SURFACE MODELING OF MULTI-POINT, MULTI-FLUTE

CUTTING TOOLS

Puneet Tandon

PDPM- Indian Institute of Information Technology, Design and Manufacturing Jabalpur, India, 482 011

Phalguni Gupta, Sanjay G. Dhande

Department of Computer Science & Engineering, Indian Instiute of Technology Kanpur, India

Keywords: Surface modeling, geometric modeling, fluted cutters, tool geometry, mapping.

Abstract: Cutting tools are usually represented by two-dimensional representation schema(s). The two-dimensional

nomenclatures have their inherent limitations. This paper outlines a detailed surface based modeling

paradigm for a variety of multi-point, multi-flute cutting tools. The work presents the generic biparametric

surface based models of slab mills, end mills and drills. The flutes are modeled as helicoidal surfaces. The

relations to map proposed three-dimensional (3D) rotational angles that generate 3D geometric models to

conventional angles (forward mapping) and their reverse relations (inverse mapping) are also developed.

The new paradigm offers immense technological advantages through numerous downstream applications.

1 INTRODUCTION

Traditionally, the geometry of cutting tools has been

defined using the principles of projective geometry.

Such definitions are two-dimensional (2D) in nature.

The developments in the field of Computer Aided

Geometric Design (CAGD) now provide a designer

to specify the cutting tool geometry in terms of

biparametric surface patches. Such an approach

provides the comprehensive three-dimensional (3D)

definitions of the cutting tools. The surface model of

a cutting tool can be converted into a solid model

and then may be used for down stream applications.

The existing 2D representation schemes are unable

to directly provide the necessary data for such

applications. The 2D modeling data need to be

converted into 3D models before they can be used.

The primary goal of this work is to outline surface

models of multi-point, multi-fluted cutters.

A wide range of cutters used in practice are

multi-point and multi-fluted in geometry (Drodza,

1983). Considerable work has been done in the area

of geometric modeling of the drill (Armarego, 1998,

Hsieh, 2002, Wang, 2001), helical milling cutters

(Sheth, 1990) and end mills (Chen, 2001); however,

the works are not in the direction of development of

unified representation schemes. Tandon et al. have

proposed the unified modeling schemes for slab

mills (Tandon, 2004) and end mills (Tandon, 2005).

In the present work, mathematical models of the

complex geometry of the fluted cutters are

formulated as a combination of surface patches. The

orientation of the surface patches is defined by 3D

angles, termed as rotational angles. Relations to

calculate the conventional two-dimensional (2D)

tool angles and the 3D rotational angles from one to

other are developed. Finally, graphics output in the

form of rendered image of the cutter is shown for

verification of the methodology.

2 SLAB MILLING CUTTERS

Slab mills produces flat surfaces parallel to the axis

of the spindle. The cutting edges can be straight or

helical and are on the circumference (Drodza, 1983).

2.1 Modeling the Surface of Slab Mill

The flutes of the slab mill is made up of five surface

patches and they are face (Σ

1

), land (Σ

2

), flank (Σ

3

),

tooth back (Σ

4

) and fillet (Σ

5

). The tooth geometry is

172

Tandon P., Gupta P. and G. Dhande S. (2007).

SURFACE MODELING OF MULTI-POINT, MULTI-FLUTE CUTTING TOOLS.

In Proceedings of the Second International Conference on Computer Graphics Theory and Applications - GM/R, pages 172-175

DOI: 10.5220/0002077501720175

Copyright

c

SciTePress

completed by left and right hand planar surfaces

(Σ

6

/Σ

7

). The flute is modeled as a sweep surface. For

this, a generic composite curve perpendicular to the

axis and a sweeping rule is required. When the

sweeping is linear, the straight tooth cutter is formed

and when the sweep is a combination of rotational

and parallel sweep, then the resultant surface is

helicoidal surface and the cutter is helical slab mill.

The composite curve in XY plane is composed of

vertices V

1

…V

7

(Figure 1). Let D, D

R

and l

1

be

cutter diameter, root circle diameter and width of

primary land respectively and α

i

, β

i

, γ

i

be the angles

of rotation of surface patch ‘i’ about X, Y, Z axis

respectively.

γ

V

γ

γ

V

V

V

V

V

V

V

D

D

2

3

1

γ

0

1

2

3

4

5

6

7

O

X

Y

ψ

R

c

1

Figure 1: Sectional Curve of a Slab Mill.

The cross-section curve V

1

V

7

is composed of five

parametric curve segments defined in terms of

parameter ‘s’. The curve segments p

1

(s), p

2

(s), p

3

(s)

and p

5

(s) are straight lines between vertices V

1

V

2

,

V

2

V

3

, V

3

V

4

and V

6

V

7

respectively, while the curve

p

4

(s) is a circular arc of radius R. The sweep

surfaces Σ

1

to Σ

5

are formed as p

i

(s, φ) = p

i

(s).T

s

.

For helical mill's tooth, if φ is the rotational angle

and P the pitch of the helical cutter, then the

transformation matrix T

s

is

⎥

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎢

⎣

⎡

−

−

=

0

2

00

0100

00cossin

00sincos

π

φ

φφ

φφ

P

S

T

;

P

L

P

L

π

φ

π

≤≤

−

(1)

The body of the slab mill has left end surface

(Σ

50

), right end surface (Σ

51

), bore surface (Σ

52

) and

the keyway. The body of slab mill has eight

transitional surfaces in the form of chamfers. The

chamfer between surface patch Σ

i

and Σ

j

is denoted

as σ

i,j

, where i = 50, 51 and j = 52, 53, 54, 55.

2.2 Mapping Relations for Slab Mills

The process of conversion of tool angles given in

one nomenclature to other is known as mapping.

When 3D rotational angles developed in this work

are converted to conventional 2D nomenclatures, the

mapping is called the forward mapping. On the

contrary, when angles defined as per existing

standards are mapped to proposed rotational angles,

the process is called the inverse mapping. The angles

for a slab mill are shown in Figure 2.

Radial Rake Angle (γ

R

) is formed by face Σ

1

with

ZX plane and viewed on projection to XY plane. To

find γ

R

, the normal to Σ

1

is projected on XY plane

and dot product of the projected normal with the unit

vector along Y axis is taken. This gives cosγ

R

=

cos(γ

1

+φ). For straight tooth cutter, angle φ is zero,

while for helical cutters, γ

R

is evaluated on z=0

plane. This gives γ

R

= γ

1

Tooth thickness

Tooth face

Cutting edge

Back of tooth

Flute

Land

Fillet (radius R)

Center of cutter

Lip angle

2nd clearance angle

Relief angle

1st clearance angle

Gash angle

Root circle

Radial rake angle

Figure 2: Conventional Tool Geometry of Slab Mill.

In the same fashion, relief angle (α

P

) is formed

by land Σ

2

about YZ plane when projected on XY

plane and is given as α

P

= γ

2

. Surface patch Σ

3

(Flank) of tooth forms first clearance angle (α

1P

)

with YZ plane on projection to the XY plane and the

angle α

1P

is expressed by the following relation:

()

()

⎥

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎢

⎣

⎡

−+

⎟

⎠

⎞

⎜

⎝

⎛

+−

−+

⎟

⎠

⎞

⎜

⎝

⎛

+−

=

−

2

213

2

213

213213

1

1

cossin

2

coscossinsin

2

cos

γγ

φγφγ

α

lVl

D

V

lVl

D

V

yx

yx

P

Second Clearance Angle (α

2P

) is formed by patch

Σ

4

and can be expressed as α

2P

= γ

4

. Gash angle (δ)

is the angle of fillet and is 908+γ

1

-γ

4

+(2π/N), where

N is the number of teeth of the slab mill. Lip angle is

the angle formed by the tooth solid side at the

cutting edge. In terms of rotational angles the lip

angle is evaluated as υ

la

= 908+γ

1

-γ

2

.

SURFACE MODELING OF MULTI-POINT, MULTI-FLUTE CUTTING TOOLS

173

3 MODELING OF END MILL

End mills are multi-point cutters with cutting edges

both on the end face and the circumference (Drodza,

1983). Figure 3 shows the projected geometry of a

flat end mill. The geometry of an end mill consists

of geometry of fluted shank and end geometry.

Figure 3: Two-Dimensional Geometry of End Mill.

3.1 Geometry of Fluted Shank

A single tooth of the end mill is modeled with nine

surface patches, labeled Σ

1

to Σ

9

(Table 1). Surfaces

Σ

1

to Σ

6

are the surfaces on the fluted shank. These

surfaces are formed as helicoidal surfaces. The

composite sectional curve (P

1

…P

8

) is composed of

six segments (Figure 4). Three segments of the

composite curve are straight lines and correspond to

the three land widths, namely peripheral land, heel

and face. While the other three segments are circular

arcs of radii r

3

, r

2

and R and correspond to fillet,

back of tooth and blending surface.

Table 1: Surface Patches of End Mill.

Symbol Surface Patch

Name

Symbol Surface

Patch Name

Σ

1

Σ

2

Σ

3

Σ

4

Σ

5

Face

Peripheral Land

Heel

Blending Surface

Back of Tooth

Σ

6

Σ

7

Σ

8

Σ

9

Fillet

Face Land

Minor Flank

Rake Face

Extension

Figure 4: Modeling of an End Mill Tooth.

The helicoidal surface for fluted shank is described

as p(s,φ) = p(s).T

S

, where

In the above equation, L is the length of fluted

shank. The length may be equal to L

1

for flat end

mills and (L

1

-D

c/

/2) for ball end mills. Three

different sweeping rules can be formulated for the

fluted shank and the end profile of the cutter. These

rules are for cylindrical, conical and hemispherical

helical path.

3.2 Mapping Relations for End Mill

Mapping guide table (Table 2) shows the planes that

forms the conventional angles. The forward mapping

relations are summarized in Table 3.

Table 2: Mapping Guide Table for End Mill.

Conventional

Angles

Formed

by

About

the Plane

Plane of

Projection

γ

R

α

R

α

1R

φ

e

α

A

Σ

1

Σ

2

Σ

3

Σ

7

Σ

7

ZX

YZ

YZ

XY

XY

XY

XY

XY

ZX

YZ

Table 3: Forward Mapping Relations for End Mill.

Conventional Angles Rotational Angles

Radial Rake Angle, ±γ

R

=

1

γ

m

Radial Relief Angle, α

R

=

2

γ

Radial Clearance Angle, α

1R

=

3

γ

Axial Relief Angle, α

A

=

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

+

−

7

2

7

2

1

2

7

1

cossincos

cos

cos

ααγ

α

End Cutting Edge , φ

e

=

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

+

−

7

2

7

2

1

2

7

1

cossinsin

cos

cos

ααγ

α

4 MODELING OF TWIST DRILL

Drills are rotary cutting tools used for the generation

of holes (Drodza, 1983). In this paper, modeling of a

two-flute, right-cut, straight shank type of solid twist

drill is presented. This is the most commonly used

drill for originating holes.

Geometrically a drill is made of (i) drill body and

(ii) shank. Drill body is the portion responsible for

material removal and the part by which drill is held

and driven in a drilling machine is shank. The drill

body may be segmented into (i) flute and (ii) end

geometry. The flute is the cutting portion of the drill.

The end of the drill is the portion that facilitates

entry of the drill into the workpiece. The

conventional two-dimensional projected geometry of

a twist drill is shown in Figure 5.

⎥

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎢

⎣

⎡

π

φ

φφ−

φφ

=

1

2

P

00

0100

00cossin

00sincos

s

T

for

P

L

π

φ

2

0 ≤≤

(2)

GRAPP 2007 - International Conference on Computer Graphics Theory and Applications

174

L

L

L

D

S

W

Land

Shank

Flute

Face

Body

2

β

Minor cutting edge

Major flank

λ

ψ

Flute

Land

Lip relief angle

α

4

2

1

Figure 5: 2D Projected Geometry of a Twist Drill.

The sectional geometry of the fluted shank has

three segments, out of which one is a straight line

and forms land (Σ

2

). The other two segments are

circular in geometry and on sweeping form flank

(Σ

3

) and face (Σ

1

) respectively. The drill end is made

of as many surface patches as the number of flutes.

For a two-flute drill, two surface patches form the

drill end. They are labeled as Σ

4

and Σ

5

and known

as lip relief surfaces. The lip relief surfaces can be

planar, cylindrical, conical and helicoidal. For a

drill, the forward mapping relations are:

Half Point Angle,

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

+

=

−

4

2

4

2

4

2

44

1

coscossin

cossin

cos

βγβ

γβ

β

(3)

Chisel Edge Angle, ψ = 90˚ - γ

4

. (4)

Relief Angle,

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

+

=

−

4

2

4

2

4

2

4

1

cossinsin

cos

cos

βγβ

β

α

(5)

Helix Angle,

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

=

−

c

D

P

π

λ

1

tan (6)

Peripheral Relief Angle, α

p

= γ

2

(7)

5 VALIDATION

This section presents an example on 3D modeling of

an end mill. The parameters used to construct the

model of end mill are referred in ANSI/ASME

B94.19-1985 standards. The resultant cutter is

rendered (Figure 6) in OGL environment.

Figure 6: Rendering of an End Mill.

6 CONLUSIONS

The present work has covered the 3D modeling of

the multi-point fluted cutters (slab mills, end mill

and drills) by mathematically expressing the

geometry of the cutting tools in terms of various

biparametric surface patches. Four rotational angles

(γ

1

, γ

2

, γ

3

, γ

4

) are used to define the geometry of a

slab mill along with other dimensional parameters.

Similarly, four rotational angles (γ

1

, γ

2

, γ

3

, α

7

) and

three rotational angles (γ

2

, β

4

, γ

4

) are defined to

model an end mill and drill respectively. The

mathematical definitions of the surfaces have been

used to obtain the standard 2D tool angles from

these proposed rotational angles. The inverse

relationships to obtain the rotational angles from the

conventional angles are also obtained. The entire

exercise attempts to recast the method of defining a

cutting tool in terms of 3D geometric models.

REFERENCES

Armarego, E.J.A., Kang; D.,1998. Computer-Aided

Modeling of the Fluting Process for Twist Drill

Design and Manufacture, Annals of the CIRP, Vol.47,

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Constant Pitch Helical Grooves, International Journal

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175