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An Analysis of Conical Drill Point Grinding — the Generation Process and Effects of Setting Errors
An Analysis of Conical Drill Point Grinding - the Generation Process and Effects of Setting Errors J. D. Wright, Engineer, Government Aircraft Factory, Melbourne; E. J. A. Armarego (1). University of Melbourne/Australia
The f i n a l d r i l l p o i n t geometry g e n e r a t i o n p r o c e s s f o r g e n e r a l p'urpose t w i s t d r i l l s u s i n g t h c p o p u l a r c o n i c a l g r i n d i n g method i s d e s c r i b e d and m o d e l l e d . Based on t h e " i d e a l " c a s e a n a l y s i s pre*--iously r e p o r t e d , t h e fundamental c l e m e n t s of t h i s p r o c e s s a r e i d e n t i f i e d and a n a l y s e d . Far t h i s " i d e a l " ( d e s i g n ) c a s e , t h e g r i n d e r d e s c r i b e d c m producc a l l t h e recommended s p e c i f i e d d r i l l p o i n t f e a t u r e v a l u e s f o r a r a n g e of g r i n d i r g c o n e a n g l e s . The i n t r o d u c t i o n of g r i n d e r s e t t i n g a c v i a t i o n s from t h e " i d e a l " c a s e ( s e t t i n g e r r o r s ) s e v e r e l y c o m p l i c a t e s t h e a n a l y s i s and t h e g e n e r a t e d curved l i p d r i l l p o i n t s h a p e . N e w d e f i n i t i o n s f o r t h e geom e t r i c a l f e a t u r e s and t h e " a c c e p t a b l e g r i n d e r c r i t e r i a " had t o b e e s t a b l i s h e d . The e f f e c t s o f d e v i a t i o n s i n s i x g r i n d e r s e t t i n g s and two d r i l l f l u t e - f e a t u r e s have been s t u d i e d f o r 8 0 g e n e r a l p u r p o s e c o m b i n a t i o n s o f d r i l l p o i n t f e a t u r e v a l u e s . I t is shown t h a t even w i t h s m a l l d e v i a t i o n s , u n a c c e p t a b l e d r i l l shapes occur f o r 26 combinations. Fcr t h e remaining 54 combinations e s s e n t i a l l y s t r a i g h t d r i l l l i p s a r e o b t a i n e d a l t h o u g h t h e r a n g e s between t h e d e s i g n and g e n e r a t e d f e a t u r e s can d i f f e r s u b s t a n t i a l l y . The dominant s e t t i n g e r r o r s and t h c s u s c e p t i b l e g e n e r a t e d f e a t u r e s a r e i d e n t i f i e d . T h i s s t u d y h i g h l i g h t s t h e d i f f i c u l t i e s ic a c h i e v i n g and c o n t r o l l i n g t h e s p e c i f i e d geometry i n p r a c t i c e . INTRODUCTION The g e n e r a l g e o n e t r y o f t h e c o n v e n t i o n a l t w i s t d r i l l h a s been d e s c r i b e d i n numerous t e x t b o o k s and handbooks E1-31. C o n s i d e r a b l e i n t e r n a t i o n a l agreement on t h e n o m e n c l a t u r e and s p e c i f i c a t i o n o f t h e s a l i e n t g e o m e t r i c a l f e a t u r e s of t w i s t d r i l l s i s a l s o e v i d e n t i n handbooks and S t a n d a r d s [2-71. F u r t h e r , i t h a s been r e c o g n i s e d t h a t t h e geometry a t t h e d r i l l p o i n t , where c u t t i n g o c c u r s , a f f e c t s t h e i m p o r t a n t machining p e r formance c h a r a c t e r i s t i c s such a s f o r c e s , power and d r i l l l i f e . Through d e c a d e s of developn.ent and e x p e r i m e n t a t i o n , recommendations f o r t h e v a l u e s o f s p e c i f i e d d r i l l p o i n t f e a t u r e s f o r optimum d r i l l p e r formance when machining d i f f e r e n t work m a t e r i a l s have been g i v e n i n handbooks [ 2 , 3 ] . Recommendations f o r g e n e r a l p u r p o s e d r i l l s , r e p r e s e n t i n g a compromise optimum geometry f o r machining a r a n g e of common w o r k m a t e r i a l s have a l s o been q u o t e d [ 2 , 3 , 7 ] and compared [81. D e s p i t e t h e a c c u m u l a t e d knowledge of t w i s t d r i l l s t h e r e is r e l a t i v e l y l i t t l e d e t a i l e d information about t h e p r e c i s e geometry of t h e d r i l l p o i n t and t h e a s s o c i a t e d s h a r p e n i n g method. There i s e v i d e n c e t o s u g g e s t t h a t t h e c o n i c a l g r i n d i n g method i s commonly used i n p r a c t i c e [ 9 ] a l t h o u g h o t h e r methods have been n o t e d i n t h e l i t e r a t u r e [lo-121. I n a d d i t i o n t h e ' a s measured' g e o m e t r i c a l v a r i a b i l i t y o f n o m i n a l l y i d e n t i c a l manufa c t u r e d d r i l l s h a s been shown t o b e e x c e s s i v e [13-161 and c o n t r i b u t e s s i g n i f i c a n t l y t o t h e s c a t t e r i n t h e d r i l l i n g f o r c e s [ 1 4 , 1 5 ] and d r i l l l i f e [1G,17]. The s e l e c t i o n of t h e d r i l l p o i n t s h a r p e n i n g method, t h e d r i l l p o i n t f e a t u r e v a l u e s and t h e q u a l i t y c o n t r o l procedures f o r g e n e r a l purpose d r i l l s h a s been t h e r e s p o n s i b i l i t y of t h e d r i l l m a n u f a c t u r e r s . In recent y e a r s , r e s e a r c h w o r k e r s have d e v e l o p e d a keen i n t e r e s t i n t h e s e i m p o r t a n t a s p e c t s o f t w i s t d r i l l s . A number of i n v e s t i g a t i o n s o f t h e d r i l l g e o m e t r i c a l v a r i a b i l i t y o b t a i n e d d u r i n g manufacture have h i g h l i g h t e d t h e s o u r c e s of v a r i a t i o n s and t h e s i g n i f i c a n t r e d u c t i o n s i n v a r i a b i l i t y p o s s i b l e t h r o u g h improved q u a l i t y A s e r i e s o f s t u d i e s of c o n t r o l p r o c e d u r e s [18-20]. t h e p o p u l a r c o n i c a l and some o t h e r d r i l l p o i n t s h a r p e n i n g methods have a l s o been made i n an a t t e m p t t o a r r i v e a t a n optimum d r i l l p o i n t geometry [13,16,21-231. The i n v e s t i g a t i o n s o f d r i l l p o i n t s h a r p e n i n ? methods c a r r i e d o u t a t t h e U n i v e r s i t y o f Melbourne have been aimed a t o b t a i n i n g a d e e p e r u n d e r s t a n d i n g of t h e g e n e r a l p u r p o s e d r i l l geometry and p l a u s i b l e d r i l l p o i n t g r i n d e r s capable of achieving a d e s i r e d optimal geometry w i t h a view t o improve t h e v a r i a b i l i t y of n o m i n a l l y i d e n t i c a l d r i l l s [24-271. I n t h i s paper an a n a l y t i c a l i n v e s t i g a t i o n of t h e f i n a l d r i l l p o i n t geometry g e n e r a t i o n p r o c e s s f o r g e n e r a l p u r p o s e d r i l l s i s p r e s e n t e d . The a n a l y s i s i s a f u r t h e r development of e a r l i e r s t u d i e s b a s e d on t h e " i d e a l " c o n i c a l g r i n d i n g method. D e t a i l e d a n a l y s e s of g r i n d i n g methods have shown t h e c o n i c a l g r i n d i n g method t o be most p r o m i s i n g a s a means o f g e n e r a t i n g t h e reconmended r a n g e o f d r i l l p o i n t f e a t u r e v a l u e s f o r g e n e r a l p u r p o s e d r i l l s [26]. In t h i s present a n a l y s i s t h e g e n e r a t e d s h a p e and d r i l l p o i n t f e a t u r e v a l u e s w i l l be considered both i n t h e absence ( i . e . ' i d e a l ' c a s e ) and p r e s e n c e o f g r i n d e r s e t t i n g errors.
-FINAL
s e t t i n g of t h e d r i l l i n a g r i n d e r ; t h e g r i n d e r a c t i o n d u r i n g which t h e i n i t i a l geometry i s p r o g r e s s i v e l y removed; and t h e s e l e c t i o n o f an end p o i n t a t which t h e d e s i r e d f i n a l d r i l l p o i n t geometry i s g e n e r a t e d ( i . e . " i d e a l " f i n a l l o c a t i o n ) . For e a c h of t h e s e e l e m e n t s o f t h e g e n e r a t i o n p r o c e s s , m a t h e m a t i c a l r e l a t i o n s h i p s need to be e s t a b l i s h e d which l i n k t h e g r i n d e r s e t t i n g p a r a meters t o t h e g e n e r a t e d d r i l l p o i n t g e o m e t r i c f e a t u r e s . The common a p p r o a c h i n e a r l i e r a n a l y t i c a l s t u d i e s [ 1 3 , 21,22,24-261 h a s been t o d e v e l o p t h e s e t y p e s o f mathe m a t i c a l r e l a t i o n s h i p s when t h e d r i l l i s p o s i t i o n e d i n t h e " i d e a l " f i n a l - l o c a t i o n and t h e i n i t i a l geometry h a s been f u l l y removed. These a n a l y s e s have p r o v i d e d i m p o r t a n t i n s i g h t s i n t o t h e p o t e n t i a l of t h e g r i n d i n g methods a s a s s e s s e d by " a c c e p t a b l e g r i n d e r c r i t e r i a " [2G]. Thus t h e y r e p r e s e n t an e s s e n t i a l s t a r t i n g p o i n t for t h e broader a n a l y s i s i n c o r p o r a t i n g a l l elements o f the generation process. The g e n e r a l geometry o f t h e c o n i c a l g r i n d i n g method wh'en t h e d r i l l i s i n t h e ' i d e a l ' f i n a l l o c a t i o n i s shown i n F i g . 1. I n a d d i t i o n t o assuming t h a t t h e i n i t i a l geometry i s f u l l y removed, i t h a s a l s o been assumed [ 2 4 , 2 6 ] t h a t : !a) The d r i l l geometry i s s y m m e t r i c a l a b o u t t h e d r i l l axis.
T C* 1
SECTION A-A
DRILL P O I N T GEOMETRY GENERATION_
PROCESS
-
IDEAL CASE
I n developing an a n a l y s i s of t h e f i n a l d r i l l p o i n t geometry g e n e r a t i o n p r o c e s s , i t i s n e c e s s a r y t o cons i d e r t h e g e o m e t r i c a l i n t e r a c t i o n between t h e d r i l l and t h e g r i n d i n g s u r f a c e . From a p r a c t i c a l p o i n t of view, t h i s i n v o l v e s f o u r major e l e m e n t s t h e s e b e i n g t h e s e l e c t i o n o f a n i n i t i a l d r i l l p o i n t geometry: t h e
Annals of the ClRP Vol. 32/1/1983
1
( b ) The p o r t i o n o f t h e f l u t e s u r f a c e which f o r m s each l i p is a "ruled" s u r f a c e (developable annular helecoid). ( c ) The d i s t a n c e b e t w e e n e a c h s t r a i g h t l i n e f l u t e g e n e r a t o r i n ( b ) and t h e d r i l l a x i s is h a l f t h e web t h i c k n e s s (W) a n d t h e a c u t e a n g l e b e t w e e n t h e skew g e n e r a t o r and t h e d r i l l a x i s i s h a l f t h e point angle ( p ) . F o r t h i s g r i n d i n g method, t h e d r i l l f l a n k s a r e gene r a t e d by two r i g h t c i r c u l a r g r i n d i n g c o n e s s y m m e t r i c a l l y l o c a t e d a b o u t t h e d r i l l a x i s w i t h e a c h c o n e gene r a t i n g one flank. T h e c o n e a n d d r i l l axes l i e i n p a r a l l e l p l a n e s w i t h t h e c o n e a x e s d i s p l a c e d by a d i s t a n c e C from t h e d r i l l a x i s . The a c u t e a n g l e s b e t ween t h g skew d r i l l and c o n e a x e s a r e e a c h y w h i l e t h e s e m i - c o n e a n g l e f o r e a c h cone i s b. I f t h e generated l i p is t o be s t r a i g h t , t h e ' i d e a l ' f i n a l location of t h e d r i l l must be such t h a t a s t r a i g h t l i n e g e n e r a t o r o f e a c h f l u t e ( t a n g e n t i a l t o t h e d r i l l core d i a m e t e r ) is coincident w i t h a corresponding s t r a i g h t l i n e cone generator. The a n g l e between t h e p r o j e c t i o n s o f t h e cone g e n e r a t o r t h r o u g h t h e l i p and t h e c o n e a x i s i n t h e p l a n e normal t o t h e d r i l l a x i s i s A . The cone a p i c e s , a t o r i g i n s 0 a n d 0 are d i s p l a c e d by C , C a n d C w i t h r e s p e c t to o r i g i n 0 on t h e d r i l l a x r s as shdwn i n F i g . 1. A number o f t h e g r i n d i n g p a r a meters ? , ~ , A , V , C ~ , , C z a r e i n t e r r e l a t e d by [ 2 4 ] ,
Y
tanv =
sinxcosxsec'e-
g'sec2i
(cos2i(sec'1:-1) + s i n ' x s e c 2
(1-cos? ysec29 )
Cx = C tanA
Y
-
,2
(1) (2)
W/cos;i
.
and Cz = C t a n v (3) Y Thus from t h e g e o m e t r y i n F i g . 1 and t h e a b o v e e q u a t ions, four grinding parameters a r e s u f f i c i e n t to f u l l y d e s c r i b e t h e geometry w i t h t h e d r i l l i n t h e ' i d e a l ' f i n a l l o c a t i o n [26]. A s w i l l be n o t e d l a t e r , t h e f o u r g r i n d i n g p a r a m e t e r s o , x , i and Cx p r o v i d e a u s e f u l practical selection. T h e e q u a t i o n s f o r g r i n d i n g c o n e 1 w i t h respect t o t h e o r i g i n 0 o n t h e d r i l l a x i s h a s b e e n e x p r e s s e d as ~241
(4) A similar e q u a t i o n f o r cone 2 c a n also h e found [24]. The e q u a t i o n s f o r t h e l i p s ( f l u t e g e n e r a t o r s tang e n t i a l t o t h e d r i l l cone d i a m e t e r ) p r o j e c t e d i n t h e x-y p l a n e a r e : for Lip 1 : for Lip 2 :
x = y tani
- W/COSA
x = y t a n i + W/COSA
The r e l e v a n t p o r t i o n o f t h e c h i s e l e d g e e q u a t i o n i n t h e x-y p l a n e c a n b c f o u n d by s G l v i n g t h e two c o n e e q u a t i o n s w i t h r e s p e c t t o o r i g i n 0 on t h e d r i l l a x i s f r o m which [ 2 4 ] i
, = sin-l(w/r) A t the outer corner,
(11)
r = D/2 and
.
The r e m a i n i n g e l e m e n t s o f t h e g e n e r a t i o n p r o c e s s need t o b e c o n s i d e r e d i n o r d e r t o p h y s i c a l l y p o s i t i o n t h e d r i l l i n t h e ' i d e a l ' f i n a l l o c a t i o n shown i n F i g . 1 and t h u s a c h i e v e t h e d e m o n s t r a t e d p o t e n t i a l o f t h i s g r i n d i n q method f o r g e n e r a l p u r p o s e d r i l l s h a r p e n i n g . The g r i n d e r shown i n F i g . 2 , h a s b e e n d e s i g n e d f o r t h i s purpose. I t is noted t h a t t h e g r i n d e r allows f o r t h e f a c t t h a t t h e i n i t i a l s e t t i n g s w i l l o c c u r when t h e d r i l l i s d i s p l a c e d from i t s f i n a l l o c a t i o n w i t h r e s p e c t to t h e g r i n d i n g s u r f a c e . T h u s , a l t h o u g h o n l y f o u r g r i n d i n g p a r a m e t e r s are n e e d e d t o d e s c r i b e t h e g e o m e t r y i n t h e " i d e a l " f i n a l l o c a t i o n , more g r i n d e r s e t t i n g p a r a m e t e r s may b e n e e d e d t o a c h i e v e t h i s condition. N e v e r t h e l e s s , a n a t t e m p t h a s b e e n made t o maximize t h e u t i l i z a t i o n o f t h e g r i n d i n g p a r a m e t e r s i n t h e above a n a l y s i s . For example, t h e g r i n d e r s e t t i n g p a r a m e t e r s 9 a n d <: are d i r e c t p h y s i c a l r e p r e s e n t a t i o n o f t h e cor-res o n d i n g r i n d i n g p a r a m e t e r s i n t h e a n a l y s i s . S i m i l a r P y by m J i n g t h e v e e g r o o v e
* o, coneapex
F o r t h e c u r v e d c h i s e l e d g e , the c h i s e l e d g e a n g l e is b a s e d on t h e t a n g e n t t o t h e c u r v e i n t h e x-y p l a n e a t x=y=O, t h u s f r o m F i g . 1 $ = 1800
tan$' =
-
-
A
-
(7)
$ 1
[ s i n 2 9 - s i n x c o s x L i n 2 ci - ( cx/c
) ( cod
-cog
(CX/C )( c o s ~ e - c o s 7 x )
3 ( 8)
Y
The r e m a i n i n g s p e c i f i e d d r i l l p o i n t f e a t u r e s , i . e . t h e p o i n t a n g l e 2 p a n d l i p c l e a r a n c e a n g l e C P are that
also r e l a t e d to t h e g r i n d i n g parameters [24] cospcosx and
+
-
sinpsinxcos~
20
(9)
cose = 0
ab"G, bc-cp
ac = cq
t a n c s = -cos ( & # - A ) [ t a n A + t a n ( ~ - 1 )( c o s 2 x - s i n 7 Xtan2 9 )
-tanvtan(w-i)sinycosxsec20] ( s i n 2 x -cos2 q, t a n 2 o ) - s i n x c o s X s e c 2 w h e r e t h e web a n g l e w is d e f i n e d a s i[ t a n "
2
D
I
( 10)
= .,,
E q u a t i o n s (1) t o (11) p r o v i d e t h e m a t h e m a t i c a l r e l a t i o n s h i p s b e t w e e n t h e g r i n d i n q p a r a m e t e r s and t h e commonly s p e c i f i e d d r i l l p o i n t f e a t u r e s 2 p , Cv , ' v , During p o i n t sharpening t h e t h r e e s p e c i f i e d 2W a n d D . f e a t u r e s 2 p . C! a n d ':i a r e g e n c r a t e d on :he f l u t e d A s noted earlier body w i t h k n o w n ' f e a t u r e s 2W and D . a n d e v i d e n t from E q u a t i o n s ( 1 ) t o (11) t h e f o u r nece s s a r y g r i n d i n g p a r a m e t e r s s u c h a s '1, Y , and C h a v e t o b e f o u n d f r o m t h e t h r e e e q u a t i o n s i n v o l v i n g t#e g e n e r a t e d f e a t u r e s 2 p , Ci a n d I ( E q u a t i o n s ( 7 ) , (9) a n d (10). Thus tine d r i l l O g e o m e t r y i s u n d e r s p e c i f i e d a l t h o u g h a l l t h e d r i l l p o i n t f e a t u r e s c a n b e independe n t l y s e l e c t e d a n d g e n e r a t e d by t h i s g r i n d i n g method p r o v i d e d one g r i n d i n g p a r a m e t e r s u c h a s '4 i s p r e s e l e c t e d [24,26]. F u r t h e r it h a s previously been shown t h a t t h e c o n d i t i o n $1 .. '< m u s t a p p l y t o e n s u r e t h a t t h e d r i l l r e m a i n s w i t h i n t h e g r i n d i n g c o n e and hence can b e f u l l y ground [24]. I n a more r e c e n t work by t h e a u t h o r s [26] a c o m p r e h e n s i v e s e t o f " a c c e p t a b l e g r i n d e r criteria" have been e s t a b l i s h e d I t was and a p p l i e d to t h e c o n i c a l . g r i n d i n g m e t h o d . f o u n d t h a t a t a l l c o m b i n a t i o n s o f t h e recommended d r i l l p o i n t f e a t u r e values f o r g e n e r a l purpose d r i l l s , t h e d e s i r e d g e o m e t r y was g e n e r a t e d a n d t h e a v a i l a b l e range w i t h i n which ( ' u ) c o u l d b e s e l e c t e d was o n l y m a r g i n a l l y r e s t r i c t e d [26].
$
where
, so
that the l i p c l e a r a n c e a n q l e C: c a n b e f o u n d f?om E q u a t i o n (101 by u s i n y , f r o m EquatTon (11). ,:
FIGURE 2
i n which t.he d r i l l i s h e l d s y n m e t r i c a l w i t h r e s p e c t t o t h e p l a n e p a r a l l e l t o t h e d r i l l and g r i n d i n g cone a x i s , t h e g r i n d i n g p a r a m e t e r C can b e d i r e c t l y set on t h e g r i n d e r ( F i g . 2 ) . The d i H e c t U S E of t h e g r i n d i n g p a r a meters C z and C a s g c i n d e r s e t t i n g p a r a m e t e r s p r e s e n t problems s i n c e These a r e d i s t a n c e s between o r i g i n s 0: and 0 i n s p a c e when t h e d r i l l i s i n t h e ' i d c a l ' f i n a l l o c a t i o n . By a n t . i c i p a t i n g t h e f i n a l l o c a t i o n o f t h e o u t e r c o r n e r of t h e d r i l l w i t h r e s p e c t t o t h e p i v o t p o i n t on t h e g r i n d e r a t t a c h m e n t , t h e s e t t i n g p a r a m e t e r d i s t a n c e L a l o n g t h e d r i l l a x i s can he e s t a b l i s h e d . T'ne d i s t a n g e normal t o t h e g r i n d i n g wheel s u r f a c e L may b e used t o d e s c r i b e t h e r e l a t i o n s h i p between t h g g r i n d e r p i v o t p o i n t and t h e g r i n d i n g s u r f a c e i n t h e 'ideal fins1 location. These s e t t i n g p a r a m e t e r s , shown i n F i g . 2 , may b e u s e a i n s t D a d of C z and C and a r e e x p r e s s e d by [ 1 2 ] , Y L~ =
c
(cot%-tan.,) Y
and
t
(D/2) (cotycosecv- tan.Jcosic,,-.)) (12)
Ls = ( C + ( D / 2 s i n ~ ) ) s i n ~ i c o s ~ c o s e c ' ~ (13) Y i s h a l f t h e vee g r o o v e a n g l e . The i n i t i a l wherc n n q u l a r s e t t i n g of t h e d r i l l i n t h e v e e g r o o v e t o a c h i e v e t h e g r i n d i n g p a r m e t e r 1. d e p e n d s on t h e i n i t i a l d r i l l p o i n t geometry and i t s r e l a t i o n t o t h e f l u t e g e n e r a t o r i n t e n d e d t o f o r n t h e l i p when t h e d r i l l i s i n its 'ideal' f i n a l position. This angular grinder s e t t i n g p a r a m e t e r may b e made e q u a l t c t h e g r i n d i n g p a r a m e t e r i by c a r e f u l l y s e l e c t i n g t h e i n i t i a l d r i l l p o i n t geometry.
The d e s i g n o f t h e i n i t i a l d r i l l p o i n t qenmetry c o n s t i t u t e s t h e f i r s t i m p o r t a n t e l e m e n t of t h e f i n a l d r i l l point generation process. The i n i t i a l d r i l l p o i n t g e o r , e t r y s h o u l d s i m p l i f y t h e s e t t i n g of t h e d r i l l i n t h e g r i n d e r (second e le m e n t) , should ensure t h a t t h e i n i t i a l d r i l l p o i n t geometry i s f u l l y removed a t t h e g r i n d e r a c t i o n end p o i n t ( e l e m e n t f o u r ) , s h o u l d n o t i n v o l v e e x c e s s i v e m a t e r i a l removal d u r i n g p o i n t g r i n d i n g and s h o u l d h e a n e a s y s h a p e t o be produced. The i n i t i a l p o i n t geometry c o n s i s t i n g of a r i g h t c i r c u l a r cone c o a x i a l with t h e d r i l l a x i s can s a t i s f y a l l t h e above c r i t e r i a when a s u i t a b l e semi-cone I t is a p p a r e n t t h a t t h i s a n g l e '3 i s s e l e c t e d . i n i t i a l D p o i n t geometry i s e a s y t o p r o d u c e and h a s a geometry which r o u g h l y a p p r o x i m a t e s t n e f i n a l p o i n t georretry so t h a t t h e amount of m a t e r i a l t o be removed is n o t eitcessive. Further t h e circumferential cleara n c e a n g l e Ct [ 2 4 , 2 6 ] i s z e r o on t h e i n i t i a l c o n i c a l f l a n k s so t h a f e x c e s s m a t e r i a l i s a v a i l a b l e a t t h e 0 on t h e s e s u r f a c e s fcr t h e f i n a l flank since Ce p o i n t geometryCO F u r t h e r t h e s o l i d cone i n t h e web ( c o r e ) r e g i m a l s o p r o v i d e s e x c e s s m a t e r i a l which e n a b l e s t h e c h i s e l e d g e t o b e formed by t h e g e n e r a t i o n process. The c o n d i t i o n s i n t h e v i c i n i t y o f t h e e v e n t u a l l i p c a n b e s t u d i e d from t h e y r o m e t r y o f t h e i n L e r s e c t i o n o f t h e i n i t i a l d r i l l p o i n t c o n e and t h e f l u t e I t i s found t h a t vhen 9 is s e l e c t e d s u c h surface. t h a t a s i n g l e flute g e n e r a t o r i n t g r s e c t s t w o c o n e g e n e r a t o r s a t t h e o u t e r and core r a d i i , r e s p e c t i v e l y , t h e i n t e r s e c t i o n c u r v e h a s a n e x t r e m e l y s m a l l convex c u r v a t u r e when viewed i n a p l a n e normal t o t h e d r i l l axis (x-y p l a n e ) . Thus t h e i n i t i a l c o n e - f l u t e i n t e r s e c t i o n c u r v e ve r y c l o s e l y approximates t h e s i n g l e f l u t e g e n e r a t o r c o n s i d e r e d a l t h o u g h a s l i g h t amount of m a t e r i a l i s a v a i l a b l e f o r removal when t h e f l u t e g e n e r a t o r i s chosen a s t h e e v e n t u a l l i p o f t h e s h a r p ened d r i l l . The r e q u i r e d i n i t i a l d r i l l semi-cone s i n g l e 4 is found t o depend on t h e p o i n t a n g l e 2p and web t h i c a n e s s t o d i a m e t e r r a t i o 2W/D [27] i . e .
From t h e above e q u a t i o n i t i s c l e a r t h a t 5 w i l l b e less t h a n t h e " i n t u i t i v e " c h o i c e o f 9 = pp With t h i s i n i t i a l geometry, t h e i n t e r s e c t i o n o f D t h e i n i t i a l d r i l l cone and f l u t e may b e u s e d t o i d e n t i f y t h e f i n a l l i p and o u t e r c o r n e r which s i m p l i f i e s t h e i n i t i a l s e t t i n g of t h e d r i l l i n t h e g r i n d e r , i . e . s e t t i n g s Lc and 1 . The g e n e r a t i o n p r o c e s s t h e r e f o r e c o n s i s t s o f g r i n d ing t h e i n i t i a l p o i n t geometry a t t h e a p p r o p r i a t e 0 ( E q u a t i o n (14)) and s e t t i n g O , x , C ,L and , Ion t h e The d r i l l i s x t h & p o s i t i o n e d i n grinder attachment. i t s ' i d e a l ' f i n a l l o c a t i o n by g r a d u a l l y i n c r e m e n t i n g t h e g r i n d e r attachment i n t h e L d i r e c t i o n whi l e O s c i l l a t i n g a b o u t t h e g r i n d i n g done a x i s u n t i l t h e i n i t i a l geometry is f u l l y removed i . e . L i s s a t i s fied T h i s a n a l y s i s h a s demonstrated t h a t t h e study of t h e d r i l l p o i n t g e n e r a t i o n p r o c e s s i s more i n t r i c a t e than the analysis f o r t h e d r i l l i n t h e ' i d e a l ' f i n a l l o c a t i o n d e p i c t e d by F i g . 1. However by a d e t a i l e d c o n s i d e r a t i o n of t h e v a r i o u s e l e m e n t s t h e g e n e r a t i o n p r o c e s s c a n b e s i m p l i f i e d and t h e p o t e n t i a l s c o p e of t h i s g r i n d i n g method n o t e d e a r l i e r [ 2 6 ] a c h i e v e d when s e t t i n g e r r o r s a r e ignored.
.
F I N A L IIRILI. POINT GEOMETRY
GEWERATION PROCESS - SETTING ERRORS I n attempting to achieve the desired f i n a l d r i l l p c i n t geometry, e r r o r s a r e p o s s i b l e i n s e t t i n q e a c h of t > e f i v e main g r i n d e r p a r a m e t e r s ( i . e . a , A , . + , C , and L 1. A d d i t i o n a l l y , t h e i n i t i a l geometry may c 8 n t a i n f f r o r s i n 2W and i , i r h i l e t h e d r i l l may be moved Loo f a r t o w a r d s t h e g r i n d i n g wheel by an amount dt, d u r i n g the g r i n d e r a c t i o n e l e m e n t o f t h e g e n e r a t i o n p r o c e s s . With such numerous p o t e n t i a l s o i i r c e s o f e r r o r , i t i s d o u b t f u l i n o r a c t i c e i f a l l t h e s e e r r o r s can b e e l i m i n a t e d . T h e i r n e t e f f e c t w i l l b e t o make i t i m p r a c t i c a l tr) p o s i t i o n t h e d r i l l i n t h e " i d e a l " f i n a l l o c a t ion. So although t h e f i n a l generated d r i l l point geometry w i l l have c o n i c a l f l a n k s i t s l i p s must b e curved because u n l i k e t h e " i d e a l " c a s e i l l u s t r a t e d i n F i g . 1, t h e l i p s a r e n o t formed by a s i n g l e g r i n d i n g c o n e g e n e r a t o r . Thus t h e need i s c r e a t e d f o r an e x t e n d e d a n a l y s i s t o s t u d y t h e i n f l u e n c e of e r r o r s on t h e g e n e r a t e d geometry. The a n a l y s i s p r e s e n t e d i s based on t h e " i d e a l " case and accommodates t h e g r i n d e r s e t t i n q , i n i t i a l geometry, and g r i n d e r a c t i o n errsrs d i s c u s s e d above. I t a l s o i n c l u d e s a s u i t a b l e d e f i n i t i o n of t h e g r i n d e r a c t i o n rnd point. Further, the "acceptable grinder c r i t e r i a " e s t a b l i s h e d f o r t h e " i d e a l " c a s e [ 2 6 ] a r e reviewed and m o d i f i e d . F i n a l l y , r e v i s e d d e f i n i t i o n s of t h e s p e c i f i e d d r i l l p o i n t f e a t u r e s a r e d e v e l o p e d and a means of m e a s u r i n g t h e amount of l i p c u r v a t u r e d e t e r m i n e d . For t h e p u r p o s e Of t h e a n a l y s i s , t h e o r i g i n i s t a k e n t o b e t h e p o i n t 0 on t h e d r i l l a x i s a s shown i n F i g . 1. The i n t r o d u c t i o n of s e t t i n g and g r i n d e r a c t i o n e r r o r s w i l l l e a d t o t h e d i s p l a c e m e n t of b o t h t h e d r i l l and c o n e a x e s away from t h e " i d e a l " f i n a l location. When t h e g r i n d e r j i g i s p o s i t i o n e d a d i s t a n c e L t r . away from t h e g r i n d i n g wheel p l a n e , t h e e q u a t i o j l s f o r t h e d i s p l a c e m e n t between t h e o r i g i n 0 on t h ? d r i l l a x i s and t h e cone apex become: CX'
'
C
cx
=
= C
Y
t dCx
(15)
sin(d0tdx) + C
cos(d*+dr)
Y 2C s i n [ % ] c o s [ $ + 5]+ 2C s i n [ Z dB! c o s [ x + d i : - . ' t ~ ] P 2 2 q (G,singI- L ) ( c o t $ - c o t ( 1 j t d 9 ) )s i n ! H t i A d a t d u )
Z
t
-
t s i n ( y t dx)cosec(e + do).\ Cz' = C z c o s ( d 0 + d i ) - C s i n ( d . 1 + dX) Y
(16)
(20)
I t i s n o t e d t h a t i n t h e " i d e a l " c a s e when d o , d r , dCx, dLc and A e o u a l z e r o , E q u a t i o n s ( 1 5 ) - ( 1 7 ) r e v e r t , a s e x p e c t e d , t o C x ' = C x , C '=C and C z ' = C z . Y Y The e q u a t i o n s f o r a p o i n t x , y , z on t h e g r i n d i n g cone 1 c a n t h e n b e d e t g r m i s e d from:
-
x = (y, + c ')tavA' cx' (22) Y (23) zc = (yc + C y ' ) t a n v ' CZ' where t h e a n g l e s > ' and Y ' c a n b e i n t e r r e l a t e d u s i n g a m o d i f i e d v e r s i o n o f t h e " i d e a l " c a s e E q u a t i o n (1) and s u b e t i t u t i n g 4 + dq and x + d i f o r 9 and x respectively. A t a radius r , a point x on a g e n e r a l f l u t e g e n e r a t o r may b e d e t e g h i % %om t h e f o l l o w i n g e x p r e s s i o n s ( w i t h r e f e r e n c e t o o r i g i n 0 on t h e d r i l l axis) :
xf
=
r sin(i t 9
+
dh
yf = r cos()i t h t d i
zf
= (Wtanh
where,
-
6)')
(24)
iu')
(25)
+ r c o s , , , ' ) c o t p + [*$]cot(6,t = W 7 t d W-
dS)
(26)
sinw'
and G r e p r e s e n t s t h e i n c l i n a t i o n between t h e p r o j e c t i o n s of t h e g e n e r a l and r e f e r e n c e f l u t e g e n e r a t o r s i n a p l a n e normal t o t h e d r i l l a x i s . The r e f e r e n c e f l u t e generator i n turn represents the generator inclined by t h e error a n g l e dA t o t h e g e n e r a t o r which f o r m s t h e
s t r a i g h t l i p when t h e d r i l l i s p o s i t i o n e d i n t h e "ideal" f i n a l location. Finally, a point x , on t h e g e n e r a l i n i t i a l d r i l l cone g e n e r a t o r t R e e u a t o n s t a k e t h e form:
c a s e v a l u e s c . g . $',+dt. f n r t i . F i n a l l y , the l i p curvatu r e was d e t e r m i n e d i n t h e x-y p l a n e (,',xy) and y-z p l a n e (!.yz) by e s t a b l i s h i n g t h e amount o f d e p a r t u r e of t h e curved l i p from t h e " a v e r a g e " l i p .
xD = r s i n e
Scope of C o n i c a l G r i n d i n g w i t h Errors
'8' 'f
yD = CoSE zD = r c c t 9
D
+ WtanAcotp + n2 ( c o s h : c o t p - c o t c D )
(30)
where i~.' i s found by s u b s t i t u t i n g r = D i n E q u a t i o n ( 2 7 ) and E i s t h e a n g l e between t h e p r o j e c t i o n s of t h e g e n e r a l i n i t i a l d r i l l cone g e n e r a t o r and t h e ya x i s i n a p l a n e normal t o t h e d r i l l a x i s . Using E q u a t i o n s ( 1 5 ) - ( 3 0 ) , t h e g r i n d e r a c t i o n end p o i n t can b c e s t a b l i s h e d and t h e r e s u l t a n t f i n a l d r i l l I t is apparent t h a t these p o i n t geometry found. e q u a t i o n s a r e f a i r l y complex and i n v o l v e v a r i a b l e s which a r e d i f f i c u l t t o s e p a r a t e . Hence i t was n e c e s s a r y t o use i t e r a t i v e s o l u t i o n search techniques t o d e t e r m i n e t h e g r i n d e r a c t i o n end p o i n t . The l a r g e number o f c a l c u l a t i o n s i n v o l v e d was e x p e d i t e d by d e v e l o p i n g and u s i n g a s u b s t a n t i a l computer programme ~271. I n b r o a d terms, t h e method of d e t e r m i n i n g t h e g r i n d e r a c t i o n end p o i n t i s t o f i r s t select v a l u e s of t h e commonly s p e c i f i e d d r i l l p o i n t f e a t u r e s t o g e t h e r w i t h s i z e s of t h e g r i n d e r s e t t i n g , i n i t i a l g e o m e t r y , and g r i n d e r a c t i o n errors. The " i d e a l " c a s e g r i n d i n g and g r i n d e r p a r a m e t e r v a l u e s a r e t h e n c a l c u l a t e d from E q u a t i o n s ( 1 ) - ( 1 3 ) and t h e i n i t i a l d r i l l semi-cone Equations a n g l e 8 determined u sin g Equation ( 1 4 ) . ( 2 8 ) - ( 3 B ) a r e used t o s e l e c t a p o i n t on t h e i n i t i a l d r i l l cone. The p o i n t on t h e g r i n d i n g cone a t which x =x and y =y i s t h e n found from E q u a t i o n s ( 1 5 ) - ( 2 3 ) . Tfiesg two g h B s o f e q u a t i o n s a r e i t e r a t e d f o r . ? t i l l t h e c o n d i t i o n z =z i s s a t i s f i e d . By r e p e a t i n g t h e above s t e p s f o r C a Eomprehensive g r i d of p o i n t s on t h e i n i t i a l d r i l l c o n e a minimum v a l u e of.! c a n be e s t a b lished. T h i s r e p r e s e n t s t h e l a s t p o i n t on t h e i n i t i a l d r i l l cone removed by t h e g r i n d i n g cone. S u b t r a c t i n g dA from t h e minimum h v a l u e g i v e s t h e g r i n d e r a c t i o n end p o i n t when errors o c c u r . The f i n a l d r i l l p o i n t geometry g e n e r a t e d a t t h e g r i n d e r a c t i o n end p o i n t m u s t , i n a s i m i l a r f a s h i o n t o t h e " i d e a l " c a s e , m e e t c e r t a i n shape c o n s t r a i n t s i f i t s g e n e r a l appearance i s t o be considered accepta b l e . With e r r o r s p r e s e n t , t h e s e v e n " a c c e p t a b l e g r i n d e r criteria" e s t a b l i s h e d 1261 a r e r e d u c e d t o t h e following f i v e c r i t e r i a : 1. The d r i l l p o i n t r e g i o n s h o u l d b e s y m m e t r i c a l a b o u t the d r i l l axis. 2. The d r i l l f l a n k s u r f a c e bounded by t h e l i p , c h i s e l e d g e , d r i l l p o i n t h e e l , and f l u t e d l a n d s h o u l d b e a continuous surface. 3 . The c h i s e l edge s h o u l d j o i n t h e l i p a t t h e c h i s e l edge c o r n e r . 4. The c l e a r a n c e a t a l l p o i n t s on t h e a n g u l a r f l a n k r e g i o n between t h e o u t e r c o r n e r and t h e c h i s e l edge corner should be s u f f i c i e n t t o prevent interference with the transient surface during d r i l l i n g . 5. C o n d i t i o n s (1) t o ( 4 ) s h o u l d b e s a t i s f i e d when a t t e m p t i n g t o a c h i e v e t h e recommended r a n g g s of t h e s p e g i f i e 8 d r i l l g o i n t f e a t u r %s i.e. 2p=118 ; Q = 120 -135 : CLn=8 -16'; 6,,=2C -32O; 2W/D=12%-20%.
-
The modes of f a i l u r e o f t h e above c r i t e r i a a r e i d e n t i c a l t o t h e " i d e a l " c o n i c a l g r i n d i n g method [26] w i t h t h e e x c e p t i o n s t h a t c r i t e r i o n 3 may b e f a i l e d by t h e c h i s e l e d g e i n t e r s e c t i n g t h e d r i l l p o i n t h e e l and c r i t e r i o n 4 may n o t b e m e t d u e t o t h e l i p c l e a r r n c e angrle becoming n e g a t i v e . I n d e s c r i b i n g t h e f i n a l d r i l l p o i n t - g e o m e t r y gene r a t e d a t t h e g r i n d e r a c t i o n end p o i n t , c o - o r d i n a t e s f o r p o i n t s on t h e d r i l l f l a n k were found from Equati o n s ( 1 5 ) - ( 2 3 ) . Combining t h e s e e q u a t i o n s w i t h E q u a t i o n s ( 2 4 ) - ( 2 7 ) f o r t h e d r i l l f l u t e and i t e r a t i n g f o r $ t i l l t h e c o n d i t i o n z =z w a s s a t i s f i e d , t h e Finally, the d r i l l l i p co-ordinates we& established. c h i s e l e d g e c o - o r d i n a t e s were found from E q u a t i o n ( 2 3 ) and a m o d i f i e d v e r s i o n of t h e " i d e a l " c a s e E q u a t i o n ( 6 ) w i t h C x ' , C I , R + d B , and x+dx s u b s t i t u t e d f o r a , and y r e s p e c t i v e l y . Cx, C Y' With t h e g e n e r a t e d l i p s b e i n g c u r v e d , r e v i s e d d e f i n i t i o n s w e r e needed f o r t h e g e n e r a t e d p o i n t f e a t u r e s s u c h as t h e p o i n t a n g l e . T h r e e r e v i s e d d e f i n i t i o n s were d e r i v e d u s i n g t h e c o n c e p t of a n " a v e r a g e " d r i l l l i p formed by j o i n i n g t h e o u t e r and c h i s e l edge c o r n e r s w i t h a s t r a i g h t l i n e . The a v e r a g e p o i n t a n g l e 2p and a v e r a g e l i p s p a c i n g 2WA were d i r e c t l y b a s e d on t h e " a v e r a g e " l i p w h i l e t h e a v e r a g e c h i s e l e d g e a n g l e $ was a l s o r e q u i r e d u s e of m o d i f i e d v e r s i o n s o f E q u a t i o h s ( 7 ) and ( 8 ) d e v e l o p e d f o r t h e " i d e a l " c a s e . S i m i l a r l y , t h e c l e a r a n c e a n g l e C Z ' was o b t a i n e d u s i n g a m o d i f i e d v e r s i o n of t h e " i d e a l " c a s e E q u a t i o n (10). I n b o t h c a s e s , t h e m o d i f i c a t i o n s i n v o l v e d s u b s t i t u t i n g t h e error c a s e f o r t h e " i d e a l "
4
The " a c c e p t a b l c g r i n d e r c r i t e r i a " w e r e used t o a s s e s s t h e g e n e r a t e d f i n a l d r i l l p o i n t geometry f o r a l a r g e number of g e n e r a l p u r p o s e c o m b i n a t i o n s of d r i l l p o i n t f e a t u r e v a l u e s . A s i n d i c a t e d i n t a b l e 1, 80 c o m b i n a t i o n s o f 0 , CI , 2W/D and h e l i x a n g l e x . were te8ted together with 9 0 values of 2 over the range 31 -50° ( i . e . lo i n t e r v a l s ) . T h i s g a v e a t o t a l of 80120 = 1 6 0 0 c o m b i n a t i o n s which were examined. The combined e f f e c t s o f t h e 8 e r r o r s a<,, dw, d ; , dC , dL , d a , dW and d i , were s t u d i e d by s i m u l t a n e o u s l y a5plyifig t h c worst c a s e l i m i t s f o r e a c h error. The f i g u r e s s e l e c t e d f o r t h e 8 errors were dil, d:? t .5O: d l ! 2O; dCx, dL I .002", dh + .002"; dW 1 .0013": and d t : 1C28O. The g r i n d e r s e t t i n p and p o s i t i o n i n g ergors were d e r i v c d f r o n c o n s i d e r a t i o n of t h e r e l e v a n t s e t t i n g mechanisms w h i l e t h e i n i t i a l geometry e r r o r s wcre o b t a i n e d from a n e a r l i e r p r o c e s s c a p a b i l i t y s t u d y 1191. From t a b l e 1 i t i s a p p a r e n t t h a t i n cont r a s t t o t h e " i d e a l " c a s e ( i . e . z e r o e r r o r s ) , a t 26 ( o r 32%) of t h e 80 g e o m e t r i c c o m b i n a t i o n s , no a c c e p t a b l e f i n a l d r i l l p o i n t geometry c o u l d be produced t h u s f a i l i n g c r i t e r i o n 5. For t h e r e m a i n i n g combinati o n s , u p p e r l i m i t s f o r fi were g e n e r a l l y due t o t h e f a i l u r e of c r i t e r i o n 4 i n t h a t t h e c i r c u m f e r e n t i a l c l e a r a n r e a n g l e C 1 [ 2 4 ] was i n a d e q u a t e w h i l e t h e l o w e r l i m i t s w e r e &own t o r e s u l t from e i t h e r no or a second c h i s e l e d g e c o r n e r b e i n g g e n e r a t e d Thus n o t m e e t i n g c r i t e r i o n 3. I n t o t a l , of t h e 1600 combinata t some 6 8 8 (or 435) t h e d r i l l ions considered, p o i n t geometry g e n e r a t e d was u n a c c e p t a b l e . Hence t h e p r e s e n c e o f r e l a t i v e l y s m a l l e r r o r s h a s been demons t r a t e d t o s u b s t a n t i a l l y reduce t h e scope of coni cal g r i n d i n g and p l a c e s i g n i f i c a n t r e s t r i c t i o n s on t h e v a l u e of .? which may b e s e l e c t e d . L i p C u r v a t u r e and P r o m i n e n t D r i l l P o i n t F e a t u r e s The combined e f f e c t s of errors on t h e g e n e r a t e d l i p c u r v a t u r e and d r i l l p o i n t f e a t u r e v a l u e s have been examined f o r s e v e r a l c o m b i n a t i o n s o f t h e s p e c i f i e d d r i l l p o i n t f e a t u r e s and g r i n d e r semi-cone a n g l e 4 a t which t h e " a c c e p t a b l e g r i n d e r c r i t e r i a " were satisfied. A sample of t h e s e r e s u l t s is i l l u s t For e a c h o f t h e 11 g e o m e t r i c rated i n table 2. c o m b i n a t i o n s l i s t e d , t h e combined w o r s t c a s e l i m i t s
TABLE 1.
Combined e f f e c t s of d e v i a t i o n s on s c o p e of c o n i c a l d r i l l F o i n t g r i n d e r modelled. ( 2 p = 118O i n a l l c a s e s ) IDEAL CASE ( i . e .
z e r o errors)
D = 1"
D = 1/2"
$\cyo 8 10 1 2 1 4 120 * * r-T--T* * 125 * * * 130 135 ,34 .33 ,31 *
16
*
D = 1/4"
g\"'o
8
* 120 * 125 130 * 135 i33
10
12
14
16
*
* *
*
* *
*
*
*
*
* * *
* COMBINED EFFECTS D = 1/2"
D = 1"
,>v'o
12 1 4 16 1 2 0 <43 <42 e42 d 4 0 e35 125 <48 <49 -50 * * 130 n 34-37* * * 1 3 5 n n n n n
-
8
10
b\"o 120 125 130 135
10 11 1 4 16 .45 ~ 4 5<45 e45 ~ 4 2 -50 ~ 5 0 * * * n 33-42* * * n n n n n 8
D = 1/4"
D = 1/8"
---
v\'*o 8 1 0 12 14 16 b\"o 8 10 1 2 14 16 1 2 0 *46 .46 <47 .47 *47 120 ~ 4 5.46 *47 .48 ~ 4 9 125
-
When D = 1". 2W/D = 1 2 % and F = 2W/D = 14% and 6 " = f o r D = k " , 2W/D = 17% and 6 " = 2W/D = 20% and"b3 D =
-
32O- f o r D = $", 30°: 26O: and f o r = 2bo.
o f t h e 8 e r r o r s c o n s i d e r c d i n t h i s i n v e s t i g a t i o r were a p p l i e d a t t h e levels l i s t e d p r e v i o u s l y . From t a b l e 2 i t c a n b e s e e n t h a t t h e l i p c u r v a t u r c ? i s i n a l l cases q u i t e small. A s t h e maxim!im m a g n i t u d e o f t h e c u r v a t u r c c o m p o n e n t s w a s J mere . 0 0 3 ( f o r ' > x y / D ) , t h e g e n e r a t e d l i p s can f o r p r a c t i c a l p u r p o s e s b e r e g a r d e d as s t r a i g h t l i n e s . Gi7Jen t h a t a l l t h e " a c c e p t a b l c g r i n d e r criteria' are a l s o s a t i s f i e d , t h e s e r e s u l t s s u g g e s t t h a t t h e g e n e r a l a p p e a r a n c e of t h e g e n e r a t e d p o i n t g e o m e t r y i s n o t a r e l i a b l e i n d i c a t o r o f t h e p r e s e n c e o r e f f e c t s of g r i n d e r s e t t i n g errors. The l i m i t e d Lip c u r v a t u r e a l s o i m p l i e s t h a t t h e g e n e r a t e d p o i n t f e a t u r e s c a n b e t r e a t e d as p h y s i c a l r e p r e s e n t a t i o n s of t h e i r " i d e a l " case c o u n t e r p a r t s . From t a b l e 2 i t i s a p p a r e n t t h a t t h e c o m b i n e d errors s u b s t a n t i a l l y a f f e c t t h e v a l u e s of t h e generate? point features. F o r i n s t a n c e i n a l l 11 cases, t h e r a n g e o f 2 p v a l u e s e x c e e d s 46 w i t h t h e maximum r a n g e S i m i l a r l y , & h e maximum r a n g e o f C . ' r e a c h i n g 6.9'. v a l u e s is a s i g n i f i c a n t 4.8 Although t h e rangesoof ' i D a r e f a i r l y moderate ( i . e . 2 . 6 % max.; t h e a n g l e :mWAseems p a r t i c u l a r l y s e n s i t i v e t o e r r o r s . I n 3 of the 11 cases i i l u s t r a t e d , t h e g e n e r a t e d r a n g e o f i exceeded t h e r e c o m n e n d e d q e n e r a l p u r p o s e r a n g e f o r A t h i s f e a t u r e o f 15' (12Oo-13So) w h i l e t h e naximum Of t h e 8 e r r o r s r a n g e of v w a s a s h i g h as 16.6O. c o n s i d e r e d ? ,, w a s f o u n d t o b e , a s e x p e c t e d , s e n s i t i v e t o d . b u t s u r ~ r i s i n g l yi t w a s e v e n more s e n s i t i v e t o d;. S i m i l a r l y , d ; a n d d; h a d t h e a n t i c i p a t e d major i n f l u e n c e o n 2 p w h i l e dW s i g n i f i c a n t l y a f f e c t e d 2W ' . T h e r e s u l t s i n A t a b l e 2 show t h a t t h e g e n e r a t e d d r i P l point feature ranges straddle the corresponding "ideal" specified values. However, d i f f i c u l t i e s are l i k e l y t o b e e n c o u n t e r e d i n t h e a c h i c v e m e n t and c o n t r o l o f a p a r t i c u l a r s e t of recommended g e n e r a l purpose f e a t u r e values. F u r t h e r when t h e n o m i n a l d e s i g n v a l u e s d i f f e r f r o m b a t c h t o b o t c h t.he t o t a l scatter i n t h e qenerated p o i n t f c a t u r e v a l u e s can b e extremcly wide. F o r e x a n p l e , f r o g tab&' 2 , when I i s s e l e c t e d w i t h i n t h c l i m i t s 1 2 0 -130 the ggneratcd a value of shown t o v a r y f r o m 1 1 2 . 4 - 1 3 8 . 7 r a n g e o f 28.3 I t should be n o t c d t h a t a s f a i r l y l o w l e v e l s of errors have been c o n s i d e r e d i n t h i s i n v e s t i g a t i o n , t h e s u b s t a n t i a l p o i n t f e a t u r e variat.i o n s shown i n t a b l e 2 may w e l l b e c o n s e r v a t i v e w h i l e t h e g e n e r a l p u r p o s e scope i l l u s t r a t e d i n t a b l e 1 c o u l d be e v e n f u r t h e r r e d u c e d .
.
is
3.)
.
co~cL,vsIor;s The f i n a l d r i l l p o i n t geometry g e n e r a t i o n p r o c e s s for g e n e r a l p u r p o s e t w i s t d r i l l s u s i n g t h e p o p u l a r c o n i c a l g r i n d i n g m e t h o d h a s 'm3en a n a l y s c d b o t h i n t h e a b s e n c e ! i . e . " i d e a l " case) a n d p r e s e n c e o f g r i n d e r setting errors. I t i s shown t h a t t h e a n a l y s i s f o r r e l a t i n y the specified dril.1 point features to t h e g r i n d i n g p a r a m c t e r s when t h e d r i l l i s i n t h e " i d e a l " f i n a l 1.otation represer.ts only part of t h e g e n e r a t i o n process. The i n i t i a l d r i l l p o i n t geometry, t h e s e t t i n g of the d r i l l i n the grinder, the grinder a c t i o n d u r i n g w h i c h t h e i n i t i a l g e o m e t r y i s removed and t h e i d e n t i f i c a t i o n of t h e g r i n d e r a c t i o n end p o i n t c o n s t i t u t e t h e f o u r major e l e m e n t s which c o n t r i b u t e t o t h e c o m p l e x i t y 3f t h e d r i l l p o i n t g e n e r a t i o n process. T h e d e t a i l e d a n a l y s i s o f t h e s e e l e m e n t s r e s u l t e d i n a n i n i t i a l d r i l l p o i n t geometry and g r i n d e r d e s i g n c a p a b l e o f p r o d u c i n g a l l t h e recorntende d s p e c i f i e d d r i l l p o i n t f e a t u r e xvalues f o r t h e " i d e a l " case of z e r o s e t t i n q errors. T h e nore c o m p l e x a n a l y s i s a n d g e n e r a t e d d r i l l point s h a p e due to t h e i n t r o d u c t i o n of s e t t i n g e r r o r s n e c e s s i t a t e d r e a p p r a i s a l of " a c c e p t a b l e y r i n d c r criteria" and d r i l l p o i n t f e a t u r e d e f i n i t i o n s . The p r e s e n c e o f r e l a t i v e l y small g r i n d e r s e t t i n o errors l e a d t o a s i g n i f i c a n t r e d u c t i o n i n t h e s c o p e of c o n i c a l g r i n d i n g f o r g e n e r a l p u r p o s e d r i l l s . Of t h e 80 g e n e r a l p u r p o s e c o m b i n a t i o n s of d r i l l p o i n t f e a t u r e v a l u e s ?.n a c c e p t a b l e s h a p e i s o n l y g e n e r a t e d a t 54 ( i . e . 6 8 % ) c o m b i n a t i o n s . Within t h e reduced scope, t h e amoullt of l i p c u r v a t u r e f o r a c c e p t a b l e d r i l l p o i n t s h a p e s was shown t o b e n e g l i g i b l e a l t h o u g h s m a l l s e t t i n g errors r e s u l t e d i n w i d e v a r i a t i o n s i n t h e v a l u e s o f t h e g e n e r a t e d f e a t u r e s ! e g 1.6.6" for A range). ilence i t h a s been d e m o n s t r a t e d t h a t i n i t s e l f t h e q e n e r a l a p p e a r a n c e of t h e g e n e r a t e d p o i n t geometry is a n u n r e l i a b l e i n d i c a t o r o f t h e p r e s e n c e or e f f e c t s o f s e t t i n g e r r o r s f o r u s e i n q u a l i t y control. Further, thP generated p o i n t f e a t u r e s have b e e n shown t o b e p a r t i c u l a r l y s e n s i t i v e to a number of s e t t i n g errors. This study has highlighted t h e complexities of t h e g e n e r a t i o n p r o c e s s and t h e a t t e n d a n t d i f f i c u l t i e s i n a c h i e v i n g and c o n t r o l l i n g t h e s p e c i f i e d d r i l l p o i n t geornctry n o t e d i n p r a c t i c e . REFERENCES
1. 2. Combined e f f e c t s of d e v i a t i o n s o n s a l i e n t f e a t u r e s of f i n a l d r i l l p o i n t g e o m r t r y .
TABLE 2 .
MAX D
2p
1,
C
-
!
*y/3
PlAX fyz/D
2pA
2WI;/D
,
A
3. 4. 5.
DONALDSON, G . H . L e CAIN a n d V.C. GOULD, " T o o l D e s i g p " , McGraw-Hill, N e w J e r s e y ( 1 9 6 9 ) A . S .T.M.E., " T o o l E n g i n e e r s ' IIandbook" , McGraw-11111, N e w Y o r k , ( 1 9 5 8 ) . METAL CUTTING TOOL INSTITUTE, "Metal C u t t i n q T o o l Handbook", ( 1 9 6 9 ) . AMERICAN STANDARD, USAS, H94-11-1967. BRITISH STANDARD INSTITUTION, B . S . 3 2 8 , P a r t I 1959. AUSTRALIAN STANDARD, A.S.2438-1981. G.M.H. STANDARD, " C u t t i n g T o o l s " , ( 1 9 6 8 ) . E.J.A. ARMAREGO a n d J . D . WRIGHT. J . E n g g . P r o d . , 2 .. 1.. ( 1 9 7 8 ) . ROHLKE, W e r k s t a t t s t e c h n i k a n d M a s c h i n e n b a u , 47, 5 , ( 1 9 5 7 ) . M.D. KINMAN, Machinery, (1963). W . D . ARNOT, M e c h a n i c a l World, 1 1 4 , March ( 1 9 5 2 ) . E.J.A. ARMAREGO a n d A. ROTENBERG, 1 n t . J . M a c h . T o o l D e s . R e s . , 13, 1 8 3 , ( 1 9 7 3 ) . D.F. GALLOWAY, T r a n s . Amer. SOC. Mech. E n g r s . , 79, 191, (1957). G.F. MICflELETTI a n d R . LEVI, P r o c . o f 8 t h I n t . M.T.D.R. C o n f . , U n i v e r s i t y of N a n c h e s t e r , Sept., (1967). fl. MALKIEWTCZ, M.Enq.Sc. T h e s i s , U n i v e r s i t v o f Melbourne ( 1 9 7 3 ) . S . KALDOR a n d E . LENZ, A n n a l s o f t h e CIRP, 2, 1, ( 1 9 8 0 ) . C . S . OXFORD J N R , A.S.T.E. c o l l e c t e d papers, 2, (1959). J . D . WRIGHT, M.Eng.Sc. T h e s i s , University of Melbourne, (1575). E.J.A. ARMAREGO a n d J . D . WRIGHT, J . Cngg. P r o d . 2 , 2, (1978). I. MUSHTAQ, M.Ena.Sc. T h e s i s , U n i v e r s i t y of Melbourne ( 1 9 8 0 ) . S. F U J I I , M.L. DeVRIES a n d S.M. W U , J. o f Eng. I n d . , 92, ( 1 9 7 0 ) . S. F U J I I , M.L. DCVRIES a n d S.M. WU, J. o f Eng. Ind., 93, (1971). W.D. TSAI a n d S.M. WU, I n t . J . Mach. T o o l D e s . Res., 19, 95, (1979). E.J.A.-jiRMAREGO a n d A . ROTENBERG, I n t . J. Mach. T o o l Des. R e s . , 2, 1 5 5 a n d 1 6 5 , ( 1 9 7 3 ) . F.K.T. Y A N , M . E n g . S c . T h e s i s , U n i v e r s i t y of Helbourne (1974). E . J . A . ARMAREGO a n d J.D. WRIGHT, A n n a l s of t h e CIHP, 2, 1, ( 1 9 8 0 ) . J.D. WRIGHT, Ph.D. T h e s i s , i i n i v e r s i t y o f Melbourne ( 1 9 8 1 ) . C.
.
_ I _ _ l l _ _ _ l
f 118 120
8
f 118 120 1 6
$j
118 130 16
42 H i . 0 0 1 L-.OOl R 3 1 H+.001 L-.OO2 R
-
120.1 115.7 4.4 120.1 115.7 4.4
128.2 9.5 112.4 6.9 15.8 2.6 126.5 lG.2 113.7 6.d 12.9 3.8
18.2 15.9 2.3 18.1 15.8 2.3
42 Hi.002 +.001 L - . 0 0 2 -.001 R 31 Hi.002 L-.002 - . 0 0 1 R
120.5 115.0 5.5 120.5 114.9 5.6
125.7 114.2 11.5 124.7 115.1 9.6
17.4 14.7 2.7 17.3 1 5 .0 2.3
18.2 15.8 2.4 18.2 15.8 2.4
10. 11. 12.
4 2 H+.COl L-.002 -.001 R 3 1 H+.OOl L-.002 - . 0 0 1 R
120.5 114.9 5.6 120.5 114.8 5.7
138.4 121.9 16.5 136.8 123.5 13.3
18.0 14.4 3.5 18.7 14.1 4.6
18.3 15.8 2.5 18.3 15.7 2.6
14.
120.3 115.7 4.6 120.4 115.7 4.7
127.6 9.4 113.3 6 . 8 1 4 . 3 2.6 125.3 10.2 115.0 6.0 10.3 4.2
12.3 11.7 .6 12.3 11.7 .6
+.001 1 2 0 . 7 1 2 3 . 6 1 7 .O - . 0 0 1 1 1 4 . 6 116.0 1 5 . 2 1.8 6.1 7.6
12.3 11.7 .6
-
-
1 118 120
8
42 €1+.002 L-.002 R
-
3 1 H+.001 L-.002 -.OOl R 1 118 120 16
3 1 Hc.002 L-.003 R
1 118 1 3 0 1 6
42 H+.OOZ +.001 1 2 0 . 6 L-.003 -.001 114.6 R 6.0 3 1 H+.OO2 +.OOl 1 2 0 . 8 L-.003 -.OOl 1 1 4 . 6 6.2 R
138.7 122.1 16.6 136.4 124.0 12.4
17.8 14.4 3.4 18.6 13.8 4.8
12.3 11.7 .6 12.3 11.7 .6
6. 7. 8. 9.
13.
15. 16. 17. 18. 19.
20. 21. 22. 23. 24. 25.
Note: When D = f " , D=l", H L R
-
2W/D=171 a n d A =26O a n d when 2W/D=12?: a n d 4::=32O
maximum v a l u e o f g e n e r a t e d f e a t u r e minimum v a l u e o f g e n e r a t e d f e a t u r r r a n q e of n e n e r a t e d f c ~ 3 t u r c s .
26. 27.
c.
102,
5
The f i n a l d r i l l p o i n t geometry g e n e r a t i o n p r o c e s s f o r g e n e r a l p'urpose t w i s t d r i l l s u s i n g t h c p o p u l a r c o n i c a l g r i n d i n g method i s d e s c r i b e d and m o d e l l e d . Based on t h e " i d e a l " c a s e a n a l y s i s pre*--iously r e p o r t e d , t h e fundamental c l e m e n t s of t h i s p r o c e s s a r e i d e n t i f i e d and a n a l y s e d . Far t h i s " i d e a l " ( d e s i g n ) c a s e , t h e g r i n d e r d e s c r i b e d c m producc a l l t h e recommended s p e c i f i e d d r i l l p o i n t f e a t u r e v a l u e s f o r a r a n g e of g r i n d i r g c o n e a n g l e s . The i n t r o d u c t i o n of g r i n d e r s e t t i n g a c v i a t i o n s from t h e " i d e a l " c a s e ( s e t t i n g e r r o r s ) s e v e r e l y c o m p l i c a t e s t h e a n a l y s i s and t h e g e n e r a t e d curved l i p d r i l l p o i n t s h a p e . N e w d e f i n i t i o n s f o r t h e geom e t r i c a l f e a t u r e s and t h e " a c c e p t a b l e g r i n d e r c r i t e r i a " had t o b e e s t a b l i s h e d . The e f f e c t s o f d e v i a t i o n s i n s i x g r i n d e r s e t t i n g s and two d r i l l f l u t e - f e a t u r e s have been s t u d i e d f o r 8 0 g e n e r a l p u r p o s e c o m b i n a t i o n s o f d r i l l p o i n t f e a t u r e v a l u e s . I t is shown t h a t even w i t h s m a l l d e v i a t i o n s , u n a c c e p t a b l e d r i l l shapes occur f o r 26 combinations. Fcr t h e remaining 54 combinations e s s e n t i a l l y s t r a i g h t d r i l l l i p s a r e o b t a i n e d a l t h o u g h t h e r a n g e s between t h e d e s i g n and g e n e r a t e d f e a t u r e s can d i f f e r s u b s t a n t i a l l y . The dominant s e t t i n g e r r o r s and t h c s u s c e p t i b l e g e n e r a t e d f e a t u r e s a r e i d e n t i f i e d . T h i s s t u d y h i g h l i g h t s t h e d i f f i c u l t i e s ic a c h i e v i n g and c o n t r o l l i n g t h e s p e c i f i e d geometry i n p r a c t i c e . INTRODUCTION The g e n e r a l g e o n e t r y o f t h e c o n v e n t i o n a l t w i s t d r i l l h a s been d e s c r i b e d i n numerous t e x t b o o k s and handbooks E1-31. C o n s i d e r a b l e i n t e r n a t i o n a l agreement on t h e n o m e n c l a t u r e and s p e c i f i c a t i o n o f t h e s a l i e n t g e o m e t r i c a l f e a t u r e s of t w i s t d r i l l s i s a l s o e v i d e n t i n handbooks and S t a n d a r d s [2-71. F u r t h e r , i t h a s been r e c o g n i s e d t h a t t h e geometry a t t h e d r i l l p o i n t , where c u t t i n g o c c u r s , a f f e c t s t h e i m p o r t a n t machining p e r formance c h a r a c t e r i s t i c s such a s f o r c e s , power and d r i l l l i f e . Through d e c a d e s of developn.ent and e x p e r i m e n t a t i o n , recommendations f o r t h e v a l u e s o f s p e c i f i e d d r i l l p o i n t f e a t u r e s f o r optimum d r i l l p e r formance when machining d i f f e r e n t work m a t e r i a l s have been g i v e n i n handbooks [ 2 , 3 ] . Recommendations f o r g e n e r a l p u r p o s e d r i l l s , r e p r e s e n t i n g a compromise optimum geometry f o r machining a r a n g e of common w o r k m a t e r i a l s have a l s o been q u o t e d [ 2 , 3 , 7 ] and compared [81. D e s p i t e t h e a c c u m u l a t e d knowledge of t w i s t d r i l l s t h e r e is r e l a t i v e l y l i t t l e d e t a i l e d information about t h e p r e c i s e geometry of t h e d r i l l p o i n t and t h e a s s o c i a t e d s h a r p e n i n g method. There i s e v i d e n c e t o s u g g e s t t h a t t h e c o n i c a l g r i n d i n g method i s commonly used i n p r a c t i c e [ 9 ] a l t h o u g h o t h e r methods have been n o t e d i n t h e l i t e r a t u r e [lo-121. I n a d d i t i o n t h e ' a s measured' g e o m e t r i c a l v a r i a b i l i t y o f n o m i n a l l y i d e n t i c a l manufa c t u r e d d r i l l s h a s been shown t o b e e x c e s s i v e [13-161 and c o n t r i b u t e s s i g n i f i c a n t l y t o t h e s c a t t e r i n t h e d r i l l i n g f o r c e s [ 1 4 , 1 5 ] and d r i l l l i f e [1G,17]. The s e l e c t i o n of t h e d r i l l p o i n t s h a r p e n i n g method, t h e d r i l l p o i n t f e a t u r e v a l u e s and t h e q u a l i t y c o n t r o l procedures f o r g e n e r a l purpose d r i l l s h a s been t h e r e s p o n s i b i l i t y of t h e d r i l l m a n u f a c t u r e r s . In recent y e a r s , r e s e a r c h w o r k e r s have d e v e l o p e d a keen i n t e r e s t i n t h e s e i m p o r t a n t a s p e c t s o f t w i s t d r i l l s . A number of i n v e s t i g a t i o n s o f t h e d r i l l g e o m e t r i c a l v a r i a b i l i t y o b t a i n e d d u r i n g manufacture have h i g h l i g h t e d t h e s o u r c e s of v a r i a t i o n s and t h e s i g n i f i c a n t r e d u c t i o n s i n v a r i a b i l i t y p o s s i b l e t h r o u g h improved q u a l i t y A s e r i e s o f s t u d i e s of c o n t r o l p r o c e d u r e s [18-20]. t h e p o p u l a r c o n i c a l and some o t h e r d r i l l p o i n t s h a r p e n i n g methods have a l s o been made i n an a t t e m p t t o a r r i v e a t a n optimum d r i l l p o i n t geometry [13,16,21-231. The i n v e s t i g a t i o n s o f d r i l l p o i n t s h a r p e n i n ? methods c a r r i e d o u t a t t h e U n i v e r s i t y o f Melbourne have been aimed a t o b t a i n i n g a d e e p e r u n d e r s t a n d i n g of t h e g e n e r a l p u r p o s e d r i l l geometry and p l a u s i b l e d r i l l p o i n t g r i n d e r s capable of achieving a d e s i r e d optimal geometry w i t h a view t o improve t h e v a r i a b i l i t y of n o m i n a l l y i d e n t i c a l d r i l l s [24-271. I n t h i s paper an a n a l y t i c a l i n v e s t i g a t i o n of t h e f i n a l d r i l l p o i n t geometry g e n e r a t i o n p r o c e s s f o r g e n e r a l p u r p o s e d r i l l s i s p r e s e n t e d . The a n a l y s i s i s a f u r t h e r development of e a r l i e r s t u d i e s b a s e d on t h e " i d e a l " c o n i c a l g r i n d i n g method. D e t a i l e d a n a l y s e s of g r i n d i n g methods have shown t h e c o n i c a l g r i n d i n g method t o be most p r o m i s i n g a s a means o f g e n e r a t i n g t h e reconmended r a n g e o f d r i l l p o i n t f e a t u r e v a l u e s f o r g e n e r a l p u r p o s e d r i l l s [26]. In t h i s present a n a l y s i s t h e g e n e r a t e d s h a p e and d r i l l p o i n t f e a t u r e v a l u e s w i l l be considered both i n t h e absence ( i . e . ' i d e a l ' c a s e ) and p r e s e n c e o f g r i n d e r s e t t i n g errors.
-FINAL
s e t t i n g of t h e d r i l l i n a g r i n d e r ; t h e g r i n d e r a c t i o n d u r i n g which t h e i n i t i a l geometry i s p r o g r e s s i v e l y removed; and t h e s e l e c t i o n o f an end p o i n t a t which t h e d e s i r e d f i n a l d r i l l p o i n t geometry i s g e n e r a t e d ( i . e . " i d e a l " f i n a l l o c a t i o n ) . For e a c h of t h e s e e l e m e n t s o f t h e g e n e r a t i o n p r o c e s s , m a t h e m a t i c a l r e l a t i o n s h i p s need to be e s t a b l i s h e d which l i n k t h e g r i n d e r s e t t i n g p a r a meters t o t h e g e n e r a t e d d r i l l p o i n t g e o m e t r i c f e a t u r e s . The common a p p r o a c h i n e a r l i e r a n a l y t i c a l s t u d i e s [ 1 3 , 21,22,24-261 h a s been t o d e v e l o p t h e s e t y p e s o f mathe m a t i c a l r e l a t i o n s h i p s when t h e d r i l l i s p o s i t i o n e d i n t h e " i d e a l " f i n a l - l o c a t i o n and t h e i n i t i a l geometry h a s been f u l l y removed. These a n a l y s e s have p r o v i d e d i m p o r t a n t i n s i g h t s i n t o t h e p o t e n t i a l of t h e g r i n d i n g methods a s a s s e s s e d by " a c c e p t a b l e g r i n d e r c r i t e r i a " [2G]. Thus t h e y r e p r e s e n t an e s s e n t i a l s t a r t i n g p o i n t for t h e broader a n a l y s i s i n c o r p o r a t i n g a l l elements o f the generation process. The g e n e r a l geometry o f t h e c o n i c a l g r i n d i n g method wh'en t h e d r i l l i s i n t h e ' i d e a l ' f i n a l l o c a t i o n i s shown i n F i g . 1. I n a d d i t i o n t o assuming t h a t t h e i n i t i a l geometry i s f u l l y removed, i t h a s a l s o been assumed [ 2 4 , 2 6 ] t h a t : !a) The d r i l l geometry i s s y m m e t r i c a l a b o u t t h e d r i l l axis.
T C* 1
SECTION A-A
DRILL P O I N T GEOMETRY GENERATION_
PROCESS
-
IDEAL CASE
I n developing an a n a l y s i s of t h e f i n a l d r i l l p o i n t geometry g e n e r a t i o n p r o c e s s , i t i s n e c e s s a r y t o cons i d e r t h e g e o m e t r i c a l i n t e r a c t i o n between t h e d r i l l and t h e g r i n d i n g s u r f a c e . From a p r a c t i c a l p o i n t of view, t h i s i n v o l v e s f o u r major e l e m e n t s t h e s e b e i n g t h e s e l e c t i o n o f a n i n i t i a l d r i l l p o i n t geometry: t h e
Annals of the ClRP Vol. 32/1/1983
1
( b ) The p o r t i o n o f t h e f l u t e s u r f a c e which f o r m s each l i p is a "ruled" s u r f a c e (developable annular helecoid). ( c ) The d i s t a n c e b e t w e e n e a c h s t r a i g h t l i n e f l u t e g e n e r a t o r i n ( b ) and t h e d r i l l a x i s is h a l f t h e web t h i c k n e s s (W) a n d t h e a c u t e a n g l e b e t w e e n t h e skew g e n e r a t o r and t h e d r i l l a x i s i s h a l f t h e point angle ( p ) . F o r t h i s g r i n d i n g method, t h e d r i l l f l a n k s a r e gene r a t e d by two r i g h t c i r c u l a r g r i n d i n g c o n e s s y m m e t r i c a l l y l o c a t e d a b o u t t h e d r i l l a x i s w i t h e a c h c o n e gene r a t i n g one flank. T h e c o n e a n d d r i l l axes l i e i n p a r a l l e l p l a n e s w i t h t h e c o n e a x e s d i s p l a c e d by a d i s t a n c e C from t h e d r i l l a x i s . The a c u t e a n g l e s b e t ween t h g skew d r i l l and c o n e a x e s a r e e a c h y w h i l e t h e s e m i - c o n e a n g l e f o r e a c h cone i s b. I f t h e generated l i p is t o be s t r a i g h t , t h e ' i d e a l ' f i n a l location of t h e d r i l l must be such t h a t a s t r a i g h t l i n e g e n e r a t o r o f e a c h f l u t e ( t a n g e n t i a l t o t h e d r i l l core d i a m e t e r ) is coincident w i t h a corresponding s t r a i g h t l i n e cone generator. The a n g l e between t h e p r o j e c t i o n s o f t h e cone g e n e r a t o r t h r o u g h t h e l i p and t h e c o n e a x i s i n t h e p l a n e normal t o t h e d r i l l a x i s i s A . The cone a p i c e s , a t o r i g i n s 0 a n d 0 are d i s p l a c e d by C , C a n d C w i t h r e s p e c t to o r i g i n 0 on t h e d r i l l a x r s as shdwn i n F i g . 1. A number o f t h e g r i n d i n g p a r a meters ? , ~ , A , V , C ~ , , C z a r e i n t e r r e l a t e d by [ 2 4 ] ,
Y
tanv =
sinxcosxsec'e-
g'sec2i
(cos2i(sec'1:-1) + s i n ' x s e c 2
(1-cos? ysec29 )
Cx = C tanA
Y
-
,2
(1) (2)
W/cos;i
.
and Cz = C t a n v (3) Y Thus from t h e g e o m e t r y i n F i g . 1 and t h e a b o v e e q u a t ions, four grinding parameters a r e s u f f i c i e n t to f u l l y d e s c r i b e t h e geometry w i t h t h e d r i l l i n t h e ' i d e a l ' f i n a l l o c a t i o n [26]. A s w i l l be n o t e d l a t e r , t h e f o u r g r i n d i n g p a r a m e t e r s o , x , i and Cx p r o v i d e a u s e f u l practical selection. T h e e q u a t i o n s f o r g r i n d i n g c o n e 1 w i t h respect t o t h e o r i g i n 0 o n t h e d r i l l a x i s h a s b e e n e x p r e s s e d as ~241
(4) A similar e q u a t i o n f o r cone 2 c a n also h e found [24]. The e q u a t i o n s f o r t h e l i p s ( f l u t e g e n e r a t o r s tang e n t i a l t o t h e d r i l l cone d i a m e t e r ) p r o j e c t e d i n t h e x-y p l a n e a r e : for Lip 1 : for Lip 2 :
x = y tani
- W/COSA
x = y t a n i + W/COSA
The r e l e v a n t p o r t i o n o f t h e c h i s e l e d g e e q u a t i o n i n t h e x-y p l a n e c a n b c f o u n d by s G l v i n g t h e two c o n e e q u a t i o n s w i t h r e s p e c t t o o r i g i n 0 on t h e d r i l l a x i s f r o m which [ 2 4 ] i
, = sin-l(w/r) A t the outer corner,
(11)
r = D/2 and
.
The r e m a i n i n g e l e m e n t s o f t h e g e n e r a t i o n p r o c e s s need t o b e c o n s i d e r e d i n o r d e r t o p h y s i c a l l y p o s i t i o n t h e d r i l l i n t h e ' i d e a l ' f i n a l l o c a t i o n shown i n F i g . 1 and t h u s a c h i e v e t h e d e m o n s t r a t e d p o t e n t i a l o f t h i s g r i n d i n q method f o r g e n e r a l p u r p o s e d r i l l s h a r p e n i n g . The g r i n d e r shown i n F i g . 2 , h a s b e e n d e s i g n e d f o r t h i s purpose. I t is noted t h a t t h e g r i n d e r allows f o r t h e f a c t t h a t t h e i n i t i a l s e t t i n g s w i l l o c c u r when t h e d r i l l i s d i s p l a c e d from i t s f i n a l l o c a t i o n w i t h r e s p e c t to t h e g r i n d i n g s u r f a c e . T h u s , a l t h o u g h o n l y f o u r g r i n d i n g p a r a m e t e r s are n e e d e d t o d e s c r i b e t h e g e o m e t r y i n t h e " i d e a l " f i n a l l o c a t i o n , more g r i n d e r s e t t i n g p a r a m e t e r s may b e n e e d e d t o a c h i e v e t h i s condition. N e v e r t h e l e s s , a n a t t e m p t h a s b e e n made t o maximize t h e u t i l i z a t i o n o f t h e g r i n d i n g p a r a m e t e r s i n t h e above a n a l y s i s . For example, t h e g r i n d e r s e t t i n g p a r a m e t e r s 9 a n d <: are d i r e c t p h y s i c a l r e p r e s e n t a t i o n o f t h e cor-res o n d i n g r i n d i n g p a r a m e t e r s i n t h e a n a l y s i s . S i m i l a r P y by m J i n g t h e v e e g r o o v e
* o, coneapex
F o r t h e c u r v e d c h i s e l e d g e , the c h i s e l e d g e a n g l e is b a s e d on t h e t a n g e n t t o t h e c u r v e i n t h e x-y p l a n e a t x=y=O, t h u s f r o m F i g . 1 $ = 1800
tan$' =
-
-
A
-
(7)
$ 1
[ s i n 2 9 - s i n x c o s x L i n 2 ci - ( cx/c
) ( cod
-cog
(CX/C )( c o s ~ e - c o s 7 x )
3 ( 8)
Y
The r e m a i n i n g s p e c i f i e d d r i l l p o i n t f e a t u r e s , i . e . t h e p o i n t a n g l e 2 p a n d l i p c l e a r a n c e a n g l e C P are that
also r e l a t e d to t h e g r i n d i n g parameters [24] cospcosx and
+
-
sinpsinxcos~
20
(9)
cose = 0
ab"G, bc-cp
ac = cq
t a n c s = -cos ( & # - A ) [ t a n A + t a n ( ~ - 1 )( c o s 2 x - s i n 7 Xtan2 9 )
-tanvtan(w-i)sinycosxsec20] ( s i n 2 x -cos2 q, t a n 2 o ) - s i n x c o s X s e c 2 w h e r e t h e web a n g l e w is d e f i n e d a s i[ t a n "
2
D
I
( 10)
= .,,
E q u a t i o n s (1) t o (11) p r o v i d e t h e m a t h e m a t i c a l r e l a t i o n s h i p s b e t w e e n t h e g r i n d i n q p a r a m e t e r s and t h e commonly s p e c i f i e d d r i l l p o i n t f e a t u r e s 2 p , Cv , ' v , During p o i n t sharpening t h e t h r e e s p e c i f i e d 2W a n d D . f e a t u r e s 2 p . C! a n d ':i a r e g e n c r a t e d on :he f l u t e d A s noted earlier body w i t h k n o w n ' f e a t u r e s 2W and D . a n d e v i d e n t from E q u a t i o n s ( 1 ) t o (11) t h e f o u r nece s s a r y g r i n d i n g p a r a m e t e r s s u c h a s '1, Y , and C h a v e t o b e f o u n d f r o m t h e t h r e e e q u a t i o n s i n v o l v i n g t#e g e n e r a t e d f e a t u r e s 2 p , Ci a n d I ( E q u a t i o n s ( 7 ) , (9) a n d (10). Thus tine d r i l l O g e o m e t r y i s u n d e r s p e c i f i e d a l t h o u g h a l l t h e d r i l l p o i n t f e a t u r e s c a n b e independe n t l y s e l e c t e d a n d g e n e r a t e d by t h i s g r i n d i n g method p r o v i d e d one g r i n d i n g p a r a m e t e r s u c h a s '4 i s p r e s e l e c t e d [24,26]. F u r t h e r it h a s previously been shown t h a t t h e c o n d i t i o n $1 .. '< m u s t a p p l y t o e n s u r e t h a t t h e d r i l l r e m a i n s w i t h i n t h e g r i n d i n g c o n e and hence can b e f u l l y ground [24]. I n a more r e c e n t work by t h e a u t h o r s [26] a c o m p r e h e n s i v e s e t o f " a c c e p t a b l e g r i n d e r criteria" have been e s t a b l i s h e d I t was and a p p l i e d to t h e c o n i c a l . g r i n d i n g m e t h o d . f o u n d t h a t a t a l l c o m b i n a t i o n s o f t h e recommended d r i l l p o i n t f e a t u r e values f o r g e n e r a l purpose d r i l l s , t h e d e s i r e d g e o m e t r y was g e n e r a t e d a n d t h e a v a i l a b l e range w i t h i n which ( ' u ) c o u l d b e s e l e c t e d was o n l y m a r g i n a l l y r e s t r i c t e d [26].
$
where
, so
that the l i p c l e a r a n c e a n q l e C: c a n b e f o u n d f?om E q u a t i o n (101 by u s i n y , f r o m EquatTon (11). ,:
FIGURE 2
i n which t.he d r i l l i s h e l d s y n m e t r i c a l w i t h r e s p e c t t o t h e p l a n e p a r a l l e l t o t h e d r i l l and g r i n d i n g cone a x i s , t h e g r i n d i n g p a r a m e t e r C can b e d i r e c t l y set on t h e g r i n d e r ( F i g . 2 ) . The d i H e c t U S E of t h e g r i n d i n g p a r a meters C z and C a s g c i n d e r s e t t i n g p a r a m e t e r s p r e s e n t problems s i n c e These a r e d i s t a n c e s between o r i g i n s 0: and 0 i n s p a c e when t h e d r i l l i s i n t h e ' i d c a l ' f i n a l l o c a t i o n . By a n t . i c i p a t i n g t h e f i n a l l o c a t i o n o f t h e o u t e r c o r n e r of t h e d r i l l w i t h r e s p e c t t o t h e p i v o t p o i n t on t h e g r i n d e r a t t a c h m e n t , t h e s e t t i n g p a r a m e t e r d i s t a n c e L a l o n g t h e d r i l l a x i s can he e s t a b l i s h e d . T'ne d i s t a n g e normal t o t h e g r i n d i n g wheel s u r f a c e L may b e used t o d e s c r i b e t h e r e l a t i o n s h i p between t h g g r i n d e r p i v o t p o i n t and t h e g r i n d i n g s u r f a c e i n t h e 'ideal fins1 location. These s e t t i n g p a r a m e t e r s , shown i n F i g . 2 , may b e u s e a i n s t D a d of C z and C and a r e e x p r e s s e d by [ 1 2 ] , Y L~ =
c
(cot%-tan.,) Y
and
t
(D/2) (cotycosecv- tan.Jcosic,,-.)) (12)
Ls = ( C + ( D / 2 s i n ~ ) ) s i n ~ i c o s ~ c o s e c ' ~ (13) Y i s h a l f t h e vee g r o o v e a n g l e . The i n i t i a l wherc n n q u l a r s e t t i n g of t h e d r i l l i n t h e v e e g r o o v e t o a c h i e v e t h e g r i n d i n g p a r m e t e r 1. d e p e n d s on t h e i n i t i a l d r i l l p o i n t geometry and i t s r e l a t i o n t o t h e f l u t e g e n e r a t o r i n t e n d e d t o f o r n t h e l i p when t h e d r i l l i s i n its 'ideal' f i n a l position. This angular grinder s e t t i n g p a r a m e t e r may b e made e q u a l t c t h e g r i n d i n g p a r a m e t e r i by c a r e f u l l y s e l e c t i n g t h e i n i t i a l d r i l l p o i n t geometry.
The d e s i g n o f t h e i n i t i a l d r i l l p o i n t qenmetry c o n s t i t u t e s t h e f i r s t i m p o r t a n t e l e m e n t of t h e f i n a l d r i l l point generation process. The i n i t i a l d r i l l p o i n t g e o r , e t r y s h o u l d s i m p l i f y t h e s e t t i n g of t h e d r i l l i n t h e g r i n d e r (second e le m e n t) , should ensure t h a t t h e i n i t i a l d r i l l p o i n t geometry i s f u l l y removed a t t h e g r i n d e r a c t i o n end p o i n t ( e l e m e n t f o u r ) , s h o u l d n o t i n v o l v e e x c e s s i v e m a t e r i a l removal d u r i n g p o i n t g r i n d i n g and s h o u l d h e a n e a s y s h a p e t o be produced. The i n i t i a l p o i n t geometry c o n s i s t i n g of a r i g h t c i r c u l a r cone c o a x i a l with t h e d r i l l a x i s can s a t i s f y a l l t h e above c r i t e r i a when a s u i t a b l e semi-cone I t is a p p a r e n t t h a t t h i s a n g l e '3 i s s e l e c t e d . i n i t i a l D p o i n t geometry i s e a s y t o p r o d u c e and h a s a geometry which r o u g h l y a p p r o x i m a t e s t n e f i n a l p o i n t georretry so t h a t t h e amount of m a t e r i a l t o be removed is n o t eitcessive. Further t h e circumferential cleara n c e a n g l e Ct [ 2 4 , 2 6 ] i s z e r o on t h e i n i t i a l c o n i c a l f l a n k s so t h a f e x c e s s m a t e r i a l i s a v a i l a b l e a t t h e 0 on t h e s e s u r f a c e s fcr t h e f i n a l flank since Ce p o i n t geometryCO F u r t h e r t h e s o l i d cone i n t h e web ( c o r e ) r e g i m a l s o p r o v i d e s e x c e s s m a t e r i a l which e n a b l e s t h e c h i s e l e d g e t o b e formed by t h e g e n e r a t i o n process. The c o n d i t i o n s i n t h e v i c i n i t y o f t h e e v e n t u a l l i p c a n b e s t u d i e d from t h e y r o m e t r y o f t h e i n L e r s e c t i o n o f t h e i n i t i a l d r i l l p o i n t c o n e and t h e f l u t e I t i s found t h a t vhen 9 is s e l e c t e d s u c h surface. t h a t a s i n g l e flute g e n e r a t o r i n t g r s e c t s t w o c o n e g e n e r a t o r s a t t h e o u t e r and core r a d i i , r e s p e c t i v e l y , t h e i n t e r s e c t i o n c u r v e h a s a n e x t r e m e l y s m a l l convex c u r v a t u r e when viewed i n a p l a n e normal t o t h e d r i l l axis (x-y p l a n e ) . Thus t h e i n i t i a l c o n e - f l u t e i n t e r s e c t i o n c u r v e ve r y c l o s e l y approximates t h e s i n g l e f l u t e g e n e r a t o r c o n s i d e r e d a l t h o u g h a s l i g h t amount of m a t e r i a l i s a v a i l a b l e f o r removal when t h e f l u t e g e n e r a t o r i s chosen a s t h e e v e n t u a l l i p o f t h e s h a r p ened d r i l l . The r e q u i r e d i n i t i a l d r i l l semi-cone s i n g l e 4 is found t o depend on t h e p o i n t a n g l e 2p and web t h i c a n e s s t o d i a m e t e r r a t i o 2W/D [27] i . e .
From t h e above e q u a t i o n i t i s c l e a r t h a t 5 w i l l b e less t h a n t h e " i n t u i t i v e " c h o i c e o f 9 = pp With t h i s i n i t i a l geometry, t h e i n t e r s e c t i o n o f D t h e i n i t i a l d r i l l cone and f l u t e may b e u s e d t o i d e n t i f y t h e f i n a l l i p and o u t e r c o r n e r which s i m p l i f i e s t h e i n i t i a l s e t t i n g of t h e d r i l l i n t h e g r i n d e r , i . e . s e t t i n g s Lc and 1 . The g e n e r a t i o n p r o c e s s t h e r e f o r e c o n s i s t s o f g r i n d ing t h e i n i t i a l p o i n t geometry a t t h e a p p r o p r i a t e 0 ( E q u a t i o n (14)) and s e t t i n g O , x , C ,L and , Ion t h e The d r i l l i s x t h & p o s i t i o n e d i n grinder attachment. i t s ' i d e a l ' f i n a l l o c a t i o n by g r a d u a l l y i n c r e m e n t i n g t h e g r i n d e r attachment i n t h e L d i r e c t i o n whi l e O s c i l l a t i n g a b o u t t h e g r i n d i n g done a x i s u n t i l t h e i n i t i a l geometry is f u l l y removed i . e . L i s s a t i s fied T h i s a n a l y s i s h a s demonstrated t h a t t h e study of t h e d r i l l p o i n t g e n e r a t i o n p r o c e s s i s more i n t r i c a t e than the analysis f o r t h e d r i l l i n t h e ' i d e a l ' f i n a l l o c a t i o n d e p i c t e d by F i g . 1. However by a d e t a i l e d c o n s i d e r a t i o n of t h e v a r i o u s e l e m e n t s t h e g e n e r a t i o n p r o c e s s c a n b e s i m p l i f i e d and t h e p o t e n t i a l s c o p e of t h i s g r i n d i n g method n o t e d e a r l i e r [ 2 6 ] a c h i e v e d when s e t t i n g e r r o r s a r e ignored.
.
F I N A L IIRILI. POINT GEOMETRY
GEWERATION PROCESS - SETTING ERRORS I n attempting to achieve the desired f i n a l d r i l l p c i n t geometry, e r r o r s a r e p o s s i b l e i n s e t t i n q e a c h of t > e f i v e main g r i n d e r p a r a m e t e r s ( i . e . a , A , . + , C , and L 1. A d d i t i o n a l l y , t h e i n i t i a l geometry may c 8 n t a i n f f r o r s i n 2W and i , i r h i l e t h e d r i l l may be moved Loo f a r t o w a r d s t h e g r i n d i n g wheel by an amount dt, d u r i n g the g r i n d e r a c t i o n e l e m e n t o f t h e g e n e r a t i o n p r o c e s s . With such numerous p o t e n t i a l s o i i r c e s o f e r r o r , i t i s d o u b t f u l i n o r a c t i c e i f a l l t h e s e e r r o r s can b e e l i m i n a t e d . T h e i r n e t e f f e c t w i l l b e t o make i t i m p r a c t i c a l tr) p o s i t i o n t h e d r i l l i n t h e " i d e a l " f i n a l l o c a t ion. So although t h e f i n a l generated d r i l l point geometry w i l l have c o n i c a l f l a n k s i t s l i p s must b e curved because u n l i k e t h e " i d e a l " c a s e i l l u s t r a t e d i n F i g . 1, t h e l i p s a r e n o t formed by a s i n g l e g r i n d i n g c o n e g e n e r a t o r . Thus t h e need i s c r e a t e d f o r an e x t e n d e d a n a l y s i s t o s t u d y t h e i n f l u e n c e of e r r o r s on t h e g e n e r a t e d geometry. The a n a l y s i s p r e s e n t e d i s based on t h e " i d e a l " case and accommodates t h e g r i n d e r s e t t i n q , i n i t i a l geometry, and g r i n d e r a c t i o n errsrs d i s c u s s e d above. I t a l s o i n c l u d e s a s u i t a b l e d e f i n i t i o n of t h e g r i n d e r a c t i o n rnd point. Further, the "acceptable grinder c r i t e r i a " e s t a b l i s h e d f o r t h e " i d e a l " c a s e [ 2 6 ] a r e reviewed and m o d i f i e d . F i n a l l y , r e v i s e d d e f i n i t i o n s of t h e s p e c i f i e d d r i l l p o i n t f e a t u r e s a r e d e v e l o p e d and a means of m e a s u r i n g t h e amount of l i p c u r v a t u r e d e t e r m i n e d . For t h e p u r p o s e Of t h e a n a l y s i s , t h e o r i g i n i s t a k e n t o b e t h e p o i n t 0 on t h e d r i l l a x i s a s shown i n F i g . 1. The i n t r o d u c t i o n of s e t t i n g and g r i n d e r a c t i o n e r r o r s w i l l l e a d t o t h e d i s p l a c e m e n t of b o t h t h e d r i l l and c o n e a x e s away from t h e " i d e a l " f i n a l location. When t h e g r i n d e r j i g i s p o s i t i o n e d a d i s t a n c e L t r . away from t h e g r i n d i n g wheel p l a n e , t h e e q u a t i o j l s f o r t h e d i s p l a c e m e n t between t h e o r i g i n 0 on t h ? d r i l l a x i s and t h e cone apex become: CX'
'
C
cx
=
= C
Y
t dCx
(15)
sin(d0tdx) + C
cos(d*+dr)
Y 2C s i n [ % ] c o s [ $ + 5]+ 2C s i n [ Z dB! c o s [ x + d i : - . ' t ~ ] P 2 2 q (G,singI- L ) ( c o t $ - c o t ( 1 j t d 9 ) )s i n ! H t i A d a t d u )
Z
t
-
t s i n ( y t dx)cosec(e + do).\ Cz' = C z c o s ( d 0 + d i ) - C s i n ( d . 1 + dX) Y
(16)
(20)
I t i s n o t e d t h a t i n t h e " i d e a l " c a s e when d o , d r , dCx, dLc and A e o u a l z e r o , E q u a t i o n s ( 1 5 ) - ( 1 7 ) r e v e r t , a s e x p e c t e d , t o C x ' = C x , C '=C and C z ' = C z . Y Y The e q u a t i o n s f o r a p o i n t x , y , z on t h e g r i n d i n g cone 1 c a n t h e n b e d e t g r m i s e d from:
-
x = (y, + c ')tavA' cx' (22) Y (23) zc = (yc + C y ' ) t a n v ' CZ' where t h e a n g l e s > ' and Y ' c a n b e i n t e r r e l a t e d u s i n g a m o d i f i e d v e r s i o n o f t h e " i d e a l " c a s e E q u a t i o n (1) and s u b e t i t u t i n g 4 + dq and x + d i f o r 9 and x respectively. A t a radius r , a point x on a g e n e r a l f l u t e g e n e r a t o r may b e d e t e g h i % %om t h e f o l l o w i n g e x p r e s s i o n s ( w i t h r e f e r e n c e t o o r i g i n 0 on t h e d r i l l axis) :
xf
=
r sin(i t 9
+
dh
yf = r cos()i t h t d i
zf
= (Wtanh
where,
-
6)')
(24)
iu')
(25)
+ r c o s , , , ' ) c o t p + [*$]cot(6,t = W 7 t d W-
dS)
(26)
sinw'
and G r e p r e s e n t s t h e i n c l i n a t i o n between t h e p r o j e c t i o n s of t h e g e n e r a l and r e f e r e n c e f l u t e g e n e r a t o r s i n a p l a n e normal t o t h e d r i l l a x i s . The r e f e r e n c e f l u t e generator i n turn represents the generator inclined by t h e error a n g l e dA t o t h e g e n e r a t o r which f o r m s t h e
s t r a i g h t l i p when t h e d r i l l i s p o s i t i o n e d i n t h e "ideal" f i n a l location. Finally, a point x , on t h e g e n e r a l i n i t i a l d r i l l cone g e n e r a t o r t R e e u a t o n s t a k e t h e form:
c a s e v a l u e s c . g . $',+dt. f n r t i . F i n a l l y , the l i p curvatu r e was d e t e r m i n e d i n t h e x-y p l a n e (,',xy) and y-z p l a n e (!.yz) by e s t a b l i s h i n g t h e amount o f d e p a r t u r e of t h e curved l i p from t h e " a v e r a g e " l i p .
xD = r s i n e
Scope of C o n i c a l G r i n d i n g w i t h Errors
'8' 'f
yD = CoSE zD = r c c t 9
D
+ WtanAcotp + n2 ( c o s h : c o t p - c o t c D )
(30)
where i~.' i s found by s u b s t i t u t i n g r = D i n E q u a t i o n ( 2 7 ) and E i s t h e a n g l e between t h e p r o j e c t i o n s of t h e g e n e r a l i n i t i a l d r i l l cone g e n e r a t o r and t h e ya x i s i n a p l a n e normal t o t h e d r i l l a x i s . Using E q u a t i o n s ( 1 5 ) - ( 3 0 ) , t h e g r i n d e r a c t i o n end p o i n t can b c e s t a b l i s h e d and t h e r e s u l t a n t f i n a l d r i l l I t is apparent t h a t these p o i n t geometry found. e q u a t i o n s a r e f a i r l y complex and i n v o l v e v a r i a b l e s which a r e d i f f i c u l t t o s e p a r a t e . Hence i t was n e c e s s a r y t o use i t e r a t i v e s o l u t i o n search techniques t o d e t e r m i n e t h e g r i n d e r a c t i o n end p o i n t . The l a r g e number o f c a l c u l a t i o n s i n v o l v e d was e x p e d i t e d by d e v e l o p i n g and u s i n g a s u b s t a n t i a l computer programme ~271. I n b r o a d terms, t h e method of d e t e r m i n i n g t h e g r i n d e r a c t i o n end p o i n t i s t o f i r s t select v a l u e s of t h e commonly s p e c i f i e d d r i l l p o i n t f e a t u r e s t o g e t h e r w i t h s i z e s of t h e g r i n d e r s e t t i n g , i n i t i a l g e o m e t r y , and g r i n d e r a c t i o n errors. The " i d e a l " c a s e g r i n d i n g and g r i n d e r p a r a m e t e r v a l u e s a r e t h e n c a l c u l a t e d from E q u a t i o n s ( 1 ) - ( 1 3 ) and t h e i n i t i a l d r i l l semi-cone Equations a n g l e 8 determined u sin g Equation ( 1 4 ) . ( 2 8 ) - ( 3 B ) a r e used t o s e l e c t a p o i n t on t h e i n i t i a l d r i l l cone. The p o i n t on t h e g r i n d i n g cone a t which x =x and y =y i s t h e n found from E q u a t i o n s ( 1 5 ) - ( 2 3 ) . Tfiesg two g h B s o f e q u a t i o n s a r e i t e r a t e d f o r . ? t i l l t h e c o n d i t i o n z =z i s s a t i s f i e d . By r e p e a t i n g t h e above s t e p s f o r C a Eomprehensive g r i d of p o i n t s on t h e i n i t i a l d r i l l c o n e a minimum v a l u e of.! c a n be e s t a b lished. T h i s r e p r e s e n t s t h e l a s t p o i n t on t h e i n i t i a l d r i l l cone removed by t h e g r i n d i n g cone. S u b t r a c t i n g dA from t h e minimum h v a l u e g i v e s t h e g r i n d e r a c t i o n end p o i n t when errors o c c u r . The f i n a l d r i l l p o i n t geometry g e n e r a t e d a t t h e g r i n d e r a c t i o n end p o i n t m u s t , i n a s i m i l a r f a s h i o n t o t h e " i d e a l " c a s e , m e e t c e r t a i n shape c o n s t r a i n t s i f i t s g e n e r a l appearance i s t o be considered accepta b l e . With e r r o r s p r e s e n t , t h e s e v e n " a c c e p t a b l e g r i n d e r criteria" e s t a b l i s h e d 1261 a r e r e d u c e d t o t h e following f i v e c r i t e r i a : 1. The d r i l l p o i n t r e g i o n s h o u l d b e s y m m e t r i c a l a b o u t the d r i l l axis. 2. The d r i l l f l a n k s u r f a c e bounded by t h e l i p , c h i s e l e d g e , d r i l l p o i n t h e e l , and f l u t e d l a n d s h o u l d b e a continuous surface. 3 . The c h i s e l edge s h o u l d j o i n t h e l i p a t t h e c h i s e l edge c o r n e r . 4. The c l e a r a n c e a t a l l p o i n t s on t h e a n g u l a r f l a n k r e g i o n between t h e o u t e r c o r n e r and t h e c h i s e l edge corner should be s u f f i c i e n t t o prevent interference with the transient surface during d r i l l i n g . 5. C o n d i t i o n s (1) t o ( 4 ) s h o u l d b e s a t i s f i e d when a t t e m p t i n g t o a c h i e v e t h e recommended r a n g g s of t h e s p e g i f i e 8 d r i l l g o i n t f e a t u r %s i.e. 2p=118 ; Q = 120 -135 : CLn=8 -16'; 6,,=2C -32O; 2W/D=12%-20%.
-
The modes of f a i l u r e o f t h e above c r i t e r i a a r e i d e n t i c a l t o t h e " i d e a l " c o n i c a l g r i n d i n g method [26] w i t h t h e e x c e p t i o n s t h a t c r i t e r i o n 3 may b e f a i l e d by t h e c h i s e l e d g e i n t e r s e c t i n g t h e d r i l l p o i n t h e e l and c r i t e r i o n 4 may n o t b e m e t d u e t o t h e l i p c l e a r r n c e angrle becoming n e g a t i v e . I n d e s c r i b i n g t h e f i n a l d r i l l p o i n t - g e o m e t r y gene r a t e d a t t h e g r i n d e r a c t i o n end p o i n t , c o - o r d i n a t e s f o r p o i n t s on t h e d r i l l f l a n k were found from Equati o n s ( 1 5 ) - ( 2 3 ) . Combining t h e s e e q u a t i o n s w i t h E q u a t i o n s ( 2 4 ) - ( 2 7 ) f o r t h e d r i l l f l u t e and i t e r a t i n g f o r $ t i l l t h e c o n d i t i o n z =z w a s s a t i s f i e d , t h e Finally, the d r i l l l i p co-ordinates we& established. c h i s e l e d g e c o - o r d i n a t e s were found from E q u a t i o n ( 2 3 ) and a m o d i f i e d v e r s i o n of t h e " i d e a l " c a s e E q u a t i o n ( 6 ) w i t h C x ' , C I , R + d B , and x+dx s u b s t i t u t e d f o r a , and y r e s p e c t i v e l y . Cx, C Y' With t h e g e n e r a t e d l i p s b e i n g c u r v e d , r e v i s e d d e f i n i t i o n s w e r e needed f o r t h e g e n e r a t e d p o i n t f e a t u r e s s u c h as t h e p o i n t a n g l e . T h r e e r e v i s e d d e f i n i t i o n s were d e r i v e d u s i n g t h e c o n c e p t of a n " a v e r a g e " d r i l l l i p formed by j o i n i n g t h e o u t e r and c h i s e l edge c o r n e r s w i t h a s t r a i g h t l i n e . The a v e r a g e p o i n t a n g l e 2p and a v e r a g e l i p s p a c i n g 2WA were d i r e c t l y b a s e d on t h e " a v e r a g e " l i p w h i l e t h e a v e r a g e c h i s e l e d g e a n g l e $ was a l s o r e q u i r e d u s e of m o d i f i e d v e r s i o n s o f E q u a t i o h s ( 7 ) and ( 8 ) d e v e l o p e d f o r t h e " i d e a l " c a s e . S i m i l a r l y , t h e c l e a r a n c e a n g l e C Z ' was o b t a i n e d u s i n g a m o d i f i e d v e r s i o n of t h e " i d e a l " c a s e E q u a t i o n (10). I n b o t h c a s e s , t h e m o d i f i c a t i o n s i n v o l v e d s u b s t i t u t i n g t h e error c a s e f o r t h e " i d e a l "
4
The " a c c e p t a b l c g r i n d e r c r i t e r i a " w e r e used t o a s s e s s t h e g e n e r a t e d f i n a l d r i l l p o i n t geometry f o r a l a r g e number of g e n e r a l p u r p o s e c o m b i n a t i o n s of d r i l l p o i n t f e a t u r e v a l u e s . A s i n d i c a t e d i n t a b l e 1, 80 c o m b i n a t i o n s o f 0 , CI , 2W/D and h e l i x a n g l e x . were te8ted together with 9 0 values of 2 over the range 31 -50° ( i . e . lo i n t e r v a l s ) . T h i s g a v e a t o t a l of 80120 = 1 6 0 0 c o m b i n a t i o n s which were examined. The combined e f f e c t s o f t h e 8 e r r o r s a<,, dw, d ; , dC , dL , d a , dW and d i , were s t u d i e d by s i m u l t a n e o u s l y a5plyifig t h c worst c a s e l i m i t s f o r e a c h error. The f i g u r e s s e l e c t e d f o r t h e 8 errors were dil, d:? t .5O: d l ! 2O; dCx, dL I .002", dh + .002"; dW 1 .0013": and d t : 1C28O. The g r i n d e r s e t t i n p and p o s i t i o n i n g ergors were d e r i v c d f r o n c o n s i d e r a t i o n of t h e r e l e v a n t s e t t i n g mechanisms w h i l e t h e i n i t i a l geometry e r r o r s wcre o b t a i n e d from a n e a r l i e r p r o c e s s c a p a b i l i t y s t u d y 1191. From t a b l e 1 i t i s a p p a r e n t t h a t i n cont r a s t t o t h e " i d e a l " c a s e ( i . e . z e r o e r r o r s ) , a t 26 ( o r 32%) of t h e 80 g e o m e t r i c c o m b i n a t i o n s , no a c c e p t a b l e f i n a l d r i l l p o i n t geometry c o u l d be produced t h u s f a i l i n g c r i t e r i o n 5. For t h e r e m a i n i n g combinati o n s , u p p e r l i m i t s f o r fi were g e n e r a l l y due t o t h e f a i l u r e of c r i t e r i o n 4 i n t h a t t h e c i r c u m f e r e n t i a l c l e a r a n r e a n g l e C 1 [ 2 4 ] was i n a d e q u a t e w h i l e t h e l o w e r l i m i t s w e r e &own t o r e s u l t from e i t h e r no or a second c h i s e l e d g e c o r n e r b e i n g g e n e r a t e d Thus n o t m e e t i n g c r i t e r i o n 3. I n t o t a l , of t h e 1600 combinata t some 6 8 8 (or 435) t h e d r i l l ions considered, p o i n t geometry g e n e r a t e d was u n a c c e p t a b l e . Hence t h e p r e s e n c e o f r e l a t i v e l y s m a l l e r r o r s h a s been demons t r a t e d t o s u b s t a n t i a l l y reduce t h e scope of coni cal g r i n d i n g and p l a c e s i g n i f i c a n t r e s t r i c t i o n s on t h e v a l u e of .? which may b e s e l e c t e d . L i p C u r v a t u r e and P r o m i n e n t D r i l l P o i n t F e a t u r e s The combined e f f e c t s of errors on t h e g e n e r a t e d l i p c u r v a t u r e and d r i l l p o i n t f e a t u r e v a l u e s have been examined f o r s e v e r a l c o m b i n a t i o n s o f t h e s p e c i f i e d d r i l l p o i n t f e a t u r e s and g r i n d e r semi-cone a n g l e 4 a t which t h e " a c c e p t a b l e g r i n d e r c r i t e r i a " were satisfied. A sample of t h e s e r e s u l t s is i l l u s t For e a c h o f t h e 11 g e o m e t r i c rated i n table 2. c o m b i n a t i o n s l i s t e d , t h e combined w o r s t c a s e l i m i t s
TABLE 1.
Combined e f f e c t s of d e v i a t i o n s on s c o p e of c o n i c a l d r i l l F o i n t g r i n d e r modelled. ( 2 p = 118O i n a l l c a s e s ) IDEAL CASE ( i . e .
z e r o errors)
D = 1"
D = 1/2"
$\cyo 8 10 1 2 1 4 120 * * r-T--T* * 125 * * * 130 135 ,34 .33 ,31 *
16
*
D = 1/4"
g\"'o
8
* 120 * 125 130 * 135 i33
10
12
14
16
*
* *
*
* *
*
*
*
*
* * *
* COMBINED EFFECTS D = 1/2"
D = 1"
,>v'o
12 1 4 16 1 2 0 <43 <42 e42 d 4 0 e35 125 <48 <49 -50 * * 130 n 34-37* * * 1 3 5 n n n n n
-
8
10
b\"o 120 125 130 135
10 11 1 4 16 .45 ~ 4 5<45 e45 ~ 4 2 -50 ~ 5 0 * * * n 33-42* * * n n n n n 8
D = 1/4"
D = 1/8"
---
v\'*o 8 1 0 12 14 16 b\"o 8 10 1 2 14 16 1 2 0 *46 .46 <47 .47 *47 120 ~ 4 5.46 *47 .48 ~ 4 9 125
-
When D = 1". 2W/D = 1 2 % and F = 2W/D = 14% and 6 " = f o r D = k " , 2W/D = 17% and 6 " = 2W/D = 20% and"b3 D =
-
32O- f o r D = $", 30°: 26O: and f o r = 2bo.
o f t h e 8 e r r o r s c o n s i d e r c d i n t h i s i n v e s t i g a t i o r were a p p l i e d a t t h e levels l i s t e d p r e v i o u s l y . From t a b l e 2 i t c a n b e s e e n t h a t t h e l i p c u r v a t u r c ? i s i n a l l cases q u i t e small. A s t h e maxim!im m a g n i t u d e o f t h e c u r v a t u r c c o m p o n e n t s w a s J mere . 0 0 3 ( f o r ' > x y / D ) , t h e g e n e r a t e d l i p s can f o r p r a c t i c a l p u r p o s e s b e r e g a r d e d as s t r a i g h t l i n e s . Gi7Jen t h a t a l l t h e " a c c e p t a b l c g r i n d e r criteria' are a l s o s a t i s f i e d , t h e s e r e s u l t s s u g g e s t t h a t t h e g e n e r a l a p p e a r a n c e of t h e g e n e r a t e d p o i n t g e o m e t r y i s n o t a r e l i a b l e i n d i c a t o r o f t h e p r e s e n c e o r e f f e c t s of g r i n d e r s e t t i n g errors. The l i m i t e d Lip c u r v a t u r e a l s o i m p l i e s t h a t t h e g e n e r a t e d p o i n t f e a t u r e s c a n b e t r e a t e d as p h y s i c a l r e p r e s e n t a t i o n s of t h e i r " i d e a l " case c o u n t e r p a r t s . From t a b l e 2 i t i s a p p a r e n t t h a t t h e c o m b i n e d errors s u b s t a n t i a l l y a f f e c t t h e v a l u e s of t h e generate? point features. F o r i n s t a n c e i n a l l 11 cases, t h e r a n g e o f 2 p v a l u e s e x c e e d s 46 w i t h t h e maximum r a n g e S i m i l a r l y , & h e maximum r a n g e o f C . ' r e a c h i n g 6.9'. v a l u e s is a s i g n i f i c a n t 4.8 Although t h e rangesoof ' i D a r e f a i r l y moderate ( i . e . 2 . 6 % max.; t h e a n g l e :mWAseems p a r t i c u l a r l y s e n s i t i v e t o e r r o r s . I n 3 of the 11 cases i i l u s t r a t e d , t h e g e n e r a t e d r a n g e o f i exceeded t h e r e c o m n e n d e d q e n e r a l p u r p o s e r a n g e f o r A t h i s f e a t u r e o f 15' (12Oo-13So) w h i l e t h e naximum Of t h e 8 e r r o r s r a n g e of v w a s a s h i g h as 16.6O. c o n s i d e r e d ? ,, w a s f o u n d t o b e , a s e x p e c t e d , s e n s i t i v e t o d . b u t s u r ~ r i s i n g l yi t w a s e v e n more s e n s i t i v e t o d;. S i m i l a r l y , d ; a n d d; h a d t h e a n t i c i p a t e d major i n f l u e n c e o n 2 p w h i l e dW s i g n i f i c a n t l y a f f e c t e d 2W ' . T h e r e s u l t s i n A t a b l e 2 show t h a t t h e g e n e r a t e d d r i P l point feature ranges straddle the corresponding "ideal" specified values. However, d i f f i c u l t i e s are l i k e l y t o b e e n c o u n t e r e d i n t h e a c h i c v e m e n t and c o n t r o l o f a p a r t i c u l a r s e t of recommended g e n e r a l purpose f e a t u r e values. F u r t h e r when t h e n o m i n a l d e s i g n v a l u e s d i f f e r f r o m b a t c h t o b o t c h t.he t o t a l scatter i n t h e qenerated p o i n t f c a t u r e v a l u e s can b e extremcly wide. F o r e x a n p l e , f r o g tab&' 2 , when I i s s e l e c t e d w i t h i n t h c l i m i t s 1 2 0 -130 the ggneratcd a value of shown t o v a r y f r o m 1 1 2 . 4 - 1 3 8 . 7 r a n g e o f 28.3 I t should be n o t c d t h a t a s f a i r l y l o w l e v e l s of errors have been c o n s i d e r e d i n t h i s i n v e s t i g a t i o n , t h e s u b s t a n t i a l p o i n t f e a t u r e variat.i o n s shown i n t a b l e 2 may w e l l b e c o n s e r v a t i v e w h i l e t h e g e n e r a l p u r p o s e scope i l l u s t r a t e d i n t a b l e 1 c o u l d be e v e n f u r t h e r r e d u c e d .
.
is
3.)
.
co~cL,vsIor;s The f i n a l d r i l l p o i n t geometry g e n e r a t i o n p r o c e s s for g e n e r a l p u r p o s e t w i s t d r i l l s u s i n g t h e p o p u l a r c o n i c a l g r i n d i n g m e t h o d h a s 'm3en a n a l y s c d b o t h i n t h e a b s e n c e ! i . e . " i d e a l " case) a n d p r e s e n c e o f g r i n d e r setting errors. I t i s shown t h a t t h e a n a l y s i s f o r r e l a t i n y the specified dril.1 point features to t h e g r i n d i n g p a r a m c t e r s when t h e d r i l l i s i n t h e " i d e a l " f i n a l 1.otation represer.ts only part of t h e g e n e r a t i o n process. The i n i t i a l d r i l l p o i n t geometry, t h e s e t t i n g of the d r i l l i n the grinder, the grinder a c t i o n d u r i n g w h i c h t h e i n i t i a l g e o m e t r y i s removed and t h e i d e n t i f i c a t i o n of t h e g r i n d e r a c t i o n end p o i n t c o n s t i t u t e t h e f o u r major e l e m e n t s which c o n t r i b u t e t o t h e c o m p l e x i t y 3f t h e d r i l l p o i n t g e n e r a t i o n process. T h e d e t a i l e d a n a l y s i s o f t h e s e e l e m e n t s r e s u l t e d i n a n i n i t i a l d r i l l p o i n t geometry and g r i n d e r d e s i g n c a p a b l e o f p r o d u c i n g a l l t h e recorntende d s p e c i f i e d d r i l l p o i n t f e a t u r e xvalues f o r t h e " i d e a l " case of z e r o s e t t i n q errors. T h e nore c o m p l e x a n a l y s i s a n d g e n e r a t e d d r i l l point s h a p e due to t h e i n t r o d u c t i o n of s e t t i n g e r r o r s n e c e s s i t a t e d r e a p p r a i s a l of " a c c e p t a b l e y r i n d c r criteria" and d r i l l p o i n t f e a t u r e d e f i n i t i o n s . The p r e s e n c e o f r e l a t i v e l y small g r i n d e r s e t t i n o errors l e a d t o a s i g n i f i c a n t r e d u c t i o n i n t h e s c o p e of c o n i c a l g r i n d i n g f o r g e n e r a l p u r p o s e d r i l l s . Of t h e 80 g e n e r a l p u r p o s e c o m b i n a t i o n s of d r i l l p o i n t f e a t u r e v a l u e s ?.n a c c e p t a b l e s h a p e i s o n l y g e n e r a t e d a t 54 ( i . e . 6 8 % ) c o m b i n a t i o n s . Within t h e reduced scope, t h e amoullt of l i p c u r v a t u r e f o r a c c e p t a b l e d r i l l p o i n t s h a p e s was shown t o b e n e g l i g i b l e a l t h o u g h s m a l l s e t t i n g errors r e s u l t e d i n w i d e v a r i a t i o n s i n t h e v a l u e s o f t h e g e n e r a t e d f e a t u r e s ! e g 1.6.6" for A range). ilence i t h a s been d e m o n s t r a t e d t h a t i n i t s e l f t h e q e n e r a l a p p e a r a n c e of t h e g e n e r a t e d p o i n t geometry is a n u n r e l i a b l e i n d i c a t o r o f t h e p r e s e n c e or e f f e c t s o f s e t t i n g e r r o r s f o r u s e i n q u a l i t y control. Further, thP generated p o i n t f e a t u r e s have b e e n shown t o b e p a r t i c u l a r l y s e n s i t i v e to a number of s e t t i n g errors. This study has highlighted t h e complexities of t h e g e n e r a t i o n p r o c e s s and t h e a t t e n d a n t d i f f i c u l t i e s i n a c h i e v i n g and c o n t r o l l i n g t h e s p e c i f i e d d r i l l p o i n t geornctry n o t e d i n p r a c t i c e . REFERENCES
1. 2. Combined e f f e c t s of d e v i a t i o n s o n s a l i e n t f e a t u r e s of f i n a l d r i l l p o i n t g e o m r t r y .
TABLE 2 .
MAX D
2p
1,
C
-
!
*y/3
PlAX fyz/D
2pA
2WI;/D
,
A
3. 4. 5.
DONALDSON, G . H . L e CAIN a n d V.C. GOULD, " T o o l D e s i g p " , McGraw-Hill, N e w J e r s e y ( 1 9 6 9 ) A . S .T.M.E., " T o o l E n g i n e e r s ' IIandbook" , McGraw-11111, N e w Y o r k , ( 1 9 5 8 ) . METAL CUTTING TOOL INSTITUTE, "Metal C u t t i n q T o o l Handbook", ( 1 9 6 9 ) . AMERICAN STANDARD, USAS, H94-11-1967. BRITISH STANDARD INSTITUTION, B . S . 3 2 8 , P a r t I 1959. AUSTRALIAN STANDARD, A.S.2438-1981. G.M.H. STANDARD, " C u t t i n g T o o l s " , ( 1 9 6 8 ) . E.J.A. ARMAREGO a n d J . D . WRIGHT. J . E n g g . P r o d . , 2 .. 1.. ( 1 9 7 8 ) . ROHLKE, W e r k s t a t t s t e c h n i k a n d M a s c h i n e n b a u , 47, 5 , ( 1 9 5 7 ) . M.D. KINMAN, Machinery, (1963). W . D . ARNOT, M e c h a n i c a l World, 1 1 4 , March ( 1 9 5 2 ) . E.J.A. ARMAREGO a n d A. ROTENBERG, 1 n t . J . M a c h . T o o l D e s . R e s . , 13, 1 8 3 , ( 1 9 7 3 ) . D.F. GALLOWAY, T r a n s . Amer. SOC. Mech. E n g r s . , 79, 191, (1957). G.F. MICflELETTI a n d R . LEVI, P r o c . o f 8 t h I n t . M.T.D.R. C o n f . , U n i v e r s i t y of N a n c h e s t e r , Sept., (1967). fl. MALKIEWTCZ, M.Enq.Sc. T h e s i s , U n i v e r s i t v o f Melbourne ( 1 9 7 3 ) . S . KALDOR a n d E . LENZ, A n n a l s o f t h e CIRP, 2, 1, ( 1 9 8 0 ) . C . S . OXFORD J N R , A.S.T.E. c o l l e c t e d papers, 2, (1959). J . D . WRIGHT, M.Eng.Sc. T h e s i s , University of Melbourne, (1575). E.J.A. ARMAREGO a n d J . D . WRIGHT, J . Cngg. P r o d . 2 , 2, (1978). I. MUSHTAQ, M.Ena.Sc. T h e s i s , U n i v e r s i t y of Melbourne ( 1 9 8 0 ) . S. F U J I I , M.L. DeVRIES a n d S.M. W U , J. o f Eng. I n d . , 92, ( 1 9 7 0 ) . S. F U J I I , M.L. DCVRIES a n d S.M. WU, J. o f Eng. Ind., 93, (1971). W.D. TSAI a n d S.M. WU, I n t . J . Mach. T o o l D e s . Res., 19, 95, (1979). E.J.A.-jiRMAREGO a n d A . ROTENBERG, I n t . J. Mach. T o o l Des. R e s . , 2, 1 5 5 a n d 1 6 5 , ( 1 9 7 3 ) . F.K.T. Y A N , M . E n g . S c . T h e s i s , U n i v e r s i t y of Helbourne (1974). E . J . A . ARMAREGO a n d J.D. WRIGHT, A n n a l s of t h e CIHP, 2, 1, ( 1 9 8 0 ) . J.D. WRIGHT, Ph.D. T h e s i s , i i n i v e r s i t y o f Melbourne ( 1 9 8 1 ) . C.
.
_ I _ _ l l _ _ _ l
f 118 120
8
f 118 120 1 6
$j
118 130 16
42 H i . 0 0 1 L-.OOl R 3 1 H+.001 L-.OO2 R
-
120.1 115.7 4.4 120.1 115.7 4.4
128.2 9.5 112.4 6.9 15.8 2.6 126.5 lG.2 113.7 6.d 12.9 3.8
18.2 15.9 2.3 18.1 15.8 2.3
42 Hi.002 +.001 L - . 0 0 2 -.001 R 31 Hi.002 L-.002 - . 0 0 1 R
120.5 115.0 5.5 120.5 114.9 5.6
125.7 114.2 11.5 124.7 115.1 9.6
17.4 14.7 2.7 17.3 1 5 .0 2.3
18.2 15.8 2.4 18.2 15.8 2.4
10. 11. 12.
4 2 H+.COl L-.002 -.001 R 3 1 H+.OOl L-.002 - . 0 0 1 R
120.5 114.9 5.6 120.5 114.8 5.7
138.4 121.9 16.5 136.8 123.5 13.3
18.0 14.4 3.5 18.7 14.1 4.6
18.3 15.8 2.5 18.3 15.7 2.6
14.
120.3 115.7 4.6 120.4 115.7 4.7
127.6 9.4 113.3 6 . 8 1 4 . 3 2.6 125.3 10.2 115.0 6.0 10.3 4.2
12.3 11.7 .6 12.3 11.7 .6
+.001 1 2 0 . 7 1 2 3 . 6 1 7 .O - . 0 0 1 1 1 4 . 6 116.0 1 5 . 2 1.8 6.1 7.6
12.3 11.7 .6
-
-
1 118 120
8
42 €1+.002 L-.002 R
-
3 1 H+.001 L-.002 -.OOl R 1 118 120 16
3 1 Hc.002 L-.003 R
1 118 1 3 0 1 6
42 H+.OOZ +.001 1 2 0 . 6 L-.003 -.001 114.6 R 6.0 3 1 H+.OO2 +.OOl 1 2 0 . 8 L-.003 -.OOl 1 1 4 . 6 6.2 R
138.7 122.1 16.6 136.4 124.0 12.4
17.8 14.4 3.4 18.6 13.8 4.8
12.3 11.7 .6 12.3 11.7 .6
6. 7. 8. 9.
13.
15. 16. 17. 18. 19.
20. 21. 22. 23. 24. 25.
Note: When D = f " , D=l", H L R
-
2W/D=171 a n d A =26O a n d when 2W/D=12?: a n d 4::=32O
maximum v a l u e o f g e n e r a t e d f e a t u r e minimum v a l u e o f g e n e r a t e d f e a t u r r r a n q e of n e n e r a t e d f c ~ 3 t u r c s .
26. 27.
c.
102,
5