Kinematic Geometry of Surface Machinin Episode Episode 8 pdf

Third, the configuration of the defined clearance surface C s of the form-cutting tool must be of the sort for which the surface C s makes the optimal clearance angle a with the generati

220 Kinematic Geometry of Surface Machining

which the surface C s passes through the computed cutting edge of the

form-cutting tool Third, the configuration of the defined clearance surface C s of

the form-cutting tool must be of the sort for which the surface C s makes the

optimal clearance angle a with the generating surface T of the form-cutting

tool at a given point of the cutting edge

It is easy to see that clearance surfaces of the cutting tool cannot always

be shaped in the form that is convenient for manufacturing the cutting tool

The cutting edge of a precision form-cutting tool can be considered as a line

of intersection of the three surfaces, say of the generating surface T of the

cutting tool, of the rake surface R s , and of the clearance surface C s This

requirement is compliant with three surfaces T, R s , and C s being the

sur-faces through the common line, say through the cutting edge of the cutting

tool, that could impose strong constraints on the actual shape of the

clear-ance surface of the cutting tool Under such restrictions, the clearclear-ance

sur-face C s usually cannot allow sliding over itself However, the desired surface

C s can be approximated by a surface that allows sliding over itself; thus,

the approximation could be more convenient for design and manufacture

of the form-cutting tool This means that in certain cases of implementation

of the first method, approximation of the desired clearance surface C s with

a surface that features another geometry can be unavoidable The

approxi-mation of the desired surface C s results in the surface P being generated

not with the precise surface T, but with an approximated surface T g of the

cutting tool The approximated surface T g deviates from the desired surface

T The deviation δT is measuring along the unit normal vector nT to the

surface T at a corresponding surface point Application of the form-cutting

tool having approximated the generated surface is allowed if and only if the

resultant deviation δT is within the corresponding tolerance [ ]δT — that is,

when the inequality δT≤[ ] is valid.δT

Summarizing, one can come up with the following generalized procedure

for designing the form-cutting tool in compliance with the first method:

1 Determination of the generating surface T of the form-cutting tool

(see Chapter 5)

2 Determination of the rake surface R s: The rake surface is selected

within surfaces that are technologically convenient (a kind of

rea-sonably practical surface) Configuration of the rake surface is

spec-ified by the rake angle of the desired value

3 Determination of the cutting edge: The cutting edge is represented

with the line of intersection of the generating surface T of the

form-cutting tool by the rake surface R s

4 Construction of the clearance surface C s that passes through the

cutting edge and makes the clearance angle of the desired value

with the surface of the cut The clearance surface is selected within

surfaces that are technologically convenient (a kind of reasonably

practical surface)

The Geometry of the Active Part of a Cutting Tool 221

For the practicality, a normal cross-section of the clearance surface must be

determined as well

Figure6.1 illustrates an example of implementation of the first method for

the transformation of the generating body of the form-cutting tool into the

workable-edge cutting tool For illustrative purposes, the round form cutter

for external turning of the part is chosen

In the case under consideration, the part surface P is represented by two

separate portions P1 and P2 An axial profile of the part is specified by the

composite line through points a p , b p , and c p For the particular case shown

in Figure6.1, the generating surface T of the form cutter is congruent with the

part surface P being machined This statement easily follows from the

con-sideration that is based on the analysis of kinematics of the machining

opera-tion (see Chapter 2) Thus, the identity T ≡ P is observed (to be more exact,

two identities T1≡ P1 and T2≡ P2 are valid) Ultimately, the generating surface

T of the form cutter is also represented with two portions T1 and T2

The concept of the first method for the transformation of the generating body of the

form-cutting tool into the workable edge form-cutting tool.

222 Kinematic Geometry of Surface Machining

Due to the identity T ≡ P observed, the axial profile of the generating

sur-face T of the form cutter is composed of two segments through the points

a T , b T , and c T (not labeled in Figure 6.1); and the identities aT≡ a p , b T≡ b p,

and c T≡ c p are valid

Then, a plane is chosen as the rake surface R s of the round form cutter

Definitely, the plane allows for sliding over itself It is convenient to machine

the plane in cutting tool production The plane is parallel to the axis of

rotation O T≡ O p of the generating surface T of the form cutter It makes the

rake angle γc perpendicular to the surface T at the base point a T≡a c

The piecewise line of intersection a b c c c c of the generating surface T of the round

form cutter by the rake surface R s serves as the cutting edge of the form cutter

The clearance surface C s of the form cutter is shaped in the form of a

surface of revolution All surfaces of revolution allow for sliding over

them-selves The surface of revolution C s is represented with two separate portions

C s.1 and C s.2 The clearance surface of the round form cutter can be generated

as a series of consecutive positions of the cutting edge a b c c c c when rotating

the cutting edge a b c c c c about the surface C s axis of rotation O c For practical

needs, the axial profile of the clearance surface C s must be determined

The considered example (Figure6.1) illustrates implementation of the first

method for the transformation of the generating body of the cutting tool into the

workable edge cutting tool This method has been known for many decades

The first method for the transformation of the generating body of the

cut-ting tool into the workable edge cutcut-ting tool is widely used in many

indus-tries Form edge cutting tools of most designs can be designed in compliance

with the first method When the first method is employed, this yields

design-ing of the form-cuttdesign-ing tools that are convenient in manufacturdesign-ing and in

application

The major disadvantages of the first method are twofold First, in most

cases of application of the first method, no optimal values of the

geo-metrical parameters of the form-cutting tool at every point of the

cut-ting edge can be ensured Optimization of the geometrical parameters at

every point of the cutting edge is a challenging problem The solution to

the problem of optimization of the geometrical parameters of the cutting

edge (if any) is often far from practical needs: It could be feasible, but it is

often not practical Second, unavoidable deviations of the actual

approxi-mated generated surface of the cutting tool from its desired shape often

cannot be eliminated when the first method is employed to the design of

the form-cutting tool

6.1.2 The Second Method for the Transformation of the Generating

Body of the Cutting Tool into the Workable Edge Cutting Tool

Consider another scenario under which the generating surface of the cutting

tool is also determined (see Chapter 5) The cutting tool clearance surface is

chosen within the surfaces that are convenient for machining of the surface

The Geometry of the Active Part of a Cutting Tool 223

when manufacturing the form-cutting tool, for inspection purposes, and

so forth In most cases, the clearance surface C s is within the surfaces that

allow for sliding over themselves (see Section 2.4)

The clearance surface C s is properly oriented with respect to the

generat-ing surface T of the form-cuttgenerat-ing tool It makes an optimal clearance angle

with the surface T at a given point.

Following the second method for the transformation of the generating

body of the cutting tool into the workable-edge cutting tool, cutting edges of

the form-cutting tool are defined as the line of intersection of the generating

surface T of the cutting tool by the clearance surface C s

Once the cutting edge is constructed, then the rake surface R s can be

con-structed in compliance with the following routing First, the rake surface is

selected within the surfaces that allow for sliding over themselves (see

Sec-tion 2.4) This requirement is highly desirable but not mandatory Actually,

any surface having reasonable geometry could serve as the rake surface of

the form-cutting tool Second, parameters of the chosen surface R s must be

computed in compliance with the requirement under which the surface R s

passes through the computed cutting edge of the form-cutting tool Third,

the configuration of the defined rake surface R s of the form-cutting tool must

be of the sort for which the surface R s makes the optimal rake angle γ with

respect to the perpendicular to the generating surface T of the form-cutting

tool at a given point of the cutting edge

It is easy to understand that rake surfaces of a cutting tool cannot always

be shaped in the form that is convenient for manufacturing the cutting tool

The cutting edge of a precision form-cutting tool can be considered as a line

of intersection of three surfaces: of the generating surface T of the cutting

tool, of the rake surface R s , and of the clearance surface C s The requirement

that three surfaces T, R s , and C s be the surfaces through the common line,

say through the cutting edge of the cutting tool, could impose strong

con-straints on the actual shape of the clearance surface of the cutting tool Under

such restrictions, the rake surface R s usually cannot allow sliding over

itself However, the desired surface R s can be approximated by a surface

that allows for sliding over itself; thus, the approximation could be more

con-venient for the design and manufacture of the form-cutting tool This means

that in certain cases of implementation of the second method, approximation

of the desired rake surface R s with a surface featuring another geometry

can be unavoidable The approximation of the desired surface R s results in

that ultimately the surface P is generated not with the precise surface T, but

with an approximated surface T g of the cutting tool The approximated

sur-face T g deviates from the desired surface T The deviation δT is measuring

along the unit normal vector nT to the surface T at a corresponding surface

point Application of the form-cutting tool having approximated the

gener-ated surface is allowed if and only if the resultant deviation δT is within

the corresponding tolerance [ ]δT — that is, when the inequality δT ≤[ ] is δT

valid

224 Kinematic Geometry of Surface Machining

Summarizing, one can come up with the following generalized procedure

for designing of the form cutting tool in compliance with the second method:

1 Determination of the generating surface T of the form-cutting tool

(see Chapter 5)

2 Determination of the clearance surface C s: The clearance surface is

selected within surfaces that are technologically convenient (a kind

of reasonably practical surface) Configuration of the clearance

sur-face is specified by the clearance angle of the desired value

3 Determination of the cutting edge: The cutting edge is represented

with the line of intersection of the generating surface T of the

form-cutting tool by the clearance surface C s

4 Construction of the rake surface R s that passes through the cutting

edge and makes the rake angle of the desired value perpendicular to

the surface of the cut The rake surface is selected within surfaces that

are technologically convenient (a kind of reasonably practical surface)

For practicality, a typical cross-section of the clearance surface must be

deter-mined as well

Figure6.2 illustrates an example of implementation of the second method

for the transformation of the generating body of the form-cutting tool into

the workable edge cutting tool For illustrative purposes, the form milling

cutter for machining helical grooves is chosen

Consider that the generating surface T of the cutting tool is already

deter-mined Geometry of the chosen clearance surface C s is predetermined by

The concept of the second method for the transformation of the generating body of the

form-cutting tool into the workable edge form-cutting tool.

The Geometry of the Active Part of a Cutting Tool 225

kinematics of the operation of relieving the milling cutter teeth The cutting

edge is represented as the line of intersection of the generating surface T of

the milling cutter by the clearance surface C s

Further, the constructed cutting edge is used for the generation of the rake

face R s For this purpose, either the cutting edge or its projection onto the

transverse plane moves along the milling cutter axis of rotation O c In the

case under consideration, the face surface R s is represented as the locus of

consecutive positions of the cutting edge in its motion along the axis O c The

rake surface R s is shaped in the form of a general cylinder

The considered example (Figure 6.2) illustrates implementation of the

sec-ond method for the transformation of the generating body of the cutting tool

into the workable edge cutting tool This method is not as widely used in

industry as is the first method

The second method for the transformation of the generating body of the

cutting tool into the workable edge cutting tool does not have wide

imple-mentation in industry Form edge cutting tools of most designs can be

designed in compliance with the second method When the second method

is employed, this yields designing of the form-cutting tools that are

conve-nient in manufacturing and in application

The major disadvantages of the second method are twofold: First, in most

cases of implementation of the second method, no optimal values of the

geo-metrical parameters of the form-cutting tool at every point of the cutting

edge can be ensured Optimization of the geometrical parameters at every

point of the cutting edge is a challenging problem The solution to the

prob-lem of optimization of the geometrical parameters of the cutting edge (if any)

is often far from practical: It could be feasible, but it is often not practical

Sec-ond, unavoidable deviations of the actual approximated generated surface of

the cutting tool from its desired shape often cannot be eliminated when the

first method is employed to design of the form-cutting tool

It is important to stress that both methods for the transformation of the

generating body of the cutting tool into the workable edge cutting tool

fea-ture a common disadvantage This disadvantage results in the incapability

of designing a form-cutting tool that has optimal value of the angle of

incli-nation l The actual value of the angle l at a current point within the cutting

edge is a function of shape, of parameters, and of location of the rake R s or

the clearance C s surfaces of the form-cutting tool with respect to the

gener-ating surface T of the form-cutting tool Due to this, the optimal values λopt

of angle of inclination of the cutting tool edge become impractical due to

significant difficulties in manufacturing the form-cutting tool

6.1.3 The Third Method for the Transformation of the Generating

Body of the Cutting Tool into the Workable Edge Cutting Tool

Ultimately, consider the third scenario under which the generating surface

of the cutting tool is also determined (see Chapter 5) However, in this case,

neither the rake surface nor the clearance surface of a desired geometry is

226 Kinematic Geometry of Surface Machining

selected at the beginning, but the cutting edge is selected instead Such an

approach allows for optimization of the angle of inclination at every point

of the cutting edge of the form-cutting tool The method considered below is

proposed by Radzevich [14,15,19]

For implementation of the third method, it is necessary to construct a

spe-cial family of lines within the generating surface of the form-cutting tool

Lines of this family of lines represent the assumed trajectories of motion of

the cutting edge points over the surface of the cut when the work is

machin-ing (Figure6.3) Below this family of lines within the generating surface T

of the cutting tool is referred to as the primary family of lines Analysis of a

particular machining operation allows for analytical representation of the

family of lines within the surface T.

Further, after the primary family of lines is defined, it is necessary to

construct a secondary family of lines The secondary family of lines is also

within the generating surface T, and it is isogonal to the primary family of

lines At every point of intersection of the lines of the primary and of the

secondary families, the angle between the lines is equal to (90° -λopt) Here,

λopt designates the optimal value of the angle of inclination of the cutting

edge Therefore, the angle of inclination is at its optimal value at every point

of the cutting edge This is due to the primary family of lines within the

gen-erating surface T of the tool being isogonal to the secondary family of lines

at every point of the cutting edge

Different segments of the cutting edge of a form-cutting tool are at different

distances from the axis of the tool rotation Because of this, they work with

dif-ferent cutting speeds This results in the optimal value of the angle of

inclina-tion being different for different porinclina-tions of the cutting tool edge Under such

The concept of the third method for the transformation of the generating body of the

form-cutting tool into the workable edge form-cutting tool.

The Geometry of the Active Part of a Cutting Tool 227

a scenario, the actual value of the angle of inclination could be either constant

within the cutting edge (and thus equal to its average value), or the desired

variation of the angle of inclination can be ensured In the last case, the

prob-lem of design of a form-cutting tool becomes more sophisticated

An appropriate number of lines from the second family of lines can be

selected to serve as the cutting edges of the form-cutting tool to be designed

These lines are uniformly distributed and are at a certain distance t from

one another The distance t is equal to the tooth pitch of the form-cutting

tool

Rake surface R s is a surface through the cutting edge of the form-cutting

tool The surface R s makes the rake angle g with the perpendicular nc to

the surface of cut Actually the perpendicular to the surface of cut deviates

from the perpendicular nT to the generating surface T of the form cutting

tool Fortunately, this deviation is of negligibly small value For practical

needs of design of the form-cutting tool, the perpendicular to the surface of

cut nc is not used, but the corresponding perpendicular nT to the

generat-ing surface T is used instead.

The clearance surface C s is also a surface through the cutting edge of the

form-cutting tool The surface C s makes the clearance angle α with the

generating surface T.

It is possible to formulate the problem of design of the form-cutting tool

in the way following which the rake angle g, as well as the clearance angle

a, could be of optimal value at every point of the cutting edge of the

form-cutting tool In order to satisfy this requirement, both the rake surface R s

and the clearance surface C s must be of special geometry This problem

could be solved analytically

When deriving equations of the surfaces R s and C s, it is necessary to

ensure optimal values γopt, αopt, and λopt for the parameters g, a, and l for

the new form-cutting tool, as well as for the cutting tool after it is reground

The optimal values γopt, αopt, and λopt for the new form-cutting tool and for

the reground cutting tool are not necessarily the same

Summarizing, one can come up with the following generalized procedure

for designing the form-cutting tool in compliance with the third method:

1 Determination of the generating surface T of the form-cutting tool

(See Chapter 5)

2 Determination of the cutting edge: The cutting edge is at the angle

of inclination of an optimal value with respect to the direction of

speed of the resultant motion of the cutting edge point relative to

the surface of the cut

3 Construction of the rake surface R s , and the clearance surface C s

simultaneously: The rake surface passes through the cutting edge

and makes the rake angle of the desired value perpendicular to

the surface of the cut The clearance surface also passes through

the cutting edge and makes the clearance angle of the desired

value with the generating surface of the cutting tool Both the

228 Kinematic Geometry of Surface Machining

rake surface and the clearance surface are selected based on their

technological convenience and property of sliding over

them-selves (a kind of reasonably practical surface) See Section 2.4 for

more detail

The third method for the transformation of the generating body of the

cut-ting tool into the workable edge cutcut-ting tool is a completely novel method

[14,15,19] It has not yet been comprehensively investigated Therefore, more

detailed explanation of the method is important

Consider the generating surface T that is shaped in the form of a

sur-face of revolution This assumption is practical, because, for example,

milling cutters of all designs have the generating surface T in the form of

a surface of revolution Using the third method, it is easy to come up with

an understanding that the cutting edge of milling cutters of all designs

must be shaped in the form of loxodroma By definition, loxodroma is a line

that makes equal angles with a given family of lines on a surface

Actu-ally, loxodroma can be easily defined with respect to coordinate lines on

the surface [2]

In the case under consideration, loxodroma having special shape

param-eters is of particular interest The loxodroma that makes the angle (90° -λopt)

with the primary family of lines on the generating surface T can be employed

as the cutting edge of the form-cutting tool

In a particular case, when parameterization of the generating surface T of

the form-cutting tool yields the expression

φ1.T⇒d S T2=dU T2+G U dV T( T) T2 (6.1)

for the first fundamental form φ1.T, then the cutting edge having optimal

value of the angle of inclination λopt at every point can be described by the

Equation (6.2) of the cutting edge is expressed in terms of UT and VT

param-eters of the generating surface T of the form-cutting tool Using

conven-tional mathematical methods, Equation (6.2) can be converted to a Cartesian

coordinates

Example 6.1

Consider a ball-nose milling cutter of radius r T (Figure 6.4) The milling cutter

is used for machining a sculptured surface on a multi-axis numerical control

230 Kinematic Geometry of Surface Machining

for the unit tangent vector ce( , )ϕ θ can be derived from Equation (6.3):

where d S T denotes the differential of the arc segment of the cutting edge

Particularly, when θ θ= c =Const, then Equation (6.4) for the unit tangent

vector ce reduces to

c c

( , )

cos coscos sinsin

where ℘ designates a certain angle

After integration of Equation (6.7) is accomplished, one can come up with

the solution

tanϕ (θ )

where q= ±cot℘ and C is an arbitrary constant value.

Implementation of the trivial trigonometric formulae

tan , cos

tantan

ϕ

ϕ

ϕϕ

=

+

-221

2

1

21

The Geometry of the Active Part of a Cutting Tool 231

Ultimately, under the assumption θc=λopt, the above analysis yields an

equation for position vector rce of a point of the cutting edge of the ball-nose

milling cutter (Figure 6.4a):

cos

sin(

λλλλ

The angle of inclination for the milling cutter having a cutting edge that is

shaped in compliance with Equation (6.11) is constant At every point of the

cutting edge, it is equal to its optimal value λopt

The entire loxodroma (see Equation 6.11) is not used for design of the

cut-ting edge of the ball-nose milling cutter Only the arc segment AB is used for

this purpose (Figure6.4b)

Other methods for derivation of Equation (6.11) can be implemented as

well [14,15,19]

Consider another approach for derivation of an analytical description of

the cutting edge of the ball-nose milling cutter In this particular case, when

parameterization of the equation of the generating surface T of the milling

cutter yields the expression

φ1.T⇒dS2T=r dU T2( T2+cos2U dV T T2) (6.12)

for the first fundamental form φ1.T, then the cutting edge having optimal

value of the angle of inclination λopt at every point can be described by the

It is important to focus on the shape of the loxodroma The loxodroma makes

an infinite number of revolutions about its pole. It approaches the pole infinitely

close This curve approaches the pole similar to an asymptotic point The last

can cause some inconveniences while manufacturing cutting tools However,

several methods are developed for avoiding the inconveniences [15]

Feasibility of the optimization of the angle of inclination λopt is not

lim-ited to ball-nose milling cutters Form-cutting tools having the generating

 The loxodroma’s pole is located at the point of intersection of the generating surface T of the

milling cutter by the axis of rotation of the cutting tool.

232 Kinematic Geometry of Surface Machining

surface T of any feasible shape can be designed with the optimal value of the

angle of inclination The last statement encompasses composite generating

surfaces T of the form-cutting tools as well.

As an example, consider the optimization of the angle of inclination of a

filleted-end milling cutter (Figure6.5) The generating surface of the

filleted-end milling cutter is composed of three portions: the cylindrical portion T1,

the flat-end T2, and the torus surface T3

For the cylindrical portion T1 of the generating surface of the filleted-end

milling cutter, the cutting edge AB having the optimal angle of inclination

λopt=Const reduces to a helix 1 of constant pitch For the flat-end portion

T2 of the generating surface, the cutting edge is represented in the form of a

logarithmic spiral curve 2 Ultimately, the equation of the cutting edge

seg-ment BC within the portion T3 of the generating surface of the milling cutter

can be derived on the premises of Equation (6.2) This segment of the cutting

edge is represented by the arc segment 3 of the loxodroma The loxodroma is

within the torus surface T3

The generalized Equation (6.2) of the cutting edge having optimal value of

the angle of inclination is valid for edge-cutting tools of any possible design

However, in particular cases of the filleted-end milling cutter, significant

simplifications are possible

For example, for the flat-end portion T2 of the filleted-end milling cutter

(Figure6.5), an equation of the cutting edge can be derived following one

of two possible ways

In compliance with the first of them, the flat-end surface is considered as

a surface of revolution that is degenerated into the plane Further, the

equa-tion of the cutting edge can be derived for the surface of revoluequa-tion

Following the second possible way, it is preferred to employ the

dif-ferential equation for isogonal trajectories If a planar curve intersects all

the curves of the initially given single-parametric family of planar curves

C B

(b)3

B

FiGurE 6.5

The filleted-end milling cutter having an optimized value of the angle of inclination λopt.

The Geometry of the Active Part of a Cutting Tool 233

ϕ( , , )x y θ =0 at a given constant angle of intersection z, then the line satisfies

the differential equation:

For the flat-end portion T2 of the generating surface of the filleted-end

milling cutter, the initially given single-parametric family of planar curves

ϕ( , , )x y θ =0 is represented by the family of straight lines through the axis

of rotation of the milling cutter Here θ designates the angular parameter of

the family of straight lines ϕ( , , )x y θ =0

The equation of the family of straight lines can be represented in the

form

ϕ( , , )x y θ = -y xtanθ=0 (6.15)Further, assume that ς= ° -90 λopt After substituting Equation (6.15) into

Equation (6.14), the Equation (6.14) casts into an equation of the cutting edge

of the filleted-end milling cutter This equation describes a logarithmic

spi-ral curve This means that in the particular case under consideration, the

logarithmic spiral curve can be interpreted as the loxodroma for the family

of straight lines within the plane

It is convenient to represent the equation of the cutting edge in polar

coordinates:

ρ ρ= 0eϕ tan λopt, (ρ0>0, - ∞< < +∞ϕ ) (6.16)where r is the position vector of a point of the cutting edge; and r0 is the

position vector of a given point of the cutting edge, from which the angle j

is measuring

The cutting edge (see Equation 6.16) intersects all straight lines through

the point O at that same angle ς= ° -90 λopt

The pole of the logarithmic spiral curve coincides with the axis of rotation

of the milling cutter It represents an asymptotic point of this planar curve

Because of this, the cutting edge of the flat-end portion T2 of the generating

surface cannot pass through the axis of the tool rotation It is possible to design

the cutting edge in the shape of the logarithmic spiral curve (see Equation 6.16)

only within a certain portion, similar to the arc AB (Figure 6.4b) The cutting

edge cannot be shaped in the form of the logarithmic spiral curve between a

certain point A and the axis of rotation of the filleted-end milling cutter.

In the case under consideration, representation of the equation of the

cut-ting edge in the following form proved to be useful:

ρ ρ= A eϕ tan λopt, (ρA>0, ϕA= ° < <0 ϕ ϕB) (6.17)where rA is the position vector of the point A of the cutting edge AB.

234 Kinematic Geometry of Surface Machining

A few other design parameters of the cutting edge of the form-cutting tool

which can be drawn up from the geometrical analysis are as follows

Length S AB of the cutting edge AB can be computed from the equation

where ρB is the position vector of the point B of the cutting edge AB.

The radius of curvature RT at a current point of the cutting edge AB can

be computed from the equation

R T( )ρ =ρ 1+tan2λopt =ρA eϕtanλopt 1+tan2λopt = ρA ee opt

Length S AB and the radius of curvature RT of the cutting edge are often

required for optimization of performance of the filleted-end milling cutter

The third method for the transformation of the generating body of the

cut-ting tool into the workable edge cutcut-ting tool can be implemented to design

cutting tools for machining both sculptured surfaces on a multi-axis NC

machine as well as for machining parts on conventional machine tools

In addition to loxodroma possessing useful properties for a tool designer,

this curve can be evolved into two possible areas

First, a curve similar to loxodroma can be constructed on the generating

surface T of a form-cutting tool that is shaped not only in the form of a

sur-face of revolution, but also for the sursur-face T of another topology, including

surfaces T that allow for sliding over themselves.

Second, the loxodroma can be evolved to a more general area, when the

optimal value of the angle of inclination λopt varies within the cutting edge

Under such a scenario, the desired current value of the angle λopt can be

expressed in terms of curvilinear coordinates U T and V T, say by

equa-tion λopt=λopt(U V T, T) The desired function λopt=λopt(U V T, T) of

varia-tion of the optimal value of angle of inclinavaria-tion λopt can be determined

experimentally

A form-cutting tool of any kind can be designed using any of three methods

considered above The tool design engineer makes his or her own decision of

which method is preferred to use to design a particular form-cutting tool

6.2 Geometry of the Active Part of Cutting Tools

in the Tool-in-Hand System

The active part of the cutting tool is composed of two surfaces intersecting

each other to form the cutting edge The surface over which the chip is

flow-ing is known as the rake surface R s or more simply as the face And that

The Geometry of the Active Part of a Cutting Tool 235

surface, which is faced to the machined surface, is known as the clearance

surface C s or the flank In the simplest yet common case, both surfaces R s

and C s are planes The cutting edge is represented as the line of intersection

of the rake surface R s and of the clearance surface C s

Cutting edges of two kinds can be distinguished: roughing cutting edges

and finishing (clean-up) cutting edges Roughing cutting edges do not

gen-erate the surface P being machined, but finishing cutting edges do

Finish-ing cuttFinish-ing edges are always within the generatFinish-ing surface T of the cuttFinish-ing

tool Roughing cutting edges are beneath the surface T and within the

gen-erating body of the cutting tool

The generating surface T of a cutting tool can make point contact with

the part surface P Under such a scenario, roughing portions of the cutting

edges may be within the surface T as well.

Major and minor cutting edges of the cutting tool are distinguished A

whole cutting edge or its portion that is faced toward the direction of the

feed rate is referred to as the major cutting edge of the cutting tool Another

cutting edge or the rest of the whole cutting edge is referred to as the minor

cutting edge of the cutting tool The major cutting edge of a cutting tool

con-tacts the chip being cut off The minor cutting edge of a cutting tool concon-tacts

with the uncut portion of the stock The regular (and not stochastic) residual

roughness on the surface P is caused by both the major and the minor

cut-ting edges

For a form-cutting tool having curved cutting edges (for example, for

a milling cutter), an elementary cutting edge of infinitesimal length dl is

considered below Depending upon the actual problem under

consider-ation, the infinitesimal cutting edge dl is considered either as a

straight-line segment or as a circular-arc segment of the corresponding radius of

curvature

6.2.1 Tool-in-Hand reference System

A references system associated with reference surfaces of the cutting tool is

referred to as the tool-in-hand reference system This reference system is often

used when designing, manufacturing, regrinding, and inspecting the

cut-ting tool In order to accomplish design of a high-performance cutcut-ting tool,

the geometry of the active part of the cutting tool in various cross-sections of

the cutting wedge must be known

The tool-in-hand reference system is made up of planes that are tangent

to the generating surface T of the cutting tool, to the rake surface R s, and to

the clearance surface C s In particular cases, the surfaces T, R s , and C s (all or

some of them) degenerate to corresponding planes

The actual values of geometric parameters of the active part of a cutting

tool are determined in a coordinate system associated with the cutting tool

This coordinate system is referred to as the static coordinate system Various

configurations of the static coordinate system with respect to the cutting tool

are feasible