Kinematic Geometry of Surface Machinin Episode Episode 6 pptx

The Geometry of Contact of Two Smooth, Regular Surfaces 147Without going into detail, mention will be made here that for the purposes of efficient surface generation in a machining oper

The Geometry of Contact of Two Smooth, Regular Surfaces 147

Without going into detail, mention will be made here that for the purposes

of efficient surface generation in a machining operation, it is desired to

main-tain that kind of contact of the surfaces P and T which features the highest

possible rate of conformity of the generating surface T of the cutting tool to

the part surface P.

Actually, while machining a part surface, deviations in the cutting tool

relative location and orientation with respect to the surface P are always

observed The deviations of the cutting tool configuration are

unavoid-able Because of the deviations, the desired local-extremal kind of contact is

replaced with another kind of contact of the surfaces P and T Such

replace-ment can be achieved with the introduction of precalculated deviations of

the cutting tool principal radii of curvature R1.T and R2.T If the

precalcu-lated deviations are small, then instead of the desired local-extremal kinds

of contact of the surfaces, the “quasi-” kind of contact of the surfaces P and

T may occur There are several kinds of quasi- contact, including quasi-line

contact of the surfaces P and T, surface of the first kind, and

quasi-surface of the second kind contact of the quasi-surfaces P and T.

The required precomputed values of small deviations of the actual normal

curvatures from their initially computed values can be determined on the

premises of the following consideration When the maximal deviations in

the actual cutting tool configuration (location and orientation of the cutting

tool relative to the surface being machined) occur, the rate of conformity of

the generating surface T with respect to the surface P must not exceed the

rate of their conformity in one of the local-extremal kinds of surface contact

When the actual deviations of the cutting tool configuration do not exceed

the corresponding tolerances, then one of the feasible kinds of quasi-contact

of the surfaces P and T is observed, Evidently, bigger deviations in the

cut-ting tool configuration result in bigger precomputed corrections in the

nor-mal curvature of the generating surface of the cutting tool, and vice versa

In the ideal case, when there are no deviations in the cutting tool

configura-tion, it is recommended to assign normal curvatures of the values that enable

one of the local-extremal kinds of contact of the surfaces P and T Local-surface

contact of the second kind is the preferred kind of contact of the surfaces P

and T The local-surface contact of the second kind yields the minimal value

of the radius r cnf(min)=0 of the indicatrix of conformity Cnf P T R( / )

When machining an actual part surface, deviations in the cutting tool

configu-ration are unavoidable The pure surface kind of contact of the surfaces when the

equality r cnf(min)=0 is observed is not feasible Due to the deviations in the

cut-ting tool configuration, maintenance of the pure surface contact of the surfaces

P and T would unavoidably result in interference of the cutting tool beneath

the part surface P Therefore, it is recommended that a pure surface contact not

be maintained, but a kind of quasi-surface contact of the second kind be

main-tained instead A quasi-surface contact of the surfaces P and T yields avoidance

of interference of the surface T within the interior of the surface P Moreover, the

minimal radius r cnf(min) of the characteristic curve Cnf P T R( / ) could be as close to

zero as possible ( r cnf(min)>0, r cnf(min)→0, r cnf(min)≠0 )

Quasi-contact of the surfaces P and T is observed only when deviations of

the cutting tool configuration are incorporated into consideration

Definition 4.1: Quasi-line contact of the surfaces P and T is a kind of

point contact of the surfaces under which actual tangency of the surfaces

is within the true-point contact and the local-line contact, and it varies as

a function of the deviations in the cutting tool configuration.

Definition 4.2: Quasi-surface (of the first kind) contact of the surfaces

P and T is a kind of point contact of the surfaces under which actual

tangency of the surfaces is within the true-point contact and the

local-surface (of the first kind) contact, and it varies as a function of deviations

in the cutting tool configuration.

Definition 4.3: Quasi-surface (of the second kind) contact of the surfaces

P and T is a kind of point contact of the surfaces under which actual

tan-gency of the surfaces is within the true-point contact to local-surface (of

the second kind) contact, and it varies as a function of deviations in the

cutting tool configuration.

The difference between various kinds of the quasi-contact of the surfaces

P and T, as well as the difference between the corresponding kinds of

local-extremal contact of the surfaces can be recognized only under the limit

val-ues of the allowed deviations in the cutting tool configuration relative to the

part surface P In the event the actual deviations are below the tolerances,

then various possible kinds of quasi-contact of the surfaces cannot be

distin-guished from other nonquasi-kinds of their contact The only difference is in

actual location of the point K of contact of the surfaces Due to the deviations,

it shifts from the original position to a certain other location

There are only nine principally different kinds of contact of the surfaces P

and T In addition to the true-point, the true-line, and the true-surface

con-tact, the following three local-extremal kinds of contact are possible: (a) the

local-line, (b) the local-surface of the first kind, and (c) the local-surface of the

second kind Three kinds of quasi-contact of the surfaces are also possible:

the quasi-line, the quasi-surface of the first kind, and the quasi-surface of the

second kind of the surfaces P and T.

Taking into consideration that there are only 10 different kinds of local

patches of smooth, regular surfaces P and T (see Chapter 1, Figure 1.11), each

of the 9 kinds of surfaces contact can be represented in detail For this

pur-pose, a square morphological matrix of size 10 × 10 = 100 is composed This

matrix covers all possible combinations of the surfaces contact One axis of

the morphological matrix is represented with 10 kinds of local patches of

the part surface P, and the other axis is represented with 10 kinds of local

patches of the generating surface T of the cutting tool The morphological

matrix contains Σm9= 1C m9 = m 99- !m = 100 102- +10 55=

the local patches of the surfaces P and T Only 55 of them are required to be

investigated The analysis reveals that the following kinds of contact of the

surfaces P and T are feasible:

The Geometry of Contact of Two Smooth, Regular Surfaces 149

29 kinds of true-point contact

23 kinds of true-line contact

6 kinds of true-surface contact

20 kinds of local-line contact

7 kinds of local-surface (of the first kind) contact

8 kinds of local-surface (of the second kind) contact

20 kinds of quasi-line contact

7 kinds of quasi-surface (of the first kind) contact

8 kinds of quasi-surface (of the second kind) contact

This means that only 29 + 23 + 6 + 20 + 7 + 8 + 20 + 7 + 8 = 128 kinds of contact

of two smooth, regular surfaces P and T are possible in nature For some kinds

of surfaces contact, no restrictions are imposed on the actual value of the angle

m of the surfaces P and T local relative orientation For the rest of the types of

surfaces contact, a corresponding interval of the allowed value of the angle

m — [µmin]≤ ≤µ [µmax] — can be determined For particular cases of the

sur-faces contact, the only feasible value µ=[ ] is allowed.µ

On the premises of the above analysis, a scientific classification of possible

kinds of contact of the surfaces P and T is developed (Figure 4.21) The class-

ification (Figure 4.21) is potentially complete It can be further developed and

enhanced It can be used for the analysis and qualitative evaluation of the rate of

effectiveness of a machining operation The classification indicates perfect

corre-lation with the earlier developed classification (see table 3.1 on pp 230–243 in [13])

Replacement of the true-point contact of the surfaces P and T with their

local-line contact, and further with the local-surface of the first and of the

second kind, and finally with the true-surface contact results in significant

alterations of the surface P generation Only local-extremal kinds of contact of

the surfaces P and T are considered here If deviations in the cutting tool

con-figuration are considered, then the above-mentioned local-extremal kinds of

surfaces contact require being replaced with the corresponding quasi-kinds

of contact of the surfaces P ||/ T.

In order to achieve the highest possible productivity of machining of the

part surface P, it is recommended that the true-surface contact of the surfaces

P and T be maintained Under such a scenario, all portions of the surface P

are machined in one instant However, in those cases, large-scale surfaces P

cannot be machined The machining of the surface P while maintaining the

true-point contact of the surfaces is least efficient In this case, the generation

of every strip on the surface P occurs in time.

Depending on the kinds of contact of the surfaces P and T that are maintained

when machining a part surface P, all possible kinds of contact of the surface can

be ranked in the following order (from the least efficient to the most efficient):

True-point contact

Local-line or quasi-line contact

Maintenance of the true-point contact of the surface P and T results in

the highest possible agility of a machining operation The true-point contact

can also be regarded as the most general kind of surfaces contact Under

the true-point contact of the surfaces, the cutting tool can perform

five-para-metric motion with respect to the surface P Under the true-surface contact,

the cutting tool is capable of performing no motion relative to the work A

relative motion of the surfaces P and T is feasible only as an exclusion, say

when the surfaces P and T yield for sliding over themselves In those cases,

an enveloping surface P to consecutive positions of the generating surface

T of the cutting tool that is moving relative to the work, is congruent to P

Generally speaking, under such a scenario, the surfaces P and T are capable

of performing a single-parametric motion, and not higher than a

three-para-metric motion (see Chapter 2, Section 2.4)

The developed classification of all possible quasi-kinds of contact of the

surfaces P and T can be extended and represented in more detail.

References

[1] Boehm, W., Differential Geometry II In Farin, G Curves and Surfaces for

Com-puter Aided Geometric Design A Practical Guide, 2nd ed., Academic Press, Boston,

[4] Gray, A., Plücker’s Conoid In Modern Differential Geometry of Curves and Surfaces

with Mathematics, 2nd ed., CRC Press, Boca Raton, FL, 1997, pp 435–437.

[5] Hertz, H., Über die Berührung Fester Elastischer Körper (The Contact of Solid

Elastic Bodies), Journal für die Reine und Angewandte Mathematik (Journal for Pure

and Applied Mathematics), Berlin, 1981, pp 156–171; Über die Berührung Fester

Elastischer Körper und Über die Härte (The Contact of Solid Elastic Bodies and

Their Harnesses), Berlin, 1882; Reprinted in: H Hertz, Gesammelte Werke

(Col-lected Works), Vol 1, pp 155–173 and pp 174–196, Leipzig, 1895, or the English

translation: Miscellaneous Papers, translated by D.E Jones and G.A Schott, pp

146–162, 163–183, Macmillan, London, 1896.

[6] Koenderink, J.J., Solid Shape, MIT Press, Cambridge, MA, 1990.

[7] Pat No 1249787, A Method of Sculptured Surface Machining on Multi-Axis

NC Machine./S.P Radzevich, Int Cl B23c 3/16, Filed: December 27, 1984.

[8] Pat No 1185749, A Method of Sculptured Surface Machining on Multi-Axis NC

Machine./S.P Radzevich, Int Cl B23c 3/16, Filed: October 24, 1983.

[9] Plücker, J., On a New Geometry of Space, Phil Trans R Soc London, 155, 725–

791, 1865.

[10] Radzevich, S.P., A Possibility of Application of Plücker’s Conoid for

Mathemat-ical Modeling of Contact of Two Smooth Regular Surfaces in the First Order of

Tangency, Mathematical and Computer Modeling, 42, 999–1022, 2004.

5

Profiling of the Form-Cutting

Tools of Optimal Design

Generation of the part surface P practically is performed with the help of

the cutting tools of appropriate design The stock removal and generation

of the surface P are the two major functions of the cutting Profiling of the

generating surface T is required for designing a high-performance cutting

tool As shown below, the shape and parameters of the generating surface T

significantly affect the performance of the cutting Cutting edges of a

preci-sion cutting tool are within the generating surface of the tool This makes it

clear that prior to developing a practical design of the cutting tool, profiling

of the optimal generating surface is required

In this chapter, profiling of the form-cutting tools for all possible

meth-ods of part surface machining is considered The consideration begins from

the theory of profiling of the tools for machining sculptured surfaces on a

multi-axis numerical control (NC) machine This subject represents the most

complex case in the theory of profiling of cutting tools

5.1 Profiling of the Form-Cutting Tools for Sculptured

Surface Machining

The problem of profiling the form-cutting tool for machining of a sculptured

surface on a multi-axis NC machine has not yet been investigated in detail

Not profiling of the form-cutting tool of optimal design but selecting a certain

cutting tool among several available designs is often recommended instead

The selection of the cutting tool is usually based on minimizing

machin-ing time, reducmachin-ing scallop height, and so forth This yields a conclusion that

a robust mathematical method for design of the optimal form-cutting tool

for the maximally productive machining of a given sculptured surface on a

multi-axis NC machine is needed

5.1.1 Preliminary Remarks

Many advanced sources are devoted to the investigation of sculptured

sur-faces generation on multi-axis NC machines Without going into a detailed

review of previous publications in the field, mention is made of a few

monographs by Amirouch [1], Chang and Melkanoff [4], Choi and Jerard [5],

and Marciniak [11] Unfortunately, the problem of profiling the form-cutting

tools for sculptured surface machining has not yet been investigated Most

often, the generation of sculptured surfaces with the milling cutters of

con-ventional designs (Figure 5.1) is considered

The following terms (some of which are not new) are introduced below to

avoid ambiguities in later discussions:

Definition 5.1: Sculptured surface P is a smooth, regular surface, the

major parameters of local topology at a point of which are not identical

to the corresponding parameters of local topology of any other

infini-tesimally close point of the surface.

It is instructive to point out here that sculptured surface P does not allow for

sliding “over itself.”

While machining a sculptured surface, the cutting tool rotates about its axis of

rotation and moves relative to the sculptured surface P When rotating or when

performing relative motion of another kind, cutting edges of the cutting tool

generate a certain surface The surface represented by consecutive positions of

cutting edges is referred to as the generating surface of the cutting tool [18, 19, 25]:

Definition 5.2: The generating surface T of the cutting tool is a surface

that is conjugate to the surface P being machined.

 In fact, our terminology draws inspiration mostly from the Theory for Mechanisms and

Machines, and from the Theory of Conjugate Surfaces.

(e) (d)

(b)

(f)

Examples of milling cutters of conventional design for the machining of sculptured surfaces

on a multi-axis numerical control machine: cylindrical (a), conical (b), ball-end (c), filleted-end

(d), and form-shaped (e), (f).

Profiling of the Form-Cutting Tools of Optimal Design 155

An infinite number of surfaces satisfy Definition 5.2 Use of all of the

con-jugate surfaces satisfies the equation of contact n VP⋅ Σ =0 (see Chapter 2)

Although the unit normal vector nP is uniquely determined at a given

sur-face point, the number of feasible vectors VΣ is equal to infinity: All

vec-tors VΣ within the common tangent plane satisfy the equation of contact

n VP⋅ Σ =0 It is natural to assume that not all are equivalent to each other

from the standpoint of efficiency of surface generation, and some could be

preferred; moreover, an optimal direction of VΣ exists within the common

tangent plane for which the efficiency of surface machining reaches it

opti-mal rate This makes the problem of profiling the form-cutting tool indefinite

However, this indefiniteness is successfully overcome below The uniquely

determined generating surface T is used in further steps of designing an

optimal form-cutting tool for machining of a given part surface.

In most cases of surface generation, the generating surface T of the cutting

tool does not exist physically Usually, it is represented as the set of

consecu-tive positions of the cutting edges in their motion relaconsecu-tive to the stationary

coordinate system, embedded in the cutting tool

In most practical cases, the generating surface T allows for sliding over

itself The enveloping surface to consecutive positions of the surface T that

performs such a motion is congruent to the surface T.

When machining a surface P, the surface T is conjugate to the sculptured

surface P.

For simplification in programming machining operation, the APT cutting

tool is proposed (Figure 5.2) Physically, the APT cutting tool does not exist

The generating surface T of the APT cutting tool is made up of a conical

por-tion that has the cone angle a, a conical portion that has the cone angle b, and

a portion of the surface of a torus The last is specified by the radius r of the

generating circle, and by the diameter d of the directing circle The axial

loca-tion of the torus surface with respect to the conical surfaces is specified by

the parameter designated as f For a certain combination of the parameters a,

b, r, d, and f, the generating surface of the virtual APT cutting tool transforms

to the generating surface T of the actual cutting tool For example,

assum-ing a = 0°, b = 0°, and r = 0, one can come up with the generating surface T

of the cylindrical milling cutter (Figure 5.1a) If r= d2 and b = 0°, then the

 The procedure of designing a form-cutting tool usually begins from determination of the

generating surface of the cutting tool This is a common practice However, sometimes when

the geometric structure of the surface to be machined is inconsistent, another procedure

is used Relieving hob clearance surfaces, cutting bevel gears with spiral teeth, machining

noninvolute gears of the first and of the second kind are perfect examples of surface

machin-ing when the geometric structure of the surface to be machined is inconsistent Under such

circumstances, the generating surface of the cutting tool of an appropriate form is selected

Further, the actual shape and parameters of the machined part surface can be determined.

Keep in mind that the part surface to be machined is the primary element of the machining

operation, on the premises of which the determination of the optimal machining operation

is possible This includes profiling of both the optimal cutting tool and the optimal

kinemat-ics of the machining operation Otherwise, only an approximate solution to the problem of

optimal surface generation is possible.

the generating surface T of the cutting tool, and kinematics of the machining

operation are applicable

The following example illustrates the actual meaning of the criterion of

opti-mization in the sense of profiling of the form-cutting tool Consider a trivial

machining operation — a turning operation of an arbor on a lathe (Figure 5.3)

When machining, the work rotates about its axis of rotation with an angular

velocity ωP The cutting tool travels along the work axis of rotation with a feed

rate S The feed rate S is of constant magnitude in the examples considered

below A stock t is removed in the turning operation.

Use of the cutter with the tool cutting edge angle ϕa, and the tool minor (end)

cutting edge angle ϕ1a causes cusps on the machined part surface P (Figure 5.3a)

The actual height h a of the cusps must be less than the tolerance [h] on accuracy

of the surface P In order to satisfy the inequality h a ≤[ ], a corresponding rela-h

tionship between the parameters ϕa, ϕ1a , and S must be observed Otherwise,

the part cannot be machined in compliance with the part blueprint

That same arbor can be machined with the cutter having the tool cutting

edge angle ϕb<ϕa, and the tool minor cutting edge angle ϕb1<ϕa1 (Figure 5.3b)

Cusps on the machined surface P are observed Elementary computations of the

actual cusp height h b in this case reveal that the inequality h b <h a is valid

Further, that same surface P can be machined with the cutter having the

cut-ting edge that is shaped in the form of a circular arc of radius R (Figure 5.3c)

Use of the cutter with the curvilinear cutting edge results in cusps on the

machined surface P However, if the radius R is chosen properly, then the

Turning of an arbor on a lathe: the concept of profiling the optimal form-cutting tool

(From Radzevich, S.P., Mathematical and Computer Modeling, 36 (7–8), 921–938, 2002 With

permission.)

158 Kinematic Geometry of Surface Machining

actual cusp height h c can be smaller than h b In other words, the inequality

h c<h b could be observed

Ultimately, consider the turning operation of the surface P with the cutter

that has an auxiliary cutting edge (Figure 5.3d) The auxiliary cutting edge is

parallel to the direction of the feed rate S The length of the auxiliary cutting

edge exceeds the distance that the cutter travels per one revolution of the

work Geometrical parameters of the auxiliary cutting edge can be specified

by the tool cutting edge angle ϕd= °0 , and the tool minor (end) cutting edge

angle ϕ1d= °0 Under such a scenario, no cusps are observed on the machined

part surface P.

The above consideration makes possible a conclusion: An appropriate

altera-tion of shape of the cutting edge of the cutter can make possible a reducaltera-tion of

devia-tions of the machined part surface with respect to the desired part surface

This conclusion is critically important for further consideration At this

point, a more general example of surface generation that supports the above

conclusion will be considered

Consider generation of a sculptured part surface P with the form-cutting

tool having arbitrarily shaped the generation surface T The intersection of the

surfaces P and T by the plane through the unit normal vector n P is shown in

Figure 5.4 This plane section is perpendicular to the tool-path on the surface

Examples of various rates of conformity of the generating surface T of the tool to the

sculp-tured surface P in the plane section through the unit normal vector n P (From Radzevich, S.P.,

Mathematical and Computer Modeling, 36 (7–8), 921–938, 2002 With permission.)

P at point K In Figure 5.4, S T designates the width of the tool-path In all the

examples considered, the width S T of the tool-path remains identical The

radius of normal curvature R P of the surface P at point K remains the same

The scallop’s height on the machined surface P is designated h P

The surface P can be generated with the generating surface T a of the

cut-ting tool (Figure 5.4a) The radius of curvature of the generacut-ting surface T a

is of a certain positive value R T a >0 Due to point contact of the surfaces P

and T a , the desired surface P is not generated, but an approximation to it is

generated instead The generated surface has scallops For the prespecified

width S T of the tool-path, the scallop’s height is equal to a certain value h P a

The scallop’s height must be smaller than the tolerance [h] on the accuracy of

the surface P.

That same surface P can be generated with the generating surface T b of the

cutting tool (Figure 5.4b) The radius of curvature of the generating surface T b

in this case exceeds the value of the radius of curvature of the surface T a in the

previous case ( R b T>R T a ) Because surfaces P and T b are in point contact,

scal-lops on the generated surface are observed When width S T of the tool-path is

predetermined, then the scallop’s height is of a certain value h P b Under such a

scenario, a certain reduction of the scallop’s height occurs ( h P b <h P a) The

scal-lop height reduction occurs because in differential vicinity of point K, the

sur-face T b is getting closer to surface P rather than to surface T a Locally, surface

T b is more congruent to surface P than to surface T a The rate of conformity

of surface T b to surface P is greater than the rate of conformity of the surface

T a to that same surface P.

Further, that same surface P can be generated with the generating

sur-face T c of the cutting tool (Figure 5.4c) At point K, surface T c is flattened;

therefore, the radius of curvature R c T is equal to infinity ( R c T→ ∞) That value

of the radius of curvature R c T exceeds the value of the radius of curvature R b T

Again, surfaces P and T c make contact at a point; therefore, scallops on the

generated surface are observed Because R T c >R b T , the scallop’s height h P c is

smaller than the scallop’s height h P b The scallop height reduction in this case

is due to the increase of the rate of conformity of the generating surface T c of

the tool to the part surface P compared to what is observed with respect to

surfaces P and T b

Ultimately, consider generation of the surface P with the concave

generat-ing surface T d of the cutting tool (Figure 5.4d) The radius of curvature R T d

of the cutting-tool surface T d is negative (R T d <0) In this case, the rate of

conformity of the generating surface T d of the cutting tool to the part

sur-face P is the biggest of all considered cases (Figure 5.4) Thus, scallops of the

smallest height h P d are observed on the generated surface P.

Summarizing the analysis of Figure 5.4, the following conclusion can be

formulated: An increase of the rate of conformity of the generating surface T of the

cutting tool to the sculptured surface P causes a corresponding reduction of height of

the residual scallop on the machined surface P

This conclusion is of critical importance for the development of methods of

profiling of form-cutting tools, as well as for the theory of surface generation

160 Kinematic Geometry of Surface Machining

The rate of conformity of the surface T to the surface P can be used as a

mathematical criterion of efficiency of a machining operation This issue is

of prime importance in order to bypass all the major bottlenecks (imposed

by the initial indefiniteness of the problem) in designing the optimal

form-cutting tool

5.1.3 R-Mapping of the Part Surface P on the generating Surface

T of the Form-Cutting Tool

The theory of surface generation offers a method for profiling form-cutting

tools of optimal design for the machining of a given sculptured part surface

on a multi-axis NC machine

The rate of conformity of the generating surface T of the cutting tool to the

sculptured part surface P significantly affects the efficiency of the machining

operation A higher rate of conformity results in higher productivity of the

machining operation, smaller residual scallops on the machined part

sur-face, shorter machining time, and so forth [18,19,25,26]

In order to come up with the optimal design of a cutting tool, the

generat-ing surface of the tool must conform to the part surface to be machined as

much as possible For this purpose, the cutting tool surface T can be

gener-ated as a kind of mapping of the part surface P to be machined The required

kind of mapping of the surface P onto the generating surface T must ensure

the required rate of conformity of the surfaces P and T at every point of

their contact This kind of mapping of the surface P onto the surface T was

initially proposed by Radzevich [16,22,23,25] It is referred to as the

R-map-ping of the sculptured part surface P onto the generating surface T of the

cutting tool

Consider the generating surface T of a form-cutting tool that makes contact

with a sculptured part surface P at point K The unit normal vector to the

sur-face P at K is designated n P A pencil of planes can be constructed using vector

nP as the directing vector of the axis of the pencil of planes The maximal rate

of conformity of the generating surface T of the tool to the sculptured surface

P is observed when for every plane of the pencil of planes the equality

is valid (in some cases, not the equality R T= −R P, but the equivalent

equal-ity k T = −k P can be used instead)

When the equality (see Equation 5.1) is satisfied, then the surfaces P and T

make either a surface kind of contact or a kind of local-surface contact (either

local-surface contact of the first kind, or local-surface contact of the second

kind — see Chapter 4)

 Pat No 4242296/08, USSR, A Method for Designing of the Optimal Form-Cutting-Tool for

Machining of a Given Sculptured Surface on Multi-Axis NC Machine./S.P Radzevich, Filed

March 31, 1987.

Actually, deviations in the configuration of the cutting tool with respect to

the sculptured part surface P are unavoidable Because of this, Equation (5.1)

cannot be satisfied This forces the replacement of Equation (5.1) with an

equality of the sort

The function R R T( P) can be expressed in terms of deviations or tolerances

of the actual configuration of the cutting tool with respect to the work It can

be determined using either analytical or experimental methods

Ultimately, the problem of profiling a form-cutting tool of the optimal

design for machining a given sculptured part surface on a multi-axis NC

machine reduces to determination of the generating surface T that is

maxi-mally conformal to the given surface P and that satisfies Equation (5.2).

Switching from the use of the function R T= −R P to the use of a

func-tion R T =R R T( P ) results in the ideal local-extremal contact of the surfaces P

and T being replaced with a kind of quasi-kind of contact (Chapter 4) Recall

that quasi-kinds of contact of the surfaces P and T yield that same range of

agility of the machining operation as that possessed by point contact of the

surfaces Moreover, the productivity of surface generation when

maintain-ing the quasi-kind of contact of the surfaces P and T is practically identical

to the productivity of surface generation when the surface-kind of contact of

the surfaces is maintained

Further, consider a sculptured surface P that is analytically represented

by an equation of the form of Equation (1.1) An equation of the sculptured

surface yields computation of the fundamental magnitudes E P , F P , and G P of

the first order (see Equation 1.7), and of the fundamental magnitudes L P , M P,

and N P of the second order (see Equation 1.11) of the part surface P.

Using R-mapping of the surfaces, it is convenient to derive an equation

of the generating surface T of the form-cutting tool initially in its natural

parameterization (see Equation 1.19) For this purpose, it is necessary to

express the first φ1.T and the second φ2.T fundamental forms of the

generat-ing surface T in terms of the fundamental magnitudes E P , F P , G P and L P,

M P , N P of the part surface P.

The R-mapping of surfaces is capable of establishing the required

corre-spondence between points of the surfaces P and T This means that for every

point on the surface P at which the precomputed principal radii of

curva-ture are equal to R1.P and R2.P, a corresponding point on the generating

surface T of the cutting tool, with the desired principal radii of curvature

to R1.T and R2.T, can be computed (but is not mandatory vice versa) There

could be one or more points on the surface P that correspond to that same

point of the surface T.

When two surfaces P and T are given, then one can easily compute the rate

of conformity of the surfaces at the given cutter-contact-point (CC-point; see

Chapter 4) In the case under consideration, a problem of another sort arises

This problem could be interpreted as an inverse problem to the problem of

162 Kinematic Geometry of Surface Machining

computation of the actual rate of conformity of surface T to surface P in the

prescribed direction on P.

The R-mapping establishes a functional relationship between principal

radii of curvature of the surface P and of the generating surface T of the

cutting tool in differential vicinity of the CC-point K The R-mapping yields

composition of two equations for the determining of six unknown

funda-mental magnitudes E T , F T , G T of the first φ1.T and L T , M T , N T of the second

φ2.T fundamental forms of the surface T.

The equation R T =R R T( P) can be split into a set of two equations:

Here MP and MT designate the mean curvatures, and GP and GT are the

Gaussian curvatures of the surfaces P and T at a CC-point K, respectively.

Expression R T=R R T( P) gives insight into the significance of a

correla-tion between the radii of normal curvature R P and R T In order to make

the function R T =R R T( P), the rate of conformity functions F1, F2 , and F3

are implemented The rate of conformity functions F1, F2, and F3 are of

principal importance for the determination of the function R T =R R T( P) The

functions F1, F2, and F3 specify the required rate of conformity of the

gen-erating surface T of the cutting tool to the sculptured surface P at every

CC-point Because of that, they are referred to as the rate of conformity functions.

In order to satisfy the set of two equation in Equation (5.3), it is necessary

to satisfy the following equalities:

In a particular case, Equation (5.4) through Equation (5.6) can be reduced

to the following forms:

These expressions analytically describe the vital link between the

opti-mal design parameters of the form-cutting tool and the actual process of

sculptured surface machining Equation (5.4) through Equation (5.6) allow

incorporation into the design of the actual cutting tool all of the important

features of the machining operation: cutting tool performance, tool wear,

rigidity of a cutting tool, and so forth

The rate of conformity functions F1, F2, and F3 are of prime importance

for designing the form-cutting tool of optimal design for the machining of a

given sculptured part surface They can be determined, for instance, using

the proposed [14] experimental method of simulating the machining of a

sculptured surface Other approaches for determining the rate of

confor-mity functions F1, F2, and F3 can be used as well There is much room for

research in this concern

Equation (5.4) through Equation (5.6) are necessary but not sufficient for

the determination of six unknown fundamental magnitudes E T , F T , G T of

the first φ1.T and L T , M T , N T of the second φ2.T fundamental forms of the

generating surface T of the form-cutting tool The equations of compatibility

could be incorporated into the analyses in order to transform Equation (5.4)

through Equation (5.6) to a set of six equations of six unknowns

Every smooth, regular generating surface T of the form-cutting tool

man-datory satisfies Gauss’ equation of compatibility that follows from his famous

theorema egregium [7,18,19,29,32]:

T T

T T

T T

U V

E V

G U

12

G U

T T

T T

2

2

12

G U E

G

T T

T T T

212

T

164 Kinematic Geometry of Surface Machining

In Equation (5.11) and Equation (5.12), Christoffel’s symbols of the second

kind are used Christoffel’s symbols can be computed from the following

formulae [7,32]:

22

22

u u

u u

22

Equations of compatibility are necessary in order to transform the set of

three equations [Equation (5.4) through Equation (5.6)] to a set of six

equa-tions of six unknowns

The set of three equations (Equation 5.4 through Equation 5.6) together

with three equations of compatibility (Equation 5.10 and Equation 5.11)

com-pletely describe the R-mapping of two smooth, regular surfaces (that is, they

describe the R-mapping of the sculptured surface P onto the generating

sur-face T of the cutting tool of optimal design).

Thus, the fundamental magnitudes of the first f1.T and of the second f2.T

fundamental forms of the generating surface T of the form-cutting tool can

be determined using the R-mapping of the sculptured part surface P onto

the generating surface T of the cutting tool A routing procedure can be

used to solve the set of six equations of six unknowns, say of Equation

(5.4) through Equation (5.6) and Equation (5.10) and Equation (5.11) with six

unknowns E T , F T , G T and L T , M T , N T Ultimately, the determined

gen-erating surface T of the cutting tool is represented in natural

parameteriza-tion similar to Equaparameteriza-tion (1.19)

Below, the fundamental magnitudes E T , F T , G T and L T , M T , N T are

con-sidered as the known functions

5.1.4 Reconstruction of the generating Surface T of the Form-Cutting

Tool from the Precomputed Natural Parameterization

Analytical representation of the generating surface T in the form of

Equa-tion (1.19) is inconvenient for applicaEqua-tion in engineering practice when

designing form-cutting tools However, natural parameterization of the

gen-erating surface T can be converted to its parameterization in a convenient

form, say to its representation in a Cartesian coordinate system