Kinematic Geometry of Surface Machinin Episode Episode 10 potx

Conditions of Proper Part Surface Generation 293The general solution of the problem of design of a form-cutting tool for machin-ing sculptured surfaces on a multi-axis NC machine see Ch

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292 Kinematic Geometry of Surface Machining

form, say in terms of the first and the second fundamental forms of the

GMap P ( ) of the given surface P.

The first fundamental form Φ1 0.P of the GMap P( ) is given by expression [2]

Φ1 0.P ⇒ ds P0=e du P P2+2f du dv P P P+g dv P 2P (7.2)

where ds P0 is the differential of an arc of a curve on the unit sphere; e P , f P , g P,

are the first-order Gaussian coefficients of the GMap(P); and u P , v P are the

parametric coordinates of an arbitrary point of the GMap(P).

Omitting bulky derivations, one can write the following equations for the

first fundamental form Φ1 0.P of the GMap P ( ) of the part surface P:

Φ1 0.P ⇒ ds P0= Φ2.PMP− Φ1.P PG (7.3)where MP designates mean curvature of the sculptured surface P (see Equa-

tion 1.15), and GP designates Gaussian curvature of that same surface P (see

Equation 1.16)

The second fundamental form Φ2 0.P of the GMap P( ) of a given patch of

the surface P is calculated as [2]

Φ2 0.P ⇒ −drP0⋅dnP0=l du P 2P+2m du dv P P P+n dv P P2

(7.4)

and is derived in a similar manner In Equation (7.4), the values l P , m P , n P

are the second-order Gaussian coefficients of the GMap P( ) of the surface

unit sphere

Skipping the proofs, some useful properties of the GMap P ( ) and GMap T( )

can be noted:

The GMap P ( ) of an orthogonal net on a sculptured surface P for

which mean curvature MP is not equal to zero ( MP ≠0 ) is also

an orthogonal net if and only if the initial net is made up of lines

of curvature If the mean curvature MP of the surface P is equal to

zero (MP =0), then the net of coordinate lines on the GMap P( ) will

be orthogonal as well

Points on the boundaries of the surface P and on its GMap P( ) are not

necessarily in one-to-one correspondence

GMap P( ) is a many-to-one map: Each point on a smooth part

sur-face P has a corresponding point on the GMap P( ), but each point

on GMap P( ) may correspond to more than one point on the part

surface P This means that in particular cases, GMap P( ) can be

interpreted as having more than one layer GMap P( ) of this kind are

often referred to as the multilayer GMap P ( ) For example, GMap P( ) of

a torus surface is of two layers

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Conditions of Proper Part Surface Generation 293

The general solution of the problem of design of a form-cutting tool for

machin-ing sculptured surfaces on a multi-axis NC machine (see Chapter 5) reveals that

the generating surface T of the cutting tool can be as complex as a sculptured

surface P can be The generating surface T of the cutting tool of any design has

the corresponding GMap T ( ) Figure 7.3 illustrates examples of GMap T( ) of

cut-ting tools that are commonly used in industry Computation of the parameters

of the GMap T ( ) of the generating surface T of the cutting tool is similar to the

calculation of the parameters of the GMap P ( ) of a sculptured surface P.

7.1.3 The Area-Weighted Mean Normal to a Sculptured Surface P

The efficiency of machining a sculptured surface on a multi-axis NC machine

can be extremely high when the workpiece orientation is optimal It is

con-venient to calculate the parameters of the (two-criterion) optimal workpiece

orientation taking into consideration the orientation of the area-weighted

mean normal to the surface P A point on the part surface P at which the

surface normal is parallel to the area-weighted mean normal of the surface P

is referred to as the central point of the surface P.

To calculate the parameters of the area-weighted mean normal, the

sur-face P can be subdivided into a large number of reasonably small patches

S Pi=∆U Pi×∆V Pi Here “i” indexes the small patches on the surface P At the

central point M i inside of each small patch of the surface P, the parameters

of the perpendicular NPi to the surface P can be computed:

The perpendicular NPi may be considered to be an area vector element with

magnitude equal to the infinitesimal area of part surface P at a point i.

FiGure 7.3

Examples of the form-cutting tools of various designs.

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For the computation of parameters of orientation of the area-weighted

mean normal vector %NP, the following formula is employed:

S S

where n designates the number of small patches on surface P, and S P

desig-nates the area of surface P to be machined ( S P is equal to the sum of all the

areas of the separate workpiece surfaces to be machined in one setup.)

Allowing the number of small patches on the surface P to approach

U V dU dV S

(7.7)

Differential of the surface P area is dS P= E G P P−F dU dV P2 P P Accordingly,

Equation (7.7) casts into

In cases when several part surfaces P i are to be machined on a multi-axis

NC machine in one setup, Equation (7.8) yields the more general formula

Pi i k

In the latter case, the area-weighted mean normal to the part surface P is not

considered, but the area-weighted mean normal to the several surfaces P i is

considered The last is referred to as the area-weighted mean normal to all part

surfaces P i In this case, instead of a central point of the surface, a central

point of the entire part to be machined is considered Definitely, this is a

considerably more general approach

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The area-weighted mean normal to a flat portion of surface P is N Pi Pi S

This result can be used in Equation (7.9)

The parameters of the area-weighted mean normal to a sculptured

sur-face P as calculated above allow alteration of the initial orientation of the

sculptured surface P to the desired orientation in the coordinate system

of the multi-axis NC machine By rotations of the workpiece, for example,

through the angle of nutation ψ, through the angle of precession θ, and

through the angle of pure rotation ϕ, the workpiece can be reoriented to its

optimal orientation In its optimal orientation, the workpiece allows

machin-ing of all surfaces with a smachin-ingle setup

7.1.4 Optimal Workpiece Orientation

In the initial orientation of the workpiece, the angles that the area-weighted

mean normal to the surface P makes with the coordinate axes of the NC

machine are denoted α, β, γ (Figure 7.4) It is convenient to show these

angles on the GMap P ( ) of the part surface P (remembering that the

area-weighted mean normal to the part surface P has the same direction as the

position vector of the point on the GMap P( ) corresponding to the point on

surface P at which the perpendicular is erected) In the case under

consider-ation, the problem of optimal workpiece orientation reduces to a problem of

coordinate system transformation

Consider two Cartesian coordinate systems X Y Z P P P and X Y Z NC NC NC The

first coordinate system is associated with the workpiece Another is

con-nected to the multi-axis NC machine

In the initial orientation of the workpiece, orientation of the coordinate

system X Y Z P P P relative to the coordinate system X Y Z NC NC NC is defined

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by the angles α, β, and γ Actual values of these angles can be computed

using one of the above-derived Equation (7.6) through Equation (7.9) The

computed value of the area-weighted mean normal vector %NP immediately

yields computation of the angles α, β, and γ For this purpose, the

follow-ing formulae cosα = ⋅i N%P, cosβ = ⋅j N%P, and cosγ = ⋅k N%P can be used All

the computations must be performed in a common reference system Use of

the coordinate system X Y Z NC NC NC is preferred

In the optimal workpiece orientation, corresponding axes of these

coor-dinate systems are parallel to each other and are of the same direction To

put the workpiece into the optimal orientation means to make three

suc-cessive rotations, for example, by the Euler angles — that is, to rotate the

workpiece in the coordinate X Y Z NC NC NC through the angle of nutation ψ,

through the angle of precession θ, and finally, through the angle of pure

rotation ϕ

The resultant coordinate system transformation using Euler’s angles can

be analytically represented with the operator Eu ( , , )ψ θ ϕ of Eulerian

trans-formation (see Equation 3.11)

In the optimal workpiece orientation, it is possible to rotate the part

sur-face P about the area-weighted mean normal %N P Under such a rotation,

the optimal orientation of the workpiece is preserved, but the orientation

of part surface P relative to the NC machine coordinate axes changes

This feasible rotation of the surface P can be used for satisfying additional

requirements to the part surface orientation on the worktable of the

multi-axis NC machine

For example, the workspace of the multi-axis NC machine is the bounded

plane or volume within which the cutting tool and the workpiece can be

positioned and through which controlled motion can be invoked When NC

instructions are generated by a part programmer, the geometry of the

work-piece must be transformed into a coordinate system that is consistent with

the workspace origin and coordinate reference frame That is why after the

workpiece is turned to a position at which its area-weighted mean normal

has an optimal orientation, it is necessary to rotate it about the weighted

normal to a position in which the projection of the part surface P (or of the

part surfaces P i) to be machined is within the largest closed contour traced

by the cutting tool on the plane of the NC machine worktable In addition,

the vertical position of the workpiece must conform to the capabilities of the

NC machine to move in the vertical direction

Proper location of the workpiece on the worktable of a multi-axis NC

machine can be specified in terms of (a) the joint space, which is defined by a

vector whose components are the relative space displacements at every joint

of a multi-axis NC machine; (b) the working envelope, which is understood

as a surface or surfaces that bound the working space; (c) the working range,

which means the range of any variable for normal operation of a multi-axis

NC machine; and (d) the working space that includes totality of points that can

be reached by the reference point of the cutting tool

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7.1.5 expanded Gaussian Map of the Generating Surface

of the Cutting Tool

An ordinary Gauss’ map can be constructed for any generating surface of a

cutting tool Examples of the GMap T( ) of the form-cutting tools of various

designs are shown in Figure 7.3 However, when machining a sculptured

surface, the cutting tool is traveling with respect to the surface P

Corre-spondingly, the GMap T( ) moves over the surface of the unit sphere covering

in such a motion that the area exceeds the area of the original GMap T( )

Gauss’ map that is constructed for the moving generating surface T of the

cutting tool in all its feasible positions is referred to as the expanded Gauss’

map GMap T e( ) of the generating surface of the cutting tool

Actually, when machining a sculptured surface on a multi-axis NC

machine, the workpiece and the cutting tool perform certain relative motions

For further analysis, it is convenient to implement the principle of inversion

of relative motions On the premises of the principle of inversion of relative

motions, consider the resultant motion of the cutting tool relative to the

sta-tionary workpiece

At every point K of contact of the surfaces P and T, the unit normal vectors

nP and nT to these surfaces are of opposite directions (Remember that a

nor-mal to the part surface P is pointed outward from the part body, and a nornor-mal

to the generating surface T of the cutting tool is pointed outward from the

generating body of the cutting tool Therefore, the equality nP = −nT must

be satisfied.) Then, employ the concept of antipodal points [5] Those points on

the Gauss’ map are usually referred to as the antipodal points, which are the

pairs of diametrically opposed points on the unit sphere Implementation of

the antipodal points yields introduction of the centro-symmetrical image of

the GMap T ( ) of cutting tool surface T The last is referred to as the antipodal

GMap T a ( ) of the generating surface T of the cutting tool.

Analysis of possible relative positions of the GMap P( ) of the sculptured

surface P and of the antipodal GMap T a ( ) of the generating surface T of the

cutting tool yields the following intermediate conclusions:

Conclusion 7.1: If GMap P ( ) of the part surface P is entirely located within

the antipodal GMap T a ( ) of the generating surface T of the cutting tool (that

is, the GMap P ( ) contains no points outside GMap T a( ) ), then machining

of the surface P is possible.

This is the necessary but not sufficient condition for the machinability of the

part surface with the given cutting tool

Conclusion 7.2: If any portion of the GMap P( ) is located outside the

antipodal GMap T a ( ) , then machining of the surface P is impossible.

This is the sufficient condition that the part surface P cannot be machined

with the given cutting tool

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

cutting tool is capable of moving along three axes of the coordinate system

X Y Z NC NC NC, and rotating about one or more of the axes These additional

degrees of freedom (rotations) allow the antipodal indicatrix GMap T a( ) of

the generating surface T of the cutting tool to expand around the surface of

the unit sphere, while the GMap P( ) remains fixed

Similar to the spherical indicatrix GInd T ( ) of the surface T that serves as

the boundary curve for the corresponding GMap T( ), the antipodal

indica-trix GInd T a ( ) serves as the boundary curve to the antipodal GMap T a( )

For example, consider machining of a sculptured surface P on a three-axis

NC machine The antipodal GMap T a ( ) of the generating surface T of the

cutting tool occupies the fixed area ABCD (Figure 7.5) Then, assume that

one more NC-axis is added somehow to the articulation capabilities of the

NC machine The additional fourth NC-axis (say, rotation of the cutting tool

about an axis not coinciding with the axis of its cutter-speed rotation) causes

the antipodal GMap T a( ) to extend in direction 1 from the initial location

ABCD to encompass A B CD1 1 If the fifth and the sixth NC-axes are added,

then these additional NC-axes cause the antipodal GMap T a( ) to extend in

direction 2 and to rotate about an axis through the center of the unit sphere

and through a point within the antipodal GMap T a( ) of the generating

sur-face T of the cutting tool.

A surface patch on the unit sphere is covered by the antipodal GMap T a( )

such that its motion over the unit sphere is referred to as the expanded

antipo-dal GMap T ae ( ) of generating surface T of the cutting tool.

GMapa(T) GMapae(T)

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The expanded antipodal indicatrix GMap T ae( ) is a useful tool for the

investi-gation of workpiece orientation on the worktable of a multi-axis NC machine

The boundary curve of the expanded antipodal GMap T ae( ) of the

generat-ing surface T of the cuttgenerat-ing tool serves as the expanded antipodal indicatrix

GInd T ae ( ) of the tool surface T Because the surface P is considered

motion-less, the expanded antipodal indicatrix GInd T ae ( ) of the generating surface T

of the cutting tool as well as its expanded antipodal GMap T ae( ) cannot rotate

about an axis through the origin of the coordinate system X Y Z P0 P0 P0.

When the parameters of the relative motion of the given sculptured

sur-face P and of the given generating sursur-face T are known, then the parameters

of the expanded initial GInd T e ( ) and of the expanded antipodal GInd T ae( )

indicatrices of the tool surface T can be calculated using the developed

meth-ods of spherical trigonometry [5]

For machining a sculptured surface on a multi-axis (four or more axes) NC

machine, the following two statements hold:

Conclusion 7.3: If GMap P ( ) of the part surface P is contained entirely

inside the expanded antipodal GMap T ae ( ) of generating surface T of the

cutting tool, then the surface P can be machined.

This is the necessary but not sufficient condition for the machinability of a

sculptured surface on a give multi-axis NC machine with the given cutting tool

Conclusion 7.4: If GMap P ( ) of the part surface P contains at least one

point outside the expanded antipodal GMap T ae( ) of generating surface

T of the cutting tool, then machining of the surface P is not feasible.

This condition is sufficient for the sculptured surface that cannot be machined

on a given multi-axis NC machine with the given cutting tool

7.1.6 important Peculiarities of Gaussian Maps of the Surfaces P and T

In particular cases, a sculptured surface P, as well as the generating

sur-face T of the form-cutting tool can have two or more points, at which unit

normal vectors are parallel to each other and are pointed in that same

direction Points of this sort can be easily found out, for example, on the

torus surface

When parallel and similarly directed unit normal vectors are observed, then

the GMap P ( ) of the sculptured surface P becomes “multilayered.” The

num-ber of layers of the the GMap P( ) is equal to the number of points with parallel

and similarly oriented unit normal vectors For example, parallel and similarly

oriented unit normal vectors occur on the part surface P (Figure 7.6)

The surface P is bounded by the bordering line ABCDEFG Gauss’ map

GMap P ( ) for this portion of the surface P is represented by the portion

A B G D E F0 0 0 0 0 0 of the unit sphere Figure 7.6 reveals that the area B0 C0D0G0

on the unit sphere corresponds to the GMap P( ) of the portion BCDG of the

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the surface P1 occupies a portion of the unit sphere within the closed contour

ABDCFE The area-weighted mean normal n P0rP0 passes through the point

M P0 In such an orientation, the surface P1 can be machined on the

three-axis NC machine in one setup

In the second case, a sculptured surface P2 has a multilayer (two-layer)

GMap Pm ( ) The subscript m here indicates that the Gauss’ map is multilayer

Likewise, in the first case, GMap Pm( ) of the surface P2 occupies the portion

within the closed contour ABDCFE on the unit sphere In addition to the

portion ABDCFE , the GMap Pm( ) is represented with the portion CDEF

on the unit sphere (Figure 7.7) When the portion CDEF is added, then the

Gauss’ map is a two-layer map, so it is twice as heavily weighted Due to

the increase in weight of the GMap Pm( ) , the area-weighted mean normal

nP0rP0of the surface P2 turns about the center of the unit sphere through

a certain angle ε In this position, the unit normal vector passes through the

point MP 0

In such a position of the workpiece, it is infeasible to machine the

sculp-tured surface P2 on the three-axis NC machine in one setup It is

neces-sary to consider the trade-offs between the orientation of the part surface

P2 in compliance with the position of its area-weighted mean normal and

the orientation of the surface P2 to avoid shadowed areas One could

con-sider, for example, whether it is preferred to machine the sculptured part

surface P2 on a cheaper, three-axis NC machine with nonoptimal

work-piece orientation vis-à-vis cutting conditions, or to machine the part

sur-face P2 in the optimal workpiece orientation but on a more costly four (or

more) axis NC machine Generally, machining of a part surface in a single

setup with some loss of optimality of cutting condition is preferable to

machining in two or more setups Thus, the generally favored situation is

to orient the workpiece such that the difference in angle between the

area-weighted normal to the part surface to be machined and the tool axis of

rotation changes as little as possible, without requiring more setups than

necessary

After the analysis of Figure 7.7 is performed, it is important to focus again

on the properties of Gauss’ mapping of the surfaces Figure 7.6 provides a

good example to illustrate the property (b) of the GMap P( ) (see Section 7.1.2)

Gauss’ map of the bordering contour ABCDE of the surface P is represented

by the circular arc A B C D E0 0 0 0 0 (Figure 7.6) In this case, all points of the

bordering contour ABCDE of the surface P and all points of the boundary

A B C D E0 0 0 0 0 of the Gauss’ map are in one-to-one correspondence On the

other hand, Gauss’ map of the bordering contour AFE of the surface P is

represented by the circular arc A F E0 0 0 It is evident that the Gauss’ map

A F E0 0 0 of the bordering contour AFE is not a border for the GMap P( ) of

the sculptured surface P Moreover, boundary arc B G D0 0 0 of the GMap P( )

of the surface P is just an image of the curve BGD on the surface P However,

the BGD is not a boundary of the surface P This example illustrates that a

boundary of the GMap P ( ) of a sculptured surface P may or may not be a

boundary of its GMap P( ) , and vice versa

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7.1.7 Spherical indicatrix of Machinability of a Sculptured Surface

The above-considered Gauss maps of the sculptured part surface and of

the generating surface of the cutting tool provide engineers with a

power-ful analytical tool Among others, implementation of this tool is helppower-ful for

determining whether or not a given sculptured surface P can be machined

with the given cutting tool on the NC machine with the given articulation

For this purpose, spherical indicatrices GInd P ( ) and GInd T( ) can be used

It is inconvenient to treat simultaneously two separate characteristic curves

GInd P ( ) and GInd T ( ) to determine whether the part surface P can or cannot

be machined in one setup with the given cutting tool on the NC machine with

the given articulation For this purpose, a characteristic curve of another nature

is proposed This characteristic curve is referred to as the spherical indicatrix of

machinability Mch P T ( / ) of a given sculptured surface P with the given cutting

tool T on the NC machine with the given articulation For CAD/CAM

applica-tion, it is necessary to represent this characteristic curve analytically

Without loss of generality, one can consider for simplicity the machining

of a sculptured surface P with a ball-end milling cutter For this case, the

GMap T( ) of the generating surface of the cutting tool occupies a hemisphere

of the unit sphere (Figure 7.8) GMap P ( ) of the sculptured surface P is

rep-resented with a certain patch on the unit sphere The great circle of the unit

sphere serves as the GInd T ( ) of the generating surface T of the cutting tool

Ultimately, GInd P ( ) is represented by the boundary of the GMap P( )

An arbitrary point MP 0 is chosen within the GMap P ( ) of the surface P

A cross-section of the unit sphere by the plane Σi through the origin of the

coordinate system X Y ZP 0 P 0 P 0 and the chosen point MP 0 is considered The

plane Σi intersects the spherical indicatrices GInd P ( ) and GInd T( ) at the

points A0i and B0i, respectively The angle between the position vector rAi

of the point A0i and the position vector rM of the chosen point MP 0 is

des-ignated as ςAi A similar angle between the position vector rBi of the point

B0i and the position vector rM is designated as ςBi The difference of the

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angles ςAi and ςBi is equal to (ςAi−ςBi)=ςMi Angle ςMi , which determines

the location of the point M0i on the indicatrix of machinability Mch P T( / )

of the surface P using tool T, is equal to ςMi =ςMi This angle is measured

within the plane Σi through the vectors rM, rAi, rBi, either in the clockwise

direction if ςAi >ςBi (in this case, ςMi <0ο), or in the counterclockwise

direc-tion if ςAi<ςBi (in this case, ςMi <0ο)

Rotating the plane Σi about the position vector rM to positions Σ1, Σ2,

Σ3, and so on, all points of the indicatrix of machinability Mch P T( / ) can

be obtained

Conclusion 7.5: A sculptured surface P is machinable using the given

generating surface T of the cutting tool if and only if the indicatrix of

machinability Mch P T( / ) has no negative diameters i.e., it is not a

self-interesting curve on the unit sphere.

The equation of the indicatrix of machinability Mch P T( / ) immediately

follows from the analysis below

The position vector rM of the point M0 P is

rM

M M M

αβγ1

(7.10)

The equation of the GMap T ( ) of the generating surface T of the cutting

tool yields two equations The first equation is for the unit vector rAi:

rAi

Ai Ai Ai

αβγ1

(7.11)

and another equation is for the unit vector rBi:

rBi

Bi Bi Bi

αβγ1

(7.12)

The vectors rM and rAi make an angle ςAi The actual value of the angle ςAi

can be computed from the formula

⋅( ,r r ) arctan r r

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The vectors rMand rBimake an angle ςBi The actual value of the angle ςBi is

With these results, the value of the angle ςMi that is determining the

loca-tion of an arbitrary point of the indicatrix of machinability Mch P T( / ) of part

surface P using tool surface T is

Equation (7.15) specifies the location of a current point of the spherical

indi-catrix of machinability Mch P T( / ) This characteristic curve is convenient

for determining whether the sculptured part surface P can or cannot be

machined using the given generating surface T of the cutting tool.

As an illustration of implementation of the spherical indicatrix of

machin-ability Mch P T( / ) , consider machining of various cylindrical local portions

of a sculptured surface P on a three-axis NC machine (Figure 7.9) In the case

under consideration, the antipodal indicatrix GInd T a( ) of the generating

sur-face T of the cutting tool is represented with a hemisphere However, an arc

of a great circle of the unit sphere serves as the spherical indicatrix GInd P( )

of the surface P.

Analysis of the relative configuration of the spherical indicatrices GInd P( )

and GInd T ( ) shows that the surface P can be machined in the first two cases,

and it cannot be machined in the third case One can come up with that same

result via analysis of shape of the indicatrix of machinability Mch P T( / ) of

the surface P using tool surface T.

The analysis shows that in the first case, the surface P can be machined

on the three-axis NC machine Moreover, there remains some freedom in

orienting the workpiece: the surface P can be turned about its axis in

oppo-site directions by a certain angle of ξ >0

In the second case, the surface P can also be machined on the three-axis

NC machine However, in this case, no degree of freedom remains The

sur-face P cannot be turned about its axis, because the angle GMap P( ) is equal

to zero (ξ =0 )

In the third case, the surface P cannot be machined on the three-axis NC

machine The arc of a great circle through the spherical map GMap P( ) includes

points for which the angle ξ is negative (ξ <0 ) The last indicates that

nega-tive diameters of the indicatrix of machinability Mch P T( / ) are observed

For convenience in implementation, the indicatrix of machinability Mch P T( / )

can be depicted in Cartesian coordinates No formulae transformations are

required in this concern The spherical parameters must be considered as the

Cartesian coordinates

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the center of the generating circle that rotates about the Z P axis, and ris the

position vector of a point on the generating circle Due to lack of space, the

vectors R , R, and r are not depicted in Figure 7.10

Representation of the position vector R in the form R R= +r yields an

expanded equation for the surface P:

r sinϕ1

The perpendicular vector NP to the surface P at an arbitrary point M can be

calculated as NP=UP×VP, where tangent vectors UP and VP are given by

sin cossin sin00

P

M M

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Equation (7.17) and Equation (7.18) yield the formula

N

for the computation of the normal vector NP to the surface P.

The angles αP, βP, and γPwhich the vector NP makes with the axes of the

coordinate system X Y Z P P P attached to the part to be machined ( cosαP= ⋅i NP,

cosβP= ⋅j NP, and cosγP= ⋅k NP) can be calculated as

For the computation, it is convenient to consider discrete values of the

parameters θP and ϕP with certain increments δθP and δϕP Under such a

scenario, the surface P could be represented in δθP=1 and δϕP=1

incre-ments by 90 90 8100⋅ = points, which provide sufficient accuracy for the

90 89 88

j (i + 1), ( j + 1) (i + 1), j

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Thereby, the surface P is covered with nearly rectangular patches,

verti-ces of which may be enumerated in the following manner: ( , )i j, (i+1, )j ,

( ,i j+1 , and () i+1,j+1 Each patch of the surface P can be considered as )

nearly a flat patch that can be inscribed in a circle

The optimal orientation of the part surface P is calculated in the following

n

P

S S

=∑− ,∆ ,

In Equation (7.23), the perpendicular NP i j , is computed by

NP i j , =R Psin( j) r Pcos( i)sin( j)

NP i j

P P

p i j, = a i j, +b i j, +c i j, +d i j,

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