected to XY-plane is given by n ^ g4p . The scalar Chisel edge angle (wg): The unit normal n g0 ^ 4p and unit vector ^j gives the chisel edge angle wg. product of n 00 00 ^ g4 is projected on YZ-plane given by n ^ g4p . The relief angle ag is the angle between n ^ g4p and unit Relief angle (ag): n ^ vector k. Helix angle (kg): Helix angle (kg) is formed by Rg1 and satisfies the relation
kg ¼ tan1 ðpDc =PÞ: Peripheral relief angle (a
ð3:4Þ g p ):
g 1
g 1
This angle is formed by the surface patch R with XY plane. The patch R is formed by helically sweeping parametric edge formed by joining vertices V1 and V2. The edge is defined as
~ pg1 ðsÞ ¼ ½ðDc =2 þ sl sin cÞ sl cos c 0 1: Table 2 Methodology to form the conventional angles. Conventional angles
Formed by the normal to
With the plane
Plane of projection
Half point angle, bg
Rg4 Rg4 Rg4 Rg3 Rg1
YZ
ZX
Chisel edge angle, wg Relief angle, ag Helix angle, kg Peripheral relief angle, agp
ZX
XY
XY
YZ
YZ
ZX
YZ
XY
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The surface patch Rg1 formed by sweeping ~ pg1 ðsÞ, is expressed by the following relation
~ pg1 ðs; /Þ ¼ ½ððDc =2 þ sl sin cÞ cos / sl cos c sin /ÞððDc =2 þ sl sin cÞ sin / þ sl cos c cos /ÞðP/=2pÞ 1 The tangents and normal to the surface patch at an arbitrary point are:
~ pg1;s ðs; /Þ ¼ l cosðc /Þ^i þ l cosðc /Þ^j; ^ ~ pg1;/ ðs; /Þ ¼ ðDc =2: sin / sl cosðc /ÞÞ^i þ ðDc =2: cos / þ sl sinðc /ÞÞ^j þ P=2pk; 2 ^ ~ ng1 ¼ Pl=2p cosðc /Þ^i Pl=2p sinðc /Þ^j ðDc =2:l sin c sl Þk:
^ g1p ¼ cosðc /Þ^i sinðc /Þ^j. The normal on projection to XY plane is given by n g ^ ^ 1p with unit vector i provides cos agp ¼ cosðc /Þ. At z = 0 plane, / = 0 and this leads to the mapping The scalar product of n relation agp ¼ c. Thus, we obtain the generic definitions for the half point angle, chisel edge angle, relief angle, helix angle and the peripheral 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 are evaluated at the midpoint of each element, which vary from point to point. Using the matrix of the cutting elements, the ! ~ ~ cutting 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 of ~ is the angular velocity. The inclination angle (i) is obtained using the relation the midpoint of the cutting lip and x
b; cosð90 þ iÞ ¼ D^l V
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: 0
0
~ pg3;/ pg ~ ^ g3 ¼ 3;s0 ; n 0 ~g pg3;/ p3;s ~ 0
0
ð3:6Þ 0
where ~ pg3;s and ~ pg3;/ denote the derivatives of ~ pg3 ðs; /Þ along s- and /- directions, respectively. Thus normal rake angle cgn is obtained as:
cos
!
V D~l : ^ g3 ! cgn ¼ n 2 V D~l
p
ð3:7Þ
4. Illustration of the model Using the methodology presented above, geometry of the basic and the generic model are generated. The sectional geometry, 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.
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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 and chisel 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 prevalent drills. First this sectional profile was used for modeling drill point with a straight generatrix (as above), and then for a curved generatrix. For the curved generatrix, the control points for NURBS of order 3 were chosen as follows:
fð12:34; 0; 2:54Þ; ð8:82; 0; 1:06Þ; ð7:05; 0; 0Þ; ð5:29; 0; 1:91Þ; ð3:53; 0; 2:97Þ; ð1:76; 0; 3:60Þ; ð0; 0; 4:24Þg:
Fig. 8b. Flute profile in the basic model as a sweep surface.
Fig. 8c. Grinding cones as surfaces of revolution.
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Fig. 8d. Cutting lips and chisel edge for the basic model.
Table 3 Basic drill profile data. Dc = 12.5 mm Hx = 2.2 mm b0 = 59° r1 = 5.5 mm
P = 60 mm l1 = 1.5, l2 = 0.2 mm da2 ¼ 1 mm r2 = 4.37 mm
rw = 0.7 mm c = 12° d = 35° h1 = 81.82°
Table 4 Generic drill profile data. Dc = 14 mm Hx = 2.2 mm b0 = 59°
P = 68.3 mm l = 0.83 mm da2 ¼ 2 mm
rw = 0.7 mm c = 10.9° d = 50°
Using the parameters listed in Table 4, the surfaces of sweep and revolution were obtained. The cutting lips and the chisel edge were subsequently obtained through optimization algorithms using MATLAB. The flute geometry and the grinding surfaces with a generic profile are shown in Figs. 9a and 9b respectively. The cutting lips and the chisel edge obtained for the straight generatrix are shown in Fig. 9c, while those obtained for the curved generatrix are shown in Fig. 9d. It can be seen clearly 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 is described below. First, a drill grinder was selected which follows the same grinding principle as described above. It is HMT universal tool and cutter grinder GTC-28T at IIT Kanpur. The parameters inherent to the grinding process were derived by measuring the distances and angles on the machine (Fig. 10) while other parameters were set on the grinder. An HSS drill of Addison make of diameter 12.5 mm was chosen. The relevant data are shown in Table 5. Using a flat Alumina grinding wheel, the drill was ground 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 micron probe 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 cutting lip, chisel edge and heel. The same were measured with the probe and registered with the model which is shown graphically in Fig. 11d. For registration, the peak of the drill was chosen as the origin. A comparison of the two data for the primary cutting lips are presented in Table 6a, for the two heels in Table 6b while Table 6c compares the data for the chisel edge. To compare the two data, coordinates were captured at the same z-values, and the x- and y-values were investigated. The primary cutting lip and the chisel edge are in close proximity, while the lower part of the heel slightly deviates away from the analytically obtained data. This is because the model uses a fixed generatrix for the grinding surface of revolution. But practically, the generatrix provided by the grinding wheel with flat face down as the drill swivels about its axis and the line of contact recedes away with the swivel of the grinder arm.
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Fig. 9a. Flute profile of the generic model.
Fig. 9b. Grinding cone with a generic profile.
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 data points 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 parameters s0, s1, . . . sn corresponding to each data point. If the unknown control points are represented by ~ ci ; i ¼ 0 . . . n, we have
K. Sambhav et al. / Applied Mathematical Modelling 36 (2012) 2384–2403
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.
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K. Sambhav et al. / Applied Mathematical Modelling 36 (2012) 2384–2403 Table 5 Data used for validation of conical and generic drill. Dc = 12.5 mm Hx = 15 mm b0 = 59°
P = 80 mm l = 1.6 mm da2 ¼ 2 mm
rw = 1.25 mm c = 11° 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 ¼
n X
Np;pþi ðuÞ~ ci :
ð5:1Þ
i¼0
Using the mapping of the data points and the parameters, we have,
~ qk ¼ ~ cðsk Þ ¼
n X
Np;pþi ðsk Þ~ ci ;
k ¼ 0...n
i¼0
or,
3 2 32 3 ~ ~ Np;p ðs0 Þ Np;pþ1 ðs0 Þ Np;pþ2 ðs0 Þ . . . Np;pþn ðs0 Þ c0 q0 7 7 6 ~ 7 6 N ðs Þ N 6 ~ 6 q1 7 6 p;p 1 p;pþ1 ðs1 Þ N p;pþ2 ðs1 Þ . . . N p;pþn ðs1 Þ 76 c1 7 76 7 6 7 6 7 7 7 6 6 6 fQg ¼ 6 7 ¼ 6 76 7 ¼ ½NfCg 76 7 6 7 6 54 5 45 4 ~ ~ Np;p ðsn Þ Np;pþ1 ðsn Þ Np;pþ2 ðsn Þ . . . Np;pþn ðsn Þ cn qn 2
ð5:2Þ
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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 ½N1 fQ g
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 the control points. Having defined the generatrix, the coordinates on the cutting lip, heel and chisel edge were calculated using a MATLAB code. Next, the drill was ground and the drill point was generated (Fig. 12b). The coordinates on the cutting lip, heel and chisel edge were measured experimentally. The drill flute together with the curved grinding surfaces were modeled (Fig. 12c).
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Table 6a Comparison of coordinates of the two primary cutting lips for the conical drill. S. No.
Primary cutting lip 1
Primary cutting lip 2
Theoretical 1 2 3 4 5 6 7 8 9 10 11 12
0.35 1.06 1.54 2.02 2.66 3.15 3.8 4.15 4.45 4.94 5.27 5.75
Experimental 1.22 0.95 0.81 0.69 0.56 0.46 0.35 0.3 0.27 0.2 0.15 0.05
3.55 3.06 2.76 2.46 2.06 1.76 1.36 1.16 0.96 0.66 0.46 0.16
0.4 1.06 1.77 2.26 2.7 3.26 3.78 4.23 4.66 5.04 5.23 5.7
1.08 1.03 0.81 0.61 0.61 0.54 0.34 0.25 0.16 0.16 0.09 0.08
Theoretical 3.55 3.06 2.76 2.46 2.06 1.76 1.36 1.16 0.96 0.66 0.46 0.16
0.35 1.06 1.54 2.02 2.66 3.15 3.8 4.15 4.45 4.94 5.27 5.75
Experimental 1.22 0.95 0.81 0.69 0.56 0.46 0.35 0.3 0.27 0.2 0.15 0.05
3.55 3.06 2.76 2.46 2.06 1.76 1.36 1.16 0.96 0.66 0.46 0.16
0.4 1.01 1.72 2.19 2.67 3.18 3.68 4.16 4.73 4.96 5.17 5.73
1.08 0.98 0.78 0.58 0.58 0.48 0.28 0.23 0.13 0.11 0.08 0.08
3.55 3.06 2.76 2.46 2.06 1.76 1.36 1.16 0.96 0.66 0.46 0.16
Table 6b Comparison of coordinates of the two heels for the conical drill. S. No.
Heel 1
Heel 2
Theoretical 1 2 3 4 5 6 7 8
0.35 0.3 0.62 0.74 0.66 0.49 0.39 0.33
Experimental 1.22 1.8 2.35 3.26 3.8 4.32 4.56 4.69
3.55 2.91 2.51 2.06 1.86 1.66 1.56 1.52
0.4 0.36 0.66 0.86 0.79 0.66 0.61 0.41
1.08 1.71 2.21 3.01 3.61 4.01 4.41 4.81
Theoretical 3.55 2.91 2.51 2.06 1.86 1.66 1.56 1.52
0.35 0.3 0.62 0.74 0.66 0.49 0.39 0.33
Experimental 1.22 1.8 2.35 3.26 3.8 4.32 4.56 4.69
3.55 2.91 2.51 2.06 1.86 1.66 1.56 1.52
0.4 0.36 0.66 0.96 0.92 0.79 0.66 0.53
1.08 1.68 2.17 2.96 3.55 3.93 4.32 4.67
3.55 2.91 2.51 2.06 1.86 1.66 1.56 1.52
Table 6c Comparison of coordinates of the chisel edge for the conical drill. S. No.
Theoretical
1 2 3
0.35 0.00 0.35
Experimental 1.22 0.00 1.22
3.55 3.56 3.55
0.40 0.00 0.40
1.08 0.00 1.08
3.55 3.58 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 is shown in Fig. 12d. The same has been tabulated in Table 7. As the primary cutting lips are curled along z-axis here, for comparison, the data were captured for identical x-values and the y- and z-values were looked into. The comparison shows again a good proximity between the model and the generated profile, except near the lower part of the primary cutting lip and heel for similar reason as mentioned above.
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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 hole quality, prediction of drill wear and drill life, study of drill dynamics, design modification for easy chip evacuation, etc. The study of drilling performance in the micro-domain is another area where new shapes of drills need to be investigated. And each of these concerns can be used for the optimization of the drill geometry for different objectives. Such tasks have been taken up by researchers in the past too. But the drill geometries reported in the earlier work have been limited to few standard shapes. Using NURBS to model the drill geometry gives a new dimension to these tasks and gives a much higher degree of freedom for predictions or optimization tasks. This generates a new scope for research in all these areas. Working in this direction, 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 model through regression analysis of the forces as a function of the rake angle, feed and velocity.
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Fig. 12d. Comparison of experimental and theoretical results for a generic drill point.
Table 7a Comparison of coordinates of the two primary cutting lips for the generic drill. S. No.
Primary cutting lip 1
Primary cutting lip 2
Theoretical 1 2 3 4 5 6 7 8 9 10 11 12
0.35 1.01 1.42 1.94 2.29 2.71 3.21 3.93 4.31 4.73 5.88 6.22
Experimental 1.26 0.88 0.7 0.48 0.18 0.23 0.11 0.01 0.01 0.01 0.1 0.19
2.35 1.67 1.3 0.91 0.45 0.51 0.37 0.24 0.2 0.15 0.26 0.39
0.35 1.01 1.42 1.94 2.29 2.71 3.21 3.93 4.31 4.73 5.88 6.22
1.15 0.85 0.69 0.47 0.34 0.28 0.22 0.24 0.21 0.23 0.29 0.42
Theoretical 2.39 1.89 1.44 1.08 0.87 0.65 0.52 0.35 0.29 0.27 0.4 0.39
0.35 0.88 1.26 1.78 2.35 2.82 3.33 3.73 4.31 4.65 5.82 6.06
Experimental
1.26 0.95 0.77 0.54 0.34 0.2 0.09 0.03 0.01 0.01 0.08 0.15
2.35 1.79 1.45 1.01 0.67 0.48 0.34 0.28 0.2 0.16 0.23 0.33
0.35 0.88 1.26 1.78 2.35 2.82 3.33 3.73 4.31 4.65 5.82 6.06
1.15 0.73 0.63 0.35 0.15 0.03 0.05 0.1 0.03 0 0.36 0.31
2.39 1.99 1.61 1.09 0.68 0.54 0.35 0.28 0.23 0.19 0.16 0.21
Table 7b Comparison of coordinates of the heels for the generic drill. S. No.
Heel 1
Heel 2
Theoretical 1 2 3 4 5 6 7 8
0.35 0.09 0.09 0.35 0.38 0.39 0.18 0.3
Experimental 1.26 1.53 1.77 2.37 2.5 3.12 3.96 4.82
2.35 1.93 1.66 1.15 1.06 0.7 0.33 0.02
0.35 0.06 0.07 0.38 0.46 0.41 0.12 0.16
1.15 1.6 2 2.56 2.96 3.57 4.15 5.03
Theoretical 2.39 1.93 1.66 1.15 1.06 0.7 0.33 0.05
0.35 0.09 0.09 0.35 0.38 0.39 0.18 0.3
Experimental 1.26 1.53 1.77 2.37 2.5 3.12 3.96 4.82
2.35 1.93 1.66 1.15 1.06 0.7 0.33 0.02
0.35 0.07 0.06 0.41 0.43 0.44 0.16 0.19
1.15 1.54 1.94 2.41 2.82 3.48 4.06 4.93
2.39 1.93 1.66 1.15 1.06 0.7 0.33 0.13
Table 7c Comparison of coordinates of the chisel edge for the generic drill. S. No.
Theoretical
1 2 3 4
0.35 0.17 0.00 0.35
Experimental 1.26 0.58 0.00 1.26
2.35 2.43 2.52 2.35
0.35 0.15 0.08 0.35
1.15 0.46 0.11 1.15
2.39 2.45 2.49 2.39
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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 shape and cutting angles can be changed just by changing the control points or their respective weights. The cutting edges 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 been solved converting it to surface-curve intersection and employing optimization technique subsequently. The cutting lip is obtained as the intersection of a sweep surface and a surface of revolution, while the chisel edge is obtained as the intersection of two surfaces of revolution. Generic definitions of the conventional angles on the drill point are presented along the primary cutting lip and the chisel edge of the drill. The model has been illustrated in MATLAB environment and validated for a conical drill as well as a drill with curved 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 for different objectives. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
[11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
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