Kinematic Geometry of Surface Machinin Episode Episode 14 pdf

Minimal machining time is the major goal of the problem of synthesis of optimal global surface generation.. 440 Kinematic Geometry of Surface MachiningAt the current CC-point, the tool-p

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438 Kinematic Geometry of Surface Machining

the surface P However, when the chip-removal output is included in the

con-sideration, the maximal speed of the cutting tool travel is smaller when the

concave portion of the surface P is machining, and it is bigger when

machin-ing a convex portion of the surface P.

For the orthogonally parameterized surface P, Equation (10.12) for [rtpopt]

In many particular cases of sculptured surface machining, both

Equa-tion (10.12) and EquaEqua-tion (10.13) can be integrated analytically

In some particular cases of sculptured surface generation, the equation for

the optimal tool-paths simplifies to the differential equation

Equation (10.14) for the optimal tool-paths is applicable, for instance, when

machining a sculptured surface P either with a ball-nose milling cutter, or

with a flat-end milling cutter, and so forth Under such a scenario, the angle

m of the local relative orientation of surfaces P and T vanishes It is getting

indefinite: no principal directions can be identified on a sphere or on the

plane surface Therefore, the optimal tool-paths align with lines of curvature

on the surface P (see Equation 10.14).

When machining a part surface, the coordinate system X Y Z T T T associated

with the cutting tool is rotating like a rigid body This rotation is performing

about a certain instant axis of rotation The angular velocity of the rotation

of the coordinate system X Y Z T T T is equal to

| |W = k tp2 +τtp2.The axis of instant rotation aligns with Darboux’s vector

W =k tp tpt +τtpbtp

(here k tp and τtp denote curvature and torsion of the trajectory of the CC-

point, and ttp and btp are the unit tangent vector and the binormal vector

to the trajectory of the CC-point at a current point K) Darboux’s vector is

located in the rectifying plane to the trajectory of the CC-point It can be

expressed in terms of the normal vector ntp and of the tangent vector ttp to

the trajectory of the CC-point:

W = k tp2 +τtp2( costtp θ+ntpsin )θ (10.15)

where q is the angle that makes Darboux’s vector W and the tangent vector

ttp to the trajectory at the CC-point

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Synthesis of Optimal Surface Machining Operations 439

It is instructive to note that velocity |W| is a function of full curvature of

the trajectory of the CC-point

10.3 Synthesis of Optimal Surface Generation:

The Global Analysis

Synthesis of optimal global surface generation is the final subproblem of the

general problem of synthesis of optimal surface generation The solution

to the problem of optimal global surface generation is based much on the

derived solutions to the problems of optimal local and of optimal regional

surface generation

Minimal machining time is the major goal of the problem of synthesis of

optimal global surface generation In order to solve the problem under

con-sideration, it is necessary to do the following:

Minimize interference of the neighboring tool-paths of the cutting tool

over the part surface being machined

Determine the optimal parameters of placing the cutting tool into

contact with the part surface, and of its departing from the contact

This subproblem is referred to as the boundary problem of surface

generation

Determine the location of the optimal starting point of the surface

machining

10.3.1 Minimization of Partial interference of the Neighboring Tool-Paths

The actual machined part surface is represented as a set of tool-paths that

cover the nominal surface P At a current surface point, the width of the

tool-path is equal to the side-step F(sscomputed at that same CC-point (see

Equation 9.46) The tool-path width varies along the trajectory of the CC-point

over the sculptured surface, as well as across the trajectory Because of this,

neighboring tool-paths partially interfere with each other Ultimately, some

portions of the part surface P are double-covered by the tool-paths Partial

interference of the neighboring tool-paths causes reduction of the surface

generation output For the synthesis of optimal surface generation operation,

the interference of the neighboring tool-paths must be minimized

The trajectory of the CC-point over the sculptured surface is a

three-dimen-sional curve For the analysis below, it is convenient to operate with the

natu-ral parameterization of the trajectory: l tr =l r tr( ,tr τtr) Here, length ltr of the

arc of the trajectory is measured from a certain point within the trajectory

The length ltr is expressed in terms of the radius of curvature rtr at a current

trajectory point, and of torsion τtr of the trajectory at that same point

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At the current CC-point, the tool-path width can be expressed in terms of

the length of the ith trajectory:

F ss i F l

ss i tr i

During the infinitesimal time dt, the cutting tool travels along the i-th

tra-jectory at a distance dl th A sculptured surface portion

dStr( )i =F(ss( )i  ⋅l tr( )i dt (10.17)

is generated in this motion of the cutting tool

The area of a single i-th tool-path is

Str i ss i tr i l

n

ss i tr i l

Due to partial interference of the neighboring tool-paths, the total area Str

(Equation 10.19) exceeds the area Ssg of the actual generated part surface P

(i.e., the inequality Str >Ssg is always observed) The rate of interference of

the neighboring tool-paths is evaluated by the coefficient of interference Kint:

K

tr sg sg

ss i tr i

l tr i

int

( ) ( ) [( )

The coefficient of interference Kint is a function of design parameters of the

part surface being machined, of design parameters of the generating surface

of the cutting tool, and of parameters of kinematics of the surface

machin-ing operation So, it can be minimized (Kint→min) using for this purpose

conventional methods of minimization of analytical functions

10.3.2 Solution to the Boundary Problem

The generation of a part surface within the area next to the surface border

differs from that when machining of a boundless surface is investigated The

shape and parameters of the surface contour affect the efficiency of the

sur-face generation process Prior to searching for a solution to the boundary

problem, it is necessary to determine the part surface region within which

the boundary effect is significant

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Consider a sculpture surface P having tolerance [ ] h on the accuracy of the

surface machining The surface of tolerance S[ ]h is at the distance [ ]h from

the surface P The actual machined part surface is located within the interior

between the surfaces P and S[ ]h

The generating surface T of the cutting tool is contacting the nominal part

surface P at a certain point K1 (Figure10.4) The surface T intersects the

sur-face of tolerance S[ ]h The line 1 of intersection of the surfaces S[ ]h and T is

a kind of closed ellipse-like curve The curve has no common points with

the part surface boundary Therefore, no boundary effect is observed in this

location of the cutting tool

At the point K2 within the trajectory of the CC-point, the cutting tool

sur-face T also intersects the sursur-face of tolerance S[ ]h The line 2 of intersection

is also a kind of closed, ellipse-like curve However, in this location of the

cutting tool, the curve 2 makes tangency with the part surface boundary at

the point A This indicates that starting at the point K2, the boundary affects

the efficiency of surface generation The impact of the boundary is getting

stronger toward point D on the part surface boundary curve.

At a certain point K3 of the trajectory of CC-point, the line of intersection

3 of the surfaces S[ ]h and T is not a closed line It intersects the part surface

boundary at the points B and C.

Departure of the cutting tool from the interaction with the surface P is over

when the limit point L on the biggest diameter of the curve 3 reaches the

part surface boundary curve at the point L

The point K2 is constructed for the point A of the part surface boundary

curve For every point A i of the part surface boundary curve, a point K i

that is similar to the point K2 can be constructed All the points K i specify

the limit contour It is necessary to take into account the impact of the

bound-ary effect for those arcs of CC-point trajectories, which are located between

the part surface boundary curve and the limit contour

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Width bc of the part surface boundary affected region is not constant

Width W i at a current point c is measured along the perpendicular to the

part surface boundary curve The point c is the endpoint of the arc ac of the

trajectory of the CC-point The feed rate per tooth F(fr of the cutting tool

could be either constant within the arc ac of the trajectory of the CC-point, or

it can vary in compliance with the current width of the tool-path

Particular features of impact of the boundary effect could be observed:

When the stock thickness is bigger, this causes longer trajectories of the

cutting tool to enter in contact with the part surface

A bigger tolerance [ ]h on accuracy of the machined part surface results

in longer trajectories of the cutting tool to exit from contact with the

part surface

The smaller the area of the nominal part surface P, the more

signifi-cant is the impact of the boundary effect on the efficiency of the

machining of the whole part surface

The impact of the boundary effect could be more significant when

machining long surfaces.

10.3.3 Optimal Location of the Starting Point

The location of the point from which machining of the sculptured surface

begins also affects the resultant surface generation output One can conclude

from this that the optimal location of the starting point exists, and it can be

determined

Consider the machining of a sculptured part surface on a multi-axis NC

machine (Figure 10.5) The boundary of the sculptured surface P is of

arbi-trary shape The region of the boundary effect is shown as the shadowed

strip along the boundary curve

The surface P can be covered by the infinite number of optimal trajectories

of the CC-point The equation of the optimal trajectories of the CC-point is the

output of the subproblem of synthesis of optimal regional surface generation

Two of the infinite number of trajectories are tangent to the surface boundary

curve of the sculptures at the points c1 and c2 (see Figure10.5)

Another two optimal trajectories of the CC-point are at the distance 0 5 F ss

from the points c1 and c2 inward from the bounded portion of the

sculp-tured surface P These two last trajectories of the CC-point can be used as the

trajectories for the actual tool-paths when machining the sculptured surface

P They intersect the sculptured surface boundary curve at the points a1, a2

and a3, a4, respectively The rest of the trajectories of CC-point are at the limit

side-step [ ]F(ss from each other (see Equation 9.46) It is important to point out

here that length f ss of the arc through the points c1 and c2 usually is not

divisible on the limit side-step [ ]F(ss However, no big problem arises in this

concern, and it can be neglected at this point

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Then, outside the bounded portion of the sculptured surface P, two points

A1 and A2 are selected within the trajectory through the points a1 and a2

The point A1 is at the distance l en from the boundary curve of the surface P

The l en distance is sufficient for entering the cutting tool in contact with the

part surface Another point A2 is at the distance lex from the boundary curve

of the surface P The l ex distance is sufficient for exiting the cutting tool from

contact with the part surface Similarly, two more points A3 and A4 are

selected within the trajectory through the points a3 and a4 The machining

of the surface P begins at the point A1

In most cases of sculptured surface machining, the inequality l en>l ex is

observed Therefore, if one wishes to begin the surface machining not from

the point A1, but from the opposite end of the trajectory a a1 2, another four

points A1, A2, A3, and A4 (the points A1, A2, A3, and A4 are not shown in

Figure10.5) can be constructed instead The points A1, A2, A3, and A4 are

constructed in the way the points A1, A2, A3, and A4 are constructed The

only difference here is that for all the points A1, A2, A3, and A4, the arc

seg-ments of the length l en are substituted with the arc segments of the length

l ex, and vice versa

When locating at the point A1 of the trajectory a a1 2, the generating surface

T is contacting the workpiece surface W ps at the point b1 The workpiece

surface W ps is an offset surface at the distance t to the part surface P Here

t designates the thickness of the stock to be removed In the general case, a

function t t U V= ( P, P) is observed (see Chapter 9)

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When locating at the point A2 of the trajectory a a1 2, the generating surface

T is contacting the surface of tolerance S[ ]h at a point b2 The surface of

toler-ance S[ ]h is an offset surface at the distance [ ]h to the part surface P Here

[ ]h denotes the tolerance on accuracy of the machined part surface P In the

general case, a function [ ] [ ](h = h U V P, P) is observed (see Chapter 9)

The distance len that is necessary for entering the cutting tool in contact

with the sculptured part surface can be expressed in terms of thickness of

the stock t, radius of normal curvature R T of the generating surface T of the

cutting tool (here RT is measured in the direction tangent to the trajectory

a a1 2 at the point A1), and radius of curvature Rtr of the trajectory at the

point A1 through the points a1 and a2

The distance lex that is necessary for exiting the cutting tool from contact

with the sculptured part surface can be expressed in terms of tolerance [ ]h

on accuracy of the part surface, radius of normal curvature RT of the

gen-erating surface T of the cutting tool (here RT is measured in the direction

tangent to the trajectory a a1 2 at the point A2), and radius of curvature Rtr of

the trajectory at the point A2 through the points a1 and a2

Ultimately, either one of four points A1, A2, A3, A4 or one of four points

A1, A2, A3, A4 is selected as the starting point of the sculptured surface

machining Practically, both sets of points are equivalent Computation of

coordinates of the chosen point is a trivial mathematical procedure The

interested reader may wish to exercise him- or herself in doing this

Prior to beginning the machining of the given part surface, a contact point

within the generating surface T of the cutting tool is computed This is the

point local geometry of the surface T which corresponds to the local geometry

of the surface P at the point a1 Then, the cutting tool contact point is snapped

with the computed starting point, say with the point A1 Satisfaction of the

conditions of proper part surface generation (see Chapter 7) is required

Much room for investigation is left in the synthesis of optimal global part

surface generation

10.4 Rational Reparameterization of the Part Surface

The solution to the problem of optimal regional synthesis of part surface

generation returns a set of optimal trajectories of the CC-point on the

sur-face P For the purposes of development of a computer program for

sculp-tured surface machining on a multi-axis NC machine, it is convenient to use

the computed optimal trajectories as a set of curvilinear coordinates on the

sculptured surface P For this purpose, it is necessary to change the initial

parameterization of the surface P with a new parameterization — with the

parameterization by means of the optimal trajectories of the CC-point on

the surface P For the reparameterization of the surface P, known methods

[1,7,11,13] and others can be used

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10.4.1 Transformation of Parameters

Consider a part surface P that is given by vector equation r P =rP(U V P, P) It

is assumed that the surface P is a smooth, regular surface The required

addi-tional restrictions that must be imposed will be introduced later

The initial (UP,VP)–parameterization of the part surface can be

trans-formed to another parameterization The new parameterization of the

sur-face P is denoted as ( , U V P P)−parameterization In the new parameters, the

initial equation of the surface P is substituted with the equivalent equation

rP=rP(U V P, P) The new parameters UP and VP can be expressed in terms

of original parameters UP and VP:

U P =U U V P( P, P) V P =V U V P( P, P) (10.21)

One of the curvilinear parameters in Equation (10.21) (for example, UP −

coordinate curve) can be congruent to the optimal trajectories of the

CC-point (see Equation 10.12), while another curvilinear parameter VP can be

directed orthogonally to the first one

Equations for the derivatives in the new parameters are as follows:

P P

U

V U

P P

U

V V

P P

for the part surface P expressed in the new parameters, the Jacobian matrix

of transformation J must not be equal to zero:

U U

U V V U V

P P

P P P P P

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The matrix [D P] of the first derivatives of the surface P in its original

The matrices [D P] and [DP] enable computation of the first fundamental

matrix [Φ1 P. ] in the new parameters of the surface P:

[Φ1 P] [= DP] [T⋅ DP]=J T⋅[D P] [T⋅ D P]⋅ =J J T⋅[Φ1 P]⋅J (10.30)

Similarly, the equation for the second fundamental matrix [Φ2 P. ] in the

new parameters of the surface P:

The rest of the major parameters of geometry in the new parameterization of

the part surface P can be computed on the premises of the above-discussed

equa-tions, particularly on the premises of Equation (10.30) and Equation (10.31)

10.4.2 Transformation of Parameters in Connection

with the Surface Boundary Contour

Boundary contour C of the sculptured surface P is made up of four smooth

arcs C11, C12, C21, and C22 as an example (Figure 10.6) The plane P0 serves as

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are the covariant and contravariant components of the vector of the fictive

displacements The components F k and F k in Equation (10.34) must be

con-structed based on the requirements of one-to-one correspondence between

can be obtained Here, rf( )αi is the position vector of a point M f that is

mapped into the point M0 having position vector r0; rf are the reciprocal

basis vectors at the point M f ; and F1 and F2 are the components of the

vec-tor of the fictive displacements M f, those that can be constructed depend

upon the shape of the region Wf

At every point of the region W0, the constructed functions F i together with

Equation (10.33) yield computation of the following:

The position vector ri:

mapping (see Equation 10.35) In Figure 10.6 they are designated by

= ∇ and e if f = ∇i f F k For the computation of the parameter 2εik f, the

formula 2εik f = ⋅ −r ri k r ri f⋅ k f =e ik f +e ki f +a e e js f ij f ks f is used Ultimately, the

param-eter A ik0j is computed from

A ik0j =a P0jn n ik f, (10.40)

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Here, for the case under consideration, the equalities a011 a a22

0 0

= , a022 a a11

0 0

On the second step, the region Ω0 (see Equation 10.35) is mapped onto the

sculptured part surface P For the mapping, the vectorial equality

The point M0 is the projection of the point M onto the reference plane P0

Therefore, Gaussian coordinates of a certain point M f serve as the Gaussian

coordinates of the point M (see Equation 10.35).

In the region M0, for the basis vectors given by Equation (10.37), as well as

for r0i =a0ikr0k, the following two equations

If the parameters α1 and α2 in Equation (10.33) determine two families of

orthogonal coordinate curves within the plane P0, then not Equation (10.35),

but the equality

( )αi = f( )V +F( )αi f +F( )αi f (10.45)

is used instead Here, r1f and r2f are the unit vectors of the coordinate curves

α1=Const and α2=Const at the point M f

Under such a scenario, not Equation (10.37) through Equation (10.41) are

used, but the relationships

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

(10.53)

used in Equation (10.37) through Equation (10.41) are still valid

Again, in these equations, the following equalities

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For the computations, the expressions (see Equation 10.45) are substituted

into the last equations

10.5 On a Possibility of the Differential Geometry/

Computer-Aided Manufacturing (CAD/CAM) System

for Optimal Sculptured Surface Machining

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

chal-lenging engineering problem Implementation of the D G /K-method of

sur-face generation makes it possible to develop a CAD/CAM system for optimal

sculptured surface machining on a multi-axis NC machine

10.5.1 Major Blocks of the DG/K-Based CAD/CAM System

The proposed concept of the DG/K-based CAD/CAM system for optimal

sculptured surface machining on a multi-axis NC machine is composed of

seven major parts In Figure10.7, the major parts are depicted as blocks of the

DG /K-based CAD/CAM system.

Only data on part surface geometry are used as the input information for

the functioning of the DG/K-based CAD/CAM system.

The synthesis of the optimal machining operation begins from an

ana-lytical description of the sculptured part surface being machined (I)

Start I

III VI

IV V End

Figure 10.7

Principal blocks of the DG/K-based CAD/CAM system for optimal sculptured surface

machin-ing on a multi-axis numerical control machine.

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The initially given representation of the sculptured surface P is

convert-ing to the natural parameterization of the surface P, when the surface P is

expressed in terms of the first Φ1.P and of the second Φ2.P fundamental

forms (see Chapter 1) The converted analytical representation of the

sculp-tured surface is used for solution (II) to the problem of optimal orientation

of the surface P on the worktable of the NC machine (see Chapter 7) It also

yields a solution to the problem of designing the optimal cutting tool (see

Chapter 5) for machining the sculptured surface P (III) For solving the above

problems, the analytical description of the geometry of contact between the

sculptured surface P and between the generating surface T of a cutting tool

is employed (see Chapter 4)

Further, implementation of the analytical description of the geometry of

contact of the surfaces P and T enables computation of optimal parameters of

kinematics of sculptured surface machining Ultimately, this yields a

closed-form solution (IV) to the problem of optimal tool-path generation,

computa-tion of coordinates of the optimal starting point for surface machining, and

verification of satisfaction or violation of the necessary conditions of proper

part surface generation (V) The cutting tool for machining the sculptured

surface can be either designed or it can be chosen (VI) within the available

cutting tools

In particular cases of sculptured surface machining, it is allowable (VII) to

maintain the desired kind of contact of the surfaces P and T, and not make

mandatory their optimal contact

Finally, when the optimization is accomplished, the shortest possible

machining time of sculptured surface machining, as well as the lowest

pos-sible cost of the machining operation can be achieved

More detail about the major blocks of the DG/K-based CAD/CAM system

(see Figure 10.7) are disclosed in the following sections

10.5.2 representation of the input Data

A sculptured surface P to be machined is initially represented either

analyti-cally or discretely For application of the DG/K-method of surface generation,

the initial representation of the surface P is converting to the natural

repre-sentation in terms of the fundamental magnitudes E P , F P , G P of the first

order Φ1.P, and of the fundamental magnitudes L P , M P , N P of the second

order Φ2.P Conversion of the initial surface P representation to its

represen-tation in the natural form is performed by the first block (I) of the CAD/CAM

system (Figure10.7)

For analytically represented surface P, computation of the fundamental

magnitudes of the first Φ1.P and of the second Φ2.P order turns to a routing

mathematical procedure (see Chapter 1) The sequence of the required steps

of computation is as follows (Figure 10.8):

( )1 →( )2 →( )3 →( )4 →( )5 →( )6 →L (10.60)

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In case of discrete representation (7), the surface P is approximated either

by a common analytical function

Fundamental magnitudes of the first and of the second order of the

sur-face P can be determined directly from the specification of the sursur-face P in

discrete form (8) Following this method, the determined partial derivatives

(14) are used as input to

K→( )8 →( )14 →( )5 →( )6 →K (10.63)(The interested reader may wish to go to [11] for more information in this

concern.)

The derived natural representation of the surface P is the output of the first

block (1) of the DG/K-based CAD/CAM system (Figure 10.7) It is used below

as the input to the cutting tool block (III)

10.5.3 Optimal Workpiece Configuration

Computed (6) values of the fundamental magnitudes of the first Φ1.P and of

the second Φ2.P order are used (II) for the computation (15) of parameters of

the optimal orientation of the surface P on the worktable of a multi-axis NC

machine For the optimization, the approach earlier developed by the author

[3,5,11,16,17] is used (see Chapter 7) The computed parameters of optimal

orientation of the workpiece on the worktable of the multi-axis NC machine

are the output (16) of this subsystem of the CAD/CAM system (Figure 10.8)

10.5.4 Optimal Design of the Form-Cutting Tool

Design parameters of the form-cutting tool for the optimal machining of the

surface P on a multi-axis NC machine are computed in the block (III) Again,

the criterion of the optimization is the lowest possible cost of the machining

operation The key problem at that point is to determine the geometry of the

generating surface T of the cutting tool [6,8,14,15].

The equation of the surface T is derived in natural parameterization — that

is, in terms of the fundamental magnitudes E T , F T , G T of the first order Φ1.T,

and the fundamental magnitudes LT , M T , N T of the second order Φ2.T of

the surface T A method for the computation of geometry of the cutting tool

surface T is disclosed in Chapter 5 Shown in Figure10.7, the cutting tool

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