A generic mathematical model of single point cutting tools in terms of grinding parameters

Applied Mathematical Modelling 35 (2011) 5143–5164

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Applied Mathematical Modelling

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

A generic mathematical model of single point cutting tools in termsof grinding parameters

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

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

a r t i c l e i n f o

Article history:Received 23 December 2010Received in revised form 11 April 2011Accepted 15 April 2011Available online 27 April 2011

Keywords:Single point cutting toolGrinding anglesGeneric profileForward and inverse mapping

0307-904X/$ - see front matter � 2011 Elsevier Incdoi:10.1016/j.apm.2011.04.017

⇑ Corresponding author. Tel.: +91 9455680200.E-mail addresses: [email protected] (K. Sambha

1 Tel.: +91 761 2632924; fax: +91 761 2632 524.2 Tel.: +91 512 2597258/2590763; fax: +91 512 25

a b s t r a c t

The paper presents the analytical geometric details of the mathematical modeling of a sin-gle point cutting tool with a generic profile. The grinding angles and the ground depths onthe tool are allowed to vary along the tool flanks and face, altering the cutting angles frompoint to point. The surface modeling begins with the creation of a tool blank model. Thenunbounded surfaces are considered and transformed to get the cutting tool surfaces. Theintersection of these surfaces gives the complete model of the tool. Starting from the basicmodel where the tool face and flank are planar, the generalization of the geometric designhas been done in two steps to give free-form shapes to the tool surfaces, termed as the twogenerations of the generic profile. Then a forward and inverse mapping has been presentedfor the basic model and the two generations of the generic tool to relate the grinding angleswith the prevalent nomenclatures (ASA, ORS and NRS). The model has been validated andthe variation of tool angles with the grinding parameters has been illustrated with anexample.

� 2011 Elsevier Inc. All rights reserved.

1. Introduction

Design and manufacturing of cutting tools occupies a crucial role in any manufacturing system, as machining processescontinue to hold a large share of product shape and form realization activities. The mathematical modeling of any cuttingtool begins with the modeling of a single point cutting tool (SPCT). Conventionally, cutting tools have been defined usingprinciples of projective geometry and the definitions have been limited to few standard shapes of the cutting tools. Theyare not unified and find a very limited use in computer based manufacturing and engineering analysis. With more and morecomplex tools in use, and the increasing demand for customization of cutting tools for improved performance, there is a needto generate a generic definition of cutting tools which can be directly used by computer controlled grinding machines forshape generation. A generic SPCT can be created through grinding or milling. If the tool is to be created through milling, gen-erating the point cloud data for the surface is sufficient. But surfaces produced by milling do not give a good surface finishand some post processing is required. If the tool can be produced through grinding, the desired surface finish and accuracyare easier to achieve. But, to relate the geometric definition of the tool to grinding parameters, a detailed mathematical def-inition of the surfaces in terms of grinding parameters is required.

. All rights reserved.

v), [email protected] (P. Tandon), [email protected], [email protected] (S.G. Dhande).

90260/259 7790.

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Nomenclature

{} vector notation[ ] matrix notationb wedge anglegmech mechanical advantaged cutting anglea clearance anglec rake anglek inclination anglecz back rake anglecx side rake angleaz end clearance angleax side clearance angle/e end cutting edge angle/s side cutting edge angler nose radiusa0z auxiliary end clearance anglea0x auxiliary side clearance angleco orthogonal rake angleao orthogonal clearance anglea0o orthogonal auxiliary clearance anglecn normal rake anglean normal clearance anglea0n normal auxiliary clearance angle/ principal cutting edge angleL, B, H length, width and height of tool blank respectivelyui, vi, wi parameters representing lengths along x, y, z axes respectivelyP

1 auxiliary flank surfaceP2 principal flank surfaceP3 rake face surfaceP4 shoulder face surfaceP5 nose surface

ai angle of rotation about x-axisbi angle of rotation about y- axisci angle of rotation about z-axisdij translation distances~q radius vector to a point on surface[Ti], [Mi] respective transformation matrices~q0r;u3

derivative of ~q0r along u3~n0r ;~n

0p normal vectors to the rake face and principal flank respectively

~n0ri;~n0pi projections of normal vectors to the ith plane

~r0p;~r0e vectors denoting principal and end cutting edges respectively

b�2;b�1 principal and end cutting edge angle respectively

~n0pr2;~n0rr2

normal vectors rotated about Y-axis~n0pr21;~n

0rr21 above normal vectors projected on the XY-plane

ai, bi, ci, ei constants used for illustration of the model

5144 K. Sambhav et al. / Applied Mathematical Modelling 35 (2011) 5143–5164

The geometric models of cutting tools developed so far approximate the geometry of cutting tools by presenting them intwo-dimensional space [1]. The single point cutting tool projective geometry has been proposed by various tool designingbodies, some of the nomenclatures being American Standard Association (ASA), Orthogonal Rake System (ORS) or DIN/OCT-BKC/CSN/ISO (old), Normal Rake System (NRS) or ISO (new) and the Maximum Rake System (MRS) or BS. The approachfollowed here proves inadequate to visualize and understand the geometry of the complex objects, cutting tools being one ofthem. It can designate the tool uniquely but fails to be effective representation for any downstream application. Modelingcutting tool surface patches as a collection of bi-parametric surfaces would help the computer aided design, analysis andmanufacturing of the cutting tools.

The literature survey shows only few pieces of work done in the field of geometric modeling of cutting tools which couldgive a generic definition to all the cutting tools based on the grinding parameters to take into account the variation in thegeometry of the existing tools, the complexity of the surfaces making the tool geometry and difficulties in defining thesecomplex, free form surfaces mathematically. Radzevich [2] has presented a novel approach of R-mapping of a sculptured

K. Sambhav et al. / Applied Mathematical Modelling 35 (2011) 5143–5164 5145

surface onto the machining surface of a form cutting tool assuming one to one correspondence between their principal cur-vatures. In another work [3] by him, the method to design a form-cutting tool for optimum machining of a sculptured surfaceon a multi-axis NC machine has been proposed. Hseih [4] has presented the interrelationships among the tool angles, settingangles and working angles of a SPCT with the methodology to derive one set of angles from the other two sets. Zhongqing [5]has brought out the shortcomings of a 2D representation of a cutter and modeled a 3D cutter using vector representation andpresented the model of a turning cutter as an example. Deo [1] and Rajpathak [6] developed the geometry of single pointcutting tools in terms of bi-parametric surface patches which was extended to multipoint cutting tools by Tandon et al.[7] to establish a set of new three-dimensional (3D) standards for defining the cutting tool geometries. They also establishedthe forward and inverse mapping relations between 3D nomenclature and conventional specification schemes. Geometricmodels to produce multi-point cutting tools with curved cutting edges has been presented by some authors too. Stephensonand Agapiou [8] described parametrically complex point geometries of a drill but they were not related to the grindingparameters. Xiong et al. [9] has presented a methodology to design a curve-edged twist drill with cutting angles distributedarbitrarily along the cutting edges. Sambhav et al. [10] followed the CAD approach to directly represent the sectional andpoint profile of the generic definition of a drill using NURBS in terms of the grinding parameters. But a detailed geometricmodel to represent a generalized model of a SPCT to facilitate the grinding process needs to be worked out which is pre-sented in this paper.

2. Geometric modeling

The surface modeling of an SPCT begins with the creation of the tool blank model which is a cuboid with 6 faces and 12edges. Then unbounded surfaces are considered and transformed geometrically to get the cutting tool surfaces. The intersec-tion of these surfaces with the blank gives the complete model of the tool. The transformation includes rotation throughgrinding angles in a given order which is non-commutative and displacements through the ground depths. To create a gen-eric shape of the tool, where the tool surfaces need not be planes but can assume an arbitrary shape, the grinding parametersare to be varied along the surface.

When the tool surface definition is required only to generate the point cloud data, the definition of the surfaces can begiven in terms of any standard definition. It can be a composite Ferguson, Bezier, B-spline or NURBS surface dependingon the requirement. NURBS surfaces give a local control of the shapes and are more useful for the purpose. These definitionscan then be used for calculation of angles on the tool. Or, if the tool angles are provided, the surfaces can be interactivelyaltered and required surfaces can be generated.

2.1. Basic shape

Any cutting tool applied in the metal cutting process has the basic shape of a wedge. A standard tool has the form of asmooth symmetric wedge with flat faces and uniform depth (Fig. 1). The cutting wedge is oriented at certain required angleswith the work surface for the production of chips during cutting.

The reaction forces and indenting force for the wedge are related as [11]:NP ¼ 1

2 sin b2 ¼ gmech;gmech termed as the mechanical advantage is dependent on the wedge angle b. The smaller the value of

b, the greater is the gain in the force but lesser the strength of the tool.When the wedge is made up of free-form surfaces, it is the case of a generic cutting tool (Fig. 2). In this case, the angles of

orientation of the wedge surfaces forming a generic tool vary along the surface of the wedge, and the tool has a very complexgeometry and distribution of forces. The forces and angles have to be defined using tangents and normals at every point. Tobe able to generate such a geometry through grinding, the wedge geometry needs to be defined in terms of grinding param-eters, implying that the lengths and angles have to be correlated with the motion of the grinder.

N

N

P

β

Fig. 1. Wedge model of tool.

Fig. 2. Model of a generic tool.


Having generated such a complex geometry, a correspondence between the prevalent nomenclature and the generatedprofile also needs to be explored so that the prevalent understanding can be used to predict the behavior of the tool duringcutting. This has been provided using forward and inverse mapping in this paper.

Considering the tool geometry, any cutting tool has the wedge angle b, set with respect to the velocity vector ~V , a cuttingangle d and clearance angle a[11]. The rake angle c describes the inclination of top face of the wedge which is the compli-mentary of the cutting angle. If d is greater than p/2, c is negative. Another parameter called as inclination angle k describesthe orientation of the wedge with respect to the velocity vector. The design of a generic cutting tool incorporates all theseangles as the edges may be curved in space (Fig. 3). The geometric model of a generic SPCT can be used to model genericmulti-point cutting tools. The application of such designs of tools can be found in the form of a generic drill point geometry,an arbitrary shaped mill and form tools. A SPCT called as Gustin Tool [12] shown in Fig. 4 has been in use as it reduces thecutting forces and tool wear by shifting the normal and frictional stresses away from the cutting edges.

2.2. Single point cutting tools and their nomenclature

The geometry of an SPCT is quite complex. During the cutting process, apart from the cutting edge, the other parts alsohave to fulfill certain functional requirements. The research done so far by the various tool designing bodies have establishedthe following reference systems which are used widely over the world:

(1) American Standards Association (ASA)In the ASA or Machine Reference System, the tool signature stands as

cz � cx � az � ax � /e � /s � r;

where the symbols have the meaning in the following order: back rake angle, side rake angle, side clearance angle, endclearance angle, side cutting edge angle, end cutting edge angle and nose radius. The auxiliary flanks have the corre-sponding clearance angles given by a0z and a0x.

Fig. 3. Angles on a generic SPCT.

Fig. 4. Gustin tool.


(2) Orthogonal Rake System (ORS)The ORS system also called the Continental or German (DIN) or Russian or Czech system has the tool signature givenby

k� co � ao � a0o � /e � /� r;

with the symbols indicating the inclination angle, orthogonal rake angle, orthogonal principal flank angle, orthogonalauxiliary flank angle, end cutting edge angle, principal cutting edge angle and nose radius.

(3) Normal Rake System (NRS)Also called the ISO system, it gives the tool signature as:

k� cn � an � a0n � /e � /� r;

where the first four symbols stand for the inclination angle, normal rake angle, normal principal flank angle and nor-mal auxiliary flank angle. The other symbols have the same meaning as for ORS system.

(4) Maximum Rake System (MRS)This system termed as British System, was generated out of the belief that the chip flows along the direction of themaximum rake which was proved wrong after some time, and hence not being used today. The tool signature hereis given as:

/z � cmax � ao � a0o � /e � /� r;

where /z and cmax stand for inclination angle and maximum rake angle respectively.

The reference planes used for the above systems can be referred to any standard book on metal cutting.

2.3. Modeling an SPCT in terms of grinding angles

To model a single point cutting tool in terms of grinding angles, the following methodology is used:

Model the tool blank as a cuboid of chosen size

Transform the XY-plane to form the auxiliary flank

Transform the XY-plane to form the principal flank

Transform the ZX-plane to form the rake face

Transform the XY-plane to form the shoulder face

Model the nose as a ruled oblique cone

Form cutting tool as composition of the above five surface patches and six orthogonal planes

Obtain the principal and end cutting edges as surface-surface intersection curves

Using the above methodology, the tool can be modeled for the standard basic shape or generic shape. There are two gen-
erations of the generic shape presented in the paper: the first generation and the second generation.
To model the tool blank, a cuboid of size LXBXH is chosen where, L, B and H are length, width and height of the blank. Thecuboid has six planes (I–VI) bounding the block whose parametric definitions can be given in terms of the parameters ri, si

Fig. 5. Surface model of the tool.


and ti (i ? 1. . .3) (Eq. (2.1)). The blank when machined at appropriate angles yields the cutting tool (Fig. 5). The cutting toolconsists of planes forming the auxiliary flank ð

P1Þ, principal flank ð

P2Þ, rake face ð

P3Þ and shoulder face ð

P4Þ along with a

nose ðP

5Þ which is part of an oblique cone. To form the surfaces, the Cartesian planes are suitably chosen and then trans-formed. The lines of intersection of these inclined and translated unbounded planes with the planes of the blank form theboundaries of the tool.

The parametric definition of the tool blank is given as follows:

Table 1Sign co

Face

Auxi

Princ

Rake

Plane I Plane IVx ¼ r1ð�B 6 r1 6 0Þ x ¼ r1ð�B 6 r1 6 0Þ;y ¼ s1ð�H 6 s1 6 0Þ y ¼ s1ð�H 6 s1 6 0Þ;z ¼ 0 z ¼ �L;

Plane II Plane Vx ¼ 0 x ¼ �B;

y ¼ s2ð�H 6 s2 6 0Þ y ¼ s2ð�H 6 s2 6 0Þ;z ¼ t2ð�L 6 t2 6 0Þ z ¼ t2ð�L 6 t2 6 0Þ;Plane III Plane VIx ¼ r3ð�B 6 r3 6 0Þ x ¼ r3ð�B 6 r3 6 0Þ;y ¼ 0 y ¼ �H;

z ¼ t3ð�L 6 t3 6 0Þ z ¼ t3ð�L 6 t3 6 0Þ:

ð2:1Þ

2.3.1. Sign convention of anglesAs per the ASA system, for a standard right hand turning tool, the angles are taken as positive if the tool rake face slopes

away from the nose point. If the face slopes towards the nose in transverse or longitudinal direction, then that angle is re-garded as negative. The ORS and NRS systems also follow the same convention. The clearance angles are regarded as positiveif the tool flank moves away from the work surface when placed against it. The side cutting edge angle is positive when theprincipal cutting edge angle is acute and negative when obtuse. The end cutting edge angle is taken as negative when theedge recedes towards the shank as one moves away from the tool tip.

The sign convention of the rotational angles is based on right hand screw theory. The planes used for forming the auxiliaryflank principal flank and rake face are X–Y, X–Y and Z–X planes respectively. The auxiliary flank face is positioned to havepositive clearance by rotating X–Y plane about the X-axis in the positive sense and about Y-axis in the negative sense. Theprincipal flank face is given positive clearance by rotating X–Y plane about the X-axis and Y-axis in the positive sense respec-tively. The rake face is modeled for positive clearance by rotating Z–X plane about Z-axis in the positive sense and X-axis in

nvention.

Plane Rotational angles ASA ORS NRS

liary flank X–Y a1 ? +ve a0z ! þve;a0x ! þve a0o ! þve a0n ! þveb1 ? +ve /e ? �ve /e ? �ve /e ? �ve

ipal flank X–Y a2 ? +ve az ? +ve,ax ? +ve ao ? + ve an ? +veb2 ? +ve /s ? +ve / ? +ve / ? +ve

face Z–X c3 ? +ve cx ? +ve co ? +ve cn ? +vea3 ? +ve cz ? �ve k ? �ve k ? �ve


the negative sense. All the angles are taken as positive and the transformation has been carried out with respective signs. Thesign convention is presented in Table 1.

Using the above methodology, the basic geometry of an SPCT is modeled as follows:

2.3.2. Auxiliary flankA plane coinciding with the XY-plane lying at the origin of a right handed coordinate system is chosen. It is successively

rotated about the X-axis by a1 and about the Y-axis by �b1; and then translated along X-, Y- and Z- axes by distances �d11,�d12 and �d13 respectively to form the auxiliary flank ð

P1Þ. The auxiliary flank face is given by fqag ¼ ½T1�½Ry;�b1 �½Rx;a1 �fq1g ¼

½M1�fq1g where,

fq1g ¼

u1

v1

01

8>><>>:

9>>=>>;; ½T1� ¼

1 0 0 �d11

0 1 0 �d12

0 0 1 �d13

0 0 0 1

26664

37775; ½Ry;�b1 � ¼

cb1 0 �sb1 00 1 0 0

sb1 0 cb1 00 0 0 1

26664

37775; ½Rx;a1 � ¼

1 0 0 00 ca1 �sa1 00 sa1 ca1 00 0 0 1

26664

37775:

�1 6 u1 61; �1 6 v1 61; d11;d12;d13 P 0

or,

~qa ¼ ðu1cb1 � v1sa1sb1 � d11Þiþ ðv1ca1 � d12 Þjþ ðu1sb1 þ v1cb1sa1 � d13Þk ð2:2Þ

d11, d12, d13 depend upon the depth of cut in each cut and number of re-sharpenings. These distances are interrelated and canhave different sets of values. One choice for the distances is [1]

d11 ¼ B=2;d12 ¼ H=2;

d13 ¼B2� de

tan jb2j

� �tan jb1j þ

H2

sin ja1j sec jb1j:

Here, de is the depth of cut and depends on the choice of the machinist.

2.3.3. Principal flankHere too, a plane coinciding with the XY-plane lying at the origin of a right handed coordinate system is chosen. It is suc-

cessively rotated about X-axis by a2 and Y-axis by b2 and then translated along X-, Y-and Z- axes by distances �d21, �d22 and�d23, respectively to form the principal flank ð

P2Þ. Here the transformation takes place in the following manner:-

fqpg ¼ ½T2�½Ry;b2 �½Rx;a2 �fq2g ¼ ½M2�fq2g;

where,

fq2g ¼

u2

v2

01

8>><>>:

9>>=>>;; ½T2� ¼

1 0 0 �d21

0 1 0 �d22

0 0 1 �d23

0 0 0 1

26664

37775; ½Ry;b2 � ¼

cb2 0 sb2 00 1 0 0�sb2 0 cb2 0

0 0 0 1

26664

37775; ½Rx;a2 � ¼

1 0 0 00 ca2 �sa2 00 sa2 ca2 00 0 0 1

26664

37775:

�1 6 u2 61; �1 6 v2 61; d21;d22;d23 P 0

or,

~qp ¼ ðu2cb2 þ v2sa2sb2 � d21Þiþ ðv2ca2 � d22 Þjþ ð�u2sb2 þ v2cb2sa2 � d23Þk: ð2:3Þ

Here again, d21, d22, d23 depend upon the depth of cut in each cut and number of re-sharpenings. They are interrelated andone choice for the distances is [1]:

d21 ¼ B=2;d22 ¼ H=2;

d23 ¼ �B2� de

tan jb2j

� �tan jb2j þ

H2

sin ja2j sec jb2j:

2.3.4. Rake faceThe rake face ð

P3Þ is formed by transforming a plane coinciding with the ZX-plane at the origin. The transformation in-

cludes successive rotation about the Z-axis by c3, about X-axis by �a3 and then translation along X-, Y- and Z- axes by dis-tances �d31, �d32 and �d33, respectively.

P3 is represented as

fqrg ¼ ½T3�½Rx;�a3 �½Rz;c3�fq3g ¼ ½M3�fq3g;


where,

fq3g ¼

u3

0w3

1

8>>><>>>:

9>>>=>>>;; ½T3� ¼

1 0 0 �d31

0 1 0 �d32

0 0 1 �d33

0 0 0 1

26664

37775; ½Rx;�a3 � ¼

1 0 0 00 ca3 sa3 00 �sa3 ca3 00 0 0 1

26664

37775; ½Rz;c3

� ¼

cc3 �sc3 0 0sc3 cc3 0 00 0 1 00 0 0 1

26664

37775;

�1 6 v3 61; �1 6 w3 61; d31; d32;d33 P 0

or,

~qr ¼ ðu3cc3 � d31Þiþ ðu3ca3sc3 þw3sa3 � d32Þjþ ð�u3sa3sc3 þw3ca3 � d33Þk: ð2:4Þ

The values of the distances d31, d32, d33 will depend on the depth of cut given in resharpenings. If the rake angle is positive,the rake face would slope away from the nose, and when it is negative, it would slop towards the nose. The coordinates ofany point on the line of intersection the auxiliary and principal flank will give the distances. When the rake angle is negative,one choice for the distances would be that the plane passes through a point x = 0, y = �T, z = �de where the parameter T is avariable whose choice is subject to the machinist.

If the tool tip is to be positioned at a given coordinate, say (�d1,�d2,�d3), the distances can be chosen so that

d11 ¼ d21 ¼ d31 ¼ d1;

d12 ¼ d22 ¼ d32 ¼ d2;

d13 ¼ d23 ¼ d33 ¼ d3: ð2:5Þ

2.3.5. Shoulder faceA plane coinciding with the XY-plane is chosen and rotated about X-axis through �a4 and Y-axis through b4 and translated

through a distance �d43(d43 > 0) to position it in the frame of reference as for the requirements of shoulder face ðP

4Þ. Valueof a4, b4 and d43 are at the discretion of the designer. Parametrically

fqsg ¼ ½T4�½Ry;b4 �½Rx;�a4 �fq4g ¼ ½M4�fq4g;

where

fq4g ¼

u4

v4

01

8>>><>>>:

9>>>=>>>;; ½Ts� ¼

1 0 0 00 1 0 00 0 1 �d43

0 0 0 1

26664

37775; ½Ry;b4 � ¼

cb4 0 sb4 00 1 0 0�sb4 0 cb4 0

0 0 0 1

26664

37775 and; ½Rx;a4 � ¼

1 0 0 00 ca4 sa4 00 �sa4 ca4 00 0 0 1

26664

37775;

�1 6 u4 61; �1 6 v4 61; d43 P 0

or,

~qs ¼ ðu4cb4 þ v4sa4sb4Þiþ v4ca4 jþ ð�u4sb4 þ v4cb4sa4 � d43Þk: ð2:6Þ

Fig. 6. Nose formation.


2.3.6. NoseThe nose ð

P5Þ can be modeled as an oblique cone, the simplest being the ruled oblique cone, comprising of one curved

edge (directrix) and two straight lines (generators). The curved edge of the nose of the tool can be modeled parametrically asa curve. Two points, one along principal cutting edge and another along auxiliary cutting edge are found based on the valueof the nose radius. Third point is obtained by ensuring C1 continuity of the nose curve with the cutting edges. The vertex ofthe cone P⁄ is obtained as the point of intersection of principal flank, auxiliary flank and the base surface. A model of nose inthe form of a ruled surface has been shown in Fig. 6.

The nose is defined as:

qnðu;vÞ ¼ pðu;0Þð1� vÞ þ P�v ; ð2:7Þ

where p(u,0) is the equation of the curve of the nose.The cutting tool is now the composition of these five surface patches (

P1 to

P5Þ and six orthogonal planes. The edges and

corners on the tool are given by the points of intersection of the surface patches (Fig. 5). A typical tool will have 13 cornerpoints.

2.3.7. Cutting edgesThe two cutting edges engaged in the cutting process are the principal and end cutting edges. The principal edge is ob-

tained by finding the intersection of the principal flank with the rake face. The end cutting edge is obtained by finding theintersection of the rake face with the auxiliary flank.

Principal Cutting EdgeTo obtain the equation for the principal cutting edge, the x-, y- and z- components of the principal flank and the rake faceare equated.

u3cc3 � d31 ¼ u2cb2 þ v2sa2sb2 � d21;

u3ca3sc3 þw3sa3 � d32 ¼ v2ca2 � d22;

� u3sa3sc3 þw3ca3 � d33 ¼ �u2sb2 þ v2cb2sa2 � d23: ð2:8Þ

Solving the simultaneous equations, we get

u3 ¼w3ðca3ca2cb2 � sa3sa2s2b2 � sa3c2b2sa2Þ � ðd31 � d21Þca2sb2 þ ðd32 � d22Þsa2s2b2 � ðd33 � d23Þca2cb2

s2b2ca3sc3sa2 þ ca3c2b2sa2sc3 þ sa3sc3cb2ca2 � sb2cc3ca2

� �;

which gives the desired equation for the principal cutting edge.For the case given by Eq. (2.5), the equation of the principal cutting edge is given as

~rp ¼ ðvpw3cc3 � d1 Þiþ ðvpw3ca3sc3 þw3sa3 � d2Þjþ ð�vpw3sa3sc3 þw3ca3 � d3Þk; ð2:9Þ

where,

vp ¼ ca3ca2cb2 � sa3sa2s2b2 � sa3c2b2sa2

s2b2ca3sc3sa2 þ ca3c2b2sa2sc3 þ sa3sc3cb2ca2 � sb2cc3ca2

� �:

In this case, the parameters will be related as:

u3 ¼ vpw3;

u2 ¼vpcc3ca2 � sa2sb2ðvpca3sc3 þ sa3Þ

ca2cb2

� �w3;

v2 ¼vpca3sc3 þ sa3

ca2

� �w3: ð2:10Þ

End Cutting EdgeAs for the principal cutting edge, the equation of the end cutting edge for the case presented by Eq. (2.5) is given by

~re ¼ ðvew3cc3 � d1Þiþ ðvew3ca3sc3 þw3sa3 � d2 Þjþ ð�vew3sa3sc3 þw3ca3 � d3Þk; ð2:11Þ

where,

ve ¼ ca3ca1cb1 � sa3sa1s2b1 � sa3c2b1sa1

s2b1ca3sc3sa1 þ ca3c2b1sa1sc3 þ sa3sc3cb1ca1 þ sb1cc3ca1

� �:

The parameters can be related in a similar manner.


2.4. First generation of generic SPCT

The above model can be generalized by allowing the angles to vary along the surface. This is possible if the grinder and thetool rotate about their axes while the grinding of surfaces takes place. This generalization has been termed as the first gen-eration of a generic SPCT. Here, the geometric model of the tool faces will be given as follows:

2.4.1. Auxiliary flankThe angles a1 and b1 are replaced by a1(v1) and b1(u1), respectively. The auxiliary flank face is now given by

q0a� �

¼ ½T1�½Ry;�b1ðu1Þ�½Rx;a1ðv1Þ�fq1g ¼ M01

� fq1g

where,

½Ry;�b1ðuÞ� ¼

cb1ðu1Þ 0 �sb1ðu1Þ 00 1 0 0

sb1ðu1Þ 0 cb1ðu1Þ 00 0 0 1

26664

37775; ½Rx;a1ðvÞ� ¼

1 0 0 00 ca1ðv1Þ �sa1ðv1Þ 00 sa1ðv1Þ ca1ðv1Þ 00 0 0 1

26664

37775

or,

~q0a ¼ ðu1cb1ðu1Þ � v1sa1ðv1Þsb1ðu1Þ � d11Þiþ ðv1ca1ðv1Þ � d12Þjþ ðu1sb1ðu1Þ þ v1cb1ðu1Þsa1ðv1Þ � d13Þk: ð2:12aÞ

2.4.2. Principal flankThe angles a2 and b2 are replaced by a2(v2) and b2(u2). Now the principal flank is given by

q0pn o

¼ ½T2�½Ry;b2ðu2Þ�½Rx;a2ðv2Þ�fq2g ¼ M02

� fq2g

where,

½Ry;b2ðu2Þ� ¼

cb2ðu2Þ 0 sb2ðu2Þ 00 1 0 0

�sb2ðu2Þ 0 cb2ðu2Þ 00 0 0 1

26664

37775; ½Rx;a2ðv2Þ� ¼

1 0 0 00 ca2ðv2Þ �sa2ðv2Þ 00 sa2ðv2Þ ca2ðv2Þ 00 0 0 1

26664

37775

or,

~q0p ¼ ðu2cb2ðu2Þ þ v2sa2ðv2Þsb2ðu2Þ � d11Þiþ ðv2ca2ðv2Þ � d12Þjþ ð�u2sb2ðu2Þ þ v2cb2ðu2Þsa2ðv2Þ � d13Þk: ð2:12bÞ

2.4.3. Rake faceReplacing c3 by c3(u3) and a3 by a3(w3), the rake face given by

q0r� �

¼ ½T3�½Rx;�a3ðw3Þ�½Rz;c3ðu3Þ�fq3g ¼ M03

� fq3g;

where,

½Rx;�a3ðw3Þ� ¼

1 0 0 00 ca3ðw3Þ sa3ðw3Þ 00 �sa3ðw3Þ ca3ðw3Þ 00 0 0 1

26664

37775; ½Rz;c3ðu3Þ� ¼

cc3ðu3Þ �sc3ðu3Þ 0 0sc3ðu3Þ cc3ðu3Þ 0 0

0 0 1 00 0 0 1

26664

37775

or,

~q0r ¼ ðu3cc3ðu3Þ � d13Þiþ ðu3ca3ðw3Þsc3ðu3Þ þw3sa3ðw3Þ � d23 Þjþ ð�u3sa3ðw3Þsc3ðu3Þ þw3ca3ðw3Þ � d33Þk: ð2:13Þ

2.4.4. Shoulder faceAs the shoulder face is not directly involved in the cutting, there is no need to redesign it.

2.4.5. NoseThe nose can again be modeled as a ruled oblique cone, comprising of one curved edge and two straight lines. The two end

points on the curve have to be located along the curved cutting edges and the third point has to ensure C1 continuity of thecurve with the edges which can be obtained iteratively. The vertex of the cone can be any point on the line of intersection ofthe principal flank and the auxiliary flank.


2.4.6. Cutting edgesThe cutting edges are now the intersection of two free-form surfaces. The cutting edges will now be given by Eqs. (2.9)

and (2.11) where

~rp ¼ ðvpw3cc3ðvpw3Þ � d1 Þiþ ðvpw3ca3ðw3Þsc3ðvpw3Þ þw3sa3ðw3Þ � d2 Þjþ ð�vpw3sa3ðw3Þsc3ðvpw3Þ þw3ca3ðw3Þ � d3Þk;ð2:14Þ

~re ¼ ðvew3cc3ðvew3Þ � d1 Þiþ ðvew3ca3ðw3Þsc3ðvew3Þ þw3sa3ðw3Þ � d2 Þjþ ð�vew3sa3ðw3Þsc3ðvew3Þ þw3ca3ðw3Þ � d3Þk:ð2:15Þ

Here, vp and ve are no longer constants and are defined from point to point which can be derived iteratively.To obtain vp or rather, the points of intersection of the intersecting 3D rake and flank surfaces, Timmer’s algorithm for

surface-surface intersection may be employed.To simplify the intersection problem, it can be conceived as the intersection of a curve and a surface, and the points of

intersection of curves traced out at small increments of a selected parameter for one of the surfaces can be obtained usinga technique that forms the basis for the hunting phase of Timmer’s algorithm. The formulation to achieve the intersectionpoints is given below:

Select any one parameter out of w3, a3, a2 and b2.For a given value of any one parameter, say w3, get the equation to a curve along the rake surface.At a point of intersection,

~djw3¼~q0r jw3

�~q0p ¼ 0; ð2:16Þ

where ~q0r jw3is the curve traced on ~q0r at any w3. The desired point of intersection can be obtained by iterating over the fol-

lowing equations:

u3;iþ1 ¼ u3;i �~q0p;u2

� ~q0p;v2� d1

!jw3 ;i

� �D

;

u2;iþ1 ¼ u2;i þ~q0p;v2

� ~q0r;u3� ~d1jw3 ;i

�D

;

v2;iþ1 ¼ v2;i �~q0r;u3

� ~q0p;u2� ~d1jw3 ;i

�D

;

where,

D ¼~q0r;u3� ~q0p;u2

�~q0p;v2

�: ð2:17Þ

Another method to obtain the cutting edges is to use the optimization methods. The problem of intersection is formulated asan optimization problem as given below:

f1 ¼ u3cc3 � u2cb2 � v2sa2sb2;

f2 ¼ u3ca3sc3 þw3sa3 � v2ca2;

f3 ¼ �u3sa3sc3 þw3ca3 þ u2sb2 � v2cb2sa2: ð2:18Þ

The points of intersection are given by the points at which

f 21 þ f 2

2 þ f 23 ¼ 0: ð2:19Þ

This is obtained by minimizing the quantity

sum ¼ f 21 þ f 2

2 þ f 23 :

2.5. Second generation of generic SPCT

The cutting tool surface generated by grinding process can be made more generic if the distances used in the transforma-tion vary along the surface. This is accomplished by rotating the grinding wheel and the tool during the grinding processwhile the tool and the grinder are also moved along the X, Y and Z-axes. This will give even more generic shape of the tool.In this formulation, it is assumed that the tool is to be rotated only about a transverse axis and not about the longitudinal axisto generate any surface of revolution (which is not desired for an SPCT).

d11 ¼ d11ðu1;v1Þ; d12 ¼ d12ðu1;v1Þ; d13 ¼ d13ðu1;v1Þ;d21 ¼ d21ðu2;v2Þ; d22 ¼ d22ðu2;v2Þ; d23 ¼ d23ðu2;v2Þ;d31 ¼ d31ðw3;u3Þ; d12 ¼ d12ðw3;u3Þ; d13 ¼ d13ðw3;u3Þ:

ð2:20Þ


Thus,

Auxiliary flank : q00a� �

¼ T 001� ½Ry;�b1ðu1Þ�½Rx;a1ðv1Þ�fq1g ¼ M00

1

� fq1g; ð2:21Þ

Principal flank : q00pn o

¼ T 002� ½Ry;b2ðu2Þ�½Rx;a2ðv2Þ�fq2g ¼ M00

2

� fq2g; ð2:22Þ

Rake face : q00r� �

¼ T 003� ½Rx;�a3ðw3Þ�½Rz;c3ðu3Þ�fq3g ¼ M00

3

� fq3g; ð2:23Þ

where,

T 001�

¼

1 0 0 �d11ðu1;v1Þ0 1 0 �d12ðu1;v1Þ0 0 1 �d13ðu1;v1Þ0 0 0 1

26664

37775; T 002

� ¼

1 0 0 �d21ðu2;v2Þ0 1 0 �d22ðu2;v2Þ0 0 1 �d23ðu2;v2Þ0 0 0 1

26664

37775; T 003

� ¼

1 0 0 �d31ðw3;u3Þ0 1 0 �d32ðw3;u3Þ0 0 1 �d33ðw3;u3Þ0 0 0 1

26664

37775:

Another way of achieving a generic shape is by successive (and not simultaneous) motions to the grinder and the tool duringgrinding. Then the transformation will be as given below:

Auxiliary flank:

q000a� �

¼ TR001�

TR01�

q1f g ¼ M0001

� fq1g; ð2:24Þ

where,

TR01�

¼

1 0 0 �d11ðv1Þ0 ca1ðv1Þ �sa1ðv1Þ �d12ðv1Þ0 sa1ðv1Þ ca1ðv1Þ �d13ðv1Þ0 0 0 1

26664

37775; TR001

� ¼

cb1ðu1Þ 0 �sb1ðu1Þ �d011ðu1Þ0 1 0 �d012ðu1Þ

sb1ðu1Þ 0 cb1ðu1Þ �d013ðu1Þ0 0 0 1

26664

37775:

Principal flank:

q000pn o

¼ TR002�

TR02�

fq2g ¼ M0002

� fq2g; ð2:25Þ

TR02�

¼

1 0 0 �d21ðv2Þ0 ca2ðv2Þ �sa2ðv2Þ �d22ðv2Þ0 sa2ðv2Þ ca2ðv2Þ �d23ðv2Þ0 0 0 1

26664

37775; TR002

� ¼

cb2ðu2Þ 0 sb2ðu2Þ �d021ðu2Þ0 1 0 �d022ðu2Þ

�sb2ðu2Þ 0 cb2ðu2Þ �d023ðu2Þ0 0 0 1

26664

37775:

Rake face:

q000r� �

¼ TR003�

TR03�

fq3g ¼ M0003

� fq3g; ð2:26Þ

TR03�

¼

cc3ðu3Þ �sc3ðu3Þ 0 �d31ðu3Þsc3ðu3Þ cc3ðu3Þ 0 �d32ðu3Þ

0 0 1 �d33ðu3Þ0 0 0 1

26664

37775; TR003

� ¼

1 0 0 �d031ðw3Þ0 ca3ðw3Þ sa3ðw3Þ �d032ðw3Þ0 �sa3ðw3Þ ca3ðw3Þ �d033ðw3Þ0 0 0 1

26664

37775

The equations for the flanks, face and cutting edges can be given accordingly.

3. Mapping

For the machinist to grind the tool faces, the grinding angles are the prime requirements. But for calculation of forces andstresses, the angles defined by the prevalent nomenclatures have to be calculated. Hence a mapping is required between thegrinding angles and the angles given by the ASA, ORS and NRS systems. Forward mapping presents the conventional anglesin terms of the grinding angles, and the inverse mapping presents the grinding angles in terms of the conventional angles.

3.1. Forward mapping

Using definitions of the surface patches, the forward mapping can be created for the first generation of the generic modeland the basic model becomes a special case. The mapping for the second generation is similar to the first generation. Thederivation of conventional angles requires finding the tangents and normals to the patches along the cutting edges, and pro-jecting them on the planes of concern. The angles of the projections with the axes give the conventional angles with the signconvention in place.


3.1.1. Forward mapping for first generation of generic cutting toolIn the first generation of the generalization, the angles vary along the edges and surfaces. Hence, the tangents and nor-

mals have to be calculated treating them as variables. The forward mapping is presented below.

3.1.1.1. ASA angles. The tangent vectors to the rake face are given in the u- and w-directions as:

~q0r;u3¼ ðcc3 � u3sc3 � c3;u3

Þiþ ðca3sc3 þ u3ca3cc3 � c3;u3Þj� ðsa3sc3 þ u3sa3cc3 � c3;u3

Þk; ð3:1Þ~q0r;w3

¼ ð�u3sa3sc3a3;w3 þ sa3 þw3ca3a3;w3 Þjþ ð�u3ca3sc3a3;w3 þ ca3 �w3sa3a3;w3 Þk; ð3:2Þ

where, c3 implies c3(u3), c3;u3implies @c3ðu3Þ

@u3;a3 stands for a3(w3) and a3;w3 for @a3ðw3Þ

@w3. This notation will be followed throughout

now for simplicity.The normal vector perpendicular to the rake face

~n0r ¼~q0r;w3�~q0r;u3

¼ nrxiþ nryjþ nrzk

¼ ðu3s2a3 � a3;w3 s2c3 þ u23s2a3 � a3;w3 sc3cc3 � c3;u3

� s2a3sc3 � u3s2a3cc3 � c3;u3þ u3c2a3 � a3;w3 s2c3

þ u23c2a3 � a3;w3 sc3cc3c3;u3

� c2a3sc3 � u3c2a3cc3 � c3;u3Þiþ ð�u3cc3ca3 � a3;w3 sc3 þ cc3ca3 �w3cc3sa3 � a3;w3

þ u23s2c3 � c3;u3

ca3 � a3;w3 � u3sc3 � c3;u3ca3 þ u3w3sc3 � c3;u3

sa3 � a3;w3 Þjþ ðu3cc3sa3 � a3;w3 sc3 � cc3sa3

�w3cc3ca3 � a3;w3 � u23s2c3 � c3;u3

sa3 � a3;w3 þ u3sc3 � c3;u3sa3 þ u3w3sc3 � c3;u3

ca3 � a3;w3 Þk: ð3:3Þ

Back Rake Angle cz

The vector ~n0r is projected on the YZ-plane and the projected unit normal vector is given by

n0r2 ¼nryjþ nrzkffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

n2ry þ n2

rz

q : ð3:4Þ

Taking dot product with unit vector j

cz ¼ cos�1 nryffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2

ry þ n2rz

q0B@

1CA: ð3:5Þ

Side Rake Angle cx

The unit normal vector projected on the XY-plane is given by

n0r1 ¼nrxiþ nryjffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

n2rx þ n2

ry

q : ð3:6Þ

Thus the side rake angle

cx ¼ cos�1 nryffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2

rx þ n2ry

q0B@

1CA: ð3:7Þ

Clearance Angles ax and az

The tangents for the principal flank in the u- and v-directions are given by:

~q0p;u2¼ ðcb2 � u2sb2 � b2;u2

þ v2sa2cb2 � b2;u2Þi� ðsb2 þ u2cb2 � b2;u2

þ v2sa2sb2 � b2;u2Þk; ð3:8Þ

~q0p;v2¼ ðsa2sb2 þ v2ca2 � a2;v2 þ v2sa2cb2sb2 � b2;u2

Þiþ ðca2 � v2sa2 � a2;v2 Þjþ ðcb2sa2 þ v2cb2ca2 � a2;v2 Þk: ð3:9Þ

The normal vector is given by

~n0p ¼~q0p;u2�~q0p;v2

¼ npxiþ npyjþ npzk

¼ ðsb2ca2 � v2sb2sa2 � a2;v2 þ u2cb2 � b2;u2ca2 � u2v2cb2 � b2;u2

sa2 � a2;v2 þ v2sa2sb2 � b2;u2ca2

� v22s2a2sb2 � b2;u2

a2;v2 Þi� ðc2b2sa2 þ v2c2b2ca2 � a2;v2 þ v2s2a2c2b2 � b2;u2þ v2

2sa2c2b2 � b2;u2ca2 � a2;v2

þ sa2s2b2 þ v2s2a2s2b2 � b2;u2þ v2ca2 � a2;v2 s2b2 þ v2

2ca2 � a2;v2 s2b2sa2b2;u2Þj

þ ðcb2ca2 � v2cb2sa2 � a2;v2 � u2sb2 � b2;u2ca2 þ u2v2sb2 � b2;u2

sa2 � a2;v2 þ v2sa2cb2 � b2;u2ca2

� v22s2a2cb2 � b2;u2

a2;v2 Þk ð3:10Þ


Thus,

az ¼ cos�1 npzffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2

py þ n2pz

q0B@

1CA ð3:11Þ

and,

ax ¼ cos�1 npxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2

px þ n2py

q0B@

1CA: ð3:12Þ

Similarly, the clearance angles on the auxiliary flank are to be obtained.Side and End Cutting Edge AnglesThe angle made by the principal (side) cutting edge when projected on the base plane, with the Z-axis is called as the side

cutting edge angle (/s). Similarly, the angle made by the end cutting edge when projected on the base plane with the X-axis iscalled the end cutting edge angle (/e). For the generic single point cutting tool, the side cutting edge angle and the end cut-ting edge angle vary along the cutting edges. At each point on the side cutting edge, the tangent to the projection of the sidecutting edge on the base plane is evaluated. When projected on the ZX-plane, the cutting edge gives

~r0p3 ¼ ðvpw3cc3ðvpw3Þ � d1Þiþ ð�vpw3sa3ðw3Þsc3ðvpw3Þ þw3ca3ðw3Þ � d3Þk: ð3:13Þ

Taking tangent to the cutting edge,

~r0p3;w3¼@~r0p3

@w3¼ rp3;w3 xiþ rp3;w3zk:

The angle made by~r0p3;w3with the Z-axis gives /s as

/s ¼ cos�1 rp3;w3zffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2

p3;w3x þ r2p3;w3z

q0B@

1CA: ð3:14Þ

Similarly, for the end cutting edge, we have

~r0e3 ¼ ðvew3cc3ðvew3Þ � d1Þiþ ð�vew3sa3ðw3Þsc3ðvew3Þ þw3ca3ðw3Þ � d3Þk: ð3:15Þ

The angle made by~r0e3;w3with the X-axis gives /e as

b�1 ¼ /e ¼ cos�1 re3;w3xffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2

e3;w3x þ r2e3;w3z

q0B@

1CA: ð3:16Þ

3.1.1.2. ORS angles. In ORS system, the base plane is the same as the one in the ASA system, but the cutting plane is along theprincipal cutting edge, and the orthogonal plane is perpendicular to these two planes. Although the principal cutting edgelies completely in the principal flank which is inclined at an angle b2 with the X-axis, the principal cutting edge angle isslightly different and depends on the principal flank angle a2 and the number of resharpenings, i.e. the position of the tooltip. The positive or negative value of a2 will decide whether it will be greater or smaller than b2. It is given by b�2. But forpractical purposes, the angle of direction of the principal and end cutting edges to find their clearance angles in the ORSnomenclature can be taken as the rotational angles b2 and b1, respectively. This produces a small difference in the clearanceangles which can be neglected.

As we move along the curved cutting edges, the orientation of the cutting plane varies. To obtain the angles in the ORSsystem, the principal cutting edge has to be divided into small straight cutting elements. A local coordinate system XoYoZo hasto be created whose origin is attached to the midpoint of the small element and the axes are rotated. While Yo remains par-allel to the Y-axis, Xo and Zo get rotated by 90o � b�2

�clockwise.

Principal and End Cutting Edge AngleThe principal cutting edge angle b�2 is the angle made by the projection of the principal cutting edge on the base plane

with the X-axis. It is the complimentary of the angle /s (Eq. (3.14)) while the definition of the end cutting edge angle is givenby Eq. (3.16).

Orthogonal Clearance AnglesTo obtain the orthogonal clearance angle ao at any point, the normal vector is rotated anticlockwise about Y-axis through

90o � b�2 �

. It gives

~n0pr2¼ npxsb�2 þ npzcb�2 �

iþ npyjþ �npxcb�2 þ npzsb�2

�k: ð3:17Þ


When projected on the XY-plane, we get

~n0pr21 ¼ npxsb�2 þ npzcb�2 �

iþ npyj: ð3:18Þ

The value of ao is given by

ao ¼ cos�1 npxsb�2 þ npzcb�2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðnpxsb�2 þ npzcb�2Þ

2 þ n2py

q0B@

1CA: ð3:19Þ

Similarly, a0o can be obtained as

a0o ¼ cos�1 nexsb�1 þ nezcb�1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðnexsb�1 þ nezcb�1Þ

2 þ n2ey

q0B@

1CA; ð3:20Þ

where nex, ney, nez have corresponding definitions.Orthogonal Rake Angle and Inclination AngleThe normal vector to the rake face (Eq. (3.3)) when rotated by an angle 90o � b�2

�about the Y-axis anticlockwise gives

~n0rr2¼ nrxsb�2 þ nrzcb

�2

�iþ nryjþ �nrxcb�2 þ nrzsb

�2

�k: ð3:21Þ

When projected on the XY-plane, it gives

~n0rr21 ¼ nrxsb�2 þ nrzcb�2 �

iþ nryj: ð3:22Þ

The orthogonal rake angle co is the angle made by ~nrr21 with the Y-axis. Hence,

co ¼ cos�1 nryffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffinrxsb�2 þ nrzcb�2 �2 þ n2

ry

q0B@

1CA: ð3:23Þ

Similarly, projecting the rotated normal vector on the YZ-plane and taking the dot product with j gives the inclination angleas

k ¼ cos�1 nryffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið�nrxcb�2 þ nrzsb�2Þ

2 þ n2ry

q0B@

1CA: ð3:24Þ

To obtain the inclination angle along the end cutting edge, the normal to the rake face on the edge is rotated by b�1 anticlock-wise and then projected on the XY-plane. The angle with the Y-axis gives the inclination angle as

ke ¼ cos�1 nryffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðnrxcb�1 þ nrzsb

�2Þ

2 þ n2ry

q0B@

1CA: ð3:25Þ

3.1.1.3. NRS angles. The angles / and /e will be the same as b�2 and b�1, respectively. The inclination angle will bear the samedefinition too as for ORS nomenclature. The normal rake angle (cn) and the normal clearance angle (an) can be calculatedusing the existing standard tool angle relationships between the ORS and the NRS systems. The angles are given by:

cn ¼ tan�1½tan co cos k�; ð3:26Þ

an ¼ tan�1 tan ao

cos k

� �: ð3:27Þ

The normal clearance angle on the auxiliary flank is given by

a0n ¼ tan�1 tan a0ocos k

� �: ð3:28Þ

3.1.2. Forward mapping for the basic modelThe basic model is a special case of the generic model when the angles are constant. Using the method outlined above, the

ASA, ORS and NRS angles have been presented below [7]:


3.1.2.1. ASA angles. The normal to the rake face stands as

Bac

Sid

~nr ¼ �sc3 iþ ca3cc3 j� sa3cc3k; ð3:29Þk Rake Angle cz ¼ a3 ð3:30Þ

e Rake Angle cx ¼ cos�1 ca3cc3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2c3 þ c2a3c2c3

p" #

: ð3:31Þ

Clearance Angles ax and az

az ¼ cos�1 ca2cb2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2a2 þ c2a2c2b2

p" #

; ð3:32Þ

ax ¼ cos�1 ca2sb2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2a2 þ c2a2s2b2

p" #

: ð3:33Þ

In a similar manner, the clearance angles on the auxiliary flank are given by

a0z ¼ cos�1 ca1cb1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2a1 þ c2a1c2b1

p" #

; ð3:34Þ

a0x ¼ cos�1 ca1sb1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2a1 þ c2a1s2b1

p" #

: ð3:35Þ

Side and End Cutting Edge AnglesFor the principal cutting edge, the projection on the ZX-plane is given by:

~rp3 ¼ ðvpw3cc3 � d1 Þiþ ð�vpw3sa3sc3 þw3ca3 � d3Þk: ð3:36Þ

Thus,

/s ¼ cos�1 �vpsa3sc3 þ ca3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðvpcc3Þ

2 þ ð�vpsa3sc3 þ ca3Þ2q264

375: ð3:37Þ

Similarly, for the end cutting edge, projection on the ZX-plane is given by:

~re3 ¼ ðvew3cc3 � d1 Þiþ ð�vew3sa3sc3 þw3ca3 � d3Þk ð3:38Þ

and,

/e ¼ cos�1 vecc3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðvecc3Þ

2 þ ð�vesa3sc3 þ ca3Þ2q264

375: ð3:39Þ

3.1.2.2. ORS angles. Principal and End Cutting Edge Angle

b�2 ¼ p=2� /s ¼ cos�1 vpcc3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðvpcc3Þ

2 þ ð�vpsa3sc3 þ ca3Þ2q264

375: ð3:40Þ

The end cutting edge angle b�1 is the same as /e in the ASA nomenclature.Orthogonal Clearance Angles

ao ¼ a2: ð3:41Þ

Similarly,

a0o ¼ a1: ð3:42Þ

Orthogonal Rake Angle and Inclination AngleThe orthogonal rake angle co is given by:

co ¼ cos�1 ca3cc3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðsc3sb�2 þ sa3cc3cb�2Þ

2 þ ðca3cc3Þ2

q0B@

1CA: ð3:43Þ


The inclination angle is given as

k ¼ cos�1 ca3cc3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðsc3cb�2 � sa3cc3sb�2Þ

2 þ ðca3cc3Þ2

q0B@

1CA: ð3:44Þ

The inclination angle along the end cutting edge is obtained as

ke ¼ cos�1 ca3cc3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðsc3cb�1 þ sa3cc3sb�1Þ

2 þ ðca3cc3Þ2

q0B@

1CA: ð3:45Þ

3.1.2.3. NRS angles. Eqs. (3.26)–(3.28) can be used to find the normal rake angle (cn) and the normal clearance angle (an).

3.1.3. Forward mapping for the second generation of generic SPCTThe method to calculate the cutting edges and angles in all the nomenclatures will bear the same method, with the only

difference that here the distances (d11,d21,d31) will vary with the parameters and their variation has to be taken into account.

3.2. Inverse mapping

The inverse problem of finding rotation angles from the standard nomenclatures is important for generating/sharpeningthe angles. The purpose of the inverse mapping is that if the user specifies the desired conventional angles along the cuttingedge, the 3D rotational angles can be obtained.

3.2.1. Inverse mapping for the basic modelThe inverse mapping for the basic model has been presented below [7].

3.2.1.1. ASA angles. From Eq. (3.30), we have

a3 ¼ cz: ð3:46Þ

From the expression of cx; c2cx ¼c2a3c2c3

s2c3þc2a3c2c3.

Thus,

c3 ¼ tan�1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic2cz � c2czc2cx

c2cx

s" #: ð3:47Þ

Squaring the expressions for az and ax (Eqs. (3.32) and (3.33)), we have

ð1� C1Þpqþ C1p� C1 ¼ 0; ð3:48Þð1� C2Þpq� pþ C2 ¼ 0; ð3:49Þ

where, C1 = c2az, C2 = c2ax, p = c2a2 and q = c2b2.Solution of the above equations gives the angles a2 and b2. Similarly, substituting C3 ¼ c2a0z; C4 ¼ c2a0x; p0 ¼ c2a1 and

q0= c2b1 in Eqs. (3.48) and (3.49), we get

ð1� C3Þp0q0 þ C3p0 � C3 ¼ 0; ð3:50Þð1� C4Þp0q0 � p0 þ C4 ¼ 0: ð3:51Þ

These equations can be used to obtain the angles a1 and b1, respectively. If the value of any of the clearance angles is zero, nodefinite solution can be obtained.

3.2.1.2. ORS angles. The angle of rotation of the principal flank about the Y-axis is equivalent to the angle made by normal tothe principal flank with the X-axis and is given by

b2 ¼ /� tan�1ðtan k tan aoÞ: ð3:52Þ

Similarly the rotational angle b1 is given by

b1 ¼ /e � tan�1 tan ke tan a0o �

: ð3:53Þ

The value of ke is found by using the standard formula

tan ke ¼ sinð/� /eÞ tan co � cosð/� /eÞ tan k: ð3:54Þ


The values of a3 and c3 are found simultaneously using Newton–Raphson technique. The two non-linear equations are

1� C25

�ðca3cc3Þ

2 � C25ðK1sa3cc3 � K2sc3Þ

2 ¼ 0; ð3:55Þ

1� C26

�ðca3cc3Þ

2 � C26ðK2sa3cc3 þ K1sc3Þ

2 ¼ 0; ð3:56Þ

where, C5 ¼ cco; C6 ¼ ck; K1 ¼ cb�2 and K2 ¼ sb�2. While solving the equations, the initial starting point is considered to be(co,0) if both inclination angle and orthogonal rake angle are positive or both negative. If one is negative and the other ispositive then initial point is taken as (0,k).

3.2.1.3. NRS angles. The Eqs. (3.27) and (3.28) along with Eqs. (3.41), (3.42), (3.44) and (3.45) are used to evaluate the rota-tional angles a1 and a2, given by

a1 ¼ tan�1 tan a0n cos ke �

; ð3:57Þa2 ¼ tan�1ðtan an cos kÞ: ð3:58Þ

Angles b2 and b1 are the same as given by Eq. (3.52) and (3.53). To find the rake face angles a3 and c3, nonlinear Eqs. (3.44)and (3.26) are solved to get

1� C26

�ðca3cc3Þ

2 � C26ðK2sa3cc3 þ K1sc3Þ

2 ¼ 0; ð3:59Þ

C7 � C6 tan co ¼ 0; ð3:60Þ

where, C7 = tancn

3.2.2. Inverse mapping for the first generation of generic SPCTThe inverse mapping for the first generation starts with finding out the grinding parameters for a given set of points along

the curved cutting edge.

3.2.2.1. Equation of the cutting edges. The inverse mapping of the first generation of the generic SPCT has an added degree ofcomplexity as the angles are varying. To have the inverse mapping, first the equation of the principal cutting edge has to beobtained in terms of the points specified on the cutting edge.

Assuming that the axes of the grinders do not have angular accelerations, the angles are allowed to have only linear rela-tionship with the parameters.

Let

c3 ¼ a1 þ a2u3; a3 ¼ b1 þ b2w3; a2 ¼ c1 þ c2v2; b2 ¼ e1 þ e2u2: ð3:61Þ

Hence,

c3;u3¼ a2; a3;w3 ¼ b2; a2;v2 ¼ c2; b2;u2

¼ e2: ð3:62Þ

Let the points on the cutting edge be specified as ~P0;~P1 . . .~Pn where ~P0 is the coordinate of the tool tip.It is required to find out the value of n to generate the equation of the principal cutting edge. The unknowns to be found

out are a1, a2, b1, b2, c1, c2, e1 and e2.Now, it is observed that P0x = �d1, P0y = �d2, P0z = �d3 taking u0

3 ¼ 0;w03 ¼ 0 where, the superscript ‘0’ denotes the tool tip.

Each new point gives 6 equations and adds 4 unknowns.

Thus number of unknowns = 8 + 4nNo. of equations = 6n.Equating the two, n = 4.

Thus a total of 4 points (excluding~Po) will be required to derive the 24 unknowns. They are obtained by solving 24 simul-taneous equations. The unknowns will be

a1; a2; b1; b2; c1; c2; e1; e2;ui3;w

i3;u

i2;v i

2; i ¼ 1 . . . 4: ð3:63Þ

The simultaneous equations can again be solved using optimization technique as mentioned earlier.vp can now be obtained in terms of the derived parameters and the equation of the principal cutting edge can be obtained.In a similar manner, the equation to the end cutting edge can be derived.

3.2.2.2. ASA angles. If the ASA angles are specified on the cutting edge, the number of points to be provided will be deriveddepending on the number of angles given. Each angle will constrain the choice of points, and hence the total number ofpoints required gets fixed.

If the angles are specified at the tool tip, equations can be framed as in the case of the basic model.Here, c3 = a1, a3 = b1, a2 = c1, b2 = e1.


3.2.2.3. ORS angles. As with ASA angles, when the ORS angles are given at the tool tip, the inverse mapping will be performedin a similar manner as in the case of the basic model. For any other point, the additional point on the cutting edge and theORS angle have to be provided.

For example, if the orthogonal rake angle is given at a point on the cutting edge, it is in terms of u3;w3; b�2; a1; a2; b1; b2. As-

sume b�2 to be b2 and solve the equations. Each angle will add one equation. The number of points can be decided accordingly.

3.2.2.4. NRS angles. The method used with the ORS angles has to be applied for NRS angles too.

3.2.3. Inverse mapping for the second generation of generic SPCTThe approach to be adopted for the inverse mapping for the second generation is similar to that of the first generation,

with the only difference that the variation of the ground depths too has to be taken into account. This adds an extra degree ofcomplexity to the system of algebraic equations.

Fig. 7. Face and flanks for the illustrated case.

Fig. 8a. Rendered model of the generic cutting tool.

Table 2Points on the principal and auxiliary cutting edge.

S. No. Principal cutting edge Auxiliary cutting edge

X Y Z X Y Z

1 �4.30 �2.31 �5.50 �7.56 �2.71 �2.842 �3.22 �1.97 �8.99 �10.22 �3.62 �2.733 �2.65 �1.65 �10.68 �12.45 �3.84 �2.704 �1.99 �1.19 �12.54 �14.74 �4.53 �2.72


4. Illustration of the proposed methodology

The above methodology has been illustrated and validated through the following example.Let the angles in radians be given as follows:

c3 ¼p

180ð8� 0:5u3Þ; a3 ¼

p180ð5þw3Þ; a2 ¼

p180ð5� 0:5v2Þ; b2 ¼

p180ð75� 0:2u2Þ; a1

¼ p180ð5� 0:5v1Þ; b1 ¼

p180ð5� 0:2u1Þ:

Let the tool tip be positioned at (�5,�2.3,�3).For the given data, a MATLAB code was developed to generate the face and flank surfaces as shown in Fig. 7. The model

was converted into IGES format and the surfaces were rendered in the Rapidform XOR solid modeling environment. Surface

Fig. 8b. Illustrated points on the principal and auxiliary cutting edges.

Fig. 9a. (a) Variation of c3 with a2 and z. (b) Variation of a3 with a2 and z.

Fig. 9b. (c) Variation of b�2 with a2 and z. (d) Variation of k with a2 and z.

Fig. 9c. (e) Variation of co with a2 and z. (f) Variation of cn with a2 and z.

Fig. 9d. (g) Variation of az with a2 and z. (h) Variation of ax with a2 and z.

Fig. 9e. (i) Variation of ao with a2 and z. (j) Variation of an with a2 and z.


trimming and Boolean operation gave the complete model of the cutting tool (Fig. 8a).The rendered model shows the prin-cipal and auxiliary cutting edges.

4.1. Validation of the model

Using another MATLAB code, the principal and end cutting edges were obtained analytically. Any four points on the prin-cipal and auxiliary cutting edges other than the tool tip were obtained which are presented in Table 2. The same points wereobtained on the rendered model corresponding to the chosen parameters of variation. This validated the mathematical mod-el. The points are shown in Fig. 8b.


Following the inverse mapping method presented in Section 3.2.2, the above points can also be used to obtain the un-knowns given in Eq. (3.63).

4.2. Studying the variation of angles with variation of parameters

The variation of angles against the variation of any of the parameters can be plotted and studied. The plot can be used tochoose the parameters for desired variation of the angles.

For example, if c3 is allowed to vary from p180 ð8� 0:5u3Þ to p

180 ð8� 2:5u3Þ, the variation of angles in ASA, ORS and NRS canbe plotted (Fig. 9). The MATLAB code plotted the variation of angles as follows (x-axis shows the z-coordinate, y-axis showsthe constant a2 and z-axis shows the respective angle).

The above plots help the designer to select the parameters prudently to obtain the desired version of a generic singlepoint cutting tool.

5. Summary and conclusions

The presented work details upon the geometric modeling of a single point cutting tool with a generic profile. In a standardSPCT, the face and flanks are plane sections, but if the tool angles are required to vary along the cutting edges, the tool gets afree-form shape. To be able to generate such a shape through grinding, an analytical modeling of the tool geometry in termsof the varying grinding angles and ground depths is required, which has been explained in this work with illustrations. Thework can be summed up as follows:

1. The geometric modeling of the basic profile of an SPCT has been presented, which has been further extended to two gen-erations of a generic profile.

2. A mapping of the tool angles in the prevalent nomenclatures in terms of the grinding angles has been presented andtermed as the forward mapping. It has first been carried out for the first generation of the generic profile and the basicmodel becomes a special case of the mapping.

3. The grinding angles have been calculated in terms of the ASA, ORS and NRS angles and this mapping has been termed asthe inverse mapping. For the basic model, the inverse mapping is simple, but for the generic model, first the cutting edgecoordinates have to be provided and the grinding parameters have to be evaluated solving 24 simultaneous equations.When the desired angles on the cutting edge are provided, the equations get modified accordingly.

4. The presented methodology has been illustrated and validated through an example showing surface model of tool for agiven set of angles. Subsequently, the variation of tool angles against the variation of grinding angles has been studiedthrough surface plots.

As a conclusion, the paper presents a generic mathematical model of single point cutting tools and establishes the feasi-bility of grinding single point cutting tools with free-form rake face and flanks. The forward and inverse mapping helps theuser to understand the variation of tool angles with varying grinding angles. This work can be further extended to multi-point cutting tools of generic shape.

References

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Math. Model. 34 (2010) 2738–2748.[5] W. Zhongqing, Research on forming theory of a 3D body by the constructing cutting geometry method (CCG), J. Mater. Process. Technol. 139 (2003)

539–542.[6] T.S. Rajpathak, Geometric modeling of single point cutting tools for grinding and sharpening, M.Tech. Thesis, IIT Kanpur, 1996.[7] P. Tandon, P. Gupta, S.G. Dhande, Geometric modeling of single point cutting tool surfaces, Int. J. Adv. Manufac. Technol. 22 (2003) 101–111.[8] D.A. Stephenson, J.S. Agapiou, Calculation of main cutting edge forces and torque for drills with arbitrary point geometries, Int. J. Mach. Tools Manufac.

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tool cutting edge, Int. J. Mach. Tools Manufac. 49 (2009) 667–677.[10] K. Sambhav, S.G. Dhande, P. Tandon, CAD based mechanistic modeling of forces for generic drill point geometry, Comput. Aided Design Appl. 7 (6)

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A generic mathematical model of single point cutting tools in terms of grinding parameters