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A mathematical model for a short cut of the NC tool path in window-frame operation
Journal of Materials Processing Technology, 40 (1994) 465-475 Elsevier
465
A mathematical model for a short cut of the NC tool path in window-frame operation R o n g - H o r n g S u n 1 a n d Y i n g - C h i e n Tsai Department of Mechanical Engineering, National Sun Yat-Sen University, Kaohsiung, Taiwan 80424, ROC (Received October 22, 1992; accepted March 31, 1993)
Industrial S u m m a r y A mathematical model is developed in this paper to represent a short cut for generating NC tool paths in window-frame operation. The paths for face milling of flat convex polygon surfaces are considered. The short cut of the tool path is found to be a function of the cutter radius, the position of the starting point, and the magnitude of the vertex angle at the starting point of the path. Combining the concept of a short cut, the length of tool path will be cut down and the machining will be more efficient. An example and some results are adduced to explain and verify the model.
1. I n t r o d u c t i o n T h e m a c h i n i n g t i m e is a p r i m a r y f a c t o r a f f e c t i n g the efficiency of c u t t i n g , h o w to r e d u c e the m a c h i n i n g t i m e a l w a y s b e i n g a n i m p o r t a n t topic in m a c h i n ing. W i n d o w - f r a m e o p e r a t i o n is one of the p a t h - g e n e r a t i n g m e t h o d s for the face milling of c o n v e x p o l y g o n a l s u r f a c e s [1]. In the w i n d o w - f r a m e t y p e of milling process, the c u t t e r s t a r t s a t the p e r i p h e r y of t h e faces, p r o c e e d s s p i r a l l y i n w a r d s a n d t h e n cuts p r o g r e s s i v e l y t o w a r d s the c e n t e r w i t h the p a t h p a r a l l e l to the edges u n t i l the e n t i r e face is m a c h i n e d . F i g u r e 1 shows t h e c o m m o n pathg e n e r a t i n g method. W h e n the c u t m o t i o n of the p r e s e n t cycle h a s b e e n finished, the tool a l w a y s goes b a c k to the s t a r t i n g p o i n t of the cycle a n d t h e n m o v e s i n w a r d s to the s t a r t i n g p o i n t of the n e x t cycle [1-4]. F o r s u c h k i n d of c u t t i n g process, a few s e g m e n t s of the p a t h will be d u p l i c a t e d u n n e c e s s a r i l y . A modified tool p a t h w i t h a s h o r t cut a t the end of e a c h cycle, a s s h o w n in Fig. 2, is derived in this p a p e r to r e d u c e the u n n e c e s s a r i l y d u p l i c a t e d path. T h e c o n c e p t
Correspondence to: Professor Y.C. Tsai, Department of Mechanical Engineering, National Sun Yat-Sen University, Kaohsiung, Taiwan 80424, ROC. 1Also with Department of Mechanical Engineering, Cheng-Shiu Junior College. 0924-0136/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved. SSDI 0924-0136(93) E0047-K
466
R.-H. Sun and Y.-C. Tsai/A mathematical model /
~
J
-~
/i---k"
\.
pa,I,
'J_J_ \ \ \
cutter
•//(2z
/I
A",\\
;
t
.I3
Fig. 1. The tool path generated by the commonly employed approach.
2
materia W ~ [ / ; a ~ ~
cu.tr s,,:_ I
//
~/
path
.,,,\ \
Fig. 2. The tool path generated by the approach with a short cut.
of a short cut will reduce the total length of the tool path and provide a more economical tool-path planning t ha n the existing common path-generating methods. The purpose of this paper is to develop a mathematical model for describing a short cut th at will avoid the unnecessary duplicate tool path, which model could give a detailed calculation of the length of the short cut and find the direction and stop position of the cutter accurately. The model for the first, the middle, and the end cycle of the path has a slight discrepancy and will be discussed in detail. An example and several results are adduced to explain and verify the model.
R.-H. Sun and ¥.-C. Tsai / A mathematical model
467
2. Model c o n s t r u c t i o n As defined in the common path-generating method, the paths are parallel to the edges of the polygon. In order to let the path be as short as possible, the starting point is selected to locate on the periphery of the feasible region that is a sector of a circle, as shown in Fig. 3. The radius of the tool is just big enough to cut the vertex of the polygon. The radius of the cutter is specified as the practical radii of the tool multiplied by a safety factor of 0.9 to ensure a complete cut. The major contents of the model include three parts. Part 1 expounds the general formulae, part 2 clarifies the supplement of the first cycle, and part 3 describes the modification of the end cycle. The detailed narration is given as follows. 2.1. G e n e r a l f o r m u l a e
Specifying the starting point on one of the vertices of a polygon at the periphery of the feasible region, as shown in Fig. 3, the path sc' is a short cut t h a t can replace segments of path sc and cc'. From Asca, the relationship of sc with R, 0, and ~ can be obtained as (sc) sin 0 - R = R sin ( 0 - a)
(1)
where R, 0, and ~ represent the radius of the cutter, the vertex angle, and the angle for indicating the position of the starting point, respectively. Rearranging
material
1,
cutter _
J/
path l
,Y
.,~
s/
.Z" ,,"
_
Y
/
o
o: s
S
Fig. 3. The short cut of the tool path (I).
)<,
R.-H. Sun and Y.-C. Tsai/A mathematical model
468
eqn. (1) yields
sc =
R [1 + s i n ( 0 - a)] sin0
(2)
Let the coordinates of point c and c' be (x~, Yc) and (xe, Ye). Thus,
xe = xc + (cb) cos (0/2) + R cos
(3)
Yc, =Yc + (cb) sin (0/2) + R sin
(4)
where
cb = R/(sin(O/2) )
(5)
The length of cc' can be obtained as
cc'= [(xc,-xc) 2 + (Yc'-Yc)2 ] 1/2
(6)
S u b s t i t u t i n g eqns. (3)-(5) into eqn. (6) and letting xc =Yc = 0 yields
cc'= RIcot2(O/2)+ 2cot(O/2)cosa + 2sina + 211/2
(7)
Similarly, the segment of p a t h sc' can be obtained as
(8)
SC'= [(X c, --SC) 2 + yc,2] 1/2
Substituting eqns. (2)-(5) into eqn. (8) yields
sc'=R
[(
cot(0/2)+cos~
l+sin(O-a) sinO
j
+(l+sin~)271/2
(9)
From eqns. (2), (7) and (9), it is clear t h a t path segments sc, cc' and the short cut sc' are all functions of R, 0, and • only, and are u n c o n c e r n e d with the edge number of the polygon. T h a t is, if R, 0, and ~ of the polygon are specified, paths sc', sc and cc' will be unique.
2.2. Supplement of the first cycle For the case shown in Fig. 4, the tool starts from point c, cuts t h r o u g h the first cycle, moves back to point s, and t h e n goes directly toward the starting point, c', of the second cycle: the area shown by oblique lines won't be cut. Comparing Fig. 4 with Fig. 3, two cases are distinguished, in t h a t the point e locates inside or outside the material. For 0 < (~/2), if point e locates outside the material, none of area will be left behind and line segment cd should be
R.-H. Sun and Y.-C. Tsai / A mathematical model
469
smaller t h a n or equal to R. From Fig. 3, cd can be obtained as R sin cd = sin (½~- 0)
(10)
Further, if cd > R, see Fig. 5, or 0 ~>(1t/2), see Fig. 6, then there exists a piece of uncut area. However, the above cases occur only on the first cycle. In order to secure a complete cut, two methods are given to modify such cases. Method 1: Instead of moving from point s directly to point c', a supplemental path is derived for the tool to move from point s to point s' then return to point s, and then go straight to point c', where ss' = s c - 1 2 R cosal
(11)
Method 2: Along the direction of the cutter extends a line segment to a position ahead of the original path from the starting point c. A different length is obtained for different cases, that can be categorized into three types. The equations for calculating the length of the extended path segments, denoted by l, and their applicable condition are obtained as follows: Type 1, for 0 < (~/2) and cd > R, from Fig. 4, l should be equal to line segment el, where l = ( c d - R) tan (½~ - 0)
(12)
and line segment cd is obtained from eqn. (10).
. \
Ltb,.x
pam
,tts S .,':,Z/ \
/ ~11
/
Fig. 4. The short cut of the too] path (II).
\
~- " I
470
R.-H. Sun and Y.-C. Tsai/A mathematical model
path material
~
'~'
~
cutter ~
X
Fig. 5. The short cut of the tool path (III).
t
\
n~i%
~
/i
f',
\
\\
Fig. 6. The short cut of the tool path (IV). T y p e 2, for 0~>@/2), a~<(~/2), f r o m Fig. 5, l = e f = f g + e g . l = ch + eg, w h e r e
As f g = c h ,
then
Rsin~ ch = . sin (~ - - 0 )
(13)
R eg = t a n (~ -
(14) 0)
R.-H. Sun and Y.-C. Tsai/A mathematical model
471
Combining eqn. (13) with eqn. (14) yields
cos(n-O)-sina s~(n(~-----~ )
l= R (
(15)
Type 3, for 0/> (~/2), ~ > @/2), from Fig. 6, l= c h - b h . From Aach
ch = R sin (0 - a)
(16)
sin (~ - O) From Ahbi,
bh= R s i n ( ~ - O - fl)
(17)
sin0 where
f l = s i n - l [ ( h i ~ inO]
(18)
and hi = a i - ah, where
ai = s c - I2R cos ~1
(19)
R sin sin (~ - 0)
(20)
ah = .
In general, method 1 is easier, but method 2 lets the cutter move more smoothly. If the objective is aimed at determining which method enables the shortest extended path to be obtained, the simplest manner is to compare the length of them, and then the shortest path will be found.
2.3. Modification in the end cycle Acab in Fig. 2 shows the end cycle of the tool path. For the common cutting approach, the cutter will move from point c through point a and point b, and finally go back to point c. In other words, point c is the end point of the tool path. However, as shown in the figure, the cutter just needs to stop at point s, and the material will be cut completely. Therefore, the dashed line segment sc is duplicated and can be reduced. Because the path will not move forward, point c' is not existent. Thus, the length of short cut sc' will be equal to zero, and the decision of the duplicate path sc is in accordance with the length of path segment ca, ab and bc. It is derived as follows,
f sc if sc < bc bc if bc <,sc < ab + bc sc = ab + bc if ab + bc <,sc < ca + ab + bc 0
if sc>~ca+ab+bc
(21)
R.-H. Sun and Y.-C. Tsai/A mathematical model
472
3. V e r i f i c a t i o n a n d d i s c u s s i o n
C o n s i d e r the t r i a n g u l a r flat s u r f a c e 123 s h o w n in Fig. 7. The edges of the t r i a n g l e are ll = 1 0 cm, /2 =8.7939 cm, a n d / 3 =6.527 cm. The v e r t e x angles are 0 1 = 6 0 ° , 0 2 = 4 0 ° , 0 3 = 8 0 ° . L e t the r a d i u s of the c u t t e r be R = I cm. S e v e r a l r e s u l t s are discussed, as follows: (1) F i g u r e 8 shows the s h o r t cut and the d u p l i c a t e p a t h s for v a r i a t i o n a l a n g l e ~. Evidently, the l a r g e r the a n g l e ~, the g r e a t e r the l e n g t h of the s h o r t
39cm
I-
6.527 cm
- I
Fig. 7. V e r i f i c a t i o n e x a m p l e for t h e s h o r t c u t .
lO 9
g
duplicate ..................
~
7
~
5
~
4
,.,1
short
cut
3 2
...................... , ................. , . .....................................................
1
20
i
i
i
30
40
50
° ...............
60
0t ( d e g . )
Fig. 8. T h e s h o r t c u t a n d t h e d u p l i c a t e p a t h s o f t h e e x a m p l e for v a r y i n g v a l u e s o f a n g l e ~.
R.-H. Sun and Y.-C. Tsai / A mathematical model
473
cut. Contrarily, the length of the duplicate path, line segments sc plus cc', as shown in Fig. 3, is decreasing gradually and so is the reduction ratio, as shown in Fig. 9, defined as reduction r a t i o -
duplicate p a t h - s h o r t cut 100% duplicate path
(22)
Because of the variation in the end cycle, the length of the duplicate paths for angle ~ from 50 ° to 55 ° appears to be increased. (2) Figure 10 shows the comparison of the total length for the paths with or without a short cut w.r.t, variational angle ~. The shortest path without a short
8O
70
60
50 20
30
40
50
60
((leg.)
Fig. 9. The reduction ratio of the tool path for v a r y i n g values of angle ~.
34
without short cut with short cut
32
......... e~
30
's az
28 26
o
24 22 20
i
30
i
40 a (deg.)
i
50
60
Fig. 10. The influence of the short cut on the total length of the tool path for v a r y i n g values o f a n g l e ~.
474
R.-H. Sun and Y.-C. Tsai/A mathematical model
cut is o b t a i n e d w h e n ~ = 50 °, but t h a t w i t h a s h o r t cut is o b t a i n e d w h e n a = 43~: t h e y are not c o i n c i d e n t for the v a r i a t i o n of the s h o r t cut. Also an i n f e r e n c e is g i v e n t h a t the h i g h e s t r e d u c t i o n r a t i o is not n e c e s s a r i l y o b t a i n e d for the p a t h with the s h o r t e s t length. (3) T h e s u p p l e m e n t a l p a t h s derived by m e t h o d 1 a n d m e t h o d 2 a r e c o m p a r e d in Fig. 11. A n g l e a l o c a t e d b e t w e e n 20 ° ~ 30 ° is the case s h o w n in Fig. 3, and t h a t b e t w e e n 30 ° ~ 6 0 ° is the case s h o w n in Fig. 4, w h e r e the l e n g t h by m e t h o d 1 is s h o r t e r t h a n t h a t by m e t h o d 2 and the length is i n c r e a s i n g g r a d u a l l y . (4) Specifying a n g l e a to be 50 °, the c o r r e s p o n d e n c e of the s h o r t cut a n d the d u p l i c a t e p a t h s w.r.t, the v a r i a t i o n a l r a d i u s of the c u t t e r is s h o w n in Fig. 12. Clearly, the c u r v e s are divided into two r e g i o n s by R = 1.5 cm, the r e a s o n b e i n g t h a t w h e n R > 1.5 cm the c u t t i n g cycles will be r e d u c e d f r o m two to one in this
0.5 0.4
method 1
£
j~,~.J
................... method2
0.3
/.7/-"
,,,,,,1
0.2
/
w
0.1 • 0.0 2O
f
f
30
J
~
40
50
60
0~ ( d e g . )
Fig. 11. The comparison of supplement paths by method 1 and method 2.
8
6 ~
0
~ ~
1
duplicate shortcut
2
3
radius of cutter (cm)
Fig. 12. The short cut and duplicate paths of the example for varying values of the cutter.
R.-H. Sun and Y.-C. Tsai/A mathematical model
475
example. If the first cycle is simultaneously the end cycle, the short cut just remains part of the supplement. Therefore, the length is cut down, evidently. 4. C o n c l u s i o n s
A mathematical model representing a short cut of tool path for the face milling of fiat convex polygonal surfaces by window-frame type operation has been developed, the model including the equations of the general formula, the supplement of the first cycle, and the modification of the end cycle. Using the model, the length of the short cut can be calculated very easily. A more efficient tool-path planning will be made by combining the concept of a short cut. References
[1] H.P. Wang, T.C. Chang, R.A. Wysk and A. Chandawarkar, ASME J. Eng. Ind., 109 (1987) 370. [2] I.H. Kral, Numerical Control Programming in APT, Prentice-Hall Inc., Englewood Cliffs, NJ, 1986. [3] C.H. Chang and M.A. Melkanoff, NC Machine Programming and Software Design, Prentice-Hall Inc., Englewood Cliffs, NJ, 1989. [4] W.S. Seames, Computer Numerical Control: Concept and Programming, Delmar Publishers Inc., Albany, NY, 1990.
465
A mathematical model for a short cut of the NC tool path in window-frame operation R o n g - H o r n g S u n 1 a n d Y i n g - C h i e n Tsai Department of Mechanical Engineering, National Sun Yat-Sen University, Kaohsiung, Taiwan 80424, ROC (Received October 22, 1992; accepted March 31, 1993)
Industrial S u m m a r y A mathematical model is developed in this paper to represent a short cut for generating NC tool paths in window-frame operation. The paths for face milling of flat convex polygon surfaces are considered. The short cut of the tool path is found to be a function of the cutter radius, the position of the starting point, and the magnitude of the vertex angle at the starting point of the path. Combining the concept of a short cut, the length of tool path will be cut down and the machining will be more efficient. An example and some results are adduced to explain and verify the model.
1. I n t r o d u c t i o n T h e m a c h i n i n g t i m e is a p r i m a r y f a c t o r a f f e c t i n g the efficiency of c u t t i n g , h o w to r e d u c e the m a c h i n i n g t i m e a l w a y s b e i n g a n i m p o r t a n t topic in m a c h i n ing. W i n d o w - f r a m e o p e r a t i o n is one of the p a t h - g e n e r a t i n g m e t h o d s for the face milling of c o n v e x p o l y g o n a l s u r f a c e s [1]. In the w i n d o w - f r a m e t y p e of milling process, the c u t t e r s t a r t s a t the p e r i p h e r y of t h e faces, p r o c e e d s s p i r a l l y i n w a r d s a n d t h e n cuts p r o g r e s s i v e l y t o w a r d s the c e n t e r w i t h the p a t h p a r a l l e l to the edges u n t i l the e n t i r e face is m a c h i n e d . F i g u r e 1 shows t h e c o m m o n pathg e n e r a t i n g method. W h e n the c u t m o t i o n of the p r e s e n t cycle h a s b e e n finished, the tool a l w a y s goes b a c k to the s t a r t i n g p o i n t of the cycle a n d t h e n m o v e s i n w a r d s to the s t a r t i n g p o i n t of the n e x t cycle [1-4]. F o r s u c h k i n d of c u t t i n g process, a few s e g m e n t s of the p a t h will be d u p l i c a t e d u n n e c e s s a r i l y . A modified tool p a t h w i t h a s h o r t cut a t the end of e a c h cycle, a s s h o w n in Fig. 2, is derived in this p a p e r to r e d u c e the u n n e c e s s a r i l y d u p l i c a t e d path. T h e c o n c e p t
Correspondence to: Professor Y.C. Tsai, Department of Mechanical Engineering, National Sun Yat-Sen University, Kaohsiung, Taiwan 80424, ROC. 1Also with Department of Mechanical Engineering, Cheng-Shiu Junior College. 0924-0136/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved. SSDI 0924-0136(93) E0047-K
466
R.-H. Sun and Y.-C. Tsai/A mathematical model /
~
J
-~
/i---k"
\.
pa,I,
'J_J_ \ \ \
cutter
•//(2z
/I
A",\\
;
t
.I3
Fig. 1. The tool path generated by the commonly employed approach.
2
materia W ~ [ / ; a ~ ~
cu.tr s,,:_ I
//
~/
path
.,,,\ \
Fig. 2. The tool path generated by the approach with a short cut.
of a short cut will reduce the total length of the tool path and provide a more economical tool-path planning t ha n the existing common path-generating methods. The purpose of this paper is to develop a mathematical model for describing a short cut th at will avoid the unnecessary duplicate tool path, which model could give a detailed calculation of the length of the short cut and find the direction and stop position of the cutter accurately. The model for the first, the middle, and the end cycle of the path has a slight discrepancy and will be discussed in detail. An example and several results are adduced to explain and verify the model.
R.-H. Sun and ¥.-C. Tsai / A mathematical model
467
2. Model c o n s t r u c t i o n As defined in the common path-generating method, the paths are parallel to the edges of the polygon. In order to let the path be as short as possible, the starting point is selected to locate on the periphery of the feasible region that is a sector of a circle, as shown in Fig. 3. The radius of the tool is just big enough to cut the vertex of the polygon. The radius of the cutter is specified as the practical radii of the tool multiplied by a safety factor of 0.9 to ensure a complete cut. The major contents of the model include three parts. Part 1 expounds the general formulae, part 2 clarifies the supplement of the first cycle, and part 3 describes the modification of the end cycle. The detailed narration is given as follows. 2.1. G e n e r a l f o r m u l a e
Specifying the starting point on one of the vertices of a polygon at the periphery of the feasible region, as shown in Fig. 3, the path sc' is a short cut t h a t can replace segments of path sc and cc'. From Asca, the relationship of sc with R, 0, and ~ can be obtained as (sc) sin 0 - R = R sin ( 0 - a)
(1)
where R, 0, and ~ represent the radius of the cutter, the vertex angle, and the angle for indicating the position of the starting point, respectively. Rearranging
material
1,
cutter _
J/
path l
,Y
.,~
s/
.Z" ,,"
_
Y
/
o
o: s
S
Fig. 3. The short cut of the tool path (I).
)<,
R.-H. Sun and Y.-C. Tsai/A mathematical model
468
eqn. (1) yields
sc =
R [1 + s i n ( 0 - a)] sin0
(2)
Let the coordinates of point c and c' be (x~, Yc) and (xe, Ye). Thus,
xe = xc + (cb) cos (0/2) + R cos
(3)
Yc, =Yc + (cb) sin (0/2) + R sin
(4)
where
cb = R/(sin(O/2) )
(5)
The length of cc' can be obtained as
cc'= [(xc,-xc) 2 + (Yc'-Yc)2 ] 1/2
(6)
S u b s t i t u t i n g eqns. (3)-(5) into eqn. (6) and letting xc =Yc = 0 yields
cc'= RIcot2(O/2)+ 2cot(O/2)cosa + 2sina + 211/2
(7)
Similarly, the segment of p a t h sc' can be obtained as
(8)
SC'= [(X c, --SC) 2 + yc,2] 1/2
Substituting eqns. (2)-(5) into eqn. (8) yields
sc'=R
[(
cot(0/2)+cos~
l+sin(O-a) sinO
j
+(l+sin~)271/2
(9)
From eqns. (2), (7) and (9), it is clear t h a t path segments sc, cc' and the short cut sc' are all functions of R, 0, and • only, and are u n c o n c e r n e d with the edge number of the polygon. T h a t is, if R, 0, and ~ of the polygon are specified, paths sc', sc and cc' will be unique.
2.2. Supplement of the first cycle For the case shown in Fig. 4, the tool starts from point c, cuts t h r o u g h the first cycle, moves back to point s, and t h e n goes directly toward the starting point, c', of the second cycle: the area shown by oblique lines won't be cut. Comparing Fig. 4 with Fig. 3, two cases are distinguished, in t h a t the point e locates inside or outside the material. For 0 < (~/2), if point e locates outside the material, none of area will be left behind and line segment cd should be
R.-H. Sun and Y.-C. Tsai / A mathematical model
469
smaller t h a n or equal to R. From Fig. 3, cd can be obtained as R sin cd = sin (½~- 0)
(10)
Further, if cd > R, see Fig. 5, or 0 ~>(1t/2), see Fig. 6, then there exists a piece of uncut area. However, the above cases occur only on the first cycle. In order to secure a complete cut, two methods are given to modify such cases. Method 1: Instead of moving from point s directly to point c', a supplemental path is derived for the tool to move from point s to point s' then return to point s, and then go straight to point c', where ss' = s c - 1 2 R cosal
(11)
Method 2: Along the direction of the cutter extends a line segment to a position ahead of the original path from the starting point c. A different length is obtained for different cases, that can be categorized into three types. The equations for calculating the length of the extended path segments, denoted by l, and their applicable condition are obtained as follows: Type 1, for 0 < (~/2) and cd > R, from Fig. 4, l should be equal to line segment el, where l = ( c d - R) tan (½~ - 0)
(12)
and line segment cd is obtained from eqn. (10).
. \
Ltb,.x
pam
,tts S .,':,Z/ \
/ ~11
/
Fig. 4. The short cut of the too] path (II).
\
~- " I
470
R.-H. Sun and Y.-C. Tsai/A mathematical model
path material
~
'~'
~
cutter ~
X
Fig. 5. The short cut of the tool path (III).
t
\
n~i%
~
/i
f',
\
\\
Fig. 6. The short cut of the tool path (IV). T y p e 2, for 0~>@/2), a~<(~/2), f r o m Fig. 5, l = e f = f g + e g . l = ch + eg, w h e r e
As f g = c h ,
then
Rsin~ ch = . sin (~ - - 0 )
(13)
R eg = t a n (~ -
(14) 0)
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Combining eqn. (13) with eqn. (14) yields
cos(n-O)-sina s~(n(~-----~ )
l= R (
(15)
Type 3, for 0/> (~/2), ~ > @/2), from Fig. 6, l= c h - b h . From Aach
ch = R sin (0 - a)
(16)
sin (~ - O) From Ahbi,
bh= R s i n ( ~ - O - fl)
(17)
sin0 where
f l = s i n - l [ ( h i ~ inO]
(18)
and hi = a i - ah, where
ai = s c - I2R cos ~1
(19)
R sin sin (~ - 0)
(20)
ah = .
In general, method 1 is easier, but method 2 lets the cutter move more smoothly. If the objective is aimed at determining which method enables the shortest extended path to be obtained, the simplest manner is to compare the length of them, and then the shortest path will be found.
2.3. Modification in the end cycle Acab in Fig. 2 shows the end cycle of the tool path. For the common cutting approach, the cutter will move from point c through point a and point b, and finally go back to point c. In other words, point c is the end point of the tool path. However, as shown in the figure, the cutter just needs to stop at point s, and the material will be cut completely. Therefore, the dashed line segment sc is duplicated and can be reduced. Because the path will not move forward, point c' is not existent. Thus, the length of short cut sc' will be equal to zero, and the decision of the duplicate path sc is in accordance with the length of path segment ca, ab and bc. It is derived as follows,
f sc if sc < bc bc if bc <,sc < ab + bc sc = ab + bc if ab + bc <,sc < ca + ab + bc 0
if sc>~ca+ab+bc
(21)
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3. V e r i f i c a t i o n a n d d i s c u s s i o n
C o n s i d e r the t r i a n g u l a r flat s u r f a c e 123 s h o w n in Fig. 7. The edges of the t r i a n g l e are ll = 1 0 cm, /2 =8.7939 cm, a n d / 3 =6.527 cm. The v e r t e x angles are 0 1 = 6 0 ° , 0 2 = 4 0 ° , 0 3 = 8 0 ° . L e t the r a d i u s of the c u t t e r be R = I cm. S e v e r a l r e s u l t s are discussed, as follows: (1) F i g u r e 8 shows the s h o r t cut and the d u p l i c a t e p a t h s for v a r i a t i o n a l a n g l e ~. Evidently, the l a r g e r the a n g l e ~, the g r e a t e r the l e n g t h of the s h o r t
39cm
I-
6.527 cm
- I
Fig. 7. V e r i f i c a t i o n e x a m p l e for t h e s h o r t c u t .
lO 9
g
duplicate ..................
~
7
~
5
~
4
,.,1
short
cut
3 2
...................... , ................. , . .....................................................
1
20
i
i
i
30
40
50
° ...............
60
0t ( d e g . )
Fig. 8. T h e s h o r t c u t a n d t h e d u p l i c a t e p a t h s o f t h e e x a m p l e for v a r y i n g v a l u e s o f a n g l e ~.
R.-H. Sun and Y.-C. Tsai / A mathematical model
473
cut. Contrarily, the length of the duplicate path, line segments sc plus cc', as shown in Fig. 3, is decreasing gradually and so is the reduction ratio, as shown in Fig. 9, defined as reduction r a t i o -
duplicate p a t h - s h o r t cut 100% duplicate path
(22)
Because of the variation in the end cycle, the length of the duplicate paths for angle ~ from 50 ° to 55 ° appears to be increased. (2) Figure 10 shows the comparison of the total length for the paths with or without a short cut w.r.t, variational angle ~. The shortest path without a short
8O
70
60
50 20
30
40
50
60
((leg.)
Fig. 9. The reduction ratio of the tool path for v a r y i n g values of angle ~.
34
without short cut with short cut
32
......... e~
30
's az
28 26
o
24 22 20
i
30
i
40 a (deg.)
i
50
60
Fig. 10. The influence of the short cut on the total length of the tool path for v a r y i n g values o f a n g l e ~.
474
R.-H. Sun and Y.-C. Tsai/A mathematical model
cut is o b t a i n e d w h e n ~ = 50 °, but t h a t w i t h a s h o r t cut is o b t a i n e d w h e n a = 43~: t h e y are not c o i n c i d e n t for the v a r i a t i o n of the s h o r t cut. Also an i n f e r e n c e is g i v e n t h a t the h i g h e s t r e d u c t i o n r a t i o is not n e c e s s a r i l y o b t a i n e d for the p a t h with the s h o r t e s t length. (3) T h e s u p p l e m e n t a l p a t h s derived by m e t h o d 1 a n d m e t h o d 2 a r e c o m p a r e d in Fig. 11. A n g l e a l o c a t e d b e t w e e n 20 ° ~ 30 ° is the case s h o w n in Fig. 3, and t h a t b e t w e e n 30 ° ~ 6 0 ° is the case s h o w n in Fig. 4, w h e r e the l e n g t h by m e t h o d 1 is s h o r t e r t h a n t h a t by m e t h o d 2 and the length is i n c r e a s i n g g r a d u a l l y . (4) Specifying a n g l e a to be 50 °, the c o r r e s p o n d e n c e of the s h o r t cut a n d the d u p l i c a t e p a t h s w.r.t, the v a r i a t i o n a l r a d i u s of the c u t t e r is s h o w n in Fig. 12. Clearly, the c u r v e s are divided into two r e g i o n s by R = 1.5 cm, the r e a s o n b e i n g t h a t w h e n R > 1.5 cm the c u t t i n g cycles will be r e d u c e d f r o m two to one in this
0.5 0.4
method 1
£
j~,~.J
................... method2
0.3
/.7/-"
,,,,,,1
0.2
/
w
0.1 • 0.0 2O
f
f
30
J
~
40
50
60
0~ ( d e g . )
Fig. 11. The comparison of supplement paths by method 1 and method 2.
8
6 ~
0
~ ~
1
duplicate shortcut
2
3
radius of cutter (cm)
Fig. 12. The short cut and duplicate paths of the example for varying values of the cutter.
R.-H. Sun and Y.-C. Tsai/A mathematical model
475
example. If the first cycle is simultaneously the end cycle, the short cut just remains part of the supplement. Therefore, the length is cut down, evidently. 4. C o n c l u s i o n s
A mathematical model representing a short cut of tool path for the face milling of fiat convex polygonal surfaces by window-frame type operation has been developed, the model including the equations of the general formula, the supplement of the first cycle, and the modification of the end cycle. Using the model, the length of the short cut can be calculated very easily. A more efficient tool-path planning will be made by combining the concept of a short cut. References
[1] H.P. Wang, T.C. Chang, R.A. Wysk and A. Chandawarkar, ASME J. Eng. Ind., 109 (1987) 370. [2] I.H. Kral, Numerical Control Programming in APT, Prentice-Hall Inc., Englewood Cliffs, NJ, 1986. [3] C.H. Chang and M.A. Melkanoff, NC Machine Programming and Software Design, Prentice-Hall Inc., Englewood Cliffs, NJ, 1989. [4] W.S. Seames, Computer Numerical Control: Concept and Programming, Delmar Publishers Inc., Albany, NY, 1990.