Kinematic Geometry of Surface Machinin Episode Episode 7 pot

The auxiliary coordinate system X Y1 1 is rigidly connected to plane T Generation of a screw involute surface as an enveloping surface to consecutive positions of a plane that performs

generates the plane P21( ) 1 and the circular cylinder P21( ) 2 The plane portion T22

of the generating surface generates the plane P22( ) 1 and the plane P22( ) 2

Knowing the dimensions of the milling cutters and the parameters of

their motion relative to the work, the above formulae can be used to derive

equations of all of the machined part surfaces Similarly, the above formulae

work when necessary to determine the generating surface of a cutting tool

for machining of a given part surface

5.2.2 Kinematical Method for the Determining of enveloping Surfaces

For engineering applications, one more method for the determination of

enveloping surfaces is helpful This method is referred to as the kinematical

method for the determination of enveloping surfaces Initially, this method

was proposed by Shishkov as early as in 1951 [31]

The kinematical method is based on the particular location of the vector

V1 2− of relative motion of the moving surface and of the enveloping surface

Vector V1 2− is located within the common tangent plane to the surfaces This

condition immediately follows from the following consideration Motions of

only two kinds are feasible for the moving surface and the enveloping

sur-face The surfaces can roll over each other, and they can slide over each other

The component of the resultant relative motion V1 2− in the direction

perpen-dicular to the surfaces is always equal to zero (Figure 5.18)

The cutting tool performs a certain motion relative to the work The part

surface P is generated as an enveloping surface to consecutive positions of

the generating surface T of the cutting tool Points of three different kinds

can be distinguished on the moving surface T.

Consider points of the first kind, for example, the point A (Figure 5.18) The

vector of resultant motion of the cutting tool with respect to the work at point

A is designated as VΣ( )A Projection Prn V( )ΣA of the vector VΣ( )A onto the unit

C

P T

(C)

VΣ

B A

Profiling of the Form-Cutting Tools of Optimal Design 187

normal vector nT( )A to the generating surface T is pointed to the interior of

the work body ( Prn VΣ( )A >0) Therefore, in the vicinity of point A, the cutting

tool penetrates the work body In this way, roughing portions of the

tool-cut-ting edges cut out the stock

Further, consider points of the second kind, for example, the point B

(Figure 5.18) The vector of resultant motion of the cutting tool with respect to the

work at point B is designated as VΣ( )B Projection Prn VΣ( )B of the vector V( )ΣB onto

the unit normal vector nT( )B to the generating surface T is perpendicular to this

unit normal vector — that is, it is tangential to the part surface P ( Prn V( )ΣB =0 )

Therefore, in the vicinity of the point B, the cutting tool does not penetrate the

part body The tool-cutting edges do not cut out stock The generating surface T

of the cutting tool generates the part surface P in the vicinity of the point B.

Ultimately, consider points of the third kind, for example, the point C

(Fig-ure 5.18) The vector of resultant motion of the cutting tool with respect to the

work at point C is designated VΣ( )C Projection Prn VΣ( )C of the vector VΣ( )C onto

the unit normal vector nT( )C to the generating surface T is pointed outside the

part body ( Prn V( )ΣC <0) Therefore, in the vicinity of point C, the cutting tool

departs from the machined part surface P In the vicinity of points of the

third kind, the tool-cutting edges do not cut out stock, and the generating

surface T of the cutting tool does not generate the part surface P.

The considered example unveils the nature of the kinematical method for

the determination of the enveloping surface Apparently, this method can be

employed to solve problems of both kinds, to profile form-cutting tools for

machining a given part surface, and to solve the inverse problem of the theory of

surface generation As an example, generation of the plane P with the cylindrical

grinding wheel having the generating surface T is considered in Figure 5.19

When machining a plane P, the grinding wheel rotates about its axis of

rotation with a certain angular velocity ωT Simultaneously, the grinding

wheel travels with a feed rate ST At each of the points A, B, and C on the

generating surface T of the grinding wheel, the speed of the resultant relative

motion of the cutter with respect to the work is designated VΣ( )A, VΣ( )B, and

VΣ( )C

, respectively The speed of the resultant motion at each point A, B, and C

is equal to the vector sum of the feed rate ST and the speed of cutting Vcut( )A,

Vcut( )B, and Vcut( )C The feed rate ST is the same value for all points A, B, and C

The velocities Vcut( )A, Vcut( )B, and Vcut( )C are equal to the linear speed of rotation

necessary to perform cutting

It is easy to see that the vectors Vcut( )A, Vcut( )B, and Vcut( )C are of the same

magni-tude (|Vcut( )A| |V( )| |V( )|

cut B cut C

= = ) They differ from each other only by directions

However, the vectors VΣ( )A, VΣ( )B, and VΣ( )C are of different magnitude (|VΣ( )A|≠

|VΣ( )B|≠|VΣ( )C|), and they have different directions VΣ( )A ≠ VΣ( )B ≠ VΣ( )C

Projections of the vectors VΣ( )A, VΣ( )B, and VΣ( )C onto the unit normal vectors

nT( )A, nT( )B, and nT( )C are as follows: Prn VΣ( )A >0, Prn V( )ΣB =0, and Prn VΣ( )C <0

Therefore, in the vicinity of point A, the grinding wheel cuts the stock off; in

the vicinity of point B, the part surface P is generating; and in the vicinity of

point C, the cutting tool departs from the machined plane P Similarly,

gen-eration of the plane surface P can be performed with the cylindrical milling

Ultimately, the equation of the characteristic curve E

1 1

V Z

U

X V

When using the kinematical method, the sufficient condition for the

exis-tence of the enveloping surface can be obtained in the following way:

Con-sider a smooth, regular surface r1 that is given in a Cartesian coordinate

system X Y Z1 1 1 The equation of the surface r1 is represented in the form

r1=r1( ,U1 V1)∈C2 The family r1ω of these surfaces in a Cartesian

coor-dinate system X Y Z2 2 2 is given in the form r1ω =r1ω( ,U1 V1, )ω , where the

inequality ω(min)≤ ≤ω ω(max) is observed Then, if the conditions

1

( )

( )

( )( )

( )[

ω

ω

ωω

ωω

f V

1

1

2 1

f U

f V

1 1

1 1

1 1

190 Kinematic Geometry of Surface Machining

are satisfied at a certain point, then the enveloping surface exists and can be

represented by equations in the forms r1=r1( , , )U V1 1ω and

Methods of determining enveloping surfaces based on the implementation

of methods developed in differential geometry make it possible to determine

points of local tangency of the moving surface with the enveloping surface

under fixed values of w However, for a certain value of w = Const, global

interference of the surfaces could occur

Differential methods for determining enveloping surfaces can be employed

only when the equation of the moving surface is differentiable Because

sur-faces in engineering applications are not infinite and could be represented

by patches, and so forth, the part surface P can also be generated by special

points on the surfaces

In the general theory of enveloping surfaces, the family of surfaces that

changes their shapes is considered as well Results of the research in this area

can be used in the theory of surface generation, particularly for generation

of surfaces with the cutting tools that have a changeable generating surface

T [15,20,21,30]

Example 5.4

Consider a plane T that has a screw motion The plane T makes a certain angle

τb with X0 axis of the Cartesian coordinate system X Y Z0 0 0 The reduced pitch

p of the screw motion is given Axis X0 is the axis of the screw motion

The auxiliary coordinate system X Y1 1 is rigidly connected to plane T

Generation of a screw involute surface as an enveloping surface to consecutive positions of a

plane that performs a screw motion.

The equation of plane T can be represented in the form

The auxiliary coordinate system X Y Z1 1 1 is performing the screw motion

together with the plane T with respect to the motionless coordinate system

X Y Z0 0 0 In the coordinate system X Y Z1 1 1, the unit normal vector nT to the

plane T can be represented as

X Y Z

Determining the characteristic E direction of vM is important, but its

magnitude is not of interest Because of that, it can be assumed that | |ω =1

192 Kinematic Geometry of Surface Machining

The dot product of the unit normal vector nT and of the speed vM equals

n vT⋅ M= ⋅p tanτb−Z1=0 (5.55)Thus, the equation of contact can be represented in the form

The above equation of contact together with the equation of plane T

repre-sents the characteristic E.

b t

y t p

(5.57)

where t designates the parameter of the characteristic E.

The characteristic E is the straight line of intersection of these two planes

It is parallel to the coordinate plane X Z1 1 and distanced at p⋅tanτb

For a given screw motion, the characteristic E remains at its location within

the plane T in the initial coordinate system X Y Z0 0 0

The angle of rotation of the coordinate system X Y Z1 1 1 about the X0 axis

is designated as e The translation of the coordinate system X Y Z1 1 1 with

respect to X Y Z0 0 0 that corresponds to the angle e is equal to p ∙ e This yields

composition of the equation of coordinate system transformation:

In order to represent analytically the enveloping surface P, it is necessary

to consider the equation rE( ) of the characteristic E together with the opera- t

tor Rs(1→0) of coordinate system transformation:

Consider the intersection of the enveloping surface P by the plane

X0=X1+ ⋅ =p ε 0 The last equation yields X1= − ⋅p ε Therefore,

(5.60)

Equation (5.60) represents the involute of a circle

The radius of the base circle of the involute curve is

Therefore, the enveloping surface to consecutive positions of a plane T

having a screw motion is a screw involute surface The reduced pitch of the

involute screw surface equals p, and the radius of the base cylinder equals

r b= ⋅p tanτb The screw involute surface intersects the base cylinder The

line of intersection is a helix The tangent to the helix makes the angle ωb

with the axis of screw motion:

tanωb r b

p

From this, tanωb=tanτb, and ωb =τb The straight line characteristic E is

tangent to the helix of intersection of the enveloping surface P with the base

cylinder This means that if a plane A is tangent to the base cylinder, a straight

line E within a plane A makes the angle τb with the axis of the screw motion,

and the plane A is rolling over the base cylinder without sliding, then the

enveloping surface P can be represented as a locus of consecutive positions

of the straight line E that rolls without sliding over the base cylinder together

with the plane A The enveloping surface is a screw involute surface.

The obtained screw involute surface (Figure 5.20) is as that shown in

Figure 1.2 and as that analytically described by Equation (1.20)

Another solution to the problem of determining the envelope of a plane

having screw motion is given by Cormac [6]

5.3 Profiling of the Form-Cutting Tools for Machining Parts

on Conventional Machine Tools

Cutting tools for machining parts on conventional machine tools feature a

property that allows the generating surface of the cutting tool to slide over

itself Those surfaces that allow for sliding over themselves are currently the

194 Kinematic Geometry of Surface Machining

most widely used surfaces in industry This feature is of importance for the

theory of surface generation It can be used for simplification of the solution

to the problem of profiling of the form-cutting tool Certain simplification is

feasible because in the case of surfaces that allow for sliding over themselves,

it is not necessary to determine the entire generating surface of the cutting

tool It is sufficient to determine either the profile of the generating surface of

the cutting tool or the characteristic line along which the generating surface

of the cutting tool makes contact with the machined part surface

5.3.1 Two Fundamental Principles by Theodore Olivier

A solution to the problem of profiling the form-cutting tool for machining

a part surface on a conventional machine tool can be derived much easier

when using the fundamental principles of surface generation proposed by

T Olivier [12] as early as 1842

The R-mapping-based method for the profiling of form-cutting tools (see

Section 5.1) is general It is a powerful tool for solving the most general

prob-lems of cutting-tool profiling However, in particular cases, simpler methods

of profiling form-cutting tools are practical It is common practice to design

form-cutting tools on the premises of one of two Olivier’s principles:

with an auxiliary generating surface The generating surface in this case

differs from both conjugate surfaces.

congruent to one of the conjugate surfaces.

Prior to solving a problem of profiling of a certain form-cutting tool,

geom-etry of the part surface P (see Chapter 1) and the kinematics of the surface P

generation (see Chapter 2) must be predefined The operators of the

coordi-nate system transformation (see Chapter 3) are used for the representation

of all elements of the surface-generation process in a common coordinates

system, use of which is preferred for a particular consideration If we are

not just to develop a workable cutting tool, but also to develop the design of

the optimal cutting tool, then the methods of analytical description of the

geometry of contact of the part surface P and of the generating surface T of

the cutting tool (see Chapter 4) are also employed

Design of a form-cutting tool can be developed on the premises of its

gen-erating surface T Derivation of the gengen-erating surface T is the starting point

of designing the form-cutting tool

For generation of a given surface P, the cutting tool performs certain motions

with respect to the work The part surface P is given, and the generating

surface T of the cutting tool is not yet known Therefore, at the beginning,

the actual relative motions of the surfaces P and T are not considered, but

the corresponding motions of the part surface P and of a certain coordinate

system X Y Z T T T are analyzed After being determined, the generating

sur-face of the form-cutting tool would be described analytically in the

coordi-nate system X Y Z T T T

5.3.2 Profiling of the Form-Cutting Tools for Single-Parametric

Kinematic Schemes of Surface generation

Kinematic schemes of surface generation, those that feature just one relative

motion of the part surface P and of the generating surface T of the cutting tool

are referred to as the single-parametric kinematic schemes of surface generation.

In the case under consideration, it is convenient to begin consideration of

the procedure of profiling of a form-cutting tool for the generation of a

sur-face P in the form of circular cylinder When machining a circular cylinder

of radius RP (Figure 5.21a) [28,29], the work rotates about its axis OP with a

certain angular velocity ωP A coordinate system X Y Z T T T is rotating with a

certain angular velocity ωT Axis of this rotation OT crosses at a right angle

the axis of rotation OP of surface P Simultaneously, the coordinate system

X Y Z T T T travels along the axis OP with a feed rate ST The generating surface

T of the cutting tool in this case can be represented as an enveloping surface

to consecutive positions of the surface P in the coordinate system X Y Z T T T

Remember that the coordinate system X Y Z T T T is the reference system at

which the generating surface T could be determined.

After being determined, the generating surface T of the cutting tool and

the part surface P become tangent along the characteristic line E In the case

under consideration, the characteristic line E is represented with a circular

arc ∪ABC of the radius RP.

C A

OT

O P

| R T |> R P

FiguRe 5.21

Example of a single-parametric kinematic scheme for derivation of the generating surface of

the cutting tool.

196 Kinematic Geometry of Surface Machining

Use of a single-parametric kinematic scheme of surface generation allows

a simplification The generating surface T of the cutting tool can be generated

not only as an enveloping surface to consecutive positions of the part surface

P in the coordinate system X Y Z T T T, but also as a family of the characteristic

lines E that rotate about the axis O T

In the example, the generating surface T of the cutting tool is shaped in

the form of a torus surface Radius R T of the generating circle of the torus

surface T is equal to the radius R P of the surface P (R T=R P) The radius

of the directing circle of the torus surface T is equal to the closest distance

of approach H of the axes O P and O T The determined torus surface T

(Figure 5.21a) can be used for the designing of various cutting tools for the

machining of the surface P: milling cutters, grinding wheels, and so forth.

The considered example of implementation of the single-parametric

kine-matic scheme of surface generation (see Figure 5.21a) returns a qualitative

(not quantitative) solution to the problem of profiling a form-cutting tool No

optimal parameters of the kinematic scheme of surface generation are

deter-mined at this point The closest distance of approach H of the axes O P and

O T , and the optimal value of the cross-axis angle c are those parameters of

interest (Figure 5.21b) Optimal values of the parameters H and c can be

com-puted on the premises of analysis of the geometry of contact of the surfaces P

and T The optimal values of the parameters H and c can be drawn from the

desired degree of conformity of the surface T to the surface P.

Actually, in any machining operation, deviations of the actual cutting-tool

configuration with respect to the desired configuration are unavoidable

Because of the deviations, it is practical to introduce appropriate alterations

to the rate of conformity of surface T to surface P.

Figure 5.21c illustrates an example when the rate of conformity of surface T

to surface P is reduced Reduction of the rate of conformity causes point contact

of the surfaces P and T, instead of their line contact in the ideal case of surface

generation (Figure 5.21a) The schematic (Figure 5.21c) allows for analysis of the

impact of the rate of conformity of the surfaces T and P onto the accuracy and

quality of surface generation Varying values of the parameters H and c

(includ-ing the case when c ≠ 90°), one can come up with the solution under which

vul-nerability of the machining process to the resultant deviations of configuration

of the cutting tool with respect to surface P is the smallest possible.

Use of single-parametric kinematic schemes of surface generation (Figure 5.21)

is a perfect example of implementation of the second Olivier’s principle for

the purposes of profiling form-cutting tools for the machining of a given part

surface

5.3.3 Profiling of the Form-Cutting Tools for Two-Parametric

Kinematic Schemes of Surface generation

Kinematic schemes of surface generation, those that feature two relative

motions of the part surface P and of the generating surface T of the cutting tool,

are referred to as the two-parametric kinematic schemes of surface generation.

The resultant motion of the cutting tool with respect to the part surface

P can be of a complex nature For simplification, it can be decomposed on

two elementary motions The elementary motions are usually represented

with a rotational motion, and with a translational motion No one of these

motions caused sliding of the surface over itself Under such a scenario, it is

convenient to implement two-parametric schemes of surface generation for

profiling the form-cutting tool

Similar to the analysis of implementation of single-parametric kinematic

schemes of surface generation for the case of machining a surface of a

circu-lar cylinder (Figure 5.22)

The work is rotating about the axis O P (Figure 5.22a) with a certain

angu-lar velocity ωP The cutting-tool coordinate system X Y Z T T T is rotating with a

certain angular velocity ωT about the axis O T The axes of rotations O P and

O T are at a right angle The resultant generating motion VΣ is decomposed

on two elementary motions V1 and V2 (Here the equality VΣ =V1+V2 is

observed.)

At the beginning, an auxiliary generating surface R must be determined

The auxiliary surface R is an enveloping surface to consecutive positions of

the part surface P in its motion with the velocity V1 of the first elementary

motion Further, the generating surface T of the cutting tool is represented

in the coordinate system X Y Z T T T with the enveloping surface to consecutive

positions of the auxiliary generating surface R in its motion with the

veloc-ity V2 of the second elementary motion Here, vector V2 designates linear

velocity of rotational motion of the auxiliary surface R about the axis OP

with the angular velocity ωT

The generating surface T of the cutting tool, which is determined

follow-ing the two-parametric approach, usually makes point contact with the part

surface P This is observed because of the following reason: The line E1 is the

characteristic line for the part surface P and the auxiliary generating surface

R The auxiliary generating surface R and the generating surface T of the

cutting tool make line contact, and the line E2 is the characteristic line to the

surfaces R and T The characteristic lines E1 and E2 are within the auxiliary

generating surface R Generally speaking, the characteristic lines E1 and E2

intersect at a point that is the characteristic point At least one common point

of the characteristic lines E1 and E2 is necessary — otherwise surfaces P and

T cannot make contact Point K is the point of intersection of the

character-istic lines E1 and E2 In particular cases, the characteristic lines E1 and E2

can be congruent to each other This results in the surfaces P and T making

line contact instead of point contact

When the surfaces P and T move relative to each other with the velocity

of V1, then the characteristic line E1 occupies a stationary location on the

part surface P, but it travels over the generating surface T of the cutting tool

When the plane R rotates about the axis OT , the characteristic curve E2 on

the auxiliary surface R is stationary, but it is traveling over the

generat-ing surface T of the cuttgenerat-ing tool Therefore, in the general case of surface

schemes (Figure 5.21), consider implementation of two-parametric kinematic

Profiling of the Form-Cutting Tools of Optimal Design 199

It is necessary that the cusp height hΣ be less than the tolerance [h] on the

accu-racy of generation of the surface P The inequality hΣ ≤[ ] must be satisfied.h

Location of the point K and the shape of the generating surface T of the

cutting tool depend upon the direction of the motion V2 Depending on V2,

the surface T can be shaped in the form of a cylinder (Figure 5.22a), a cone

(Figure 5.22b), or a plane (Figure 5.22c)

In particular cases, surfaces P and T make line contact Line contact of the

surfaces P and T is observed when the characteristic lines E1 and E2 are

congruent No cusps are observed on the machined part surface P when the

surfaces P and T are in line contact.

The auxiliary generating surface R is tangent to the part surface P

There-fore, the surface R can be employed not only as an auxiliary surface, but also

as the generating surface T of the cutting tool In the last case, it is necessary

to consider the surface R as the generating surface of the cutting tool that is

derived using the single-parametric scheme of surface generation In order

to generate the generating surface T of the cutting tool (Figure 5.22d), the

generating surface T1 can be determined as the enveloping surface to

con-secutive positions of the part surface P in its motion relative to the coordinate

system X Y Z T T T By doing this, the approach shown in Figure 5.22a can then

be followed Another motion V2 of the determined generating surface T1 is

then considered [28,29] Ultimately, the generating surface T2 of the cutting

tool can be determined as the enveloping surface to consecutive positions of

the surface T1 that is performing the motion V2

In the example shown in Figure 5.22d, the generating surface T2 of the

cutting tool is shaped in the form of a noncircular cylinder The generating

surface T2 and the part surface P make point contact Point of contact K of

the surfaces P and T2 is traveling over both surfaces: over the part surface

P and over the generating surface T2 The closed three-dimensional curve l

represents the tool-path on the part surface P.

The solution to the problem of profiling the form-cutting tool using

two-parametric kinematic schemes of surface generation yields qualitative (not

quantitative) results No optimal parameters of the kinematic scheme of

sur-face generation or of the optimal cutting tool can be derived from the

con-sidered approach

An analytical solution to the problem of determining optimal parameters

of the kinematic scheme of surface generation and of optimal parameters

of the geometry of the generating surface of the form-cutting tool can be

obtained on the premises of the comprehensive analysis of the geometry of

contact of the surfaces P and T (see Chapter 4)

The considered example (Figure 5.22) is insightful It gives impetus to the

investigation of impact of the rate of conformity of the surfaces P and T on

the accuracy and quality of the machined part surface It is also practical for

the analysis of vulnerability of the surface-generation process to the

devia-tions of configuration of the cutting tool with respect to the work

The use of two-parametric kinematic schemes of surface generation

(Figure 5.22) is a perfect example of implementation of the first Olivier’s

principle for the purposes of profiling form-cutting tools for machining a

given part surface

5.3.4 Profiling of the Form-Cutting Tools for Multiparametric

Kinematic Schemes of Surface generation

Kinematic schemes of surface generation, those that feature more than two

relative motions of part surface P and generating surface T of the cutting tool

are referred to as the multiparametric kinematic schemes of surface generation.

The resultant motion of the cutting tool with respect to the part surface P

can be of a complex nature It is not a common practice to implement

kine-matic schemes of surface generation that have more than two elementary

motions However, as an example, methods of hob relieving composed of up

to six elementary motions are known

Speed VΣ of the resultant relative motion of the surfaces P and T can be

decomposed on a finite number of elementary motions Vi (where i is an

integer number, and i > 2 is considered): VΣ =Σi n= 1Vi The total number of

elementary motions is designated as n There are no principal restrictions on

the number n of elementary relative motions V i

In the event the kinematic scheme of surface generation is composed of n

elementary relative motions, then it is possible to generate (n − 1) auxiliary

generating surfaces R1, R2, … , Rn− 1 The last auxiliary generating surface

Rn is congruent to the generating surface T of the cutting tool (R n≡T)

Extension of Olivier’s principles in this direction is supported by the proven

feasibility of the generation of conjugate surfaces by means of two auxiliary

generating surfaces [18,19] This achievement can be recognized as the third

principle of generation of conjugate surfaces

If the multiparametric kinematic scheme of surface generation is used,

then the equation of the generating surface T of the form-cutting tool can be

derived by solving the set of two equations:

Σ

(5.63)

In this equation, the resultant motion VΣ is represented in the summa

VΣ =V1+V2+ +K Vi+ +K Vn However, this does not mean that the

par-ticular equalities n VP⋅ 1=0, n VP⋅ 2=0, … , n VP⋅ i=0 and others must

be satisfied Satisfaction of the particular equalities is the sufficient, but not

necessary, condition The elementary relative motions Vi are not

manda-tory within the common tangent plane Location of the resultant motion VΣ

within the common tangent plane is the only mandatory requirement in this

concern

 Pat No 965.728, USSR, A Method of Hob Relieving./S.P Radzevich, Filed January 21, 1980.

Profiling of the Form-Cutting Tools of Optimal Design 201

However, an increase in the number of elementary motions Vi causes

kinematic schemes of surface generation to become more complex This is

the major reason that the generating surface T of the form-cutting tool is

generated in most cases by using either single-parametric or two-parametric

kinematic schemes of surface generation

The problem of derivation of the equation of the generating surface T of

the form-cutting tool when the implemented kinematic scheme of surface

generation is multiparametric is not complex in principle However, it often

causes technical problems when performing routing computations

5.4 Characteristic Line E of the Part Surface P and of the

Generating Surface T of the Cutting Tool

In this section, an advantage of implementing kinematic schemes of surface

generation is provided Because sliding of the generating surface of the

cut-ting tool over itself is allowed, then it is necessary to derive not the equation

of the entire surface T, but the equation of its profile instead Such

substitu-tion results in significant simplificasubstitu-tion of the problem of profiling the

form-cutting tool

When the part surface P is given by equation r P =rP(U V P, P), then

the equation of a family of the surfaces P can be represented in the form

rP fm r

= ( , ,ω ω1, 2,K,ω ,K,ω ) By definition, the unit normal vector

nP to surface P is equal to n P=uP×vP Then, the equation of the

character-istic line E can be derived after solving the set of equations:

of all the equations at the bottom (see Equation 5.64) must be equal to zero

However, it is not mandatory that each expression equal zero

Another way of deriving the characteristic line is based on consideration of

instant screw motion of the part surface P relative to the coordinate system

X Y Z T T T to which the generating surface T will be connected The surface P

performs the screw motion about the part axis of the instant screw motion

(Figure 5.23) This axis of the screw motion yields analytical representation

in the form

is uniquely perpendicular to both axes at the same time No other segment

between the axes possesses this property — that is, the unit direction vector c

is uniquely perpendicular to the line direction vectors p and t This is

equiva-lent to the vector c satisfying the two equations p c⋅ =0 and t c⋅ =0 In order

to satisfy these equations, the unit vector c must be equal to c = p × t.

Ultimately, for the computation of the closest distance of approach of the

part axis and of the tool axis, Equation (5.67) can be employed The direction

of the line segment through the closest distance of approach is specified by

the unit vector c This yields computation of coordinates of points of the

characteristic line E.

The characteristic line E can be interpreted as the projection of the axis

of the instant relative screw motion onto the part surface P At every point

of the characteristic line E, the unit normal vector n P to the part surface P

makes a certain angle e with the axis of the screw motion (see Equation 5.67)

Following the above consideration, the equation of the characteristic line E

can be derived

Another approach to the determinination of the characteristic of a surface

having a certain motion can be found in the monograph by Cormac [6]

Parameters of the kinematic scheme of surface generation can be time

dependent This means that the parameters of the characteristic line E could

also be time dependent Methods of surface machining with a cutting tool

when the characteristic line E changes it shape in time are known [8].

5.5 Selection of the Form-Cutting Tools of Rational Design

For machining sculptured surfaces, cutting tools of standard design are used

Form-cutting tools of the most widely used designs for sculptured surface

machining a given sculptured surface, the user usually follows conventional

rules This means that the user will want to select the cutting tool design that

satisfies two requirements: First, any and all regions of the part surface must be

reachable by the cutting tool without mutual interference of the part surface P

and of the generating surface T of the cutting tool In order to achieve the

high-est possible productivity of surface machining, in compliance with the second

requirement, the diameter of the cutting tool must be the biggest feasible

Versatility of known designs of form-cutting tools for sculptured surface

machining on a multi-axis NC machine is not limited to the designs

sche-be implemented

For machining sculptured surfaces, form-milling cutters are used The

machining surface T of the milling cutter is shaped in the form of a

sur-face of revolution For better performance of the milling cutter, it is highly

desired to have the principal radii of curvature of the generating surface of

machining are shown in Figure 5.1 When selecting a certain cutting tool for

matically shown in Figure 5.1 Other designs of form-cutting tools can also