Kinematic Geometry of Surface Machinin Episode Episode 13 ppsx

et al., On the Optimization of Parameters of Sculptured Surface Machining on Multi-Axis NC Machine, In Investigation into the Surface Generation, UkrNIINTI, Kiev, No... The local analy

At the point of intersection P of the straight line rh (see Equation 8.50) and

to the torus surface Tr P (see Equation 8.51) the equality rh(θTr P. ,ϕTr P. ) [= rTr P K. ]( )

is observed In expended form, this equality casts into

which yields the computation of three parameters θTr P. , ϕTr P. , and t h that

specify coordinates of the point P Finally, an analytical expression for the

vector rP can be derived

The computed vectors rQ and rP yield computation of the resultant

sur-face deviation hΣ:

With the above analysis, the final conclusion can be made with respect

to the principle of superposition of the elementary surface deviations: The

principle of superposition of the elementary surface deviations h fr and h ss is

valid if and only if the inequality hΣ −hΣ ≤[∆hΣ] is observed.

Here, [∆hΣ] designates the tolerance on accuracy of computation of the

height of the resultant cusps

References

[1] doCarmo, M.P., Differential Geometry of Curves and Surfaces, Prentice Hall,

Engle-wood Cliffs, NJ, 1967.

[2] Monge, G., Application de l’analyse à la géométrie, Bachelier, 1850.

[3] Radzevich, S.P., Conditions of Proper Sculptured Surface Machining,

Computer-Aided Design, 34 (10), 727–740, 2002.

[4] Radzevich, S.P., Differential-Geometric Method of Surface Generation, DrSci thesis,

Tula Polytechnic Institute, 1991.

[5] Radzevich, S.P., Fundamentals of Surface Generation, Rastan, Kiev, 2001.

[6] Radzevich, S.P., Generation of Actual Sculptured Part Surface on Multi-Axis NC

Machine Part 1, Izvestiya VUZov Mashinostroyeniye, 5, 138–142, 1985.

[7] Radzevich, S.P., Generation of Actual Sculptured Part Surface on Multi-Axis NC

Machine Part 2, Izvestiya VUZov Mashinostroyeniye, 9, 141–146, 1985.

[8] Radzevich, S.P., Sculptured Surface Machining on Multi-Axis NC Machine, Vishcha

Shkola, Kiev, 1991.

[9] Radzevich, S.P et al., On the Optimization of Parameters of Sculptured Surface

Machining on Multi-Axis NC Machine, In Investigation into the Surface Generation,

UkrNIINTI, Kiev, No 65-Uk89, pp 57–72.

Application

9

Selection of the Criterion of Optimization

For machining a part surface, a machine tool, a cutting tool, a fixturing, and

so forth are necessary All these elements together are referred to as the

tech-nological system For lubrication and for cooling purposes, liquids and gas

substances are often used The coolants and the lubricants create the

techno-logical environment When a surface machining process is designed properly,

then capabilities of both the technological system and of the technological

environment are used most completely When capabilities of the

technologi-cal system and of the technologitechnologi-cal environment are used the most

com-pletely, manufacturing processes of this kind are usually called the extremal

manufacturing processes Ultimately, use of the extremal manufacturing

pro-cesses results in the most economical machining of part surfaces

The Differential Geometry/Kinematics (DG/K )-method of surface

genera-tion (disclosed in previous chapters) is capable of synthesizing extremal

meth-ods of machining of sculptured surfaces on a multi-axis numerical control (NC)

machine, as well as synthesizing extremal methods of machining surfaces that

have relatively simple geometry on conventional machine tools [5,6,10]

Machining the part surface in the most economical way is the main goal

when designing a manufacturing process For synthesizing the most efficient

machining operation, appropriate input information is required

Capabili-ties of a theoretical approach can be estimated by the amount of input

infor-mation the approach requires for its implementation, and by the amount of

output information the method is capable of creating A more powerful

theo-retical approach requires less input information to solve a problem, and use

of it enables more output information in comparison with the less powerful

theoretical approach

The DG/K-method requires a minimum of input information: just the

geo-metrical information on the part surface to be machined The geogeo-metrical

information on the part surface to be machined is the smallest possible input

information for solving a problem of synthesis of the optimal machining

operation Based only on the geometrical information on the part surface P,

use of the DG/K-method yields computation of the optimal parameters of the

machining process No selection of parameters of the machining operation is

required when the DG/K-method is used This makes it possible to conclude

that the DG/K-method of surface generation is the most powerful theoretical

method capable of solving problems of synthesis of optimal machining

oper-ations on the premises of the smallest possible input information No other

theoretical method is capable of solving problems of this sort on only the

premises of geometrical information on the part surface to be machined

The selection of appropriate criterion of optimization is critical for the

implementation of the DG/K-method of surface generation.

9.1  Criteria of the Efficiency of Part Surface Machining

The design of a sculptured surface machining process is an example of a

problem having a multivariant solution In order to solve a problem of this

sort, a criterion of optimization is necessary

Various criteria of optimization are used in industry for the optimization

of parameters of surface machining The productivity of surface machining,

tool life, accuracy, and quality of the machined surface are among them

Other criteria of optimization of parameters of machining operations are

used as well

Economical criteria of optimization are the most general and the most

pre-ferred criteria of optimization of machining processes However, analytical

description of the economical criteria of optimization is complex and makes

them very inconvenient for practical computations For particular cases of

surface machining, equivalent criteria of optimization of significantly

sim-pler structure can be proposed

The productivity of surface machining and productivity of surface

genera-tion are the important criteria of optimizagenera-tion Both are often used for

creat-ing more general criteria of optimization of surface machincreat-ing Therefore, it

is reasonable to use the productivity of surface machining as the criterion of

optimization for the purpose of demonstration of the potential capabilities of

the DG/K-method of surface generation Results of the synthesis of optimal

surface machining operations can be generalized for the case of

implementa-tion of another criterion of optimizaimplementa-tion

There are many ways to increase the productivity of surface machining

on machine tools Here, mostly geometrical and kinematical aspects of the

optimization of surface machining are considered

In the theory of surface generation, three aspects of the surface generation

process are distinguished: the local surface generation, the regional surface

generation, and the global surface generation [5,6,10]

The local analysis of the part surface generation encompasses generation

of the surface P in differential vicinity of the point K of contact of the part

surface P and of the generating surface T of the cutting tool Generation of

the part surface within a single tool-path is investigated from the

perspec-tive of the regional surface generation Ultimately, partial interference of

the neighboring tool-paths, coordinates of the starting point for the surface

machining, and impact of shape of the contour of the surface P patch are

investigated from the perspective of the global surface generation

Conse-quently, three kinds of productivity of surface machining are distinguished:

local productivity of surface generation, regional productivity of surface

generation, and global productivity of surface generation

9.2  Productivity of Surface Machining

Productivity of surface generation reflects the intensity of generation of the

nominal part surface in time It can be used for the purpose of synthesis of

optimal surface machining operations (for example, of machining a

sculp-tured part surface on a multi-axis NC machine)

9.2.1 Major Parameters of Surface Machining Operation

It is natural to begin the investigation of major parameters of the surface

machining operation from the local surface generation

When machining a sculptured surface on a multi-axis NC machine, all

major parameters of the machining operation and the instantaneous

pro-ductivity of surface generation vary in time This makes reasonable

consid-eration of instantaneous (current) values of the surface-genconsid-eration process

Instantaneous productivity of surface generation Psg( ) is determined by t

current values of the feed-rate F(fr and of the side-step F(ss (here t

desig-nates time) Usually, the vector Ffr and the vector Fss are orthogonal to each

other ( Ffr⊥Fss) In particular cases, the vectors Ffr and Fss are at a certain

angle θ to each other

Instantaneous productivity of surface generation can be computed by the

following formula [7,8]:

P ( ) |t = Ffr×F ss| (9.1)Equation (9.1) casts into [7,8]

P ( )t =F F(fr⋅(ss⋅sinθ (9.2)Here, F(fr is equal to | |Ffr , and F(ss is equal to | |Fss

Equation (9.1) and Equation (9.2) reveal that an increase of the feed-rate

(

F fr, and an increase of the side-step F(ss lead to an increase of the

instanta-neous productivity of surface generation P ( )t Deviation of the angle θ from

θ =90 results in a corresponding reduction of the instantaneous

productiv-ity of surface generation P ( )t

At a current point K of contact of the part surface P and of the generation

surface T of the cutting tool, optimal values of the parameters F(fr, F(ss,

and θ depend upon local geometrical (differential) characteristics of the surfaces

P and T, and upon the tolerance on accuracy [ ] h of the machined part surface

The value of the tolerance on accuracy [ ]h of surface machining is usually

constant within the patch of the surface P However, in a more general case

of surface machining, the current value of the tolerance [ ]h can vary within

the surface patch:

Within certain portions of a surface patch, the tolerance can be bigger, and

within other portions, it can be smaller depending on the functional

require-ments of the actual part surface

Because the resultant cusp height hΣ is made up of two components h fr

and h ss, it is necessary to split the tolerance [ ]h on two corresponding

por-tions: on the portion [ ]h fr for the elementary deviation h fr, and on the

por-tion [ ]h ss for the elementary deviation h ss The equality

Here, coefficients a h and b h are within the intervals 0≤a h≤1 and 0≤b h≤1

The coefficients a h and b h can be determined at a current point K of the

sculptured surface P.

At a current point K on the part surface P having coordinates U P and V P,

the current values of the coefficients a h and b h also depend on coordinates

of the point K on the surface P (that is, depend on U T and V T parameters)

and on the angle µ of the local relative orientation of surfaces P and T at the

point K This relationship is expressed by two formulae:

a h=a U V U V h( P, P, T, T, )µ (9.7)

b h=b U V U V h( P, P, T, T, )µ (9.8)

Values of the feed-rate F(fr per tooth of the cutting tool, and of the side-step

[ ]h ss at a current point K depend on the partial tolerances [ ] h fr and [ ]h ss One

can immediately conclude from the above that both the feed-rate F(fr and the

side-step [ ]h ss are functions of coordinates of the point K on the surface P, of

coordinates of the point K on the surface T, of the angle µ of the local relative

orientation of surfaces P and T, and of the direction of motion of the surface

T with respect to the surface P The following expressions

F fr=F h fr([ ])fr =F U V U V fr( P, P, T, T, , )µ ϕ (9.9)

F ss =F h ss([ ])ss =F U V U V ss( P, P, T, T, , )µ ϕ (9.10)

reveal this relationship Here the angle that specifies the direction of the

feed-rate vector Ffr is designated as ϕ

Substituting Equation (9.9) and Equation (9.10) in Equation (9.2), it is easy to

conclude that productivity of surface generation Psg also depends on

coor-dinates of the current point of contact K on both the surfaces P and T, on the

angle µ of the local relative orientation of surfaces P and T, and on the

direc-tion of the relative modirec-tion of the surfaces P and T at point K:

Psg =Psg(U V U V P, P, T, T, , )µ ϕ (9.11)

Certainly, not just the tolerance [ ]h , but also the partial tolerances [ ] h fr

and [ ]h ss can be constant within the part surface patch or can vary within

the sculptured surface P In the first case, the actual values of the tolerances

[ ]h , [ ] h fr , and [ ]h ss must be given In the second case, the following

func-tions must be known:

[ ] [ ](h = h U V U V P, P, T, T) (9.12)[ ] [ ](h fr = h U V U V fr P, P, T, T, )µ (9.13)[ ] [ ](h ss = h U V U V ss P, P, T, T, )µ (9.14)

The principal radii of curvature R1.P and R2.P of the part surface are the

functions of parameters U P and V P of the sculptured surface P, while

the principal radii of curvature R1.T and R2.T of the generating surface T of

the cutting tool are the functions of the parameters U T and V T

In special cases of sculptured surface machining, when, for example,

elas-tic deformation is applied to the work for technological purposes as shown

in Figure 2.3, or for a special-purpose cutting tool with changeable

generat-ing surface that is used for the machingenerat-ing [3, 9, 13], then in addition to the

parameters U P , V P , U T , V T, µ, ϕ some more parameters have to be

incorpo-rated into Equation (9.11) for the computation of the productivity of surface

generation Psg (see Chapter 8 in [6] for details)

9.2.2 Productivity of Material Removal

When machining a part surface, the intensity of stock removal is evaluated by

the productivity of material removal The productivity of material removal is

equal to the amount of stock removed from the work in a unit of time

9.2.2.1 Equation of the Workpiece Surface

For the analytical description of productivity of material removal in terms of

parameters of the machining operation, an equation of the workpiece surface

W ps must be derived Equation rwp of the surface W ps can be composed on

the premises of numerical data obtained from measurements of the actual

workpiece

The stock to be removed b can be of constant value, or its value can vary

within the surface patch In the first case, the thickness of the stock b must be

known In the second case, it is necessary to know the function of the stock

distribution b U V( P, P)

In the event the equation of the workpiece surface W ps is obtained on the

basis of the surface measurements, then the equation rP of the part surface

P together with the equation rwp of the workpiece surface W ps yields a

com-putation of the stock-distribution function b U V( P, P) :

b U V( P, P) |= rwp−rP| (9.15)

When the stock-distribution function b U V( P, P) is given, then the

equa-tion of the workpiece surface W ps can be derived analytically For this

pur-pose, an equation of the nominal part surface rP=rP(U V P, P) is employed

(Figure 9.1):

rwp =rP+nP⋅b U V( P, P) (9.16)

In Figure 9.1, point M wp on the surface of the workpiece W ps is shown at a

distance b U V( P, P ) from the point M on the nominal part surface P.

Elements of local topology of the workpiece surface W ps (say, the first

Φ1.ps and the second Φ2.ps fundamental forms of the workpiece surface W ps)

can be expressed in terms of the elements of local topology of the nominal

part surface P These derivations are similar to the derivations of major elements

of local topology of the characteristic R1-surfaces (see Section 7.3.2.2.2 for

details)

For the computation of the productivity of material removal, equation

rP=rP(U V P, P ) of the part surface P and Equation (9.16) of the workpiece

sur-face W ps must be represented in a common reference system When

neces-sary, an appropriate operator Rs(W psaP) of the resultant coordinate system

transformations can be composed for this purpose (see Chapter 3 for details)

Modified Equation (9.16) can also be helpful for the computation of

param-eters of uncut chip

Similar to Equation (9.16), the equation of the surface of tolerance S[ ]h can

be immediately written:

r[ ]h =rP+nP⋅[ ](h U V P, P) (9.17)

Equation (9.17) is used for the computation of parameters of the critical

val-ues of the feed rate and of the side step

Elements of analysis of machine tool performance can be found in [12]

9.2.2.2 Mean Chip-Removal Output

For the computation of the chip-removal output, vectorial equations of the

part surface to be machined P and of workpiece surface W ps are necessary

Mean chip-removal output is used for the analysis of efficiency of a global

machining operation, say for the whole part surface P The mean

chip-removal output %Pmr can be used as an index By definition [5,6,10,12],

where V mr is the total volume of the stock to be removed, and tΣ is the total

time required for the stock removal

9.2.2.3 Instantaneous Chip-Removal Output

For the local analysis of efficiency of a machining operation, instantaneous

chip-removal output is used The instantaneous chip-removal output Pmr

can also be used as an index By definition [5,6,10,12],

Pmr t d v mr

d t

The volume of chip d v mr to be removed in an instant of time is

where dFP is the vector area element, and b is the vector of the stock

thick-ness (here, b=b U V( P, P)⋅nP, see Equation (9.15) for details; | |b =MM wp in

Figure 9.1)

From another viewpoint, the following equation can be used for

computa-tion of the vector b:

Generally, the curvilinear coordinates U P and V P depend upon time t

according to the relations

where w is a new variable Here, the Jacobian matrix of transformation J P for the

implementation of Equation (9.24) and Equation (9.25) is as follows [1,5,6,10]:

J

U w

U t V

w

V t

The region of the part surface within which the sign of the Jacobi

transforma-tion matrix J P is maintained constant is the one under consideration

Further, an expression for the computation of the instantaneous

chip-removal output [5,6,10,12]

P

P P

can be obtained Here, w t1( ) and w t2( ) are the boundary values of the

vari-able w on the coordinate curve t Const= that corresponds to the boundaries

of the part surface P (Figure 9.2).

An alternative approach for the computation of the instantaneous

chip-removal output can be used

For this purpose, consider the surface of a cut The surface of cut S c is

gener-ated by the cutting edge of the cutting tool in its motion with respect to the work

The surface of cut S c can be considered as a set of consecutive positions of the

cutting edge of the cutting tool that is moving relative to the work Such a

con-sideration yields an equation for the position vector rsc of the surface of the cut

Implementation of a corresponding operator of the resultant coordinate system

transformations could be helpful when performing derivations of this kind

Two different kinds of analytical representation of the instantaneous

chip-removal output Pmr( ) can be derived in this case The first kind of analyti-t

cal representation of the instantaneous chip-removal output Pmr( ) relates t

to implementation of the cutting tools having the whole generating surface T In

other words, it relates to implementation of grinding wheels, shaving

cut-ters, and so forth In this case, the equation of the surface of cut S c can be

represented in the form

Curves t = Const

FiguRe 9.2

On the definition of the chip-removal output Pmr( ).t

Here, U T and V T denote curvilinear coordinates on the generating surface T

of the cutting tool, and t i is a fixed moment of time

The second kind of analytical representation of the instantaneous

chip-removal output Pmr( ) relates to the implementation of the cutting tools t

having a discrete generating surface T (for example, it relates to the

imple-mentation of milling cutters, etc.) In this case, the equation of the surface of

cut S c can be represented in the form

where U ce designates a coordinate along the cutting edge of the cutting

tool

Vector area element d F of the surface of cut S sc c can be computed either

from the formula

d

sc

sc P

sc P

sc ce

Equation (9.30) is valid for the first kind, and Equation (9.31) is applicable

for the second kind of analytical representation of the instantaneous

chip-removal output

When machining a part surface, the vector area element dFsc is traveling

with a certain velocity w through the stock to be removed In this way, a

vol-ume of the stock d v mr= ⋅ω dF is removed in a unit of time.sc

Hence, for the computation of the instant chip-removal output, the

follow-ing formula can be used:

When surface machining is performed with a cutting tool having multiple

cutting edges, Equation (9.32) acquires the form [5,6,10,12]

S i

where N is the total number of cutting edges of the cutting simultaneously

taking part in the chip-removal process, and dFsc i. is the vector area element

of the surface of cut S c i. that is created by the i-th cutting edge.

9.2.3 Surface generation Output

When machining a part surface, the rate of increase of the machined surface

area reflects the surface generation output The mean surface generation

out-put %Psg can be analytically expressed by the following formula:

where S sg designates the machined part surface area

Instantaneous surface generation output Psg is another characteristic of

surface machining performance By definition, the instantaneous surface

This means that the instantaneous surface generation output Psg can be

of constant value In this case, it is equal to the mean surface generation

out-put %Psg Generally speaking, the instantaneous surface generation output

is time dependent An expression for the computation of Psg( ) in this case t

can be derived in the following way: For the computation of area S sg of the

surface P patch, a formula

P P

This equation yields

for the computation of the instantaneous surface generation output %Psg can

be obtained (see Figure 9.2)

9.2.4 Limit Parameters of the Cutting Tool Motion

When the part surface P and the generating surface T of the cutting tool are

in point contact with each other, then the feed-rate motion and the side-step

motion of the cutting tool must both be performed when machining a

sculp-tured surface on a multi-axis NC machine

The maximal allowed displacements of the cutting tool are constrained

by the corresponding limit values [ ]F(fr and [ ]F(ss These limits specify many

parameters of the elementary surface cell on the machined part surface

The limit values [ ]F(fr and [ ]F(ss of the cutting tool displacements F(fr and

(

F ss can be computed For this purpose, the tolerance [ ]h on accuracy of

sur-face machining has been taken into consideration In compliance with [4], it is

assumed below that the maximal resultant height of cusps hΣ is within the

tol-erance [ ]h This means that the inequality hΣ ≤[ ] is valid, and, moreover, the h

elementary surface deviation δP (see Equation 8.37) is not investigated here

9.2.4.1 Computation of the Limit Feed-Rate Shift

Milling cutters are widely used for machining sculptured part surfaces on

multi-axis NC machines The use of milling cutters causes waviness of the

machined surface P It is necessary to keep the waviness height h fr under

the corresponding portion [ ]h fr of the total tolerance [ ]h The limit feed-rate

displacement [ ]F(fr strongly depends on the allowed value of the partial

toler-ance [ ]h fr

In order to compute the limit feed-rate displacement [ ]h fr , it is necessary to

investigate the topography of the machined part surface

In the direction of vector Ffr of the feed-rate motion, the cusps profile is

shaped in the form of prolate cycloids. The elementary machined surface

 In special cases of surface machining, the profile of the machined surface in the direction of

the feed-rate motion of the cutting tool can be shaped in the form of pure cycloid and even in

the form of curtate cycloid.