Formulation of Shape Generation Processes of Machine Tools

CopYTight © IFAC Control Science and Technology (8th Triennial World Congress) Kyoto. Japan. 1981

FORMULATION OF SHAPE GENERATION PROCESSES OF MACHINE TOOLS N. Sugimura*, K. Iwata* and F. Oba** -Faculty of Engineering, Kobe University, N ada, Kobe, Japan - -College of Engineering, University of Osaka Prefecture, Sakai, J apan

Abstract. A description model for the shape generation processes of machine tools is proposed, which enables us to realize a CAD system for machine tools. The proposed model plays a key role to determine the structures of machine tools at the functional design stage. The model is a description of the relative motion between the workpiece and the cutting edge which has a specified geometry. The shape generation processes are classified, and related to the resulting surfaces, based on the model proposed here. Keywords . Computer-aided design; functional design; machine tools; machining; models.

specified geometry, and the workpiece .

INTRODUCTION In promoting automation of CAM systems in mach i ne shops , machine tools are being required to have more flexible functions in order to adapt to various kinds of machinary parts and products. The objective of this research is to establish a systematic procedure for designing adequate machine tools which satisfy specified functions.

Constructing a Model of the Relative Motion between the Cutting Edge and the Workpiece Cons i der two rectangular coordinate systems: 0w- xwYwzw which has the origine 0w' and xw'

y

W

There are a few research works concerning systematic procedures for machine tool design (Salje , 1973, 1980; Ito, 1979; Saito, 1980) . This paper , as the first step of this research, deals with a mathematical model which describes the shape generation processes of machi ne tools . The model proposed here plays a key role to determine the structures of machine tools at the functional design stage in a CAD system for machine tools. The shape generation process is defined as the relative motion between the workpiece and the cutting edge which has a specified geometry. The model is then simplified by excluding redundancy in parameters, to describe the relative motion . The shape generation processes are classified, and related to the resulting surfaces, based on the model proposed here.

W

°c - xc yc z c

fixed on the cutting edge(hereafter referred to as O(C)) . The point P , which constructs c the cutting edge, is given by a position vector x =(x y z)T of O(C), where T c c' c ' c stands for the transpose of matrix. The point P is also identified by a position vector

X:=(xw'

Yw' zw)T of O(W). The relationship

between Xc and Xw is expressed as follows:

w wc Xc + bwc

X =A

(1)

where, A = raU:~ is a 3 by 3 orthogonal wc C'l"jJ matrix , which gives the rotation of the axis of O(C) against O(W). The indices i =1,2 and 3 of a element aU:~ correspond to the x , y and 'l"J W W Z - axes of O(W) , and j =1 , 2 and 3 correspond W

to the Xc' Yc and z - axes of O(C), respecc wc tively. The j - th column vector of A ' a j = wc WC wc ( a , a , awc)T , is the direction cosine of 3j 2j 1j j - axis of O(C) with respect to o(w).

CONSTRUCTING A MODEL OF THE SHAPE GENERATION PROCESSES The shape of a part is formed through the relative motion between the cutting tool and the workpiece in a machining process . The shape generation processes of machine tools , therefore, can be described by the relative motion between the cutting edge, which has a

wC bWC b~c) T is the vector O'b wc =(b l ' 2 ' " wc

b

The relative motion between the cutting edge and the workpiece is given by the following

2035 eS T" _ G O

and z - axis fixed on the workpiece(here-

after referred to as O(W)) , and

2036

N. Sugimura, K. Iwata and F. Oba

equation.

CUTTII'G

xw(t)=Awc (t)x c+ bwc (t)

(2)

The relative motion in a machining process is, in general, composed of two kinds of motions, namely the cutting motion and the feed motion. The cutting edge forms the trace on the workpiece, such as a straight line or a circular cylinder, depending on the cutting edge geometry and the mode of cutting motion (linear or revolutionary). The trace of the cutting edge obtained here is called a virtual generating surface. The feed motion translates and/or modifies the virtual generating surface to generate the specified surface. Typical cutting motions, feed motions and virtual generating surfaces are shown in Fig. 1. (a) is the case where a straight line is generated by the cutting motion and a plane surface is obtained by the feed motion. (b) corresponds to the case where a circle is generated by the cutting motion and a circular cylinder is obtained by the feed motion. (c) corresponds to the case where a circular cylinder is generated by the cutting motion and two rectangular plane surfaces are obtained by the feed motion. The cutting motion and feed motion can be described by using four rectangular coordinate systems as shown in Fig. 2. The cutting motion in this figure is defined as the motion of O(Q) against O(P). The motion of O(C) against O(Q) is the feed motion of the tool side and the motion of O(P) against O(W) is the feed motion of the work side. The coordinate systems of typical machine tools are shown in Fig. 3.

@ T~~.

/{)T1~

CUTTING

..

~CUTTING I{)TI~

"+~'*--;~ FEED

~,~

/{)T1~

VIRnJAL GEl£RATlNG SlJIFACE

4--

(a)

VIRnJAL GENERATlI'G ~FACE

(b)

(c)

Fig. 1. Typical cutting motions, feed motions and virtual generating surfaces.

ClJITER)

O(C)

~

It

,

~Q) ~ r/~ Oq z,.

o
/t--

Yp

'. ,

\

zp

SIW'ER

(b)

tOll ZlM"Al TYPE MACHIN II'G CENTER

Fig. 3. Coordinate systems of typical machine tools.

Description of the Cutting Motion. Two types of cutting motions are considered; linear cutting motion and revolutionary cutting motion. The linear cutting motion is given by the linear motion of O(Q) against O(P), b (t), and represented as follows: pq

xp (t)=x q+ bpq (t) where, Xq is a point fixed on O(Q), and Xp(t) is the same point identified on O(p). The y -axis of O(Q) is set in parallel to the

q

yp -axis of O(P) and the linear cutting motion is taken in the direction of the y -axis. The p cutting motion, bpq(t) , is given as follows: b (t)=(O, b P q (t), O)T (4) 2 pq In the revolutionary case, the cutting motion is given by the following representation:

(5)

The coordinate system O(Q) is so set that the z -axis is in parallel to the z -axis of O(P)

>re

Xq

(a)

x (t)=A (t)x + e p pq q pq

'.

Zc\

~Tlrtl

FEED MJT I 00

1

rF TOOl SIDE

q

"1

CUTTING

I'DTI~

J

1

Fern IllTION

I

I

P

q

and the axis of cutting motion is taken along the z -axis. In this case, A (t) is given as

r>F "OlK SIDf

O~W)' Zw J Xw / (\Oij(pIECE) Yw

Fig. 2. Coordinate systems.

pq

follows; A (t)= ~cosCP(t) -sincp(t) 0] pq sincp(t) coscp(t) 0

o

°

(6)

1

where, CP(t)is the angle of rotation, and

_ Dq

epq-(~l

'

Dq

Dq

~2 ' ~3)

~

is the position of

o , q

which is the origine of O(Q), in O(P) Description of the Feed Motion of the Work ~. Let us consider the description of the feed motion. The feed motion of the work side, which is given by the relative motion of O(P) to O(W) in Fig. 2, is represented as follows:

Formulation of Shape Generation Processes substituting 1/1 0 for

(t)(x + b (t)) (7) UJp UJpV P P where, AUJpo=[aijO] is the initial clamping x (t) =A OA

w

tool,

A

wpv

(t) = [a~V (t)l

'J

1-J

is the initial angle between the z - axis and

q

e

the rotary feed

(b~(t), b~(t), b~(t))T the

set of linear feed motions. At a time t =O, A v(O) =E, where E is the unit matrix,

UJp

,vO

and bp(O) =bpO= (~l '

tl 0 tl0T 2'

1/1 (t)in Eq . (9), where 1/1 0

the z - axis. r 0 is also given in the z - x

attitude of the workpiece on the machine motion and bp(t) =

2037

1

q

(10)

3

b

(t) =(bqe(t) 0 1 "

b

(O) =b

qe

For example, the shaper shown in fig. 3(a) has a two - axis linear feed motion, b (t) =

q

The linear feed motion of the tool side, b (t), is given as follows: qe

qe

3)

e

plane , i. e . , r eO=( r eO , 0 , reO)T

qeO

=(b qeO 1

bqe(t))T 3 '

0

,

(11)

bqeO)T at a timp. t =O ;3

In the case of the shaper and the horizontal type machining center shown in Fig . 3(a) and (b) , there are no feed motions of the tool side because the coordinate systems O(Q) and As for the horizontal type machining center shown in Fig. 3(b) , there is a three- axis linear O(C) are identical . The lathe shown in Fig. 3(c) has the feed motions of the tool side feed motio'n, bp(t) = (~(t), ~(t), b~(t))T, consisting of a two - axis linear feed motion and a rotary feed motion around the Yw- axis and a rotary feed motion in the z - x plane . q q which is given by Description of the Relative Notion of the A (t) =[ cos8(t) 0 Sin8(t)] Cutting Edge to the Workpiece. As a result, UJpv 0 1 0 the relative motion of the cutting edge to - sin8(t) 0 cos8(t) the workpiece i s obtained by composing the where, 8(t) is the angle of rotation . In the cutting mot i on , the feed motion of the work case of the lathe shown in Fig . 3(c) , there side and the feed motion of the tool side. In are no feed motions of the work side because the case of the linear cutting motion, the the coordinate systems O(W) and O(P) are relat ive motion is represented as fol l ows by identical. composing Eqs. (3) , (7) and (8).

(b~(t),

0,

~(t))T,

and no rotary

fe~d

motions.

Descri ption of the Feed Motion of the Tool Side . The feed motion of the tool side i s given by the relative mot i on of O(C) to O(Q) . Consider a case where the feed motion of the tool side consists of the trans l ation of a point U and the rotary mot i on of the cutting edge around the poi nt U as shown in Fig . 3(c) . The feed motion is given by the fo llowing equation, X (t) =A (t)(A OX + r 0) + b (t) (8)

q

qev

qe e

e

qe

where , A 0 is the in i tial cl ampi ng att i tude

qe

of the cutting edge on the machine tool . reO is the vector DOe l ength of which is the radius of the rotary feed motion of the cutting edge around the point U. In Eq . (8), there are no feed motions in the direction of the cutting mot i on . Thus , the feed motion is consider ed only in the perpendicular directions to the cutting motion . The cutting motion is in the direction of the Y -

q

axis as shown in Eq . (4) and Eq . (6) fo r both cases of linear and revol ut i onary cutt i ng motions. Therefore , the feed motion of the tool side is defined in the z - x plane . q q

X (t) =A OA

w

UJp UJpv (t){Aqev (t)(Aqe OX e + r e 0) + bqe (t)

+ b

pq

(t) + b (t)} p

(12)

In the above equation , the y - component p of t he l inear feed motion of the work side ,

~(t), is cancelled by replacing

bp(t) = (~(t) , ~(t) , ~(t))T b~(t) = (~(t) ,

0,

~(t))T

with

and replacing

(t) = (O , tl q (t) , O)T with 2 pq b;q(t)=(O , ~q(t) + b~(t), O)T

b

In the case of the revolutionary cutting motion , the re l at i ve motion of the cutting edge to the workpiece is represented as fo llows by composing Eqs . (5), (7) and (8) . X (t) =A OA

w

wp UJpv (t)[Apq (t){Aqev (t)(Aqe OX e + r e 0) + b (t)} + e + b (t)] (13) qe pq p

Integr ation of the Feed Motions

The rotary feed motion of the tool side is given as follows for the case where the y axis is parallel to the y - axis, e

The feed motions in Eq . ( 12) or (13) can be i nt e grated between the tool side and work side .

A

Fir st , l et us consider the case of l inear cutting mot i on . In Eq. (12) , the rotary feed mot i on, A A (t)A (t)A O, composed of the

q

qev

(t) =[ cosjJ (t) 0 s i ll/! (t)]

0

1

0

- sinj! (t) 0 cosjJ (t )

where , 1/1 (t) is the angle of rotation and 1/1(0) =0 at a time t =O. A 0 is obtai ned by

qe

UJpO UJpv

qev

qe

mot i ons of the tool side and work side can be rewritten as f ol lows.

2038

N. Su gimura , K. Iwat a and F . Oba (14)

AwpOAwpv(t)Aqev(t)Aqeo=A!roA!rV(t) where , Al

=A

A

wpO wpO qeO

A!rv(t) =

A;~oAwpv(t)Aqev(t)Aqeo

pq

where , - 1 stands for the inverse of the matrix .

Al 0 and Al (t) are hereafter regarded as wp wpv the initial clamping attitude of the workpiece and the rotary feed motion of the work side , respectively . It follows that

A!rv(O) =A;~oAwpv(O)Aqev(O)Aqeo=E

bl(t) is hereafter regarded as the linear p

feed motion of the work side . In the above equation , it follows that the Y - componen~ of l p b (t) is equal to zero from the forms of p l' Aqeo ' Aqev(t) , r eO ' bqe(t) and bp(t) in Eqs . (9) , (10) , (ll) and (12) . (t) and bl(t) introduced

p

above , Eq . (12) is rewritten as follows: X (t) =Al OAl

w

wp wpv

(t)( x + bl (t) + bl(t))

e

pq

p

(16)

The effect of the cutt i ng motion in the above equation is invariant i . e ., the following equation holds .

l Alwp OAlwpv (t) bpq (t) , =Awp OAwpv (t) bpq (t) l

pq

qe

pq

p

(17)

(19)

The above equation can be shown by use of the corresponding terms in Eqs . (6) , (7) , (11) and (18) . The relative motion between the cutting edge and the workpiece given by Eq. (13) , can be rewritten as follows:

w(t) =Awp OAwpv (t)[Apq (t){Aqcv (t)(Aqe OX e+ r e 0)

X

qe

The linear feed motions of the work side and the tool side in Eq. (12) can be rewritten as follows: b l (t) =A- 1 r 0+ A- 1 A- 1 (t)( b (t) + bl(t)) (15) p qe 0 e qe 0 qev qe p

wpv

p

+ bP (t)} + bP(t)]

wpv (O) =Aqev (O) =E .

wp

qe

from

A

By using Al 0 ' Al

the l inear feed motion of the work side . Through the above transformation , the total effect of the linear feed motion is invariant , i . e ., the following relation holds . P A (t) b (t)+ bP(t) =A (t) b (t) + e + b (t)

Now , let velocity ly large the feed

(20)

p

us consider the case where the of the cutting motion is sufficient in comparison with the velocity of motions in Eqs . (16) and (20). In Eq .

(16) , the feed motions , Al

wpv

(t) and bl(t) , p

are taken to be constant for one stroke of the linear cutting motion , bl (t) .

pq

The trace

+ b l (t) , forms a pq e virtual generating surface in one stroke of the cutting motion . The specified surface is formed by giving the feed motion to the virtual generating surface . Simi l arly , in Eq . (20) , the feed motions , of the cutting edge ,

X

(t) , A (t) , bP (t) and bP(t) , are A WPV qev qe p taken to be constant for one revolution of the cutting motion , A (t) . The virtual gen-

pq

er ating surface is given by the trace of the cutting edge in one revolution of the cutting

pq (t)(Aqe OX e + r e 0

motion , and represented by A

+ bP (0)) at a initial time , t =O. It is mod-

qe

The above equation is easily derived from Eqs . (9) , (12) and (14).

ified by the feed motions of the tool side ,

Next , consider the case of the revolutionar y cutting motion . In Eq . (13) , the rotary motion of the cutting edge , A OA (t)A (t)A (t)A 0 ' is composed of

motions of the work side , A

the feed motion of the work side , the cutting motion and the feed motion of the tool side . It is , therefore , clear that the rotary feed motions of the work side and tool side can not be integrated without the influence on the cutting motion . On the other hand , the linear feed motion , A (t) b (t) + e + b (t) ,

The shape generation process is determined by the feed motion and the virtual generating surface which is given by the cutting motion of the cutting edge . The geometry of the cutting edge is an essential factor for the shape generation process . The description model of the cutting edge is given in the following.

wp wpv

pq

qev

pq

qe

qe

·

pq

p

can be integrated through the following trans formation . bP (t) =(bqe(t)

qe

1

"

0

O)T

b~(t) , b~(t))T (18) q , ~(t) + ~q , ~(t) + ~q+ bje(t))T

b;(t) = (bjP(t) ,

= (~(t) + bP (t)

qe

ei

is hereafter regarded as the linear

feed motion of the tool side , and bP(t) as p

Aqev(t) and b~e(t) ' and shifted by the feed wpv

(t) and bP(t) .

p

Constructing a Model for the Cutting Tool Geometry

The geometry of the cutting edge can be defined in a perpendicular plane to the direction of the cutting motion since the virtual generating surface is determined by the motion of the cutting edge along the cutting motion . Consider the case where the Ye- axis of the coordinate system O(C) fixed on the cutting edge is taken to be parallel to the Y - axis , wh i ch is the direction of the cut-

q

ting motion . The geometry of the cutting

Formulation of Shape Gene ration Processes edge is defined as a point or a line in the zc- xc plane as shown in Fig . 4. The hatched area in the figure corresponds to the interference area of the cutter , the tool holder , etc .

Zp

Yp

(b) STRAIGHT

(O) F'OINT

ClITTIJ'{; EroE

ClITTIJ'{; EroE

(C)FaHIJ

CLASSIFICATION AND ANALYSIS OF THE SHAPE GENERATION PROCESSES

Classification of the Virtual Generating Surfaces In the case of linear cutting motion, there are three kinds of virtual generating surfaces shown in Table 2 corresponding to the three kinds of geometries of cutting edge shown in Fig . 4.

Wo,

i}," m," W y,

zp

p (b)-IIjI,,=O

Yp

Zp (bl-24Jo= Tt/2

pq

(t) (A

qc

OX + r 0+ bl' (0)).

c

c

qc

Yp

Zp

(b)-4 I!\,=-Tt/2

Zp

(b)-S - Tt <%<0

4Jo~-Tt/2

( I tmRFEREI«:E I 5 I N CUTS I DE )

(b)

p

STRAIGIT OJTTI"; EtX;E

w"

~

(C)-I

Zp

Yp

1jI,,=0

Zp

'(C)- 2
INTERFERE~E ( INSIDE (C)

IS IN)

qc

c

qc

ment of A 0' is essential to characterize

qc

the virtual generating surface . As a result, eleven typical kinds of virtual generating surfaces shown in Fig. 5 are considered here for the combination of the geometry of cutting edge and the value of 1jJ O. Classification of the Feed Motions Feed motions are, in general, composed of the those of the tool side and those of the work side . In the case of linear cutting motion given in Eq . (16) , however , there are no feed motions on the tool side . While, in the case of revolutionary cutting motion in Eq . (20) , there are feed motions on both sides. In Eq . (20) , the feed motion of the tool side, bl' (t) , varies the diameter of

qc

the virtual generating surface as shown in Fig . 6(a) . The feed motion of the tool side , A (t), modifies the virtual generating

qcv

surface as shown in Fig . 6(b). In these

YP

Zp

(INTERFERENCE IS IN) OUTSIDE

FOfl"ED ClfTTtrlG frx;E

~~t)l~ '

p

~

factors which determine the virtual generatever, it can be seen from Eqs. (9), (10) and (18) that the factor './I 0' which is the ele-

tiJ"

(<) - 3
Fig. 5. Virtual generating surfaces for the case of revolutionary cutting motion .

Thus, the

ing surface are A 0' r 0 and bl' (0) . How-


$" ~"

In the case of revolutionary cutting motion, the virtual generating surface is given by A

Zp

(b)-J o<
(ItmRFEREI«:E IS IN INSIDE)

Yp

The shape generation process of machine tool is characterized by the virtual generating surfaces and the feed motions . Therefore , the classification of the virtual generating surfaces and the feed motions are utilized to classify the shape generation processes .

YP Zp (0)- 3 4Jo=-Tt/2

POINT 0JTl1J'{; EroE

ClITTIJ'{; EroE

Fig. 4. Description model of cutting tool geometry .

Zp

(0)-2 4Jo = Tt/2

(0)-1 1jI,,=0

IJ )

Zc

.'"

-,; ~'~;.:~

Yp

ItmRFEROCE AAfA

Zc

2039

p _

•

!zp/

p

Yp -z~ lo)L1 tEM FEED M:lTi~

b~1tl

Ib) ROTARY FEED M:lTl~

4J1tl

Ic) L1tEM FEED f1)T1~

b~ t)

Fig . 6. Modification of virtual generating surfaces due to feed motions of Type I. cases , bl' (t) and A

qc

qcv

(t) are determined by

bjc(t) and 1jJ(t) from Eqs. (18) and (9) , respectively . Fig . 6(c) shows the feed motion along z - axis , bjP(t), which is the z -comp p l' ponent of b (t) in Eq . (18). These three p kinds of feed motions do not move the axis of the cutting motion , z - axis . These are p referred to as feed motions of type I . The combination of these feed motions yields seven feed motions of Type I shown in Table 1. TABLE 1

:-.m:

Feed Motions of Type I 11 12 13 14 15 16 17

FEED ~(t)

1I NEAR FEED b'~t) '")TION J ROTARY FEED

I1OTlON

."m

0

0 0 0 0 0 0 0 0 0 0 0

N. Sugimur a , K. Iwat a and F. Oba

20 40

On the other hand , all the feed motions of

CONCLUSION

the work side except the z - component of bP(t) p

A description model for the shape generation processes of machine tools is proposed which enables us to realize a CAD systems for machine tools.

p

move the axis of cutting motion , which is given by y - axis for the case of linear p cutting motion, and z -axis for the revolup

tionary one. These are referred to as feed motions of Type II. Seven kinds of feed motions of Type II gi ven in Table 2 , II 1 through II 7, show the typical moving pat terns of the axis of the cutting motion .

The model is a description of the relative motion betweer. t he workpiece and the cutting edge wh i ch has a specified geomet r y . The shape generation processes are classified and r e l ated t o the resulting surfaces based on the model pr oposed here .

Shape Generation processes and Result ing Surfaces

REFERENCE

In the case of linear cutting motion , the shape generati on processes are determined by the combination of the virtual generating surfaces and the feed motions of Type II. Table 2 shows the resulting surfaces which are generated by these combinations .

Salje , E., J . Bockem , H. Dopcke , K. Puttkarnrner and W. Redeker ( 1973) . Eine Systematik fur Re l ativbewegungen bei spanender Bear be i tung in Abhangigkeit von Werkstuck und We r kzeug . ZwF , 68 , 404- 408 . Sal je, E. and W. Redeker (1980) . Design of machine tools (cutting machines) by means o f computer aided design . Proc. of 4th I CPE , 386- 393 .

As for the case of revolutionar y cutting motion , the shape generation processes are determined by the combinat i on of the v i rtual generatin g surfaces and the feed motions of Type I and Type II . Table 3 shows the resulting surfaces which are generated by combinations of the feed motions of Type I and Type II for a typical virtual gene r ating surface . In Tabl es 2 and 3, I ~ and II ~ mean t hat the corresponding feed motions a r e not used . Symbol * means the complicated free formed surfaces .

I to , Y. and Y. Yoshida (1979). Design conception of hierarchical modular const ruction - manufacturi ng different kinds of machine tools by using common modul es . Proc. of 19th Inte r . MTDR Conf ., 147- 153 . Saito , Y., Y. Ito and T. Ot suka (1980) . Automat is i erte Darstellung von Entwurfs ze i chnungen fur Werkzeug machinenKonstruktionen . ZwF , 12, 492- 495 .

It is possibl e to design the machine tool s with specified functions , based on the shape generation processes for the spec i f i ed s urfaces.

TABLE 2 VIRTUAL fiEMERATIN6 UAFACE

FEED JIIIOTI

OF TYPE D

Shape Generation Processes and Resul ting Surfaces (Linear Cutting Motion)

.~~ ~ z.

~ z.

PI..NE SURfACE

CYLI NOLI CAL SURfACE

Y.

Zp

Yp

D~

Dl~

m, D2/~

D~

ITS

06

7

•et»

~A:~

;~-

TABLE 3

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