Kinematic Geometry of Surface Machinin Episode Episode 12 pps

Two major reasons often cause surface deviation: When machining a part surface, the entire generating surface of the cutting tool does not actually exist.. Point contact of the part surf

8

Accuracy of Surface Generation

Accuracy of the machined part surfaces is a critical issue for many reasons

Deviations of the actual part surface from the desired part surface are

inves-tigated in this chapter from the prospective of capabilities of the theory of

surface generation

Two major reasons often cause surface deviation:

When machining a part surface, the entire generating surface of the

cutting tool does not actually exist In all cases of

implementa-tion of wedge cutting tools, the generating surface of the

cut-ting tool is not represented entirely but by a limited number of

cutting edges In other words, the generating surface of the

cut-ting tool is represented discretely The discrete representation of

the surface T of the cutting tool causes deviations of the actual

nomi-nal) part surface P nom

Point contact of the part surface and of the generating surface of the

cutting tool is usually observed when machining a sculptured

sur-face on a multi-axis numerical control (NC) machine When the

surfaces make point contact, then articulation capabilities of the

multi-axis NC machine can be utilized in full From this

prospec-tive, point contact of the surfaces can be considered as the most

general kind of surface contact However, point contact of the

sur-faces P and T also causes deviations of the actual machined part

Ultimately, when the generating surface T of a cutting tool is represented

discretely, and the surfaces P and T make point contact, then the deviations

are getting bigger

Sources for the deviations of the machined part surface from the desired

part surface are limited to two major reasons only in a simplified case of

surface machining In the simplified cases of surface machining, no

devia-tions in the surfaces P and T configuration are observed Deviadevia-tions in the

configuration of surfaces P and T are unavoidable Therefore, the impact of

deviations of the configuration of surfaces P and T onto the resultant

8.1 Two Principal Kinds of Deviations of the Machined

Surface from the Nominal Part Surface

The discrete representation of the generating surface of the cutting tool as

well as point contact of the surfaces P and T result in that during a certain

limited period of time, it is impossible to generate the part surface precisely,

without deviations of the actual machined part surface from the desired part

surface

8.1.1 Principal Deviations of the First Kind

For proper generation of a part surface, the entire generating surface T must

be represented by the cutting tool Actually, the surface T of a cutting tool

is represented as a certain number of cutting edges The number of cutting

edges of the cutting tools of conventional design is limited, and the total

number could be easily counted The generated surface T of a cutting tool of

this type is discontinuous

The number of cutting edges of grinding wheels and of other abrasive tools

is also limited However, it is not that easy to count all the cutting edges of

a grinding wheel as can be done with respect to wedge cutting tools

There-fore, in most cases of surface machining, the generating surface of abrasive

cutting tools can be considered as a continuous surface T.

When machining a part surface, for example, with a milling cutter

(Figure 8.1), the cutting tool is rotating about its axis O T with a certain angular

contact-ing the nominal part surface P at a point K The actual machined part surface

Usu-ally, the trajectories can be represented by prolate cycloids In particular cases,

the trajectories are represented by pure cycloids and even by curtate cycloids In

If the part surface to be machined and the generating surface of the cutting

of the milling cutter For milling cutters of most conventional designs, those

be eliminated from the analysis of the surface P accuracy The elementary

370 Kinematic Geometry of Surface Machining

8.1.3 The resultant Deviation of the Machined Part Surface

deviations h fr and h ss

portion of the surface is bounded by two neighboring arc segments m and

segments m and ( m+1 is equal to the feed rate per tooth ) F(fr of the cutting tool,

and n, ( n+1 is referred to as the elementary surface cell of the part surface P.)

The major parameters h fr, F(fr , h ss, and F(ss of the elementary surface cell are

not constant within the part surface P They vary in certain intervals within

the sculptured surface Current values of the major parameters of the

of the surface P; (b) the principal radii or curvature P1.T , P2.T of the surface T;

Deviation of the machined part surface P ac from the desired part surface P nom that is caused by

point kind of contact of the surfaces P and T.

The maximal resultant deviation hmaxΣ of the surface P ac from the surface

part surface

It is widely recognized that in sculptured surface machining on a

multi-axis NC machine, the principle of superposition of the elementary deviations

fur-ther investigation

the following equation can be used:

con-stants a h and b h are within the intervals 0≤a h≤1 and 0≤b h≤1

Generally, the function hΣ =h h hΣ( ,fr ss) is complex

In compliance with the sixth necessary condition of proper part surface

the surface machining It is recommended that an operation of a sculptured

of the surface P ac from the surface P nom is equal to the tolerance [ ]h A

is satisfied within the entire part surface P being machined.

surface generation

8.2 Local Approximation of the Contacting Surfaces P and T

elemen-tary surface cell In order to solve the problem, an analytical local

representa-tion of the surfaces is helpful

Accuracy of Surface Generation 373

The nominal part surface is given Locally, the surface P is specified by

torsion τP

congru-ent to the surface of the cut When machining a part, the cutting edge of the

cutting tool moves relative to the work Consecutive positions of the moving

that is located within the elementary surface cell is congruent to the actual

torsion τc For the computation of the parameters R1.c , R2.c, and τc, the

premises of the geometry of the cutting edge of the cutting tool, the

kinemat-ics of the relative motion of the cutting edge with respect to the work, and the

close to the generating surface T of the cutting tool as long as the elementary

param-eters R1.T , R2.T, and τT of the generating surface T of the cutting tool can be

computed instead

8.2.1 Local Approximation of the Surfaces P

and T by Portions of Torus Surfaces

Actual surfaces P and T can be given in a complex analytical form that is not

convenient for computations of the major parameters of the surfaces

Solu-tions to many geometrical problems can be more easily derived from local

consideration of the surfaces rather than from consideration of the entire

surfaces

For the local analysis, the surfaces are often represented by quadrics

As shown in our previous works [4,5,8], from the perspective of local

approximation of surface patches, helical canal surfaces feature important

advantages over other candidates

Monge was the first to investigate the class of surfaces formed by sweeping a

sphere, in 1850 [2] He named them canal surfaces In the particular case when

the path on which the sphere is swept along is a helix, and the sphere has

constant radius, the surface swept out is referred to as a helical canal surface A

surface of this kind is of particular interest for engineers

A canal surface is the envelope of a one-parametric family of spheres The

envelope is defined as the union of all circles of intersection of infinitesimally

neighboring pairs of spheres These circles are referred to as the composing

circles Helical canal surfaces can fit the principal curvatures and torsion of

the local patch of sculptured surfaces, as well as of the generating surfaces

of cutting tools

A torus surface can be expressed in terms of radius r tr of its generating

the torus radius R tr in this case is equal to the difference R tr =R2 P−R1 P

For another ratio between the radii r tr and R tr , the equalities r tr =R2 P−R1 P

In the coordinate system X Y Z tr tr tr associated with the torus surface (Figure 8.6),

the position vector rtr( ,θ ϕtr tr) of a point of the approximating torus

sur-face can be represented in the following way: rtr( ,θ ϕtr tr)=R( )θtr +r( ,θ ϕtr tr)

the position vector of the center of the generating circle, which rotates about

Accuracy of Surface Generation 377

[R tr C. =(R tr+ ⋅r tr cosθtr)R tr C. ] of the directing circle of the torus surface (Here,

circle of radius r tr.)

Note that all ten kinds of local patches of smooth, regular surfaces (see

illus-trates this important property of the torus surface

Consider points on the surface Tr that occupy various positions M1, M2, M3,

either within the convex surface Tr or within the concave surface Tr, all ten kinds

of local patches of smooth, regular surface can be found on the torus surface Tr.

The major advantage of implementation of the torus surface for local

approximation of the sculptured surface is due to a patch of the torus surface

being capable of providing perfect approximation for bigger surface area

compared to the approximation by quadrics, use of which is valid just within

a differential vicinity of the surface point

C

A C

The Darboux trihedron is implemented here for the purpose of construction

origin at the point K.

Configuration of the sculptured surface P as well as configuration of the

the machine tool is known Therefore, the corresponding operators of the

the resultant coordinate systems transformation can be composed

the consequent coordinate systems transformations are composed for the

generating surface T of the cutting tool Ultimately, the operators of the

380 Kinematic Geometry of Surface Machining

systems transformation to a closed loop of the coordinate systems

The derived operators of the coordinate systems transformations yield

geom-etry in a common coordinate system Implementation of the local coordinate

8.3 Computation of the Elementary Surface Deviations

the formula hΣ =a h h⋅ fr+ ⋅b h h ss (see Equation 8.4) For the computation of

necessary to investigate both elementary deviations separately It is sufficient

to investigate just one of them, and afterwards to write similar equations for

the computation of another

8.3.1 Waviness of the Machined Part Surface

Fig-ure 8.10 illustrates a cross-section of a sculptured part surface P by a

It is convenient to mention here that the rate of conformity of the

when the higher rate of conformity of the surface T to the surface P observes,

then the higher accuracy of the machined part surface and vice versa

equa-tion is derived by Radzevich [6,7]:

where radii of normal curvature of the surfaces P and T are designated as

R P fr. and R T fr. , respectively, and the arc segment F(fr designates the feed rate

per tooth of the cutting tool The radii R P fr. and R T fr. are measured in the

An equation similar to Equation (8.8) is derived in [6,7] for the computation

assumed in Equation (8.9) that the radius of normal curvature of the surface

Figure 8.10

Computation of the elementary deviation h fr (the waviness) on the sculptured part surface P.

382 Kinematic Geometry of Surface Machining

of cut is approximately equal to the corresponding radius of normal

curva-ture of the generating surface T of the cutting tool.

In particular cases, Equation (8.7) can be significantly simplified For

example, when a flat portion of a part surface P is machined with the milling

Equation (8.10) is well known from practice

Derivation of the equation for the computation of the elementary deviation

details of derivation, the final equation for the computation of the

In Equation (8.11), radii of normal curvature of the surfaces P and T are

des-ignated as R P ss. and R T ss. , respectively, and the arc segment F(ss designates

8.3.3 An Alternative Approach for the Computation

of the elementary Surface Deviations

Reasonable assumptions yield simplification of equations for the

computa-tion of elementary surface deviacomputa-tions As an example, an alternative approach

Figure 8.11

Elementary analysis of Figure 8.11 yields computation of coordinates of

centers O T( )1 and O T( )2 in two consecutive positions of the cutting tool relative

of the surface P waviness is expended in the Taylor’s series Ultimately, this

yields the approximate equation

reader may wish to go to [5] for details on the derivation of Equation (8.14)

h fr T

384 Kinematic Geometry of Surface Machining

The considered approach can be enhanced to the situation when radii of

normal curvature of the part surface P and of the generating surface T of the

cutting tool are significantly different between two consequent tool-passes

8.4 Total Displacement of the Cutting Tool

with Respect to the Part Surface

No absolute accuracy is observed in machining sculptured surfaces on a

multi-axis NC machine Both the NC machine and the cutting tool are the

major sources of unavoidable deviations of the machined part surface from

the desired sculptured surface Actual relative motion of the cutting tool

is performed with certain deviations of its parameters with respect to the

desired relative motion of the cutting tool The last is also a source of

signifi-cant surface deviations

Displacements of the generating surface T of the cutting tool with respect

to the desired part surface P are unavoidable Problems of two kinds arise in

this concern First, it is important to compute how much the displacement of

a cutting tool donates to the resultant deviation of the actual machined part

surface from the desired part surface Second, in order to avoid the cutter

penetration into the part surface P, it is of critical importance to determine the

maximal allowed dimensions of the cutting tool in order to avoid violation of

For solving problems of both kinds, computation of the closest distance of

approach (CDA) of the surfaces P and T is necessary The minimal separation

between objects is a fundamental problem that has application in a variety

of arenas The problem of computation of the CDA of two surfaces is

sophis-ticated However, it can be solved using methods developed in the theory of

surface generation

8.4.1 Actual Configuration of the Cutting Tool

with respect to the Part Surface

It is convenient to begin the analysis from the ideal case, when the surfaces P

case of surface generation, the closed loop of consequent coordinate systems

is used for the construction of the left-hand-oriented local Cartesian

equality Rs(K T aK P)=Rs− 1(K PaK T) is always observed The operators

opera-tors of the coordinate systems transformation to the closed loop of the

Equation rP=rP(U V P, P) and equation rT =rT(U V T, T ) of the surfaces P

and T together with the above-mentioned operators of the coordinate

common coordinate system Below, the local Cartesian coordinate system

x y z P P P is used for this purpose

When the generating surface T of the cutting tool is in proper tangency with

surfaces P and T In reality, the surfaces P and T do not make proper contact

Actually, the surfaces are either slightly apart, or the surface T penetrates into

the surface P This is due to the unavoidable deviations of configuration of

the cutting tool with respect to the part surface P The deviations cause a

the actual position x y z T T T* * * Again, deviations of this kind are unavoidable

part surface P can be expressed in terms of the elementary linear

displace-ments δx, δy, and δz of the cutting tool along the axes x P , y P , z P:

δ

δδδ

1

(8.15)

angu-lar displacements θx, θy, and θz of the local coordinate system x y z T T T with

to the part surface P can be expressed in terms of the elementary angular

axes x P , y P , z P:

θ

θθθ

1

( 8.16)

tool moves to a position x y z T T T* * *

Accuracy of Surface Generation 387

between local patches of the surfaces P and T occurs, or the surfaces cause P

and T to interfere with each other.

expressed in terms of the corresponding elementary displacements of all the

x y z T T T* * * No closed loop of the consequent coordinate systems

transforma-tions can be constructed at this point The loop of the consequent coordinate

systems transformations is not closed yet In order to make the loop close, it is

of the inverse coordinate systems transformation For the composing of the

In order to solve the problem, the CDA between the surfaces P and T must

be computed

In the ideal case of surface generation when no displacement of the surface

at a point K Actually, it is allowed to interpret the ideal surfaces contact

cutting tool surface T are snapped into a common point K Therefore, the

of the surfaces in the ideal case of surfaces generation, but the designation K

is used instead

approach between the surfaces P and T is identical to the closest distance

is identical to zero The closest distance of approach between the surfaces

valid

In reality, the generating surface T of the cutting tool is displaced with

respect to the part surface P The total linear displacement of the surface T

Equation 8.15) The total angular displacement of the surface T with respect

The closest distance of approach of the surfaces P and T is not equal to zero

It can be positive or negative In the first case, the cutting tool surface T is

located apart from the part surface P In the second case, the cutting tool

sur-face T interferes with the part sursur-face P.